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[ "h Parameter or Hybrid Parameter of Two Port Network\n\nh parameter of two port network is a square matrix of order 2×2. Basically it is a way to represent a two port network. It is also known as hybrid parameter. In this form of representation, voltage of input port and current of the output...\n\nWhat is Power Factor? – Definition and Formula\n\nPower factor is an important parameter for the calculation of active and reactive power in electrical circuit. It has significance only for AC circuit. For DC circuit, it is not considered for power calculation.Its value is unity for DC circuit and may vary in between...\n\nZ Parameter of Two Port Network\n\nZ parameter is a factor by which input voltage and current & output voltage and current of two port network is related with. For any two port network, input voltage V1 and output voltage V2 can be expressed in terms of input current I1 and...\n\nPower Triangle\n\nPower Triangle is a right angled triangle whose sides represent the active, reactive and apparent power. Base, Perpendicular and Hypogenous of this right angled triangle denotes the Active, Reactive and Apparent power respectively. This triangle is not a new concept but it is just the...\n\nActive, Reactive and Apparent Power\n\nActive Power: Active power is the real power consumed in an electrical circuit. It is the useful power which can be converted into other form of energy like heat energy in heater, light energy in bulb etc. It is also known as true or real...\n\nFinal Value Theorem and Its Application\n\nFinal Value Theorem is used for determining the final value of a Laplace domain function F(s). Though we can always transform a time domain function into Laplace domain to apply Final Value Theorem. According to Final Value Theorem, final value of a function i.e. value...\n\nInitial Value Theorem\n\nInitial value Theorem is a very useful tool for transient analysis and calculating the initial value of a function f(t). This theorem is often abbreviated as IVT. The limiting value of a function in frequency domain when time tends to zero i.e. initial value can...\n\n4 Most Asked Numerical Questions", null, "In job interview often interviewer asks some numerical this is quit tricky and easy to answer without going into much calculation. In this post, I am going to discuss 3 most asked numerical questions in job interviews. These 3 questions are picked from my personal..." ]

[ null, "https://electricalbaba.com/wp-content/uploads/2018/01/STAR-connected-Load-150x150.jpg", null ]

[ "## Drawing orthogons with an SMT solver\n\nI’m long overdue to finish up my post series on orthogons as promised. First, a quick recap:\n\nRecall that we defined a “good” orthogon drawing as one which satisfies three criteria:\n\n1. No two edges intersect or touch, other than adjacent edges at their shared vertex.\n2. The length of each edge is a positive integer.\n3. The sum of all edge lengths is as small as possible.\n\nWe know that good drawings always exist, but our proof of this fact gives us no way to find one. It’s also worth noting that good drawings aren’t unique, even up to rotation and reflection. Remember these two equivalent drawings of the same orthogon?", null, "It turns out that they are both good drawings; as you can verify, each has a perimeter of 14 units. (I’m quite certain no smaller perimeter is possible, though I’m not sure how to prove it off the top of my head.)\n\nSo, given a list of vertices (convex or concave), how can we come up with a good drawing? Essentially this boils down to picking a length for each edge. The problem, as I explained in my previous post, is that this seems to require reasoning about all the edges globally. I thought about this off and on for a long time. Each new idea I had seemed to run up against this same local-global problem. Finally, I had an epiphany: I realized that the problem of making a good orthogon drawing can be formulated as an input appropriate for an SMT solver.\n\nWhat is am SMT solver, you ask? SMT stands for “satisfiability modulo theories”. The basic idea is that you give an SMT solver a proposition, and it tries to satisfy it, that is, find values for the variables which make the proposition true. Propositions are built using first-order logic, that is, things like “and”, “or”, “not”, implication, as well as “for all” (", null, "$\\forall$) and “there exists” (", null, "$\\exists$). The “modulo theories” part means that the solver also supports various theories, that is, collections of extra functions and relations over certain sets of values along with axioms explaining how they work. For example, a solver might support the theory of integers with addition, multiplication, and the", null, "$\\geq$ relation, as well as many other specialized theories.\n\nSometimes solvers can even do optimization: that is, find not just any solution, but a solution which gives the biggest (or smallest) value for some other function. And this is exactly what we need: we can express all the requirements of an orthogon as a proposition, and then ask the solver to find a solution which minimizes the perimeter length. SMT solvers are really good at solving these sorts of “global optimization” problems.\n\nSo, how exactly does this work? Suppose we are given an orthobrace, like XXXXXXVVXV, and we want to turn it into a drawing. First, let’s give names to the coordinates of the vertices:", null, "$(x_0, y_0), (x_1, y_1), \\dots, (x_{n-1}, y_{n-1})$. (Note these have to be integers, which enforces the constraint that all edge lengths are integers.) Our job is to now specify some constraints on the", null, "$x_i$ and", null, "$y_i$ which encode all the rules for a valid orthogon.\n\nWe might as well start by assuming the first edge, from", null, "$(x_0, y_0)$ to", null, "$(x_1, y_1)$, travels in the positive", null, "$x$ direction. We can encode this with two constraints:\n\n•", null, "$y_1 = y_0$\n•", null, "$x_1 > x_0$", null, "$y_1 = y_0$ means the edge is horizontal;", null, "$x_1 > x_0$ expresses the constraint that the second endpoint is to the right of the first (and since", null, "$x_0$ and", null, "$x_1$ are integers, the edge must be at least 1 unit long).\n\nWe then look at the first corner to see whether it is convex or concave, and turn appropriately. Let’s assume we are traveling counterclockwise around the orthogon; hence the first corner being convex means we turn left. That means the next edge travels “north”, i.e. in the positive", null, "$y$ direction:\n\n•", null, "$x_2 = x_1$\n•", null, "$y_2 > y_1$\n\nAnd so on. We go through each edge (including the last one from", null, "$(x_{n-1}, y_{n-1})$ back to", null, "$(x_0, y_0)$), keeping track of which direction we’re facing, and generate two constraints for each.\n\nThere’s another very important criterion we have to encode, namely, the requirement that no non-adjacent edges touch each other at all. We simply list all possible pairs of non-adjacent edges, and for each pair we encode the constraint that they do not touch each other. I will leave this as a puzzle for you (I will reveal the solution next time): given two edges", null, "$(x_{11},y_{11})\\text{---}(x_{12},y_{12})$ and", null, "$(x_{21},y_{21})\\text{---}(x_{22},y_{22})$, how can we logically encode the constraint that the edges do not touch at all? You are allowed to use “and” and “or”, addition, comparisons like", null, "$>$,", null, "$<$,", null, "$\\geq$ or", null, "$\\leq$, and the functions", null, "$\\mathrm{max}$ and", null, "$\\mathrm{min}$ (which take two integers and tell you which is bigger or smaller, respectively). This is tricky in the general case, but made much easier here by the assumption that the edges are always either horizontal or vertical. It turns out it is possible without even using multiplication.\n\nFinally, we write down an expression giving the perimeter of the orthogon (how?) and ask an SMT solver to optimize it subject to the constraints. I used the Z3 solver via the sbv Haskell library. It outputs specific values for all the", null, "$x_i$ and", null, "$y_i$ which satisfy the constraints and give a minimum perimeter; from there it’s a simple matter to draw them by connecting the points.\n\nTo see this in action, just for fun, let’s turn the orthobrace XVXXVXVVXVVXXVVXVVXXXXXXVVXXVX (which I made up randomly) into a drawing. This has 30 vertices (and 30 edges); hence we are asking the solver to find values for 60 variables. We give it two constraints per edge plus one (more complex) constraint for every pair of non-touching edges, of which there are 405 (each of the 30 edges is non-adjacent to 27 others, so", null, "$30 \\times 27 = 810$ counts each pair of edges twice); that’s a total of 465 constraints. Z3 is able to solve this in 15 seconds or so, yielding this masterpiece (somehow it kind of reminds me of Space Invaders):", null, "", null, "" ]

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[ "## 13279540\n\n13,279,540 (thirteen million two hundred seventy-nine thousand five hundred forty) is an even eight-digits composite number following 13279539 and preceding 13279541. In scientific notation, it is written as 1.327954 × 107. The sum of its digits is 31. It has a total of 4 prime factors and 12 positive divisors. There are 5,311,808 positive integers (up to 13279540) that are relatively prime to 13279540.\n\n## Basic properties\n\n• Is Prime? No\n• Number parity Even\n• Number length 8\n• Sum of Digits 31\n• Digital Root 4\n\n## Name\n\nShort name 13 million 279 thousand 540 thirteen million two hundred seventy-nine thousand five hundred forty\n\n## Notation\n\nScientific notation 1.327954 × 107 13.27954 × 106\n\n## Prime Factorization of 13279540\n\nPrime Factorization 22 × 5 × 663977\n\nComposite number\nDistinct Factors Total Factors Radical ω(n) 3 Total number of distinct prime factors Ω(n) 4 Total number of prime factors rad(n) 6639770 Product of the distinct prime numbers λ(n) 1 Returns the parity of Ω(n), such that λ(n) = (-1)Ω(n) μ(n) 0 Returns: 1, if n has an even number of prime factors (and is square free) −1, if n has an odd number of prime factors (and is square free) 0, if n has a squared prime factor Λ(n) 0 Returns log(p) if n is a power pk of any prime p (for any k >= 1), else returns 0\n\nThe prime factorization of 13,279,540 is 22 × 5 × 663977. Since it has a total of 4 prime factors, 13,279,540 is a composite number.\n\n## Divisors of 13279540\n\n12 divisors\n\n Even divisors 8 4 4 0\nTotal Divisors Sum of Divisors Aliquot Sum τ(n) 12 Total number of the positive divisors of n σ(n) 2.78871e+07 Sum of all the positive divisors of n s(n) 1.46075e+07 Sum of the proper positive divisors of n A(n) 2.32392e+06 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 3644.11 Returns the nth root of the product of n divisors H(n) 5.71428 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors\n\nThe number 13,279,540 can be divided by 12 positive divisors (out of which 8 are even, and 4 are odd). The sum of these divisors (counting 13,279,540) is 27,887,076, the average is 2,323,923.\n\n## Other Arithmetic Functions (n = 13279540)\n\n1 φ(n) n\nEuler Totient Carmichael Lambda Prime Pi φ(n) 5311808 Total number of positive integers not greater than n that are coprime to n λ(n) 663976 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 864783 Total number of primes less than or equal to n r2(n) 16 The number of ways n can be represented as the sum of 2 squares\n\nThere are 5,311,808 positive integers (less than 13,279,540) that are coprime with 13,279,540. And there are approximately 864,783 prime numbers less than or equal to 13,279,540.\n\n## Divisibility of 13279540\n\n m n mod m 2 3 4 5 6 7 8 9 0 1 0 0 4 1 4 4\n\nThe number 13,279,540 is divisible by 2, 4 and 5.\n\n• Arithmetic\n• Abundant\n\n• Polite\n\n## Base conversion (13279540)\n\nBase System Value\n2 Binary 110010101010000100110100\n3 Ternary 220222200002211\n4 Quaternary 302222010310\n5 Quinary 11344421130\n6 Senary 1152343204\n8 Octal 62520464\n10 Decimal 13279540\n12 Duodecimal 4544b04\n20 Vigesimal 42jih0\n36 Base36 7wmk4\n\n## Basic calculations (n = 13279540)\n\n### Multiplication\n\nn×i\n n×2 26559080 39838620 53118160 66397700\n\n### Division\n\nni\n n⁄2 6.63977e+06 4.42651e+06 3.31988e+06 2.65591e+06\n\n### Exponentiation\n\nni\n n2 176346182611600 2341796185838046664000 31097976121683774196454560000 412966817826944546792786187702400000\n\n### Nth Root\n\ni√n\n 2√n 3644.11 236.807 60.3665 26.585\n\n## 13279540 as geometric shapes\n\n### Circle\n\n Diameter 2.65591e+07 8.34378e+07 5.54008e+14\n\n### Sphere\n\n Volume 9.80929e+21 2.21603e+15 8.34378e+07\n\n### Square\n\nLength = n\n Perimeter 5.31182e+07 1.76346e+14 1.87801e+07\n\n### Cube\n\nLength = n\n Surface area 1.05808e+15 2.3418e+21 2.30008e+07\n\n### Equilateral Triangle\n\nLength = n\n Perimeter 3.98386e+07 7.63601e+13 1.15004e+07\n\n### Triangular Pyramid\n\nLength = n\n Surface area 3.05441e+14 2.75983e+20 1.08427e+07\n\n## Cryptographic Hash Functions\n\nmd5 4fee1f8ce1c94c3ec638163b449954b9 599c7e329f17a7c1ef7eb86c5f1921813b8a8822 aee82a2b1d1ef84e18907a908fe95a19f7baa08190bc8eb4909fd8a8a29a1e01 a7e84e0a42de10537e9455af98777e780b3a5a4ce9637d2b60f9ee8bb9e3a4ef6147d4b3785e9732260125e4507b0504f3c4fa98aa6d72c99382bf904153eef4 ba7238c57c442137d90ce1416734158f60636876" ]

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[ "# A new interpretation of Jensen's inequality and geometric properties of ϕ-means\n\n## Abstract\n\nWe introduce a mean of a real-valued measurable function f on a probability space induced by a strictly monotone function φ. Such a mean is called a φ-mean of f and written by M φ (f). We first give a new interpretation of Jensen's inequality by φ-mean. Next, as an application, we consider some geometric properties of M φ (f), for example, refinement, strictly monotone increasing (continuous) φ-mean path, convexity, etc.\n\nMathematics Subject Classification (2000): Primary 26E60; Secondary 26B25, 26B05.\n\n## 1. Introduction\n\nWe are interested in means of real-valued measurable functions induced by strictly monotone functions. These means are somewhat different from continuously differentiable means, i.e., C1-means introducing by Fujii et al. , but they include many known numerical means. Here we first give a new interpretation of Jensen's inequality by such a mean and we next consider some geometric properties of such means, as an application of it.\n\nThroughout the paper, we denote by (Ω, μ), I and f a probability space, an interval of and a real-valued measurable function on Ω with f(ω) I for almost all ω Ω, respectively. Let C(I) be the real linear space of all continuous real-valued functions defined on I. Let ${C}_{sm}^{+}\\left(I\\right)$ (resp. ${C}_{sm}^{-}\\left(I\\right)$) be the set of all φ C(I) which is strictly monotone increasing (resp. decreasing) on I. Then ${C}_{sm}^{+}\\left(I\\right)$ (resp. ${C}_{sm}^{-}\\left(I\\right)$) is a positive (resp. negative) cone of C(I). Put ${C}_{sm}\\left(I\\right)={C}_{sm}^{+}\\left(I\\right)\\cup {C}_{sm}^{-}\\left(I\\right)$. Then C sm (I) denotes the set of all strictly monotone continuous functions on I.\n\nLet C sm , f (I) be the set of all φ C sm (I) with φ f L1 (Ω, μ). Let φ be an arbitrary function of C sm , f (I). Since φ(I) is an interval of and μ is a probability measure on Ω, it follows that\n\n$\\int \\left(\\phi \\circ f\\right)\\mathsf{\\text{d}}\\mu \\in \\phi \\left(I\\right).$\n\nThen there exists a unique real number M φ (f) I such that $\\int \\left(\\phi \\circ f\\right)\\mathsf{\\text{d}}\\mu =\\phi \\left({M}_{\\phi }\\left(f\\right)\\right)$. Since φ is one-to-one, it follows that\n\n${M}_{\\phi }\\left(f\\right)={\\phi }^{-1}\\left(\\int \\left(\\phi \\circ f\\right)\\mathsf{\\text{d}}\\mu \\right).$\n\nWe call M φ (f) a φ-quasi-arithmetic mean of f with respect to μ (or simply, φ-mean of f). A φ-mean of f has the following invariant property:\n\n${M}_{\\phi }\\left(f\\right)={M}_{a\\phi +b}\\left(f\\right)$\n\nfor each a, b with a ≠ 0.\n\nAssume that μ(Ω\\{ω1, ..., ω n }) = 0 for some ω1, ..., ω n Ω, f is a positive measurable function on Ω and I = . Then M φ (f) will denote a weighted arithmetic mean, a weighted geometric mean, a weighted harmonic mean, etc. of {f(ω1), ..., f(ω n )} if φ(x) = x, φ(x) = log x, $\\phi \\left(x\\right)=\\frac{1}{x}$, etc., respectively.\n\nIn Section 2, we prepare some lemmas which we will need in the proof of our main results.\n\nIn Section 3, we first see that a φ-mean function: φ M φ (f) is order-preserving as a new interpretation of Jensen's inequality (see Theorem 1). We next see that there is a strictly monotone increasing φ-mean (continuous) path between two φ-means (see Theorem 2). We next see that the φ-mean function is strictly concave (or convex) on a suitable convex subset of C sm , f (I) (see Theorem 3). We also observe a certain boundedness of φ-means, more precisely,\n\n$\\underset{s\\ge 0}{sup}{M}_{\\left(1-s\\right)\\phi +s\\psi }\\left(f\\right)={M}_{\\psi -\\phi }\\left(f\\right)$\n\nunder some conditions (see Theorem 4).\n\nIn Section 4, we treat a special φ-mean in which φ is a C2-functions with no stationary points.\n\nIn Section 5, we will give a new refinement of the geometric-arithmetic mean inequality as an application of our results.\n\n## 2. Lemmas\n\nThis section is devoted to collecting some lemmas which we will need in the proof of our main results. The first lemma is to describe geometric properties of convex function, but this will be standard, so we will omit the proof (cf. [, (13.34) Exercise: Convex functions].\n\nLemma 1. Let φ be a real-valued function on I. Then the following three assertions are pairwise equivalent:\n\n1. (i)\n\nφ is convex (resp. strictly convex).\n\n2. (ii)\n\nFor any c I°, a function λ c,φ defined by\n\n${\\lambda }_{c,\\phi }\\left(x\\right)=\\frac{\\phi \\left(x\\right)-\\phi \\left(c\\right)}{x-c}\\phantom{\\rule{1em}{0ex}}\\left(x\\in I\\\\left\\{c\\right\\}\\right)$\n\nis monotone increasing (resp. strictly monotone increasing) on I\\{c}.\n\n1. (iii)\n\nFor any c I°, there is a real constant m c such that\n\n$\\phi \\left(x\\right)-\\phi \\left(c\\right)-{m}_{c}\\left(x-c\\right)\\ge 0\\phantom{\\rule{1em}{0ex}}\\left(resp.>0\\right)$\n\nfor all x I\\{c}, i.e., the line through (c, φ(c)) having slope m c is always below or on (resp. below) the graph of φ.\n\nHere I° denotes the interior of I.\n\nFor φ, ψ C sm (I) and c I°, put\n\n${\\lambda }_{c,\\phi ,\\psi }\\left(x\\right)=\\frac{\\psi \\left(x\\right)-\\psi \\left(c\\right)}{\\phi \\left(x\\right)-\\phi \\left(c\\right)}\\phantom{\\rule{1em}{0ex}}\\left(x\\in I\\\\left\\{c\\right\\}\\right).$\n\nThis function has the following invariant property:\n\n${\\lambda }_{c,\\phi ,\\psi }={\\lambda }_{c,a\\phi +b,a\\psi +b}$\n\nfor each a, b with a ≠ 0. In this case, we have the following\n\nLemma 2. Let $\\phi ,\\psi \\in {C}_{sm}^{+}\\left(I\\right)$. Then, the following three assertions are pairwise equivalent:\n\n1. (i)\n\nFor any c I°, λ c,φ,ψ is monotone increasing (resp. strictly monotone increasing) on I\\{c}.\n\n2. (ii)\n\nFor any c I°, there is a real constant m c such that\n\n$\\psi \\left(x\\right)-\\psi \\left(c\\right)-{m}_{c}\\left(\\phi \\left(x\\right)-\\phi \\left(c\\right)\\right)\\ge 0\\phantom{\\rule{1em}{0ex}}\\left(resp.>0\\right)$\n\nfor all x I\\{c}.\n\n1. (iii)\n\nψ φ -1 is convex (resp. strictly convex) on φ(I).\n\nProof. (i) (ii). Fix c I° arbitrarily. For any x I\\{c}, put u = φ(x) and then\n\n$\\left({\\lambda }_{c,\\phi ,\\psi }\\circ {\\phi }^{-1}\\right)\\left(u\\right)=\\frac{\\left(\\psi \\circ {\\phi }^{-1}\\right)\\left(u\\right)-\\left(\\psi \\circ {\\phi }^{-1}\\right)\\left(\\phi \\left(c\\right)\\right)}{u-\\phi \\left(c\\right)}.$\n(1)\n\nIf λ c,φ,ψ is monotone increasing (resp. strictly monotone increasing) on I\\{c}, then λ c,φ,ψ φ-1 is also monotone increasing (resp. strictly monotone increasing) on φ(I)\\{φ(c)} and hence by (1) and Lemma 1, we can find a real constant m c which is independent of x such that\n\n$\\left(\\psi \\circ {\\phi }^{-1}\\right)\\left(u\\right)-\\left(\\psi \\circ {\\phi }^{-1}\\right)\\left(\\phi \\left(c\\right)\\right)-{m}_{c}\\left(u-\\phi \\left(c\\right)\\right)\\ge 0\\phantom{\\rule{1em}{0ex}}\\left(resp.>0\\right).$\n\nSince u = φ(x), we have\n\n$\\psi \\left(x\\right)-\\psi \\left(c\\right)-{m}_{c}\\left(\\phi \\left(x\\right)-\\phi \\left(c\\right)\\right)\\ge 0\\phantom{\\rule{1em}{0ex}}\\left(resp.>0\\right).$\n1. (ii)\n\n(iii). Take u φ(I) and d (φ(I)) arbitrarily. Put x = φ -1 (u) and c = φ -1 (d). Then x I and c I°. If we can find a real constant m c which is independent of u such that\n\n$\\psi \\left(x\\right)-\\psi \\left(c\\right)-{m}_{c}\\left(\\phi \\left(x\\right)-\\phi \\left(c\\right)\\right)\\ge 0\\phantom{\\rule{1em}{0ex}}\\left(resp.>0\\right),$\n\nthen\n\n$\\left(\\psi \\circ {\\phi }^{-1}\\right)\\left(u\\right)-\\left(\\psi \\circ {\\phi }^{-1}\\right)\\left(d\\right)-{m}_{c}\\left(u-d\\right)\\ge 0\\phantom{\\rule{1em}{0ex}}\\left(resp.>0\\right),$\n\nand hence ψ φ-1 is convex (resp. strictly convex) on φ(I) by Lemma 1.\n\n1. (iii)\n\n(i). Take c I° and x I\\{c} arbitrarily. Put u = φ(x) and d = φ(c).\n\nThen u φ(I)\\{d} and d (φ(I)), hence\n\n$\\left({\\lambda }_{c,\\phi ,\\psi }\\circ {\\phi }^{-1}\\right)\\left(u\\right)=\\frac{\\left(\\psi \\circ {\\phi }^{-1}\\right)\\left(u\\right)-\\left(\\psi \\circ {\\phi }^{-1}\\right)\\left(d\\right)}{u-d}.$\n(2)\n\nIf ψ φ-1 is convex (resp. strictly convex) on φ(I), then by (2) and Lemma 11, λ c,φ,ψ φ-1 and hence λ c,φ,ψ is monotone increasing (resp. strictly monotone increasing) on I\\{c}. □\n\nFor each φ C sm (I), t [0, 1] and x, y I, put\n\n$x{\\nabla }_{t,\\phi }y={\\phi }^{-1}\\left(\\left(1-t\\right)\\phi \\left(x\\right)+t\\phi \\left(y\\right)\\right).$\n\nThis can be regarded as a φ-mean of {x, y} with respect to a probability measure which represents a weighted arithmetic mean (1-t) x + ty.\n\nFor each φ C sm (I), denote by φ a three variable real-valued function:\n\n$\\left(t,x,y\\right)\\to x{\\nabla }_{t,\\phi }y$\n\non (0, 1) × {(x, y) I2 : xy}. For each φ, ψ C sm (I), we write φ ψ (resp. φ < ψ ) if\n\nfor all t (0, 1) and x, y I with xy.\n\nRemark. The continuity of φ implies that φ ψ (resp. φ < ψ ) if and only if\n\nfor all x, y I with xy.\n\nThese order relations \"≤\" and \"<\" play an important role in our discussion.\n\nLemma 3. Let φ, ψ C sm (I). Then\n\n1. (i)\n\nφ = ψ holds if and only if ψ = + b for some a, b with a ≠ 0.\n\n2. (ii)\n\nIf $\\psi \\in {C}_{sm}^{+}\\left(I\\right)$, then φ ψ (resp. φ < ψ ) holds if and only if ψ φ -1 is convex (resp. strictly convex) on φ(I).\n\n3. (iii)\n\nIf $\\psi \\in {C}_{sm}^{-}\\left(I\\right)$, then φ ψ (resp. φ < ψ ) holds if and only if ψ φ -1 is concave (resp. strictly concave) on φ(I).\n\nProof. (i) Suppose that φ = ψ holds. Take u, v φ(I) with uv arbitrarily and put x = φ-1(u) and y = φ-1(v), hence xy. By hypothesis,\n\n$\\psi \\left({\\phi }^{-1}\\left(\\left(1-t\\right)u+tv\\right)\\right)=\\left(1-t\\right)\\psi \\left({\\phi }^{-1}\\left(u\\right)\\right)+t\\psi \\left({\\phi }^{-1}\\left(v\\right)\\right)$\n\nfor all t (0, 1). This means that ψ φ-1 is convex and concave on φ(I) and hence we can write ψ(φ-1(u)) = au + b for all u φ(I) and some a, b . Therefore, ψ(x) = (x) + b for all x I. Since ψ is non-constant, it follows that a ≠ 0.\n\nThe reverse assertion is straightforward.\n\n1. (ii)\n\nAssume that ψ is monotone increasing. Take u, v φ(I) with uv arbitrarily and put x = φ -1(u) and y = φ -1(v), hence xy. If φ ψ holds, then\n\n$\\psi \\left({\\phi }^{-1}\\left(\\left(1-t\\right)\\phi \\left(x\\right)+t\\phi \\left(y\\right)\\right)\\right)\\le \\left(1-t\\right)\\psi \\left(x\\right)+t\\psi \\left(y\\right)$\n\nand hence\n\n$\\psi \\left({\\phi }^{-1}\\left(\\left(1-t\\right)u+tv\\right)\\right)\\le \\left(1-t\\right)\\psi \\left({\\phi }^{-1}\\left(u\\right)\\right)+t\\psi \\left({\\phi }^{-1}\\left(v\\right)\\right)$\n\nfor all t (0, 1). This means that ψ φ-1 is convex.\n\nConversely, if ψ φ-1 is convex, we see that φ ψ holds by observing the reverse of the above proof.\n\nAlso a similar observation implies that φ < ψ holds if and only if ψ φ-1 is strictly convex on I.\n\n1. (iii)\n\nAssume that ψ is monotone decreasing. Then -ψ is monotone increasing. Hence, by (ii), we have that φ -ψ(resp. φ < ) holds if and only if (-ψ) φ -1 is convex (resp. strictly convex) on φ(I). However, since ψ = -ψholds by (i) and (-ψ) φ -1 is convex (resp. strictly convex) on φ(I) iff ψ φ -1 is concave (resp. strictly concave) on φ(I), we obtain the desired result. □\n\nLemma 4. Let$\\phi ,\\psi \\in {C}_{sm}^{+}\\left(I\\right)$(or${C}_{sm}^{-}\\left(I\\right)$) with φ < ψ . For each s [0, 1], define ξ s = (1 - s) φ + . Then\n\n1. (i)\n\nEach ξ s belongs to ${C}_{sm}^{+}\\left(I\\right)$ (resp. ${C}_{sm}^{-}\\left(I\\right)$ ) when $\\phi ,\\psi \\in {C}_{sm}^{+}\\left(I\\right)$ (resp. ${C}_{sm}^{-}\\left(I\\right)$ ).\n\n2. (ii)\n\nFor each t (0, 1) and x, y I with xy, a function $s\\to x{\\nabla }_{t,{\\xi }_{s}}y$ is strictly monotone increasing on [0, 1].\n\nProof. (i) Straightforward.\n\n1. (ii)\n\nAssume $\\phi ,\\psi \\in {C}_{sm}^{+}\\left(I\\right)$ with φ < ψ . Take t (0, 1) and x, y I with xy arbitrarily. To show that a function $s\\to x{\\nabla }_{t,{\\xi }_{s}}y$ is strictly monotone increasing on [0, 1], let 0 ≤ s 1 < s 2 ≤ 1. Take c I arbitrarily. Since φ < ψ holds, it follows from Lemmas 2 and 3 that λ c,φ,ψ is strictly monotone increasing on I\\{c}. Moreover, we have\n\n$\\begin{array}{lll}\\hfill {\\lambda }_{c,{\\xi }_{{s}_{1}},{\\xi }_{{s}_{2}}}\\left(x\\right)& =\\frac{{\\xi }_{{s}_{2}}\\left(x\\right)-{\\xi }_{{s}_{2}}\\left(c\\right)}{{\\xi }_{{s}_{1}}\\left(x\\right)-{\\xi }_{{s}_{1}}\\left(c\\right)}\\phantom{\\rule{2em}{0ex}}& \\hfill \\text{(1)}\\\\ =\\frac{{s}_{2}\\left(\\psi \\left(x\\right)-\\psi \\left(c\\right)\\right)+\\left(1-{s}_{2}\\right)\\left(\\phi \\left(x\\right)-\\phi \\left(c\\right)\\right)}{{s}_{1}\\left(\\psi \\left(x\\right)-\\psi \\left(c\\right)\\right)+\\left(1-{s}_{1}\\right)\\left(\\phi \\left(x\\right)-\\phi \\left(c\\right)\\right)}\\phantom{\\rule{2em}{0ex}}& \\hfill \\text{(2)}\\\\ =\\frac{{s}_{2}{\\lambda }_{c,\\phi ,\\psi }\\left(x\\right)+1-{s}_{2}}{{s}_{1}{\\lambda }_{c,\\phi ,\\psi }\\left(x\\right)+1-{s}_{1}}.\\phantom{\\rule{2em}{0ex}}& \\hfill \\text{(3)}\\\\ \\hfill \\text{(4)}\\end{array}$\n\nfor each x I\\{c}. Therefore, we have\n\n${\\lambda }_{c,{\\xi }_{{s}_{1}},{\\xi }_{{s}_{2}}}\\left(x\\right)={s}_{2}{\\lambda }_{c,\\phi ,\\psi }\\left(x\\right)+1-{s}_{2}\\phantom{\\rule{1em}{0ex}}\\left({s}_{1}=0\\right)$\n(3)\n\nand\n\n${\\lambda }_{c,{\\xi }_{{s}_{1}},{\\xi }_{{s}_{2}}}\\left(x\\right)=\\frac{{s}_{2}}{{s}_{1}}-\\frac{{s}_{2}-{s}_{1}}{{s}_{1}^{2}}\\frac{1}{{\\lambda }_{c,\\phi ,\\psi }\\left(x\\right)-\\frac{{s}_{1}-1}{{s}_{1}}}\\phantom{\\rule{1em}{0ex}}\\left({s}_{1}\\ne 0\\right)$\n(4)\n\nfor each x I\\{c}. If s1 = 0, then it is trivial by (3) that ${\\lambda }_{c,{\\xi }_{{s}_{1}},{\\xi }_{{s}_{2}}}$ is strictly monotone increasing on I\\{c}. If s1 ≠ 0, then\n\n$\\frac{{s}_{1}-1}{{s}_{1}}<0<{\\lambda }_{c,\\phi .\\psi }\\left(x\\right)$\n\nfor all x I\\{c}. So, by (4), ${\\lambda }_{c,{\\xi }_{{s}_{1}},{\\xi }_{{s}_{2}}}$ is also strictly monotone increasing on I\\{c}. Hence we see that ${\\nabla }_{{\\xi }_{{s}_{1}}}<{\\nabla }_{{\\xi }_{{s}_{2}}}$ holds by (i), Lemmas 2 and 3. This implies that $x{\\nabla }_{t,{\\xi }_{{s}_{1}}}y. Then a function $s\\to x{\\nabla }_{t,{\\xi }_{s}}y$ is strictly monotone increasing on [0, 1], as required.\n\nFor the case of $\\phi ,\\psi \\in {C}_{sm}^{-}\\left(I\\right)$, since $-\\phi ,-\\psi \\in {C}_{sm}^{+}\\left(I\\right)$, it follows from the above discussion that a function $s\\to x{\\nabla }_{t,-{\\xi }_{s}}y$ is strictly monotone increasing on [0, 1]. However, by Lemma 3-(i), $x{\\nabla }_{t,-{\\xi }_{s}}y=x{\\nabla }_{t,{\\xi }_{s}}y$, where t (0, 1) and x, y I with xy, and then we obtain the desired result. □\n\nLemma 5. Let φ and ψ be two functions on I such that ψ - φ is strictly monotone increasing (resp. decreasing) on I and ψ is convex (resp. concave) on I. Then\n\n$\\left(1-t\\right)\\phi \\left(x\\right)+t\\psi \\left(y\\right)-\\left(\\left(1-t\\right)\\phi +t\\psi \\right)\\left(\\left(1-t\\right)x+ty\\right)>0\\phantom{\\rule{0.3em}{0ex}}\\left(resp.<0\\right)$\n\nholds for all t (0, 1) and x, y I with x < y.\n\nProof. Let x, y I with x < y and t (0, 1). Put z = (1 - t)x + ty. Then, we must show that (1 - t) φ(x) + (y) - ((1 - t)φ + )(z) > 0 (resp. < 0). Since x < z < y and ψ- φ is strictly monotone increasing (resp. decreasing) on I, it follows that\n\n$\\psi \\left(z\\right)-\\psi \\left(x\\right)-\\phi \\left(z\\right)+\\phi \\left(x\\right)>0\\phantom{\\rule{0.3em}{0ex}}\\left(resp.<0\\right).$\n\nAlso since ψ is convex (resp. concave) on I, it follows from Lemma 1 that λz,ψis monotone increasing (resp. decreasing). Therefore, we have\n\n$\\begin{array}{c}\\left(1-t\\right)\\phi \\left(x\\right)+t\\psi \\left(y\\right)-\\left(\\left(1-t\\right)\\phi +t\\psi \\right)\\left(z\\right)\\\\ \\phantom{\\rule{1em}{0ex}}=t\\left(\\psi \\left(y\\right)-\\psi \\left(z\\right)\\right)-\\left(1-t\\right)\\left(\\phi \\left(z\\right)-\\phi \\left(x\\right)\\right)\\\\ \\phantom{\\rule{1em}{0ex}}>t\\left(\\psi \\left(y\\right)-\\psi \\left(z\\right)\\right)-\\left(1-t\\right)\\left(\\psi \\left(z\\right)-\\psi \\left(x\\right)\\right)\\\\ \\phantom{\\rule{1em}{0ex}}\\left(resp.<\\right)\\\\ \\phantom{\\rule{1em}{0ex}}=t\\left(1-t\\right)\\left(y-x\\right)\\left(\\frac{\\psi \\left(y\\right)-\\psi \\left(z\\right)}{\\left(1-t\\right)\\left(y-x\\right)}-\\frac{\\psi \\left(z\\right)-\\psi \\left(x\\right)}{t\\left(y-x\\right)}\\right)\\\\ \\phantom{\\rule{1em}{0ex}}=t\\left(1-t\\right)\\left(y-x\\right)\\left(\\frac{\\psi \\left(y\\right)-\\psi \\left(z\\right)}{y-z}-\\frac{\\psi \\left(x\\right)-\\psi \\left(z\\right)}{x-z}\\right)\\\\ \\phantom{\\rule{1em}{0ex}}=t\\left(1-t\\right)\\left(y-x\\right)\\left({\\lambda }_{z,\\psi }\\left(y\\right)-{\\lambda }_{z,\\psi }\\left(x\\right)\\right)\\\\ \\phantom{\\rule{1em}{0ex}}\\ge 0\\\\ \\phantom{\\rule{1em}{0ex}}\\left(resp.\\le 0\\right),\\end{array}$\n\nso that (1 - t) φ(x) + (y) - ((1 - t)φ + )(z) > 0 (resp. < 0), as required. □\n\nThe following lemma gives an equality condition of Jensen's inequality. For the sake of completeness, we will give a proof.\n\nLemma 6. Let δ be a strictly convex or strictly concave function on I. Suppose that g is a real-valued integrable function on Ω such that g(ω) I for almost all ω Ω and δ g L1 (Ω, μ). Then $\\delta \\left(\\int gd\\mu \\right)=\\int \\left(\\delta \\circ g\\right)d\\mu$ if and only if g is a constant function.\n\nProof. We first consider the strictly convex case. Put $c=\\int g\\mathsf{\\text{d}}\\mu$. If c = inf I, then cg(ω) for almost all ω Ω and so g(ω) = c must hold for almost all ω Ω. Similarly, if c = max I, then g(ω) = c for almost all ω Ω. Therefore, we can without loss of generality assume that c belongs to I. Since δ is strictly convex, we can from Lemma 1 find a real constant m c such that\n\n$\\delta \\left(x\\right)>{m}_{c}\\left(x-c\\right)+\\delta \\left(c\\right)$\n(5)\n\nfor all x I\\{c}. Replacing x by g(ω) in (5), we obtain\n\n$\\delta \\left(g\\left(\\omega \\right)\\right)\\ge {m}_{c}\\left(g\\left(\\omega \\right)-c\\right)+\\delta \\left(c\\right)$\n\nfor almost all ω Ω. Integrating both sides of this equation, we have\n\n$\\int \\left(\\delta \\circ g\\right)\\mathsf{\\text{d}}\\mu \\ge \\int \\left({m}_{c}\\left(g-c\\right)+\\delta \\left(c\\right)\\right)\\mathsf{\\text{d}}\\mu =\\delta \\left(c\\right)=\\delta \\left(\\int g\\mathsf{\\text{d}}\\mu \\right).$\n(6)\n\nNow assume that $\\delta \\left(\\int g\\mathsf{\\text{d}}\\mu \\right)=\\int \\left(\\delta \\circ g\\right)\\mathsf{\\text{d}}\\mu$. Then (6) implies that\n\n$\\delta \\left(g\\left(\\omega \\right)\\right)={m}_{c}\\left(g\\left(\\omega \\right)-c\\right)+\\delta \\left(c\\right)$\n\nfor almost all ω Ω. If μ({gc}) > 0, then we can find ω c Ω such that δ(g(ω c )) = m c (g(ω c ) - c) + δ(c) and g(ω c ) ≠ c. This contradicts (5) and hence g(ω) = c for almost all ω Ω.\n\nConversely, assume that g is a constant function on Ω. Then it is trivial that $\\delta \\left(\\int g\\mathsf{\\text{d}}\\mu \\right)=\\int \\left(\\delta \\circ g\\right)\\mathsf{\\text{d}}\\mu$.\n\nFor the strictly concave case, since -δ is strictly convex on I, it follows from the above discussion that $-\\delta \\left(\\int g\\mathsf{\\text{d}}\\mu \\right)=\\int \\left(-\\delta \\circ g\\right)\\mathsf{\\text{d}}\\mu$ iff g is a constant function on Ω. However, since $-\\delta \\left(\\int g\\mathsf{\\text{d}}\\mu \\right)=\\int \\left(-\\delta \\circ g\\right)\\mathsf{\\text{d}}\\mu$ iff $\\delta \\left(\\int g\\mathsf{\\text{d}}\\mu \\right)=\\int \\left(\\delta \\circ g\\right)\\mathsf{\\text{d}}\\mu$, we obtain the desired result. □\n\nLemma 7. Suppose that f is non-constant and φ, ψ C sm,f (I). Then\n\n1. (i)\n\nIf either ψ φ -1 is convex (resp. strictly convex) on φ(I) and $\\psi \\in {C}_{sm}^{+}\\left(I\\right)$ or ψ φ -1 is concave (resp. strictly concave) on φ(I) and $\\psi \\in {C}_{sm}^{-}\\left(I\\right)$, then\n\n${M}_{\\phi }\\left(f\\right)\\le {M}_{\\psi }\\left(f\\right)\\phantom{\\rule{1em}{0ex}}\\left(resp.\\phantom{\\rule{0.3em}{0ex}}{M}_{\\phi }\\left(f\\right)<{M}_{\\psi }\\left(f\\right)\\right)$\n\nholds.\n\n1. (ii)\n\nIf either ψ φ -1 is convex (resp. strictly convex) on φ(I) and $\\psi \\in {C}_{sm}^{-}\\left(I\\right)$ or ψ φ -1 is concave (resp. strictly concave) on φ(I) and $\\psi \\in {C}_{sm}^{+}\\left(I\\right)$, then\n\n${M}_{\\phi }\\left(f\\right)\\ge {M}_{\\psi }\\left(f\\right)\\phantom{\\rule{1em}{0ex}}\\left(resp.\\phantom{\\rule{0.3em}{0ex}}{M}_{\\phi }\\left(f\\right)>{M}_{\\psi }\\left(f\\right)\\right)$\n\nholds.\n\nProof. (i) Put δ = ψ φ-1 and g = φ f. Assume that g is convex on φ(I) and $\\psi \\in {C}_{sm}^{+}\\left(I\\right)$. Since g and δ g are integrable functions on Ω, we have\n\n$\\delta \\left(\\int g\\mathsf{\\text{d}}\\mu \\right)\\le \\int \\left(\\delta \\circ g\\right)\\mathsf{\\text{d}}\\mu$\n(7)\n\nby Jensen's inequality. This means M φ (f) ≤ M ψ (f) because ψ is monotone increasing on I.\n\nNext assume that g is concave on φ(I) and $\\psi \\in {C}_{sm}^{-}\\left(I\\right)$. Then\n\n$\\delta \\left(\\int g\\mathsf{\\text{d}}\\mu \\right)\\ge \\int \\left(\\delta \\circ g\\right)\\mathsf{\\text{d}}\\mu$\n(8)\n\nby Jensen's inequality. This also means M φ (f) ≤ M ψ (f) because ψ is monotone decreasing on I.\n\nFor the strict case, since g is a non-constant function on Ω, we obtain the desired results from (7), (8), Lemma 6 and the above argument. □\n\n1. (ii)\n\nSimilarly.\n\n## 3. Main results\n\nIn this section, we first give a new interpretation of Jensen's inequality by φ-mean. Next, as an application, we consider some geometric properties of φ-means of a real-valued measurable function f on Ω.\n\nThe first result asserts that a φ-mean function: φ M φ (f) is well defined and order preserving, and this assertion simultaneously gives a new interpretation of Jensen's inequality. However, this assertion also teaches us that a simple inequality yields a complicated inequality.\n\nTheorem 1. Suppose that f is non-constant and φ, ψ C sm,f (I). Then\n\n1. (i)\n\nIf φ ψ holds, then M φ (f) ≤ M ψ (f).\n\n2. (ii)\n\nIf φ < ψ holds, then M φ (f) < M ψ (f).\n\nProof. (i) Suppose that φ ψ holds. If ψ is monotone increasing on I, then ψ φ-1 is convex on φ(I) by Lemma 3-(ii). Therefore, we have M φ (f) ≤ M ψ (f) by Lemma 7-(i). If ψ is monotone decreasing on I, then ψ φ-1 is concave on φ(I) by Lemma 3-(iii). Therefore, we have M φ (f) ≤ M ψ (f) by Lemma 7-(i).\n\n1. (ii)\n\nSimilarly. □\n\nLet φ, ψ C sm,f (I) and t (0, 1). Then, we can easily see that if either both φ and ψ are monotone increasing or both φ and ψ are monotone decreasing, then (1 - t)φ + is also an element of C sm,f (I) [cf. Lemma 4-(i)]. The next result asserts that there is a strictly monotone increasing φ-mean (continuous) path between two φ-means.\n\nTheorem 2. Suppose that f is non-constant and φ, ψ C sm,f (I) with φ < ψ .\n\n1. (i)\n\nIf $\\phi ,\\psi \\in {C}_{sm}^{+}\\left(I\\right)$ [or ${C}_{sm}^{-}\\left(I\\right)$ ], then a function: s → M (1-s)φ+(f) is strictly monotone increasing on [0, 1].\n\n2. (ii)\n\nIf $\\phi ,\\psi -\\phi \\in {C}_{sm}^{+}\\left(I\\right)$ [resp. ${C}_{sm}^{-}\\left(I\\right)$] and ψ(x) - φ(x) ≥ 0 (resp. ≤ 0) for all x I, then a function: s → M (1-s)φ+(f) is strictly monotone increasing and continuous on [0.1].\n\nProof. (i) Suppose that $\\phi ,\\psi \\in {C}_{sm}^{+}\\left(I\\right)$[or${C}_{sm}^{-}\\left(I\\right)$]. For each s [0, 1], define ξs = (1 - s)φ + s ψ . Let 0 ≤ s1 < s2 ≤ 1. Then, we must show that ${M}_{{\\xi }_{{s}_{1}}}\\left(f\\right)<{M}_{{\\xi }_{{s}_{2}}}\\left(f\\right)$. By Lemma 4-(ii), a function $s\\to x{\\nabla }_{t,{\\xi }_{s}}y$ is strictly monotone increasing on [0, 1] for each t (0, 1) and x, y I with xy, and hence we see that ${\\nabla }_{{\\xi }_{{s}_{1}}}<{\\nabla }_{{\\xi }_{{s}_{2}}}$ holds. Therefore, we have from Theorem 1-(ii) that ${M}_{{\\xi }_{{s}_{1}}}\\left(f\\right)<{M}_{{\\xi }_{{s}_{2}}}\\left(f\\right)$, as required.\n\n1. (ii)\n\nSuppose that $\\phi ,\\psi -\\phi \\in {C}_{sm}^{+}\\left(I\\right)$ and φ(x) ≤ ψ(x) for all x I. Since ψ = φ + (ψ- φ), it follows that $\\psi \\in {C}_{sm}^{+}\\left(I\\right)$. For each s [0, 1], put α s = M(1-s)φ+ (f). Then, we must show that a function sα s is continuous on [0, 1]. To do this, take 0 ≤ s < t ≤ 1 arbitrarily. By (i), we have α s < α t . Note that\n\n$\\left(1-t\\right)\\phi \\left({\\alpha }_{t}\\right)+t\\psi \\left({\\alpha }_{t}\\right)=\\left(1-t\\right)\\int \\left(\\phi \\circ f\\right)\\mathsf{\\text{d}}\\mu +t\\int \\left(\\psi \\circ f\\right)\\mathsf{\\text{d}}\\mu$\n\nand\n\n$\\left(1-s\\right)\\phi \\left({\\alpha }_{s}\\right)+s\\psi \\left({\\alpha }_{s}\\right)=\\left(1-s\\right)\\int \\left(\\phi \\circ f\\right)\\mathsf{\\text{d}}\\mu +s\\int \\left(\\psi \\circ f\\right)\\mathsf{\\text{d}}\\mu .$\n\nTherefore, we have\n\n$\\phi \\left({\\alpha }_{t}\\right)-\\phi \\left({\\alpha }_{s}\\right)+t\\left(\\psi -\\phi \\right)\\left({\\alpha }_{t}\\right)-s\\left(\\psi -\\phi \\right)\\left({\\alpha }_{s}\\right)=\\left(t-s\\right)\\left(\\int \\left(\\left(\\psi -\\phi \\right)\\circ f\\right)\\mathsf{\\text{d}}\\mu \\right).$\n(9)\n\nSince $\\phi ,\\psi -\\phi \\in {C}_{sm}^{+}\\left(I\\right)$ and φ(x) ≤ ψ(x) for all x I by hypothesis, it follows that\n\n$\\phi \\left({\\alpha }_{t}\\right)-\\phi \\left({\\alpha }_{s}\\right)>0\\phantom{\\rule{1em}{0ex}}and\\phantom{\\rule{1em}{0ex}}t\\left(\\psi -\\phi \\right)\\left({\\alpha }_{t}\\right)-s\\left(\\psi -\\phi \\right)\\left({\\alpha }_{s}\\right)>0.$\n\nHence, after taking the limit with respect to s in the Eq. (9), we obtain\n\n$\\underset{s\\to t-0}{lim}\\phi \\left({\\alpha }_{s}\\right)=\\phi \\left({\\alpha }_{t}\\right)\\phantom{\\rule{1em}{0ex}}and\\phantom{\\rule{1em}{0ex}}\\underset{s\\to t-0}{lim}s\\left(\\psi -\\phi \\right)\\left({\\alpha }_{s}\\right)=t\\left(\\psi -\\phi \\right)\\left({\\alpha }_{t}\\right).$\n\nHowever, since φ-1 is continuous on φ(I), we conclude that\n\n$\\underset{s\\to t-0}{lim}{\\alpha }_{s}={\\alpha }_{t}.$\n\nSimilarly, after taking the limit with respect to t in the Eq. (9), we obtain\n\n$\\underset{t\\to s+0}{lim}{\\alpha }_{t}={\\alpha }_{s}.$\n\nThese observations imply that a function sα s is continuous on [0, 1], as required.\n\nFor the case that $\\phi ,\\psi -\\phi \\in {C}_{sm}^{-}\\left(I\\right)$ and φ(x) ≥ ψ(x) for all x I, a similar argument above implies that a function sα s is also continuous on [0, 1]. □\n\nThe next result asserts that the φ-mean function is strictly concave (or convex) on a suitable convex subset of C sm,f (I).\n\nTheorem 3. Suppose that f is non-constant and φ, ψ C sm,f (I) with φ < ψ . Then\n\n1. (i)\n\nIf $\\phi ,\\psi -\\phi \\in {C}_{sm}^{+}\\left(I\\right)$ (resp. ${C}_{sm}^{-}\\left(I\\right)$ ) and ψ is convex (resp. concave) on I, then\n\n$\\left(1-t\\right){M}_{\\phi }\\left(f\\right)+t{M}_{\\psi }\\left(f\\right)<{M}_{\\left(1-t\\right)\\phi +t\\psi }\\left(f\\right)$\n\nholds for all t (0, 1).\n\n1. (ii)\n\nIf $\\psi ,\\phi -\\psi \\in {C}_{sm}^{-}\\left(I\\right)$ (resp. ${C}_{sm}^{+}\\left(I\\right)$ ) and ψ is convex (resp. concave) on I, then\n\n$\\left(1-t\\right){M}_{\\phi }\\left(f\\right)+t{M}_{\\psi }\\left(f\\right)>{M}_{\\left(1-t\\right)\\phi +t\\psi }\\left(f\\right)$\n\nholds for all t (0, 1).\n\nProof. (i) Suppose that $\\phi ,\\psi -\\phi \\in {C}_{sm}^{+}\\left(I\\right)$[resp. ${C}_{sm}^{-}\\left(I\\right)$] and ψ is convex [resp. concave] on I. Since ψ = φ + (ψ- φ), it follows from hypothesis that $\\psi \\in {C}_{sm}^{+}\\left(I\\right)$ [resp. ${C}_{sm}^{-}\\left(I\\right)$]. Put x = M φ (f) and y = M ψ (f), and so x < y by Theorem 1-(ii). Also, we have from definition that\n\n$\\phi \\left(x\\right)=\\int \\left(\\phi \\circ f\\right)\\mathsf{\\text{d}}\\mu \\phantom{\\rule{1em}{0ex}}and\\phantom{\\rule{1em}{0ex}}\\psi \\left(y\\right)=\\int \\left(\\psi \\circ f\\right)\\mathsf{\\text{d}}\\mu .$\n\nLet 0 < t < 1 and put u = M(1-t)φ+(f). Then, we have\n\n$\\left(\\left(1-t\\right)\\phi +t\\psi \\right)\\left(u\\right)=\\int \\left(\\left(\\left(1-t\\right)\\phi +t\\psi \\right)\\circ f\\right)\\mathsf{\\text{d}}\\mu$\n\nby definition. Therefore,\n\n$\\begin{array}{lll}\\hfill \\left(1-t\\right)\\phi \\left(x\\right)+t\\psi \\left(y\\right)& =\\left(1-t\\right)\\int \\left(\\phi \\circ f\\right)\\mathsf{\\text{d}}\\mu +t\\int \\left(\\psi \\circ f\\right)\\mathsf{\\text{d}}\\mu \\phantom{\\rule{2em}{0ex}}& \\hfill \\text{(1)}\\\\ =\\int \\left(\\left(\\left(1-t\\right)\\phi +t\\psi \\right)\\circ f\\right)\\mathsf{\\text{d}}\\mu \\phantom{\\rule{2em}{0ex}}& \\hfill \\text{(2)}\\\\ =\\left(\\left(1-t\\right)\\phi +t\\psi \\right)\\left(u\\right).\\phantom{\\rule{2em}{0ex}}& \\hfill \\text{(3)}\\\\ \\hfill \\text{(4)}\\end{array}$\n\nPut z = (1 - t)x + ty. Then, by the above equality and Lemma 5, we have\n\n$\\left(\\left(1-t\\right)\\phi +t\\psi \\right)\\left(z\\right)<\\left(resp.>\\right)\\left(1-t\\right)\\phi \\left(x\\right)+t\\psi \\left(y\\right)=\\left(\\left(1-t\\right)\\phi +t\\psi \\right)\\left(u\\right).$\n\nSince (1 - t)φ + is strictly increasing (resp. decreasing), it follows that z < u, that is,\n\n$\\left(1-t\\right)x+ty\n\nThis means that (1 - t)M φ (f) + tM ψ (f) < M(1-t)φ+(f).\n\n1. (ii)\n\nSimilarly.\n\nRemark. It seems that Theorem 3 is slightly related to [3, 4] which discuss a comparison between a convex linear combination of the arithmetic and geometric means and the generalized logarithmic mean.\n\nThe following result describes a certain boundedness of φ-means.\n\nTheorem 4. Suppose that f is non-constant and φ, ψ C sm,f (I) with φ < ψ .\n\n1. (i)\n\nIf $\\phi ,\\psi -\\phi \\in {C}_{sm}^{+}\\left(I\\right)$ [or ${C}_{sm}^{-}\\left(I\\right)$ ], then a function: sM (1-s)φ+(f) is strictly monotone increasing on [0, ∞) and\n\n$\\underset{s\\to \\infty }{lim}{M}_{\\left(1-s\\right)\\phi +s\\psi }\\left(f\\right)={M}_{\\psi -\\phi }\\left(f\\right).$\n2. (ii)\n\nIf $\\phi ,\\psi -\\phi \\in {C}_{sm}^{+}\\left(I\\right)$ [resp. ${C}_{sm}^{-}\\left(I\\right)$ ] and ψ(x) - φ(x) ≥ 0 (resp. ≤ 0) for all x I, then a function: sM (1-s)φ+(f) is strictly monotone increasing and continuous on [0, ∞).\n\nProof. (i) Suppose that $\\phi ,\\psi -\\phi \\in {C}_{sm}^{+}\\left(I\\right)$. For each s ≥ 1, put ξ s = (1 - s)φ + . Since ξ s = φ + s(ψ - φ), it follows from hypothesis that each ξ s is in ${C}_{sm}^{+}\\left(I\\right)$, and then ξ s C sm,f (I). Since ψ = φ + (ψ- φ), it follows from hypothesis that ψ is also in ${C}_{sm}^{+}\\left(I\\right)$. Then by Lemmas 2 and 3, we have that λ c,φ,ψ is strictly monotone increasing on I\\{c} for any c I°. Let 1 ≤ s1 < s2 < ∞ and take c I° arbitrarily. In this case, we obtain the equality (4), as observe in the proof of Lemma 4-(ii). Note that\n\n$\\frac{{s}_{1}-1}{{s}_{1}}<1<{\\lambda }_{c,\\phi ,\\psi }\\left(x\\right)$\n\nfor all x I\\{c}. So, by (4), ${\\lambda }_{c,{\\xi }_{{s}_{1}},{\\xi }_{{s}_{2}}}$is also strictly monotone increasing on I\\{c}. Then by Lemmas 2 and 3, we conclude that ${\\nabla }_{{\\xi }_{{s}_{1}}}<{\\nabla }_{{\\xi }_{{s}_{2}}}$. Therefore, we have from Theorem 1-(ii) that ${M}_{{\\xi }_{{s}_{1}}}\\left(f\\right)<{M}_{{\\xi }_{{s}_{2}}}\\left(f\\right)$ and then a function: sM(1-s)φ+(f) is strictly monotone increasing on [1, ∞) and hence [0, ∞) by Theorem 2-(i).\n\nMoreover, we can easily see that\n\n${\\lambda }_{c,{\\xi }_{s},\\psi -\\phi }\\left(x\\right)=\\frac{1}{s}-\\frac{1}{{s}^{2}}\\frac{1}{{\\lambda }_{c,\\phi ,\\psi }\\left(x\\right)-\\frac{s-1}{s}}$\n\nand\n\n$\\frac{s-1}{s}<1<{\\lambda }_{c,\\phi ,\\psi }\\left(x\\right)$\n\nfor all s ≥ 1, x I\\{c} and c I°. This implies that ${\\lambda }_{c,{\\xi }_{s},\\psi -\\phi }$ is strictly monotone increasing on I\\{c} for each s ≥ 1 and c I°. Then by Lemmas 2 and 3, we conclude that ${\\nabla }_{{\\xi }_{s}}<{\\nabla }_{\\psi -\\phi }$ for each s ≥ 1. Therefore, we have from Theorem 1-(ii) that ${M}_{{\\xi }_{s}}\\left(f\\right)<{M}_{\\psi -\\phi }\\left(f\\right)$ for each s ≥ 1.\n\nNow take s ≥ 1 arbitrarily and put ${\\alpha }_{s}={M}_{{\\xi }_{s}}\\left(f\\right)$ and α = Mψ-φ(f), so α s < α. Since $\\phi ,\\psi -\\phi \\in {C}_{sm}^{+}\\left(I\\right)$, it follows that φ(α s ) < φ(α) and (ψ- φ)(α s ) < (ψ- φ)(α). By definition, we have\n\n$\\left(\\psi -\\phi \\right)\\left(\\alpha \\right)=\\int \\left(\\left(\\psi -\\phi \\right)\\circ f\\right)\\mathsf{\\text{d}}\\mu .$\n\nAlso since ${\\xi }_{s}=s\\left(\\frac{1}{s}\\phi +\\psi -\\phi \\right)$, it follows from an invariant property of φ-mean that ${\\alpha }_{s}={M}_{\\frac{1}{s}\\phi +\\psi -\\phi }\\left(f\\right)$ and then\n\n$\\frac{1}{s}\\phi \\left({\\alpha }_{s}\\right)+\\left(\\psi -\\phi \\right)\\left({\\alpha }_{s}\\right)=\\int \\left(\\frac{1}{s}\\phi +\\psi -\\phi \\right)\\circ f\\mathsf{\\text{d}}\\mu$\n\nTherefore, we have\n\n$\\begin{array}{lll}\\hfill 0& <\\left(\\psi -\\phi \\right)\\left(\\alpha \\right)-\\left(\\psi -\\phi \\right)\\left({\\alpha }_{s}\\right)\\phantom{\\rule{2em}{0ex}}& \\hfill \\text{(1)}\\\\ =\\frac{1}{s}\\left(\\phi \\left({\\alpha }_{s}\\right)-\\int \\left(\\phi \\circ f\\right)\\mathsf{\\text{d}}\\mu \\right)\\phantom{\\rule{2em}{0ex}}& \\hfill \\text{(2)}\\\\ <\\frac{1}{s}\\left(\\phi \\left(\\alpha \\right)-\\int \\left(\\phi \\circ f\\right)\\mathsf{\\text{d}}\\mu \\right).\\phantom{\\rule{2em}{0ex}}& \\hfill \\text{(3)}\\\\ \\hfill \\text{(4)}\\end{array}$\n\nHence, after taking the limit with respect to s, we obtain\n\n$\\underset{s\\to \\infty }{lim}\\left(\\psi -\\phi \\right)\\left({\\alpha }_{s}\\right)=\\left(\\psi -\\phi \\right)\\left(\\alpha \\right).$\n\nHowever, since (ψ- φ)-1 is continuous on (ψ- φ)(I), we conclude that\n\n$\\underset{s\\to \\infty }{lim}{\\alpha }_{s}=\\alpha ,$\n\nthat is,\n\n$\\underset{s\\to \\infty }{lim}{M}_{\\left(1-s\\right)\\phi +s\\psi }\\left(f\\right)={M}_{\\psi -\\phi }\\left(f\\right).$\n\nFor the decreasing case, replacing φ and ψ by -φ and -ψ, respectively, apply the above discussion for the increasing case.\n\n1. (ii)\n\nRefer to the Proof of Theorem 2-(ii). □\n\n## 4. φ-means by C2-functions\n\nIn this section, we treat a special φ-mean in which φ is a C2-functions with no stationary points. For each real-valued measurable function f on Ω, let ${C}_{sm*,f}^{2}\\left(I\\right)$ be the set of all C2-functions φ in C sm,f (I) with no stationary points, that is, φ'(t) ≠ 0 for all t I.\n\nLemma 8. Let$\\phi ,\\psi \\in {C}_{sm*,f}^{2}\\left(I\\right)$. Then\n\n1. (i)\n\nThe following two statements are equivalent:\n\n2. (a)\n\nψ φ -1 is convex (resp. concave) on φ(I).\n\n3. (b)\n\n$\\left(\\frac{{\\psi }^{″}\\left(x\\right)}{{\\psi }^{\\prime }\\left(x\\right)}-\\frac{{\\phi }^{″}\\left(x\\right)}{{\\phi }^{\\prime }\\left(x\\right)}\\right){\\psi }^{\\prime }\\left(x\\right)\\ge 0$ (resp. ≤ 0) for all x I°.\n\n4. (ii)\n\nThe following two statements are equivalent:\n\n5. (c)\n\nψ φ -1 is strictly convex (resp. strictly concave) on φ(I).\n\n6. (d)\n\n$\\left(\\frac{{\\psi }^{″}\\left(x\\right)}{{\\psi }^{\\prime }\\left(x\\right)}-\\frac{{\\phi }^{″}\\left(x\\right)}{{\\phi }^{\\prime }\\left(x\\right)}\\right){\\psi }^{\\prime }\\left(x\\right)>0$ (resp. < 0) for all x I°.\n\nProof. (i) Define τ(u) = ψ((φ-1(u)) for each u φ(I). Then a simple calculation yields that\n\n${\\tau }^{″}\\left(u\\right)=\\frac{\\left(\\frac{{\\psi }^{″}\\left(x\\right)}{{\\psi }^{\\prime }\\left(x\\right)}-\\frac{{\\phi }^{″}\\left(x\\right)}{{\\phi }^{\\prime }\\left(x\\right)}\\right){\\psi }^{\\prime }\\left(x\\right)}{{\\left({\\phi }^{\\prime }\\left(x\\right)\\right)}^{2}}$\n\nfor all u (φ(I))°, where x = φ-1(u). This equation implies that (a) and (b) are equivalent.\n\n1. (ii)\n\nSimilarly. □\n\nLemma 9. Let φ and ψ be C1-functions on I. Then,\n\n1. (i)\n\nIf φ'(x) < ψ'(x) for all x I° and ψ' is monotone increasing on I, then\n\n$\\left(\\left(1-t\\right)\\phi +t\\psi \\right)\\left(\\left(1-t\\right)x+ty\\right)<\\left(1-t\\right)\\phi \\left(x\\right)+t\\psi \\left(y\\right)$\n\nholds for all x, y I with x < y and t (0, 1).\n\n1. (ii)\n\nIf φ'(x) > ψ'(x) for all x I° and ψ' is monotone decreasing on I, then\n\n$\\left(\\left(1-t\\right)\\phi +t\\psi \\right)\\left(\\left(1-t\\right)x+ty\\right)>\\left(1-t\\right)\\phi \\left(x\\right)+t\\psi \\left(y\\right)$\n\nholds for all x, y I with x < y and t (0, 1).\n\nProof. (i) Suppose that φ'(x) < ψ'(x) for all x I° and ψ' is monotone increasing on I. Let x, y I with x < y and t (0, 1). Put z = (1 - t)x + ty. Then, we must show that ((1 - t)φ + )(z) < (1 - t)φ(x) + (y). By the mean value theorem, we have\n\n$\\begin{array}{c}\\left(1-t\\right)\\phi \\left(x\\right)+t\\psi \\left(y\\right)-\\left(\\left(1-t\\right)\\phi +t\\psi \\right)\\left(z\\right)\\\\ \\phantom{\\rule{1em}{0ex}}=t\\left(\\psi \\left(y\\right)-\\psi \\left(z\\right)\\right)-\\left(1-t\\right)\\left(\\phi \\left(z\\right)-\\phi \\left(x\\right)\\right)\\\\ \\phantom{\\rule{1em}{0ex}}=t{\\psi }^{\\prime }\\left(z+\\left(y-z\\right)\\theta \\right)\\left(y-z\\right)-\\left(1-t\\right){\\phi }^{\\prime }\\left(x+\\left(z-x\\right){\\theta }^{\\prime }\\right)\\left(z-x\\right)\\\\ \\phantom{\\rule{1em}{0ex}}=t\\left(1-t\\right)\\left({\\psi }^{\\prime }\\left(z+\\left(y-z\\right)\\theta \\right)-{\\phi }^{\\prime }\\left(x+\\left(z-x\\right){\\theta }^{\\prime }\\right)\\right)\\left(y-x\\right)\\\\ \\phantom{\\rule{1em}{0ex}}\\ge t\\left(1-t\\right)\\left({\\psi }^{\\prime }\\left(x+\\left(z-x\\right){\\theta }^{\\prime }\\right)-{\\phi }^{\\prime }\\left(x+\\left(z-x\\right){\\theta }^{\\prime }\\right)\\right)\\left(y-x\\right)\\end{array}$\n\nfor some θ, θ' (0, 1) because z + (y- z)θx + (z- x)θ' and hence ψ'(z + (y- z)θ) ≥ ψ'( x + (z- x)θ') by hypothesis. Since x + (z- x)θ' I°, it follows from hypothesis that\n\n${\\psi }^{\\prime }\\left(x+\\left(z-x\\right){\\theta }^{\\prime }\\right)>{\\phi }^{\\prime }\\left(x+\\left(z-x\\right){\\theta }^{\\prime }\\right)$\n\nand so (1 - t)φ(x) + (y) - ((1 - t)φ + )(z) > 0 from the preceding inequalities. Therefore, we obtain the desired inequality.\n\n1. (ii)\n\nSimilarly. □\n\nCorollary 1. Suppose that f is non-constant and$\\phi ,\\psi \\in {C}_{sm*,f}^{2}\\left(I\\right)$. Then\n\n1. (i)\n\nIf $\\frac{{\\phi }^{″}\\left(x\\right)}{{\\phi }^{\\prime }\\left(x\\right)}\\le \\frac{{\\psi }^{″}\\left(x\\right)}{{\\psi }^{\\prime }\\left(x\\right)}$ for all x I°, then M φ (f) ≤ M ψ (f).\n\n2. (ii)\n\nIf $\\frac{{\\phi }^{″}\\left(x\\right)}{{\\phi }^{\\prime }\\left(x\\right)}<\\frac{{\\psi }^{″}\\left(x\\right)}{{\\psi }^{\\prime }\\left(x\\right)}$ for all x I° then M φ (f) < M ψ (f).\n\nProof. (i) Suppose that $\\frac{{\\phi }^{″}\\left(x\\right)}{{\\phi }^{\\prime }\\left(x\\right)}\\le \\frac{{\\psi }^{″}\\left(x\\right)}{{\\psi }^{\\prime }\\left(x\\right)}$ for all x I°. If ψ is monotone increasing on I, then ψ'(x) > 0 for all x I° and hence ψ φ-1 is convex on φ(I) by Lemma 8-(i). Therefore, by Lemma 3-(ii), φ ψ holds and then M φ (f) ≤ M ψ (f) by Theorem 1-(i). If ψ is monotone decreasing on I, then ψ'(x) < 0 for all x I° and hence ψ φ-1 is concave on φ(I) by Lemma 8-(i). Therefore, by Lemma 3-(iii), φ ψ also holds and then M φ (f) ≤ M ψ (f) by Theorem 1-(i).\n\n1. (ii)\n\nSimilarly. □\n\nRemark. Let (Ω, μ) be a probability space, 0 < p < q < ∞ and let f be a non-constant real-valued function in Lq (Ω, μ). Then the well-known inequality: ||f|| p < ||f|| q follows immediately from Corollary 1 (ii), by considering a family {φ r : r > 0} of functions on +, where φ r (x) = xr (x > 0).\n\nLet $\\phi ,\\psi \\in {C}_{sm*,f}^{2}\\left(I\\right)$ and let t (0, 1). Then, we can easily see that if either both φ and ψ are monotone increasing on I or both φ and ψ are monotone decreasing on I, then $\\left(1-t\\right)\\phi +t\\psi \\in {C}_{sm*,f}^{2}\\left(I\\right)$. In this case, we have the following\n\nCorollary 2. Suppose that f is non-constant and$\\phi ,\\psi \\in {C}_{sm*,f}^{2}\\left(I\\right)$. If$\\frac{{\\phi }^{″}\\left(x\\right)}{{\\phi }^{\\prime }\\left(x\\right)}<\\frac{{\\psi }^{″}\\left(x\\right)}{{\\psi }^{\\prime }\\left(x\\right)}$and φ'(x)ψ'(x) > 0 for all x I°, then a function: sM(1-s)φ+(f) is strictly increasing on [0, 1].\n\nProof. Suppose that $\\frac{{\\phi }^{″}\\left(x\\right)}{{\\phi }^{\\prime }\\left(x\\right)}<\\frac{{\\psi }^{″}\\left(x\\right)}{{\\psi }^{\\prime }\\left(x\\right)}$ and φ'(x)ψ'(x) > 0 for all x I°. We define ξ(s, x) = (1 - s)φ(x) + (x) for each s (0, 1). We can easily see that\n\n$\\frac{\\frac{{\\partial }^{2}}{\\partial {x}^{2}}\\xi \\left(s,x\\right)}{\\frac{\\partial }{\\partial x}\\xi \\left(s,x\\right)}-\\frac{{\\phi }^{″}\\left(x\\right)}{{\\phi }^{\\prime }\\left(x\\right)}=\\frac{s{\\psi }^{\\prime }\\left(x\\right)\\left(\\frac{{\\psi }^{″}\\left(x\\right)}{{\\psi }^{\\prime }\\left(x\\right)}-\\frac{{\\phi }^{″}\\left(x\\right)}{{\\phi }^{\\prime }\\left(x\\right)}\\right)}{\\left(1-s\\right){\\phi }^{\\prime }\\left(x\\right)+s{\\psi }^{\\prime }\\left(x\\right)}>0$\n\nfor each s (0, 1) and x I°. Then, we have from Corollary 1-(ii) that M φ (f) < M(1-s)φ+(f) for all s (0, 1). Similarly, we can see that M(1-s)φ+(f) < M ψ (f) for all s (0, 1). Now put\n\n$A\\left(s,x\\right)=\\frac{\\frac{{\\partial }^{2}}{\\partial {x}^{2}}\\xi \\left(s,x\\right)}{\\frac{\\partial }{\\partial x}\\xi \\left(s,x\\right)}$\n\nfor each s (0, 1) and x I°. Then a simple calculation implies that\n\n$\\frac{\\partial }{\\partial s}A\\left(s,x\\right)=\\frac{\\left(\\frac{{\\psi }^{″}\\left(x\\right)}{{\\psi }^{\\prime }\\left(x\\right)}-\\frac{{\\phi }^{″}\\left(x\\right)}{{\\phi }^{\\prime }\\left(x\\right)}\\right){\\phi }^{\\prime }\\left(x\\right){\\psi }^{\\prime }\\left(x\\right)}{{\\left(\\left(1-t\\right){\\phi }^{\\prime }\\left(x\\right)+t{\\psi }^{\\prime }\\left(x\\right)\\right)}^{2}}>0$\n\nfor each s (0, 1) and x I°. Therefore, for a fixed x I°, a function: sA(s, x) is strictly increasing on (0, 1). Therefore, Corollary 1-(ii) implies that a function: sM(1-s)φ+(f) is strictly increasing on (0, 1) and hence [0, 1]. □\n\nCorollary 3. Suppose that f is non-constant and that$\\phi ,\\psi \\in {C}_{sm*,f}^{2}\\left(I\\right)$is such that$\\frac{{\\phi }^{″}\\left(x\\right)}{{\\phi }^{\\prime }\\left(x\\right)}<\\frac{{\\psi }^{″}\\left(x\\right)}{{\\psi }^{\\prime }\\left(x\\right)}$for for all x I°. Then\n\n1. (i)\n\nIf either 0 < φ' < ψ' and ψ\" ≥ 0 on I° or ψ' < φ' < 0 and ψ\" ≤ 0 on I°, then\n\n$\\left(1-t\\right){M}_{\\phi }\\left(f\\right)+t{M}_{\\psi }\\left(f\\right)<{M}_{\\left(1-t\\right)\\phi +t\\psi }\\left(f\\right)$\n\nholds for all t (0, 1).\n\n1. (ii)\n\nIf either φ' < ψ' < 0 and ψ\" ≥ 0 on I° or 0 < ψ' < φ' and ψ\" ≤ 0 on I°, then\n\n$\\left(1-t\\right){M}_{\\phi }\\left(f\\right)+t{M}_{\\psi }\\left(f\\right)>{M}_{\\left(1-t\\right)\\phi +t\\psi }\\left(f\\right)$\n\nholds for all t (0, 1).\n\nProof. (i) Suppose that 0 < φ' < ψ' and ψ\" ≥ 0 on I°. Put x = M φ (f) and y = M ψ (f), and so x < y by Corollary 1-(ii). Take t with 0 < t < 1 arbitrarily. By hypothesis, (1 - t)φ + is strictly monotone increasing on I. Put u = M(1-t)φ+(f). As observe in the proof of Theorem 3-(i), we have\n\n$\\left(1-t\\right)\\phi \\left(x\\right)+t\\psi \\left(y\\right)=\\left(\\left(1-t\\right)\\phi +t\\psi \\right)\\left(u\\right).$\n(10)\n\nPut z = (1 - t)x + ty. Then, by (10) and Lemma 9-(i), we have\n\n$\\left(\\left(1-t\\right)\\phi +t\\psi \\right)\\left(z\\right)<\\left(1-t\\right)\\phi \\left(x\\right)+t\\psi \\left(y\\right)=\\left(\\left(1-t\\right)\\phi +t\\psi \\right)\\left(u\\right),$\n\nand then z < u, that is, (1 - t)M φ (f) + tM ψ (f) < M(1-t)φ+(f).\n\nIn the case of ψ' < φ' < 0 and ψ ≤ 0 on I°, we apply Lemma 9-(ii).\n\n1. (2)\n\nSimilarly. □\n\n## 5. Remarks\n\n1. (i)\n\nLet I = +. Put $\\phi \\left(x\\right)=\\frac{1}{x}$ and ψ(x) = x for each x I. Of course, these functions belong to C sm (I). The harmonic-arithmetic mean inequality asserts that φ < ψ . Take a non-constant positive measurable function f on a probability space (Ω, μ) such that φ f and ψ f are in L 1(Ω, μ). Then, we have from Theorem 1-(ii) that M φ (f) < M ψ (f). Observe that this inequality means\n\n$1<\\left(\\int \\frac{1}{f}\\mathsf{\\text{d}}\\mu \\right)\\left(\\int f\\mathsf{\\text{d}}\\mu \\right).$\n\nThis is a special case of Jensen's inequality (or Schwarz's inequality). We note that if 0 < mfM, then $\\left(\\int \\frac{1}{f}\\mathsf{\\text{d}}\\mu \\right)\\left(\\int f\\mathsf{\\text{d}}\\mu \\right)\\le \\frac{{\\left(m+M\\right)}^{2}}{4mM}$. The right side of this inequality is called a Kantorovich constant (cf. ).\n\n1. (ii)\n\nA similar consideration for the geometric-arithmetic mean inequality yields that\n\n$\\int logf\\mathsf{\\text{d}}\\mu\n\nThis is also a special case of Jensen's inequality. We note that if 0 < mfM, then $log\\int f\\mathsf{\\text{d}}\\mu -\\int logf\\mathsf{\\text{d}}\\mu \\le {h}^{\\frac{1}{h-1}}{\\left(elog{h}^{\\frac{1}{h-1}}\\right)}^{-1}$, where $h=\\frac{M}{m}$. The right side of this inequality is called Specht's ratio (cf. ).\n\n1. (iii)\n\nFor each t [0, 1], put log[t] x = (1 - t)log x + tx(x > 0). Then log[t]is a strictly monotone increasing real-valued continuous function on +. Denote by exp[t]the inverse function of log[t]. Let x 1, ..., x n > 0 and p 1, ..., p n > 0 with ${\\sum }_{k=1}^{n}{p}_{k}=1$. Then Theorem 2-(i) (or Corollary 2) implies that $t\\to {exp}_{\\left[t\\right]}\\left({\\sum }_{k=1}^{n}{p}_{k}{log}_{\\left[t\\right]}{x}_{k}\\right)$ is strictly monotone increasing on [0, 1]. Note that ${exp}_{\\left[0\\right]}\\left({\\sum }_{k=1}^{n}{p}_{k}{log}_{\\left[0\\right]}{x}_{k}\\right)={\\prod }_{k=1}^{n}{x}_{k}^{{p}_{k}}$ and ${exp}_{\\left[1\\right]}\\left({\\sum }_{k=1}^{n}{p}_{k}{log}_{\\left[1\\right]}{x}_{k}\\right)={\\sum }_{k=1}^{n}{p}_{k}{x}_{k}$. Therefore, we obtain that\n\n$\\prod _{k=1}^{n}{x}_{k}^{{p}_{k}}\\le {exp}_{\\left[t\\right]}\\left(\\sum _{k=1}^{n}{p}_{k}{log}_{\\left[t\\right]}{x}_{k}\\right)\\le \\sum _{k=1}^{n}{p}_{k}{x}_{k}\\left(0\\le t\\le 1\\right).$\n\nThis is a new refinement of geometric-arithmetic mean inequality (cf. ).\n\n## References\n\n1. Fujii JI, Fujii M, Miura T, Takagi H, Takahasi S-E: Continuously differentiable means. J Inequal Appl 2006, 2006: 15. (Art. ID75941)\n\n2. Hewitt E, Stromberg K: Real and Abstract Analysis. Springer, Berlin; 1969.\n\n3. Long BY, Chu YM: Optimal inequalities for generalized logarithmic, arithmetic, and geometric means. J Inequal Appl 2010, 2010: 10. (Article ID 806825)\n\n4. Matejicka L: Proof of one optimal inequality for generalized logarithmic, arithmetic, and geometric means. J Inequal Appl 2010, 2010: 5. (Article ID 902432)\n\n5. Kantorovich LV: Functional analysis and applied mathematics. Uspechi Mat Nauk 1948, 3: 89–185. (in Russian)\n\n6. Ptak V: The Kantorovich inequality. Am Math Month 1995, 102: 820–821. 10.2307/2974512\n\n7. Tsukada M, Takahasi S-E: The best possiblity of the bound for the Kantorovich inequality and some remarks. J Inequal Appl 1997, 1: 327–334. 10.1155/S1025583497000222\n\n8. Fujii M, Tominaga M: An estimate of Young type operator inequality and Specht ratio RIMS Kôkyûroku. 2002, 1259: 173–178.\n\n9. Hara T, Uchiyama M, Takahasi S-E: A refinement of various mean inequalities. J Inequal Appl 1998, 2: 387–395. 10.1155/S1025583498000253\n\n## Acknowledgements\n\nThe authors are grateful to the referee, for the careful reading of the paper and for the helpful suggestions and comments. Also, we would like to thank Professor Masatoshi Fujii for his helpful informations of and Specht's ratio. S.-E. Takahasi is partially supported by Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science.\n\n## Author information\n\nAuthors\n\n### Corresponding author\n\nCorrespondence to Sin-Ei Takahasi.\n\n### Competing interests\n\nThe authors declare that they have no competing interests.\n\n### Authors' contributions\n\nYN carried out the design of the study and performed the analysis. KK conceived of the study, and participated in its design and coordination. ST participated in the sequence alignment and drafted the manuscript. All authors read and approved the final manuscript.\n\n## Rights and permissions\n\nReprints and Permissions\n\nNakasuji, Y., Kumahara, K. & Takahasi, SE. A new interpretation of Jensen's inequality and geometric properties of ϕ-means. J Inequal Appl 2011, 48 (2011). https://doi.org/10.1186/1029-242X-2011-48\n\n• Accepted:\n\n• Published:\n\n• DOI: https://doi.org/10.1186/1029-242X-2011-48\n\n### Keywords\n\n• Jensen's inequality\n• Mean\n• Refinement\n• Convexity\n• Concavity", null, "" ]

[ null, "https://journalofinequalitiesandapplications.springeropen.com/track/article/10.1186/1029-242X-2011-48", null ]

[ "USING OUR SERVICES YOU AGREE TO OUR USE OF COOKIES\n\n# What Is The Square Root Of 16?\n\n• The simplified root of √16 is number 4\n• All radicals are now simplified. The radicand no longer has any square factors.\n\n## Determine The Square Root Of 16?\n\n• The square root of sixteen √16 = 4\n\n## How To Calculate Square Roots\n\n• In mathematics, a square root of a number a is a number y such that y² = a, in other words, a number y whose square (the result of multiplying the number by itself, or y * y) is a. For example, 4 and -4 are square roots of 16 because 4² = (-4)² = 16.\n• Every non-negative real number a has a unique non-negative square root, called the principal square root, which is denoted by √a, where √ is called the radical sign or radix. For example, the principal square root of 9 is 3, denoted √9 = 3, because 32 = 3 ^ 3 = 9 and 3 is non-negative. The term whose root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this example 9.\n• The justification for taking out the square root of any number is this theorem to help simplify √a*b = √a * √b. The square root of a number is equal to the number of the square roots of each factor.\n\n## What is a square root?\n\nA square root of a number is a number that, when it is multiplied by itself (squared), gives the first number again. For example, 2 is the square root of 4, because 2x2=4. Only numbers bigger than or equal to zero have real square roots. A number bigger than zero has two square roots: one is positive (bigger than zero) and the other is negative (smaller than zero). For example, 4 has two square roots: 2 and -2. The only square root of zero is zero. A whole number with a square root that is also a whole number is called a perfect square. The square root radical is simplified or in its simplest form only when the radicand has no square factors left. A radical is also in simplest form when the radicand is not a fraction." ]

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[ "# Sciweavers\n\nShare\nDM\n1998\n•", null, "", null, "Email", null, "discuss", null, "report\n91views more  DM 1998»\n\n# On some relations between 2-trees and tree metrics\n\n10 years 1 months ago\nA tree function (TF) t on a ÿnite set X is a real function on the set of the pairs of elements of X satisfying the four-point condition: for all distinct x; y; z; w ∈ X; t(xy)+t(zw)6 max{t(xz)+ t(yw); t(xw) + t(yz)}. Equivalently, t is representable by the lengths of the paths between the leaves of a valued tree Tl. TFs are a straightforward generalization of the tree dissimilarities and tree metrics of the literature. A graph is a 2-tree if it belongs to the following class Q: an edge-graph belongs to Q: if ∈ Q and yz is an edge of , then the graph obtained by the addition to of a new vertex x adjacent to y and z belongs to Q. These graphs, and the more general k-trees, have been studied in the literature as generalizations of trees. It is ÿrst explicited here how to make a TF t ; d correspond to any positively valued 2-tree d on X . Then, given a tree dissimilarity t, the set Q(t) of the 2-trees such that t = t ; t is studied. Any element of Q(t) gives a way of summarizing t b...", null, "" ]

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[ "Shortcuts\n\n# torch.normal¶\n\ntorch.normal(mean, std, *, generator=None, out=None)Tensor\n\nReturns a tensor of random numbers drawn from separate normal distributions whose mean and standard deviation are given.\n\nThe mean is a tensor with the mean of each output element’s normal distribution\n\nThe std is a tensor with the standard deviation of each output element’s normal distribution\n\nThe shapes of mean and std don’t need to match, but the total number of elements in each tensor need to be the same.\n\nNote\n\nWhen the shapes do not match, the shape of mean is used as the shape for the returned output tensor\n\nNote\n\nWhen std is a CUDA tensor, this function synchronizes its device with the CPU.\n\nParameters\n• mean (Tensor) – the tensor of per-element means\n\n• std (Tensor) – the tensor of per-element standard deviations\n\nKeyword Arguments\n\nExample:\n\n>>> torch.normal(mean=torch.arange(1., 11.), std=torch.arange(1, 0, -0.1))\ntensor([ 1.0425, 3.5672, 2.7969, 4.2925, 4.7229, 6.2134,\n8.0505, 8.1408, 9.0563, 10.0566])\n\ntorch.normal(mean=0.0, std, *, out=None)Tensor\n\nSimilar to the function above, but the means are shared among all drawn elements.\n\nParameters\n• mean (float, optional) – the mean for all distributions\n\n• std (Tensor) – the tensor of per-element standard deviations\n\nKeyword Arguments\n\nout (Tensor, optional) – the output tensor.\n\nExample:\n\n>>> torch.normal(mean=0.5, std=torch.arange(1., 6.))\ntensor([-1.2793, -1.0732, -2.0687, 5.1177, -1.2303])\n\ntorch.normal(mean, std=1.0, *, out=None)Tensor\n\nSimilar to the function above, but the standard deviations are shared among all drawn elements.\n\nParameters\n• mean (Tensor) – the tensor of per-element means\n\n• std (float, optional) – the standard deviation for all distributions\n\nKeyword Arguments\n\nout (Tensor, optional) – the output tensor\n\nExample:\n\n>>> torch.normal(mean=torch.arange(1., 6.))\ntensor([ 1.1552, 2.6148, 2.6535, 5.8318, 4.2361])\n\ntorch.normal(mean, std, size, *, out=None)Tensor\n\nSimilar to the function above, but the means and standard deviations are shared among all drawn elements. The resulting tensor has size given by size.\n\nParameters\n• mean (float) – the mean for all distributions\n\n• std (float) – the standard deviation for all distributions\n\n• size (int...) – a sequence of integers defining the shape of the output tensor.\n\nKeyword Arguments\n\nout (Tensor, optional) – the output tensor.\n\nExample:\n\n>>> torch.normal(2, 3, size=(1, 4))\ntensor([[-1.3987, -1.9544, 3.6048, 0.7909]])", null, "## Docs\n\nAccess comprehensive developer documentation for PyTorch\n\nView Docs\n\n## Tutorials\n\nGet in-depth tutorials for beginners and advanced developers\n\nView Tutorials" ]

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[ "# When and where was the first logic-grid-based puzzle published?\n\nLogic puzzles with a set of conditions and a \"logic grid\" are very popular. More than one publisher is currently producing puzzle books with these types of puzzles.\n\nA Wikipedia article claims Charles Lutwidge Dodgson, (Lewis Carroll, author of Alice's Adventures in Wonderland.) was the instigator of what later became these type of puzzles.\n\nWhen and where was the first logic puzzle using a logic grid published? Here is an example of a logic grid from Wikipedia:", null, "• Hmm... many puzzles not designed for a logic grid could potentially have been solved with one. Do you mean to ask about puzzles solved specifically using a logic grid, or puzzles which have the capacity to be?\n– user20\nMay 21, 2014 at 23:27\n• @Emrakul, \"puzzles solved specifically using a logic grid\", I am assuming the capacity developed then the use. But no reason not to answer both if you can do it. May 21, 2014 at 23:36\n• Actually, believe it or not, logic grids are one approach among many to solve this class of reduction puzzles, which is why I ask. Though, I suppose it's true - we could just let the answers come in. I can't answer myself, though.\n– user20\nMay 21, 2014 at 23:37\n• Also see puzzling.stackexchange.com/q/89\n– SQB\nMay 23, 2014 at 11:13\n\nSo while not explicitly a logic grid, Carrol's book The Game Of Logic popularized logic \"puzzles\" of this nature (his weren't exactly puzzles so much as complicated logical statements) with answers represented in a notational style of his own creation involving squares and counters.\n\nThese problems were later refined in magazines and articles into the modern variety of what we call the logic puzzle: a single unique combination of facts which could be deduced through a provided set of statements. These became increasingly common during the 1940s and 1950s. When printed in magazines, some would suggest the form of an array or grid to assist with the problem.\n\nThe first actual printed grid I can find is in Clarence Raymond Wylie, Jr's 1957 publication 101 Puzzles in Thought and Logic.\n\nThe sort of logic puzzles James Jenkins refers to involve n-valued logic where n is always at least 3, and, in my experience, almost always at least 4. The grid in James Jenkins's OP is for a problem in 4-valued logic.\n\nBy contrast, the logic problems Lewis Carroll invented are all in Boolean or 2-valued logic. His main work on the subject is Symbolic Logic (pub. 1896). His The Game of Logic (pub. 1886) has a narrower focus, namely syllogisms, where a conclusion is to be deduced from two premisses. Three boolean variables (x, y and m, in Carroll's notation) are involved. One premiss involves x and m; the other, y and m; a conclusion in x and y is to be deduced. For example:\n\n• No exciting books suit feverish patients;\n• Unexciting books make one drowsy.\n• Therefore, No books suit feverish patients, except such as make one drowsy.\n\n(question 13, p.52 & p.74)\n\nIn Symbolic Logic, Carroll progresses from the syllogism to the sorites, which has n premisses involving a total of n+1 variables. As before, the variables are Boolean and each premiss involves two of them. A sorites is really a chain of syllogisms.\n\nIt is true that Carroll taught how to solve his logic problems using diagrams. But these diagrams were not like the grid in the OP. They were like Venn diagrams in purpose, but using rectangles instead of curves. In Symbolic Logic, Appendix, section 6, p.174, Carroll mentions Venn's method of diagrams, and reproduces the designs which Venn suggests, for up to 5 variables. Carroll presents his own diagrams for up to 8 variables in section 7 of the Appendix, p.176-179.\n\nCarroll's Symbolic Logic is just Part I (Elementary) of what Carroll planned as a three-part work. He worked on Part II (Advanced), but did not complete it, as he died in January 1898, slightly more than one year after Part I was published. But in Symbolic Logic, Appendix, section 10 (p.185-194), he gives 8 problems \"as a taste of what is coming in Part II\". Each of these has the nature of a 3-SAT problem, except that a conclusion is to be deduced (whereas in 3-SAT the task is merely to find out whether or not the premisses can all be satisfied), and some of the premisses involve four variables, not three. Each of these problems is in Boolean logic, except number 1, which is in three-valued logic. That problem is on Google Books here: Problem 1.\n\nThough Carroll did not complete Part II, a lot of manuscripts for his work on Part II survive, and an incomplete version of Part II, edited by William Warren Bartley III, was published by Harvester Press in 1977 (ISBN 0-85527-974-5). This shows Carroll's methods for solving his logic problems, one method using trees and one using symbols. There is nothing like the grids in the OP, or even Carroll's diagrams mentioned above.\n\n• Fascinating, great job! May 25, 2016 at 15:39" ]

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[ "", null, "# Ready-To-Use Random Number Generator Template\n\nRandom Number Generator in Excel, OpenOffice Calc & Google Sheet with 9 random generators to generate random integers, decimals, date, time, alphabets, names, teams, etc.\n\nMoreover, Random Number Generator generates singular or multiple random numbers with or without duplicates.\n\nIn addition to that, the Random Name and Team Generator generates a random name from a given list. It also generates random teams from a given set of names.\n\nFurthermore, it consists of the Random Date and Time Generator that helps you generate random dates and times between two given ranges.\n\n## Download Random Number Generator (Excel, OpenOffice Calc & Google Sheet)\n\nWith the help of VBA coding and using Functions, we have created a simple and easy 9 different Random Generators.\n\nJust download the file, insert the lower range and upper range and the template will automatically generate the random for you.\n\n## What are Random Numbers?\n\nA random number is a number chosen as if by chance from some specified distribution such that the selection of a large set of these numbers reproduces the underlying distribution. Almost always, such numbers are also required to be independent, so that there are no correlations between successive numbers.\n\nSource: www.mathworld.wolfarm.com.\n\nA random number is chosen from a large set of numbers. Such numbers are also required to be independent so that there are no correlations between successive numbers.\n\n## Contents of Random Number Generator Template\n\nThis template consists of 9 Random Generators:\n\n1. Number Generator (With Duplicates)\n2. Integer Generator (Without Duplicates)\n3. Decimal Generator\n4. Date Generator\n5. Time Generator\n6. Alphabet Generator\n7. Name Generator\n8. Team Generator\n9. Lottery Ticket Winner Generator\n\nLet us understand the functioning of each in detail.\n\n### Random Number Generator (With Duplicates)", null, "This template generates a single Random number and a set of random numbers from a given range. Specify the lower range in cell D6 and upper range in cell F6. It will automatically generate random numbers for you between that range.\n\nThis section uses the RANDBETWEEN function to generate random numbers within a given range. The only limitation with the RANDBETWEEN function is that when applied to multiple cells it generates some duplicates numbers between that range.\n\nYou can use this to generate such numbers for kids to practice mathematical functions like addition, subtraction, multiplication, division, etc.\n\nEvery time you make changes to the sheet these random numbers will change. If you want to use those numbers, kindly copy and paste using “Paste Special” selecting the “Values” option to another location in the sheet.\n\n### Random Integer Generator (Without Duplicates)", null, "This Random Number Generator displays multiple random numbers/integers without duplicates. It generates 100 random numbers.\n\nIn this section, it is necessary that the difference between the lower range and the upper range should be 100. Then only the generator works properly. Otherwise, it will display the following message:\n\nMoreover, the template uses a VBA code to generate random numbers between a range. The code is explained at the end of the article so that you can use it according to your requirements.\n\n### Random Decimal Generator", null, "Random Decimal Generator generates random decimal numbers between a range. Insert the lower limit and the upper limit to generate the decimal numbers.\n\nThis section uses the RANDBETWEEN function to generate random numbers within a given range. When applied to multiple cells it generates some duplicates numbers between that range.\n\nFurthermore, if you want to use those numbers, kindly copy and paste using “Paste Special” selecting the “Values” option to another location in the sheet.\n\nIt uses the following formula:\n\n=RANDBETWEEN(\\$D\\$6*100,\\$F\\$6*100)/100\n\nYou can use this to generate such numbers for kids to practice mathematical functions.\n\n### Random Date Generator", null, "Date Generator 5 different dates between a given range of dates. Insert the start date and end date. This section also uses RANDBETWEEN Function with DATEVALUE Function.\n\nIt uses the following formula:\n\n=RANDBETWEEN(DATEVALUE(“1-Jan-21”),DATEVALUE(“15-Jan-21”))\n\n### Random Time Generator", null, "Time Generator displays random time between the time range. Insert start time and end time and it will generate 5 different times between the range. This section uses the TIME VALUE Function with RAND Function.\n\nIt uses the following formula:\n\n=TIMEVALUE(“12:00 AM”) + RAND() * (TIMEVALUE(“6:00 PM”) – TIMEVALUE(“12:00 AM”))\n\n### Random Alphabet Generator", null, "Alphabet Generator display alphabets between a given range. Insert the start alphabet between A To Z and it will display the random alphabet for you.\n\nIt uses RANDBETWEEN Function with a combination of the CHAR Function and CODE Function. It uses the following formula:\n\n=CHAR(RANDBETWEEN(CODE(\\$D\\$22),CODE(\\$F\\$22)))\n\n### Random Name Generator", null, "Usually, this section is useful to a teacher or school to generate a random name from a given list of students. Insert names of the students or employees in a single column and apply the formula.\n\nIf the names are not in a single column it will not generate the random name and displays a #Value error.\n\nThis section uses RANDBETWEEN Function along with the INDEX Function. It uses the following formula:\n\n=INDEX(B31:B42,RANDBETWEEN(1,12))\n\n### Random Team Generator", null, "Team Generator generates a team from a list of names. Insert the names in a single column and apply the formula. As in the above generator, if the names are not in a single column it will not generate the random name and displays a #Value error.\n\nThis section uses RANDBETWEEN Function along with the INDEX Function. It uses the following formula:\n\n=INDEX(\\$B\\$31:\\$B\\$42,RANDBETWEEN(1,12))\n\n### Random Lottery Ticket Winner Generator", null, "This section is similar to the above section. Insert the lottery number in series in the column and apply the formula. This section also uses RANDBETWEEN Function along with the INDEX Function.\n\n## VBA Code to Generate Random Numbers Without Duplicates\n\nThe Random Number Generator Without Duplicates uses the following VBA Code:\n\nThis file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.\n\n Public Sub generateRandNum() 'Define your variabiles lowerbound = Range(\"D22\") upperbound = Range(\"F22\") Set randomrange = Range(\"B24:F43\") randomrange.ClearContents For Each rng1 In randomrange counter = counter + 1 Next If counter > upperbound – lowerbound + 1 Then MsgBox (\"Number of cells > number of unique random numbers\") Exit Sub End If For Each Rng In randomrange randnum = Int((upperbound – lowerbound + 1) * Rnd + lowerbound) Do While Application.WorksheetFunction.CountIf(randomrange, randnum) >= 1 randnum = Int((upperbound – lowerbound + 1) * Rnd + lowerbound) Loop Rng.Value = randnum Next End Sub\n\nview raw\n\nRNG\n\nhosted with ❤ by GitHub\n\nSource: www.access-excel.tips\n\n## Frequently Asked Questions\n\n### What are the different methods to generate random numbers?\n\nThere are two methods of generating random numbers: True-Random Method and Psuedo-Random Method.\n\n### What is True Random Method?\n\nTrue random numbers use physical phenomena like atmospheric noise, thermal noise, and other quantum phenomena. It generates true random numbers.\n\n### What is Pseudo-Random Method?\n\nA pseudo-random method uses an algorithm for generating a sequence of numbers. Usually, computer-based random number generators always use the pseudo-random method. Moreover, the numbers generated by the pseudo-random method are not truly random.\n\nIf you like this article, kindly share it on different social media platforms. So that your friends and colleagues can also benefit from the same. Sharing is Caring.\n\nMoreover, send us your queries or suggestions in the comment section below. We will be more than happy to assist you." ]

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[ "### Rewrite `org-indent-drawer' and `org-indent-block'\n\n`* lisp/org.el (org-indent-block, org-indent-drawer): Rewrite functions.`", null, "Nicolas Goaziou 6 years ago\nparent\ncommit\n7515066d94\n1 changed files with 34 additions and 41 deletions\n1. 34 41\nlisp/org.el\n\n#### + 34 - 41 lisp/org.el View File\n\n ``@@ -22382,47 +22382,6 @@ Also align node properties according to `org-property-format'.\"`` `` (org--align-node-property)`` `` (org-move-to-column column)))))))))`` `` `` ``-(defun org-indent-drawer ()`` ``- \"Indent the drawer at point.\"`` ``- (interactive)`` ``- (let ((p (point))`` ``- (e (and (save-excursion (re-search-forward \":END:\" nil t))`` ``- (match-end 0)))`` ``- (folded`` ``- (save-excursion`` ``- (end-of-line)`` ``- (when (overlays-at (point))`` ``- (member 'invisible (overlay-properties`` ``- (car (overlays-at (point)))))))))`` ``- (when folded (org-cycle))`` ``- (indent-for-tab-command)`` ``- (while (and (move-beginning-of-line 2) (< (point) e))`` ``- (indent-for-tab-command))`` ``- (goto-char p)`` ``- (when folded (org-cycle)))`` ``- (message \"Drawer at point indented\"))`` ``-`` ``-(defun org-indent-block ()`` ``- \"Indent the block at point.\"`` ``- (interactive)`` ``- (let ((p (point))`` ``- (case-fold-search t)`` ``- (e (and (save-excursion (re-search-forward \"#\\\\+end_?\\\\(?:[a-z]+\\\\)?\" nil t))`` ``- (match-end 0)))`` ``- (folded`` ``- (save-excursion`` ``- (end-of-line)`` ``- (when (overlays-at (point))`` ``- (member 'invisible (overlay-properties`` ``- (car (overlays-at (point)))))))))`` ``- (when folded (org-cycle))`` ``- (indent-for-tab-command)`` ``- (while (and (move-beginning-of-line 2) (< (point) e))`` ``- (indent-for-tab-command))`` ``- (goto-char p)`` ``- (when folded (org-cycle)))`` ``- (message \"Block at point indented\"))`` ``-`` `` (defun org-indent-region (start end)`` `` \"Indent each non-blank line in the region.`` `` Called from a program, START and END specify the region to`` ``@@ -22520,6 +22479,40 @@ assumed to be significant there.\"`` `` (set-marker element-end nil))))`` `` (set-marker end nil))))`` `` `` ``+(defun org-indent-drawer ()`` ``+ \"Indent the drawer at point.\"`` ``+ (interactive)`` ``+ (unless (save-excursion`` ``+ (beginning-of-line)`` ``+ (org-looking-at-p org-drawer-regexp))`` ``+ (user-error \"Not at a drawer\"))`` ``+ (let ((element (org-element-at-point)))`` ``+ (unless (memq (org-element-type element) '(drawer property-drawer))`` ``+ (user-error \"Not at a drawer\"))`` ``+ (org-with-wide-buffer`` ``+ (org-indent-region (org-element-property :begin element)`` ``+ (org-element-property :end element))))`` ``+ (message \"Drawer at point indented\"))`` ``+`` ``+(defun org-indent-block ()`` ``+ \"Indent the block at point.\"`` ``+ (interactive)`` ``+ (unless (save-excursion`` ``+ (beginning-of-line)`` ``+ (let ((case-fold-search t))`` ``+ (org-looking-at-p \"[ \\t]*#\\\\+\\\\(begin\\\\|end\\\\)_\")))`` ``+ (user-error \"Not at a block\"))`` ``+ (let ((element (org-element-at-point)))`` ``+ (unless (memq (org-element-type element)`` ``+ '(comment-block center-block example-block export-block`` ``+ quote-block special-block src-block`` ``+ verse-block))`` ``+ (user-error \"Not at a block\"))`` ``+ (org-with-wide-buffer`` ``+ (org-indent-region (org-element-property :begin element)`` ``+ (org-element-property :end element))))`` ``+ (message \"Block at point indented\"))`` ``+`` `` `` `` ;;; Filling`` `` ``" ]

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[ "Science and technology\n\n# Try quantum computing with this open supply software program improvement package", null, "Classical computing relies on bits. Zeros and ones. This is not as a result of there’s some inherent benefit to a binary logic system over logic techniques with extra states—and even over analog computer systems. But on-off switches are simple to make and, with fashionable semiconductor expertise, we will make them very small and really low-cost.\n\nBut they are not with out limits. Some issues simply cannot be effectively solved by a classical laptop. These are typically issues the place the associated fee, in time or reminiscence, will increase exponentially with the dimensions (`n`) of the issue. We say such issues are `O(2n)` in Big O notation.\n\nMuch of contemporary cryptography even will depend on this attribute. Multiplying two, even massive, prime numbers collectively is pretty low-cost computationally (`O(n2)`). But reversing the operation takes exponential time. Use massive sufficient numbers, and decryption that will depend on such a factoring assault is infeasible.\n\n## Enter quantum\n\nAn in depth primer on the mathematical and quantum mechanical underpinnings of quantum computing is past the scope of this text. However, listed below are some fundamentals.\n\nA quantum laptop replaces bits with qubits—controllable items of computing that show quantum properties. Qubits are sometimes made out of both an engineered superconducting part or a naturally occurring quantum object resembling an electron. Qubits may be positioned right into a “superposition” state that may be a complicated mixture of the zero and 1 states. You generally hear that qubits are each zero and 1, however that is probably not correct. What is true is that, when a measurement is made, the qubit state will collapse right into a zero or 1. Mathematically, the (unmeasured) quantum state of the qubit is some extent on a geometrical illustration referred to as the Bloch sphere.\n\nWhile superposition is a novel property for anybody used to classical computing, one qubit by itself is not very fascinating. The subsequent distinctive quantum computational property is “interference.” Real quantum computer systems are primarily statistical in nature. Quantum algorithms encode an interference sample that will increase the likelihood of measuring a state encoding the answer.\n\nWhile novel, superposition and interference do have some analogs within the bodily world. The quantum mechanical property “entanglement” does not, and it is the true key to exponential quantum speedups. With entanglement, measurements on one particle can have an effect on the end result of subsequent measurements on any entangled particles—even ones not bodily related.\n\n## What can quantum do?\n\nToday’s quantum computer systems are fairly small by way of the variety of qubits they comprise—tens to lots of. Thus, whereas algorithms are actively being developed, the hardware wanted to run them sooner than their classical equivalents does not exist.\n\nBut there’s appreciable curiosity in quantum computing in lots of fields. For instance, quantum computer systems could provide a great way to simulate pure quantum techniques, like molecules, whose complexity quickly exceeds the flexibility of classical computer systems to mannequin them precisely. Quantum computing can also be tied mathematically to linear algebra, which underpins machine studying and plenty of different fashionable optimization issues. Therefore, it is cheap to assume quantum computing might be a great match there as nicely.\n\nOne generally cited instance of an current quantum algorithm prone to outperform classical algorithms is Shor’s algorithm, which may do the factoring talked about earlier. Invented by MIT mathematician Peter Shor in 1994, quantum computer systems cannot but run the algorithm on bigger than trivially sized issues. But it has been demonstrated to work in polynomial `O(nthree)` time relatively than the exponential time required by classical algorithms.\n\n## Getting began with Qiskit\n\nAt this level, chances are you’ll be pondering: “But I don’t have a quantum computer, and I really like to get hands-on. Is there any hope?”\n\nEnter an open supply (Apache 2.zero licensed) mission referred to as Qiskit. It’s a software program improvement package (SDK) for accessing each the quantum computing simulators and the precise quantum hardware (without spending a dime) within the IBM Quantum Experience. You simply want to enroll in an API key.\n\nCertainly, a deep dive into Qiskit, to say nothing of the associated linear algebra, is much past what I can get into right here. If you search such a dive, there are many free resources online, together with an entire textbook. However, dipping a toe within the water is easy, requiring just some surface-level information of Python and Jupyter Notebooks.\n\nTo provide you with a style, let’s do a “Hello, World!” program solely throughout the Qiskit textbook.\n\nBegin by putting in some instruments and widgets particular to the textbook:\n\n``pip set up git+https://github.com/qiskit-community/qiskit-textbook.git#subdirectory=qiskit-textbook-src``\n\nNext, do the imports:\n\n`from qiskit import QuantumCircuit, assemble, Aerfrom math import pi, sqrtfrom qiskit.visualization import plot_bloch_multivector, plot_histogram`\n\nAer is the native simulator. Qiskit consists of 4 parts: Aer, the Terra basis, Ignis for coping with noise and errors on actual quantum techniques, and Aqua for algorithm improvement.\n\n`# Let's do an X-gate on a |zero> qubitqc = QuantumCircuit(1)qc.x(zero)qc.draw()`\n\nAn X-gate in quantum computing is much like a Not gate in classical computing, though the underlying math really entails matrix multiplication. (It is, in reality, typically referred to as a NOT-gate.)\n\nNow, run it and do a measurement. The result’s as you’d count on as a result of the qubit was initialized in a `|zero>` state, then inverted, and measured. (Use `|zero>` and `|1>` to tell apart from basic bits.)\n\n`# Let's see the outcomesvsim = Aer.get_backend('statevector_simulator')qobj = assemble(qc)state = svsim.run(qobj).outcome().get_statevector()plot_bloch_multivector(state)`\n\n## Conclusion\n\nIn some respects, you’ll be able to consider quantum computing as a form of unique co-processor for classical computer systems in the identical method as graphics processing items (GPUs) and field-programmable gate arrays (FPGAs). One distinction is that quantum computer systems shall be accessed virtually solely as community sources for the foreseeable future. Another is that they work basically in a different way, which makes them not like most different accelerators you could be accustomed to. That’s why there’s a lot curiosity in algorithm improvement and vital sources stepping into to find out the place and when quantum matches greatest. It would not damage to see what all of the fuss is about.\n\nbreakingExpress.com features the latest multimedia technologies, from live video streaming to audio packages to searchable archives of news features and background information. The site is updated continuously throughout the day." ]

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[ "# How to get the maximum value of an implicit function?\n\n8 views (last 30 days)\nEungkyun Kim on 4 Feb 2020\nCommented: Matt J on 5 Feb 2020\nI have the following implicit function:\nI-Ipv+Io.*exp(((V+Rs.*I)/(Vt.*a))-1)-(V+Rs.*I)/Rp = 0\nwhere I and V are unknowns and everything else is known. From this equation, I am trying to acheive the maximum value of P where P = V*I. To acheive this, I wrote the following codes:\nfun = @(V,I) I-Ipv+Io.*exp(((V+Rs.*I)/(Vt.*a))-1)-(V+Rs.*I)/Rp;\nf = fimplicit(fun,[0,40,0,9])\nvoltage = f.XData;\ncurrent = f.YData;\npower = voltage.*current\nmaximum_power = max(power)\nI believe this method does not allow me to change the number of points in X, therefore the calculated maximum power is not as accurate.\nI would appreciate any suggestion regarding calculating a maximum power.\nThank you.\n\nJohn D'Errico on 4 Feb 2020\nIf you have the otimization toolbox, then just formulate it as a nonlinear optimization. fmincon will be the correct tool. That is, minimize -V*I, subject to a nonlinear equality constraint, as you have written. So two variables, one equality constraint.\n\nEungkyun Kim on 5 Feb 2020\nThanks. Do you mind elaborating on how to use fmincon to find the maximum value of V*I?\nEungkyun Kim on 5 Feb 2020\nI think I got it actually. Thanks.\nMatt J on 5 Feb 2020\n\nMatt J on 5 Feb 2020\nEdited: Matt J on 5 Feb 2020\nI believe this method does not allow me to change the number of points in X,\nIt does, e.g.,\nf = fimplicit(fun,[0,40,0,9],'MeshDensity',1000)\nIf it were me, though, I would probably use this method to initialize fmincon, as recommended by John.\n\n#### 1 Comment\n\nEungkyun Kim on 5 Feb 2020\nThanks! I didn't know I could do that." ]

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[ "#", null, "Rudloff, Birgit, Ulus, Firdevs, Vanderbei, Robert. 2017. A parametric simplex algorithm for linear vector optimization problems. Mathematical Programming 163 (1), 213-242.\n\nBibTeX\n\n## Abstract\n\nIn this paper, a parametric simplex algorithm for solving linear vector optimization problems (LVOPs) is presented. This algorithm can be seen as a variant of the multi-objective simplex (Evans-Steuer) algorithm . Different from it, the proposed algorithm works in the parameter space and does not aim to find the set of all efficient solutions. Instead, it finds a solution in the sense of Löhne , that is, it finds a subset of efficient solutions that allows to generate the whole frontier. In that sense, it can also be seen as a generalization of the parametric self-dual simplex algorithm, which originally is designed for solving single objective linear optimization problems, and is modified to solve two objective bounded LVOPs with the positive orthant as the ordering cone in Ruszczynski and Vanderbei . The algorithm proposed here works for any dimension, any solid pointed polyhedral ordering cone C and for bounded as well as unbounded problems. Numerical results are provided to compare the proposed algorithm with an objective space based LVOP algorithm (Benson algorithm in ), that also provides a solution in the sense of , and with Evans-Steuer algorithm . The results show that for non-degenerate problems the proposed algorithm outperforms Benson algorithm and is on par with Evan-Steuer algorithm. For highly degenerate problems Benson's algorithm excels the simplex-type algorithms; however, the parametric simplex algorithm is for these problems computationally much more efficient than Evans-Steuer algorithm.\n\n## Tags\n\nPress 'enter' for creating the tag\n\n## Publication's profile\n\nStatus of publication Published WU Journal article Mathematical Programming SCI A FIN-A, INF-A, STRAT-A, WH-A English A parametric simplex algorithm for linear vector optimization problems 163 1 2017 213 242 Y http://arxiv.org/pdf/1507.01895.pdf http://dx.doi.org/10.1007/s10107-016-1061-z\n\n## Associations\n\nPeople\nRudloff, Birgit (Details)\nExternal\nUlus, Firdevs (Bilkent University, Turkey)\nVanderbei, Robert (Princeton University, United States/USA)\nOrganization\nInstitute for Statistics and Mathematics IN (Details)\nResearch areas (ÖSTAT Classification 'Statistik Austria')\n1118 Probability theory (Details)\n1137 Financial mathematics (Details)\n5361 Financial management (Details)" ]

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[ "# Plus Two Math's Solution Ex 1.3 Chapter1 Relations and Function\n\nNCERT Solutions for Plus Two Maths Chapter 1 Relations and Functions Ex 1.3, NCERT Solutions for Plus Two Maths is available here. Plus Two Math's solutions are given here as a PDF file which is easily downloadable free of cost by the students of Plus Two. If you want to learn more about NCERT Solutions then visit www.keralanotes.com; this website provides complete solutions for Plus Two mathematics\n\nPractice the NCERT Solutions PDF Ex 1.3 to solve examples related to Relations and Functions. This NCERT Solutions for Plus Two Maths Chapter helps you understand what are Relations and Functions.\n\n Board SCERT, Kerala Text Book NCERT Based Class Plus Two Subject Math's Textbook Solution Chapter Chapter 1 Exercise Ex 1.3 Chapter Name Relations and Function Category Plus Two Kerala\n\n## Kerala Syllabus Plus Two Math's Textbook Solution Chapter  1 Relation and Function Exercises 1.3\n\n### Chapter  1  Relations and Function Solution\n\nKerala plus two maths NCERT textbooks, we provide complete solutions for the exercise and answers provided at the end of each chapter. We also cover the entire syllabus given by the Board of secondary education, Kerala state.\n\n### Chapter  1  Relations and Function Exercise   1.3\n\nLet f: {1, 3, 4} → {1, 2, 5} and g: {1, 2, 5} → {1, 3} be given by = {(1, 2), (3, 5), (4, 1)} and = {(1, 3), (2, 3), (5, 1)}. Write down gof.\n\nThe functions f: {1, 3, 4} → {1, 2, 5} and g: {1, 2, 5} → {1, 3} are defined as\n\n= {(1, 2), (3, 5), (4, 1)} and = {(1, 3), (2, 3), (5, 1)}.", null, "Let fg, and h be functions from to R. Show that", null, "To prove:", null, "", null, "", null, "", null, "", null, "Find goand fog, if\n\n(i)", null, "(ii)", null, "(i)", null, "", null, "(ii)", null, "", null, "If", null, ", show that f f(x) = x, for all", null, ". What is the inverse of f?\n\nIt is given that", null, ".", null, "Hence, the given function f is invertible and the inverse of f is f itself.\n\nState with reason whether following functions have inverse\n\n(i) f: {1, 2, 3, 4} → {10} with\n\nf = {(1, 10), (2, 10), (3, 10), (4, 10)}\n\n(ii) g: {5, 6, 7, 8} → {1, 2, 3, 4} with\n\ng = {(5, 4), (6, 3), (7, 4), (8, 2)}\n\n(iii) h: {2, 3, 4, 5} → {7, 9, 11, 13} with\n\nh = {(2, 7), (3, 9), (4, 11), (5, 13)}\n\n(i) f: {1, 2, 3, 4} → {10}defined as:\n\nf = {(1, 10), (2, 10), (3, 10), (4, 10)}\n\nFrom the given definition of f, we can see that f is a many one function as: f(1) = f(2) = f(3) = f(4) = 10\n\nis not one-one.\n\nHence, function does not have an inverse.\n\n(ii) g: {5, 6, 7, 8} → {1, 2, 3, 4} defined as:\n\ng = {(5, 4), (6, 3), (7, 4), (8, 2)}\n\nFrom the given definition of g, it is seen that g is a many one function as: g(5) = g(7) = 4.\n\nis not one-one,\n\nHence, function g does not have an inverse.\n\n(iii) h: {2, 3, 4, 5} → {7, 9, 11, 13} defined as:\n\nh = {(2, 7), (3, 9), (4, 11), (5, 13)}\n\nIt is seen that all distinct elements of the set {2, 3, 4, 5} have distinct images under h.\n\n∴Function h is one-one.\n\nAlso, h is onto since for every element y of the set {7, 9, 11, 13}, there exists an element x in the set {2, 3, 4, 5}such that h(x) = y.\n\nThus, h is a one-one and onto function. Hence, h has an inverse.\n\nShow that f: [−1, 1] → R, given by", null, "is one-one. Find the inverse of the function f: [−1, 1] → Range f.\n\n(Hint: For y ∈Range fy =", null, ", for some x in [−1, 1], i.e.,", null, ")\n\nf: [−1, 1] → R is given as", null, "Let f(x) = f(y).", null, "∴ f is a one-one function.\n\nIt is clear that f: [−1, 1] → Range f is onto.\n\n∴ f: [−1, 1] → Range f is one-one and onto and therefore, the inverse of the function:\n\nf: [−1, 1] → Range exists.\n\nLet g: Range f → [−1, 1] be the inverse of f.\n\nLet y be an arbitrary element of range f.\n\nSince f: [−1, 1] → Range f is onto, we have:", null, "Now, let us define g: Range f → [−1, 1] as", null, "gof =", null, "and fo", null, "", null, "f−1 = g\n\n⇒", null, "Consider fR → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.\n\nfR → R is given by,\n\nf(x) = 4x + 3\n\nOne-one:\n\nLet f(x) = f(y).", null, "∴ f is a one-one function.\n\nOnto:\n\nFor y ∈ R, let y = 4x + 3.", null, "Therefore, for any ∈ R, there exists", null, "such that", null, "∴ f is onto.\n\nThus, f is one-one and onto, and therefore, f−1 exists.\n\nLet us define gR→ R by", null, ".", null, "", null, "Hence, f is invertible and the inverse of f is given by", null, "Consider fR→ [4, ∞) given by f(x) = x2 + 4. Show that f is invertible with the inverse f−1 of given by", null, ", where R+ is the set of all non-negative real numbers.\n\nfR+ → [4, ∞) is given as f(x) = x2 + 4.\n\nOne-one:\n\nLet f(x) = f(y).", null, "∴ f is a one-one function.\n\nOnto:\n\nFor y ∈ [4, ∞), let y = x2 + 4.", null, "Therefore, for any ∈ R, there exists", null, "such that", null, ".\n\n∴ f is onto.\n\nThus, f is one-one and onto, and therefore, f−1 exists.\n\nLet us define g: [4, ∞) → Rby,", null, "", null, "", null, "Hence, f is invertible and the inverse of f is given by", null, "Consider fR+ → [−5, ∞) given by f(x) = 9x2 + 6x − 5. Show that f is invertible with", null, ".\n\nfR+ → [−5, ∞) is given as f(x) = 9x2 + 6x − 5.\n\nLet y be an arbitrary element of [−5, ∞).\n\nLet y = 9x2 + 6− 5.", null, "f is onto, thereby range f = [−5, ∞).\n\nLet us define g: [−5, ∞) → R+ as", null, "We now have:", null, "", null, "", null, "and", null, "Hence, f is invertible and the inverse of f is given by", null, "Let fX → Y be an invertible function. Show that f has unique inverse.\n\n(Hint: suppose g1 and g2 are two inverses of f. Then for all y ∈ Y,\n\nfog1(y) = IY(y) = fog2(y). Use one-one ness of f).\n\nLet fX → Y be an invertible function.\n\nAlso, suppose f has two inverses (say", null, ").\n\nThen, for all y ∈Y, we have:", null, "Hence, f has a unique inverse.\n\nConsider f: {1, 2, 3} → {abc} given by f(1) = af(2) = b and f(3) = c. Find f−1 and show that (f−1)−1 = f.\n\nFunction f: {1, 2, 3} → {abc} is given by,\n\nf(1) = af(2) = b, and f(3) = c\n\nIf we define g: {abc} → {1, 2, 3} as g(a) = 1, g(b) = 2, g(c) = 3, then we have:", null, "", null, "and", null, ", where X = {1, 2, 3} and Y= {abc}.\n\nThus, the inverse of exists and f−1= g.\n\nf−1: {abc} → {1, 2, 3} is given by,\n\nf−1(a) = 1, f−1(b) = 2, f-1(c) = 3\n\nLet us now find the inverse of f−1 i.e., find the inverse of g.\n\nIf we define h: {1, 2, 3} → {abc} as\n\nh(1) = ah(2) = bh(3) = c, then we have:", null, "", null, ", where X = {1, 2, 3} and Y = {abc}.\n\nThus, the inverse of exists and g−1 = h ⇒ (f−1)−1 = h.\n\nIt can be noted that h = f.\n\nHence, (f−1)−1 = f.\n\nLet fX → Y be an invertible function. Show that the inverse of f−1 is f, i.e.,\n\n(f−1)−1 = f.\n\nLet fX → Y be an invertible function.\n\nThen, there exists a function gY → X such that gof = IXand fo= IY.\n\nHere, f−1 = g.\n\nNow, gof = IXand fo= IY\n\n⇒ f−1of = IXand fof−1= IY\n\nHence, f−1Y → X is invertible and f is the inverse of f−1\n\ni.e., (f−1)−1 = f.\n\nIf f→ be given by", null, ", then fof(x) is\n\n(A)", null, "(B) x3 (C) x (D) (3 − x3)\n\nfR → R is given as", null, ".", null, "Let", null, "be a function defined as", null, ". The inverse of f is map g: Range", null, "(A)", null, "(B)", null, "(C)", null, "(D)", null, "It is given that", null, "Let y be an arbitrary element of Range f.\n\nThen, there exists x ∈", null, "such that", null, "", null, "Let us define g: Range", null, "as", null, "Now,", null, "", null, "", null, "", null, "Thus, g is the inverse of f i.e., f−1 = g.\n\nHence, the inverse of f is the map g: Range", null, ", which is given by", null, "#### Chapter 1: Relations and Function EX 1.3 Solution- Preview\n\nFeel free to comment and share this article if you found it useful. Give your valuable suggestions in the comment session or contact us for any details regarding HSE Kerala Plus Two syllabus, Previous year question papers, and other study materials.\n\n### Plus Two Math's Related Links\n\nWe hope the given HSE Kerala Board Syllabus Plus Two Math's Solution Chapter Wise Pdf Free Download in both English Medium and Malayalam Medium will help you.\n\nIf you have any query regarding Higher Secondary Kerala Plus Two NCERT syllabus, drop a comment below and we will get back to you at the earliest.\n\nKeralanotes.com      Keralanotes.com      Keralanotes.com      Keralanotes.com      Keralanotes.com" ]

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[ "# scipy.signal.chebwin¶\n\nscipy.signal.chebwin(M, at, sym=True)[source]\n\nReturn a Dolph-Chebyshev window.\n\nParameters: M : int Number of points in the output window. If zero or less, an empty array is returned. at : float Attenuation (in dB). sym : bool, optional When True (default), generates a symmetric window, for use in filter design. When False, generates a periodic window, for use in spectral analysis. w : ndarray The window, with the maximum value always normalized to 1\n\nNotes\n\nThis window optimizes for the narrowest main lobe width for a given order M and sidelobe equiripple attenuation at, using Chebyshev polynomials. It was originally developed by Dolph to optimize the directionality of radio antenna arrays.\n\nUnlike most windows, the Dolph-Chebyshev is defined in terms of its frequency response:\n\n$W(k) = \\frac {\\cos\\{M \\cos^{-1}[\\beta \\cos(\\frac{\\pi k}{M})]\\}} {\\cosh[M \\cosh^{-1}(\\beta)]}$\n\nwhere\n\n$\\beta = \\cosh \\left [\\frac{1}{M} \\cosh^{-1}(10^\\frac{A}{20}) \\right ]$\n\nand 0 <= abs(k) <= M-1. A is the attenuation in decibels (at).\n\nThe time domain window is then generated using the IFFT, so power-of-two M are the fastest to generate, and prime number M are the slowest.\n\nThe equiripple condition in the frequency domain creates impulses in the time domain, which appear at the ends of the window.\n\nReferences\n\n [R188] C. Dolph, “A current distribution for broadside arrays which optimizes the relationship between beam width and side-lobe level”, Proceedings of the IEEE, Vol. 34, Issue 6\n [R189] Peter Lynch, “The Dolph-Chebyshev Window: A Simple Optimal Filter”, American Meteorological Society (April 1997) http://mathsci.ucd.ie/~plynch/Publications/Dolph.pdf\n [R190] F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transforms”, Proceedings of the IEEE, Vol. 66, No. 1, January 1978\n\nExamples\n\nPlot the window and its frequency response:\n\n>>> from scipy import signal\n>>> from scipy.fftpack import fft, fftshift\n>>> import matplotlib.pyplot as plt\n\n>>> window = signal.chebwin(51, at=100)\n>>> plt.plot(window)\n>>> plt.title(\"Dolph-Chebyshev window (100 dB)\")\n>>> plt.ylabel(\"Amplitude\")\n>>> plt.xlabel(\"Sample\")\n\n>>> plt.figure()\n>>> A = fft(window, 2048) / (len(window)/2.0)\n>>> freq = np.linspace(-0.5, 0.5, len(A))\n>>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))\n>>> plt.plot(freq, response)\n>>> plt.axis([-0.5, 0.5, -120, 0])\n>>> plt.title(\"Frequency response of the Dolph-Chebyshev window (100 dB)\")\n>>> plt.ylabel(\"Normalized magnitude [dB]\")\n>>> plt.xlabel(\"Normalized frequency [cycles per sample]\")", null, "", null, "#### Previous topic\n\nscipy.signal.boxcar\n\n#### Next topic\n\nscipy.signal.cosine" ]

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[ "trailer << /Size 58 /Info 18 0 R /Root 20 0 R /Prev 79313 /ID[<5d8c460fc1435631a11a193b53ccf80a><5d8c460fc1435631a11a193b53ccf80a>] >> startxref 0 %%EOF 20 0 obj << /Type /Catalog /Pages 7 0 R /JT 17 0 R >> endobj 56 0 obj << /S 91 /Filter /FlateDecode /Length 57 0 R >> stream A disconnection of a set A in a metric space (X,d) consists of two nonempty sets A1, A2 whose disjoint union is A and each is open relative to A. 1. The Overflow Blog Ciao Winter Bash 2020! Product Spaces 201 6.1. 0000009681 00000 n To partition a set means to construct such a cover. Theorem. Connectedness of a metric space A metric (topological) space X is disconnected if it is the union of two disjoint nonempty open subsets. 1.2 Open Sets (in a metric space) Now that we have a notion of distance, we can define what it means to be an open set in a metric space. Connectedness is a topological property quite different from any property we considered in Chapters 1-4. A video explaining the idea of compactness in R with an example of a compact set and a non-compact set in R. %PDF-1.2 %���� b.It is easy to see that every point in a metric space has a local basis, i.e. 0000009660 00000 n In this section we relate compactness to completeness through the idea of total boundedness (in Theorem 45.1). Connectedness in topological spaces can also be defined in terms of chains governed by open coverings in a manner that is more reminiscent of path connectedness. Let X = {x ∈ R 2 |d(x,0) ≤ 1 or d(x,(4,1)) ≤ 2} and Y = {x = (x 1,x 2) ∈ R 2 | − 1 ≤ x 1 ≤ 1,−1 ≤ x 2 ≤ 1}. 4.1 Compact Spaces and their Properties * 81 4.2 Continuous Functions on Compact Spaces 91 4.3 Characterization of Compact Metric Spaces 95 4.4 Arzela-Ascoli Theorem 101 5 Connectedness 106 5.1 Connected Spaces • 106 5.2 Path Connected spaces 115 Theorem. D. Kreider, An introduction to linear analysis, Addison-Wesley, 1966. Connectedness 1 Motivation Connectedness is the sort of topological property that students love. The metric spaces for which (b))(c) are said to have the \\Heine-Borel Property\". Related. Exercises 194 6. 0000005357 00000 n 252 Appendix A. So far so good; but thus far we have merely made a trivial reformulation of the definition of compactness. 0000027835 00000 n Define a subset of a metric space that is both open and closed. Let be a Cauchy sequence in the sequence of real numbers is a Cauchy sequence (check it!). @�6C׏�'�:,V}a���m؅G�a5v��,8��TBk\\u-}��j���Ut�&5�� ��fU��:uk�Fh� r� ��. Let X be a connected metric space and U is a subset of X: Assume that (1) U is nonempty. 0000001816 00000 n (I originally misread your question as asking about applications of connectedness of the real line.) 1. Watch Queue Queue Introduction. $��2�d��@���@�����f�u�x��L�|)��*�+���z�D� �����=+'��I�+����\\E�R)OX.�4�+�,>[^- x��Hj< F�pu)B��K�y��U%6'���&�u���U�;�0�}h���!�D��~Sk� U�B�d�T֤�1���yEmzM��j��ƑpZQA��������%Z>a�L! Our space has two different orientations. This volume provides a complete introduction to metric space theory for undergraduates. 0000011092 00000 n Metric Spaces A metric space is a set X that has a notion of the distance d(x,y) between every pair of points x,y ∈ X. Connectedness and path-connectedness. Its de nition is intuitive and easy to understand, and it is a powerful tool in proofs of well-known results. 3. 0000008983 00000 n Metric Spaces Notes PDF. A ball B of radius r around a point x ∈ X is B = {y ∈ X|d(x,y) < r}. Defn. 0000002498 00000 n The set (0,1/2) È(1/2,1) is disconnected in the real number system. Arcwise Connectedness 165 4.4. Browse other questions tagged metric-spaces connectedness or ask your own question. The purpose of this chapter is to introduce metric spaces and give some definitions and examples. X and ∅ are closed sets. Compact Sets in Special Metric Spaces 188 5.6. Let X be a metric space. (6) LECTURE 1 Books: Victor Bryant, Metric spaces: iteration and application, Cambridge, 1985. Compactness in Metric Spaces 1 Section 45. A disconnection of a set A in a metric space (X,d) consists of two nonempty sets A 1, A 2 whose disjoint union is A and each is open relative to A. Other Characterisations of Compactness 178 5.3. 0000005929 00000 n A set is said to be connected if it does not have any disconnections. Compactness in Metric Spaces Note. a sequence fU ng n2N of neighborhoods such that for any other neighborhood Uthere exist a n2N such that U n ˆUand this property depends only on the topology. Continuous Functions on Compact Spaces 182 5.4. 0000001127 00000 n 4. Let (x n) be a sequence in a metric space (X;d X). Metric spaces are generalizations of the real line, in which some of the theorems that hold for R remain valid. 0000007441 00000 n §11 Connectedness §11 1 Definitions of Connectedness and First Examples A topological space X is connected if X has only two subsets that are both open and closed: the empty set ∅ and the entire X. Theorem. The set (0,1/2) ∪(1/2,1) is disconnected in the real number system. We define equicontinuity for a family of functions and use it to classify the compact subsets of C(X,Rn) (in Theorem 45.4, the Classical Version of Ascoli’s Theorem). H�|SMo�0��W����oٻe�PtXwX|���J렱��[�?R�����X2��GR����_.%�E�=υ�+zyQ���ck&���V�%�Mť���&�'S� }� Request PDF | Metric characterization of connectedness for topological spaces | Connectedness, path connectedness, and uniform connectedness are well-known concepts. Conversely, the only topological properties that imply “ is connected” are very extreme such as “ 1” or “\\ l\\lŸ\\ has the trivial topology.”. Given a subset A of X and a point x in X, there are three possibilities: 1. Informally, a space Xis path-connected if, given any two points in X, we can draw a path between the points which stays inside X. 0000008396 00000 n (a)(Characterization of connectedness in R) A R is connected if it is an interval. Otherwise, X is connected. (2) U is closed. A set is said to be connected if it does not have any disconnections. Finally, as promised, we come to the de nition of convergent sequences and continuous functions. 0000002477 00000 n 0000005336 00000 n PDF. We present a unifying metric formalism for connectedness, … (III)The Cantor set is compact. It is possible to deform any \"right\" frame into the standard one (keeping it a frame throughout), but impossible to do it with a \"left\" frame. (iii)Examples and nonexamples: (I)Any nite set is compact, including ;. 19 0 obj << /Linearized 1 /O 21 /H [ 1193 278 ] /L 79821 /E 65027 /N 2 /T 79323 >> endobj xref 19 39 0000000016 00000 n 0000054955 00000 n 0000001471 00000 n About this book. So X is X = A S B and Y is Are X and Y homeomorphic? Locally Compact Spaces 185 5.5. This video is unavailable. Firstly, by allowing ε to vary at each point of the space one obtains a condition on a metric space equivalent to connectedness of the induced topological space. There exists some r > 0 such that B r(x) ⊆ A. d(x,y) = p (x 1 − y 1)2 +(x 2 −y 2)2, for x = (x 1,x 2),y = (y 1,y 2). metric space X and M = sup p2X f (p) m = inf 2X f (p) Then there exists points p;q 2X such that f (p) = M and f (q) = m Here sup p2X f (p) is the least upper bound of ff (p) : p 2Xgand inf p2X f (p) is the greatest lower bounded of ff (p) : p 2Xg. Proposition 2.1 A metric space X is compact if and only if every collection F of closed sets in X with the finite intersection property has a nonempty intersection. with the uniform metric is complete. H�bfY������� �� �@Q���=ȠH�Q��œҗ�]���� ���Ji @����|H+�XD������� ��5��X��^aP/������ �y��ϯ��!�U�} ��I�C � V6&� endstream endobj 57 0 obj 173 endobj 21 0 obj << /Type /Page /Parent 7 0 R /Resources 22 0 R /Contents [ 26 0 R 32 0 R 34 0 R 41 0 R 43 0 R 45 0 R 47 0 R 49 0 R ] /MediaBox [ 0 0 612 792 ] /CropBox [ 0 0 612 792 ] /Rotate 0 >> endobj 22 0 obj << /ProcSet [ /PDF /Text ] /Font << /F2 37 0 R /TT2 23 0 R /TT4 29 0 R /TT6 30 0 R >> /ExtGState << /GS1 52 0 R >> >> endobj 23 0 obj << /Type /Font /Subtype /TrueType /FirstChar 32 /LastChar 121 /Widths [ 250 0 0 0 0 0 0 0 0 0 0 0 0 0 250 0 0 0 0 0 0 0 0 0 0 0 333 0 0 0 0 0 0 722 0 722 722 667 0 0 0 389 0 0 667 944 722 0 0 0 0 556 667 0 0 0 0 722 0 0 0 0 0 0 0 500 0 444 556 444 333 0 556 278 0 0 278 833 556 500 556 0 444 389 333 0 0 0 500 500 ] /Encoding /WinAnsiEncoding /BaseFont /DIAOOH+TimesNewRomanPS-BoldMT /FontDescriptor 24 0 R >> endobj 24 0 obj << /Type /FontDescriptor /Ascent 891 /CapHeight 0 /Descent -216 /Flags 34 /FontBBox [ -28 -216 1009 891 ] /FontName /DIAOOH+TimesNewRomanPS-BoldMT /ItalicAngle 0 /StemV 133 /FontFile2 50 0 R >> endobj 25 0 obj 632 endobj 26 0 obj << /Filter /FlateDecode /Length 25 0 R >> stream Finite unions of closed sets are closed sets. 0000055751 00000 n Path Connectedness Given a space,1 it is often of interest to know whether or not it is path-connected. (II)[0;1] R is compact. 0000009004 00000 n Definition 1.2.1. 0000008375 00000 n 0000003439 00000 n 2. 0000055069 00000 n 0000001193 00000 n Roughly speaking, a connected topological space is one that is \\in one piece\". (IV)[0;1), [0;1), Q all fail to be compact in R. Connectedness. 0000011071 00000 n Introduction to compactness and sequential compactness, including subsets of Rn. Watch Queue Queue. Arbitrary intersections of closed sets are closed sets. 0000003208 00000 n d(f,g) is not a metric in the given space. The hyperspace of a metric space Xis the space 2X of all non-empty closed bounded subsets of it, endowed with the Hausdor metric. A path-connected space is a stronger notion of connectedness, requiring the structure of a path.A path from a point x to a point y in a topological space X is a continuous function ƒ from the unit interval [0,1] to X with ƒ(0) = x and ƒ(1) = y.A path-component of X is an equivalence class of X under the equivalence relation which makes x equivalent to y if there is a path from x to y. {����-�t�������3�e�a����-SEɽL)HO |�G�����2Ñe���|��p~L����!�K�J�OǨ X�v �M�ن�z�7lj�M�E��&7��6=PZ�%k��KG����VÈa���n�����0H����� �Ї�n�C�yާq���RV(ye�>��|m3,����8}A���m�^c���1s�rS��! 0000008053 00000 n Already know: with the usual metric is a complete space. In compact metric spaces uniform connectedness and connectedness are well-known to coincide, thus the apparent conceptual difference between the two notions disappears. 0000001677 00000 n Theorem 1.1. Compact Spaces 170 5.1. 3. Suppose U 6= X: Then V = X nU is nonempty. Finite and Infinite Products … 1 Distance A metric space can be thought of as a very basic space having a geometry, with only a few axioms. 4.1 Connectedness Let d be the usual metric on R 2, i.e. De nition (Convergent sequences). Note. (3) U is open. Date: 1st Jan 2021. Example. 0000001450 00000 n If a metric space Xis not complete, one can construct its completion Xb as follows. A partition of a set is a cover of this set with pairwise disjoint subsets. Bounded sets and Compactness 171 5.2. 0000007259 00000 n 0000064453 00000 n 0000007675 00000 n Exercises 167 5. A metric space is called complete if every Cauchy sequence converges to a limit. m5†Ôˆ7Äxì }á ÈåœÏÇcĆ8 \\8\\\\µóå. Our purpose is to study, in particular, connectedness properties of X and its hyperspace. 1 Metric spaces IB Metric and Topological Spaces Example. Featured on Meta New Feature: Table Support. 0000010397 00000 n Local Connectedness 163 4.3. Then U = X: Proof. Since is a complete space, the sequence has a limit. A connected space need not\\ have any of the other topological properties we have discussed so far. M. O. Searc oid, Metric Spaces, Springer Undergraduate Mathematics Series, 2006. Second, by considering continuity spaces, one obtains a metric characterisation of connectedness for all topological spaces. 2. 0000010418 00000 n 11.A. The next goal is to generalize our work to Un and, eventually, to study functions on Un. For a metric space (X,ρ) the following statements are true. Otherwise, X is disconnected. 3.1 Euclidean n-space The set Un is an extension of the concept of the Cartesian product of two sets that was studied in MAT108. Metric Spaces Joseph Muscat2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 0000002255 00000 n We do not develop their theory in detail, and we leave the verifications and proofs as an exercise. For example, a disc is path-connected, because any two points inside a disc can be connected with a straight line. Example. 0000011751 00000 n METRIC SPACES and SOME BASIC TOPOLOGY Thus far, our focus has been on studying, reviewing, and/or developing an under-standing and ability to make use of properties of U U1. Sn= fv 2Rn+1: jvj= 1g, the n-dimensional sphere, is a subspace of Rn+1. PDF | Psychedelic drugs are creating ripples in psychiatry as evidence accumulates of their therapeutic potential. 0000003654 00000 n In these “Metric Spaces Notes PDF”, we will study the concepts of analysis which evidently rely on the notion of distance.In this course, the objective is to develop the usual idea of distance into an abstract form on any set of objects, maintaining its inherent characteristics, and the resulting consequences. Proof. Metric Spaces, Topological Spaces, and Compactness sequences in X;where we say (x ) ˘ (y ) provided d(x ;y ) ! Metric Spaces: Connectedness . 0000004269 00000 n A metric space with a countable dense subset removed is totally disconnected? Let an element ˘of Xb consist of an equivalence class of Cauchy 251. Let (X,ρ) be a metric space. 0000004663 00000 n yÇؕK÷”Ñ0öÍ7qiÁ¾’KÖ\"•æ¤Gпb^~˜ÇW\\Ú²Ž9A¶q$ýám9%Š*9de‹•yY̒ÆØJ\"ýa¶—>c8L‰Þë'”ˆ¸Y0䘔ìl¯Ã•g=Ö ±k¾ŠzB49Ä¢5Ž²Óû ‰þƒŒ2åW3Ö8叁=~Æ^jROpk\\Š4 -Òi|˜÷=%^U%1fAW\\à}€Ì¼³ÜÎ_ՋÕDÿEF϶]¡+\\:[½5?kãÄ¥Io´!rm¿…¯©Á#èæÍމoØÞ¶æþYŽþ5°Y3*̂q£Uík9™ÔÒ5ÙÅؗLô­‹ïqéÁ€¡ëFØw{‘ F]ì)Hã@Ù0²½U.j„/–*çÊJ‰ƒ ]î3²þ×îSõւ~âߖ¯Åa‡×8:xü.Në(c߄µÁú}h˜ƒtl¾àDoJ 5N’’êãøÀ!¸F¤£ÉÌA@2Tü÷@䃾¢MÛ°2vÆ\"Aðès.Ÿl&Ø'‰•±†B‹Ÿ{²”Ðj¸±SˆœH9¡ˆ?ŽÝåb4( 0000004684 00000 n Metric Spaces: Connectedness Defn. Chapter 8 Euclidean Space and Metric Spaces 8.1 Structures on Euclidean Space 8.1.1 Vector and Metric Spaces The set K n of n -tuples x = ( x 1;x 2:::;xn) can be made into a vector space by introducing the standard operations of addition and scalar multiplication Swag is coming back! 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[ "# Common Core Grade 7 Math (Worksheets, Homework, Lesson.\n\nCommon Core Grade 4 Math Browse through the list of common core standards for Grade-4 Math. Click on the common core topic title to view all available worksheets. (4-g-1) Geometry: Draw And Identify Lines And Angles, And Classify Shapes By Properties Of Their Lines And Angles.", null, "In order to assist educators with the implementation of the Common Core, the New York State Education Department provides curricular modules in P-12 English Language Arts and Mathematics that schools and districts can adopt or adapt for local purposes. The full year of Grade 4 Mathematics curriculum is available from the module links. Additional Materials: Grades Pre-K-Grade 5 Math Curriculum.", null, "Looking for video lessons that will help you in your Common Core Grade 5 math classwork or homework? Looking for Common Core Math Worksheets and Lesson Plans that will help you prepare lessons for Grade 5 students? The following lesson plans and worksheets are from the New York State Education Department Common Core-aligned educational.", null, "EngageNY math 4th grade 4 Eureka, worksheets, Place Value of Multi-Digit Whole Numbers, Comparing and rounding Multi-Digit Whole Numbers, Multi-Digit Whole Number Addition and Subtraction, Common Core Math, by grades, by domains, examples and step by step solutions.", null, "Common Core Grade 5 Math. Browse through the list of common core standards for Grade-5 Math. Click on the common core topic title to view all available worksheets. (5-g-1) Geometry: Graph Points On The Coordinate Plane To Solve Real-World And Mathematical Problems.", null, "Common Core Grade 2 Math. Browse through the list of common core standards for Grade-2 Math. Click on the common core topic title to view all available worksheets. (2-g-1) Geometry: Reason With Shapes And Their Attributes. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces.5 Identify triangles, quadrilaterals, pentagons.", null, "Common Core Grade 3 Math. Browse through the list of common core standards for Grade-3 Math. Click on the common core topic title to view all available worksheets. (3-g-1) Geometry: Reason With Shapes And Their Attributes. Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can.", null, "Grade 4 Mathematics Module 1: Topic D Lessons 11-12 - Zip File of Individual Documents (8.41 MB) Grade 4 Mathematics Module 1: Topic E Lessons 13-16 - Zip File of Individual Documents (17.05 MB) Grade 4 Mathematics Module 1: Topic F Lessons 17-19 - Zip File of Individual Documents (7.31 MB) Grade 4 Mathematics Module 1: Arabic - Zip Folder of.", null, "Mathematics Introduction to Common Core Math Engage New York Mathematics Math Homework Help 7-12 Mathematics ENY Math Homework Help, Video Lockers Math Test Writing Support Introduction to Writing Support Writing Samples Kindergarten Writing Samples 1st Grade Writing Samples.", null, "Lesson NYS COMMON CORE MATHEMATICS CURRICULUM 12 Homework 2 Name Date 1. Partition the rectangles in 2 different ways to show equal shares. a. 2 halves b. 3 thirds c. 4 fourths d. 2 halves e. 3 thirds f. 4 fourths 2. Cut out the square at the bottom of this page. a. Cut the square in half to make 2 equal-size rectangles. Shade 1 half using your.", null, "Skills available for Common Core fourth-grade math standards Standards are in black and IXL math skills are in dark green. Hold your mouse over the name of a skill to view a sample question. Click on the name of a skill to practice that skill.", null, "This course is meant to teach the teacher, parent or student the fundamentals of Common Core Math at the 4th grade level. The lectures are organized by subject, grade level, module, lesson and topic. Students can easily navigate to the specific lesson in this course and review the day's work that your student very likely just covered in class.", null, "Lesson 12 Summary In this lesson, you learned about a summary measure for a set of data called the median. To find a median you first have to order the data. The median is the midpoint of a set of ordered data; it separates the data into two parts with the same number of values below as above that point.", null, "This packet consists of 198 worksheets that are designed to review the standards taught in chapters 1 - 12 of Harcourt’s Go Math for first grade; however, they can be used with any CCSS first grade math series. These math sheets are great to use as a review before introducing the next lesson, as morning work, and as homework.\n\n## Common Core Grade 4 Math (Homework, Lesson Plans.\n\nFinding the Unit and Lesson Numbers. Everyday Mathematics is divided into Units, which are divided into Lessons. In the upper-left corner of the Study Link, you should see an icon like this: The Unit number is the first number you see in the icon, and the Lesson number is the second number. In this case, the student is working in Unit 5, Lesson.CCSS.ELA-Literacy.RL.4.10 By the end of the year, read and comprehend literature, including stories, dramas, and poetry, in the grades 4-5 text complexity band proficiently, with scaffolding as needed at the high end of the range.Common Core Grade 4 Math. Browse through the list of common core standards for Grade-4 Math. Click on the common core topic title to view all available worksheets. (4-g-1) Geometry: Draw And Identify Lines And Angles, And Classify Shapes By Properties Of Their Lines And Angles.\n\nHarcourt Go Math Common Core Daily Spiral Review for 1st Grade - Chapter 3. Need more spiral review in your math curriculum? This packet consists of 24 worksheets that are designed to review the standards taught in chapters 2 and 3 of Harcourt’s Go Math for first grade; however, they can be used with any CCSS 1st grade math series.This daily Math homework not only reviews students on the current skills the are working on, but it also loops back to things they have learned in the past. Each lesson reviews skills that may have even been taught at the beginning of the year. These homeworks have been proven to be effective in assisting students in passing standardized tests. The majority of my most previous class was made.\n\nEssay Coupon Codes Updated for 2021 Help With Accounting Homework Essay Service Discount Codes Essay Discount Codes" ]

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[ "## Synchronizing non-deterministic finite automata\n\nMar 23, 2017 - Henk Don1 and Hans Zantema2,3. 1. Department of ... Eindhoven, The Netherlands, email: [email protected]. 3Radboud University Nijmegen ...\n\nSynchronizing non-deterministic finite automata Henk Don1 and Hans Zantema2,3\n\narXiv:1703.07995v1 [math.CO] 23 Mar 2017\n\n1\n\nDepartment of Mathematics, Vrije Universiteit Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands, email: [email protected] 2 Department of Computer Science, TU Eindhoven, P.O. Box 513,, 5600 MB Eindhoven, The Netherlands, email: [email protected] 3 Radboud University Nijmegen, P.O. Box 9010, , 6500 GL Nijmegen, The Netherlands\n\nMarch 24, 2017\n\nAbstract In this paper, we show that every D3-directing CNFA can be mapped uniquely to a DFA with the same synchronizing word length. This imˇ plies that Cern´ y’s conjecture generalizes to CNFAs and that the general upper bound for the length of a shortest D3-directing word is equal to the Pin-Frankl bound for DFAs. As a second consequence, for several classes of CNFAs sharper bounds are established. Finally, our results allow us to detect all critical CNFAs on at most 6 states. It turns out that only very few critical CNFAs exist.\n\n1\n\nIntroduction and preliminaries\n\nIn this paper we study synchronization of non-deterministic finite automata (NFAs). As is the case for deterministic finite automata (DFAs), symbols define functions on the state set Q. However, in an NFA symbols are allowed to send a state to a subset of Q, rather than to a single state. An NFA is called complete if these subsets are non-empty. This basically says that in every state, every symbol has at least one out-going edge. Formally, a complete non-deterministic finite automaton (CNFA) A over a finite alphabet Σ consists of a finite set Q of states and a map δ : Q × Σ → 2Q \\ {∅}. We denote the number of states by |A| or by |Q|. A DFA is called synchronizing if there exists a word that sends every state ˇ to the same fixed state. In 1964 Cern´ y conjectured that a synchronizing DFA on n states always admits a synchronizing (or directing, reset) word of length at most (n − 1)2 . He gave a sequence Cn of DFAs in which the shortest synchronizing word attains this bound. In this paper, we denote\n\n1\n\nthe maximal length of a shortest synchronizing word in an n-state DFA by d(n). The best known bounds for d(n) are (n − 1)2 ≤ d(n) ≤\n\nn3 − n . 6\n\n(1)\n\nFor a proof of the upper bound, we refer to . A DFA on n states is critical if its shortest synchronizing word has length (n − 1)2 ; it is superˇ critical if its shortest synchronizing word has length > (n − 1)2 . So Cern´ y’s conjecture states that no super-critical DFAs exist. It turns out that there are not too many critical DFAs. Investigation of all critical DFAs with less than 7 states but unrestricted alphabet was recently completed [8, 6]. DFAs without copies of the same symbol and without the identity are called basic. For n = 3, 4, 5, 6, only 31 basic critical DFAs exist up to isomorphism. So critical DFAs are very infrequent, as the total number of basic DFAs on n n states is 2n −1 , including isomorphisms. For n ≥ 7, the only known examples ˇ are from Cern´ y’s sequence. For S ⊆ Q and w ∈ Σ∗ , let Sw be the set of all states where one can end when starting in some state q ∈ S and reading the symbols in w consecutively. Write qw for {q} w. Formal definitions will be given in Section 1.1. A DFA is synchronizing if there exists w ∈ Σ∗ and qs ∈ Q such that qw = qs for all q ∈ Q. There are several ways to generalize this concept of synchronization to CNFAs, see . In this paper, we study CNFAs known in the literature as D3-directing. This notion is defined as follows: Definition 1. A CNFA (Q, Σ, δ) is called D3-directing if there exists a word w ∈ Σ∗ and a state qs such that qs ∈ qw for all q ∈ Q. The word w is called a D3-directing word. An example of a D3-directing CNFA is depicted below. There exist several D3-directing words of length four, but no shorter ones. An example is w = baba, which gives 1w = {1, 3}, 2w = {1, 2} and 3w = {1, 2, 3}. The synchronizing state for this word is 1. Another D3-directing word is v = aabb, for which 1v = 3v = {1, 2, 3} and 2v = 2. Here the synchronizing state is 2. a 1 b\n\na a\n\nb 2 a\n\nb\n\n2\n\n3 b\n\nA word is D3-directing if starting in any state q, there exists a path labelled by w that ends in qs . For DFAs this notion coincides with a synchronizing word. If a CNFA A is D3-directing, a natural question is to find the length of a shortest D3-directing word. We denote this length by d3 (A). Furthermore, we denote by cd3 (n) the worst case, i.e. we let CDir(3) be the collection of all D3-directing CNFAs and define cd3 (n) = max {d3 (A) : A ∈ CDir(3), |A| = n} .\n\n(2)\n\nIn it is shown that for all n ≥ 1, 1 (n − 1)2 ≤ cd3 (n) ≤ n(n − 1)(n − 2) + 1. 2\n\n(3)\n\nThe lower bound follows from the fact that every DFA is also a CNFA and that for DFAs the notions of synchronization and D3-directability coincide. As far as we are aware, these bounds are still the sharpest known for CNFAs, although sharper results were recently obtained for the essentially equivalent problem of bounding lengths of column-primitive products of matrices . Analogous to DFAs, a D3-directing CNFA is called critical if its shortest D3-directing word has length (n − 1)2 , and super-critical if it has length > (n − 1)2 . In the current paper, we will prove that in fact cd3 (n) = d(n), which immediately sharpens the upper bound for cd3 (n) to (n3 − n)/6. Our ˇ result also implies that Cern´ y’s conjecture is equivalent to the following: Conjecture 1. Every D3-directing CNFA with n states admits a D3-directing word of length at most (n − 1)2 . The main ingredient to prove that cd3 (n) = d(n) is a splitting transformation Split that maps a CNFA to a DFA. Every D3-directing CNFA A is transformed into a synchronizing DFA Split(A), preserving the shortest ˇ D3-directing word length. For several classes of DFAs, the Cern´ y conjecture has been established, or sharper bounds than the general bounds have been proven, see for example [1, 2, 7, 9, 10, 13, 16]. If Split(A) satisfies the properties for one of these classes, then the sharper results for Split(A) also apply to the CNFA A. This observation gives rise to generalize several properties of DFAs into notions for CNFAs and to check if these generalized properties are preserved under Split. In this way, we derive sharper upper bounds on the maximal D3-directing word length for several classes of CNFAs. Finally, in this paper we search for examples of critical D3-directing CNFAs. Note that the number of CNFAs without identical symbols on n states is n n huge, namely 2(2 −1) when we include isomorphisms. Therefore an exhaustive search is problematic, even for small n. However, since we know that every critical CNFA can be transformed into a critical DFA, we can try to find critical examples by reversing the transformation Split. Since all critical DFAs on ≤ 6 states are known, this approach allows us to identify all critical CNFAs on ≤ 6 states. Applying this strategy to the other known critical 3\n\nˇ DFAs, the only critical CNFAs we find are small modifications of Cern´ y’s sequence.\n\n1.1\n\nPreliminaries\n\nIn this section we present our formal definitions and notation which will be slightly different from the traditional notation, as we avoid the use of the transition function. A symbol (or letter, label) a in a CNFA will be a function a : Q → 2Q \\ {∅}, and we denote a(q) by qa. A symbol extends Q Q (denoting S the extension by a as well) to a function a : 2 → 2 \\ {∅} by Sa = q∈S qa. The set of all letters on Q that can be obtained in this way is denoted T (Q), which is a strict subset of the set of all functions from 2Q to 2Q \\ {∅}. The set of all possible symbols in a DFA on its turn is a subset of T (Q): T d (Q) = {a ∈ T (Q) : ∀q ∈ Q |qa| = 1} . A CNFA A is defined to be a pair (Q, Σ), where Σ ⊆ T (Q). Similarly a DFA is a pair (Q, Σ) with Σ ⊆ T d (Q). Note that these definitions do not allow for two symbols that act exactly in the same way: if a is a possible symbol, then either a ∈ Σ or a 6∈ Σ. A symbol a ∈ T (Q) induces a directed graph Ga with vertex set Q and can therefore be viewed as a subset of Q × Q: a = {(q, p) : q ∈ Q, p ∈ qa} , so a is identified with the set of all edges in Ga . This point of view is used to define set relations and operations like inclusion and union on T (Q). For example, if a, b ∈ T (Q), then a ∪ b = {(q, p) : q ∈ Q, p ∈ qa ∪ qb} . Suppose A = (Q, Σ) is a CNFA and a, b ∈ Σ are such that a ⊆ b. If A is D3-directing, then the automaton (Q, Σ \\ {a}) is D3-directing as well with the same shortest synchronizing word length. Also the identity symbol has no influence on synchronization. Therefore, a CNFA is called basic if it has no identity symbol and no symbol is contained in another one. For DFAs this coincides with the existing notion of basic. If A = (Q, Σ) and B = (Q, Γ) are CNFAs, we say that B is contained in A and write B ⊆ A if for all b ∈ Γ there exists a ∈ Σ such that b ⊆ a. Alternatively, we say that A is an extension of B. If B ⊆ A and B = 6 A, we say that A strictly contains (or is a strict extension of) B. A critical CNFA is minimal if it is not the strict extension of another critical CNFA; it is maximal if it does not admit a basic critical strict extension. Finally, for w = w1 . . . wk ∈ Σ∗ and S ⊆ Q, define Sw inductively by S\u000f = S and Sw = (Sw1 . . . wk−1 )wk . So a word also is a function on 2Q , being the composition of the transformations by each of its letters. Therefore also the transition monoid Σ∗ is contained in T (Q). 4\n\n2\n\nTransforming a CNFA into a DFA, preserving D3-directing word length\n\nIn this section we present the transformation Split and explore some of its properties. We note that similar but less explicit ideas were recently used in [3, 11] to give bounds on the length of a positive product in a primitive set of matrices. We start by introducing a parametrized version of our transformation: Definition 2. Let A = (Q, Σ) be a CNFA. Fix qsplit ∈ Q, a ∈ Σ and denote the set qsplit a by {q1 , . . . , qm }. Define new symbols a1 , . . . , am on Q as follows: qsplit ai := qi ,\n\nand\n\nqai := qa\n\nfor q 6= qsplit ,\n\nand a new alphabet Γ = {a1 , . . . , am } ∪ (Σ \\ {a}). The CNFA (Q, Γ) will be denoted Split(A, qsplit , a). The idea of this transformation is that we want to make a CNFA A ‘more deterministic’. If |qsplit a| ≥ 2, then multiple outgoing edges in the state qsplit are labelled by the symbol a. So we could say that a offers a choice in qsplit . For each possible choice, we introduce a new symbol that is deterministic in qsplit and behaves as a in all other states. If |qsplit a| = 1, then A is not changed by the transformation. This definition immediately implies the following properties: Lemma 1. Let A = (Q, Σ) be a CNFA. Fix qsplit ∈ Q, as ∈ Σ and let (Q, Γ) be Split(A, qsplit , as ). Let c ∈ T d (Q) be an arbitrary deterministic symbol. Then 1. for all b ∈ Γ, there exists a ∈ Σ such that b ⊆ a, 2. (there exists a ∈ Σ such that c ⊆ a) ⇐⇒ (there exists b ∈ Γ such that c ⊆ b). Proof. Let b ∈ Γ, we will find a with the property claimed in the lemma. If b ∈ Γ ∩ Σ, take a = b. If b ∈ Γ \\ Σ, then b is one of the new symbols a1 , . . . am . In this case take a = as . Then qb = qa for q 6= qsplit and qb ∈ qa for q = qsplit . This proves the first statement. Suppose a ∈ Σ such that c ⊆ a, so qc ∈ qa for all q. Then qsplit c ∈ qsplit a. By Definition 2, there exists b ∈ Γ such that qsplit b = qsplit c and qb = qa for q 6= qsplit . This means c ⊆ b. Now suppose b ∈ Γ such that c ⊆ b. By statement 1 of the lemma, there exists a ∈ Σ such that b ⊆ a, which implies c ⊆ a. The parametrized Split preserves synchronization properties, as is shown in the next lemma.\n\n5\n\nLemma 2. Let A = (Q, Σ) be a CNFA and let B = Split(A, qsplit , a) for some qsplit ∈ Q, a ∈ Σ. Then 1. A is D3-directing if and only if B is D3-directing, 2. If A and B are D3-directing, then d3 (A) = d3 (B). Proof. Let Γ be the alphabet of B and denote the new labels by a1 , . . . , am . First assume that A is D3-directing. There exist w ∈ Σ∗ and qs ∈ Q such that qs ∈ qw for all q ∈ Q = {q1 , . . . , qn }. From each state q there exists a path labelled by w = w1 . . . w|w| that ends in qs : w\n\nw\n\nw\n\nw\n\nw\n\nw\n\n1 2 q1 = q10 −→ q11 −→ 1 2 q2 = q20 −→ q21 −→ .. . 1 2 qn = qn0 −→ qn1 −→\n\nw|w|\n\n|w|\n\nw|w|\n\n|w|\n\n...\n\n−→ q1 = qs\n\n...\n\n−→ q2 = qs\n\n...\n\n−→ qn|w| = qs\n\nw|w|\n\nWe will construct a word w ˜=w ˜1 . . . w ˜|w| ∈ Γ∗ which follows the same paths. We may assume that paths do not diverge again once they have met, i.e. if qit = qjt , then qit+1 = qjt+1 . \b Let Qt = q1t , . . . , qnt and suppose wt = a for some 1 ≤ t ≤ |w|. If qsplit 6∈ Qt−1 , define w ˜t to be a1 . Then qwt = q w ˜t for all q ∈ Qt−1 . If qsplit ∈ Qt−1 , then qit−1 = qsplit for some 1 ≤ i ≤ n. This means qit ∈ qsplit a, so there exists a ˜ ∈ Γ such that qit = qsplit a ˜. Define w ˜t to be a ˜. Finally, for all wt 6= a, let w ˜t = wt . Then qs ∈ q w ˜ for all q ∈ Q, so w ˜ is a D3-directing word for B. Now assume that B is D3-directing with D3-directing word w ∈ Γ∗ and synchronizing state qs . By repeated application of Lemma 1 (replacing every symbol of w that is not in Σ by a), it follows that there exists w ˜ ∈ Σ∗ such that that qw ⊆ q w ˜ for all q. Therefore qs ∈ q w ˜ and the word w ˜ is D3directing for A. The above arguments prove the first statement of the lemma. Clearly, rewriting a D3-directing word from Σ∗ to Γ∗ and vice versa preserves the length. This implies the second statement. Next we investigate the result of applying consecutive parametrized Split transformations to all non-deterministic symbols in a CNFA A. We will show that this terminates and that the result is a uniquely defined DFA. Lemma 3. Let A0 be a CNFA, and repeat the following. If Ak = (Q, Σk , δ) is not a DFA, choose ak ∈ Σk and qk ∈ Q for which |qk ak | ≥ 2. Let Ak+1 = Split(Ak , qk , ak ). Then 1. There exists k ≥ 0 where this process ends such that Ak is a DFA. 6\n\n2. The resulting DFA does not depend on the choices of ak and qk . Proof. If Ak is a DFA, then |qa| = 1 for all q ∈ Q and a ∈ Σk . If Ak is not a DFA, then Y Y Y Y |qa| > |qa| ≥ 1. q∈Q a∈Σk+1\n\nq∈Q a∈Σk\n\nAs this integer sequence is strictly decreasing, it ends in 1, i.e. there exists k ≥ 0 such that the kth term is equal to 1. This is equivalent to the first claim of the lemma. For the second claim, choose k such that Ak is a DFA. Let c ∈ T d (Q) be an arbitrary deterministic symbol. By repeated application of the second statement of Lemma 1, it follows that c ∈ Σk if and only if there exists a ∈ Σ0 for which c ⊆ a. Therefore the DFA Ak does not depend on the splitting choices, proving the second statement. Definition 3. Let A = (Q, Σ) be a CNFA. The unique DFA that is produced by repeated application of the parametrized Split will be called Split(A). Lemma 3 guarantees that Split(A) is well-defined. Extension of Lemma 1 leads to the following characterization: Lemma 4. Let A = (Q, Σ) be a CNFA. Denote the DFA Split(A) by (Q, Γ). Let b ∈ T d (Q) be an arbitrary deterministic symbol. Then b ∈ Γ if and only if there exists a ∈ Σ such that b ⊆ a. Proof. If b ∈ Γ, repeatedly apply the first statement of Lemma 1. If there exists a ∈ Σ such that b ⊆ a, then repeated application of the second statement of Lemma 1 proves existence of b0 ∈ Γ such that b ⊆ b0 . Since both b and b0 are deterministic symbols, b = b0 , so b ∈ Γ. Moreover, we have the following: Corollary 1. If A = (Q, Σ) is a D3-directing CNFA, then d3 (A) = d(Split(A)). Proof. This is an immediate consequence of Lemma 2. Now also the main result of this section is straightforward: Theorem 1. The maximal shortest D3-directing word length for DFAs is the same as for CNFAs, i.e. d(n) = cd3 (n). Proof. Since every DFA is also a CNFA and the notions of synchronization and D3-directedness coincide for DFAs, it follows that d(n) ≤ cd3 (n). By Corollary 1 every CNFA A has a corresponding DFA Split(A) with the same shortest D3-directing word length. Therefore cd3 (n) ≤ d(n). 7\n\nThis theorem establishes equivalence of Cern´ y’s conjecture to Conjecture 1. It also implies the following sharpening of the upper bound for cd3 (n): Corollary 2. cd3 (n) ≤\n\n3\n\nn3 − n . 6\n\nSharper bounds for several classes of CNFAs\n\nˇ For several classes of DFAs the Cern´ y conjecture has been settled, or at least better upper bounds than the cubic one for the general case have been obtained. If the Split transform reduces a CNFA to a DFA that belongs to one of these classes, then as a direct consequence we obtain improved bounds for the D3-directing length in the CNFA. In this section we present a couple of results of this type. The general pattern of the arguments in this section is as follows. First we give the definition of a property P for DFAs, together with references to the best known upper bound uP for synchronization lengths in DFAs satisfying P. Then we give a natural extension of P to the class of CNFAs. Finally, we show that every CNFA A satisfying P is reduced to a DFA Split(A) satisfying P. Corollary 1 then guarantees that the length of the shortest D3-directing word in A is at most uP .\n\n3.1\n\nCyclic automata\n\nA DFA A = (Q, Σ) is cyclic if one of the letters in Σ acts as a cyclic permutation on Q. Definition 4. A DFA A = (Q, Σ) with |Q| = n is called cyclic if there exists a ∈ Σ such that for all q ∈ Q qan = q\n\nand\n\nqak 6= q\n\nfor\n\n1 ≤ k ≤ n − 1.\n\nEquivalently, the states can be indexed in such a way that qi a = qi+1 for 1 ≤ i ≤ n − 1, and qn a = q1 . Examples of cyclic automata include the ˇ well-known sequence Cn discovered by Cern´ y. Dubuc proved that a synchronizing cyclic DFA has a synchronizing word ˇ of length at most (n − 1)2 , as predicted by Cern´ y’s conjecture. We define non-deterministic cyclic automata as follows. Definition 5. A CNFA A = (Q, Σ) is called cyclic if there exists a ∈ Σ and an indexing of the states such that qi+1 ∈ qi a\n\nfor\n\n1 ≤ i ≤ n − 1,\n\nand\n\nq1 ∈ qn a.\n\nNote that with this definition, a CNFA is cyclic if and only if it is the extension of a cyclic DFA. 8\n\nProposition 1. If A is a D3-directing cyclic CNFA, then A has a shortest D3-directing word of length at most (n − 1)2 . Proof. Denote Split(A) by (Q, Γ). Choose a ∈ Σ and an indexing of the states as in Definition 5. Define b such that qi b = qi+1 for 1 ≤ i ≤ n − 1, and qn b = q1 . Then b ⊆ a so Lemma 4 gives b ∈ Γ. Therefore Split(A) is a cyclic DFA and the result follows.\n\n3.2\n\nOne-cluster automata\n\nA DFA A = (Q, Σ) is called one-cluster if for some letter a ∈ Σ, there is only one cycle (possibly a self-loop) labelled a. For every q ∈ Q, the path qaaa . . . eventually ends in this cycle. One way to formally define this is: Definition 6. A DFA A = (Q, Σ) is called one-cluster if there exists a ∈ Σ and p ∈ Q such that for all q ∈ Q qak = p\n\nfor some\n\nk ∈ N.\n\nNote that cyclic DFAs are contained in the class of one-cluster automata. B´eal, Berlinkov and Perrin proved that in a synchronizing one-cluster DFA, the length of the shortest synchronizing word is at most 2n2 − 7n + 7. We define one-cluster CNFAs in the following way. Definition 7. A CNFA A = (Q, Σ) is called one-cluster if there exists a ∈ Σ and p ∈ Q such that for all q ∈ Q p ∈ qak\n\nfor some\n\nk ∈ N.\n\nWith this definition, cyclic CNFAs are a special case of one-cluster CNFAs. Like for the cyclic case, the CNFA is one-cluster if and only if it it an extension of a one-cluster DFA. Proposition 2. If A is a D3-directing one-cluster CNFA, then A has a shortest D3-directing word of length at most 2n2 − 7n + 7. Proof. Let Split(A) = (Q, Γ). Choose a and p as in Definition 7 and denote the states by q1 , . . . , qn . For i = 1, . . . , n, let ki ≥ 1 be the smallest integer such that p ∈ qi aki (note that this can also be done if qi = p). Choose qi0 ∈ qi a such that p ∈ qi0 aki −1 . Define a symbol b by qi b = qi0 for all i. Then b ⊆ a and qi bki = p. By Lemma 4, b ∈ Γ and therefore Split(A) is a cyclic DFA. Remark 1. To see if a CNFA is one-cluster, it is sufficient to check pairs of states. A CNFA A = (Q, Σ) is one-cluster if and only if there exists a ∈ Σ such that for any q, q 0 ∈ Q there exist r, s ∈ N for which qar ∩ q 0 as 6= ∅.\n\n9\n\n3.3\n\nMonotonic automata\n\nDefinition 8. A DFA A = (Q, Σ) is called monotonic if Q admits a linear order < such that for each a ∈ Σ the map a : Q → Q preserves the order 1. After adding the identity id its graph consists of a single edge {b, id}, yielding one more basic critical CNFA on n states: Cn to which a b-self-loop is added to 1, yielding 1b = {1, 2}. Summarizing, for n ≤ 6 we have up to isomorphism the following numbers of basic critical DFAs and CNFAs: nr of states nr of DFAs nr of CNFAs 2 4 20 3 15 50 4 12 24 5 2 3 6 2 3 while for n > 6 the only known basic critical DFA is Cn , to be extended to exactly one more basic critical CNFA. 19\n\n5\n\nConclusions\n\nThe central result of this paper is that every D3-directing CNFA can be transformed to a synchronizing DFA with the same synchronizing word length. In this paper we present this Split transformation and explore its properties. An immediate consequence is that the maximal shortest D3directing length for CNFAs is equal to the maximal shortest synchronizing ˇ length for DFAs, which means that the famous Cern´ y conjecture for DFAs extends to CNFAs. For several classes of DFAs with some additional properties, tighter bounds for synchronization lengths have been established. If a CNFA is transformed into a DFA belonging to such a class, then the tighter bound also applies to the CNFA. This observation is used to define properties for CNFAs that guarantee improvements over the general cubic bound. In the last part of the paper, we investigate critical CNFAs. The tight connection between critical DFAs and critical CNFAs, combined with the fact that critical DFAs are extremely rare, implies that also a very small fraction of CNFAs is critical. All critical DFAs on at most 6 states are known. By essentially inverting Split, we identify all critical CNFAs on at most 6 states. Furthermore, for all n ≥ 3 there is exactly one critical CNFA which is a ˇ strict extension of Cern´ y’s DFA Cn .\n\nReferences D.S. Ananichev and M.V. Volkov. Synchronizing monotonic automata. Theoretical Computer Science, 327:225–239, 2004. M.-P. B´eal, M.V. Berlinkov, and D. Perrin. A quadratic upper bound on the size of a synchronizing word in one-cluster automata. International Journal of Foundations of Computer Science, 22:277–288, 2011. V. D. Blondel, R. M. Jungers, and A. Olshevsky. On primitivity of sets of matrics. Automatica, 61(C):80–88, 2015. ˇ J. Cerny. Pozn´ amka k homog´ennym experimentom s koneˇcn´ ymi automatmi. Matematicko-fyzik´ alny ˇcasopis, Slovensk. Akad. Vied, 14(3):208–216, 1964. P. Chevalier, J. M. Hendrickx, and R. M. Jungers. Reachability of consensus and synchronizing automata. 54th IEEE Conference on Decision and Control, pages 4139–4144, 2015. M. de Bondt, H. Don, and H. Zantema. DFAs and PFAs with long shortest synchronizing word length. Available at http://arxiv.org/abs/1703.07618, 2017.\n\n20\n\nˇ H. Don. The Cern´ y conjecture and 1-contracting automata. Electronic Journal of Combinatorics, 23(3):P3.12, 2016. H. Don and H. Zantema. Finding DFAs with maximal shortest synchronizing word length. In Drewes F., Mart´ın-Vide C., and Truthe B., editors, Language and Automata Theory and Applications, volume 10168 of Springer Lecture Notes in Computer Science. Springer, Cham, 2017. ˇ L. Dubuc. Sur les automates circulaires et la conjecture de Cern´ y. RAIRO Inform. Theor. Appl., 32:21–34, 1998. D. Eppstein. Reset sequences for monotonic automata. SIAM Journal on Computing, 19:500–510, 1990. B. Gerencs´er, V. V. Gusev, and R. M. Jungers. Primitive sets of nonnegative matrices and synchronizing automata. Available at https://arxiv.org/abs/1602.07556, 2016. B. Imreh and M. Steinby. Directable non-deterministic automata. Acta Cybernetica, 14:105–115, 1999. J. Kari. Synchronizing finite automata on eulerian digraphs. Theoretical Computer Science, 295(1–3):223–232, 2003. J.-E. Pin. On two combinatorial problems arising from automata theory. Annals of Discrete Mathematics, 17:535–548, 1983. A. N. Trahtman. An efficient algorithm finds noticeable trends and ˇ examples concerning the Cern´ y conjecture. In Rastislav Kr´aloviˇc and Pawel Urzyczyn, editors, Mathematical Foundations of Computer Science 2006: 31st International Symposium, MFCS 2006, pages 789–800. Springer Berlin Heidelberg, 2006. M.V. Volkov. Synchronizing automata preserving a chain of partial orders. Theoretical Computer Science, 410(37):3513–3519, 2009.\n\n21" ]

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[ "Explore BrainMass\nShare\n\n# Mathematics - Finite Mathematics\n\nThis content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!\n\nPlease see the attached file for the fully formatted problem(s).\n\nNote the similarities in the following parallel treatments of a frequency distribution and a probability distribution.\n\nFrequency Distribution\nComplete the table below for the following data. (Recall that x is the midpoint for the interval.)\n14,7,1,11,2,3,11,6,10,13,11,11,16,12,9,11,9,10,7,12,9,6,4,5,9,16,12,12, 11,10,14,9,13,10,15,11,11,1,12,12,6,7,8,2,9,12,10,15,9,3\nInterval x Tally f x*f f*x2\n1-3 2 |||| | 6 12 24\n4-6\n7-9\n10-12\n13-15\n16-18\nTotals\nProbability Distribution\nA binomial distribution has n = 10 and p = .5. Complete the following table.\nx P(x) x*P(x)\n0 .001\n1 .010\n2 .044\n3 .117\n4\n5\n6\n7\n8\n9\n10\nTotals\n\na. Find the mean (or expected value) for each distribution.\nb. Find the standard deviation for each distribution.\nc. (Extra credit) Use the normal approximation of the binomial probability distribution to find the interval that contains 95.44% of that distribution." ]

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[ "# Thread: If? And? Or? Or something else?\n\n1. ## If? And? Or? Or something else?\n\nI'd really appreciate some help with this, please.\n\nI'm trying to put a value into a target cell. It looks at two other cells to decide what value should be there. There are three possible values, D, M and E.\n\nPut D in the target cell if any of these conditions are true:\nH3=2 and N3<3\nH3=3 and N3<6\nH3=4 and N3<9\nH3=5 and N3<12\nH3=6 and N3<15\n\nPut M in the target cell if any of these conditions are true:\nH3=2 and N3<4\nH3=3 and N3<7\nH3=4 and N3<10\nH3=5 and N3<13\nH3=6 and N3<16\n\nPut E in the target cell if any of these conditions are true:\nH3=2 and N3<5\nH3=3 and N3<8\nH3=4 and N3<11\nH3=5 and N3<14\nH3=6 and N3<17\n\nI am completely stumped with this one. I can get it to put D in the cell if H3=5 and N3 < 12 using =IF(AND(H3=5,N3<12),\"D\"). I tried using OR to add more conditions and it just falls over.", null, "", null, "Reply With Quote\n\n2. Try\n\n=IF(AND(H3>=2,H3<=6),IF(N3<3*(H3-1),\"D\",IF(N3<3*(H3-1)+1,\"M\",IF(N3<3*(H3-1)+2,\"E\",\"No match\"))),\"No match\")", null, "", null, "Reply With Quote\n\n#### Tags for this Thread\n\nif and or", null, "####", null, "Posting Permissions\n\n• You may not post new threads\n• You may not post replies\n• You may not post attachments\n• You may not edit your posts\n•" ]

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[ "Vol. 30 No. 1 ORIGINAL ARTICLE\n\n# Axial-length computation using corneal dimensions and A-scan biometry\n\nRey Matanguihan, MD, James Co Shu Ming, MD, Processo Surrell, MD\n\nTHE INTRODUCTION of new technologies such as ultrasound and the keratometer has made it possible to measure the axial length and corneal curvature for estimating intraocular-lens (IOL) power. Subsequent largescale population studies have led to the development of linear regression formulas, the most popular of which is the SRK II formula named after its authors Sanders, Retzlaff, and Kraff. These new technologies are hardly available in many areas in the Philippines. For instance, most of the 700 or so cataract surgeries performed in outreach surgical missions by the Department of Ophthalmology of the East Avenue Medical Center were done without the use of automated biometry machines. Many of these surgeries used IOLs within 20 to 22 D. La Nauze showed in a study of 346 Vietnamese adults that the mean IOL power for emmetropia was 20.72 D for males and 21.94 D for females.1 About 95% of their study population were within 5.5 D of these mean results. No similar study on Filipinos has been published. The need for a system of determining a suitable IOL without relying on biometry machines was addressed by Processo Surrell in a 2002 pilot study where he used a formula for computing IOL power based on measurements of the corneal diameter and height.2 He based his computations on two assumptions: the eyeball is composed of two unequal spheres (corneal and sclera) fused at the limbus; the sclera covers 300 degrees while the cornea covers the rest. He then devised a formula for axial-length computation based on a system of triangles and principles of the Pythagorean system as follows:", null, "In a study of 30 eyes, Surrell found no significant statistical difference between biometry readings and the axial length, keratometry, and IOL power derived from this formula. Determination of axial length is an important step in IOL estimation. As early as 1992, Olsen had already noted in a study of 584 IOL implantations that as much as 54% of the error was attributed to axial-length discrepancies.3 Since then, few studies have explored the relationship of the cornea and axial length. Touzeau compared biometric parameters measured with Orbscan with the subjective spherical equivalent of 190 eyes of 95 patients.4 Corneal biometric parameters did not correlate with refractive error but a strong correlation between corneal radius and axial length was seen. Yebra-Pimentel found a direct relationship between the axial-length-to-corneal-radius (AL/CR) ratio and the refractive error in hyperopes, emmetropes and low myopias.5 This study compared computed axial length using Surrell’s formula with readings retrieved from automated biometry measurements using the A-scan. We hypothesized that the results of the two methods would be the same and may then be used in subsequent studies of IOL determination using clinical measurements.\n\nMETHODOLOGY\nThis is a comparative, nonrandomized study of patients from the outpatient department of a tertiary-care academic medical institution who were screened for cataract surgery from April 1, 2004 to September 1, 2004. Patients with a history of eye trauma and ocular surgery and those previously diagnosed with gross abnormalities of size and shape of the cornea and sclera were excluded. Main outcome measures were corneal diameter, corneal slope, and axial length. The corneal diameter and slope were measured at the outpatient department. The patients were made to lie supine on a flat bed. Topical anesthesia was administered and a lid retractor placed. The horizontal and vertical corneal diameters were measured (Figure 2) using the anterior edge of the gray transition zone between the clear cornea and limbus as landmarks at the 3 and 9 o’clock and 6 and 12 o’clock positions under a microscope with a plastic Vernier caliper (Hangzhou Jinnan Tools and Measure Co. Ltd., China). The caliper is accurate up to 0.05 millimeter. The corneal slope is the distance measured from the center of the cornea to the landmarks of the gray transition zone at the 3, 6, 9, and 12 o’clock positions (Figure 3). Corneal measurements were done 3 times by a single examiner and results were encoded in MS Excel (Microsoft Corporation, Redmond, Washington, USA). The axial lengths were computed using Surrell’s formula for eachof the 4 meridians and the average axial length was labeled as the “Computed AXL.” Corresponding axial-length values were measured three consecutive times by another masked observer using Ophthasonic A-Scan III machine (Teknar Ophthasonic A-scan III, Teknar, St. Louis, MO, USA) and the values were recorded as “Measured AXL.” All data were recorded up to one-hundredth of a millimeter. The precision of computed axial length was defined as standard deviation of the difference between computed and measured axial-length values. Student t test compared the average of the “Computed AXL” values versus the average “Measured AXL” at p < 0.05 for statistical significance, using SPSS 10.0 for Windows (SPSS Inc., Chicago, IL, USA).\n\nRESULTS\nA total of 105 eyes (96 patients; 47 males, 49 females; ages 20 to 78) were included in the study. Computed axial lengths ranged from 21.10 to 24.41 mm, with a mean of 22.94 mm ± 0.65 mm (Table 1). Measured axial lengths ranged from 20.81 to 26.43 mm, with a mean of 23.06 mm ± 0.99 mm. Figure 4 shows a plot of the computed axial lengths with the corresponding measured axial length in ascending order.", null, "The mean difference between measured and computed axial was +0.12mm ±0.66 mm (range of -1.62 to +2.08 mm). Figure 5 shows a scatter graph plotting the difference between measured and computed axial length values. A positive or negative value means that the computed axial length was either overestimated or underestimated compared with the corresponding A-scan measurement. The difference between computed and measured axial lengths for the whole study population was not statistically significant (p = 0.08, 95% CI). Neither was the difference statistically significant for axial lengths less than 22.00 mm (p = 0.64), 22.00 to 22.99 mm (p = 0.11), 23.00 to 23.99 mm (p = 0.81). However, the difference was significant for axial length of 24.00 mm or greater (p = 0.03) (Table 1).", null, "", null, "DISCUSSION\nThe results indicate that the axial length obtained from measurements of the corneal diameter and slope approximates that obtained by A-scan biometry. The results are important in relation to the determination of IOL power. When applying the SRK II formula to compute for the difference in IOL power using the mean computed and measured axial lengths for the whole subject population, the 0.12 mm difference translates to a 0.30 diopter IOL discrepancy. The diopter IOL discrepancy was 0.175 (from mean difference of 0.07 mm) in the group with computed axial length of < 22.00 mm, 0.325 (from mean difference of 0.14 mm) in the group with computed axial length of 22.00 to 22.99 mm, 0.075 (from mean difference of 0.03 mm) in the group with computed axial length of 23.00 to 23.99 mm, and 1.875 (from mean difference of 0.75 mm) in eyes with computed axial length of 24.00 mm and greater. Although the methodology is simple, interobserver\nvariability may exist and alter the computed axial length due to differences in identifying corneal/limbal landmarks, the positioning of the caliper, and reading the measurements on the scale. Surell’s formula assumes that the cornea and the globe are two unequal perfect spheres. In reality, imperfections in the actual shape of the eyeball become sources of error when calculating for the axial length this way. This is obviated with direct ultrasound measurements in the A-scan method. In summary, this study showed that axial-length computation using the Surell formula can reliably approximate the actual axial length measured by A-scan and, in cases where an ultrasound machine is not available, may be used to determine the correct IOL power for\npatients about to undergo cataract surgery.\n\nReferences\n1. La Nauze J, Kim Thanh T, et al. Intraocular-lens-power prediction in a Vietnamese population: a study to establish a limited set of intraocular-lens powers that could be used in a Vietnamese population. Ophthalmic Epidemiology 1999; 6:147-158.\n2. Surrell P, Co Shu Ming J, Del Mundo J. Formulas based on geometric properties of the eye in intraocular-lens-power calculation. Phillippine Academy of Ophthalmology. http://www.pao.org.ph/main.php?fid=annual/abs2002bt7.\n3. Olsen T. Sources of error in intraocular-lens-power calculation. J Cataract Refract Surg 1992; 18: 125-129.\n4. Touzeau O, Allouch C, Borderie V, et al. Correltion entre la refraction et la biometrie oculaire. J Fr Ophthalmol 2003; 26:355-363.\n5. Yebra-Pimentel E, Giraldez MJ, Glez-Meijome JM, et al. Variacion de la ratio longitud axial/radio corneal (LA/CR) con el estado refractivo ocular. Relacion con los\ncomponentes oculares. Arch Soc Esp Oftalmol 2004; 79: 317-324.\n6. Bron AJ, et al. Wolff’s Anatomy of the Eye and Orbit, 8th ed. Arnold Publishing, 1997; 211-214.\n7. Sayegh FN. The correlation of corneal refractive power, axial length, and the refractive power of the emmetropizing intraocular lens in cataractous eyes. Ger J Ophthalmol 1996; 5: 328-331.\n8. Raj PS, Ilango B, Watson A. Measurement of axial length in the calculation of intraocular-lens power. Eye 1998; 12: 227-229." ]

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[ "WOLFRAM SYSTEMMODELER\n\n# GearType1\n\nMotor inertia and gearbox model for r3 joints 1,2,3", null, "# Wolfram Language\n\nIn:=", null, "`SystemModel[\"Modelica.Mechanics.MultiBody.Examples.Systems.RobotR3.Components.GearType1\"]`\nOut:=", null, "# Information\n\nThis information is part of the Modelica Standard Library maintained by the Modelica Association.\n\nModels the gearbox used in the first three joints with all its effects, like elasticity and friction. Coulomb friction is approximated by a friction element acting at the \"motor\"-side. In reality, bearing friction should be also incorporated at the driven side of the gearbox. However, this would require considerable more effort for the measurement of the friction parameters. Default values for all parameters are given for joint 1. Model relativeStates is used to define the relative angle and relative angular velocity across the spring (=gear elasticity) as state variables. The reason is, that a default initial value of zero of these states makes always sense. If the absolute angle and the absolute angular velocity of model Jmotor would be used as states, and the load angle (= joint angle of robot) is NOT zero, one has always to ensure that the initial values of the motor angle and of the joint angle are modified correspondingly. Otherwise, the spring has an unrealistic deflection at initial time. Since relative quantities are used as state variables, this simplifies the definition of initial values considerably.\n\n# Parameters (6)\n\ni Value: -105 Type: Real Description: Gear ratio Value: 43 Type: Real (N·m/rad) Description: Spring constant Value: 0.005 Type: Real (N·m·s/rad) Description: Damper constant Value: 0.4 Type: Torque (N·m) Description: Viscous friction torque at zero velocity Value: 0.13 / 160 Type: Real (N·m·s/rad) Description: Viscous friction coefficient (R=Rv0+Rv1*abs(qd)) Value: 1 Type: Real Description: Maximum static friction torque is peak*Rv0 (peak >= 1)\n\n# Connectors (2)\n\nflange_a flange_b", null, "Type: Flange_a Description: Flange of left shaft", null, "Type: Flange_b Description: Flange of right shaft\n\n# Components (3)\n\ngear spring", null, "Type: IdealGear", null, "Type: SpringDamper", null, "Type: BearingFriction" ]

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[ "# Molar masses of compounds\n\nWrite a program that takes in a compound made solely of elements with an atomic number less than or equal to 92 (Uranium), and outputs the molar mass of the compound in grams/mole.\n\n## Rules and Restrictions\n\n• You may not use a function that directly calculates molar masses for you.\n• Your program must be able to run on an offline machine.\n• You MAY use a separate data file. Contents of this file must be provided.\n• Your score is the length of your program in bytes, plus the the length of your data file in bytes, should you choose to use one.\n• This is , therefore, lowest score wins.\n\n## Input\n\nA string containing the compound. This string may be read through STDIN, passed as an argument, or set to a variable (please specify which). The following are all valid inputs:\n\n• CH2 (one Carbon, two Hydrogens)\n• (CH2)8 (8 Carbons, 16 Hydrogens)\n• U (1 Uranium)\n\nYour program is not required to function for embedded parentheses (i.e. ((CH3)2N4)4), nor for any compound that contains an element with atomic number greater than 92. If your program does not function for either of the above two cases, it must output Invalid formula for such cases - no undefined behavior.\n\n## Output\n\nYou should output, to STDOUT or to a variable, the molar mass of the compound in grams/mole. A list of atomic masses of elements is available here (Wikipedia). Your answer should be accurate to the decimal place for compounds containing fewer than 100 atoms (rounded).\n\nIf the input is invalid, Invalid formula must be outputted.\n\nInformation on calculating the molar mass of a compound in grams/mole (Wikipedia).\n\n## Examples\n\nInput Output\nH2O 18.015\nO 15.999 (16 acceptable)\nC6H2(NO2)3CH3 227.132\nFOOF 69.995\nC6H12Op3 Invalid formula\nNp 237 (or Invalid formula)\n((C)3)4 144.132 (or Invalid formula)\nCodeGolf Invalid formula\n\n• \"to the decimal point\" - for how large molecules? The mass of U1000000 is harder to determine to the decimal point than the mass of U2 – John Dvorak Aug 2 '14 at 11:09\n• to the decimal point for molecules with less than 100 atoms. Added this to the question statement. – es1024 Aug 2 '14 at 11:16\n• I also assume I'm allowed to output 13 if the atomic mass is 12.999? – John Dvorak Aug 2 '14 at 11:16\n• That doesn't sound right @es1024. 13, ok, but 12? – RubberDuck Aug 2 '14 at 12:16\n• \"Your answer should be accurate to the decimal place\" does that mean to one decimal place or to the nearest integer? – user16402 Aug 2 '14 at 13:04\n\n# Bash, 978 708 675 673 650 636 632 631 598 594\n\n211 bytes for the program and 382 bytes for the data.\n\nInput is on STDIN, output is on STDOUT. WARNING: creates files called g and a, if they exist, they will be overwritten!\n\nzcat y>g 2>a\ns=(sed 's/[A-Z][a-z]*/& /g;q'<g)\np=1.008\nv=sed 's/[0-9]\\+/*&/g;s/(/+(0/g'\nr=(tail -1 g)\nfor n in {0..91};{\np=bc<<<$p+${r[n]}\nv=${v//${s[n]}/+$p} } o=bc<<<0$v 2>a\necho ${o:-Invalid formula} ## Data file This requires a file called y, which is the zopfli-compressed form of this data (no trailing newline). zopfli is a gzip-compatible compression algorithm and can be decompressed with standard gzip tools. It was run with 1024 iterations (this is probably too much). The last 8 bytes were then removed. HHeLiBeBCNOFNeNaMgAlSiPSClKArCaScTiVCrMnFeNiCoCuZnGaGeAsSeBrKrRbSrYZrNbMoTcRuRhPdAgCdInSnSbITeXeCsBaLaCePrNdPmSmEuGdTbDyHoErTmYbLuHfTaWReOsIrPtAuHgTlPbBiPoAtRnFrRaAcPaThU 0 2.995 2.937 2.072 1.798 1.201 1.996 1.992 2.999 1.182 2.81 1.315 2.677 1.103 2.889 1.086 3.39 3.648 .850 .130 4.878 2.911 3.075 1.054 2.942 .907 2.848 .24 4.613 1.834 4.346 2.904 2.292 4.038 .944 3.894 1.670 2.152 1.286 2.318 1.682 3.054 2.04 3.07 1.836 3.514 1.448 4.543 2.407 3.892 3.050 5.144 .696 3.693 1.612 4.422 1.578 1.211 .792 3.334 .758 5.36 1.604 5.286 1.675 3.575 2.43 2.329 1.675 4.120 1.913 3.523 2.458 2.892 2.367 4.023 1.987 2.867 1.883 3.625 3.788 2.82 1.78 .02 1 12 1 3 1 4.036 1.002 5.991 The base64 of y is (use base64 -d to reproduce the original file): H4sIAAAAAAACAwTB226DMAwA0G9LvEJQIbVi70LfHPBoJAiSaR729zsnBB2LVw/x0UWNMm1up4IE +90ZCC1cvsCm2mkscEJ71l56dRept7ulTDY/Lebp5CW19MLVbbAOlSrlgfVH4fIyCihaXPGg49b6 lfPHXzhvxsecxxZ+Wb6TPq7B8O1a2HjH7Aue7p1qZ0ncgsKvz/8WxuRGoigGgfcfxYvA8r7kn9iA ei6ohAt/+lzuihmD1PFnMrdIV0PeNfOczh3Ylrw8hnHaM6w1WC8V3X4hcYjOfbKlTyz0pewsP5nh plOUK9mkPzkd4HLiCbktIGyQI5uaUvZzNBrwLhOf9hJij+Jo5WBf6mHLfh2OFqeaxOHbaGAZl5mL h5UBI3Hlx99GX4llPumDjgw+NIee7uCaRbrZkzluIhJYi2E0ZU2gb5OnYBTSJQMRfv91irmCz4KK B5Va5J7T7IGjHnR22PeAd+m3F3KW/voz4BMFPGNgxHE0Loq65M6+Pw== The md5sum is d392b0f5516033f2ae0985745f299efd. ## Explanation The numbers in the file are increments of relative atomic mass (so the relative atomic mass of lithium is 1.008 + 0 + 2.995 + 2.937). This script works by converting the chemical formula into an arithmetical expression with + and *, replacing each symbol with its relative atomic mass, then feeding the expression to bc. If the formula contains invalid symbols, bc will give a syntax error and output nothing to STDOUT; in that case, the output is Invalid formula. If STDIN is empty, the output is 0. Nested brackets are supported. zcat y>g 2>a # Unzip the file y and output result to the file g. Send standard error to file a (there will be an error because the last 8 bytes are missing) s=(sed 's/[A-Z][a-z]*/& /g;q'<g) # Read from g to sed; append a space after any capital letter optionally followed by a lowercase letter; exit at the end of the first line so the atomic masses are not processed in this way; parse result as space-separated array and save to variable$s\np=1.008 # In the upcoming loop, $p is the value obtained in the previous iteration v=sed 's/[0-9]\\+/*&/g;s/(/+(0/g' # Read from standard input to sed; prepend a * after any sequence of digits; replace ( with +(0; save to$v\nr=(tail -1 g) # Get the last line of file g; parse as space-separated array; save to $r for n in {0..91};{ # Loop once for each number from 0 to 91; set$n to the current number each iteration\np=bc<<<$p+${r[n]} # Concatenate $p, + and the next number from$r; send to bc and evaluate as arithmetic expression; save to $p (i.e. add the next increment from the file to$p)\nv=${v//${s[n]}/+$p} # Replace every occurence of the current element symbol with + and$p (i.e. replace each symbol with a + and its relative atomic mass\n} # end loop\no=bc<<<0$v 2>a # Prepend 0 to$v; send to bc and evaluate as arithmetic expression; redirect any errors on standard error to the file a; save result to $o echo${o:-Invalid formula} # If $o is empty (if there was a syntax error), output Invalid formula; otherwise output$o\n\n\n## Example\n\nC6H2(NO2)3CH3 #input\nC*6H*2+(0NO*2)*3CH*3 #after line 3\n+12.011*6+*1.008*2+(0+14.007+15.999*2)*3+12.011+1.008*3 #after the loop in lines 4-6\n0+12.011*6+*1.008*2+(0+14.007+15.999*2)*3+12.011+1.008*3 #given to bc in line 7\n227.132 #output as evaluated by bc\n\n• You could save a lot by storing the differences between the masses of the elements and calculating on the fly. The minimum and maximmm differences are -1.002 and +6.993 (either side of Pa.) By rearanging the elements in atomic mass order instead of atomic number order, the range becomes 0 to 5.991. When the periodic table was devised, there was much debate about which order was better. (Obviously, from a Chemistry point of view, the atomic number order is better, but they took some time deciding that.) – Level River St Aug 2 '14 at 14:11\n• @steveverrill I thought about that when writing the code, I'll look into it soon – user16402 Aug 2 '14 at 14:38\n• @steveverrill Done, the saving was about 33 chars – user16402 Aug 2 '14 at 16:09\n\n# Perl - 924\n\nIt uses a series of regex substitution operations on the inputted formula to expand the subscripted elements and groups, replace the elements with atomic weights, and transform it into a sequence of additions which it then evaluates.\n\n$_=<>;chop;@g='H1008He4003Li6940Be9012B10810C12011N14007O15999F18998Ne20179Na22989Mg24305Al26981Si28085P30974S32060Cl35450Ar39948K39098Ca40078Sc44955Ti47867V50942Cr51996Mn54938Fe55845Co58933Ni58693Cu63546Zn65380Ga69723Ge72630As74922Se78960Br79904Kr83798Rb85468Sr87620Y88906Zr91224Nb92906Mo95960Tc98Ru101070Rh102906Pd106420Ag107868Cd112411In114818Sn118710Sb121760Te127600I126904Xe131293Cs132905Ba137327La138905Ce140116Pr140907Nd144242Pm145Sm150360Eu151964Gd157250Tb158925Dy162500Ho164930Er167259Tm168934Yb173054Lu174967Hf178490Ta180948W183840Re186207Os190230Ir192217Pt195084Au196967Hg200592Tl204380Pb207200Bi208980Po209At210Rn222Fr223Ra226Ac227Th232038Pa231036U238028'=~/(\\D+)([\\d\\.]+)/g;for$b(0..91){$d=2*$b+1;$g[$d]=$g[$d]>999?$g[$d]/1000:$g[$d]}%h=@g;for$a('($$((?>[^()]|(?1))*)$$)(\\d+)','()([A-Z][a-z]?)(\\d+)'){for(;s/$a/$2x$3/e;){}}s/([A-Z][a-z]?)/($h{$1}||_).'+'/ge;if(/_/){print'Invalid formula';exit}$_.=0;print eval; # Mathematica 9 - 247 227 This is clearly cheating as I'm using a function that directly calculates atomatic masses (but not molar masses!): r=StringReplace;f[s_]:=Check[ToExpression@r[r[r[s,x:RegularExpression[\"[A-Z][a-z]*\"]:>\"ElementData[\\\"\"<>x<>\"\\\",\\\"AtomicWeight\\\"]+\"],x:DigitCharacter..:>\"*\"<>x<>\"+\"],{\"+*\"->\"*\",\"+\"~~EndOfString->\"\",\"+)\"->\")\"}],\"Invalid formula\"] Usage: Call the function f with a string containing the formula, the output will be the mass. Test: f[\"H2O\"] (* => 18.0153 *) f[\"O\"] (* => 15.9994 *) f[\"C6H2(NO2)3CH3\"] (* => 227.131 *) f[\"FOOF\"] (* => 69.9956 *) f[\"C6H12Op3\"] (* => Invalid formula *) f[\"Np\"] (* => 237 *) f[\"((C)3)4\"] (* => 144.128 *) f[\"CodeGolf\"] (* => Invalid formula *) Mathematica 10 doesn't output a raw number, but a number with a unit, so that may not be acceptable. • @MartinBüttner Thanks, I basically just copied a function from my init file and golfed it a little bit. The golfing could certainly be improved. – Tyilo Aug 2 '14 at 15:59 # Javascript, 1002 Input is in q and output is in a. I was unsure of what the rules for rounding were, so I truncated to 3 places after the decimal (or fewer, if digits were unavailable from Wikipedia). H=1.008 He=4.002 Li=6.94 Be=9.012 B=10.812 C=12.011 N=14.007 O=15.999 F=18.998 Ne=20.179 Na=22.989 Mg=24.305 Al=26.981 Si=28.085 P=30.973 S=32.06 Cl=35.45 Ar=39.948 K=39.098 Ca=40.078 Sc=44.955 Ti=47.867 V=50.941 Cr=51.996 Mn=54.938 Fe=55.845 Co=58.933 Ni=58.693 Cu=63.546 Zn=65.38 Ga=69.723 Ge=72.630 As=74.921 Se=78.96 Br=79.904 Kr=83.798 Rb=85.467 Sr=87.62 Y=88.905 Zr=91.224 Nb=92.906 Mo=95.96 Tc=98 Ru=101.07 Rh=102.905 Pd=106.42 Ag=107.868 Cd=112.411 In=114.818 Sn=118.710 Sb=121.760 Te=127.60 I=126.904 Xe=131.293 Cs=132.905 Ba=137.327 La=138.905 Ce=140.116 Pr=140.907 Nd=144.242 Pm=145 Sm=150.36 Eu=151.964 Gd=157.25 Tb=158.925 Dy=162.500 Ho=164.930 Er=167.259 Tm=168.934 Yb=173.054 Lu=174.966 Hf=178.49 Ta=180.947 W=183.84 Re=186.207 Os=190.23 Ir=192.217 Pt=195.084 Au=196.966 Hg=200.592 Tl=204.38 Pb=207.2 Bi=208.980 Po=209 At=210 Rn=222 Fr=223 Ra=226 Ac=227 Th=232.038 Pa=231.035 U=238.028 try{a=eval(q.replace(/(\\d+)/g,'*$1').replace(/(\\w)(?=[A-Z\\(])/g,'$1+'))}catch(e){a=\"Invalid formula\"} • +1 for the idea of using simple variables ... now I can leave my overcomplex answer for future shame – edc65 Aug 2 '14 at 14:38 • Input is actually in q, though everything else seems fine. +1 – es1024 Aug 2 '14 at 21:20 • You could save some bytes by deleting the trailing zeros: 121.760 = 121.76 – Fels Aug 20 '14 at 8:32 # Javascript (E6) 1231 As a function with the input as argument and returning the output. Precision: 3 decimal digits Use regexp to transform the chemical formula in a simple arithmethic expression, with sums and products, replacing: • ( with +( • any numeric sequence with '*', then the numeric sequence • any Capital letter followed by letters with '+', then the atomic mass of the corresponding element (if found) Then the expression is evaluated and the value returned. In case of errors or if value is NaN (or zero) then function returns 'Invalid formula' Now I see that all other answers use the same method ... oh well here is the javascript version F=f=>{ T={H:1.008,He:4.002,Li:6.94,Be:9.012,B:10.81,C:12.011 ,N:14.007,O:15.999,F:18.998,Ne:20.179,Na:22.989,Mg:24.305 ,Al:26.981,Si:28.085,P:30.973,S:32.06,Cl:35.45,Ar:39.948 ,K:39.098,Ca:40.078,Sc:44.955,Ti:47.867,V:50.941,Cr:51.996,Mn:54.938 ,Fe:55.845,Co:58.933,Ni:58.693,Cu:63.546,Zn:65.38,Ga:69.723,Ge:72.630 ,As:74.921,Se:78.96,Br:79.904,Kr:83.798,Rb:85.467,Sr:87.62,Y:88.905,Zr:91.224 ,Nb:92.906,Mo:95.96,Tc:98,Ru:101.07,Rh:102.905,Pd:106.42,Ag:107.868,Cd:112.411 ,In:114.818,Sn:118.710,Sb:121.760,Te:127.60,I:126.904,Xe:131.293 ,Cs:132.905,Ba:137.327,La:138.905,Ce:140.116,Pr:140.907,Nd:144.242,Pm:145 ,Sm:150.36,Eu:151.964,Gd:157.25,Tb:158.925,Dy:162.500,Ho:164.930,Er:167.259 ,Tm:168.934,Yb:173.054,Lu:174.966,Hf:178.49,Ta:180.947,W:183.84,Re:186.207 ,Os:190.23,Ir:192.217,Pt:195.084,Au:196.966,Hg:200.592,Tl:204.38,Pb:207.2 ,Bi:208.980,Po:209,At:210,Rn:222,Fr:223,Ra:226,Ac:227,Th:232.038,Pa:231.035 ,U:238.028,Np:237,Pu:244,Am:243,Cm:247,Bk:247,Cf:251,Es:252,Fm:257,Md:258 ,No:259,Lr:266,Rf:267,Db:268,Sg:269,Bh:270,Hs:269,Mt:278 ,Ds:281,Rg:281,Cn:285,Uut:286,Fl:289,Uup:289,Lv:293,Uus:294,Uuo:294}; e='Invalid formula'; try{return eval(f.replace(/([A-Z][a-z]*)|(\\d+)|(\\()/g,(f,a,b,c)=>c?'+(':b?'*'+b:a='+'+T[a]))||e} catch(x){}return e } • I used almost the same method, but I created the function at least a year ago, so at least I didn't copy you. ;) – Tyilo Aug 2 '14 at 16:10 ## PHP — 793 (583 + 210) Largely outdistanced by professorfish's answer, who is using a similar method, but hey… The symbols and the masses are gzip-compressed in the file a obtained with the following code: $symbolsList = ['H', 'He', 'Li', 'Be', 'B', 'C', 'N', 'O', 'F', 'Ne', 'Na', 'Mg', 'Al', 'Si', 'P', 'S', 'Cl', 'Ar', 'K', 'Ca', 'Sc', 'Ti', 'V', 'Cr', 'Mg', 'Fe', 'Co', 'Ni', 'Cu', 'Zn', 'Ga', 'Ge', 'As', 'Se', 'Br', 'Kr', 'Rb', 'Sr', 'Y', 'Zr', 'Nb', 'Mo', 'Tc', 'Ru', 'Rh', 'Pd', 'Ag', 'Cd', 'In', 'Sn', 'Sb', 'Te', 'I', 'Xe', 'Cs', 'Ba', 'La', 'Ce', 'Pr', 'Nd', 'Pm', 'Sm', 'Eu', 'Gd', 'Tb', 'Dy', 'Ho', 'Er', 'Tm', 'Yb', 'Lu', 'Hf', 'Ta', 'W', 'Re', 'Os', 'Ir', 'Pt', 'Au', 'Hg', 'Tl', 'Pb', 'Bi', 'Po', 'At', 'Rn', 'Fr', 'Ra', 'Ac', 'Th', 'Pa', 'U'];\n$massesList = [1.008, 4.003, 6.94, 9.012, 10.81, 12.011, 14.007, 15.999, 18.998, 20.18, 22.99, 24.305, 26.982, 28.085, 30.974, 32.06, 35.45, 39.948, 39.098, 40.078, 44.956, 47.867, 50.942, 51.996, 54.938, 55.845, 58.933, 58.6934, 63.546, 65.38, 69.726, 72.630, 74.922, 78.96, 79.904, 83.798, 85.468, 87.62, 88.906, 91.224, 92.906, 95.96, 98, 101.07, 102.906, 106.42, 107.868, 112.411, 114.818, 118.710, 121.760, 127.60, 126.904, 131.293, 132.905, 137.327, 138.905, 140.116, 140.908, 144.242, 145, 150.36, 151.964, 157.25, 158.925, 162.5, 164.93, 167.259, 168.934, 173.054, 174.967, 178.49, 180.948, 183.84, 186.207, 190.23, 192.217, 195.084, 196.967, 200.592, 204.38, 207.2, 208.98, 209, 210, 222, 223, 226, 227, 232.038, 231.036, 238.029];$fileArrayContent = [$symbolsList,$massesList];\n$fileStringContent = json_encode($fileArrayContent);\n\n$file = gzopen('a', 'w9'); gzwrite($file, $fileStringContent); gzclose($file);\n\n\nThe formula should be stored in the $f variable: $a=json_decode(gzfile('a'));$r=@eval('return '.str_replace($a,$a,preg_replace(['#(?<=(?!\\().)(\\(|'.implode('|',$a).')#','#\\d+#'],['+${1}','*$0'],$f)).';');echo error_get_last()?'Invalid formula':$r;\n\n\nHere is the ungolfed and commented version:\n\n// Recover the data stored in the compressed file\n$fileStringContent = gzfile('a');$fileArrayContent = json_decode($fileStringContent);$symbolsList = $fileArrayContent;$massesList = $fileArrayContent;$formula = preg_replace('#(?<=(?!\\().)(\\(|'. implode('|', $symbolsList) .')#', '+${1}', $formula); // Add a \"+\" before each opening paranthesis and symbol not at the beginning of the string and not preceded by an opening paranthesis$formula = preg_replace('#\\d+#', '*$0',$formula); // Add a \"*\" before each number\n\n$formula = str_replace($symbolsList, $massesList,$formula); // Replace each symbol with its corresponding mass\n\n$result = @eval('return '.$formula .';'); // Evaluate the value without showing the errors\necho error_get_last() ? 'Invalid formula' : $result; // Print the value, or \"Invalid formula\" if there was an error • I'm beating you by one byte now, I have 673 bytes – user16402 Aug 2 '14 at 16:19 • @professorfish Indeed! Besides, I've miscounted the file size, which is larger that what I've said before. You're by far the winner, currently ^^! – Blackhole Aug 2 '14 at 16:50 # Scala, 1077 I see all your solutions in dynamically typed languages with cop-outs like eval or built-in atomic mass function and raise you a statically-typed language solution: object M{type S=String;type Z=(Int,S) val d=\"H *dHe JVLi inBe!!rB !5\\\"C !AiN !W!O !l3F \\\".*Ne\\\":_Na\\\"XUMg\\\"fUAl#%#Si#0iP #OOS #[&Cl$!,Ar$P|K$GxCa$RBSc%(7Ti%G5V %gwCr%s.Mn&4JFe&>)Co&^yNi&\\\\ECu'2\\\"Zn'ERGa'seGe(4^As(M#Se(x Br)$$Kr)MLRb)_5Sr)v,Y *%kZr*>LNb*PBMo*ppTc+(TRu+I4Rh+\\\\ePd,$,Ag,3RCd,cqIn,}LSn-HrSb-i>Te.IJI .B$Xe.peCs/#sBa/RwLa/ccCe/pXPr/y!Nd0>NPm0FTSm1!VEu12\\\\Gd1jrTb1|aDy2DdHo2^VEr2wATm3+0Yb3W Lu3k@Hf42nTa4L{W 4kfRe5&wOs5QdIr5fqPt6'BAu6;DHg6azTl7,8Pb7J8Bi7]2Po7]FAt7h$Rn9+bFr96@Ra9V8Ac9tTh:8NPa:-mU :x4\".grouped(5).map{s=>(s.take(2).trim,s(2)*8836+s(3)*94+s(4)-285792)}.toMap\ndef k(n:S):Z={val x=n.takeWhile(_.isDigit);(if(x==\"\")1 else x.toInt,n drop x.length)}\ndef o(m:S):Z={if(m(0)==40){val(i,s)=a(m.tail);if(s(0)!=41)???;val(j,t)=k(s.tail);(j*i,t)}else{val l=if(m.size>1&&m(1).isLower)2 else 1;val(i,s)=d(m.take(l))->m.drop(l);val(j,t)=k(s);(j*i,t)}}\ndef a(m:S)={var(r,s)=(m,0);do{val(y,z)=o(r);r=z;s+=y}while(r!=\"\"&&r(0)!=41);s->r}\ndef main(q:Array[S]){println(try{val(m,s)=a(io.Source.stdin.getLines.next);if(s!=\"\")???;m/1e3}catch{case _=>\"Invalid formula\"})}}\n\n\nI should consider some compression for the data, but for now let's just have the atomic masses in base 94, uncompressed.\n\n• Gives me molarmass.scala:5: error: ';' expected but identifier found. def a(m:S)={var(r,s)=(m,0);do{val(y,z)=o(r);r=z;s+=y}while(r!=\"\"&&r(0)!=41)s->r}` – es1024 Aug 2 '14 at 22:20\n• Fixed. Also I found out Scala's parser is weird. – Karol S Aug 3 '14 at 11:53" ]

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[ "Related Search\nWeb Results\nwww.calculatorsoup.com/calculators/conversions/weight.php\n\nConvert among weight (mass) units. Convert to grams, miligrams, kilograms, metric tons, ounces, pounds, and tons.\n\nwww.rapidtables.com/convert/weight/kg-to-gram.html\n\nKilograms (kg) to Grams (g) weight conversion calculator and how to convert.\n\nwww.expii.com/t/grams-to-kilograms-g-to-kg-conversion-practice-9103\n\nIt also means 1 kilogram and 1000 grams are defined as being equal. Traditionally, grams are referred to as the base unit. That means we usually start in grams, ...\n\nwww.extension.iastate.edu/agdm/wholefarm/html/c6-80.html\n\nMetric units can be modified by adding a prefix to simplify expressions of ... lbs) and then converted to kilograms (11,200 lbs * .4536 kg/lb = 5,080 kg).\n\nwww.lenntech.com/calculators/mass-weight/mass-weight.htm\n\nUse this conversion program to convert units like kilogram to carat of pound or stone as alternative for a metric conversion table / chart.\n\nm.convert-me.com/en/convert/weight/kilogram.html\n\nWeight And Mass Converter / Metric / Kilogram [kg] Online converter page for a specific unit. Here you can make instant conversion from this unit to all ...\n\nKilogram (kilo) is the metric system base unit of mass. 1 Kilogram = 2.2046226218 Pounds. The symbol is \"kg\". Please visit weight and mass units conversion tool ...\n\ncalculator-converter.com/milliliters-to-kilograms.htm\n\n1 ml (milliliter) = 1/1000 L (Liter, the official SI unit of volume). Weight of 1 milliliter (ml) of pure water at temperature 4 °C = 0.001 kilogram (kg) ." ]

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[ "“One equals two” growled the mass of zombies in the distance. “One equals two.”\n\nThe first put down the shotgun. “I’ve got this one,” he said, picking up the megaphone.\n\n“If you’re sure,” said the second.\n\n“I’M SURE.”\n\nThe second covered his ears.\n\n“SORRY. I mean, sorry.” The first redirected the megaphone towards the horde. “PROVE IT!”\n\n“Let $x=y$” they grumbled.\n\n“CONSIDER IT LET”\n\n“Multiply by $x$, so $x^2 = xy$.”\n\n“AGREED”\n\n“Take away $y^2$, so $x^2 - y^2 = xy - y^2$.”\n\n“NOLO CONTENDERE.”\n\n“Nolo contendere?” said the second.\n\n“I don’t dispute it,” said the first.\n\n“Now factorise,” groaned the zombies. “$(x+y)(x-y) = y(x-y)$.”\n\n“Divide by $(x-y)$…”" ]

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[ "+0\n\nimage is question\n\n+1\n283\n1", null, "help\n\nMay 4, 2018\n\n#1\n+1\n\nThe pylon  cosists of a cylinder with a radius of 1 ft  and a height of 3 ft topped by a cone with a radius of 1ft  and a height of 2 ft\n\nSo.....the total volume is\n\npi * r^2 * h  +  pi * r^2 * h  /3  =\n\npi  [ 1^2 * 3   +  1^2 * 2 / 3 ]  =\n\npi [ 3 + 2/3 ]  =\n\n[ 3 + 2/3] pi   ft^3", null, "", null, "", null, "May 4, 2018" ]

[ null, "https://web2.0calc.com/api/ssl-img-proxy", null, "https://web2.0calc.com/img/emoticons/smiley-cool.gif", null, "https://web2.0calc.com/img/emoticons/smiley-cool.gif", null, "https://web2.0calc.com/img/emoticons/smiley-cool.gif", null ]

[ "# Majority Vote Algorithm\n\nQuestion: How we can determine which element in an array is majority element? (A majority element is the one which occurs most of the time compared with the other elements)\n\nFor example in the string: ABCAGRADART - A is the majority element.\n\nA naive solution of this problem will be to traverse the string (the array) and count the number of occurrences of every character. - In that case we will store a of of unnecessary information.\n\nAnother approach is to put every element in a hash-map and then after all the elements are in the map to traverse the elements in the map and see which has the biggest value in its key/value pair.\n\nBoth of the above solutions can be used for solving the problem but there is more effective way of doing that.\n\nIf we store a pair of the current candidate for majority element and the number of occurences. When we move through the array we compare the next element with the current one...\n\npublic class MajorityVotingAlgorithm {\npublic static void main(String[] args) {\nchar[] votes = { 'A', 'A', 'A', 'C', 'C', 'C', 'B', 'B', 'C', 'C', 'C',\n'B', 'C', 'C' };\n\nint counter = 0;\nchar candidate = ' ';\nfor (char v : votes) {\nif (counter == 0) {\ncandidate = v;\ncounter++;\n} else {\nif (candidate == v)\ncounter++;\nelse\ncounter--;\n}\n}\n\nSystem.out.println(\"Majority Element is: \" + candidate + \" (counter = \"\n+ counter + \")\");\n}\n}" ]

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[ "## Key\n\n• This line was added.\n• This line was removed.\n• Formatting was changed.\nComment: First published version\n##### Coming soon . . .\n\nExcerpt\nhidden true\n\nDonna Curry\n—(2010) The Math Practitioner, Vol.15, #3.\n\nDonna Curry\n\n—(2010) The Math Practitioner, Vol.15, #3.\n\n### Summary\n\nWe know that division is the inverse of multiplication. We also know that the answer to a division problem can sometimes be found by repeated subtraction e.g., the answer to 18 ÷ 3, determined by seeing how many 3’s can be subtracted from 18.\n\nMany of us, though, have not been taught that there are two ways to look at division: partitive and quotitive. Understanding these two different models of division may help your students visualize division of fractions. Having students understand that there are two ways to look at division does not mean that students need to be taught the terminology, but rather that they understand that division can be represented by different models." ]

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[ "# 零件参数设计\n\n(全国竞赛1997年A题)\n\n$y$ 的目标值(记作 $y_0$)为1.50。当 $y$ 偏离 $y_0\\pm0.1$ 时，产品为次品，质量损失为1000元；当 $y$ 偏离 $y_0\\pm0.3$ 时，产品为废品，质量损失为9000元。\n\n 标定值容许范围 C等 B等 A等 $x_1$ [0.075,0.125] / 25 / $x_2$ [0.225,0.375] 20 50 / $x_3$ [0.075,0.125] 20 50 200 $x_4$ [0.075,0.125] 50 100 500 $x_5$ [1.125,1.875] 50 / / $x_6$ [12,20] 10 25 100 $x_7$ [0.5625,0.935] / 25 100" ]

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[ "# PHYSICAL CHEMISTRY III\n\n1. Oxidizing power of element depend upon its\n\na. ionization energy\n\nb. electron affinity\n\nc. electrode potential\n\nd. oxidation potential\n\nCorrect Option: c\n\n2. when atom reacts chemically and gains one or more electron, it is said to be\n\na. catalyzed\n\nb. reduced\n\nc. oxidized\n\nd. decomposed\n\nCorrect Option: b\n\n3. oxidation takes place always on\n\na. anode\n\nb. cathode\n\nc. at both electrodes\n\nd. none of these\n\nCorrect Option: a\n\n4. electrolysis is used for\n\na. electroplating\n\nb. manufacture of Na\n\nc. mnufacture of Al\n\nd. all o these\n\nCorrect Option: d\n\n5. At equilibrium standard cell potential will  be\n\na. minimum\n\nb. zero\n\nc. greater than zero\n\nd. lesser than zero\n\nCorrect Option: b\n\n6. in electrolytic cell current move from\n\na. anode to cathode in external circuit\n\nb. cathode to anode in external circuit\n\nc. anode to cathode in internal\n\nd. none of these\n\nCorrect Option: a\n\n7. in electrolytic cell reaction always will be\n\na. fast\n\nb. slow\n\nc. non spontaneous\n\nd. spontaneous\n\nCorrect Option: c\n\n8. in Galvanic cell ∆E can never be\n\na. zero\n\nb. constant\n\nc. negative\n\nd. positive\n\nCorrect Option: c\n\n9. the reduction potential of Al is -1.66v and Sn is -0.14v when these are connected which one act as cathode\n\na. depend upon Conc.\n\nb. tin electrode\n\nc. Al electrode\n\nd. depend upon temp\n\nCorrect Option: b\n\n10. overall positive value of cell potential predicts that process is\n\na. not possible\n\nb. cannot be predicted\n\nc. feasible\n\nd. not feasible\n\nCorrect Option: c\n\n11. Standard electrode potential of x, y, z are -1.2 v, +0.5v and -3.0V respectively . Predict decreasing order of reducing power sequence\n\na. Y> X> Z\n\nb. Z > X> Y\n\nc. Y> Z> X\n\nd. X > Y> Z\n\nCorrect Option: b\n\n12. two Half cell of galvanic cell are  connected through salt bridge which\n\na. carries out electrolysis\n\nb. carries ions from one half cell to other\n\nc. carries electrons from one Half cell to other\n\nd. indicate the value of emf\n\nCorrect Option: b\n\n13. reaction on galvanic cell mostly will be\n\na. depend upon emf\n\nb. sontaneous\n\nc. non spontaneous\n\nd. depend upon conc.\n\nCorrect Option: b\n\n14. Saturated solution of KNO3 is used to make ‘salt-bridge’ because\n\na. KNO3 is highly soluble in water\n\nb. velocities of both K+ and are nearly the same\n\nc. velocity of K+ is greater than that of NO3\n\nd. ) velocity of NO3 ion is greater than that of K+\n\nCorrect Option: b\n\n15. the strongest reducing agent is\n\na. Mn+2\n\nb. Cr+3\n\nc. Cr\n\nd. Cl-\n\nCorrect Option: c\n\n16. In the electrolytic cell, flow of electrons is from\n\na. cathode to anode in solution\n\nb. cathode to anode through internal supply\n\nc. cathode to anode through external supply\n\nd. anode to cathode through internal supply\n\nCorrect Option: b\n\n17. based on data strong oxidizing agent is\n\na. Cr+3\n\nb. Cl-1\n\nc. MnO42-\n\nd. Mn+2\n\nCorrect Option: c\n\n18. Reaction will spontaneous if\n\na. ∆E equal to zero\n\nb. ∆E smaller than zero\n\nc. ∆E remains constant\n\nd. ∆E greater than zero\n\nCorrect Option: d\n\n19. Fuel cell efficiency is upto\n\na. 40%\n\nb. 60%\n\nc. 20%\n\nd. 90%\n\nCorrect Option: d\n\n20. Efficiency of heat engine is upto\n\na. 60%\n\nb. 70%\n\nc. 90%\n\nd. 40%\n\nCorrect Option: d\n\n21. liquid junction potential is\n\na. Affect efficiency of cell\n\nb. Does not contribute into anything\n\nc. useful\n\nd. not useful\n\nCorrect Option: a\n\n22. Determination of cell potential under non standard conditions is done through\n\na. galvanic cell\n\nb. cannot done\n\nc. Nerst equation\n\nd. electrolytic cell\n\nCorrect Option: c\n\n23. in IUPAC system negative sign is for\n\na. Reduction\n\nb. oxidation potential\n\nc. depend upon condition\n\nd. none of these\n\nCorrect Option: a\n\n24. Asymmetric and Electrophoretic effect are not shown by\n\na. Dilute solution\n\nb. Concentrated solution\n\nc. non ideal solution\n\nd. none of these\n\nCorrect Option: a\n\n25. based on data strong oxidizing agent is\n\na. Cr+3\n\nb. Cl-1\n\nc. MnO42-\n\nd. Mn+2\n\nCorrect Option: c\n\n26. The gas X at 1 atm is bubbled through a solution containing a mixture of 1 M Y- and I MZ- at 25°C. If the order of reduction potential is Z> Y > X, then\n\na. Y will reduce both X and Z\n\nb. Y will oxidise X and not Z\n\nc. Y will oxidise Z and not X\n\nd. Y will oxidise both X and Z\n\nCorrect Option: b\n\n27. Debye theory was postulated to cover\n\na. Interionic repulsion\n\nb. Interionic repulsion in dilution solution\n\nc. Interionic repulsion in conc. Solution\n\nd. Interionic attraction\n\nCorrect Option: d\n\n28. Effective Concentration accounts for\n\na. Non ideal behavior\n\nb. Ideal behavior\n\nc. dilute solution\n\nd. none of these\n\nCorrect Option: a\n\n29. Activity coefficient value is unity when\n\na. Solution is dilute\n\nb. solution is concentrated\n\nc. solution is non ideal\n\nd. none of these\n\nCorrect Option: a\n\n30. Corrosion is prevented by\n\na. Barrier protection\n\nb. cathodic protection\n\nc. sacrificial protection\n\nd. all o these\n\nCorrect Option: d\n\n31. Electrolyte’s solution shows deviation from ideal behavior due\n\na. pressure\n\nb. volume\n\nc. concentration\n\nd. tamperature\n\nCorrect Option: c\n\n32. Can we calculate electrode potential of electrode individually\n\na. only in the presence of standard electrode\n\nb. yes\n\nc. no\n\nd. none  of these\n\nCorrect Option: a\n\n33. Rate of electron transfer is calculated by method of\n\na. Debye huckle theory\n\nb. collision theory\n\nc. solution theory\n\nd. Transition state theory\n\nCorrect Option: d\n\n34. Schrodinger equation describe\n\na. Motion of electron wave and particle\n\nb. Motion of electron particle\n\nc. motion of electron wave\n\nd. none of these\n\nCorrect Option: a\n\n35. Calculation of motion of electron in three dimensional box gives\n\na. wave function of electron\n\nb. quantum numbers\n\nc. energy of electron\n\nd. all of these\n\nCorrect Option: d\n\n36. Eign value is\n\na. Numerical value\n\nb. imaginary value\n\nc. theoretical value\n\nd. none of these\n\nCorrect Option: a\n\n37. In quantum, lowest energy level is\n\na. highest\n\nb. discrete\n\nc. not zero\n\nd. zero\n\nCorrect Option: c\n\n38. for rigid rotator quantum mechanical calculation\n\na. variable bond length\n\nb. variable energy\n\nc. negligible bond length\n\nd. Vibration kept neligible\n\nCorrect Option: d\n\n39. what is Cn number in water\n\na. C2\n\nb. C1\n\nc. C4\n\nd. none of these\n\nCorrect Option: a\n\n40. what is Cn number in BF3\n\na. C4\n\nb. C3\n\nc. C1\n\nd. C2\n\nCorrect Option: b\n\n41. what is main axis in symmetry\n\na. y axis\n\nb. principle axis\n\nc. subsidiary axis\n\nd. x-axis\n\nCorrect Option: b\n\n42. what is Cn in staight line structure\n\na. C4\n\nb. C∞ (infinity)\n\nc. C2\n\nd. C3\n\nCorrect Option: b\n\n43. symmetry is repitition of\n\na. faces\n\nb. edges\n\nc. corners\n\nd. all of these\n\nCorrect Option: d\n\n44. σ v is the plane\n\na. parallel to priciple axis\n\nb. perpendicular to principle axis\n\nc. dihedral to priciple axis\n\nd. none of these\n\nCorrect Option: a\n\n45. σ h is the plane which is\n\na. perpendicular to priciple axis\n\nb. parallel to principle axis\n\nc. diherdral to principle axis\n\nd. none of these\n\nCorrect Option: a\n\n46. at higher atomic number quntum theory follows rules of\n\na. classical theory\n\nb. quantum theory\n\nc. planks theory\n\nd. none of these\n\nCorrect Option: a\n\n47. σ  v number is either zero or\n\na. 2\n\nb. 3\n\nc. equal to priciple axis number\n\nd. subsidiary axis number\n\nCorrect Option: c\n\n48. reduction is\n\na. gain of oxygen\n\nb. loss of hydrogen\n\nc. gain of electron\n\nd. loss of electron\n\nCorrect Option: c\n\n49. Number of C2 in BF3\n\na. 2\n\nb. 1\n\nc. 0\n\nd. 3\n\nCorrect Option: d\n\n50. Cn in benzene\n\na. 5\n\nb. 2\n\nc. 1\n\nd. 6\n\nCorrect Option: d\n\n———————-\n\nMCQs Part-2\n\n———————\n\n1.What is Corrosion?\na) Destruction or deterioration of a material\nb) Conversion of metal atoms to metallic ions\nc) Conversion of metal ions to metal atoms\nd) Destruction of materials involving in the conversion of metal atoms into metal ions\n\n2.Which of the following materials will undergo Corrosion?\na) Metals only\nb) Metals and Non-metals\nc) Metals, Non-metals, Ceramics and Plastics\nd) Metals, Non-metals, Ceramics, Plastics and Rubbers\n\n3.Corrosion of material by furnace gases is classified as _____;\na) wet corrosion\nb) dry corrosion\nc) galvanic corrosion\nd) crevice corrosion\n\n4.Which of the following is an example of wet corrosion?\na) Corrosion of metal in the water\nb) Corrosion of iron in the presence of anhydrous calcium chloride\nc) Corrosion of titanium in dry chlorine\nd) Corrosion due to furnace gases\n\n5.What is the reason for corrosion?\na) Stability of a metal ion\nb) Stability of a metal atom\nc) Passivation\nd) Use of coatings\n\n6.Which of the following are necessary in the process of corrosion?\na) Anode\nb) Cathode\nc) Electrolyte\nd) Anode, Cathode and Electrolyte\n\n7.Corrosion involves _______ reactions.\na) oxidation\nb) reduction\nc) displacement\nd) both oxidation and reduction\n\n8.Which of the following is considered as high corrosive resistant material?\na) Mild steel\nb) Cast iron\nc) Zinc\nd) Stainless steel\n\n9.Which of the following is an incorrect statement?\na) Corrosion is an irreversible process\nb) Corrosion is a non-spontaneous process\nc) Corrosion is a degradation process\nd) Corrosion is a spontaneous process\n\n10.Which of the following is an example of corrosion?\na) Rusting of iron\nb) Tarnishing of silver\nc) Liquefaction of ammonia\nd) Rusting of iron and tarnishing of silver\n\n11.In wet corrosion ____ are formed at the cathodic areas.\na) Organic compounds\nb) Metallic ions\nc) Non-metallic ions\nd) Inorganic compounds\n\n12.Rusting of iron in neutral aqueous solution of electrolyte occurs in the presence of oxygen with the evolution of _____;\na) Nitrogen\nb) Chloride\nc) Sulphide\nd) Hydrogen\n\n13.Select the incorrect statement about the wet corrosion from the following option.\na) It involves the setting up of large number of galvanic cells\nb) It is explained by absorption mechanism\nc) It occurs only on heterogeneous metal surface\nd) It is a fast process\n\n14.Which of the following factor influences the rate and extent of corrosion?\na) Nature of metal only\nb) Nature of the environment only\nc) Both nature of metal and environment\nd) Nature of reaction\n\n15.What is the other name of galvanic corrosion?\na) Bi-metallic corrosion\nb) Mono-metallic corrosion\nc) Localized corrosion\nd) Mono-metallic and localized corrosion\n\n16.Which of the following is the driving force in galvanic corrosion?\na) Conductivity of electrolyte\nb) Crystal structure of metals\nc) The potential difference between the two metals\nd) Temperature of electrolyte\n\n17.Which of the following is the primary characteristic of an electrolyte to form corrosion?\na) Electrical resistivity\nb) Thermal resistivity\nc) Thermal conductivity\nd) Electrical conductivity\n\n18.Which of the following is the most corrosion-resistant metal at room temperature?\na) Titanium\nb) Platinum\nc) Gold\nd) Tantalum\n\n19.Pitting is a form of extremely localized attack that results in holes in the metal.\na) It involves the setting up of large number of galvanic cells\nb) It is explained by absorption mechanism\nc) It occurs only on heterogeneous metal surface\nd) It is a fast process\n\n20.Which of the following form of corrosion is more destructive and insidious in nature?\na) Uniform corrosion\nb) Intergranular corrosion\nc) Pitting corrosion\nd) Galvanic corrosion\n\n21.Which of the following are the reasons that make it difficult to detect pits?\na) Small size\nb) Varying depths\nc) Pits covered with corrosion products\nd) Small size, varying depths and covered with corrosion products\n\n22.Which of the following corrosion form is autocatalytic in nature?\na) Pitting and crevice corrosion\nb) Crevice corrosion only\nc) Pitting corrosion only\nd) Pitting and intergranular corrosion\n\n23.Which of the following corrosion form is/are autocatalytic in nature?\na) Pitting and crevice corrosion\nb) Crevice corrosion only\nc) Pitting corrosion only\nd) Pitting and intergranular corrosion\n\n24.Which of the following corrosion test is most reliable to know the extent of pitting corrosion?\na) To measure the average depth of pits\nb) To measure the maximum depth of a pit\nc) Weight loss method\nd) Weight gain method\n\n25.Which of the following are the destructive effects of corrosion?\na) Contamination of product\nb) Effect on safety\nc) Reliability\nd) Contamination of product, effect on safety and reliability\n\n26.Which of the following method is adopted for preventing corrosion by acids?\na) Deaeration\nb) Removal by using ion-exchange resin\nc) Neutralisation with lime\nd) Dehumidification\n\n27.Which of the following method is adopted for preventing corrosion by salts?\na) Deaeration\nb) Removal by using ion-exchange resin\nc) Neutralisation with lime\nd) Dehumidification\n\n28.Which of the following method is adopted for preventing corrosion by moisture?\na) Deaeration\nb) Removal by using ion-exchange resin\nc) Neutralisation with lime\nd) Dehumidification\n\n29.The organic or inorganic substances which when added to the environment are able to reduce the rate of corrosion are called ______\na) Inhibitors\nb) Stimulants\nc) Insulators\nd) Stipulator\n\n30.The process by which Ta, Nb and Ti combine with carbon to form respective carbides and prevent corrosion is known as _______\na) Passivation\nb) Neutralisation\nc) Inhibition\nd) Stimulation\n\n———————-\n\nMCQs Part-3\n\n———————\n\n1. Rigid rotator tells us about….\n• Rotational quantum number\n• Vibrational quantum number\n1. The operator ∇ is called _______ operator\n• Hamiltonian (b) Laplacian\n\n(c ) Poisson    (d) vector\n\n1. In a rigid rotator distance between two particles is _______\n\n(a) Variable   (b) Zero\n\n(c) Infinite    (d) constant\n\n1. The balancing of a rigid rotor can be achieved by appropriately placing balancing masses in\n\n(a) Single plane   (b) two planes\n\n(c) Three planes  (d) four planes\n\n1. The energy gap between two adjacent quantum states for the rigid rotor system\n• decrease as the quantum number becomes larger\n• increase as the quantum number becomes larger\n• is a constant\n• neither\n\n1. The main condition for the rigid body is that the distance between various particles of the body does change.\na) True\nb) False\n2. The potential energy of a rigid rotator will be….\n• Zero\n• Constant\n• 1\n• Both a & b\n1. Wave function is a product of two functions” is this statement true?\n• True\n• false\n1. while considering the rotational spectra of diatomic molecule quantum number is written as?\n• J\n• L\n• P\n• I\n1. The solution of the Schrödinger equation for the rigid rotor system resulted in degenerated states for a nonzero angular momentum quantum number . For the energy level, the degeneracy is:\n• 5\n• 2\n• 3\n• 1\n1. The solution of the Schrödinger equation for the rigid rotor system results in degenerated states for a nonzero angular momentum quantum number energy level, the degeneracy is:\n• 5\n• 3\n• 2\n• 1\n1. The laplacian is which of the following operator?\n• Non linear operator\n• Order statistic operator\n• Linear operator\n• None of the mentioned\n1. The laplacian operator cannot be used in which of the following?\n• Two dimensional heat equation\n• Two dimensional wave equation\n• Poison equation\n• Maxwell equation\n1. Which types of coordinates are used in study of rigid rotator?\n• Polar coordinates\n• Cartesian coordinates\n1. Ɵ & ɸ are?\n\n(a) Cartesian coordinates\n\n(b) Polar coordinates\n\n1. When does the moment of inertia of a body come into the picture?\n\n(a) When the motion is rotational\n\n(b) When the motion is linear\n\n(c) When the motion is along a curved path\n\n(d) None of the above" ]

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[ "", null, "# Computing normalisers of highly intransitive permutation groups\n\n•", null, "Mun See Chang (University of St Andrews)\n•", null, "Thursday 30 January 2020, 16:35-17:05\n•", null, "Seminar Room 1, Newton Institute.\n\nGRAW02 - Computational and algorithmic methods\n\nIn general, there is no known polynomial-time algorithm for computing the normaliser $N_{S_n}(H)$ of a given group $H \\leq S_n$. In this talk, we will consider the case when $H$ is a subdirect product of permutation isomorphic non-abelian simple groups. In contrast to the case with abelian simple groups, where only practical improvements have been made, here we show that $N_{S_n}(H)$ can be computed in polynomial time.\n\nThis talk is part of the Isaac Newton Institute Seminar Series series." ]

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[ "# Annotating a plot#\n\nThis example shows how to annotate a plot with an arrow pointing to provided coordinates. We modify the defaults of the arrow, to \"shrink\" it.\n\nFor a complete overview of the annotation capabilities, also see the annotation tutorial.\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n\nfig, ax = plt.subplots()\n\nt = np.arange(0.0, 5.0, 0.01)\ns = np.cos(2*np.pi*t)\nline, = ax.plot(t, s, lw=2)\n\nax.annotate('local max', xy=(2, 1), xytext=(3, 1.5),\narrowprops=dict(facecolor='black', shrink=0.05),\n)\nax.set_ylim(-2, 2)\nplt.show()", null, "References\n\nThe use of the following functions, methods, classes and modules is shown in this example:\n\nGallery generated by Sphinx-Gallery" ]

[ null, "https://matplotlib.org/stable/_images/sphx_glr_annotation_basic_001.png", null ]

[ "Is 16311 a prime number? What are the divisors of 16311?\n\n## Parity of 16 311\n\n16 311 is an odd number, because it is not evenly divisible by 2.\n\nFind out more:\n\n## Is 16 311 a perfect square number?\n\nA number is a perfect square (or a square number) if its square root is an integer; that is to say, it is the product of an integer with itself. Here, the square root of 16 311 is about 127.715.\n\nThus, the square root of 16 311 is not an integer, and therefore 16 311 is not a square number.\n\n## What is the square number of 16 311?\n\nThe square of a number (here 16 311) is the result of the product of this number (16 311) by itself (i.e., 16 311 × 16 311); the square of 16 311 is sometimes called \"raising 16 311 to the power 2\", or \"16 311 squared\".\n\nThe square of 16 311 is 266 048 721 because 16 311 × 16 311 = 16 3112 = 266 048 721.\n\nAs a consequence, 16 311 is the square root of 266 048 721.\n\n## Number of digits of 16 311\n\n16 311 is a number with 5 digits.\n\n## What are the multiples of 16 311?\n\nThe multiples of 16 311 are all integers evenly divisible by 16 311, that is all numbers such that the remainder of the division by 16 311 is zero. There are infinitely many multiples of 16 311. The smallest multiples of 16 311 are:\n\n## Numbers near 16 311\n\n### Nearest numbers from 16 311\n\nFind out whether some integer is a prime number" ]

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[ "Spectro-temporal reconstruction algorithms aim to achieve the following: A time series is converted into a 'series' of 'time series' containing the temporal course of amplitude&phase (or power) at a series of frequencies. Irrespective of the algorithm used, this always boils down to sliding a window through the original time series and estimating the quantities of interest at prespecified timepoints. Equivalently, one can convolve with a bandpass-filter kernel, or a wavelet. In order to get a good estimate of the quantities of interest, the sliding window should be completely filled with data (or the convolution-kernel should completely overlap with data). Either way, if the prespecified time points of interest are close to the boundaries of the original time series, NaNs will appear in the output of the TFR-analysis, because there is insufficient data at these locations. Practically this means that if your data is defined e.g. between -1 and +1 seconds, and using a cfg.t_ftimwin of 0.5 seconds, freqanalysis_mtmconvol will only give non-NaN output only between -0.75 and 0.75 seconds." ]

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[ "Home   Package List   Routine Alphabetical List   Global Alphabetical List   FileMan Files List   FileMan Sub-Files List   Package Component Lists   Package-Namespace Mapping\nSub-Field: 81.062\n\n# Sub-Field: 81.062\n\n## Information\n\nParent File Name Number Package\nCPT(#81) DESCRIPTION (VERSIONED) 81.062 CPT HCPCS Codes\n\n## Details\n\nField # Name Loc Type Details\n.01 VERSION DATE 0;1 DATE\n\n• INPUT TRANSFORM:  `S %DT=\"EX\" D ^%DT S X=Y K:2790101>X X`\n• LAST EDITED:  `MAR 19, 2004`\n• HELP-PROMPT:  `TYPE A DATE NOT EARLIER THAN JAN 01, 1979`\n• DESCRIPTION:\n`This is the date the description was first used.`\n• CROSS-REFERENCE:  `81.062^B`\n`1)= S ^ICPT(DA(1),62,\"B\",\\$E(X,1,30),DA)=\"\"`\n`2)= K ^ICPT(DA(1),62,\"B\",\\$E(X,1,30),DA)`\n`^ICPT(IEN1,62,\"B\",EFF,IEN2) - Where IEN1 is the internal entry number in the CPT-4 Procedure file #81, EFF is the effective date for the description (long text), and IEN2 is the internal entry number in the DESCRIPTION`\n`(VERSIONED) multiple where the versioned procedure description is stored.`\n• CROSS-REFERENCE:  `81^ADS^MUMPS`\n`1)= S:\\$L(\\$P(\\$G(^ICPT(DA(1),0)),\"^\",1)) ^ICPT(\"ADS\",(\\$P(\\$G(^ICPT(DA(1),0)),\"^\",1)_\" \"),+X,DA(1),DA)=\"\"`\n`2)= K:\\$L(\\$P(\\$G(^ICPT(DA(1),0)),\"^\",1)) ^ICPT(\"ADS\",(\\$P(\\$G(^ICPT(DA(1),0)),\"^\",1)_\" \"),+X,DA(1),DA)`\n`^ICPT(\"ADS\",(CODE_\" \"),EFF,IEN1,IEN2) - Where CODE is the CPT-4 Procedure Code, EFF is the effective date for the description (long text), IEN1 is the internal entry number in the CPT-4 Procedure file (#81) and IEN2 is`\n`the internal entry number in the DESCRIPTION (VERSIONED) multiple where the versioned long text is stored.`\n1 DESCRIPTION (VERSIONED) 1;0 Multiple #81.621 81.621\n\n• LAST EDITED:  `MAR 19, 2004`\n• DESCRIPTION:\n`This is the long description (multiple line).`" ]

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[ "Call Now\n\nOPSC Pre Exam Syllabus : Statistics\n\nUpdated on: Apr 18, 2013\nGroup-A:\nProbability & Probability Distributions\nUnit-I\nRandom Experiment, Sample Space, Event, Algebra Of Events, Probability On A Discrete Sample Space, Basic Theorems Of Probability And Simple Examples Based Thereon, Conditional Probability, Independent Events, Bayes’ Theorem And Its Applications, Discrete And Continuous Random Variables And Their Distributions, Expectation, Moments, Moment Generating Function, Joint Distribution Of Two Random Variables, Marginal And Conditional Distributions, Independence Of Random Variables, Covariance. Chebyshev's Inequality, Weak Law Of Large Numbers And Central Limit Theorem For Independently And Identically Distributed Random Variables With Finite Variance And Their Simple Applications.\n\nUnit-II\nDistribution Function Of Random Variables, Bernoulli, Binomial, Geometric, Negative Binomial, Hypergeometric, Poisson, Uniform, Beta, Exponential, Gamma, Cauchy, Normal, Lognormal And Bivariate Normal Distributions, Real-Life Situations Where These Distributions Provide Appropriate Models. Derivation Of Mgf And Characteristic Function Of The Distributions, Computation Of Their Moments And Pearsonian Co-Efficients.\n\nGroup-B:\nStatistical Methods\nUnit-I\nConcept Of A Statistical Population And Sample, Types Of Data, Presentation And Summarization Of Data, Measures Of Central Tendency, Dispersion, Moments, Skewness And Kurtosis, Measures Of Association And Contingency, Correlation, Rank Correlation, Intraclass Correlation, Correlation Ratio, Simple And Multiple Linear Regression, Multiple And Partial Correlations (Involving Three Variables Only)\n\nUnit-II\nCurve-Fitting And Principle Of Least Squares, Concepts Of Parameter And Statistic, Z, C2 , T And F- Statistics And Their Distributions, Properties And Applications, Distributions Of Sample Range And Median (For Continuous Distributions Only).\n\nGroup-C:\nStatistical Inference\nUnit-I\nProperties Of A Good Estimator : Unbiasedness, Consistency, Efficiency, Sufficiency, Completeness; Minimum Variance Unbiased Estimation, Rao-Blackwell Theorem, Cramer- Rao Inequality And Minimum Variance Bound Estimator, Methods Of Estimation : Moments, Maximum Likelihood, Least Squares And Minimum Chi-Square; Properties Of Maximum Likelihood Estimator, Idea Of A Random Interval, Confidence Intervals For The Parameters Of Standard Distributions, Shortest Confidence Intervals, Large-Sample Confidence Intervals.\n\nUnit-II\nSimple And Composite Hypotheses, Two Kinds Of Errors, Level Of Significance, Size And Power Of A Test, Desirable Properties Of A Good Test, Most Powerful Test, Neyman-Pearson Lemma And Its Application For Testing Simple Hypothesis, Uniformly Most Powerful Test, Likelihood Ratio Test And Its Properties And Applications. Chi-Square Test, Sign Test, Wald- Wolfowitz Run Test, Run Test For Randomness, Median Test, Wilcoxon Test And Wilcoxon- Mann-Whitney U-Test. Wald's Sequential Probability Ratio Test, Oc And Asn Functions, Application To Binomial And Normal Distributions.\n\nGroup-D :\nSampling Theory And Design Of Experiments\n\nUnit-I\nComplete Enumeration Vs Sampling, Need For Sampling, Basic Concepts Of Sampling And Sampling Design, Large-Scale Sample Surveys, Sampling And Non-Sampling Errors, Simple Random Sampling, Estimation Of Sample Size, Stratified Random Sampling, Systematic Sampling, Cluster Sampling, Ratio, Product And Regression Methods Of Estimation Under Simple And Stratified Random Sampling, Double Sampling For Ratio And Regression Methods Of Estimation, Two-Stage Sampling With Equal Size First-Stage Units.\n\nUnit-II\nAnalysis Of Variance With Equal Number Of Observations Per Cell In One, Two And Threeway Classifications, Analysis Of Covariance In One And Two-Way Classifications, Basic Principles Of Experimental Designs, Completely Randomized Design, Randomized Block Design, Latin Square Design, Missing Plot Technique, 2n Factorial Experiment, Total And Partial Confounding, 32 Factorial Experiments, Split-Plot Design And Balanced Incomplete Block Design.\nEach Group Should Have Equal Weight\n\nFor detail about more National and State Civil Services Exams" ]

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[ "I'm want to use map() of purrr package to apply filter() function to the data stored in a nested data frame.\n\nBelow is my code, where I would filter and then nest it.\n\nI have achieved my desired result by performing nest() twice i.e. once on all the complete data and the second time on filtered data.\n\n```library(tidyverse)\n\ndf <- tibble(\nx = sample(x = rep(c('a','b'),5), size = 10),\ny = sample(c(1:10)),\nz = sample(c(91:100))\n)\n#First time nested\ndf_fully_nested_dataframe <- df %>%\ngroup_by(x) %>%\nnest(.key = 'full')\n#Second time nested\ndf_filtered_nested_dataframe <- df %>%\nfilter(z >= 95) %>%\ngroup_by(x) %>%\nnest(.key = 'filtered')\n\n#The desired outcome is a single data frame with 2 nested list-columns: one full data and other with filtered data.\n## Now how to achieve this without dividing into 2 dataframes\ndf_nested <- df_fully_nested_dataframe %>%\nleft_join(df_filtered_nested_dataframe, by = 'x')```\n\nThe objects looks as follows:\n\n```> df\n# A tibble: 10 x 3\nx y z\n<chr> <int> <int>\n1 b 8 93\n2 a 9 94\n3 b 10 99\n4 a 5 97\n5 b 2 100\n6 b 3 95\n7 a 7 96\n8 b 6 92\n9 a 4 91\n10 a 1 98\n\n> df_fully_nested_dataframe\n# A tibble: 2 x 2\nx full\n<chr> <list>\n1 b <tibble [5 x 2]>\n2 a <tibble [5 x 2]>\n\n> df_filtered_nested_dataframe\n# A tibble: 2 x 2\nx filtered\n<chr> <list>\n1 b <tibble [3 x 2]>\n2 a <tibble [3 x 2]>\n\n> df_nested\n# A tibble: 2 x 3\nx full filtered\n<chr> <list> <list>\n1 b <tibble [5 x 2]> <tibble [4 x 2]>\n2 a <tibble [5 x 2]> <tibble [4 x 2]>```", null, "Apr 6, 2018 3,232 views\n\n## 1 answer to this question.\n\nYou can use map() call as follows:\n\nmap(full, ~ filter(., z >= 95))\n\nwhere . stands for individual nested tibble, to which you can apply the filter directly\n\n```df_nested_again <- df_fully_nested_dataframe %>% mutate(filtered = map(full, ~ filter(., z >= 95)))\n\nidentical(df_nested, df_nested_again)\n# TRUE\n```", null, "answered Apr 6, 2018 by\n• 6,380 points\n\n## How to filter a data frame with dplyr and tidy evaluation in R?\n\nRequires the use of map_df to run each model, ...READ MORE\n\n## How to use plyr and dplyr functions inside one source file?\n\nHi, Lakshmi, Use dplyr::function( ) or plyr::function( ). Since ...READ MORE\n\n## How to join a list of data frames using map()\n\nYou can use reduce set.seed(24) r1 <- map(c(5, 10, ...READ MORE\n\n+1 vote\n\n## How to convert a list of vectors with various length into a Data.Frame?\n\nWe can easily use this command as.data.frame(lapply(d1, \"length< ...READ MORE\n\n## How to remove NA values with dplyr::filter()\n\nTry this: df %>% filter(!is.na(col1)) READ MORE\n\n## How to use a function to repeat a set of procedures on specific set of columns in a data frame?\n\nYou can parse the strings to symbols. ...READ MORE\n\n## How can I use parallel so that it preserves the list of data frames\n\nYou can use pmap as follows: nc <- ...READ MORE" ]

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[ "Cody\n\n# Problem 115. Distance walked 1D\n\nSolution 38161\n\nSubmitted on 12 Feb 2012 by Roy Fahn\nThis solution is locked. To view this solution, you need to provide a solution of the same size or smaller.\n\n### Test Suite\n\nTest Status Code Input and Output\n1   Pass\n%% x = [7 10 6 4]; y_correct = 9; assert(isequal(your_fcn_name(x),y_correct))\n\n2   Pass\n%% x = [1 2 3 2]; y_correct = 3; assert(isequal(your_fcn_name(x),y_correct))\n\n3   Pass\n%% x = -5:5:30; y_correct = 35; assert(isequal(your_fcn_name(x),y_correct))\n\n4   Pass\n%% x = [45 -16 34 19 120]; y_correct = 227; assert(isequal(your_fcn_name(x),y_correct))\n\n### Community Treasure Hunt\n\nFind the treasures in MATLAB Central and discover how the community can help you!\n\nStart Hunting!" ]

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[ "## The Mid Ordinate Rule\n\nAlso called the midpoint rule – the mid ordinate rule is another method to numerically estimate integrals. It states:\n\nIf an area of integration is divided into n strips, the area of the strip between", null, "and", null, "is given by", null, "so that the width of the strip is multiplied by the", null, "– value at the midpoint.\n\nWe do this for all n strips obtaining", null, "If the strips are all of the same width", null, "where", null, "and", null, "are the limits of integration, then we can write,", null, "Example: Using the mid ordinate rule with 5 strips, estimate the vale of the integral", null, "to three decimal places. Evaluate the integral and compare the approximation given by the ordinate rule with the true value.", null, "Complete the table of values for", null, "Since we must estimate the value of the integral to three decimal places, we calculate values to four decimal places. The final answer is quoted to three decimal places.\n\n I 0 1 2 3 4 5", null, "0 0.2 0.4 0.6 0.8 1", null, "0.1 0.3 0.5 0.7 0.9", null, "1.1052 1.3499 1.6487 2.0138 2.4596", null, "", null, "The % error is", null, "", null, "" ]

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[ "ru / en", null, "# Bauman Moscow State Technical University\n\n### List of articles written by the authors of the organization\n\n•", null, "Description\nFor the first time we have developed a general additive-multiplicative model of the risk estimation (to estimate the probabilities of risk events). In the two-level system in the lower level the risk estimates are combined additively, on the top – in a multiplicative way. Additive-multiplicative model was used for risk estimation for (1) implementation of innovative projects at universities (with external partners), (2) the production of new innovative products, (3) the projects for creation of rocket and space equipmen\n•", null, "Description\nIn various applications it is necessary to analyze some expert orderings, ie clustered rankings of examination objects. These areas include technical studies, ecology, management, economics, sociology, forecasting, etc. The objects may make samples of the products, technologies, mathematical models, projects, job applicants and others. We obtain clustered rankings which can be both with the help of experts and objective way, for example, by comparing the mathematical models with experimental data using a particular quality criterion. The method described in this article was developed in connection with the problems of chemical safety and environmental security of the biosphere. We propose a new method for constructing a clustered ranking which can be average (in the sense, discussed in this work) for all clustered rankings under our consideration. Then the contradictions between the individual initial rankings are contained within clusters average (coordinated) ranking. As a result, ordered clusters reflects the general opinion of the experts, more precisely, the total that is contained simultaneously in all the original rankings. Newly built clustered ranking is often called the matching (coordinated) ranking with respect to the original clustered rankings. The clusters are enclosed objects about which some of the initial rankings are contradictory. For these objects is necessary to conduct the new studies. These studies can be formal mathematics (calculation of the Kemeny median, orderings by means of the averages and medians of ranks, etc.) or these studies require involvement of new information from the relevant application area, it may be necessary conduct additional scientific research. In this article we introduce the necessary concepts and we formulate the new algorithm of construct the coordinated ranking for some cluster rankings in general terms, and its properties are discussed\n•", null, "Description\nWe consider an approach to the transition from continuous to discrete scale which was defined by means of step of quantization (i.e. interval of grouping). Applied purpose is selecting the number of gradations in sociological questionnaires. In accordance with the methodology of the general stability theory, we offer to choose a step so that the errors, generated by the quantization, were of the same order as the errors inherent in the answers of respondents. At a finite length of interval of the measured value change of the scale this step of quantization uniquely determines the number of gradations. It turns out that for many issues gated it is enough to point 3 - 6 answers gradations (hints). On the basis of the probabilistic model we have proved three theorems of quantization. They are allowed to develop recommendations on the choice of the number of gradations in sociological questionnaires. The idea of \"quantization\" has applications not only in sociology. We have noted, that it can be used not only to select the number of gradations. So, there are two very interesting applications of the idea of \"quantization\" in inventory management theory - in the two-level model and in the classical Wilson model taking into account deviations from it (shows that \"quantization\" can use as a way to improve stability). For the two-level inventory management model we proved three theorems. We have abandoned the assumption of Poisson demand, which is rarely carried out in practice, and we give generally fairly simple formulas for finding the optimal values of the control parameters, simultaneously correcting the mistakes of predecessors. Once again we see the interpenetration of statistical methods that have arisen to analyze data from a variety of subject areas, in this case, from sociology and logistics. We have another proof that the statistical methods - single scientificpractical area that is inappropriate to share by areas of applications\n•", null, "Description\nNonparametric estimates of the probability distribution density in spaces of arbitrary nature are one of the main tools of non-numerical statistics. Their particular cases are considered - kernel density estimates in spaces of arbitrary nature, histogram estimations and Fix-Hodges-type estimates. The purpose of this article is the completion of a series of papers devoted to the mathematical study of the asymptotic properties of various types of nonparametric estimates of the probability distribution density in spaces of general nature. Thus, a mathematical foundation is applied to the application of such estimates in non-numerical statistics. We begin by considering the mean square error of the kernel density estimate and, in order to maximize the order of its decrease, the choice of the kernel function and the sequence of the blur indicators. The basic concepts are the circular distribution function and the circular density. The order of convergence in the general case is the same as in estimating the density of a numerical random variable, but the main conditions are imposed not on the density of a random variable, but on the circular density. Next, we consider other types of nonparametric density estimates - histogram estimates and Fix-Hodges-type estimates. Then we study nonparametric regression estimates and their application to solve discriminant analysis problems in a general nature space\n•", null, "Description\nStatistical control is a sampling control based on the probability theory and mathematical statistics. The article presents the development of the methods of statistical control in our country. It discussed the basics of the theory of statistical control – the plans of statistical control and their operational characteristics, the risks of the supplier and the consumer, the acceptance level of defectiveness and the rejection level of defectiveness. We have obtained the asymptotic method of synthesis of control plans based on the limit average output level of defectiveness. We have also developed the asymptotic theory of single sampling plans and formulated some unsolved mathematical problems of the theory of statistical control\n•", null, "Description\nIntuitively everyone understands that noise is a signal in which there no information is, or which in practice fails to reveal the information. More precisely, it is clear that a certain sequence of elements (the number) the more is the noise, the less information is contained in the values of some elements on the values of others. It is even stranger, that noone has suggested the way, but even the idea of measuring the amount of information in some fragments of signal of other fragments and its use as a criterion for assessing the degree of closeness of the signal to the noise. The authors propose the asymptotic information criterion of the quality of noise, and the method, technology and methodology of its application in practice. As a method of application of the asymptotic information criterion of noise quality, we offer, in practice, the automated systemcognitive analysis (ASC-analysis), and as a technology and software tools of ASC-analysis we offer the universal cognitive analytical system called \"Eidos\". As a method, we propose a technique of creating applications in the system, as well as their use for solving problems of identification, prediction, decision making and research the subject area by examining its model. We present an illustrative numerical example showing the ideas presented and demonstrating the efficiency of the proposed asymptotic information criterion of the quality of the noise, and the method, technology and methodology of its application in practice\n•", null, "Description\nThe mathematical theory of classification contains a large number of approaches, models, methods, algorithms. This theory is very diverse. We distinguish three basic results in it - the best method of diagnosis (discriminant analysis), an adequate indicator of the quality of discriminant analysis algorithm, the statement about stopping after a finite number of steps iterative algorithms of cluster analysis. Namely, on the basis of Neyman - Pearson Lemma we have shown that the optimal method of diagnosis exists and can be expressed through probability densities corresponding to the classes. If the densities are unknown, one should use non-parametric estimators of training samples. Often, we use the quality indicator of diagnostic algorithm as \"the probability (or share) the correct classification (diagnosis)\" - the more the figure is the better algorithm is. It is shown that widespread use of this indicator is unreasonable, and we have offered the other - \"predictive power\", obtained by the conversion in the model of linear discriminant analysis. A stop after a finite number of steps of iterative algorithms of cluster analysis method is demonstrated by the example of k-means. In our opinion, these results are fundamental to the theory of classification and every specialist should be familiar with them for developing and applying the theory of classification\n•", null, "Description\nFrom a modern point of view we have discussed Kolmogorov’s researches in the axiomatic approach to probability theory, the goodness-of-fit test of the empirical distribution with theoretical, properties of the median estimates as a distribution center, the effect of \"swelling\" of the correlation coefficient, the theory of averages, the statistical theory of crystallization of metals, the least squares method, the properties of sums of a random number of random variables, statistical control, unbiased estimates, axiomatic conclusion of logarithmic normal distribution in crushing, the methods of detecting differences in the weather-type experiments\n•", null, "Description\nWe analyze the probabilistic-statistical methods in the researches of Boris Vladimirovich Gnedenko – the academician of Ukrainian Academy of Science, which are very important for the XXI century. We have also discussed the limit theorems of probability theory, mathematical statistics, reliability theory, statistical methods of quality control and queuing theory. We give some information about the main stages of scientific career of B.V. Gnedenko, his views on the history of mathematics and teaching\n•", null, "Description\nThe correlation and determination coefficients are widely used in statistical data analysis. According to measurement theory, Pearson's linear paired correlation coefficient is applicable to variables measured on an interval scale. It cannot be used in the analysis of ordinal data. The nonparametric Spearman and Kendall rank coefficients estimate the relationship of ordinal variables. The critical value when testing the significance of the difference of the correlation coefficient from 0 depends on the sample size. Therefore, using the Chaddock Scale is incorrect. When using a passive experiment, the correlation coefficients are reasonably used for prediction, but not for control. To obtain probabilistic-statistical models intended for control, an active experiment is required. The effect of outliers on the Pearson correlation coefficient is very large. With an increase in the number of analyzed sets of predictors, the maximum of the corresponding correlation coefficients — indicators of approximation quality noticeably increases (the effect of “inflation” of the correlation coefficient). Four main regression analysis models are considered. Models of the least squares method with a determinate independent variable are distinguished. The distribution of deviations is arbitrary, however, to obtain the limit distributions of parameter estimates and regression dependences, we assume that the conditions of the central limit theorem are satisfied. The second type of model is based on a sample of random vectors. The dependence is nonparametric, the distribution of the two-dimensional vector is arbitrary. The estimation of the variance of an independent variable can be discussed only in the model based on a sample of random vectors, as well as the determination coefficient as a quality criterion for the model. Time series smoothing is discussed. Methods of restoring dependencies in spaces of a general nature are considered. It is shown that the limiting distribution of the natural estimate of the dimensionality of the model is geometric, and the construction of an informative subset of features encounters the effect of \"inflation coefficient correlation\". Various approaches to the regression analysis of interval data are discussed. Analysis of the variety of regression analysis models leads to the conclusion that there is no single “standard model”" ]

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[ "Total Visits: 3224\n\nMathematical Questions and Solutions in Continuation of the Mathematical Columns of the Educational Times., Volume 69... book\n\nMathematical Questions and Solutions in Continuation of the Mathematical Columns of the Educational Times., Volume 69... book\n\nMathematical Questions and Solutions in Continuation of the Mathematical Columns of the Educational Times., Volume 69....cAnonymous", null, "---------------------------------------------------------------\nAuthor: Anonymous\nNumber of Pages: 138 pages\nPublished Date: 30 Jan 2012\nPublisher: Nabu Press\nPublication Country: United States\nLanguage: none\nISBN: 9781273856402\nDownload Link: Mathematical Questions and Solutions in Continuation of the Mathematical Columns of the Educational Times., Volume 69...\n---------------------------------------------------------------\n\nnone\n\nRead online Mathematical Questions and Solutions in Continuation of the Mathematical Columns of the Educational Times., Volume 69... Buy and read online Mathematical Questions and Solutions in Continuation of the Mathematical Columns of the Educational Times., Volume 69... Download and read Mathematical Questions and Solutions in Continuation of the Mathematical Columns of the Educational Times., Volume 69... for pc, mac, kindle, readers Download to iPad/iPhone/iOS, B&N nook Mathematical Questions and Solutions in Continuation of the Mathematical Columns of the Educational Times., Volume 69...\n\nOther files:\n\nwwe raw serial killers\nglobal mapper 14.2 serial number\nRole of Salivary Biomarkers in Periodontal Diseases epub" ]

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[ "Hostname: page-component-594f858ff7-x2rdm Total loading time: 0 Render date: 2023-06-08T23:07:21.037Z Has data issue: false Feature Flags: { \"corePageComponentGetUserInfoFromSharedSession\": true, \"coreDisableEcommerce\": false, \"corePageComponentUseShareaholicInsteadOfAddThis\": true, \"coreDisableSocialShare\": false, \"useRatesEcommerce\": true } hasContentIssue false\n\n## Abstract", null, "The three-dimensional instability of the flow past a 5 : 1 rectangular cylinder is investigated via Floquet analysis and direct numerical simulations. A quasi-subharmonic (QS) unstable mode is detected, marking an important difference with the flow past bodies with lower aspect ratio and/or with smooth leading edge. The QS mode becomes unstable at Reynolds number (based on the cylinder thickness and free-stream velocity)", null, "$Re \\approx 480$; its spanwise wavelength is approximately three times the cylinder thickness. The structural sensitivity locates the wavemaker region over the longitudinal sides of the cylinder, indicating that the instability is triggered by the mutual inviscid interaction of vortices generated by the leading edge shear layer. ## JFM classification Type JFM Papers Information Journal of Fluid Mechanics , 10 November 2022 , A20 Copyright © The Author(s), 2022. Published by Cambridge University Press ## Access options Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.) ## References #### REFERENCES Abdalla, I.E. & Yang, Z. 2004 Numerical study of the instability mechanism in transitional separating–reattaching flow. Intl J. Heat Fluid Flow 25 (4), 593605.CrossRefGoogle Scholar Barkley, D. & Henderson, R.D. 1996 Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215241.CrossRefGoogle Scholar Bernal, L.P., Breidenthal, R.E., Brown, G.L., Konrad, J.H. & Roshko, A. 1979 On the development of three dimensional small scales in turbulent mixing layers. In 2nd Symposium on Turbulent Shear Flows (ed. L.J.S. Bradbury), pp. 8.1–8.6. Springer-Verlag.Google Scholar Blackburn, H.M. & Lopez, J.M. 2003 On three-dimensional quasiperiodic Floquet instabilities of two-dimensional bluff body wakes. Phys. Fluids 15 (8), L57L60.CrossRefGoogle Scholar Blackburn, H.M., Marques, F. & Lopez, J.M. 2005 Symmetry breaking of two-dimensional time-periodic wakes. J. Fluid Mech. 522, 395411.CrossRefGoogle Scholar Blackburn, H.M. & Sheard, G.J. 2010 On quasiperiodic and subharmonic Floquet wake instabilities. Phys. Fluids 22 (3), 031701.CrossRefGoogle Scholar Boghosian, M.E. & Cassel, K.W. 2016 On the origins of vortex shedding in two-dimensional incompressible flows. Theor. Comput. Fluid Dyn. 30 (6), 511527.CrossRefGoogle ScholarPubMed Brezzi, F. 1974 On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. ESAIM. Math. Model Numer. Anal. 8 (R2), 129151.Google Scholar Browand, F.K. & Troutt, T.R. 1980 A note on spanwise structure in the two-dimensional mixing layer. J. Fluid Mech. 97 (4), 771781.CrossRefGoogle Scholar Chandrsuda, C., Mehta, R.D., Weir, A.D. & Bradshaw, P. 1978 Effect of free-stream turbulence on large structure in turbulent mixing layers. J. Fluid Mech. 85 (4), 693704.CrossRefGoogle Scholar Chaurasia, H.K. & Thompson, M.C. 2011 Three-dimensional instabilities in the boundary-layer flow over a long rectangular plate. J. Fluid Mech. 681, 411433.CrossRefGoogle Scholar Chiarini, A., Gatti, D., Cimarelli, A. & Quadrio, M. 2022 a Structure of turbulence in the flow around a rectangular cylinder. J. Fluid Mech. 946, A35.CrossRefGoogle Scholar Chiarini, A. & Quadrio, M. 2021 The turbulent flow over the BARC rectangular cylinder: a DNS study. Flow Turbul. Combust. 107, 875899.CrossRefGoogle Scholar Chiarini, A., Quadrio, M. & Auteri, F. 2021 Linear stability of the steady flow past rectangular cylinders. J. Fluid Mech. 929, A36.CrossRefGoogle Scholar Chiarini, A., Quadrio, M. & Auteri, F. 2022 b On the frequency selection mechanism of the low-", null, "$Re\\$ flow around rectangular cylinders. J. Fluid Mech. 933, A44.CrossRefGoogle Scholar\nChoi, C.-B. & Yang, K.-S. 2014 Three-dimensional instability in flow past a rectangular cylinder ranging from a normal flat plate to a square cylinder. Phys. Fluids 26 (6), 061702.CrossRefGoogle Scholar\nCimarelli, A., Leonforte, A. & Angeli, D. 2018 On the structure of the self-sustaining cycle in separating and reattaching flows. J. Fluid Mech. 857, 907936.CrossRefGoogle Scholar\nCitro, V., Luchini, P., Giannetti, F. & Auteri, F. 2017 Efficient stabilization and acceleration of numerical simulation of fluid flows by residual recombination. J. Comput. Phys. 344, 234246.CrossRefGoogle Scholar\nGiannetti, F., Camarri, S. & Luchini, P. 2010 Structural sensitivity of the secondary instability in the wake of a circular cylinder. J. Fluid Mech. 651, 319337.CrossRefGoogle Scholar\nGiannetti, F. & Luchini, P. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167197.CrossRefGoogle Scholar\nHecht, F. 2012 New development in FreeFem++. J. Numer. Maths 20 (3–4), 251266.Google Scholar\nHourigan, K., Thompson, M.C. & Tan, B.T. 2001 Self-sustained oscillations in flows around long blunt plates. J. 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[ "# zbMATH — the first resource for mathematics\n\nComments on the core of the direct method for proving Hyers-Ulam stability of functional equations. (English) Zbl 1052.39031\nThe author formulates, in a general form, the method of proving the Hyers-Ulam stability for functional equations in several variables. This method appeared in his paper [Stochastica 4, No. 1, 23–30 (1980; Zbl 0442.39005)] and has been actually repeated in numerous papers of various authors.\nThe main result reads as follows: Assume that $$S$$ is a set, $$(X,d)$$ a complete metric space and $$G:S\\to S$$, $$H:X\\to X$$ given functions. Let $$f:S\\to X$$ satisfy the inequality $d(H(f(G(x))),f(x))\\leq\\delta(x),\\;\\;\\;x\\in S$ for some function $$\\delta:S\\to\\mathbb R_+$$. If $$H$$ is continuous and satisfies: $d(H(u),H(v))\\leq\\phi(d(u,v)),\\;\\;\\;u,v\\in X,$ for a non-decreasing subadditive function $$\\phi:\\mathbb R_+\\to\\mathbb R_+$$, and the series $$\\sum_{i=0}^{\\infty}\\phi^i(\\delta(G^i(x)))$$ is convergent for every $$x\\in S$$, then there exists a unique function $$F:S\\to X$$ – a solution of the functional equation $H(F(G(x)))=F(x),\\;\\;\\;x\\in S$ and satisfying $d(F(x),f(x))\\leq\\sum_{i=0}^{\\infty}\\phi^i(\\delta(G^i(x))).$\nMoreover, an analogous result, for a mapping $$f$$ satisfying the inequality $\\left| \\frac{H(f(G(x)))}{f(x)}-1\\right| \\leq\\delta(x)$ is considered.\n\n##### MSC:\n 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges\nFull Text:\n##### References:\n Borelli, C., On hyers – ulam stability for a class of functional equations, Aequationes math., 54, 74-86, (1997) · Zbl 0879.39009 Borelli, C.; Forti, G.L., On a general hyers – ulam stability result, Internat. J. math. math. sci., 18, 229-236, (1995) · Zbl 0826.39009 Forti, G.L., An existence and stability theorem for a class of functional equations, Stochastica, 4, 23-30, (1980) · Zbl 0442.39005 Forti, G.L., Hyers – ulam stability of functional equations in several variables, Aequationes math., 50, 143-190, (1995) · Zbl 0836.39007 Ger, R., Superstability is not natural, Rocznik nauk.-dydakt. prace mat., 13, 109-123, (1993) · Zbl 0964.39503 Hyers, D.H., On the stability of the linear functional equation, Proc. nat. acad. sci., 27, 222-224, (1941) · Zbl 0061.26403 Hyers, D.H.; Isac, G.; Rassias, Th.M., Stability of functional equations in several variables, Progress in nonlinear differential equations and their applications, vol. 34, (1998), Birkhäuser · Zbl 0894.39012 Rassias, Th.M., On the stability of linear mappings in Banach spaces, Proc. amer. math. soc., 72, 297-300, (1978) · Zbl 0398.47040 Trif, T., On the stability of a general gamma-type functional equation, Publ. math. debrecen, 60, 47-61, (2002) · Zbl 1004.39023\nThis reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching." ]

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[ "# Sorting Array of Objects Using PHP\n\nConsider you have an array or list of objects and wanted to sort it based on one attribute of objects. If the array consists of integer, it is very easy to sort it ascendingly using `sort()` function, otherwise if the array consists of objects, you have to create custom sorting function which is passed to the PHP `usort()` function.\n\nHere is an example on how to manually sort array of objects in PHP:\n\n```<?php\nclass Individual {\nprivate \\$name = null;\npublic \\$length = 0;\n\nfunction __construct(\\$name, \\$length){\n\\$this->name = \\$name;\n\\$this->length = \\$length;\n}\n}\n\nfunction psort(\\$a,\\$b){\nif(\\$a->length == \\$b->length) return 0;\nreturn (\\$a->length > \\$b->length ? 1 : -1);\n}\n\n\\$population = array();\n\\$population[] = new Individual('Ben',100);\n\\$population[] = new Individual('Deerp',50);\n\\$population[] = new Individual('Sheep',120);\n\\$population[] = new Individual('Garet',20);\n\nprint_r(\\$population);\n\nusort(\\$population, 'psort');\n\nprint_r(\\$population);```\n\nThe `psort()` is the custom sorting function to sort the array of objects based on object’s length attribute. Below is the output of the above code:\n\n```Array\n(\n => Individual Object\n(\n[name:Individual:private] => Ben\n[length] => 100\n)\n => Individual Object\n(\n[name:Individual:private] => Deerp\n[length] => 50\n)\n => Individual Object\n(\n[name:Individual:private] => Sheep\n[length] => 120\n)\n => Individual Object\n(\n[name:Individual:private] => Garet\n[length] => 20\n)\n)\nArray\n(\n => Individual Object\n(\n[name:Individual:private] => Garet\n[length] => 20\n)\n => Individual Object\n(\n[name:Individual:private] => Deerp\n[length] => 50\n)\n => Individual Object\n(\n[name:Individual:private] => Ben\n[length] => 100\n)\n => Individual Object\n(\n[name:Individual:private] => Sheep\n[length] => 120\n)\n)```\n\nThe first Array is the array before sorting and the second one after sorting has been done.\n\n## 3 Replies to “Sorting Array of Objects Using PHP”\n\n1. Download Old Music BY : Hojat Ashrafzade | Mahdokht With Text And 2 Quality 320 And 128 On Music-fa\n\n2. Download New Music BY : Alireza Barzin | Faramooshi With Text And 2 Quality 320 And 128 On Music-fa" ]

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[ "# parascends in Scrabble®\n\nThe word parascends is playable in Scrabble®, no blanks required. Because it is longer than 7 letters, you would have to play off an existing word or do it in several moves.\n\nPARASCENDS\n(144)\nPARASCENDS\n(144)\n\n## Seven Letter Word Alert: (24 words)\n\nancress, arcades, ascends, canapes, canards, dancers, escarps, panders, parades, parsecs, pranced, prances, resands, rescans, resnaps, respans, sanders, sarapes, scarped, scraped, scrapes, spacers, spaders, spreads\n\nPARASCENDS\n(144)\nPARASCENDS\n(144)\nPARASCENDS\n(96)\nPARASCENDS\n(96)\nPARASCENDS\n(72)\nPARASCENDS\n(64)\nPARASCENDS\n(60)\nPARASCENDS\n(60)\nPARASCENDS\n(60)\nPARASCENDS\n(60)\nPARASCENDS\n(60)\nPARASCENDS\n(51)\nPARASCENDS\n(48)\nPARASCENDS\n(48)\nPARASCENDS\n(46)\nPARASCENDS\n(46)\nPARASCENDS\n(46)\nPARASCENDS\n(42)\nPARASCENDS\n(40)\nPARASCENDS\n(38)\nPARASCENDS\n(38)\nPARASCENDS\n(38)\nPARASCENDS\n(36)\nPARASCENDS\n(34)\nPARASCENDS\n(34)\nPARASCENDS\n(34)\nPARASCENDS\n(34)\nPARASCENDS\n(34)\nPARASCENDS\n(32)\nPARASCENDS\n(32)\nPARASCENDS\n(32)\nPARASCENDS\n(32)\nPARASCENDS\n(30)\nPARASCENDS\n(30)\nPARASCENDS\n(27)\nPARASCENDS\n(25)\nPARASCENDS\n(20)\nPARASCENDS\n(20)\nPARASCENDS\n(20)\nPARASCENDS\n(19)\nPARASCENDS\n(19)\nPARASCENDS\n(19)\nPARASCENDS\n(19)\nPARASCENDS\n(18)\n\nPARASCENDS\n(144)\nPARASCENDS\n(144)\nSCRAPED\n(98 = 48 + 50)\nPRANCED\n(98 = 48 + 50)\nSCARPED\n(98 = 48 + 50)\nPARASCENDS\n(96)\nPARASCENDS\n(96)\nPRANCED\n(95 = 45 + 50)\nSCARPED\n(95 = 45 + 50)\nSCARPED\n(95 = 45 + 50)\nPRANCED\n(95 = 45 + 50)\nSCRAPED\n(95 = 45 + 50)\nSCRAPED\n(95 = 45 + 50)\nCANAPES\n(94 = 44 + 50)\nSPACERS\n(94 = 44 + 50)\nESCARPS\n(94 = 44 + 50)\nPARSECS\n(94 = 44 + 50)\nPRANCES\n(94 = 44 + 50)\nSCRAPES\n(94 = 44 + 50)\nSCRAPES\n(92 = 42 + 50)\nPARSECS\n(92 = 42 + 50)\nPARSECS\n(92 = 42 + 50)\nESCARPS\n(92 = 42 + 50)\nSCRAPES\n(92 = 42 + 50)\nSCRAPED\n(92 = 42 + 50)\nSCARPED\n(92 = 42 + 50)\nPRANCED\n(92 = 42 + 50)\nESCARPS\n(92 = 42 + 50)\nPRANCES\n(92 = 42 + 50)\nPRANCES\n(92 = 42 + 50)\nSPACERS\n(92 = 42 + 50)\nCANAPES\n(92 = 42 + 50)\nSPACERS\n(92 = 42 + 50)\nCANAPES\n(92 = 42 + 50)\nSPACERS\n(92 = 42 + 50)\nASCENDS\n(90 = 40 + 50)\n(90 = 40 + 50)\nDANCERS\n(90 = 40 + 50)\nPANDERS\n(90 = 40 + 50)\n(90 = 40 + 50)\nCANARDS\n(90 = 40 + 50)\n(90 = 40 + 50)\n(90 = 40 + 50)\nPRANCED\n(89 = 39 + 50)\nCANARDS\n(89 = 39 + 50)\nPRANCED\n(89 = 39 + 50)\nSCRAPED\n(89 = 39 + 50)\nDANCERS\n(89 = 39 + 50)\n(89 = 39 + 50)\nSCRAPED\n(89 = 39 + 50)\nSCRAPED\n(89 = 39 + 50)\nPRANCED\n(89 = 39 + 50)\nPRANCED\n(89 = 39 + 50)\nPRANCED\n(89 = 39 + 50)\nSCRAPED\n(89 = 39 + 50)\nSCRAPED\n(89 = 39 + 50)\n(89 = 39 + 50)\nSCARPED\n(89 = 39 + 50)\nSCARPED\n(89 = 39 + 50)\nSCARPED\n(89 = 39 + 50)\nPANDERS\n(89 = 39 + 50)\nASCENDS\n(89 = 39 + 50)\nSCARPED\n(89 = 39 + 50)\nDANCERS\n(89 = 39 + 50)\nSCARPED\n(89 = 39 + 50)\n(89 = 39 + 50)\n(89 = 39 + 50)\nPRANCES\n(86 = 36 + 50)\n(86 = 36 + 50)\nSARAPES\n(86 = 36 + 50)\nESCARPS\n(86 = 36 + 50)\nSPACERS\n(86 = 36 + 50)\nCANAPES\n(86 = 36 + 50)\nPANDERS\n(86 = 36 + 50)\nSARAPES\n(86 = 36 + 50)\nRESCANS\n(86 = 36 + 50)\nRESCANS\n(86 = 36 + 50)\nCANAPES\n(86 = 36 + 50)\nPRANCES\n(86 = 36 + 50)\nCANAPES\n(86 = 36 + 50)\nSPACERS\n(86 = 36 + 50)\nPARSECS\n(86 = 36 + 50)\nPRANCES\n(86 = 36 + 50)\nPANDERS\n(86 = 36 + 50)\nDANCERS\n(86 = 36 + 50)\n(86 = 36 + 50)\n(86 = 36 + 50)\nRESPANS\n(86 = 36 + 50)\nPRANCES\n(86 = 36 + 50)\nRESPANS\n(86 = 36 + 50)\nCANAPES\n(86 = 36 + 50)\nRESNAPS\n(86 = 36 + 50)\nRESPANS\n(86 = 36 + 50)\nSPACERS\n(86 = 36 + 50)\nRESNAPS\n(86 = 36 + 50)\nPRANCES\n(86 = 36 + 50)\nSPACERS\n(86 = 36 + 50)\nSPACERS\n(86 = 36 + 50)\nCANAPES\n(86 = 36 + 50)\nCANAPES\n(86 = 36 + 50)\nESCARPS\n(86 = 36 + 50)\nSCARPED\n(86 = 36 + 50)\nSCRAPED\n(86 = 36 + 50)\nANCRESS\n(86 = 36 + 50)\nPARSECS\n(86 = 36 + 50)\nRESCANS\n(86 = 36 + 50)\nSCRAPED\n(86 = 36 + 50)\nESCARPS\n(86 = 36 + 50)\nPARSECS\n(86 = 36 + 50)\nESCARPS\n(86 = 36 + 50)\nPARSECS\n(86 = 36 + 50)\nPRANCED\n(86 = 36 + 50)\n(86 = 36 + 50)\nASCENDS\n(86 = 36 + 50)\nSCRAPES\n(86 = 36 + 50)\nESCARPS\n(86 = 36 + 50)\nSCRAPES\n(86 = 36 + 50)\nSCRAPES\n(86 = 36 + 50)\nPARSECS\n(86 = 36 + 50)\nANCRESS\n(86 = 36 + 50)\nPRANCES\n(86 = 36 + 50)\nESCARPS\n(86 = 36 + 50)\nPARSECS\n(86 = 36 + 50)\nSCRAPES\n(86 = 36 + 50)\nSCARPED\n(86 = 36 + 50)\n(86 = 36 + 50)\nSCRAPED\n(86 = 36 + 50)\nSCRAPES\n(86 = 36 + 50)\nSCRAPES\n(86 = 36 + 50)\nCANARDS\n(86 = 36 + 50)\nPRANCED\n(86 = 36 + 50)\nSCARPED\n(86 = 36 + 50)\nESCARPS\n(84 = 34 + 50)\nSPACERS\n(84 = 34 + 50)\nSCRAPES\n(84 = 34 + 50)\nSCRAPES\n(84 = 34 + 50)\nPARSECS\n(84 = 34 + 50)\nESCARPS\n(84 = 34 + 50)\nPRANCES\n(84 = 34 + 50)\nSCARPED\n(84 = 34 + 50)\nCANAPES\n(84 = 34 + 50)\nSCRAPED\n(84 = 34 + 50)\nPRANCED\n(84 = 34 + 50)\nPANDERS\n(83 = 33 + 50)\nDANCERS\n(83 = 33 + 50)\nASCENDS\n(83 = 33 + 50)\nPARSECS\n(83 = 33 + 50)\nCANARDS\n(83 = 33 + 50)\nSPACERS\n(83 = 33 + 50)\nPANDERS\n(83 = 33 + 50)\n(83 = 33 + 50)\nASCENDS\n(83 = 33 + 50)\n(83 = 33 + 50)\n(83 = 33 + 50)\n(83 = 33 + 50)\nDANCERS\n(83 = 33 + 50)\n(83 = 33 + 50)\n(83 = 33 + 50)\nESCARPS\n(83 = 33 + 50)\nCANARDS\n(83 = 33 + 50)\nASCENDS\n(83 = 33 + 50)\n(83 = 33 + 50)\nCANAPES\n(83 = 33 + 50)\nASCENDS\n(83 = 33 + 50)\nCANARDS\n(83 = 33 + 50)\nCANARDS\n(83 = 33 + 50)\nASCENDS\n(83 = 33 + 50)\nCANARDS\n(83 = 33 + 50)\nDANCERS\n(83 = 33 + 50)\nCANARDS\n(83 = 33 + 50)\nPANDERS\n(83 = 33 + 50)\n(83 = 33 + 50)\nPANDERS\n(83 = 33 + 50)\nPANDERS\n(83 = 33 + 50)\nSCRAPES\n(83 = 33 + 50)\n(83 = 33 + 50)\nDANCERS\n(83 = 33 + 50)\n(83 = 33 + 50)\n(83 = 33 + 50)\n(83 = 33 + 50)\nASCENDS\n(83 = 33 + 50)\nDANCERS\n(83 = 33 + 50)\n(83 = 33 + 50)\n(83 = 33 + 50)\n(83 = 33 + 50)\n(83 = 33 + 50)\n(83 = 33 + 50)\n(83 = 33 + 50)\n(83 = 33 + 50)\n(83 = 33 + 50)\n(83 = 33 + 50)\n(83 = 33 + 50)\n(83 = 33 + 50)\nPRANCES\n(83 = 33 + 50)\nRESANDS\n(82 = 32 + 50)\n(82 = 32 + 50)\n(82 = 32 + 50)\n(82 = 32 + 50)\nSANDERS\n(82 = 32 + 50)\nPRANCED\n(82 = 32 + 50)\n\n# parascends in Words With Friends™\n\nThe word parascends is playable in Words With Friends™, no blanks required. Because it is longer than 7 letters, you would have to play off an existing word or do it in several moves.\n\nPARASCENDS\n(252)\n\n## Seven Letter Word Alert: (24 words)\n\nancress, arcades, ascends, canapes, canards, dancers, escarps, panders, parades, parsecs, pranced, prances, resands, rescans, resnaps, respans, sanders, sarapes, scarped, scraped, scrapes, spacers, spaders, spreads\n\nPARASCENDS\n(252)\nPARASCENDS\n(198)\nPARASCENDS\n(120)\nPARASCENDS\n(120)\nPARASCENDS\n(90)\nPARASCENDS\n(88)\nPARASCENDS\n(80)\nPARASCENDS\n(80)\nPARASCENDS\n(76)\nPARASCENDS\n(76)\nPARASCENDS\n(76)\nPARASCENDS\n(72)\nPARASCENDS\n(72)\nPARASCENDS\n(72)\nPARASCENDS\n(66)\nPARASCENDS\n(66)\nPARASCENDS\n(60)\nPARASCENDS\n(56)\nPARASCENDS\n(56)\nPARASCENDS\n(56)\nPARASCENDS\n(48)\nPARASCENDS\n(44)\nPARASCENDS\n(44)\nPARASCENDS\n(40)\nPARASCENDS\n(36)\nPARASCENDS\n(36)\nPARASCENDS\n(36)\nPARASCENDS\n(36)\nPARASCENDS\n(27)\nPARASCENDS\n(25)\nPARASCENDS\n(24)\nPARASCENDS\n(24)\nPARASCENDS\n(24)\nPARASCENDS\n(24)\nPARASCENDS\n(24)\nPARASCENDS\n(24)\nPARASCENDS\n(24)\nPARASCENDS\n(23)\nPARASCENDS\n(22)\nPARASCENDS\n(22)\nPARASCENDS\n(22)\nPARASCENDS\n(22)\nPARASCENDS\n(21)\nPARASCENDS\n(21)\nPARASCENDS\n(20)\nPARASCENDS\n(20)\n\nPARASCENDS\n(252)\nPARASCENDS\n(198)\nSPACERS\n(122 = 87 + 35)\nPARASCENDS\n(120)\nPARASCENDS\n(120)\nPRANCED\n(116 = 81 + 35)\nSCARPED\n(113 = 78 + 35)\nSCRAPED\n(113 = 78 + 35)\nCANAPES\n(113 = 78 + 35)\nPRANCED\n(110 = 75 + 35)\nCANARDS\n(107 = 72 + 35)\nPRANCES\n(107 = 72 + 35)\nCANAPES\n(107 = 72 + 35)\nSCARPED\n(107 = 72 + 35)\nPANDERS\n(107 = 72 + 35)\nSCRAPED\n(107 = 72 + 35)\nPRANCES\n(107 = 72 + 35)\nSPACERS\n(104 = 69 + 35)\nESCARPS\n(104 = 69 + 35)\nRESPANS\n(104 = 69 + 35)\nSCRAPES\n(104 = 69 + 35)\n(104 = 69 + 35)\nSCRAPES\n(104 = 69 + 35)\nESCARPS\n(104 = 69 + 35)\nRESNAPS\n(104 = 69 + 35)\nPARSECS\n(104 = 69 + 35)\nRESCANS\n(104 = 69 + 35)\nPARSECS\n(104 = 69 + 35)\nPRANCED\n(104 = 69 + 35)\nSCRAPED\n(101 = 66 + 35)\nCANAPES\n(101 = 66 + 35)\nPRANCES\n(101 = 66 + 35)\nDANCERS\n(101 = 66 + 35)\nDANCERS\n(101 = 66 + 35)\nSCARPED\n(101 = 66 + 35)\nASCENDS\n(101 = 66 + 35)\nRESCANS\n(98 = 63 + 35)\n(98 = 63 + 35)\nSPACERS\n(98 = 63 + 35)\nPARSECS\n(98 = 63 + 35)\nSPACERS\n(98 = 63 + 35)\nPRANCED\n(98 = 63 + 35)\nSPACERS\n(98 = 63 + 35)\nANCRESS\n(98 = 63 + 35)\nESCARPS\n(98 = 63 + 35)\n(98 = 63 + 35)\n(98 = 63 + 35)\nPRANCED\n(98 = 63 + 35)\nPARSECS\n(98 = 63 + 35)\nRESPANS\n(98 = 63 + 35)\nSCRAPES\n(98 = 63 + 35)\nDANCERS\n(95 = 60 + 35)\nDANCERS\n(95 = 60 + 35)\nPRANCED\n(95 = 60 + 35)\nCANARDS\n(95 = 60 + 35)\nPRANCES\n(95 = 60 + 35)\nDANCERS\n(95 = 60 + 35)\nSARAPES\n(95 = 60 + 35)\nPRANCED\n(95 = 60 + 35)\nPRANCED\n(95 = 60 + 35)\nPRANCES\n(95 = 60 + 35)\nPANDERS\n(95 = 60 + 35)\nPRANCED\n(92 = 57 + 35)\nRESCANS\n(92 = 57 + 35)\nRESPANS\n(92 = 57 + 35)\nRESPANS\n(92 = 57 + 35)\nPRANCED\n(92 = 57 + 35)\n(92 = 57 + 35)\n(92 = 57 + 35)\nRESNAPS\n(92 = 57 + 35)\n(92 = 57 + 35)\nRESCANS\n(92 = 57 + 35)\nPRANCED\n(92 = 57 + 35)\nSCARPED\n(91 = 56 + 35)\nCANAPES\n(91 = 56 + 35)\nSCRAPED\n(91 = 56 + 35)\nPRANCES\n(91 = 56 + 35)\nCANAPES\n(91 = 56 + 35)\nSCARPED\n(91 = 56 + 35)\nSCARPED\n(91 = 56 + 35)\nCANAPES\n(91 = 56 + 35)\nSCRAPED\n(91 = 56 + 35)\nPRANCES\n(91 = 56 + 35)\nSCRAPED\n(91 = 56 + 35)\nPRANCES\n(91 = 56 + 35)\nPARASCENDS\n(90)\nSCRAPED\n(89 = 54 + 35)\nCANAPES\n(89 = 54 + 35)\nPANDERS\n(89 = 54 + 35)\nSCRAPED\n(89 = 54 + 35)\nCANARDS\n(89 = 54 + 35)\nSCRAPED\n(89 = 54 + 35)\nSCARPED\n(89 = 54 + 35)\nPANDERS\n(89 = 54 + 35)\nASCENDS\n(89 = 54 + 35)\nPRANCES\n(89 = 54 + 35)\nSCARPED\n(89 = 54 + 35)\nPRANCES\n(89 = 54 + 35)\nASCENDS\n(89 = 54 + 35)\nCANAPES\n(89 = 54 + 35)\nSCARPED\n(89 = 54 + 35)\nPARASCENDS\n(88)\nSPACERS\n(87 = 52 + 35)\nESCARPS\n(87 = 52 + 35)\nESCARPS\n(87 = 52 + 35)\nPARSECS\n(87 = 52 + 35)\nSCRAPES\n(87 = 52 + 35)\nSPACERS\n(87 = 52 + 35)\nPARSECS\n(87 = 52 + 35)\nPARSECS\n(87 = 52 + 35)\nPECANS\n(87)\nESCARPS\n(87 = 52 + 35)\nSCRAPES\n(87 = 52 + 35)\nSPACERS\n(87 = 52 + 35)\nSCRAPES\n(87 = 52 + 35)\nANCRESS\n(86 = 51 + 35)\nPARSECS\n(86 = 51 + 35)\n(86 = 51 + 35)\nSCRAPES\n(86 = 51 + 35)\nSPACERS\n(86 = 51 + 35)\nPRANCED\n(86 = 51 + 35)\nSPACERS\n(86 = 51 + 35)\nPRANCED\n(86 = 51 + 35)\n(86 = 51 + 35)\nSCRAPES\n(86 = 51 + 35)\n(86 = 51 + 35)\nESCARPS\n(86 = 51 + 35)\nRESNAPS\n(86 = 51 + 35)\nESCARPS\n(86 = 51 + 35)\n(86 = 51 + 35)\nPARSECS\n(86 = 51 + 35)\nCAPERS\n(84)\nPACERS\n(84)\nPANDERS\n(83 = 48 + 35)\nSCRAPED\n(83 = 48 + 35)\nASCENDS\n(83 = 48 + 35)\nASCENDS\n(83 = 48 + 35)\nCANAPES\n(83 = 48 + 35)\nDANCERS\n(83 = 48 + 35)\nPANDERS\n(83 = 48 + 35)\nASCENDS\n(83 = 48 + 35)\nCANARDS\n(83 = 48 + 35)\nSCARPED\n(83 = 48 + 35)\nASCENDS\n(83 = 48 + 35)\nSCARPED\n(83 = 48 + 35)\nCANARDS\n(83 = 48 + 35)\nCANARDS\n(83 = 48 + 35)\nCANAPES\n(83 = 48 + 35)\nCANARDS\n(83 = 48 + 35)\nSCARPED\n(83 = 48 + 35)\nCANARDS\n(83 = 48 + 35)\nCANAPES\n(83 = 48 + 35)\nCANAPES\n(83 = 48 + 35)\nDANCERS\n(83 = 48 + 35)\nPRANCES\n(83 = 48 + 35)\nSCRAPED\n(83 = 48 + 35)\nSCARPED\n(83 = 48 + 35)\nPRANCES\n(83 = 48 + 35)\nPANDERS\n(83 = 48 + 35)\nSCRAPED\n(83 = 48 + 35)\nDANCERS\n(83 = 48 + 35)\nPANDERS\n(83 = 48 + 35)\nPANDERS\n(83 = 48 + 35)\nDANCERS\n(83 = 48 + 35)\nPRANCES\n(83 = 48 + 35)\nCANAPES\n(83 = 48 + 35)\nASCENDS\n(83 = 48 + 35)\nPANDERS\n(83 = 48 + 35)\nCANARDS\n(83 = 48 + 35)\nSCRAPED\n(83 = 48 + 35)\nDANCERS\n(83 = 48 + 35)\nPRANCED\n(81 = 46 + 35)\nPRANCED\n(81 = 46 + 35)\nRESPANS\n(80 = 45 + 35)\nANCRESS\n(80 = 45 + 35)\nRESCANS\n(80 = 45 + 35)\nANCRESS\n(80 = 45 + 35)\n(80 = 45 + 35)\nSPACERS\n(80 = 45 + 35)\nPARASCENDS\n(80)\nSCRAPES\n(80 = 45 + 35)\nSPACERS\n(80 = 45 + 35)\n(80 = 45 + 35)\nESCARPS\n(80 = 45 + 35)\n(80 = 45 + 35)\nRESNAPS\n(80 = 45 + 35)\nRESPANS\n(80 = 45 + 35)\nRESNAPS\n(80 = 45 + 35)\nESCARPS\n(80 = 45 + 35)\n(80 = 45 + 35)\n(80 = 45 + 35)\nRESNAPS\n(80 = 45 + 35)\nPARASCENDS\n(80)\nRESNAPS\n(80 = 45 + 35)\nSANDERS\n(80 = 45 + 35)\nRESCANS\n(80 = 45 + 35)\nSCRAPES\n(80 = 45 + 35)\nRESCANS\n(80 = 45 + 35)\nRESPANS\n(80 = 45 + 35)\nSPACERS\n(80 = 45 + 35)\n\n# Word Growth involving parascends\n\n## Shorter words in parascends\n\nas ascend ascends\n\nen end ascend ascends\n\nen end ends ascends\n\nas ascend parascend\n\nen end ascend parascend\n\nar par parascend\n\npa par parascend\n\n## Longer words containing parascends\n\n(No longer words found)" ]

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[ "# Lesson 9 - Does the second generator override variables x and y?\n\nHi,\n\nGoing through the notebook, I was wondering what is the purpose of two consecutive generators that assign the same variables (x and y):\n\n``````trn_ds2 = ConcatLblDataset(md.trn_ds, trn_mcs) val_ds2 = ConcatLblDataset(md.val_ds, val_mcs) md.trn_dl.dataset = trn_ds2 md.val_dl.dataset = val_ds2\n\nx,y=to_np(next(iter(md.val_dl))) x=md.val_ds.ds.denorm(x)\n\nx,y=to_np(next(iter(md.trn_dl))) x=md.trn_ds.ds.denorm(x)\n\nfig, axes = plt.subplots(3, 4, figsize=(16, 12)) for i,ax in enumerate(axes.flat): show_ground_truth(ax, x[i], y[i], y[i]) plt.tight_layout()\n``````\n\nI guess the second assignment x, y overrides the previous, so I am not sure if the purpose of first assignment was just to demonstrate something or it has to be there in order to execute the rest of the code correctly." ]

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[ "# Black Hole Entropy and Gravity Cutoff\n\nGia Dvali and Sergey N. Solodukhin\n\nTheory Group, Physics Department, CERN, CH-1211 Geneva 23, Switzerland\nCenter for Cosmology and Particle Physics, Department of Physics,\nNew York University, New York NY 10003, USA\nLaboratoire de Mathématiques et Physique Théorique CNRS-UMR 6083,\nUniversité de Tours, Parc de Grandmont, 37200 Tours, France\n###### Abstract\n\nWe study the black hole entropy as entanglement entropy and propose a resolution to the species puzzle. This resolution comes out naturally due to the fact that in the presence of species the universal gravitational cutoff is , as opposed to . We demonstrate consistency of our solution by showing the equality of the two entropies in explicit examples in which the relation between and is known from the fundamental theory.\n\nCERN-PH-TH/2008-141\n\n## 1 Introduction\n\nOne of the most important unresolved theoretical problems is the existence of black holes which quantum mechanically possess entropy even though there are no obvious degrees of freedom in the theory that would constitute this entropy statistical mechanically. Despite the numerous attempts, including successful ones proposed in string theory, it remains a mystery what produces the huge Bekenstein-Hawking (BH) entropy , \n\n SBH=M2PlanckA(Σ)  , (1)\n\nwhere , of an uncharged macroscopic 4-dimensional black hole. Among others, the most intriguing approach to attack this problem is to use entanglement entropy. Obvious advantage of this approach is its universality, the mechanism of generation of the entropy is the same for all possible types of black holes, and its geometrical nature, the fact that the BH entropy is geometric finds a simple explanation. The identification of the BH entropy with entanglement entropy, however, has a well-known difficulty: the black hole entropy is universal finite quantity while the entanglement entropy is quadratically-sensitive to the cutoff of the theory () and grows with the number of particle species . In this note we suggest that the species problem results from the use of the wrong cutoff. Namely, the mistake is in ignoring the -dependence of the cutoff of the theory. The latter dependence follows from completely independent consistency arguments, which show that there is a strict relation between , and , so that with growing number of species, decreases relative to the Planck mass by . We show, that when this is taken into the account, the puzzle of species disappears and the two entropies match to leading order.\n\n## 2 Entanglement Entropy of Black Holes\n\nEntanglement entropy is defined for a system divided by surface into two subsystems and . Typically, the total system is considered to be in a pure state (for instance, in a ground state) described by wave function so that one defines . If one does not have access to the modes in one of the subsystems, say , then one should trace over these modes and this situation is described by the reduced density matrix . Thus, provided we have access only to a part of the system then entanglement entropy defined as gives us a measure for the lack of information about the state of the total system. We could have traced over modes in subsystem A and get density matrix and respective entropy . If the total system was in a pure state then . This property of entanglement entropy means that if the entropy is non-zero then it should not depend on extensive quantities (such as volume) which characterize A or B but only on the surface that separates the two subsystems.\n\nOn the other hand, this entropy is non-vanishing because there are correlations between the subsystems. Entanglement entropy, thus, can be also viewed as some measure for these correlations. In local theories the correlations are short-distance and, hence, the entropy is expected to be determined by the geometry of the boundary dividing the two subsystems. It is clear that in order to regularize these correlations one has to introduce a short-distance cut-off so that the entropy essentially depends on the cut-off.\n\nThe natural (if not the only one possible) way to prevent the access to a part of the system inside a closed surface is to put the system into a black hole space-time so that would be a black hole horizon. In this situation the access to the inside region of is impossible in principle, so that the notion of entanglement entropy is well suited to describe the lack of information in black hole. In order to calculate the entropy then we have to start with a pure state described by a wave function of black hole (for a construction of this function see ) that is a functional of the modes inside horizon and modes outside . No observer has an access to the complete set of modes. By tracing over modes we end up with a density matrix and the corresponding entropy depends on the geometry of . Again the short-distance correlations are present in the system that should be cured by an UV cutoff . As far as the technicalities are concerned , the calculation of entanglement entropy of black hole horizon goes along the same steps as in the flat space-time. The only essential difference is that the surface in the flat spacetime may have non-trivial extrinsic geometry while the extrinsic curvature vanishes for black hole horizons. This restricts the geometric structures which may appear in the entropy. To leading order entanglement entropy is proportional to the area of . If there are fields available, then each of them equally contributes to the entropy so that the entanglement entropy of a black hole in four dimensions is given by\n\n Sent=NΛ2A(Σ)  . (2)\n\nThe fact that the entanglement entropy is proportional to the area similarly to the Bekenstein-Hawking (BH) entropy makes it an interesting candidate for the statistical origin of the BH entropy. A would be natural identification however faces a well-known problem of species (see for a review): entanglement entropy depends on the number of the species while the BH entropy is obviously universal and should not depend on . We now wish to show, that in order to resolve this puzzle we have to invoke the correct gravity cutoff in the presence of species.\n\n## 3 Gravity Cutoff\n\nIt was proposed in that in a theory with species the self-consistency of large-distance black hole physics implies the existence of the following new scale\n\n Λ=MPlanck/√N. (3)\n\nThe physical meaning of this scale is twofold. First , it sets the bound on the masses of the particle species. The same scale sets the bound on the gravitational cutoff of the theory. We shall now briefly reproduce some of the key consistency arguments from the black hole physics.\n\nFollowing , let us consider a theory with particle species, . For simplicity, we shall assume the species to be stable, and to be having equal masses, which we denote by . It is useful to monitor the “personalities” of the individual species by prescribing them parities under some gauged -symmetries, one per each species. Under a given -symmetry, only a corresponding species changes the sign, , whereas the remaining species stay invariant. We then perform the following thought experiment. Imagine an arbitrarily large classical black hole, and let us endow it with a maximal possible discrete charge. This can be done by throwing in the black hole one particle per each species. In this way, the resulting black hole will carry units of different -charges. Since the symmetries are gauged, this charge cannot be lost and must be returned back after the black hole evaporation. (The discrete gauge charges can be continuously monitored at infinity, by Aharonov-Bohm type experiments). However, irrespective of the original size, the black hole can only start giving back the charge after its Hawking temperature becomes comparable to the particle masses, . (Emission of particles before this moment is Boltzmann-suppressed and can only correct our results by factor log). At this point, the black hole mass is . After this moment, we can use the conservation of energy and immediately derive that the maximal number of quanta that can be emitted by the black hole is . This proves that the bound on particle masses is set by , defined by (3).\n\nHaving this bound on the particle masses already indicates that the cutoff of the theory also should be somewhere around the same scale . For example, assume that the species are self-interacting scalars. If cutoff were much higher than the scale , it would be hard to reconcile the lightness of the species versus the cutoff, in the view of quantum corrections to their masses that are quadratically sensitive to the cutoff. And indeed, the cutoff is at the scale .\n\nIn order to see this, we can again use as a tool the well-known properties of the black holes . Let us assume the opposite, that the gravity cutoff is much higher than the scale . We shall now see that with this assumption, we shall inevitably run into an inconsistency with the well-established macro black hole physics. If the true gravitational cutoff is much higher than the scale , then the black holes of size must behave as normal Schwarszchild black holes. Thus, consider such a black hole. In the normal case, with only few species, the lifetime of a black hole of size would be\n\n τBH∼M2PlanckΛ3. (4)\n\nFor , this lifetime would be perfectly consistent with our assumption that such a black hole is a quasi-classical object with well-defined (slowly-changing) Hawking temperature, . However, because of species, the evaporation rate is enhanced by factor of . Taking into the account (3), this enhancement reduces the black hole lifetime to\n\n τBH∼Λ−1. (5)\n\nThus, the black hole has a lifetime of order of its inverse temperature! This is a clear indication that such a black hole cannot be regarded as a quasi-classical state, with a well-defined temperature. Thus, we are inevitably lead into the contradiction with our initial assumption that black holes of size are normal Schwarzschild black holes. The only resolution of this inconsistency is that the gravity cutoff is .\n\nFinally, let us note that the above presented non-perturbative arguments exactly match the perturbative ones [7, 8, 5], which also indicate that in theory with species gravitational cutoff must be set by , and this is the scale above which the low energy perturbation theory breaks down111In this fact was used to explain largeness of the four-dimensional Planck scale relative to the fundamental high-dimensional one, in theory where four-dimensional particle species are localized on a 3-brane embedded in extra space..\n\nThus, there are number of different perturbative and non-perturbative arguments for the validity of the relation (3), all of which will not be reproduced here. For more detailed discussions the reader is referred to the above mentioned papers. The most relevant point for our present subject is the fact that the scale is the gravitational cutoff.\n\nGiven the relation (3), our strategy for the resolution of the species puzzle is clear. Since the scale (3) is the right UV cutoff, we should use and not in the entanglement entropy calculation. Then the disturbing dependence of the entanglement entropy on the number of species disappears and the entropy (2) precisely agrees with the BH entropy (1).\n\n## 4 Explicit Examples\n\n### 4.1 Entropy of High-Dimensional Black Holes\n\nWe now wish to demonstrate the consistency of our solution on explicit examples, in which the relation (3) is fixed by the fundamental theory. We then demonstrate that in such cases, the two entropies are automatically equal, despite the presence of the arbitrarily large number of species.\n\nAs such example, consider dimensional theory with space dimensions compactified on -torus, and non-compact dimensions forming Minkowskian geometry. Without affecting any of our results, for simplicity, we shall set all the radii being equal to . We shall choose the radius to be much larger than the fundamental Planck length , and otherwise keep it as a free parameter, which can be consistently taken to infinity. In this limit one recovers a dimensional Minkowski space. The fundamental high-dimensional theory has one dimensionfull parameter, the -dimensional Planck mass , which is the cutoff of the theory. We shall assume that the only species in the theory is a -dimensional graviton. From the four-dimensional point of view it is a theory of the tower of spin-2 Kaluza-Klein species. For any , the relation between the four-dimensional Planck mass and the cutoff of the theory is\n\n M2Planck=Λ2(RΛ)n (6)\n\nNotice that the factor measures the number of KK species\n\n N=(RΛ)n. (7)\n\nThus, as already noted in [4, 5], the relation (6) is a particular example of relation (3) in which has to be understood as the number of KK species.\n\nLet us now prove that a well known Bekenstein-Hawking entropy of a high-dimensional black hole, is correctly reproduced by the entanglement entropy of KK species. For this consider a high dimensional black hole of gravitational radius .\n\nFirst we consider a small black hole for which . In this regime, on one hand black hole is classical, and on the other hand the effects of compactification can be ignored on the near-horizon geometry. The black hole horizon is -sphere of radius in this case. We can then use the well known generalization of Beckenstein-Hawking entropy for a high dimensional black hole. Not surprisingly, this entropy is given by the black hole area in fundamental Planck units\n\n SBH=(Λrg)2+n. (8)\n\nNote that this is entropy of the black hole in the -dimensional theory. Let us now show that, from 4-dimensional perspective, this equation can be understood as entanglement entropy of KK species. We can do this in two ways, working with species that are either coordinate or momentum eigenstates in extra dimensions. Conventional KK expansion in the flat space is done in eigenstates of high-dimensional momentum operator, which coincides with eigenstates of four-dimensional mass operator. In accordance with the usual uncertainty relation, the wave-functions of KK states are plane waves e in the extra coordinate, forming a complete set of functions. Cutting of this infinite tower by , we get KK, species as explained above.\n\nBy forming appropriate orthogonal superpositions of the momentum eigenstates, we could form a complete set of coordinate eigenstates, each being localized at one particular point in extra dimensions, However, since we are limited by the cutoff in momentum space, the delta functions in position space will be smeared over a distance , and again we get coordinate eigenstates, localized at distance apart. For each of these localized states we can compute the entanglement entropy. For each state what will matter is the radial (in extra coordinate) distance from its localization site to the center of the black hole. In the other words, for each state localized on a -dimensional surface in dimensional space, the black hole horizon will cut out a -dimensional sphere of radius , where . Suppressing the factors of order one, the entanglement entropy for each such state will be\n\n Sind=(rΛ)2. (9)\n\nIntegrating this over all possible localization sites, we obviously get\n\n Sent= (rgΛ)2N(rg)=(rgΛ)n+2, (10)\n\nwhere is the number of distinct species localised in the extra n dimensions that see the black hole horizon of radius . This equation correctly reproduces (8). Despite the existence of arbitrarily large number of species, no puzzle appears, since dependence is automatically taken care of by the relation (6). Had we incorrectly used as a cutoff we would end up with the species puzzle.\n\nWe can do the same computation using momentum eigenstates. Each KK species sees the in four-dimensions the cut-out sphere of surface , which is simply a four-dimensional projection of the high-dimensional black hole horizon. However, because the wave-function of each KK is spread out in extra dimension, we have to take into the account the intersection probability. Since the species are the uniform plane waves in extra dimensions, whereas the cross-section of the black hole is dimensional sphere of volume , the intersection probability is equal to . Thus the individual contributions to the entanglement entropies are equal to\n\n SKK=(rgΛ)2(rgR)n. (11)\n\nSumming this over all the KK states, with total number given by (7), we get exactly (8)\n\n Sent=∑KKSKK=NSKK=SBH. (12)\n\nConsider now the case of large black hole . In this case the black hole horizon is a product of 2-sphere of radius and -dimensional torus of size . From the point of view of 4-dimensional gravity this black hole has entropy\n\n S(4)BH=(MPlanckrg)2 (13)\n\nwhich, by means of relation (6), is identical to the Bekenstein-Hawking entropy in -dimensional theory\n\n S(4+n)BH=Λn+2r2gRn, (14)\n\nwhere is the area of -dimensional horizon.\n\nThe calculation of the entanglement entropy remains the same. The entropy of a single KK state (in the coordinate representation) has entropy\n\n Sind=(rgΛ)2. (15)\n\nThe number of states in the case when does not depend on and is equal to (7), the full number of KK species. Summing over all species we get\n\n Sent=(rgΛ)2N=(rgΛ)2(RΛ)n (16)\n\nwhich exactly agrees both with (13) and (14).\n\nThat the entanglement entropy calculation gives correct entropy in two limiting cases ( and ) makes us beleive that this method should work also in the intermediate regime although to do the computation one should know the concrete profile of the black hole horizon along the extra dimensions.\n\n### 4.2 AdS/CFT Example\n\nTo further illustrate our point and provide the reader with another explicit example we consider a 3-brane in a configuration in anti-de Sitter space-time. This is the Randall-Sundrum set-up, in the framework of the AdS/CFT correspondence it has a description in terms of CFT on the brane coupled to gravity at a UV cutoff . If the brane is placed at the distance from the Anti-de Sitter boundary, one obtains that there is a dynamical gravity induced on the brane with the induced Newton constant . According to the AdS/CFT dictionary, in the theory on the brane has the meaning of the UV cut-off. Thus, one finds that , in agreement with (3). We remind that the quantum field theory defined on the brane is the super-conformal gauge theory, so that in the large limit represents the number of species. Thus, the 3-brane in the anti-de Sitter space-time gives us an example when the relation (3) holds automatically. Consider now a black hole on the 3-brane and compute its entanglement entropy due to the CFT. To leading order one finds\n\n Sent=N24πϵ2A(Σ)=N2Λ2A(Σ), (17)\n\nwhich is exactly the Bekenstein-Hawking entropy of the black hole provided one expresses the UV cut-off in terms of the induced Planck scale .\n\nThe fact that the BH entropy is correctly reproduced in the entanglement entropy calculation, in accordance to our proposal, is not limited to the above particular examples, but is much more general and relies solely on the existence of the cutoff (3).\n\n## 5 Conclusions\n\nIn this note we consider the black hole entropy as entanglement entropy and propose a resolution to the species puzzle. We suggest that the puzzle never appears provided the correct cutoff in the theory is suppressed relative to the Planck mass. This suppression follows from completely independent consistency arguments given in . We demonstrate the equality of two entropies in explicit examples in which the relation between the Planck mass and the cutoff is known from the fundamental theory. It would be interesting to verify our proposal directly in string theory. For this one would have to compute entanglement entropy of strings (for a recent work in this direction see ).\n\nIt is straightforward to generalize our proposal to arbitrary dimensions. Indeed, the BH entropy of a black hole in dimensions, , where is the Planck mass in d-dimensional gravity and is the area of -surface of the horizon, matches the entanglement entropy produced by species at cutoff if the relation between , and is given by\n\n Λd−2N=Md−2(d). (18)\n\nThis is exactly the generalized gravity cutoff in dimensions found in . Thus, our proposal resolves the species problem in arbitrary dimensions.\n\n### Acknowledgments\n\nThe work is supported in part by David and Lucile Packard Foundation Fellowship for Science and Engineering, and by NSF grant PHY-0245068. S.S. is grateful to the Theory Division at CERN for the hospitality extended to him while this work was in progress.\n\n## References\n\nWant to hear about new tools we're making? Sign up to our mailing list for occasional updates.\n\nIf you find a rendering bug, file an issue on GitHub. Or, have a go at fixing it yourself – the renderer is open source!\n\nFor everything else, email us at [email protected]." ]

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[ "## NewWaySys\n\n### Vector Art Online", null, "", null, "", null, "", null, "", null, "# Algebra Use Gauss Jordan Elimination Determine Whether N Given Vectors Rm Linearly Indepen Q\n\nThis post categorized under Vector and posted on June 22nd, 2019.", null, "This Algebra Use Gauss Jordan Elimination Determine Whether N Given Vectors Rm Linearly Indepen Q has 898 x 1024 pixel resolution with jpeg format. was related topic with this Algebra Use Gauss Jordan Elimination Determine Whether N Given Vectors Rm Linearly Indepen Q. You can download the Algebra Use Gauss Jordan Elimination Determine Whether N Given Vectors Rm Linearly Indepen Q picture by right click your mouse and save from your browser.\n\nAlgebra (a) How can we use Gauss-Jordan elimination to determine whether n given vectors in Rm are linearly independent Write down a precise answer based on a theorem from the course notes. (b) Find the value(s) of c for which the vectors (10 10) (0 12-1) 2 -2-2c) in IR are linearly independent.100 %(1)Question Algebra (a) How can we use Gauss-Jordan elimination to determine whether n given vectors in Rm sp100 %(1)Problem 277. Determine whether the following set of vectors is linearly independent or linearly dependent. If the set is linearly dependent express one vector in the set as a\n\n11.04.2013 m n (the number of vectors is greater than their graphicgth) they are linearly dependent (always). The reason in short is that any solution to the system of m x n equations is also a solution to the n x n system of equations (youre trying to solve Av0 ).03.01.2018 A short graphic outlining the process of finding the reduced row echelon form (RREF) of a given matrix.Autor Steven ClontzAufrufe 421graphiclnge 16 Min.12.07.2012 Gaussian Elimination. Here we solve a system of 3 linear equations with 3 unknowns using Gaussian Elimination. Here we solve a system of 3 linear equations with 3 unknowns using Gaussian Autor patrickJMTAufrufe 19Mgraphiclnge 9 Min.\n\nWe solve a system of linear equations by Gauss-Jordan elimination. This is similar to Gaussian elimination but we reduce a matrix to reduced row echelon form. This is similar to Gaussian elimination but we reduce a matrix to reduced row echelon form." ]

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[ "# Observation of the universal magnetoelectric effect in a 3D topological insulator\n\nV. Dziom    A. Shuvaev    A. Pimenov    G. V. Astakhov    C. Ames    K. Bendias    J. Böttcher    G. Tkachov    E. M. Hankiewicz    C.  Brüne    H Buhmann    L. W. Molenkamp Institute of Solid State Physics, Vienna University of Technology, 1040 Vienna, Austria\nPhysikalisches Institut (EP6), Universität Würzburg, 97074 Würzburg, Germany\nInstitut für Theoretische Physik und Astronomie, Universität Würzburg, 97074 Würzburg, Germany\nPhysikalisches Institut (EP3), Universität Würzburg, 97074 Würzburg, Germany\nJanuary 24, 2021\n###### Abstract\n\nThe electrodynamics of topological insulators (TIs) is described by modified Maxwell’s equations, which contain additional terms that couple an electric field to a magnetization and a magnetic field to a polarization of the medium, such that the coupling coefficient is quantized in odd multiples of per surface. Here, we report on the observation of this so-called topological magnetoelectric (TME) effect. We use monochromatic terahertz (THz) spectroscopy of TI structures equipped with a semi-transparent gate to selectively address surface states. In high external magnetic fields, we observe a universal Faraday rotation angle equal to the fine structure constant when a linearly polarized THz radiation of a certain frequency passes through the two surfaces of a strained HgTe 3D TI. These experiments give insight into axion electrodynamics of TIs and may potentially be used for a metrological definition of the three basic physical constants.\n\nMaxwell’s equations are in the foundation of modern optical and electrical technologies. In oder to apply Maxwell’s equations in conventional matter, it is necessary to specify constituent relations, describing the polarization and magnetization as a function of the applied electric and magnetic fields, respectively. Soon after the theoretical prediction Kane and Mele (2005); Bernevig et al. (2006); Fu and Kane (2007) and experimental discovery of 2D and 3D TIs König et al. (2007); Hsieh et al. (2008), it has been recognized that the constituent relations in this new phase of quantum matter contain additional cross-terms and Qi et al. (2008)\n\n Pt(B)=(N+12)α2πBMt(E)=−(N+12)α2πE. (1)\n\nHere, is an integer, and is the fine structure constant. Intriguing consequences of Eq. (1) are the universal Faraday rotation angle , when a linearly polarized electromagnetic radiation passes through the top and bottom topological surfaces Qi et al. (2008); Tse and MacDonald (2010a), and magnetic monopole images, induced by electrical charges in proximity to a topological surface Qi et al. (2009). However, experimental verification of these TME effects has been lacking.\n\nAs the modified Maxwell’s equations describing electrodynamics of TIs are applicable in the low-energy limit, optical experiments should be performed at THz or sub-THz frequencies Hancock et al. (2011); Valdés Aguilar et al. (2012); Shuvaev et al. (2013a); Wu et al. (2013). In real samples, the TME may be screened by nontopological contributions Maciejko et al. (2010); Tse and MacDonald (2010b); Tkachov and Hankiewicz (2011). In fact, quantized Faraday rotation has been detected in 2D electron gas Ikebe et al. (2010) and graphene Shimano et al. (2013) in the quantum Hall effect (QHE) regime, but the observed values are not fundamental.\n\nWe report on the observation of the universal Faraday rotation angle equal to the fine structure constant . Strained HgTe layers grown on CdTe, that are investigated in the present work, are shown to be a 3D TI Brüne et al. (2011) with surface-dominated charge transport Brüne et al. (2014). In oder to eliminate the material details, we perform measurements under constructive interference conditions, such that the transmission through the CdTe substrate is approaching 100% Shuvaev et al. (2011).\n\nThe strained HgTe film is a 58 nm thick HgTe layer embedded between two CdHgTe layers (Fig. 1a). The CdHgTe layers have a thickness of (lower layer) and (top/cap layer), respectively. The purpose of these layers is to provide the identical crystalline interface for top and bottom surface of the HgTe films as well as to protect the HgTe from oxidization and adsorption. This leads to an increase in carrier mobility with a simultaneous decrease in carrier density compared to uncaped samples Brüne et al. (2011). The transport characterization on a standard Hall bar sample shows a carrier density at gate of and a carrier mobility of . The optical measurements are carried out on a sample fitted with a thick multilayer insulator of SiO/SiN and a thick Ru film. The Ru film (oxidized in the air) is used as a semitransparent top gate electrode Shuvaev et al. (2013b).", null, "Figure 1: THz magnetooptics of a strained HgTe 3D TI. a, A scheme of the experimental setup (only one arm of the Mach-Zehnder interferometer is shown). The strained HgTe layer, which is a 3D TI, is sandwiched between (Cd,Hg)Te protecting layers. The top-gate electrode, consisting of a SiO2/Si3N4 multilayer insulator and a thin conducting Ru film, is semitransparent at THz frequencies. The THz radiation (ν=0.35THz) is linearly polarized, and the Faraday rotation (θF) and ellipticity (ηF) are measured as a function of the magnetic field B for different gate voltages UG. b, c, Transmission spectra in the parallel |tp| and crossed |tc| polarizer configurations, respectively. The gate voltage is color-coded, and the experimental curves are shifted for clarity. Notations in b and c: e denotes the CR of the topological surface states of electron character, s1 and s2 denote extra resonances with opposite phase to that of the e-CR as discussed in the text.\n\nThe transmittance experiments at THz frequencies () are carried out in a Mach-Zehnder interferometer arrangement Volkov et al. (1985); Shuvaev et al. (2012), allowing measurement of the amplitude and the phase shift of the electromagnetic radiation in a geometry with controlled polarization (Fig. 1a). The monochromatic THz radiation is provided by a backward-wave oscillator (BWO). The THz power on the sample is in between and with the focal spot of . Using wire grid polarizers, the complex transmission coefficient is obtained both in parallel (Fig. 1b) and cross (Fig. 1c) polarization geometries, providing full information about the transmitted light. External magnetic fields are applied using a split-coil superconducting magnet. The experiments are carried out in Faraday geometry, i.e with applied parallel to the propagation direction of the THz radiation. The conductivity tensor at THz angular frequency is obtained from the experimental data by inverting the Berreman equations Berreman (1972) for the complex transmission coefficient through a thin conducting film on an insulating substrate.\n\nIn general case, the light propagating along the direction can be characterized by the orthogonal and components of the electric and magnetic fields, which can be written in the form of a 4D vector . The interconnection between vectors and , corresponding to different points in space separated by a distance , is given by . Here, is a transfer matrix. For an insulating substrate of thickness and dielectric constant , this is the identity matrix provided is an integer. We find in a separate experiment on a bare CdTe substrate that this condition is fulfilled for . Therefore, all the measurements presented here are performed at this frequency to eliminate any contribution to the Faraday signal from the substrate. The corresponding photon energy of is much smaller than the energy gap in strained HgTe (above ) Brüne et al. (2011), and Eqs. (1) are a good approximation.\n\nFor normal incidence, the fields across the conducting interface are connected by the Maxwell equation . Here, the time dependence is assumed for all fields. As the wavelength of for is much larger than the HgTe layer thickness, we use the limit of thin film, and the corresponding transfer matrix is determined by the diagonal () and Hall () components of the conductivity tensor . Within the Drude-like model, these components for one type of charge carriers can be written in the form Palik and Furdyna (1970); Tse and MacDonald (2010b)\n\n σxx=σyy=1−iωτ(1−iωτ)2+(Ωcτ)2σ0, (2) σxy=−σyx=Ωcτ(1−iωτ)2+(Ωcτ)2σ0. (3)\n\nHere, is the cyclotron resonance (CR) frequency, is the conductivity, and is the scattering time. For classical conductors, the CR frequency is written as , where is the effective electron mass in the parabolic approximation.\n\nThe total transfer matrix relates vectors on both sides of the sample and hence contains full information about the transmission and reflection coefficients. Thus, when , the influence of the substrate is minimized, and the THz response is dominated by the transport properties of the HgTe layer, in accord with Eqs. (2) and (3). The calculation of the complex transmission coefficients and based on the transfer matrix formalism as well as the exact form of the transfer matrices are presented in Methods.", null, "Figure 2: Charge carriers in strained HgTe. a, The band structure of the Cd0.7Hg0.3Te/HgTe heterostructure close to the Γ-point. The chemical potential (dashed line) crosses the Dirac-like surface state in the band gap corresponding to the electron CR Ωce. b, 2D dc conductivity σ0 of different charge carriers (e and Σs=s1+s2), obtained by Drude-like fits to Eqs. (2,3) of the magnetooptical spectra. The dimensionless values are given relative to the impedance of free space Z0=1/cϵ0≈377Ohm. The inset shows the e-CR in terms of Ωce/B as a function of the gate voltage UG.\n\nMagnetic field dependence of the THz transmission is dominated by a sharp CR of surface electrons () at (Figs. 1b and 1c). Below we demonstrate their Dirac-like character and that they are responsible for the universal Faraday rotation. Remarkably, the observation of the CR both in and indicates a high purity of our HgTe layer. The scattering time is significantly longer than the inverse THz frequency , and according to Eqs. (2) and (3) the conductivity reveals a resonance-like behavior .\n\nFurther features are broad resonances at and at indicated in Fig. 1 as and , respectively. The phase of the corresponding THz transmission coefficient in the vicinity of these resonances has the opposite sign with respect to that of the -CR. Remarkably, the and resonances disappear with applying positive gate voltage (Fig. 1 and Fig. 2b). We associate them with either interband Landau level transitions or thermally activated states as discussed below.\n\nTo understand the origin of the experimentally observed resonances, we analyze the band structure of tensile strained layer as shown in Fig. 2a. It is obtained similar to Ref. [Brüne et al., 2014] within the tight binding approximation of the - Kane Hamiltonian Novik et al. (2005); Baum et al. (2014). Due to reduced point symmetry at the boundary between the and HgTe layers, an additional interface potential is allowed in the Hamiltonian Ivchenko et al. (1996). This potential is used to shift the Dirac point closer to the valence band edge, so that the tight binding results are in good agreement with recent ARPES experiments Brüne et al. (2011); Liu et al. (2015) and ab-initio calculations Wu et al. (2014) on HgTe. The Dirac-like surface states are located in the band gap between the light-hole (conduction) and heavy hole (valence) subbands (Fig. 2a). The camel back of the heavy hole band originates from coupling of this band to the electron-like valence band and is therefore a hallmark of the inverted band structure of HgTe. In accordance with previous transport data, the chemical potential crosses the topological surface states for a large range of gate voltages Brüne et al. (2014). The total electron density in Fig. 2a is , representing the experimental situation at . For simplicity, we assume here the same density at the top and bottom surfaces. Using the general formula for a classical cyclotron resonance Ashcroft and Mermin (1976) , where is the energy dispersion, is the magnetic field, and is the area enclosed by the wave vector , we calculate for the topological surface state .\n\nExperimentally, simultaneous fit of the real and imaginary parts of and allows the extraction of all transport characteristics, i.e., conductivity, charge carrier density, scattering time and CR frequency Shuvaev et al. (2011). The inset of Fig. 2b shows experimentally determined electron CR as a function of gate voltage, which perfectly agrees with the theoretical value for the topological Dirac-like surface states. Since only surface states are observed in transport experiments on the similar structures Brüne et al. (2014), a possible explanation of the appearance of additional resonances is interband Landau level transitions between heavy hole-like (HH) bulk bands and topological surface states. Such transitions are generally allowed as can be shown using the Kubo formula. Another possibility would be thermally activated transport between the camel back of the HH bulk band and the surface states. This is generally possible since the THz field may well induce heating of the carriers, resulting in a higher effective temperature compared to that of the lattice. The heating of the system would be consistent with the effective temperature of the surface states carriers , as shown later in Fig. 3.\n\nFrom the obtained scattering time and the CR positions in the magnetooptical spectra of Figs. 1b and 1c, one can calculate the mobility . The surface states demonstrate high mobility , which agrees with the transport data. Since the -CR and ,-resonances occur at different magnetic fields, their contributions to the transport can be clearly separated, as presented in Fig. 2b. The striking feature of this plot is that the conductivity of the surface states dominates at large gate voltages. In what follows, we concentrate therefore on , while remaining weak contribution from the interband Landau-level transitions/thermally activated transitions are subtracted as explained in Methods.", null, "Figure 3: THz QHE of the surface states. The real part of the THz Hall conductivity σxy in units of e2/h, obtained at UG=1.9V. The vertical solid and dashed lines indicate the positions of the Hall plateaus in two surfaces (Na and Nb), estimated from the extrema in ∂σxy/∂B (upper panel). Theoretical calculations represented by the thin line are performed as explained in the text. Inset presents the same experimental and theoretical curves in the whole magnetic field range, including the surface carrier CR at 0.4T.\n\nFigure 3 demonstrates the real part of the Hall conductivity . The overall behavior is provided by the high-field wing of the classical Drude model, i.e., Eq. (3), resulting in a rapid suppression of with growing magnetic field. In addition to the classical behaviour, regular oscillations in can be recognized, which are linear in inverse magnetic field. The slope of the linear behavior changes with gate voltage, reflecting gate dependence of the electron density in one of the two surfaces Shuvaev et al. (2013a). These QHE oscillations extrapolate to value for large magnetic fields, demonstrating Dirac character of the surface electrons Buttner et al. (2011). While the oscillations of in Fig. 3 are not clearly resolved, the visibility can be significantly improved by inserting the sample in a Fabry-Pérot resonator, as we have previously demonstrated for a similar structure Shuvaev et al. (2013a).\n\nIn magnetic fields above , the Hall conductivity clearly shows a plateau close to , corresponding to a value per surface (Fig. 3). Another plateau close to is also recognizable at a magnetic field of . The steps in loose their regularity in lower magnetic fields, as can be qualitatively explained by a finite THz frequency in magnetooptical experiments. As mentioned above, the overall behaviour of is provided by the classical curve of Eq. (3), and the real part of can be approximated as , which in the limit reduces to the expression , being a multiple of . In low magnetic fields, the CR frequency becomes comparable to the THz frequency , destroying the regularities in .\n\nSince in strained HgTe the Fermi level lies in the bulk band gap (see Fig. 2a and Ref. Brüne et al., 2014), we attribute the observed THz QHE to the formation of the 2D Landau levels at the top and bottom surfaces of the HgTe layer (Fig. 1a). This interpretation is further substantiated by our theoretical analysis of the quantum Hall conductivity calculated from the Kubo formula for both top and bottom surface states within the Dirac model Tse and MacDonald (2010b); Tkachov and Hankiewicz (2013).\n\nOur two-surface Dirac model describes well the surface carrier CR (the inset of Fig. 3). The lengths of the theoretical Hall plateaus in the high magnetic field region (Fig. 3) correlate correctly with the positions of the extrema in the derivative . However, the model predicts much sharper transitions between the QHE plateaus, as observed in the experiment. One of two possible explanations is the heating of the surface carriers by the THz field, resulting in a higher effective temperature compared to that of the lattice. Such a heating can occur due to inefficient energy relaxation in the electronic system through the emission of LO phonons at low temperatures Kiessling et al. (2012). The best fit of our experimental data is obtained with (Fig. 3). Another explanation is based on spatial fluctuations of the surface carrier densities, which are likely to occur in our samples due to their large lateral sizes compared to the typical Hall bars used in the measurements. The experimental data of Fig. 3 can alternatively be well fitted assuming cold carriers () with density fluctuations within 10% relative to their nominal values (Fig. 3). As the fits are nearly indistinguishable, we cannot quantitatively determine the contributions of both mechanisms leading to the smearing of the THz QHE plateaus.\n\nThe stronger the field, the closer the Hall conductivity to the quantized values expected for a two-surface Dirac system\n\n σxy=(Na+Nb+1)e2h,Na,b=Int(na,bΦ0|B|), (4)\n\nwhere are the integer numbers of the highest occupied Landau levels at the top and bottom surfaces, with being the corresponding carrier densities ( is the magnetic flux quantum). Upon approaching the cyclotron resonance, the Hall conductivity deviates from the quantized values in Eq. (4) due to the predominance of the intraband transitions between the Landau levels. From the fitting procedure, we extract the nominal carrier densities and . The total surface carrier density agrees well with that obtained from Drude-like fits of magnetooptical spectra. Another fitting parameter is the classical (Drude) surface conductivity . Its large value indicates high surface carrier mobility, insuring that the condition for the quantum Hall regime, , is met for T. Here, is the characteristic Landau level spacing for a Dirac system, are the scattering times of the top and bottom carriers, and is the resistance quantum.", null, "Figure 4: Quantized THz Faraday rotation of Dirac fermions. a, Faraday rotation and b, Faraday ellipticity in a 3D HgTe TI as a function of the external magnetic field for different gate voltages (color-coded). The horizontal solid line in a indicates the universal Faraday rotation angle θF=−α≈−7.3×10−3rad. c, Gate voltage dependence of the Faraday rotation in a magnetic field of 5T.\n\nHaving established that the THz response of the topological surface states in high magnetic fields is determined by the conductivity quantum (), we turn to the central result of this work, the THz Faraday effect. Owing to the TME of Eq. (1), an oscillating electric (magnetic ) field of the linearly polarized THz radiation induces in a 3D TI an oscillating magnetic (electric ) field. The generated in such a way secondary THz radiation is polarized perpendicular to the primary polarization and its amplitude is times smaller. This can be viewed as a rotation of the initial polarization by an angle . Indeed, Fig. 4a clearly demonstrates that the Faraday angle in high magnetic fields is close to this fundamental value.\n\nWe rigorously characterize the THz Faraday effect, and the Faraday ellipticity is shown in Fig. 4b. It is relatively small in high magnetic fields, but does not reach zero. This observation indicates that while the TME dominates, the interaction of TIs with THz radiation is not a completely dissipationless process in our samples. Remarkably, the universal value of the Faraday angle remains robust against the gate voltage. This is demonstrated in Fig. 4c, where for .\n\nThe observed terahertz Faraday rotation equal to the fine structure constant is a direct consequence of the topological magnetoelectric effect, confirming axion electrodynamics of 3D topological insulators. We use monochromatic terahertz spectroscopy, providing complete amplitude and phase reconstruction, which can be applied to investigate topological phenomena in various systems, including graphene, 2D electron gas, layered superconductors and recently experimentally discovered Weyl semimetals Xu et al. (2015). Picoradian angle resolution can be achieved using a balanced detection scheme Crooker et al. (2010), and the universal Faraday rotation in combination with the magnetic flux quantum and the conductivity quantum are suggested Maciejko et al. (2010) to use for a metrological definition of the three basic physical constants, , , and .\n\n## Acknowledgments\n\nThis work was supported by Austrian Science Funds (I1648-N27, W-1243, P27098-N27), as well as by the SFB 1170 ”ToCoTronics”, the ENB Graduate School on Topological Insulators, the SPP 1666 and the ERC (project 3-TOP).\n\n## Methods\n\n### Theoretical analysis of magnetooptical spectra\n\nIn general case, the light propagating along the direction can be characterized by the orthogonal components of electric (, ) and magnetic (, ) fields. These procedure closely follows the formalism described by Berreman Berreman (1972). We write the field components in the form of a 4D vector\n\n V=⎛⎜ ⎜ ⎜ ⎜⎝ExEyHxHy⎞⎟ ⎟ ⎟ ⎟⎠. (5)\n\nThe interconnection between vectors and , corresponding to different points in space separated by a distance , is given by . Here, is a transfer matrix. In case of isotropic dielectric substrate the transfer matrix is given by\n\n ^MCdTe(ℓ)=⎛⎜ ⎜ ⎜ ⎜⎝cos(kℓ)00ıZsin(kℓ)0cos(kℓ)−ıZsin(kℓ)00−ıZ−1sin(kℓ)cos(kℓ)0ıZ−1sin(kℓ)00cos(kℓ)⎞⎟ ⎟ ⎟ ⎟⎠. (6)\n\nHere, and is the wave vector. In the following is assumed.\n\nThe particular choice of complex amplitudes as the tangential components of electric and magnetic fields simplifies the treatment of interfaces in case of normal incidence. The fields across the interfaces are connected by the Maxwell equation . Here, the time dependence is assumed for all fields and is the complex conductivity tensor of the material. In the limit of thin film the transfer matrix reads\n\n ^MHgTe(^σ)=⎛⎜ ⎜ ⎜ ⎜⎝10000100Z0σyxZ0σyy10−Z0σxx−Z0σxy01⎞⎟ ⎟ ⎟ ⎟⎠. (7)\n\nHere,   is the impedance of free space. We note that matrices and as given in Eqs. (6)-(7) are fully equivalent to well known transmission and reflection expressions for a bare substrate and a thin film, respectively.\n\nThe total transfer matrix relates vectors in the air on both sides of the sample and contains full information about transmission and reflection coefficients. In order to calculate them, it is more convenient to change the basis. In the new basis, the first component of the vector is the amplitude of the linearly polarized wave () propagating in positive direction, the second is that of the wave with the same polarization propagating in negative direction, the third and the fourth components are the same but for two waves with the orthogonal linear polarization (). The propagation matrix in the new basis is , with the basis transformation matrix being\n\n ^V=⎛⎜ ⎜ ⎜⎝1100001100−111−100⎞⎟ ⎟ ⎟⎠. (8)\n\nThe complex transmission () and reflection () coefficients for a linearly polarized incident radiation could now be easily found from the following equation\n\n ⎛⎜ ⎜ ⎜⎝tc0tp0⎞⎟ ⎟ ⎟⎠=^M′⎛⎜ ⎜ ⎜⎝0rc1rp⎞⎟ ⎟ ⎟⎠. (9)\n\nHere, the and denote the complex transmittance amplitudes within parallel and crossed polarizers, respectively. The same conventions for reflectance are given by and . Eq. (9) can be inverted analytically to obtain the complex conductivity matrix from the transmission data.\n\nThe Faraday rotation and Faraday ellipticity can be directly obtained from the transmission amplitudes , and phases , as\n\n tan(2θ) = 2|tp||tc|cos(ϕp−ϕc)|tp|2−|tc|2, (10) sin(2η) = 2|tp||tc|sin(ϕp−ϕc)|tp|2+|tc|2.\n\n### Subtraction of a background contribution\n\nWe start from the directly measured spectra of Faraday rotation and ellipticity. We note, that related data on complex transmission and are known as well. Using the expression for the transmission of a film on a substrate, the Faraday rotation and ellipticity are directly inverted to obtain complex conductivities and . In addition, from the fits of the transmission data, the parameters of all charge carriers are obtained. In the approximation of independent carriers, their conductivities are additive, and a weak background contribution associated with the -resonances can be directly subtracted from and . Finally, the corrected conductivity data are used to calculate the Faraday rotation and ellipticity from electron-like surface states.\n\n## References\n\n• Kane and Mele (2005) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005).\n• Bernevig et al. (2006) B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science 314, 1757 (2006).\n• Fu and Kane (2007) L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007).\n• König et al. (2007) M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science 318, 766 (2007).\n• Hsieh et al. (2008) D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Nature 452, 970 (2008).\n• Qi et al. (2008) X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 78, 195424 (2008).\n• Tse and MacDonald (2010a) W.-K. Tse and A. H. MacDonald, Phys. Rev. Lett. 105, 057401 (2010a).\n• Qi et al. (2009) X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, Science 323, 1184 (2009).\n• Hancock et al. (2011) J. N. Hancock, J. L. M. van Mechelen, A. B. Kuzmenko, D. van der Marel, C. Brüne, E. G. Novik, G. V. Astakhov, H. Buhmann, and L. W. Molenkamp, Phys. Rev. Lett. 107, 136803 (2011).\n• Valdés Aguilar et al. (2012) R. Valdés Aguilar, A. V. Stier, W. Liu, L. S. Bilbro, D. K. George, N. Bansal, L. Wu, J. Cerne, A. G. Markelz, S. Oh, et al., Phys. Rev. Lett. 108, 087403 (2012).\n• Shuvaev et al. (2013a) A. M. Shuvaev, G. V. Astakhov, G. Tkachov, C. Brüne, H. Buhmann, L. W. Molenkamp, and A. Pimenov, Phys. Rev. B 87, 121104 (2013a).\n• Wu et al. (2013) L. Wu, M. Brahlek, R. Valdes Aguilar, A. V. Stier, C. M. Morris, Y. Lubashevsky, L. S. Bilbro, N. Bansal, S. Oh, and N. P. Armitage, Nat. Phys. 9, 410 (2013).\n• Maciejko et al. (2010) J. Maciejko, X.-L. Qi, H. D. Drew, and S.-C. Zhang, Phys. Rev. Lett. 105, 166803 (2010).\n• Tse and MacDonald (2010b) W.-K. Tse and A. H. MacDonald, Phys. Rev. B 82, 161104 (2010b).\n• Tkachov and Hankiewicz (2011) G. Tkachov and E. M. Hankiewicz, Phys. Rev. B 84, 035405 (2011).\n• Ikebe et al. (2010) Y. Ikebe, T. Morimoto, R. Masutomi, T. Okamoto, H. Aoki, and R. Shimano, Phys. Rev. Lett. 104, 256802 (2010).\n• Shimano et al. (2013) R. Shimano, G. Yumoto, J. Y. Yoo, R. Matsunaga, S. Tanabe, H. Hibino, T. Morimoto, and H. Aoki, Nat. Commun. 4, 1841 (2013).\n• Brüne et al. (2011) C. Brüne, C. X. Liu, E. G. Novik, E. M. Hankiewicz, H. Buhmann, Y. L. Chen, X. L. Qi, Z. X. Shen, S. C. Zhang, and L. W. Molenkamp, Phys. Rev. Lett. 106, 126803 (2011).\n• Brüne et al. (2014) C. Brüne, C. Thienel, M. Stuiber, J. Böttcher, H. Buhmann, E. G. Novik, C.-X. Liu, E. M. Hankiewicz, and L. W. Molenkamp, Phys. Rev. X 4, 041045 (2014).\n• Shuvaev et al. (2011) A. M. Shuvaev, G. V. Astakhov, A. Pimenov, C. Brüne, H. Buhmann, and L. W. Molenkamp, Phys. Rev. Lett. 106, 107404 (2011).\n• Shuvaev et al. (2013b) A. Shuvaev, A. Pimenov, G. V. Astakhov, M. M hlbauer, C. Br ne, H. Buhmann, and L. W. Molenkamp, Appl. Phys. Lett. 102, 241902 (2013b).\n• Volkov et al. (1985) A. A. Volkov, Y. G. Goncharov, G. V. Kozlov, S. P. Lebedev, and A. M. Prokhorov, Infrared Phys. 25, 369 (1985).\n• Shuvaev et al. (2012) A. M. Shuvaev, G. V. Astakhov, C. Brüne, H. Buhmann, L. W. Molenkamp, and A. Pimenov, Semicond. Sci. Technol. 27, 124004 (2012).\n• Berreman (1972) D. W. Berreman, J. Opt. Soc. Am. 62, 502 (1972).\n• Palik and Furdyna (1970) E. D. Palik and J. K. Furdyna, Rep. Prog. Phys. 33, 1193 (1970).\n• Novik et al. (2005) E. G. Novik, A. Pfeuffer-Jeschke, T. Jungwirth, V. Latussek, C. R. Becker, G. Landwehr, H. Buhmann, and L. W. Molenkamp, Phys. Rev. B 72, 035321 (2005).\n• Baum et al. (2014) Y. Baum, J. Böttcher, C. Brüne, C. Thienel, L. W. Molenkamp, A. Stern, and E. M. Hankiewicz, Phys. Rev. B 89, 245136 (2014).\n• Ivchenko et al. (1996) E. L. Ivchenko, A. Y. Kaminski, and U. Rössler, Phys. Rev. B 54, 5852 (1996).\n• Liu et al. (2015) C. Liu, G. Bian, T.-R. Chang, K. Wang, S.-Y. Xu, I. Belopolski, I. Miotkowski, H. Cao, K. Miyamoto, C. Xu, et al., Phys. Rev. B 92, 115436 (2015).\n• Wu et al. (2014) S.-C. Wu, B. Yan, and C. Felser, EPL (Europhysics Letters) 107, 57006 (2014).\n• Ashcroft and Mermin (1976) N. W. Ashcroft and N. D. Mermin, Solid State Physics (Harcourt College Publishers, 1976).\n• Buttner et al. (2011) B. Buttner, C. X. Liu, G. Tkachov, E. G. Novik, C. Brune, H. Buhmann, E. M. Hankiewicz, P. Recher, B. Trauzettel, S. C. Zhang, et al., Nat. Phys. 7, 418 (2011).\n• Tkachov and Hankiewicz (2013) G. Tkachov and E. M. Hankiewicz, physica status solidi (b) 250, 215 (2013).\n• Kiessling et al. (2012) T. Kiessling, J.-H. Quast, A. Kreisel, T. Henn, W. Ossau, and L. W. Molenkamp, Phys. Rev. B 86, 161201 (2012).\n• Xu et al. (2015) S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, et al., Science 349, 613 (2015).\n• Crooker et al. (2010) S. A. Crooker, J. Brandt, C. Sandfort, A. Greilich, D. R. Yakovlev, D. Reuter, A. D. Wieck, and M. Bayer, Phys. Rev. Lett. 104, 036601 (2010)." ]

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[ "# I don’t know how to solve the last pieces\n\nSo, I almost finished a rubik’s cube, but the last 3 pieces are not in their places (green on red face, blue on green face, and red on blue face). Apart from these, the other faces are done.", null, "How can I finish solving the cube?\n\n• This was closed as a duplicate of a question about a situation where just two cubelets were swapped. It is not a duplicate of that question; here the permutation is a 3-cycle. I've reopened it. Apr 29 '19 at 16:09\n\nIf the three edge pieces you want to rotate are on the top layer at front, left and right, then the move $$F^2UM^{-1}U^2MUF^2$$ will permute them cyclically where $$F$$ means rotating the front face $$90^{\\circ}$$ clockwise (so $$F^2$$ means rotating it $$180^{\\circ}$$), $$U$$ means rotating the top (upper) face $$90^{\\circ}$$ clockwise, and $$M$$ means rotating the vertical \"middle slice\" perpendicular to the front face $$90^{\\circ}$$ towards you. See here for an animation.\nThis will move front $$\\rightarrow$$ left $$\\rightarrow$$ right $$\\rightarrow$$ front which is actually the reverse of the direction you want; so either do it twice or do it backwards meaning $$F^2U^{-1}M^{-1}U^2MU^{-1}F^2$$. (Take the steps in reverse order, inverting each one, noting that e.g. $$F^2$$ is its own inverse.)" ]

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[ "#", null, "How to reorder the data table column inside the for loop", null, "• ### Question\n\n•", null, "Hi,\n\nHow to reorder the data table column inside the loop. For example:\n\nColumn1  |  Column2  |  Column3  |  Column4  |  Column5\n\n-------------------------------------------------------------------------\n\nI need to reorder like below:\n\nColumn4  |  Column3  |  Column1  |  Column2  |  Column5\n\n-------------------------------------------------------------------------\n\nI tried below but below will not Reorder because it changes the order. But I need this inside the loop since I have many dynamic column based on the table selection. I can't hardcode directly:\n\nfor (int dtInputCount = 0; dtInputCount < inpuDataTable.Columns.Count; dtInputCount++) {\n\nif (inpuDataTable.Columns[dtInputCount].ColumnName == \"Column1\") {\ncsvDataTablecopy.Columns[csvDataTable.Columns[i].ColumnName].SetOrdinal(2);\n}\nelse if (inpuDataTable.Columns[dtInputCount].ColumnName == \"Column2\") {csvDataTablecopy.Columns[csvDataTable.Columns[i].ColumnName].SetOrdinal(4);\n}\nelse if (inpuDataTable.Columns[dtInputCount].ColumnName == \"Column3\") {csvDataTablecopy.Columns[csvDataTable.Columns[i].ColumnName].SetOrdinal(1);\n}\nelse if (inpuDataTable.Columns[dtInputCount].ColumnName == \"Column4\") {csvDataTablecopy.Columns[csvDataTable.Columns[i].ColumnName].SetOrdinal(0);\n\n}\nelse if (inpuDataTable.Columns[dtInputCount].ColumnName == \"Column5\") {csvDataTablecopy.Columns[csvDataTable.Columns[i].ColumnName].SetOrdinal(4);\n\n}\n\n}\n\nMonday, July 22, 2013 1:10 PM\n\n•", null, "As you told there are many dynamic and unknown columns, Instead of reordering column in existing datatable, make use of new datatable and copy the from columns from the last to first.\n\nThanks & Regards\n\nMonday, July 22, 2013 6:13 PM\n•", null, "Generate a new array to keep the name of the column you need to reorder,then loop the new array ,adjust the position of the column\n\n``` DataTable inpuDataTable=new DataTable();\n\nDataColumn second = new DataColumn(\"second\", typeof(int));\n\nDataColumn dc5 = new DataColumn(\"Column5\", typeof(string));\n\nDataColumn dc3 = new DataColumn(\"Column3\", typeof(string));\n\nDataColumn dc1 = new DataColumn(\"Column1\", typeof(string));\n\nDataColumn first = new DataColumn(\"first\", typeof(int));\n\nDataColumn dc2 = new DataColumn(\"Column2\", typeof(string));\n\nDataColumn dc4 = new DataColumn(\"Column4\", typeof(string));\n//a new array keep the columns you need to reorder\nstring[] columnsname = new string[] { \"Column1\", \"Column2\", \"Column3\", \"Column4\", \"Column5\" };\n\nfor (int dtInputCount = 0; dtInputCount < columnsname.Length; dtInputCount ++)\n{\ninpuDataTable.Columns[columnsname[dtInputCount]].SetOrdinal(dtInputCount);\n}```\n\nTuesday, July 23, 2013 4:41 AM\n\n### All replies\n\n•", null, "As you told there are many dynamic and unknown columns, Instead of reordering column in existing datatable, make use of new datatable and copy the from columns from the last to first.\n\nThanks & Regards\n\nMonday, July 22, 2013 6:13 PM\n•", null, "Generate a new array to keep the name of the column you need to reorder,then loop the new array ,adjust the position of the column\n\n``` DataTable inpuDataTable=new DataTable();\n\nDataColumn second = new DataColumn(\"second\", typeof(int));\n\nDataColumn dc5 = new DataColumn(\"Column5\", typeof(string));\n\nDataColumn dc3 = new DataColumn(\"Column3\", typeof(string));\n\nDataColumn dc1 = new DataColumn(\"Column1\", typeof(string));\n\nDataColumn first = new DataColumn(\"first\", typeof(int));\n\nDataColumn dc2 = new DataColumn(\"Column2\", typeof(string));" ]

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[ "• A\n• A\n• A\n• ABC\n• ABC\n• ABC\n• А\n• А\n• А\n• А\n• А\nRegular version of the site\n\n## Observation of the diphoton decay of the Higgs boson and measurement of its properties\n\nThe European Physical Journal C - Particles and Fields. 2014. Vol. 74. No. 10. P. 3076.\n\nObservation of the diphoton decay mode of the recently discovered Higgs boson and measurement of some of its properties are reported. The analysis uses the entire dataset collected by the CMS experiment in proton-proton collisions during the 2011 and 2012 LHC running periods. The data samples correspond to integrated luminosities of 5.1 fb-1$\\phantom{\\rule{0ex}{0ex}}$ at s=7TeV$s=7\\phantom{\\rule{0ex}{0ex}}$ and 19.7 fb-1$\\phantom{\\rule{0ex}{0ex}}$ at 8 TeV$\\phantom{\\rule{0ex}{0ex}}$  . A clear signal is observed in the diphoton channel at a mass close to 125 GeV$\\phantom{\\rule{0ex}{0ex}}$  with a local significance of 5.7σ$5.7\\phantom{\\rule{0ex}{0ex}}$ , where a significance of 5.2σ$5.2\\phantom{\\rule{0ex}{0ex}}$ is expected for the standard model Higgs boson. The mass is measured to be 124.70±0.34GeV=124.70±0.31(stat)±0.15(syst)GeV$124.70±0.34\\phantom{\\rule{0ex}{0ex}}$, and the best-fit signal strength relative to the standard model prediction is 1.14+0.260.23=1.14±0.21(stat)+0.090.05(syst)+0.130.09(theo)$1.14-0.23+0.26=1.14±0.21\\phantom{\\rule{0ex}{0ex}}$ . Additional measurements include the signal strength modifiers associated with different production mechanisms, and hypothesis tests between spin-0 and spin-2 models." ]

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[ "", null, "Pete Freitag\n\n# JavaScript isInteger Function\n\nIt is pretty common that you need to check for an integer in javascript validation code. Here's a simple javascript isInteger function code examples.\n\n```function isInteger(s) {\nreturn (s.toString().search(/^-?[0-9]+\\$/) == 0);\n}\n```\n\nTwo things to note about our `isInteger` function. The first is that we are calling the javascript `toString()` function / method to convert the value to a string, so we can perform the regular expression. The second is that the regular expression allows for negative integers (since by definition an integer can be negative), if you want an unsigned isInt function try the following:\n\n```function isUnsignedInteger(s) {\nreturn (s.toString().search(/^[0-9]+\\$/) == 0);\n}\n```\n\n#### Like this? Follow me ↯\n\nJavaScript isInteger Function was first published on May 24, 2007." ]

[ null, "https://www.petefreitag.com/images/pete-freitag-med-opt.jpg", null ]

[ "# Price Earnings Ratio Calculator\n\nPrice/earnings ratio – also often called the price to earnings ratio or the P/E ratio – is a finance indicator that measures a company. Price  Earnings Ratio is equal to share price divided earnings per share\n\nFormula us,\n\nP/E ratio = share price / earnings per share\n\nLet us consider,\n\n• Price per share – the market price of a stock. This value heavily depends on the supply and demand on the market.\n• Earnings per share – the profit which a company gains from each outstanding share of common stock. If a company doesn’t have any net income, but only net losses, it won’t have a P/E ratio.\n\n## Calculate the price/earnings ratio\n\n1. Determine the market share price. Let’s assume that it is equal to \\$25.\n2. Determine the earnings per share over the last 12 months. In our example, we’ll set this value to \\$1.80.\n3. Use the price/earnings ratio formula:\n\nP/E ratio = share price / earnings per share\n\nP/E ratio = 25/1.80 = 13.9" ]

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[ "Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.\n\nProbably sooner than I expected, I have managed to also upload the codes for the examples in Chapter 5 of the book, which deals with doing Bayesian health economic evaluations. Basically, there are 3 examples, which sort of represent the main classes of cases (of course there are many, many other types of models that can be used and these are by no means exhaustive. But if you learn to do these, you can start working in health economics with reasonable confidence…).\n\n1. The first example assumes individual data (eg, but not necessarily, from RCTs) on the pair of variables $(e,c)$, representing a suitable measure of effectiveness (eg, but not necessarily, QALYs) and cost. The tricky part is that, as seems reasonable, these two quantities will tend to be correlated. This of course needs to be accounted for when modelling the data, with the added complication that neither can usually be associated with Normal distributions.\nSometimes it is possible to make transformations to approximately reach normality, but there is still the issue of correlation. One easy-ish way of dealing with this (especially if you’re a Bayesian) is to de-compose the joint distribution $p(e,c\\mid\\boldsymbol\\theta)$ as the product $p(c \\mid e,\\boldsymbol\\phi) p(e \\mid \\boldsymbol\\psi)$, effectively using a regression specification.\nThe examples provide the same structure for different distributional choices and I’ll soon post some variations based on mixture models to better account for the possibility that some individuals have observed null costs (for example, subjects among the controls that happen to require no extra care at all).\n\n2. The second example deals with evidence synthesis, ie the situation where there is composite evidence coming from different sources, that can be combined to estimate several parameters of interest.\n\nIn this case (see the graph), I consider a model that has $H$ studies estimating the probability of infection with influenza, which via a hierarchical model is given by $p_0$. Moreover, there are other $S$ studies investigating the effectiveness of some treatment, which again via hierarchical modelling is given by $\\rho$. Then these two can be put together to estimate the probability of infection for the subjects treated  with the prophylactic drug as $p_1=f(p_0,\\rho)$, for a suitable function $f(\\cdot)$.\n\n3. In a way this is the king of health economic evaluation methods, since increasingly often people use Markov models (MMs) to do their cost-effectiveness analysis.\n\nThe idea is to model the transitions of a cohort of individuals among a set of “states”. Initially, we can assume that everybody is healthy; then, as time progresses and according to some transition probabilities (ie the quantities that are normally to be estimated, indicated as $\\lambda_{hk}$ in the graph above, for the transition from $h$ to $k$), they start to move around and eventually (if the model is so-called “life-long”) they all die.\nThe example in the book is not too complicated and MMs can be quite tricky to run (just like the HPV model we’re trying to extend). But I think it gives a (slightly more than) basic idea of how they should work." ]

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[ "## How to enter a function\n\nHome\n\nNavigation: Entering a function | Operations | Functions | Information on the parser\n\nEntering a Function\n\nThe input of functions is similar to most computer algebra systems (such as Maple or Mathematica). It should be noted that our parser accepts expressions without an explicit multiplication sign, i.e. 3sin(x)cos(x) is a correct input.\n\n### Supported operators\n\n + addition != not equal > greater than - subtraction == equal to < less than * multiplication >= greater than or equal to ^ exponentiation / division <= less than or equal to - minus (sign) ! factoral\n\n### Supported functions\n\nwith a single argument\n\n exp exponential function asin arcsine round rounding log natural logarithm (base e) acos arccos floor floor function sin sine atan arctan ceil ceiling function cos cosine sqrt square root factorial factorial tan tangent abs absolute value sign sign\n\n### with two arguments\n\nmax maximum min minimum atan2 phase\n\nInformation on the parser\n\nThe parser used by the applets is based on a parser written by Darius Bacon. We complemented the parser with a number of features (e.g. multiplication without an explicit multiplication sign).\n\nFinancially supported by\n\nUniversity of Innsbruck: New Media and Learning Technologies\nAustrian Minister of Science and Research bm:bwk\n\nNach oben scrollen" ]

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[ " Fuzzy Based BEENISH Protocol for Wireless Sensor Network\n\nCircuits and Systems\nVol.07 No.08(2016), Article ID:67651,13 pages\n10.4236/cs.2016.78164\n\nFuzzy Based BEENISH Protocol for Wireless Sensor Network\n\nA. Devasena1, B. Sowmya2\n\n1Anna University Chennai, Dhanalakshmi College of Engineering, Chennai, India\n\n2Department of ECE, Dhanalakshmi College of Engineering, Chennai, India", null, "", null, "", null, "", null, "Received 23 March 2016; accepted 30 March 2016; published 24 June 2016\n\nABSTRACT\n\nThe main parameter to be considered in the wireless sensor network is the amount of energy that is available in each sensor node. The lifetime of the sensor node (SN) depends on it. As the SNs are deployed in remote locations, if the entire energy is consumed, it would be very difficult to replace or recharge the energy source immediately. Hence the energy consumed by each node is very important. If individual SNs send information directly to the base station (BS), then the availability of energy in such SN decreases very fast. This will lead to reduction in the life time of the SN. Instead, the SNs can send the data to the cluster head (CH), then the CH consolidates the received data. The CH sends it to the BS periodically. In this way, utilizing CH for sending the information to the BS increases the lifetime of the SN. The cluster head selection is very crucial in such networks. This paper proposes a novel fuzzy based BEENSIH protocol for CH selection.\n\nKeywords:\n\nSensor Nodes, Cluster, Energy Level, Heterogeneous Sensor Network, Fuzzy Logic, BEENISH", null, "1. Introduction\n\nWSN consists of a number of sensor nodes working together in a particular area in order to obtain data from the surroundings . Two types of sensor networks are available. Based on the types of sensors utilized in the network, the wireless sensor network may be homogeneous wireless sensor network and heterogeneous wireless sensor network. Wireless sensors are small in size when compared to the traditional sensors. The processing and computing resources are also small. The cost of wireless sensors is cheap when compared to traditional one. Sensor nodes are small in size. It consists of sensing, data processing and communicating components. A sensor network is composed of a large number of sensor nodes which are densely deployed either inside the location where the operation is being performed or very close to it. The sensor nodes are randomly deployed. Unique feature of the sensor network is the cooperative effort of sensor nodes. The SN protocols and algorithms must possess self-organizing capabilities . Sensor nodes are fitted with an onboard processor. Instead of sending raw data to the nodes responsible for fusion, they use their processing abilities to locally carry out simple computations and transmit only the required and partially processed data. Sensor nodes are prone to failures. Sensor nodes mainly use a broadcast communication paradigm and they may not have global identification. The topology of the sensor networks changes very frequently . The purpose of deploying sensor nodes is to monitor some important parameters in that area. SN consists of a sensing element for sensing, signal conditioning element for processing the sensed parameters, transmission element for conveying the processed information either to CH or to the BS, and power supply unit for the operation of the sensor node. Figure 1 shows the structure of sensor node, Figure 2 shows the formation of clustering in a Wireless Sensor Network.\n\n2. Heterogeneous Clustering Protocols\n\nThere are many types of heterogeneous clustering protocols. They are as follows.\n\nFigure 1. Structure of basic sensor node.\n\nFigure 2. Formation of clustering in a wireless sensor network.\n\n2.1. Stable Election Protocol\n\nPerformance of the clustering protocols can be analyzed with the help of the parameters like stability period, instability period, number of cluster heads per round, number of alive nodes per round. Stable and unstable region are available in any WSN. In the unstable region, if one node starts dying the remaining nodes present in the region also started dying thereby the stability of the WSN has been disturbed. But in stable region, if one node starts dyeing, the remaining nodes present in the cluster continue to be alive, thereby the stability of the cluster continues. By implementing Stable Election Protocol (SEP), the stability of the cluster has been improved with the help of the parameters like excess energy faction that is available between normal and advanced nodes (α) and the portion of advanced nodes (m). In order to achieve the stability in the cluster, the energy consumed by the SNs has to be balanced - .\n\nTwo types of nodes are available in SEP. They are normal nodes and advanced nodes. The energy level present in the advanced nodes is high when compared to the energy level present in the normal nodes. The probability of the selection of the CH depends on the amount of energy level present in the sensor node. Therefore for the advanced nodes the probability of becoming the CH is more when compared with normal nodes. SEP has been implemented for two level heterogeneous WSN, i.e. two levels of energy (high energy level, low energy level ) has been considered.\n\nIf the sensor nodes posses different energy levels (low energy, medium energy, high energy) then Distributed Energy Efficient Clustering (DEEC), Enhanced Distributed Energy Efficient clustering (EDEEC) protocols are used. For DEEC, EDEEC, the nodes have been identified as Normal nodes (low energy level), advanced nodes (medium energy level) and super energy nodes (high energy level).\n\n2.2. Distributed Energy Efficient Clustering (DEEC) Protocol\n\nThe cluster head (CH) selection in DEEC utilizes the amount of energy level available in that node. For calculating the energy level of the node, the initial energy level and residual energy level parameters have been considered. In DEEC two types of nodes are available. They are normal nodes and advanced nodes. The advanced nodes possess higher energy level when compared to the normal nodes. Therefore the possibility of selection as CH is more for advanced nodes than normal nodes. After some rounds, the advanced nodes lose some of the residual energy level. Due to this condition; the lifetime of the CH node is reduced very fast. This drawback has been overcome in Developed distributed energy efficient clustering (DDEEC) protocol .\n\n2.3. Developed Distributed Energy Efficient Clustering (DDEEC) Protocol\n\nThe nodes available in DDEEC are the same as in DEEC. In DDEEC, a threshold limit has been set for the node which is acting as a CH. After some rounds when the advanced node (which is acting as a cluster head) reaches the threshold limit (i.e. after the reduction in the amount of residual energy level) that CH node has been converted to normal node, then another advanced node has been acting as new CH. Thereby the early death of the advanced node (which acts as CH recently) has been prevented. Due to this threshold limit checking, all the advanced nodes available in that sensor network will have the chance of operating as a CH .\n\n2.4. Enhanced Developed Distributed Energy Efficient Clustering (EDDEEC) Protocol\n\nThe level of heterogeneity in EDDEEC is three, i.e. three types of nodes are available in EDDEEC. They are normal nodes, advanced nodes and super nodes. The super nodes posses’ highest energy level when compared to normal and advanced nodes. In EDDEEC, the selection of CH is based on the amount of absolute energy level (Tabsolute). If the node posses the energy level that is less than the Tabsolute, then chances are less for that node to be selected as CH. In general normal and advanced nodes posses the energy level that is below Tabsolute). Due to this reason, the chance for the selection of CH is more for super nodes when compared to other nodes. Now one super node which posses the energy level above the Tabsolute, then that node acts as CH. After some rounds the energy level of the CH node starts decreasing and at one particular stage if the energy level of the CH reaches below Tabsolute. Now the existing CH node is not eligible for acting as CH. The chance of acting as CH then goes to some other super node which posses the energy level above the Tabsolute. Thereby balance in the energy level of the super nodes continues. This leads to prevent the early die of the super node which is acting as CH .\n\n2.5. Balanced Energy Efficient Network Integrated Super Heterogeneous Clustering (BEENISH) Protocol\n\nFor four level of heterogeneity is being utilized in BEENISH protocol. Here, based on the energy level, the nodes have been classified as Normal nodes (low energy level), advanced nodes (medium energy level) super nodes (high energy level) and ultra super nodes (highest energy level) . In general ultra super nodes have the highest chance for the selection of cluster head. The first node dies very late when compared to other protocols (DEEC, DDEEC, EDEEC). Due to this reason the lifetime of the BEENISH protocol has been extended. The efficiency of this BEENISH protocol is highest when compared to DEEC, DDEEC, EDEEC protocols. The number of packets sent to the base station is also high. Let us consider ri the number of rounds for a node (SN) to become as cluster head. This is referred to a rotating epoch. The amount of energy consumed by the cluster head is more when compared to the other nodes that are available in that cluster. The CH selection in heterogeneous sensor network is different from the homogeneous sensor network based on the energy level; the ultra-super node has highest chance for the CH selection. Table 1 shows the comparison of these clustering protocols.\n\n3. Existing Cluster Head Selection in BEENISH protocol\n\nThe nodes available in BEENISH protocol include normal nodes, advanced nodes, super nodes, and ultra super nodes. Ultra super node possesses highest energy level, while normal node possesses lowest energy level. The chance for the selection of the cluster head (CH) is very high for ultra super node. The average energy of each node has been determined with the help of the following equations\n\nIn BEENISH, average energy of rth round can be obtained as follows and as supposed in DEEC:", null, ". (1)\n\nR is showing total rounds from the start of network to the all nodes die and can be estimated as in DEEC and given as under:", null, ". (2)\n\nTable 1. Comparison between the heterogeneous protocols.\n\nEround is the energy dissipated in a network during single round.\n\nTo achieve the optimal number of CH at start of each round, node si decides whether to become a CH or not based on probability threshold calculated by expression in the following equation,", null, "(3)\n\nwhere G is the set of nodes eligible to become CH. If a node si has not been CH in the most recent ni then it belongs to set G. Random number between 0 and 1 is selected by nodes belonging to set G. If the number is less than threshold T(si), the node si will be CH for that current round.\n\nThreshold is calculated for CH selection of normal, advanced, super and ultra-super nodes by putting above values in equation below:", null, ". (4)\n\n4. Proposed Cluster Head Selection in BEENISH Protocol Using Fuzzy Logic\n\n4.1. Fuzzy Logic Based Cluster Head Determination\n\nFuzzy sets had been published by Lotfi A. Zadah of university of California in the year 1965. These fuzzy sets formed the basis for fuzzy logic. With the help of fuzzy logic, it is easy to implement real time scenario. This can be done by utilizing less expensive microcontrollers. The purpose of fuzzification is to convert the input values into fuzzy values. The rule evaluation module calculates the input real values to the corresponding output fuzzy values. The defuzzification module converts the output real values into appropriate output values .\n\nBefore implementing the fuzzy logic, the following assumptions are to be made.\n\n1) The cluster head election should be done by the base station (BS).\n\n2) The base station should be in static condition. BS should not move from one location to another.\n\n3) The location of the nodes has to be conveyed to the BS.\n\n4) The channel used for propagating the information is in symmetric form.\n\n5) The energy level of the nodes also has to be intimated to the BS. This can be done during the setup phase of the cluster formation.\n\n6) The distance between the BS and the nodes should be large.\n\n7) The nodes that are present in the WSN should be homogeneous in nature (i.e. nodes have identical characteristics).\n\nThe algorithm adapted in the FL scenario is similar to the centralized approach, i.e. decision has been taken by the base station alone. In order to implement this, the base station possesses more energy, power, as well as memory capacity when compared to the other nodes . Figure 3 shows the Implementation of cluster head determination using Fuzzy Logic.\n\nMamdani method of Fuzzy Inference System (FIS) has been adapted here. The fuzzy logic control model consists of fuzzifier module, fuzzy rules module, fuzzy inference module, and a defuzzifier module. Three main input variables like energy dissipation rate of the node, packet loss and distance from the base station have been considered here for the determination of cluster head. For analysis purpose here the fuzzy subsets (related to these fuzzy main input variables) have been considered. For the first fuzzy input variable energy dissipation rate, the fuzzy subsets like low energy dissipation rate, medium energy dissipation rate and high energy dissipation rate have been considered. Similarly for the second fuzzy input variable packet loss, the fuzzy subsets like low number of packet loss, medium number of packet loss and high number of packet loss parameters are considered. We have three fuzzy subsets for the third input (distance) far distance, medium distance and near distance BS (related to BS) are considered. With the help of these fuzzy subsets, IF-THEN rules are applied in order to determine the output . Here the fuzzy output variable is considered to be selection of cluster head (CH) among\n\nFigure 3. Implementation of cluster head determination using fuzzy logic.\n\nFigure 4. Fuzzy logic designer―cluster head selection.\n\nthe nodes that are available in that cluster. For cluster head selection output variable, the fuzzy subsets like low, medium, high and highest are considered. To be a CH, the node should posses low energy dissipation rate, low number of packet losses, and distance from the base station should be near. Figure 4 shows the fuzzy logic designer for cluster head selection. Figure 5 shows the membership function for the input variable dissipation rate. Figure 6 shows the membership function for the input variable packet loss. Figure 7 shows the membership function for the input variable distance. Figure 8 shows the Rule editor for CH selection. Tables 2-4 show the Rule base for energy dissipation rate low, medium, high respectively\n\n4.2. Proposed Algorithm for the Cluster Head Determination in BEENISH Protocol\n\nStep 1: Check whether the node possess highest energy level. If yes, then go to step 5 or go to step 6.\n\nStep 2: Now check if the same node possesses low number of packet loss. If yes then go to step 5 or go to step 6.\n\nFigure 5. Membership function for the input variable-dissipation rate.\n\nFigure 6. Membership function for the input variable―packet loss.\n\nFigure 7. Membership function for the input variable―distance.\n\nFigure 8. Rule editor―cluster head selection.\n\nTable 2. Rule base with energy dissipation rate―low.\n\nTable 3. Rule base with energy dissipation rate―medium.\n\nTable 4. Rule base with energy dissipation rate―high.\n\nStep 3: Now check if the same node also possesses low energy dissipation rate. If yes, then go to step 5 or go to step 6.\n\nStep 4: Now check if the same node also possesses nearer to the base station (distance between BS and SN should be near). If yes, then go to step 5 or go to step 6.\n\nStep 5: Select the node which possesses highest energy level, low number of packet loss, low energy dissipation rate, and nearer distance to the base station as cluster head.\n\nStep 6: Need not select the node which possesses low energy level, high number of packet loss, high energy dissipation rate, and far distance from the base station as cluster head. That node will not compete for cluster head selection.\n\nStep 7: Now check the energy level of Cluster head (selected recently), then go to step 1.\n\n5. Simulation and Results\n\nThe Fuzzy based BEENISH protocol is simulated with the help of MATLAB and fuzzy logic and the corresponding results are shown below. Figure 9 shows the surface viewer for CH selection. Figure 10 shows the rule viewer for CH selection. Figure 11(a) shows the transmission between single sensor node and Base station in two hops for fuzzy based BEENISH protocol. Figure 11(b) shows the transmission between single sensor node and Base station in two hops for ordinary BEENISH protocol. Figure 12 shows the transmission between multiple sensor node and Base station in two hops for ordinary BEENISH protocol.\n\nThe comparison between ordinary BEENISH protocol and fuzzy based BEENISH protocol is shown in Table 5.\n\nThe parameters to be chosen for comparison are, packets delivered and distance travelled. As already mentioned, two hop communication has been chosen for transmission purpose. The packets delivered by the fuzzy based BEENISH protocol are comparatively higher than the ordinary BEENISH protocol for each and every transmission, and the distanced travelled is comparatively lower. As the distance travelled is less, then the path loss will be less.\n\n6. Conclusion\n\nBy adapting this fuzzy based BEENISH protocol, the lifetime of the ultra-super node gets maintained, i.e. in\n\nFigure 9. The surface viewer for cluster head selection.\n\nFigure 10. The rule viewer for cluster head selection.\n\n(a)(b)\n\nFigure 11. (a) The transmission between single sensor node and Base station in two hops for fuzzy based BEENISH protocol; (b) The transmission between single sensor node and base station in two hops for ordinary BEENISH protocol.\n\nFigure 12. The transmission between multiple sensor node and Base station in two hops for fuzzy based BEENISH protocol and ordinary BEENISH protocol.\n\nTable 5. Comparison of fuzzy based BEENISH protocol with ordinary BEENISH protocol for single transmission.\n\nthis protocol, the energy level of the CH is maintained. If the energy level of the CH node decreases, then the node with another higher energy level is considered for cluster head. Thereby the energy consumed by the CH is maintained. The fuzzy based BEENISH protocol shows better performance when compared to the ordinary BEENISH protocol based on number of packets delivered to the base station. It also enhances fast transmission.\n\nCite this paper\n\nA. Devasena,B. Sowmya, (2016) Fuzzy Based BEENISH Protocol for Wireless Sensor Network. Circuits and Systems,07,1893-1905. doi: 10.4236/cs.2016.78164\n\nReferences\n\n1. 1. Yick, J., Mukherjee, B. and Ghosal, D. (2008) Wireless Sensor Network Survey. Computer Networks, 52, 2292 -2330.\nhttp://dx.doi.org/10.1016/j.comnet.2008.04.002\n\n2. 2. Akyildiz, I.F., Su, W., Sankarasubramaniam, Y. and Cayirci, E. (2002) Wireless Sensor Networks: A Survey. Computer Networks, 38, 393-422.\nhttp://dx.doi.org/10.1016/S1389-1286(01)00302-4\n\n3. 3. Akyildiz, I.F., Su, W., Sankarasubramaniam, Y. and Cayirci, E. (2002) A survey on Wireless Sensor Networks. IEEE Communications Magazine, 40, 102-114.\nhttp://dx.doi.org/10.1109/MCOM.2002.1024422\n\n4. 4. Khan, A., et al. (2012) HSEP: Heterogeneity-Aware Hierarchical Stable Election Protocol for WSNs.\narXiv:1208.2335v1[cs.N1].\n\n5. 5. Smaragdakis, G. and Bestavros, I.M.A. SEP: A Stable Election Protocol for Clustered Heterogeneous Wireless Sensor Network.\n\n6. 6. Kashaf, A., et al. (2012) TSEP: Threshold-Sensitive Stable Election Protocol for WSNs.\nhttp://120.52.73.76/arxiv.org/pdf/1212.4092v1.pdf\n\n7. 7. Elbhiri, B., et al. (2010) Developed Distributed Energy-Efficient Clustering (DDEEC) for Heterogeneous Wireless Sensor Networks.\n\n8. 8. Khair, M.G., et al. (2011) Heterogeneous Clustering of Sensor Network. Procedia Computer Science, 5, 939-944.\nhttp://dx.doi.org/10.1016/j.procs.2011.07.132\n\n9. 9. Mishra, B.K., Dhabariya, A.S. and Jain, A. (2014) Enhanced Distributed Energy Efficient Clustering (E-DEEC) Based on Particle Swarm Optimization. International Journal of Digital Application & Contemporary Research, 2, No. 6.\n\n10. 10. Javaid, N., Qureshi, T.N., Khan, A.H., Iqbal, A., Akhtar, E. and Ishfaq, M. (2013) EDDEEC: Enhanced Developed Distributed Energy-Efficient Clustering for Heterogeneous Wireless Sensor Networks. Procedia Computer Science, 19, 914-919.\n\n11. 11. Kumar, D., Aseri, T.C. and Patel, R.B. (2009) EEHC: Energy Efficient Heterogeneous Clustered Scheme for Wireless Sensor Networks. Computer Communications, 32, 662-667.\n\n12. 12. Qureshi, T.N., Javaid, N., Khan, A.H., Iqbal, A., Akhtar, E. and Ishfaq, M. (2013) BEENISH: Balanced Energy Efficient Network Integrated Super Heterogeneous Protocol for Wireless Sensor Networks. Procedia Computer Science, 19, 920-925.\nhttp://dx.doi.org/10.1016/j.procs.2013.06.126\n\n13. 13. Gupta, I., Riordan, D. and Sampalli, S. (2005) Cluster-Head Election Using Fuzzy Logic for Wireless Sensor Networks.\n\n14. 14. Tamene, M. and Rao, K.N. (2016) Fuzzy Based Distributed Cluster Formation and Route Construction in Wireless Sensor Networks. International Journal of Computer Applications, 140, No. 5.\n\n15. 15. Gharghi, M., Parvinnia, E. and Khayami, R. (2013) Designing a Fuzzy Rule Base System to Head Cluster Election inWireless Sensor Networks. Indian Journal of Science and Technology, 6, No. 5." ]

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[ "# Discrete version of pillbox function\n\nThis is an image processing question.\n\nIn 2-D Euclidean space a pillbox function, $f(x)$ is given by:\n\n$$f(x,y) = 1\\text{ if } x^2 + y^2 < R, 0 \\text{ otherwise }$$\n\n$R$ is come constant.\n\nIf I try to approximate the top hat when I'm looking at an image matrix of points, a naive algorithm would make some of the image pixels 1 inside a given radius and the others 0 outside that radius.\n\nHowever, if I'm looking at a 4 x 4 grid of points, the shape I'd approximate would look like a square if I tried to naively create a discrete top-hat.\n\nHow would I create a discrete top-hat function that accurately approximates it's continuous counterpart, even when my image matrix is small?\n\nEDIT: I believe the comment about the edge image pixels having a value equal to the area of the pixel contained within the pillbox is correct. However, implementing this algorithm seems difficult. A reference would be appreciated.\n\nEDIT 2: Matlab has an algorithm for computing a discrete pillbox function in it's fspecial function (disk option). This is the algorithm I'm after, though I still need a reference or an explanation of the geometry, as the function is quite complicated.\n\n• I'm debating assigning a 1 to entirely inside pixels, 0 to entirely outside pixels and an intermediate value to partially in, partially out pixels. – ncRubert Sep 18 '14 at 21:51\n• I would set the coefficient equal to the area of the pixel which is enclosed by the circle, so it would be 1 for the inner pixels and some value < 1 for pixels that the circle intersects. The sum of all the coefficients would then be equal to the area of the circle. – Paul R Sep 19 '14 at 11:43\n\nI think you should first construct a smooth function of $r$ (where $r=\\sqrt{x^2+y^2}$). For example, your original solution was based on the following function of one variable:\n\n$$f_1(r) = \\begin{cases} 1 & r < R \\\\ 0 & \\text{otherwise} \\end{cases}$$\n\nbut you might try using a \"softer\" step function such as:\n\n$$f_2(r) = \\begin{cases} 1 & r<R-\\frac{s}{2} \\\\ 0 & r\\ge R+\\frac{s}{2} \\\\ \\frac{1}{2} - \\frac{r-R}{s} & otherwise \\end{cases}$$\n\nand you can create smoother variants, such as $$f_3(r) = 3f_2(r)^2-2f_2(r)^3$$ since $f_3$ is simply formed by mapping the linear ramp function of $f_2$ to smoother cubic shape.\n\nThe transition width of the ramp, $s$, should be set to $\\frac{1}{2}$ or $\\frac{1}{\\sqrt2}$ if the linear ramp, $f_2$, is used, and you may need a larger value of $s$ (say, $s=1$) if the cubic function, $f_3$ is used.\n\nNote that, strictly speaking, what you are trying to do is sample the original pillar-box function at discrete sample-points, so if you think about the function prior to discrete sampling (let's call this the pre-sampling function, $f(x,y)$ being defined over the Cartesian plane, for all real $x$,$y$) the problem with your original pre-sampling function was that it contained a step-discontinuity (a $0^{th}$order discontinuity), which contains large amounts of high-frequency energy (in 2D spatial frequency), so you need to remove some of this high-frequency energy.\n\nThe radial function, $f_2(r)$, removes the $0^{th}$order discontinuity, resulting in far less high-(spatial)-frequency energy in the pre-sampling signal (but the resulting signal still has a $1^{st}$order disconinuity, since it's first derivative is not continuous). Likewise, the cubic function, $f_3(r)$, has even more reduced high-frequency energy, since it has no $0^{th}$ or $1^{st}$order disconinuities (but it's second derivative is discontinuous).\n\nIn general, if you make sure your pre-sampling function has it's first $n-1$ derivatives being continuous (and it's $n^{th}$ is disconinuous), then the amplitude of it's high frequencies will fall of like $\\left(\\frac{1}{f}\\right)^{n+1}$\n\nI hope this helps...\n\nBuilding on your comment about assigning 1 to pixels inside the circle and 0 to pixels entirely outside, with values in between for ones that are on the edge, I would try this:\n\nCreate you circle as 10N*10N instead of N*N. Maybe even use a higher value in place of 10. Draw your circle with your naive algorithm - 1 inside, 0 outside. Now scale this image down from 10N*10N to N*N. This will automatically give you fractional values for the pixels that are on the border at the lower resolution. You could also at this point split it into 0 and 1 again with a threshold function. Doing that with 4*4 gives you something like this (colors inverted:)", null, "You will need to use continuous values if you want a better approximation on small images. As Paul R mentioned a natural choice here would be to choose the ratio of each pixel's area inside the circle to the area of the pixel.\n\nYou could perform an integral to give you the area of a pixel inside the circle, however, a Riemann sum is much less tedious. Partition each pixel into a grid of N x N equal-sized squares. If the center of a square is inside the circle then its area is added to the sum.\n\nEdit:\nThe discrete pixel coordinates represent the center of a $1 \\times 1$ square in the Cartesian coordinate system. The origin of the coordinate system is chosen to be the center of the circle.\n\nCoordinate system: pillbox-3x3-grid\nLabeled diagram: pillbox-3x3-labelled\n\nEverything is labeled using $(x,y)$ Cartesian coordinates to make the math consistent, however, since a circle is symmetric you don't need to worry about it. The relationship is of course that $(x,y)$ coordinates correspond to $[-y,x]$ pixel coordinates.\n\nFrom the diagram you can set up the appropriate integrations to find out the areas within each pixel's box. Then you just need to normalize.\n\nIf you need to create pillboxes in code then it would be simpler to use a Riemann sum albeit at the cost of computational time. From a signal processing standpoint this Riemann sum would be equivalent to:\n\n$Upsample \\rightarrow Calculation \\rightarrow Downsample$" ]

[ null, "https://i.stack.imgur.com/cHscy.png", null ]

[ "# MySQL SUBSTR Function | Extract Substring from String\n\nMySQL SUBSTR is an inbuilt string Function in MySQL. It is used to extract a substring from a string. We will discuss the MySQL SUBSTR Function in this article. Also, we will discuss a few examples to demonstrate it’s usage.\n\n## Syntax\n\nSUBSTR(stringstartlength)\n\n## Parameters\n\nThe MySQL SUBSTR Function need three parameters. Two parameters are mandatory and the third one is optional. The description of the parameters is as follows:\n\n• string: The first parameter is the string to extract from. It is a mandatory parameter.\n• start: The second parameter is the starting point of extraction. If the parameter is positive then the function extracts from the beginning of the string. Similarly, if it is negative then the function extracts from the end.\n• length: The third parameter is the length of characters to extract. However, if you don’t pass it then the function returns the whole string from start point.\n\n## Return Value\n\nThe MySQL SUBSTR Function returns a substring from a string with the given length and beginning from the starting point.\n\n## Examples\n\nLet’s discuss a few examples to use to SUBSTR Function.\n\n#### Example 1: Substring with Start Point\n\n```#Substring beginning from 10th character\nmysql> SELECT SUBSTRING('Building Knowledge Together', 10);\n+----------------------------------------------+\n| SUBSTRING('Building Knowledge Together', 10) |\n+----------------------------------------------+\n| Knowledge Together |\n+----------------------------------------------+\n\n#Substring beginning from the 8th character from the end\nmysql> SELECT SUBSTRING('Building Knowledge Together', -8);\n+----------------------------------------------+\n| SUBSTRING('Building Knowledge Together', -8) |\n+----------------------------------------------+\n| Together |\n+----------------------------------------------+\n\n```\n\nIn the above example, we pass the 10 and -8 to the MySQL SUBSTR. When start point is 10, the function slices the string from 10th character up to the end of the string. Also, when the start point is -8, the function slices the string beginning from the 8th character from the last.\n\n#### Example 2: Substring with Start Point and Length\n\nFor instance, consider the following example with start point and length.\n\n```#Start Point 10 and length is 9 characters\nmysql> SELECT SUBSTR('Building Knowledge Together', 10, 9);\n+-------------------------------------------------+\n| SUBSTR('Building Knowledge Together', 10, 9) |\n+-------------------------------------------------+\n| Knowledge |\n+-------------------------------------------------+\n1 row in set (0.00 sec)\n\n#Start Point -8 and length 8\nmysql> SELECT SUBSTR('Building Knowledge Together', -8, 8);\n+-------------------------------------------------+\n| SUBSTR('Building Knowledge Together', -8, 8) |\n+-------------------------------------------------+\n| Together |\n+-------------------------------------------------+\n1 row in set (0.00 sec)```\n\nAgain, in the above example the start points are 10 and -8 respectively. Also, when the length is passed, the MySQL SUBSTR extracts the string containing the provided number of characters.\n\n#### Example 3: Using with Concat Function\n\nIf you want to show three dots (…) after selecting some characters from the database, you can use the Substring Function along with Concat Function. This is shown in the example below.\n\n```mysql> SELECT CONCAT(SUBSTRING('Building Knowledge Together', 1, 18), '...');\n+----------------------------------------------------------------+\n| CONCAT(SUBSTRING('Building Knowledge Together', 1, 18), '...') |\n+----------------------------------------------------------------+\n| Building Knowledge... |\n+----------------------------------------------------------------+```\n\n## Conclusion\n\nWe discussed the MySQL SUBSTR in this article. You can also read more about MySQL String Functions on Concatly. Also, you can read more about SUBSTR on Official MySQL Documentation.\n\nAlso, MySQL SUBSTRING and MID Function equal to SUBSTR Function." ]

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[ "Cody\n\n# Problem 2020. Area of an Isoceles Triangle\n\nSolution 2101934\n\nSubmitted on 22 Jan 2020 by Matteo Tufi\nThis solution is locked. To view this solution, you need to provide a solution of the same size or smaller.\n\n### Test Suite\n\nTest Status Code Input and Output\n1   Pass\nx = 5; y = 8; A_correct = 12; tolerance = 1e-12; assert(abs(isocelesArea(x,y)-A_correct)<tolerance)\n\n2   Pass\nx = 2; y = 2; A_correct = sqrt(3); tolerance = 1e-12; assert(abs(isocelesArea(x,y)-A_correct)<tolerance)\n\n3   Pass\nx = 10; y = 2; A_correct = sqrt(99); tolerance = 1e-12; assert(abs(isocelesArea(x,y)-A_correct)<tolerance)" ]

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[ "# Asymptote: Graphing\n\n(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)\n\nNote These example are for the AoPS forums, If you want to use them on your computer, you must \"import graph;\".\n\nThere are two parts to a graph.\n\nThe axes' codes:\n\n```Label f;\nf.p=fontsize(6);\nxaxis(-8,8,Ticks(f, 2.0));\nyaxis(-8,8,Ticks(f, 2.0));\n```\n\nThis means ticks at an interval of 2.0, between -8 and 8 for both the x and y axes.\n\nAnd the graph code:\n\n```real f(real x)\n{\nreturn x^3;\n}\ndraw(graph(f,-2,2));```\n\nThe return", null, "$x^3$ is the function of the graph, and the draw command draws the graph from the", null, "$x$ coordinate of", null, "$x = - 2$ to", null, "$x = 2$.", null, "$[asy] import graph; size(300); Label f; f.p=fontsize(6); xaxis(-8,8,Ticks(f, 2.0)); yaxis(-8,8,Ticks(f, 2.0)); real f(real x) { return x^3; } draw(graph(f,-2,2)); [/asy]$\n\nWe can also alter the ticks code. For example, if we want intervals of 2 labeled, then 0.5 in between each one, we can use the following code:\n\n```Label f;\nf.p=fontsize(6);\nxaxis(-8,8,Ticks(f, 2.0,0.5));\nyaxis(-8,8,Ticks(f, 2.0,0.5));\n```", null, "$[asy] Label f; f.p=fontsize(6); xaxis(-8,8,Ticks(f, 2.0,0.5)); yaxis(-8,8,Ticks(f, 2.0,0.5)); [/asy]$ This graph, like with any draw command, can have several attributes fixed to it, such as color:", null, "$[asy] import graph; size(300); Label f; f.p=fontsize(6); xaxis(-8,8,Ticks(f, 2.0)); yaxis(-8,8,Ticks(f, 2.0)); real f(real x) { return x^3; } draw(graph(f,-2,2),green+linewidth(1)); [/asy]$\n\n```import graph;\nsize(300);\nLabel f;\nf.p=fontsize(6);\nxaxis(-8,8,Ticks(f, 2.0));\nyaxis(-8,8,Ticks(f, 2.0));\nreal f(real x)\n{\nreturn x^3;\n}\ndraw(graph(f,-2,2),green+linewidth(1));\n```\n\nRemember, you can still draw normal functions, so you can create lines, circles and ellipses." ]

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[ "# Using Multiple Columns in a Lookup to return a single value\n\n1. ## Using Multiple Columns in a Lookup to return a single value\n\nHi.\n\nI have two worksheets, and each of the worksheets have 5 of the same columns. I want to take the 5 values (in the 5 different columns) and then check for a match in the other worksheet, and if there is a match, then return a value from a 6th column in the 2nd worksheet.\n\nHow would you do it?\n\nTHanks", null, "", null, "Register To Reply\n\n2. I'm no pro at this but it appears to work. There has to be a better way though\n\n=IF(SUM(IF(Sheet1!A1:E1 = Sheet2!A1:E1,0,1)) = 0,Sheet2!F1,\"\")\n\nThis compares the values in A1 to E1 from Sheet1 to the values in A1 to E1 in Sheet2. If they match, it takes the value in F1.\n\nIt basically sums cells that don't match and checks to see if its > 0. If the sum is 0 (all match) then it returns F1.\n\nHTH\n\nJason", null, "", null, "Register To Reply\n\n3. Thanks. This won't work for me. The data from one sheet needs to be looked up in the second sheet. I'm trying to take 5 different values from one sheet and then use those 5 values to do a lookup in the second sheet. Can you do a lookup using multiple lookup values?", null, "", null, "Register To Reply" ]

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[ "## Topics in Theory of Surfaces in Elliptic Space – Pogorelov\n\nIn this post, we will see the book Topics In Theory Of Surfaces In Elliptic Space by A. V. Pogorelov.", null, "# About the book\n\nThis book deals with the solution of a number of problems in the theory of surfaces in elliptic space,through consideration of isometric surfaces. The principal method of investigation is comparison of a pair of isometric figures in elliptic space with a pair of isometric figures in a Euclidean space which corresponds geodesically to the elliptic space. This enables us to transpose the main difficulties in the proof to Euclidean space, where they can be overcome by the appropriate theorems.\n\nThe book was translated from Russian by Rogey and Royer Inc. and was Richard Sacksteder edited by  published in 1961.\n\nCredits to original uploader.\n\nYou can get the book here.\n\nFollow us on The Internet Archive: https://archive.org/details/@mirtitles\n\nWrite to us: [email protected]\n\nFork us at GitLab: https://gitlab.com/mirtitles/\n\nAdd new entries to the detailed book catalog here.\n\n# Contents\n\n## Chapter I. Elliptic Space 1\n\n1. Four-dimensional vector space 1\n2. The concept of elliptic space 5\n3. Curves in elliptic space 9\n4. Surfaces in elliptic space 14\n5. Fundamental equations in the theory of surfaces in elliptic space 18\n\n## Chapter II. Convex Bodies and Convex Surfaces in Elliptic Space 23\n\n1. The concept of a convex body 23\n2. Convex surfaces in elliptic space 35\n3. The deviation of a segment on a convex surface from its semitangent at its initial point 30\n4. Manifolds of curvature not less than K.A.D. Aleksandrov’s theorem 34\n\n## Chapter III. Transformation of Congruent Figures 41\n\n1. Transformation of congruent figures in elliptic space to congruent figures in Euclidean space 41\n2. Transformation of congruent figures in Euclidean space onto congruent figures in elliptic space 45\n3. Transformation by infinitesimal motions 49\n4. Transformation of straight lines and Planes 53\n\n## Chapter IV. Isometric Surfaces 59\n\n1. Transformation of isometric surfaces 59\n2. Transformation of locally convex isometric surfaces in elliptic space 63\n3. Proof of lemma 1 67\n4, Transformation of locally convex isometric surfaces in Euclidean space. 71\n5. Proof of lemma 2 74\n\n## Chapter V. Infinitesimal Deformations of Surfaces in Elliptic Space 77\n\n1. Pairs of isometric surfaces and infinitesimal deformations 77\n2. Transformation of surfaces and their infinitesimal deformations 81\n3. Some theorems on infinitesimal deformations of surfaces in elliptic space. 85\n\n## Chapter VI. Single-Value Definiteness of General Convex Surfaces in Elliptic Space 89\n\n1. A lemma on rib points on a convex surface 89\n2. Transformation of isometric dihedral angles and cones 92\n3. Local convexity of the surfaces 𝜙_1 and 𝜙_2 at smooth points 97\n4, Convexity and isometry of the surfaces 𝜙_1 and 𝜙_2 101\n5. Various theorems on uniqueness of convex surfaces in elliptic space 105\n\n## Chapter VIL. Regularity of Convex Surfaces with a Regular Metric 113\n\n1. The deformation equation for surfaces in elliptic space 113\n2. Evaluation of the normal curvatures of a regular convex cap in elliptic space 117\n3. Convex surfaces of bounded specific curvature in elliptic space 122\n4. Proof of the regularity of convex surfaces with a regular metric in elliptic space 125\n\nBibliography 131", null, "## About The Mitr\n\nI am The Mitr, The Friend\nThis entry was posted in books, mathematics, soviet and tagged , , , , , , , , , . Bookmark the permalink.\n\n### 1 Response to Topics in Theory of Surfaces in Elliptic Space – Pogorelov\n\nThis site uses Akismet to reduce spam. Learn how your comment data is processed." ]

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[ "# Maximizing general first Zagreb and sum-connectivity indices for unicyclic graphs with given independence number\n\n• Ioan Tomescu University of Bucharest, Romania\nKeywords: Unicyclic graph, independence number, general first Zagreb index, general sum-connectivity number, spider graph, Jensen inequality\n\n### Abstract\n\nIn this paper it is shown that in the class of unicyclic graphs of order n and independence number s, the spider graph SΔ(n, s) is the unique graph maximizing general first Zagreb index 0Rα(G) for α > 1 and general sum-connectivity index χα(G) for α ≥ 1.\n\nPublished\n2018-08-08\nIssue\nSection\nArticles" ]

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[ "# Match values in 2 excel sheets\n\nExcel match\n\n## Match values in 2 excel sheets\n\nIs there a formula I can use to. I was comparing MAC addresses ( for IP phones) from a list I had sheets off one server to one that had the MAC addresses with the. Look up values with VLOOKUP INDEX, MATCH. Another option excel is Index Match, which can be excel used no matter the order ( As long as the values in column a sheet 1 are unique) This is what you would put in column B on sheet 2. I have two different work sheets ( say F1 excel F2) with last name in Column A first name in Column B. Mar 15, · Matching values from 2 different sheets to return 3rd value Become a Registered Member ( free) to remove the ads that appear in thread area. table and paste it into cell A1 on excel a blank worksheet in Excel.\n\nIf it does then it will take the information from column c on sheet 2 and place it. so when I vlookup i would get the 1st number it finds next to 42 for every row that is 42 but I need to find all the different values that are next to 42. A: B refers to columns A & b on Sheet 2. I' ve been trying to read up on how to do this, but I' m not sure if I. active oldest votes. Comparing columns in two different excel sheets and if they match copying third column.\nOct 17 · Match Values on two sheets Switch to sheet 2 Check Columns A & B to see if it finds a match. Match values in 2 excel sheets. I was trying to match one value in a range of values and return an associated value from a different column of the matched value. Click here to reset your password. 98 Responses to \" How to compare two Excel files for differences\" Jaime Lechuga says: February 27, at 3: 22 pm. in column A each row is a number eg 42 43 then in column B there is a different number for excel each row eg 42- 333, 42- 345, 43, 42, 42, 42- 678 43- 999. Match values in 2 excel sheets. Vlookup is good if the reference values ( column A, sheet 1) are in ascending order. Thank you for making great content!\n\nI' ve sheets tried \" How to compare two Excel sheets for differences in values\" and \" Highlight differences between 2 sheets with conditional formatting. Lookup values across multiple worksheets: VLOOKUP & INDEX MATCH in Excel. How to excel compare two columns in Excel ( from different sheets) excel and copy values from excel a corresponding column if the first two columns match? How to find matching values between 2 excel sheets. The other approach uses INDEX & MATCH Excel Table names references. match will match the value provide by the first argument.\nexcel I want to know if the value in Column B match' s on both sheets against column A. I need to compare the names and if the names match I need to copy the corresponding email address in to F1. up vote 5 down vote. I have 2 columns A in sheet X , 2 columns A , B in sheet Y, B column A has a few different values but some are the same eg row 1= 42 row 2= 42 row 3= sheets 43. If I find a match in SP2 I need to find a value starting with ' TC_ ' from the previous rows excel , get the corresponding value in column B of SP2 paste in column F of SP1. The 2 tells Excel which column in the lookup range to return the FALSE forces an exact excel match. I need to match values in column E of SP1 with values in column A of SP2. You can get a more full explanation in my article:. Hi I have two spreadsheets SP1 SP2.\n\nthen copy corresponding values in column b from sheet 1 to column b excel in Sheet 2 ( where the column a values match). The email address are present in Column F of F2.\n\n## Excel match\n\nHow to match 2 row of data in excel? - Forum - Excel Excel if 2 columns match copy another cell - Forum - Excel Match data from two excel sheets - Forum - Excel. Match Values on two sheets So I' m trying to take the value found in Sheet 1 starting at A2. Switch to sheet 2 and Check Columns A & B to see if it finds a match.\n\n``match values in 2 excel sheets``\n\nComparing two Excel sheets and combining the unique data can be done with the help of macros. There are several macros available for free that can be downloaded from the internet and tweaked according to the requirements." ]

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[ "The MOSFET is considered one of the most powerful semiconductor devices in electronics, in this article we explain the possibility of parallel MOSFET.\n\nThe parallel MOSFET connection is the perfect way to higher power load such as motors and transformers, in this post we try to explain some main topics about the parallel MOSFET network.\n\n## How to parallel MOSFET\n\nThe paralleling of MOSFET is network connection much similar to the generators in parallel and transformers in parallel at power generation stations, at power generation the parallel network is used to share the load, on parallel MOSFET circuit have the same application which is load sharing.\n\nMultiple identical MOSFETs is been connected in parallel with the same gate driver resistor for each MOSFET, this is how a parallel MOSFET network looks like.\n\n## Parallel MOSFET circuit\n\nThe figure shows the parallel MOSFET circuit model, the drain terminal and source terminal each of MOSFETs is connected to the bus, as same as the parallel generator.\n\nEach of the MOSFETs and resistors needs to be identical because on the load exchanging purpose this must be fixed criteria.\n\n## Parallel MOSFET gate resistor\n\nFor the parallel MOSFET network, we need to connect the same resistance value to the gate terminal for gate driving purposes. Using this we can trigger all the MOSFET at the perfect same time.\n\nThe MOSFET triggering and current flowing are at the same time, this will enable lower current for each MOSFET load or evenly distributed the load.\n\nThey all have the same ON-state resistance and it will reduce the amount of heat generation at the whole circuit." ]

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[ "publicité", null, "```Jestr\nJOURNAL OF\nJournal of Engineering Science and Technology Review 10 (2) (2017) 8- 13\nEngineering Science and\nTechnology Review\nResearch Article\nwww.jestr.org\nModeling of Photovoltaic Panel by using Proteus\nSaad Motahhir*, Abdelilah Chalh, Abdelaziz El Ghzizal, Souad Sebti and Aziz Derouich\nLaboratory of Production engineering, Energy and Sustainable Development, Smart Energy Systems and Information Processing\nResearch Team, higher school of technology, USMBA University, Fez, Morocco\nReceived 16 April 2017; Accepted 15 May 2017\n________________________________________________________________________________________\nAbstract\nThis paper focuses on a Proteus Spice model of the photovoltaic Panel. This model is based on a mathematical equation\nwhich is got from the equivalent circuit of the photovoltaic Panel; it includes a photocurrent source, a diode, a series\nresistor and a shunt resistor. Next, this model is validated by comparing its data with the experimental data. In addition,\nsince Proteus provides in its library different microcontrollers and electronic boards , this model is connected to the\nArduino UNO Board through the voltage and current sensors, that in order to acquire and supervise the photovoltaic\nvoltage, current and power. And for experimental validation, a prototype using real components has been developed.\nKeywords: PV panel; Spice; Arduino UNO; Proteus; Solar Energy Measurement\n_____________________________________________________________________________________________\n1. Introduction\nThe growing concern about environmental issues and the\nstrong need of low-cost energy sources have engendered a\ngreat importance in the use of renewable energy such as the\nPV solar energy. Currently, PV solar energy has become a\nperfect promising renewable energy source due to its many\nbenefits namely; low cost maintenance, non-necessity of\nmoving and no pollution. However, the low efficiency of a\nPV panel and high cost of PV system installation may be a\ndiscouraging factor as far as its use . Moreover, the nonlinear comportment and heavy dependence of PV modules\non the solar irradiation and temperature poses important\nchallenges for researchers in PV solar energy topic [2, 3].\nTo avoid these limitations, the operation of the PV panel\nat the MPP is a requirement which can improve the\nefficiency of the PV system. For this reason MPPT\nalgorithm is used -. And in order to test the\nperformance of such MPPT algorithm, the modeling of PV\npanel should be done. Therefore several researchers have\nmodeled the PV panel either in Matlab/Simulink or PSIM\ntools -; however these tools don’t provide a\nmicrocontroller or an electronic board in which our\nalgorithm can be implemented and tested. Consequently, we\ncan’t rely on these tools to validate our algorithm. On the\nother hand, Proteus is the best simulation software for\nvarious designs with electronics and microcontroller. It is\nmainly popular because of availability of almost all\nmicrocontrollers in it. However, it does not contain a PV\npanel model. Hence, in this paper a PV panel model under\nProteus tool is proposed as an alternative, and this model is\nvalidated by comparing its data with experimental data.\nHence, by using Proteus we can implement our algorithm\nunder a real development board. Therefore, the PV panel is\nconnected to the Arduino UNO Board through the voltage\nand current sensors, that in order to acquire and supervise\n______________\nthe photovoltaic voltage, current and power. And to validate\nthe functionality and performance of the developed Solar\nEnergy Measurement System, a prototype using real\ncomponents has been developed.\n2. PV Panel model\nA PV panel is a component which can convert a solar energy\ninto direct current electricity using semiconducting\nmaterials that exhibit the PV effect. The equivalent circuit of\nthe PV panel is shown in Fig. 1[12, 13].\nFig. 1.PV cell equivalent circuit\nAs presented in fig. 1, the equivalent circuit of the PV\npanel contains a current source, a diode, a shunt resistor and\na series resistor. The current generated by the PV panel can\nbe given as [12, 13]:\n⎛\nq(V + Rs I) ⎞ (V + IRs )\nI = I ph - I 0 ⎜ exp\n-1⎟ aKTN s\nRsh\n⎝\n⎠\nIn this study the MSX-60 panel is used and table I\npresents its specification :\n(1)\nSaad Motahhir, Abdelilah Chalh, Abdelaziz El Ghzizal, Souad Sebti and Aziz Derouich/\nJournal of Engineering Science and Technology Review 10 (2) (2017) 8-13\nThe number of cells\n36\nTable 1. Specifications of the PV module Solarex MSX-60\nat STC\nCharacteristics\nMSX-60\nLight-generated current Iph\n3.8128 A\nDiode saturation current I0\n2.5245e-10 A\nMaximum power, Pmax\n60W\nIdeality factor\n0.9784\nVoltage at Pmax,Vmp\n17.1V\nShunt Resistance Rsh\n153.5644Ω\nCurrent at Pmax, Imp\n3.5A\nSeries Resistance Rs\n0.38572Ω\nShort-circuit current, Isc\n3.8A\nOpen-circuit voltage, Vco\n21.1V\nTemperature coefficient of\nopen-circuit voltage Voc, Kv\n-80mV/°C\nTemperature coefficient of\nshort-circuit current Isc, Ki\n2.4mA/°C\nIn order to model a PV panel in Proteus tool, its\nequivalent circuit is done with a controlled current source\nand a diode with modified Spice code, that in order to design\na real model of PV panel. Fig. 2 presents the Proteus model\nand its Spice code.\nFig. 2. The PV panel model under Proteus\nAs shown in Fig. 2, in order to model a PV panel in Proteus\ntool, the below steps are followed:\nis set to 35.09424 which is the multiplication\nbetween the ideality factor and number of cells.\n3. Two resistors are used to model the shunt resistor\nand the series resistor with the values mentioned\nin table I.\n4. A “DC Voltage Source” block is connected to the\nPV panel model as a variable load. Its value is\nequal to the “Sweep variable” value of the “DC\nSWEEP ANALYSIS” graph used in order to\nsimulate our model as shown in fig. 3, note that\nthe range of “Sweep variable” variable must be\nbetween 0V and the open-circuit voltage.\n1. A “Voltage Controlled Current Source” block\ncontrolled by “DC Voltage Source” block is\nused to model the Current Source. For example\nto simulate our model under STC, the value of\n“DC Voltage Source” block is set to 3.8128 V\nvalue, which is the photocurrent of MSX-60\npanel under STC.\n2. As shown in fig.2, a diode with modified spice\ncode is used in this model, because it is required\nto change the values of the saturation current Is,\nthe ideality factor, number of cells and bandgap\nenergy in the Spice code according to the\nspecification of MSX-60 panel . Note that N\nThe simulation of the PV panel in ISIS Proteus is\npresented in Fig. 3\n9\nSaad Motahhir, Abdelilah Chalh, Abdelaziz El Ghzizal, Souad Sebti and Aziz Derouich/\nJournal of Engineering Science and Technology Review 10 (2) (2017) 8-13\nFig. 3. I-V and P-V characteristics for PV panel by using Proteus\nFig. 4 shows the I–V and P-V characteristics obtained by our\nmodel and experimental data at STC. Experimental data are\nextracted from the datasheet . And as shown in fig. 4,\nthe model accurately is in accordance with the experimental\ndata both in the power and the current characteristics.\nFig 4. I–V and P-V model curves and experimental data of Solarex\nMSX-60 module\nAs presented in fig. 5, the Proteus PV model is put in a\nsub-circuit in order to make it easy to use.\nFig. 5. The sub circuit of the PV panel model under Proteus\nModeling a PV panel in Proteus tool allows controlling our\nPV system by microcontroller, microprocessor, DSP, and\nFPGA. Therefore, the performance obtained will be similar to\nthe performance obtained during real experience. One of the\naims of this study is to acquire and supervise the current,\nvoltage and power of our PV panel by using Arduino and\nProteus.\n3. Solar Energy Measurement\n3.1. Materials used\nIn order to supervise the energy of our panel, different\ncomponents are required apart PV panel such as current\nsensor, voltage sensor, development board and LCD.\nBoard: the development board used in this paper is Arduino\nUNO, in which the ATMega328 microcontroller is\nintegrated. It is a low cost board.\nVoltage sensor: it is used to reduce the PV voltage to\nanother voltage between (Vd) [0, 5] which can be supported\n10\nSaad Motahhir, Abdelilah Chalh, Abdelaziz El Ghzizal, Souad Sebti and Aziz Derouich/\nJournal of Engineering Science and Technology Review 10 (2) (2017) 8-13\nby Arduino, because Arduino cannot read voltage more than\n5 V, therefore the voltage divider circuit as presented in fig. 6\nis used.\nand the Load (fig. 8), so the current through this resistor is\nthe PV current, therefore the operational amplifier subtractor\ncircuit is used to compute the voltage across this resistor,\nand this voltage is provided to the Arduino in order to get\nthe value of PV the panel current. Note that R1 should be\nsmall in order to minimize the loss of energy (as the Ampere\nMeter).\nFig. 6. Voltage divider circuit\nThe design of resistors R3 and R4 is made as follows:\nFig. 7. Operational amplifier subtractor circuit\nFrom fig.6,\nVd = (R3 / R3 + R 4 ) *V\nLCD screen: it is used to display values of PV voltage,\ncurrent and power.\n(2)\nSince the range of PV panel voltage (V) is [0, 21.1],\ntherefore the divider's voltage ratio should be lower or equal\n0.23, hence, in order to make simple the choice of resistors, a\ndivider's voltage ratio equal to 0.2 is selected. Another\nrequirement is that the resistors should be high in order to\nminimize the loss of energy (as the voltmeter). Therefore, R3\nis selected with a value equals to 25kΩ and R4 is selected\nwith a value equals to 100kΩ.\n4. Results and Discussion\nAs shown in fig. 8, the PV panel is connected to the load and\nthe Arduino board acquires the PV voltage and current from\nvoltage and current sensors in order to use them to compute\nPV power. And then, the PV power, voltage and current are\ndisplayed on the LCD screen. And as presented in fig. 8, the\nsame values displayed on Ampere Meter and Voltmeter are\ndisplayed on LCD Screen.\nCurrent sensor: it is used to provide to the Arduino the\nimage of the PV panel current. In this study we design a\ncurrent sensor as presented in fig. 7. The design of this\nsensor consists of putting a resistor (R1) between the panel\nFig. 8. Solar Energy Measurement Using Arduino and Proteus\n11\nSaad Motahhir, Abdelilah Chalh, Abdelaziz El Ghzizal, Souad Sebti and Aziz Derouich/\nJournal of Engineering Science and Technology Review 10 (2) (2017) 8-13\nTo validate the functionality and performance of the\ndeveloped Solar Energy Measurement System, a prototype\nby using real components has been developed as shown in\nfig. 9, and the experiment is performed using the artificial\ninsolation with the help of lamps. Note that the same Arduino\ncode used in Proteus is used in the experiment, and this is the\nbenefit of using Proteus in simulation instead of PSIM and\nMatlab/Simulink, because by using PSIM and Matlab we\nmust again write the code of our algorithm once we start the\nexperiment. Another benefit is that if our system is tested by\nusing Proteus and it gives good performance, it will probably\ngive the same result in the experiment, because we use the\nsame components and Arduino code in simulation and\nexperiment.\nFig. 9. Experimental setup of the developed Solar Energy Measurement System\n5. Conclusion\nIn this paper, a Proteus Spice model of the photovoltaic\nPanel is made, and it is validated by comparing its data with\nexperimental data, hence the model is in accordance with\nexperimental data. As a result, a Solar Energy Measurement\nSystem is done by using electronic board provided by\nProteus (Arduino UNO), in which our algorithm is\nimplemented and tested. And to validate the functionality\nand performance of our system, a prototype by using real\ncomponents has been developed, and the same Arduino code\nused in Proteus is used in the experiment. That can decrease\nthe time spent in debugging runtime errors during the\nexperiment, and this is the benefit of using Proteus in\nby using PSIM and Matlab we must again write the code of\nour algorithm once we start the experiment. Another benefit\nis that if our system is tested by using Proteus and it gives\ngood performance, it will probably give the same result in\nthe experiment, because we use the same components and\nArduino code in simulation and experiment\nThis is an Open Access article distributed under the terms of the\n______________________________\nReferences\n1.\n2.\n3.\n4.\nLiserre, M., Sauter, T., Hung, J.Y.,“Future energy systems:\nintegrating renewable energy sources into the smart power grid\nthrough industrial electronics”, IEEE Industrial Electronics\nMagazine, Vol. 4, No.1, pp. 18-37, 2010\nHohm, D.P., Ropp, M.E., “Comparative study of maximum power\npoint tracking algorithms”, PROGRESS IN PHOTOVOLTAICS,\nVol. 11, No. 1, pp. 47-62, 2003\nDerouich, “Shading effect to energy withdrawn from the\nphotovoltaic panel and implementation of DMPPT using C\nlanguage”, International review of automatic control, Vol. 9, No.\n2, pp. 88-94, 2016.\nSera, D., Mathe, L., Kerekes, T., et al. “On the perturb-and-observe\nand incremental conductance MPPT methods for PV systems”,\nIEEE Journal of Photovoltaic, Vol. 3, No. 3, pp. 1070-1078, 2013\n5.\n6.\n7.\n8.\n12\nElgendy, M.A., Zahawi, B., Atkinson, D.J., “Assessment of perturb\nand observe MPPT algorithm implementation techniques for PV\npumping applications”, IEEE Transactions on Sustainable\nEnergy, Vol. 3, No. 1, pp. 21-33, 2012\nDerouich, “Proposal and Implementation of a novel perturb and\nobserve algorithm using embedded software”, IEEE International\nRenewable and Sustainable Energy Conference, pp.1-5, 2015.\nSekhar, P.C., Mishra, S., “Takagi–Sugeno fuzzy-based incremental\nconductance algorithm for maximum power point tracking of a\nphotovoltaic generating system”, IET Renewable Power\nGeneration, Vol. 8, No. 8, pp. 900-914, 2014\nLopez-Guede JM, et al., “Systematic modeling of photovoltaic\nmodules based on artificial neural networks”, International\nJournal of Hydrogen Energy, Vol. 41, No.29, pp. 12672-12687,\n2016\nSaad Motahhir, Abdelilah Chalh, Abdelaziz El Ghzizal, Souad Sebti and Aziz Derouich/\nJournal of Engineering Science and Technology Review 10 (2) (2017) 8-13\n9. Chia-Hung Lin, Cong-Hui Huang, Yi-Chun Du, Jian-Liung Chen,\nInternational Workshop on Pedagogic Approaches & E-Learning\n“Maximum photovoltaic power tracking for the PV array using\n(APEL 2015).\nthe fractional-order incremental conductance method”, Applied\n12. M. G. Villalva, J. R. Gazoli, E. Ruppert F., “MODELING AND\nEnergy, Vol. 88, pp. 4840-4847, 2011\nCIRCUIT-BASED SIMULATION OF PHOTOVOLTAIC\n10. H. Patel and V. Agarwal, “MATLAB-based modeling to study the\nARRAYS”, Brazilian Journal of Power Electronics, Vol. 14, No.\neffects of partial shading on PV array characteristics”, IEEE\n1, pp. 35-45, 2009.\nTransaction Energy Conversion., Vol. 23, No. 1, pp. 302-310,\n13. Saad MOTAHHIR, Abdelaziz El Ghzizal, Aziz Derouich,\n2008.\nModélisation et commande d’un panneau photovoltaïque dans\nl’environnement PSIM, Congrès International de Génie Industriel\nDerouich, Une ressource pédagogique pour l'enseignement par\net Management des Systèmes CIGIMS 2015.\nsimulation : cas des panneaux photovoltaïques, Proceedings of\n14. Solarex MSX60 and MSX64 photovoltaic panel, datasheet.\nNomenclatures\n•\na : diode’s ideality factor;\n•\nI : panel output current;\n•\nI0 : diode saturation current;\n•\nIph : panel photocurrent;\n•\nK : constant of Boltzmann;\n•\nNs : number of cells connected in series;\n•\nq : electron charge;\n•\nRs : series resistance;\n•\nRsh : shunt resistance;\n•\nT: junction temperature;\n•\nV : panel output voltage;\n•\nVd: output of voltage divider circuit.\nAbbreviations\n•\nSTC: standard test condition.\n•\nPV : Photovoltaic;\n•\nMPP : maximum power point;\n•\nMPPT : maximum power point tracking;\n•\nLCD: Liquid Crystal Display.\n13\nData of this paper:\nhttps://data.mendeley.com/datasets/dh8rhvkr64/4\nFor more papers and works please visit:\nMore papers and works:\n Une ressource pédagogique pour l'enseignement par simulation\n: cas des panneaux photovoltaïques:\nPaper:\nulation_cas_des_panneaux_photovolta%C3%AFques\nData:\n Proposal and implementation of a novel perturb and observe algorithm using embedded\nsoftware:\nPaper:\nhttps://www.researchgate.net/publication/301912126_Proposal_and_implementation_of_a_novel_perturb_a\nnd_observe_algorithm_using_embedded_software\nData:\nhttps://data.mendeley.com/datasets/m5yv5r2d6k/3\n MIL and SIL and PIL tests for MPPT algorithm\nPaper:\n: https://www.researchgate.net/publication/319684243_MIL_and_SIL_and_PIL_tests_for_MPPT_algorithm\nhttp://www.tandfonline.com/doi/full/10.1080/23311916.2017.1378475\nData:\nhttps://fr.mathworks.com/matlabcentral/fileexchange/64452-data-of-mil-and-sil-and-pil-tests-for-mpptalgorithm-paper\n Shading Effect to Energy Withdrawn from the Photovoltaic Panel and Implementation of\nDMPPT Using C Language\nPaper:" ]

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[ "", null, "Shortcut Component - Maple Help\n\nDocumentTools[Components]\n\n Shortcut\n generate XML for a Shortcut Component", null, "Calling Sequence Shortcut( caption, opts )", null, "Parameters\n\n caption - (optional) string:=\"\"; the caption on the component opts - (optional) ; one or more keyword options as described below", null, "Options\n\n • enabled : truefalse; Indicates whether the component is enabled. The default is true. If enabled is false then the inserted component is grayed out and interaction with it cannot be initiated.\n • height : posint; The height in pixels of the component.\n • identity : {name,string}; The reference name of the component.\n • image : {string,Matrix,Array}; Image to be displayed on the component, specified as either the name of an external image file or a Matrix or Array as recognized by commands in the ImageTools package. The caption parameter and the image option are mutually exclusive; only one can be supplied.\n • target : string; A string denotes the link target. See the section Valid Link Targets on the Shortcut Component help page for a table of allowed targets. The default is the empty string (no target prespecified).\n • tooltip : string; The text that appears when the mouse pointer hovers over the component. The default is the empty string (no tooltip).\n • visible : truefalse; Indicates whether the component is visible. The default is true.\n • width : posint; The width in pixels of the component.", null, "Description\n\n • The Shortcut command in the Component Constructors package returns an XML function call which represents a Shortcut Component.\n • The generated XML can be used with the results of commands in the Layout Constructors package to create an entire Worksheet or Document in XML form. Such a representation of a Worksheet or Document can be inserted into the current document using the InsertContent command.", null, "Examples\n\n > $\\mathrm{with}\\left(\\mathrm{DocumentTools}\\right):$\n > $\\mathrm{with}\\left(\\mathrm{DocumentTools}:-\\mathrm{Layout}\\right):$\n > $\\mathrm{with}\\left(\\mathrm{DocumentTools}:-\\mathrm{Components}\\right):$\n\nExecuting the Shortcut command produces a function call.\n\n > $S≔\\mathrm{Shortcut}\\left(\"My Shortcut\",\\mathrm{identity}=\"Shortcut0\"\\right)$\n ${S}{≔}{\\mathrm{_XML_EC-Shortcut}}{}\\left({\"id\"}{=}{\"Shortcut0\"}{,}{\"caption\"}{=}{\"My Shortcut\"}{,}{\"enabled\"}{=}{\"true\"}{,}{\"visible\"}{=}{\"true\"}\\right)$ (1)\n\nBy using commands from the Layout Constructors package a nested function call can be produced which represents a worksheet.\n\n > $\\mathrm{xml}≔\\mathrm{Worksheet}\\left(\\mathrm{Group}\\left(\\mathrm{Input}\\left(\\mathrm{Textfield}\\left(S\\right)\\right)\\right)\\right):$\n\nThat XML representation of a worksheet can be inserted directly.\n\n > $\\mathrm{InsertContent}\\left(\\mathrm{xml}\\right):$", null, "My Shortcut\n\n > $S≔\\mathrm{Shortcut}\\left(\"New Shortcut\",\\mathrm{identity}=\"Shortcut0\",\\mathrm{tooltip}=\"my shortcut\"\\right):$\n > $\\mathrm{xml}≔\\mathrm{Worksheet}\\left(\\mathrm{Group}\\left(\\mathrm{Input}\\left(\\mathrm{Textfield}\\left(S\\right)\\right)\\right)\\right):$\n\nThe previous example's call to the InsertContent command inserted a component with identity \"Shortcut0\", which still exists in this worksheet. Inserting additional content whose input contains another component with that same identity \"Shortcut0\" incurs a substitution of the input identity in order to avoid a conflict with the identity of the existing component.\n\nThe return value of the following call to InsertContent is a table which can be used to reference the substituted identity of the inserted component.\n\n > $\\mathrm{lookup}≔\\mathrm{InsertContent}\\left(\\mathrm{xml},\\mathrm{output}=\\mathrm{table}\\right)$\n ${\\mathrm{lookup}}{≔}{table}{}\\left(\\left[{\"Shortcut0\"}{=}{\"Shortcut1\"}\\right]\\right)$ (2)", null, "New Shortcut\n\n > $\\mathrm{lookup}\\left[\"Shortcut0\"\\right]$\n ${\"Shortcut1\"}$ (3)\n > $\\mathrm{GetProperty}\\left(\\mathrm{lookup}\\left[\"Shortcut0\"\\right],\\mathrm{caption}\\right)$\n ${\"New Shortcut\"}$ (4)\n\nThe next example generates and inserts a shortcut for launching help on the int command.\n\n > $L≔\\mathrm{Shortcut}\\left(\"Help for int\",\\mathrm{identity}=\"Shortcut17\",\\mathrm{target}=\"Help:int\"\\right):$\n > $\\mathrm{InsertContent}\\left(\\mathrm{Worksheet}\\left(\\mathrm{Group}\\left(\\mathrm{Input}\\left(\\mathrm{Textfield}\\left(L\\right)\\right)\\right)\\right)\\right):$\n\nThe following examples insert components with custom images.\n\n > $\\mathrm{img}≔\\mathrm{cat}\\left(\\mathrm{kernelopts}\\left(\\mathrm{datadir}\\right),\"/images/antennas.jpg\"\\right):$\n > $L≔\\mathrm{Shortcut}\\left(\"New Shortcut\",\\mathrm{identity}=\"Shortcut17\",\\mathrm{image}=\\mathrm{img},\\mathrm{width}=120,\\mathrm{height}=120\\right):$\n > $\\mathrm{InsertContent}\\left(\\mathrm{Worksheet}\\left(\\mathrm{Group}\\left(\\mathrm{Input}\\left(\\mathrm{Textfield}\\left(L\\right)\\right)\\right)\\right)\\right):$", null, "New Shortcut\n\n > $\\mathrm{img}≔\\mathrm{Array}\\left(1..60,1..40,1..3,\\left(i,j,k\\right)↦\\mathrm{if}\\left(k=3,\\frac{j}{40},0\\right),\\mathrm{datatype}=\\mathrm{float}\\left[8\\right],\\mathrm{order}=\\mathrm{C_order}\\right):$\n > $L≔\\mathrm{Shortcut}\\left(\"New Shortcut\",\\mathrm{identity}=\"Shortcut17\",\\mathrm{image}=\\mathrm{img}\\right):$\n > $\\mathrm{InsertContent}\\left(\\mathrm{Worksheet}\\left(\\mathrm{Group}\\left(\\mathrm{Input}\\left(\\mathrm{Textfield}\\left(L\\right)\\right)\\right)\\right)\\right):$", null, "New Shortcut\n\n > $L≔\\mathrm{Shortcut}\\left(\"New Shortcut\",\\mathrm{identity}=\"Shortcut17\",\\mathrm{image}=\\mathrm{img},\\mathrm{width}=70,\\mathrm{height}=20\\right):$\n > $\\mathrm{InsertContent}\\left(\\mathrm{Worksheet}\\left(\\mathrm{Group}\\left(\\mathrm{Input}\\left(\\mathrm{Textfield}\\left(L\\right)\\right)\\right)\\right)\\right):$", null, "New Shortcut", null, "Compatibility\n\n • The DocumentTools:-Components:-Shortcut command was introduced in Maple 2015." ]

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[ "## Overview\n\nVavr is a functional component library that provides persistent data types and functional control structures. I started using it recently, and really loved it! Its simplicity, immutable data types, and the functional programming concept are really remarkable. In this article, I will introduce Vavr’s List, by doing a comparison with the built-in Java List and its implementations in Java 8.\n\n• List creation\n• Get element\n• Update element\n• Delete element\n• List streaming\n• From Vavr to Java\n\nFor this article, I’m using Vavr 0.9.3 `io.vavr:vavr:0.9.3`.\n\n## List Creation\n\nIn Java 8, you can create a list by calling the constructor of any implementation of `java.util.List`. Or using a factory method which returns a list.\n\n``````// java.util.List\nList<String> animals = new ArrayList<>();\nList<String> another = new ArrayList<>(animals);\n``````\n``````List<String> animals = new LinkedList<>();\n``````\n``````List<String> animals = Arrays.asList(\"🐱\", \"🐶\");\n``````\n``````List<String> animals = Collections.singletonList(\"🐱\");\n``````\n``````List<String> animals = Collections.unmodifiableList(...);\n``````\n\nIn Vavr, you can create a list using the factory methods of interface `io.vavr.collection.List`:\n\n``````// io.vavr.collection.List\nList<String> animals = List.of(\"🐱\", \"🐶\");\nList<String> another = List.ofAll(animals);\nList<String> empty = List.empty();\n``````\n\nThere’re also other factory methods which allow you to create a list of primitives. But I won’t go into more detail here.\n\nIn Java, interface `java.util.List` defines method `add(E e)` for adding new element of type `E` into the existing list. Therefore, all the implementations of `List` must override the method `add`. The new element will be added at the end of the list.\n\n``````// java.util.List\nList<String> animals = new ArrayList<>();\n// \"🐱\", \"🐶\"\n``````\n``````List<String> animals = new LinkedList<>();\n// \"🐱\", \"🐶\"\n``````\n\nIn case of a read-only (immutable) list, an exception will throw when calling the add method, which is a side-effect. This is tricky because when using interface `List`, you don’t know if the underlying implementation is immutable.\n\n``````// java.util.List\nList<String> animals = Arrays.asList(\"🐱\", \"🐶\");\n// java.lang.UnsupportedOperationException\n``````\n``````List<String> animals = Collections.singletonList(\"🐱\");\n// java.lang.UnsupportedOperationException\n``````\n``````List<String> animals = Collections.unmodifiableList(Arrays.asList(\"🐱\", \"🐶\"));\n// java.lang.UnsupportedOperationException\n``````\n\nIn Vavr, list does not have `add()` method. It has `prepend()` and `append()`, which adds a new element respectively before and after the list, and creates a new list. It means that the original list remains unchanged.\n\n``````// io.vavr.collection.List\nList<String> animals = List.of(\"🐱\", \"🐶\");\nList<String> another = animals.prepend(\"🙂\");\n// animals: \"🐱\", \"🐶\"\n// another: \"🙂\", \"🐱\", \"🐶\"\n``````\n``````List<String> animals = List.of(\"🐱\", \"🐶\");\nList<String> another = animals.append(\"😌\");\n// animals: \"🐱\", \"🐶\"\n// another: \"🐱\", \"🐶\", \"😌\"\n``````\n\nThis is very similar to the `addFirst()` and `addLast()` methods of `java.util.LinkedList`.\n\n## Get Element\n\nIn Java, getting the element at the specified position in the list can be done using `get(int)`.\n\n``````// java.util.List\nList<String> animals = Arrays.asList(\"🐱\", \"🐶\");\nanimals.get(0)\n// \"🐱\"\n``````\n\nIn Vavr, you can get the first element using `get()` without input params, or get the element at specific position using `get(int)`. You can also get the first element using `head()` and get the last element using `last()`.\n\n``````// io.vavr.collection.List\nList<String> animals = List.of(\"🐱\", \"🐶\");\nanimals.get();\n// \"🐱\"\n// \"🐱\"\nanimals.get(1);\n// \"🐶\"\nanimals.last();\n// \"🐶\"\n``````\n\nPerformance. If you’re doing “get” operation on a list with high volume of elements, it’s important to consider the performance issue. The “get” operation with index in Vavr takes linear time to finish: O(N). While for Java lists, some implementations, like `java.util.ArrayList` takes constant time to do the same operation; and other implementations, like `java.util.LinkedList` takes linear time. If you need something faster in Vavr, you might want to consider `io.vavr.collection.Array`.\n\n## Remove Element\n\nFirstly, let’s take a look on removing element.\n\nIn Java, removing an element can be done using `List#remove(Object)`. Note that the input parameter is not parameterized `T`, but `Object`. So you can pass any object to try to remove it from the list. They don’t need to have the same type. The element will be removed if it is equal to the input object. For more detail, see Stack Overflow: Why aren’t Java Collections remove methods generic?.\n\n``````List<String> animals = Arrays.asList(\"🐱\", \"🐶\");\nList<String> animals = new ArrayList<>();\nanimals.remove(true); // remove(Object)\n// \"🐱\", \"🐶\"\nanimals.remove(\"🐱\");\n// \"🐶\"\n``````\n\nIn Vavr, removing an element can be done using `List#remove(T)`. This method is defined by `io.vavr.collection.Seq`, which removes the first occurrence of the given element. Different from Java, it requires the input object has the same type `T` as the type of elements in the list. Note that list is immutable, and a new list is returned when doing remove operation.\n\n``````// io.vavr.collection.List\nList<String> animals = List.of(\"🐱\", \"🐶\");\nList<String> another = animals.remove(\"🐱\");\n// animals: \"🐱\", \"🐶\"\n// another: \"🐶\"\n``````\n\nNow, let’s take a look at removing by index.\n\nIn Java, removing an element by index can be done using `List#remove(int)`. Note that this operation is very tricky when having a list of integer `List<Integer>` which auto-boxes the primitives.\n\n``````List<Integer> numbers = new ArrayList<>();\n// numbers: 2, 3\nnumbers.remove(Ingeter.valueOf(1)); // remove(Object)\n// numbers: 2, 3\nnumbers.remove(1); // remove(int)\n// numbers: 2\n``````\n\nIn Vavr, removing an element by index is done via another method, called `removeAt(int)`. It makes the operation more explicit, and avoids error-prone.\n\n``````List<Integer> numbers = List.of(2, 3);\nList<Integer> another = numbers.removeAt(1);\n// numbers: 2, 3\n// another: 2\n``````\n\n## Streaming API\n\nIn Java, the streaming API is very explicit. From a collection `x`, you can start a stream using `stream()` method, followed by the operation wished, then ends with the desired collections using `collect(...)`. There’s no shortcut / default options to make it simpler.\n\n``````x.stream().\\$OPERATION.collect(...);\n``````\n\nIn Vavr, the stream-like operations are more implicit. You can simply call the operation and Vavr will transform it to a collection with the same type. Then, if you need something else, you can convert it using a collector method.\n\n``````x.\\$OPERATION;\n``````\n\nFor example, in Java:\n\n``````Arrays.asList(\"🐱\", \"🐶\")\n.stream()\n.map(s -> s + s)\n.collect(Collectors.toList());\n// \"🐱🐱\", \"🐶🐶\"\n``````\n``````Arrays.asList(\"🐱\", \"🐶\")\n.stream()\n.filter(\"🐱\"::equals)\n.collect(Collectors.toList());\n// \"🐱\"\n``````\n``````List<String> cats = Arrays.asList(\"🐱\", \"🐈\");\nList<String> dogs = Arrays.asList(\"🐶\", \"🐕\");\nList<List<String>> lists = Arrays.asList(cats, dogs);\nList<String> animals = lists.stream().flatMap(Collection::stream).collect(Collectors.toList());\n// \"🐱\", \"🐈\", \"🐶\", \"🐕\"\n``````\n\nIn Vavr:\n\n``````List.of(\"🐱\", \"🐶\").map(s -> s + s);\n// \"🐱🐱\", \"🐶🐶\"\n``````\n``````List.of(\"🐱\", \"🐶\").filter(\"🐱\"::equals)\n// \"🐱\"\n``````\n``````List<String> cats = List.of(\"🐱\", \"🐈\");\nList<String> dogs = List.of(\"🐶\", \"🐕\");\nList<List<String>> lists = List.of(cats, dogs);\nList<String> list = lists.flatMap(Function.identity());\n// \"🐱\", \"🐈\", \"🐶\", \"🐕\"\n``````\n\n## From Vavr to Java\n\nVavr provides a lot of methods to convert a Vavr collection to Java collection. This is done by using syntax `toJava*`:\n\n``````toJavaSet()\ntoJavaList()\ntoJavaMap()\n...\n``````" ]

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[ "# #27ecff Color Information\n\nIn a RGB color space, hex #27ecff is composed of 15.3% red, 92.5% green and 100% blue. Whereas in a CMYK color space, it is composed of 84.7% cyan, 7.5% magenta, 0% yellow and 0% black. It has a hue angle of 185.3 degrees, a saturation of 100% and a lightness of 57.6%. #27ecff color hex could be obtained by blending #4effff with #00d9ff. Closest websafe color is: #33ffff.\n\n• R 15\n• G 93\n• B 100\nRGB color chart\n• C 85\n• M 7\n• Y 0\n• K 0\nCMYK color chart\n\n#27ecff color description : Vivid cyan.\n\n# #27ecff Color Conversion\n\nThe hexadecimal color #27ecff has RGB values of R:39, G:236, B:255 and CMYK values of C:0.85, M:0.07, Y:0, K:0. Its decimal value is 2616575.\n\nHex triplet RGB Decimal 27ecff `#27ecff` 39, 236, 255 `rgb(39,236,255)` 15.3, 92.5, 100 `rgb(15.3%,92.5%,100%)` 85, 7, 0, 0 185.3°, 100, 57.6 `hsl(185.3,100%,57.6%)` 185.3°, 84.7, 100 33ffff `#33ffff`\nCIE-LAB 85.825, -38.318, -22.085 48.877, 67.637, 105.082 0.221, 0.305, 67.637 85.825, 44.227, 209.957 85.825, -62.517, -29.903 82.242, -37.84, -18.186 00100111, 11101100, 11111111\n\n# Color Schemes with #27ecff\n\n• #27ecff\n``#27ecff` `rgb(39,236,255)``\n• #ff3a27\n``#ff3a27` `rgb(255,58,39)``\nComplementary Color\n• #27ffa6\n``#27ffa6` `rgb(39,255,166)``\n• #27ecff\n``#27ecff` `rgb(39,236,255)``\n• #2780ff\n``#2780ff` `rgb(39,128,255)``\nAnalogous Color\n• #ffa627\n``#ffa627` `rgb(255,166,39)``\n• #27ecff\n``#27ecff` `rgb(39,236,255)``\n• #ff2780\n``#ff2780` `rgb(255,39,128)``\nSplit Complementary Color\n• #ecff27\n``#ecff27` `rgb(236,255,39)``\n• #27ecff\n``#27ecff` `rgb(39,236,255)``\n• #ff27ec\n``#ff27ec` `rgb(255,39,236)``\n• #27ff3a\n``#27ff3a` `rgb(39,255,58)``\n• #27ecff\n``#27ecff` `rgb(39,236,255)``\n• #ff27ec\n``#ff27ec` `rgb(255,39,236)``\n• #ff3a27\n``#ff3a27` `rgb(255,58,39)``\n• #00c6da\n``#00c6da` `rgb(0,198,218)``\n• #00def3\n``#00def3` `rgb(0,222,243)``\n• #0eeaff\n``#0eeaff` `rgb(14,234,255)``\n• #27ecff\n``#27ecff` `rgb(39,236,255)``\n• #41eeff\n``#41eeff` `rgb(65,238,255)``\n• #5af0ff\n``#5af0ff` `rgb(90,240,255)``\n• #74f3ff\n``#74f3ff` `rgb(116,243,255)``\nMonochromatic Color\n\n# Alternatives to #27ecff\n\nBelow, you can see some colors close to #27ecff. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #27ffdc\n``#27ffdc` `rgb(39,255,220)``\n• #27ffee\n``#27ffee` `rgb(39,255,238)``\n• #27feff\n``#27feff` `rgb(39,254,255)``\n• #27ecff\n``#27ecff` `rgb(39,236,255)``\n• #27daff\n``#27daff` `rgb(39,218,255)``\n• #27c8ff\n``#27c8ff` `rgb(39,200,255)``\n• #27b6ff\n``#27b6ff` `rgb(39,182,255)``\nSimilar Colors\n\n# #27ecff Preview\n\nThis text has a font color of #27ecff.\n\n``<span style=\"color:#27ecff;\">Text here</span>``\n#27ecff background color\n\nThis paragraph has a background color of #27ecff.\n\n``<p style=\"background-color:#27ecff;\">Content here</p>``\n#27ecff border color\n\nThis element has a border color of #27ecff.\n\n``<div style=\"border:1px solid #27ecff;\">Content here</div>``\nCSS codes\n``.text {color:#27ecff;}``\n``.background {background-color:#27ecff;}``\n``.border {border:1px solid #27ecff;}``\n\n# Shades and Tints of #27ecff\n\nA shade is achieved by adding black to any pure hue, while a tint is created by mixing white to any pure color. In this example, #001213 is the darkest color, while #ffffff is the lightest one.\n\n• #001213\n``#001213` `rgb(0,18,19)``\n• #002427\n``#002427` `rgb(0,36,39)``\n• #00353b\n``#00353b` `rgb(0,53,59)``\n• #00474e\n``#00474e` `rgb(0,71,78)``\n• #005962\n``#005962` `rgb(0,89,98)``\n• #006b75\n``#006b75` `rgb(0,107,117)``\n• #007d89\n``#007d89` `rgb(0,125,137)``\n• #008f9d\n``#008f9d` `rgb(0,143,157)``\n• #00a1b0\n``#00a1b0` `rgb(0,161,176)``\n• #00b3c4\n``#00b3c4` `rgb(0,179,196)``\n• #00c5d8\n``#00c5d8` `rgb(0,197,216)``\n• #00d6eb\n``#00d6eb` `rgb(0,214,235)``\n• #00e8ff\n``#00e8ff` `rgb(0,232,255)``\n• #13eaff\n``#13eaff` `rgb(19,234,255)``\n• #27ecff\n``#27ecff` `rgb(39,236,255)``\n• #3beeff\n``#3beeff` `rgb(59,238,255)``\n• #4eefff\n``#4eefff` `rgb(78,239,255)``\n• #62f1ff\n``#62f1ff` `rgb(98,241,255)``\n• #75f3ff\n``#75f3ff` `rgb(117,243,255)``\n• #89f5ff\n``#89f5ff` `rgb(137,245,255)``\n• #9df6ff\n``#9df6ff` `rgb(157,246,255)``\n• #b0f8ff\n``#b0f8ff` `rgb(176,248,255)``\n• #c4faff\n``#c4faff` `rgb(196,250,255)``\n• #d8fcff\n``#d8fcff` `rgb(216,252,255)``\n• #ebfdff\n``#ebfdff` `rgb(235,253,255)``\n• #ffffff\n``#ffffff` `rgb(255,255,255)``\nTint Color Variation\n\n# Tones of #27ecff\n\nA tone is produced by adding gray to any pure hue. In this case, #8b9a9b is the less saturated color, while #27ecff is the most saturated one.\n\n• #8b9a9b\n``#8b9a9b` `rgb(139,154,155)``\n• #82a1a4\n``#82a1a4` `rgb(130,161,164)``\n• #7aa8ac\n``#7aa8ac` `rgb(122,168,172)``\n• #72aeb4\n``#72aeb4` `rgb(114,174,180)``\n• #69b5bd\n``#69b5bd` `rgb(105,181,189)``\n• #61bcc5\n``#61bcc5` `rgb(97,188,197)``\n• #59c3cd\n``#59c3cd` `rgb(89,195,205)``\n``#51cad5` `rgb(81,202,213)``\n• #48d1de\n``#48d1de` `rgb(72,209,222)``\n• #40d7e6\n``#40d7e6` `rgb(64,215,230)``\n• #38deee\n``#38deee` `rgb(56,222,238)``\n• #2fe5f7\n``#2fe5f7` `rgb(47,229,247)``\n• #27ecff\n``#27ecff` `rgb(39,236,255)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #27ecff is perceived by people affected by a color vision deficiency. This can be useful if you need to ensure your color combinations are accessible to color-blind users.\n\nMonochromacy\n• Achromatopsia 0.005% of the population\n• Atypical Achromatopsia 0.001% of the population\nDichromacy\n• Protanopia 1% of men\n• Deuteranopia 1% of men\n• Tritanopia 0.001% of the population\nTrichromacy\n• Protanomaly 1% of men, 0.01% of women\n• Deuteranomaly 6% of men, 0.4% of women\n• Tritanomaly 0.01% of the population" ]

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[ "Solutions by everydaycalculation.com\n\n1st number: 6 6/14, 2nd number: 2 13/16\n\n90/14 + 45/16 is 1035/112.\n\n1. Find the least common denominator or LCM of the two denominators:\nLCM of 14 and 16 is 112\n2. For the 1st fraction, since 14 × 8 = 112,\n90/14 = 90 × 8/14 × 8 = 720/112\n3. Likewise, for the 2nd fraction, since 16 × 7 = 112,\n45/16 = 45 × 7/16 × 7 = 315/112", null, "Download our mobile app and learn to work with fractions in your own time:" ]

[ null, "https://answers.everydaycalculation.com/mathstep-app-icon.png", null ]

[ "# Best Fuel for Potato Gun\n\nI have been using deodorant to propel potatoes (potato gun :D), but I want to switch over to pure butane. However, in order to do this safely I have to know how much butane is typically in deodorant (percentage). If you do know, please also include a source.\n\nSecondly, for a potato gun, is it wiser to use butane or propane?\n\nAnd could a spark ignite butane? Or is a real flame needed? Tested this already, it does.\n\nHow much butane is typically in deodorant (percentage)\n\nHere's a link to an MSDS for a deodorant; it claims the butane range is 30-60% (they don't state if this is by volume or weight). It might be cheaper to just go to a hardware store and buy a small butane cylinder, like the ones used for outdoor cooking, then you know exactly how much butane you are using.\n\nFor a potato gun, is it wiser to use butane or propane?\n\nMy first though was butane because it gives off a bit more heat per mole (or per unit volume) when it burns, than does propane.\n\n$$\\ce{C3H8 + 5 O2 -> 3 CO2 + 4 H2O + \\pu{2219 kJ/mol}}$$ $$\\ce{C4H10 + \\mathrm{6.5}~ O2 -> 4 CO2 + 5 H2O + \\pu{2878 kJ/mol}}$$\n\nHowever, according to \"Common Fuels for Combustion Spud Guns\",\n\nThe key value for comparing two fuels based on their heats of combustion is not the actual heats of combustion. Instead, the \"Heat per mole Oxygen\" should be used since the amount of energy in the combustion chamber is limited by the amount of oxygen present in the chamber. Fuel is added to match that amount of oxygen. As you can see from the table, there is relatively little difference between the various fuels based on their \"Heat per mole Oxygen\" values.\n\nBoth propane and butane give off $\\pu{105 kcal/mol}$ of oxygen, so they should perform equivalently in a spud gun. The article goes on to note that, of common fuels, only hydrogen ($\\pu{119 kcal/mol}$ of oxygen) and acetylene ($\\pu{120 kcal/mol}$ of oxygen) produce significantly more energy under these conditions than other fuels.\n\nNow going back to your first question, if you know the volume of your chamber, you can back-calculate the correct amount of air (oxygen) and butane." ]

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[ "# Binary Clock in C# And WPF\n\nIn this article, we will create a simple binary clock using C# and WPF.\n\nForeward\n\nThe project itself will serve to show some peculiarities, like the use of Tasks, how to manipulate the UI of a page, and basic data conversions.\n\nIntroduction\n\nA binary clock is a clock which displays the current time in a binary format. In the following example, we will create a set of graphical leds, each of which will represent a binary digit. Each led could be set in two statuses: on (which represents the value of 1) or off (which represent the value of zero). From right to left, our leds will represents the values of 1, 2, 4, 8, 16, 32, because we will base our conversion on 24h formatted time, and we need a number of digits that can represent up to the decimal value of 60 (for minutes and seconds).\n\nEach part of the current time (hour, minutes, seconds) will have its own row of six leds, to represent the binary conversion of the decimal value. For example, if we wish to display a time like 10:33:42, our leds must be illuminated according to the following pattern:\n\nA Binary Clock in XAML\n\nThe XAML rendering of the concept above is fairly simple. In a new XAML page, we need to create three rows of rectangles, whose border radius will be set to 50 to give them a circular shape. Other settings will refer to the filling color, shape shadows, and so on, to draw a led the way we desire. In our example, the led will be shadowed and colored according to the following XAML code:\n\n1. <Rectangle HorizontalAlignment=\"Left\" Height=\"35\" Margin=\"211,40,0,0\"\n2.     Stroke=\"#FF033805\" VerticalAlignment=\"Top\"\n4.       <Rectangle.Effect>\n6.       </Rectangle.Effect>\n7.       <Rectangle.Fill>\n12.       </Rectangle.Fill>\n13. </Rectangle>\nOnce we have finished creating each row for our UI, and have embellished everything, the XAML page will look like this:\n\nYou may refer to the download section, available at the end of the article for a complete reference of the snippet above.\n\nSource Code\n\nThe following section explains how our rectangles c be canontrolled to show a binary representation of the current time.\n\nIn our XAML Window, we've declared an Event calling -- more precisely, an event that must be fired on the Page loading (Loaded Event). In the code behind of the `Window_Loaded` routine, we execute two main operations: the first is merely graphical, and consists in setting each rectangle opacity to `0.35`, in order to give the impression of a switched-off led. The second one is the execution of the task which will execute calculations and update the UI. More on that later. First, let's see how we can identify a control declared on a XAML page.\n\nLet's take a look at the loop to set all rectangle's opacity to `0.35`:\n\n1. // Sets all rectangles opacity to 0.35\n2. foreach (var r in LogicalTreeHelper.GetChildren(MainGrid))\n3. {\n4.   if (r is Rectangle) (r as Rectangle).Fill.Opacity = 0.35;\n5. }\n\nSpeaking about identifying controls, the main difference between WinForms and WPF is that we can't refer to a container's controls by using the property `Controls()`. The WPF-way of doing that kind of operation passes through the `LogicalTreeHelper` class. Through it, we can call upon the method `GetChildren`, indicating the name of the main control for which retrieve children controls. In our case, we've executed`LogicalTreeHelper.GetChildren` on the control `MainGrid` (the name which identifies the `Grid` object of our XAML Page). Then, while traversing the controls array, we check if that particular control is a `Rectangle` and - if so - we'll proceed in setting its Opacity to the desired value.\n\nThe second set of instructions from the `Window_Loaded` event is the execution of a secondary task for calculating the binary representation of each time part, and updating the UI as well. The code is as follows:\n\n2. {\n3.     // while the thread is running...\n4.     while (true)\n5.     {\n6.         // ...get the current system time\n7.         DateTime _now = System.DateTime.Now;\n8.\n9.         // Convert each part of the system time (i.e.: hour, minutes, seconds) to\n10.         // binary, filling with 0s up to a length of 6 char each\n11.         String _binHour = Convert.ToString(_now.Hour, 2).PadLeft(6, '0');\n12.         String _binMinute = Convert.ToString(_now.Minute, 2).PadLeft(6, '0');\n13.         String _binSeconds = Convert.ToString(_now.Second, 2).PadLeft(6, '0');\n14.\n15.         // For each digit of the binary hour representation\n16.         for (int i = 0; i <= _binHour.Length - 1; i++)\n17.         {\n18.           // Dispatcher invoke to refresh the UI, which belongs to the main thread\n19.           H0.Dispatcher.Invoke(() =>\n20.           {\n21.              // Update the contents of the labels which use decimal h/m/s representation\n22.              lbHour.Content = _now.Hour.ToString(\"00\");\n23.              lbMinute.Content = _now.Minute.ToString(\"00\");\n24.              lbSeconds.Content = _now.Second.ToString(\"00\");\n25.\n26.              // Search for a rectangle which name corresponds to the _binHour current char index.\n27.              // Then, set its opacity to 1 if the current _binHour digit is 1, or to 0.35 otherwise\n28.              (MainGrid.FindName(\"H\" + i.ToString()) as Rectangle).Fill.Opacity =\n29.              _binHour.Substring(i, 1).CompareTo(\"1\") == 0 ? 1 : 0.35;\n30.              (MainGrid.FindName(\"M\" + i.ToString()) as Rectangle).Fill.Opacity =\n31.              _binMinute.Substring(i, 1).CompareTo(\"1\") == 0 ? 1 : 0.35;\n32.              (MainGrid.FindName(\"S\" + i.ToString()) as Rectangle).Fill.Opacity =\n33.              _binSeconds.Substring(i, 1).CompareTo(\"1\") == 0 ? 1 : 0.35;\n34.           });\n35.         }\n36.     }\n37. });\n\nPretty self-explanatory, the task consists in a neverending loop, which continuously retrieves the current system time. Then, it separates it in its three main parts (hours, minutes, seconds) and proceeds in converting them to their binary representation, through the use of the `Convert.ToString()` function, to which we'll pass the numeric base for conversion (in our case, `2`). Since we need three `string`s of length equal to six (we have six leds for each row), we need to pad each `string` up to six characters. So, if for example, we are converting the value of `5`, the function will produce `101` as output - a value we will pad to `000101`.\n\nA second loop which works up to the length of the binary `string` related to hours (a value that will be always six), will the provide the UI update, using the `Dispatcher` property to Invoke the `update` method of an object which runs on another thread (please refer to «VB.NET: Invoke Method to update UI from secondary threads» for further details on `Invoke` method). For each digit contained in our `string`s, we need to identify the correct`Rectangle`, to update its Opacity value.\n\nWe can accomplish this kind of task through the `FindName()` function: given a parent object (`MainGrid`, in our case), `FindName` will proceed in referring an UI control, if the passed parameter corresponds to an existent control name. Since we've named the Rectangles for each time part with a progressive number (`0` to `5`, beginning with `H` for hours, `M` for minutes, `S` for seconds), we can ask the function to retrieve the control whose name starts with a particular letter, and continues with an index equal to the current binary string index.\n\nLet's see one of those lines for the sake of clarity: with the following line:\n\n1. (MainGrid.FindName(\"H\" + i.ToString()) as Rectangle).Fill.Opacity =\n2.         _binHour.Substring(i, 1).CompareTo(\"1\") == 0 ? 1 : 0.35;\nWe are asking: retrieve from `MainGrid` a control whose name is equal to \"`H`\" + the current loop index. Consider it as a `Rectangle`, then set its `Opacity` according to the following rule: if the indexed character from the binary`string` is `1`, then the `Opacity` must be `1`, otherwise it must be set to `0.35`. Executing the program will result in what can be seen in the following video.\n\nDemonstrative Video" ]

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[ "# C++ Program to calculate the profit sharing ratio\n\nC++Server Side ProgrammingProgramming\n\nGiven with an array of amount invested by multiple person and another array containing the time period for which money is invested by corresponding person and the task is to generate the profit sharing ratio.\n\n### What is Profit Sharing Ratio\n\nIn a partnership firm, profits and losses are shared between the partners depending upon the capital invested by them in the business. On the basis of that capital investment percentage we calculate the profit sharing ratio which shows the amount of profit which is to be given to each partner of a business.\n\nFormula − Partner 1 = capital invested * time period\n\nPartner 2 = capital invested * time period\n\nPartner 3 = capital invested * time period\n\nPartner 4 = capital invested * time period .       .\n\nPerson n = capital invested * time period\n\nProfit Sharing Ratio = Partner 1 : Partner 2: Partner 3\n\n## Example\n\nInput-: amount[] = { 1000, 2000, 2000 }\ntime[] = { 2, 3, 4 }\nOutput-: profit sharing ratio 1 : 3 : 4\nInput-: amount[] = {5000, 6000, 1000}\ntime[] = {6, 6, 12}\nOutput-: profit sharing ratio 5 : 6 :2\n\nApproach we will be using to solve the given problem\n\n• Take the input as an array of capital invested by multiple partners and another array for time period for which they have invested the capital\n• Multiply the capital of one partner with his corresponding time period and repeat the same  with other partners\n• Calculate the ratio of multiplied values\n• Display the final result\n\n## ALGORITHM\n\nStart\nstep 1-> declare function to calculate GCD\nint GCD(int arr[], int size)\ndeclare int i\nDeclare int result = arr\nLoop For i = 1 and i < size and i++\nset result = __gcd(arr[i], result)\nEnd\nreturn result\nstep 2-> declare function to calculate profit sharing ratio\nvoid cal_ratio(int amount[], int time[], int size)\ndeclare int i, arr[size]\nLoop For i = 0 and i < size and i++\nset arr[i] = amount[i] * time[i]\nEnd\ndeclare int ratio = GCD(arr, size)\nLoop For i = 0 and i < size - 1 and i++\nprint arr[i] / ratio\nEnd\nprint  arr[i] / ratio\nStep 3-> In main()\ndeclare int amount[] = { 1000, 2000, 2000 }\ndeclare int time[] = { 2, 3, 4 }\ncalculate int size = sizeof(amount) / sizeof(amount)\ncall cal_ratio(amount, time, size)\nStop\n\n## Example\n\n#include <bits/stdc++.h>\nusing namespace std;\n//calculate GCD\nint GCD(int arr[], int size) {\nint i;\nint result = arr;\nfor (i = 1; i < size; i++)\nresult = __gcd(arr[i], result);\nreturn result;\n}\n//calculate profit sharing ratio\nvoid cal_ratio(int amount[], int time[], int size) {\nint i, arr[size];\nfor (i = 0; i < size; i++)\narr[i] = amount[i] * time[i];\nint ratio = GCD(arr, size);\nfor (i = 0; i < size - 1; i++)\ncout << arr[i] / ratio << \" : \";\ncout << arr[i] / ratio;\n}\nint main() {\nint amount[] = { 1000, 2000, 2000 };\nint time[] = { 2, 3, 4 }\nint size = sizeof(amount) / sizeof(amount);\ncout<<\"profit sharing ratio \";\ncal_ratio(amount, time, size);\nreturn 0;\n}\n\n## Output\n\nprofit sharing ratio 1 : 3 : 4" ]

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[ "# Transcript\n\nGo is a statically typed programming language. If you only worked with JavaScript, Ruby, Python, or other dynamically typed languages before, there are a couple of new things to learn. But even for C++ or Java programmers, there are a few aspects where Go is different.\n\n## Variable Declaration\n\nLet’s start by declaring some variables.\n\nThere are a couple of ways of declaring a variable. First, we can declare a variable with the `var` keyword, followed by the variable name, followed by the type name.\n\n```/* At package level, outside any function */\nvar a int\n```\n\nThis creates a variable named “a” of type “int” with an initial value of zero. The last part is an important detail. Go does not have the concept of undefined variables. Each data type has a well-defined zero value that is used when the variable is not explicitly initialised to something else.\n\nSo if we print out the value of a,\n\n```fmt.Println(a)\n```\n\nwe get no error even though we have not assigned anything to the new variable yet.\n\nIt is also possible to assign a non-zero value at the point of declaration.\n\n```var b int = 10\n```\n\nIf we omit the type, the Go compiler attempts to infer the type from the initial value.\n\n```var c = 10\n```\n\nIn this example, c becomes an integer variable.\n\nThere can be more than one variable in a var declaration.\n\n```var d, e, f bool\n```\n\nWe can also use the syntax that we have already seen with the import statement.\n\n```var (\ng int\nh string\ni int = 1234\nj, k, l bool\n)\n```\n\nThis leads to less typing and clearer code.\n\nWe can use all of the above also inside a function.\n\n```func main() {\nvar (\ng int\nh string\ni int = 1234\nj, k, l bool\n)\n}\n```\n\nHere we can also use an alternate style called “short variable declaration”. This actually is a statement, rather than a declaration, which is why we cannot use it outside a function. The short variable declaration is of the form\n\n```m := 1\n```\n\nand is declaration and assignment in one step. Multiple declarations are possible as well.\n\n```n, o := 2, 3\n```\n\n## Assignment\n\nTo assign a value to an existing variable, use a simple equal sign.\n\n```a = 11\n```\n\nGo also supports multiple assignments in one statement.\n\n```e, f = f, e\n```\n\nHere we effectively have swapped the values of the two variables e and f.\n\nYou can even assign to new and existing variables at the same time, using the short variable declaration operator.\n\n```a := 11\na, p := 100, 200\n```\n\n## Static Typing\n\nIf you only used dynamically typed programming languages before, be aware that a variable cannot change its type at runtime. For example, trying to assign a string to a variable that has been declared as an int produces an error at compile time.\n\n```a = \"wrong\"\n```\n\nThis may seem like a big inconvenience at first (if you are used to languages like JavaScript or Python), but keep in mind that this strict type checking may save you from having to painfully track down some erratic behaviour at runtime. And the compiler can better optimize for speed.\n\nThe Go compiler even goes one step further and also complains if a variable is declared but not used. That is, this code will not compile.\n\n```func main() {\na := 1\n}\n```\n\nTrying to run this code triggers the message, “a declared and not used”.\n\n## The Blank Identifier\n\nIn some situations, however, you need to declare a variable that you do not use later. For example, if a function returns two values but you only need one of them,\n\n```a, b := f()\ng(a)\n```\n\nthen you can use the so-called “blank identifier”. This is simply an underscore that you use instead of a real variable name.\n\n```a, _ := f()\ng(a)\n```\n\n## To Summarize\n\n• Declare variables using the var keyword.\n• The variable name comes first, then the type name.\n• Undefined variables do not exist. Each new variable gets initialized to its zero value.\n• You can declare a variable and assign a value in one step.\n• You can declare a variable without a type if you assign a value in the same step. The compiler will then deduce the type from the assigned value.\n• Inside functions, the short variable declaration operator declares a variable, assigns a value, and deduces the type of the variable in one single step.\n• Declaring and not using a variable is an error.\n• Use a blank identifier if you must declare a variable that is not used later.\n\nThere are a couple of advanced data types as well, and we will discuss each of them in an extra lecture." ]

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[ "Linear Regression in EnergyDeck: how to make the most of it", null, "Bruno Girin\n\nJuly 04, 2013\n\nIf you've been entering meter readings, you may have noticed a new tab in the Meter Details view (the view you get when clicking on any meter name in the Setup section) called Regression Analysis. This post provides a quick tour of the new functionality.", null, "The Basics\n\nWhen you click on the Regression Analysis tab, you will see a graph called a fitted curve. By default, it is drawn against heating degree days (HDD) but you can draw it against cooling degree days (CDD) too. A fitted curve shows the result of a linear regression algorithm: simply put, the system has taken consumption data and HDD data and tried to identify a linear relationship between the two. The red dots are actual measured values while the blue line is the line the dots fit most closely to. The idea is that, if the consumption for this meter is driven by how cold the weather is, which is typical of gas consumption, then the data should closely fit the line.", null, "The graph offers a quick view of the data but there are also important numbers at the top. Here is what they mean in reverse order:\n\n• R2: this is a measure of how closely the data fits the line. The closer this value is to 1, the better the fit; the closer to 0, the worse the fit. At 0.66, it means that there is correlation between this meter's consumption and HDD data but it's not perfect.\n• Base Load: this tells you how much is consumed against this meter on average when HDD is zero, or in other words what is the average load when the outside temperature is high enough that heating is not needed.\n• kWh per HDD: this is how much more is consumed against this meter above the base load for every additional HDD.\n• Base Temperature: this is the temperature below which heating is needed, in degrees celsius. EnergyDeck performs regression against a variety of temperatures and retains the one where the fit is highest. In this case, it means that on average, the building starts being heated when the temperature drops below 18°C. This is quite high: a normal base temperature for a building in the UK is considered to be 15.5°C, except for hospitals for which it is 18.5°C. A house insulated to Passive House standard could have a heating base temperature as low as 10°C.\n\nThe same metrics are available if you do regression against CDD but they are applied to cooling rather than heating so are relevant to a meter to which is connected a cooling system like air conditionning.\n\nTwo more graphs are available in addition to the fitted curve: residuals and base load. They are very similar and show two steps in the analysis of your data.\n\nThe residuals graph shows the regression residuals, that is the difference between actual data and the fitted curve. When there is strong correlation between the consumption data and HDD or CDD, this graph should be close to a horizontal line around the zero value. In this particular case, the various peaks in the middle of the graph and troughs at the extremes suggest that there are other parameters than the weather that drive the consumption on this meter.", null, "The base load value given by the regression analysis is an average, so the last graph applies this average to the residuals to show how the base load varies over time.", null, "In practice, this graph looks very similar to the residuals graph but with the deepest troughs shaved off. This highlights the two central peaks: one between mid-December to mid-February and the other one from March to May. Based on this graph, it looks like in cold weather, this office building is heated more than it needs to: could it be draughty by any chance?\n\nMaking the Most out of Regression Analysis\n\nThe data in the example above highlights the basic functionality behind this initial regression analysis implementation. However, the underlying data is test data that happens to be monthly bill data. Bill data is usually estimated rather than actual and monthly data is not very granular. The more data and the more granular that data is, the better regression analysis will work. Here are a few tips to get the most out of it:\n\n• Use actual meter readings rather than bill data, either manual readings or automated,\n• Ensure that your readings are taken at least once a week,\n• Ensure you have one year worth of readings so that the analysis can cover a whole seasonal cycle; note that regression analysis will not be performed if you have less than 10 readings.\n\nNext Steps\n\nThis first implementation of regression analysis is a first step in helping you understand your data better. There is a lot more we want to do so here is a short list of improvements to look out for in the next few months:\n\n• Integration into the Trend Analysis graphs,\n• Regression against other metrics than HDD and CDD,\n• Multi-variate regression, that is the ability to perform regression against several metrics at the same time,\n• Detection of outliers and behaviour change.\n\nTags:   product features" ]

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[ "Learning Continuum of Numeracy (Version 8.4)\n\nPage 1 of 3\n\nThis element involves students developing an understanding of the meaning of fractions and decimals, their representations as percentages, ratios and rates, and how they can be applied in real-life situations.\n\nStudents visualise, order and describe shapes and objects using their proportions and the relationships of percentages, ratios and rates to solve problems in authentic contexts. In developing and acting with numeracy, students:\n\n• interpret proportional reasoning\n• apply proportional reasoning.\n\nLevel 1a\n\nStudents:\n\nInterpret proportional reasoning\n\nrecognise a ‘whole’ and ‘parts of a whole’ within everyday contexts\n\nApply proportional reasoning\n\nLevel 1b is the starting point for this sub-element\n\nLevel 1b\n\nTypically by the end of Foundation Year, students:\n\nInterpret proportional reasoning\n\nrecognise that a whole object can be divided into equal parts\n\nApply proportional reasoning\n\nidentify quantities such as more, less and the same in everyday comparisons\n\nLevel 2\n\nTypically by the end of Year 2, students:\n\nInterpret proportional reasoning\n\nvisualise and describe halves and quarters\n\nApply proportional reasoning\n\nsolve problems using halves and quarters" ]

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[ "2019-05-19T22:09:01Z http://jmathnano.sru.ac.ir/?_action=export&rf=summon&issue=169\n2017-12-01 10.22061\nJournal of Mathematical Nanoscience J. Math. Nanosci. 2017 7 2 On borderenergetic and L-borderenergetic graphs Mardjan Hakimi-Nezhaad A graph G of order n is said to be borderenergetic if its energy is equal to 2n − 2. In this paper, we study the borderenergetic and Laplacian borderenergetic graphs. energy (of graph) adjacency matrix Laplacian matrix signless Laplacian matrix 2017 12 01 71 77 http://jmathnano.sru.ac.ir/article_513_e93b254f3bb974fd3d9d566269730f04.pdf\n2017-12-01 10.22061\nJournal of Mathematical Nanoscience J. Math. Nanosci. 2017 7 2 On the modified Wiener number Maryam Jalali Rad The Graovac-Pisanski index is defined in 1991 namely 56 years after the definition of Wiener index by Graovac and Pisanski. They called it as modified Wiener index based on the sum of distances between all the pairs α(u,α(u)) where α stands in the automorphism group of given graph. In this paper, we compute the Graovac-Pisanski index of some classes of graphs. 2017 12 01 79 83 http://jmathnano.sru.ac.ir/article_512_4cb22d3564fa9bcb9250b4c8dacf41ac.pdf\n2017-12-01 10.22061\nJournal of Mathematical Nanoscience J. Math. Nanosci. 2017 7 2 Sanskruti index of bridge graph and some nanocones K Pattabiraman Sanskruti index is the important topological index used to test the chemical properties of chemical comopounds. In this paper, first we obtain the formulae for calculating the Sanskruti index of bridge graph and carbon nanocones CNCn(k). In addition, Sanskruti index of the Line graph of CNCk[n] nanocones are obtained. Sanskruti index bridge graph carbon nanocones 2017 12 01 85 95 http://jmathnano.sru.ac.ir/article_707_cd8b6a11d8861630dce00cce8e14f068.pdf\n2017-12-01 10.22061\nJournal of Mathematical Nanoscience J. Math. Nanosci. 2017 7 2 The Wiener and Szeged indices of hexagonal cored dendrimers Abbas Heydari A topological index of a molecule graph G is a real number which is invariant under graph isomorphism. The Wiener and Szeged indices are two important distance based topological indices applicable in nanoscience. In this paper, these topological indices is computed for hexagonal cored dendrimers. Wiener index Szeged index Dendrimers nanoparticles 2017 12 01 97 101 http://jmathnano.sru.ac.ir/article_741_23f9998233252ea7874a727ab592643c.pdf\n2017-12-01 10.22061\nJournal of Mathematical Nanoscience J. Math. Nanosci. 2017 7 2 Study of inverse sum indeg index Marzieh Hasani Let \\$MG(i,n)\\$ \\$(1leq  i leq 3)\\$ denote to the class of all \\$n\\$-vertex molecular graphs with minimum degree \\$ i\\$.  The  inverse  sum  indeg  index of   a   graph  is   defined   as    \\$ISI=sum_{uvin E(G)}  d_ud_v/(d_u+d_v)\\$,   where \\$ d_{u}\\$ denotes to the degree of vertex \\$ u\\$. In this paper, we propose some extremal molecular graphs with the minimum and the maximum value of inverse sum indeg index in \\$MG(i,n)\\$. 2017 12 01 103 109 http://jmathnano.sru.ac.ir/article_748_fa5e1b4c717655e7c63778f623e327fc.pdf\n2017-12-01 10.22061\nJournal of Mathematical Nanoscience J. Math. Nanosci. 2017 7 2 A note on the entropy of graphs Samaneh Zangi A useful tool for investigation various problems in mathematical chemistry and  computational physics is graph entropy.  In this paper, we introduce a new version of graph entropy  and  then we determine it for some classes of graphs. graph eigenvalues entropy regular graph 2017 12 01 111 115 http://jmathnano.sru.ac.ir/article_749_8a904083c382def25280394c40b316b7.pdf" ]

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[ "Summer of our Discontent In Markets – Part 2\n\nThis is the second in a three-part series on the growing likelihood of a transition in major markets. Part 1 focused on Macro Event Risk and Cycles. This post discusses the application of popular complex systems Early Warning Signals (EWS) to assess a possible transition in the S&P 500.\n\nComplex Systems Theory\n\nMost people associate complexity science with natural systems such as animal populations, climate ecosystems, and earthquake prediction.\n\nFor some years now, there has been expanding research into man-made systems that exhibit similar characteristics, including social networks and financial markets. In markets, the application of complexity science is still nascent and is likely best applied in combination with other analysis toolkits.\n\nBifurcations and Critical Transitions\n\nComplex systems can have tipping points where they undergo a sudden transition to a new state or regime. These points are called bifurcations1 and the shifts that occur when a system passes bifurcations are called critical transitions2. There are many types of bifurcations, including\n\n• Catastrophic bifurcations where a system transitions from one stable state to a new stable state\n• Bifurcations in cyclic and chaotic systems where a system transitions from a stable state to an unstable state\n\nEarly Warning Signals for Critical Transitions\n\nResearchers have come to believe that there exist standard Early Warning Signals (EWS) to identify when critical transitions are approaching.\n\nVarious categories of EWS have been produced. Two popular categories are (1) signals based on time series analysis and (2) trait-based signals based on trends in the visual representation of the systems. To date, I have focused on time series signals, but believe there is potential in visual trait-based signals for financial markets.\n\nDiscussing every time-series based EWS is beyond the scope of this post. I focus on the most widely researched EWS to date, Critical Slowing Down (CSD), a phenomenon whereby a slight disturbance of the system away from a perceived equilibrium takes a long time to revert back to equilibrium.\n\nThe figure below depicts CSD in a fold bifurcation system. In (a), the system quickly recovers from the perturbation (depicted by a clear circle) to its equilibrium (the dark circle) as the slope of recovery is steep. As the system moves toward bifurcation to a new phase (b), the recovery slope is flatter and recovery takes longer. Finally, at bifurcation/phase transition, there is no recovery and the system moves to a new state.", null, "Source: Diks, Hommes, Wang; Critical slowing down as an early warning signal for financial crises? (Empirical Economics Aug 2018)\n\nMeasuring Critical Slowing Down\n\nCSD, and thus a system’s proximity to a phase transition, can be evaluated using two statistical signals: rising correlation and increasing variance.\n\nRising Correlation\n\nRising correlation is best measured in a time-series based system using autocorrelation, which per Wikipedia is:\n\n“The correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations as a function of the time lag between them.”\n\nSource: https://en.wikipedia.org/wiki/Autocorrelation\n\nWhat this means is that a given data point in a time series is impacted by a “copy of itself”, i.e. a previous data point in the time series. So, the price of the S&P 500 today can be impacted by a different point in the time series, which despite the time lag, is the same actual variable.\n\nTo identify CSD, researchers have found that an increase in the lag-1 autocorrelation (AR(1) increases as the system approaches a critical state. AR(1) can be estimated using the following autoregression model:", null, "Source: Diks, Hommes, Wang; Critical slowing down as an early warning signal for financial crises? (Empirical Economics Aug 2018)\n\nThe above should look familiar as a linear function. What makes it autoregressive is that you are using a time series value at t-1 to predict the time series value at t.\n\n• ?t is the price at time t\n• ?t-1 is the price at time t-1\n• ?−?Δ? is the AR(1) coefficient or λ\n• ?? is random noise\n\nThe key element is the AR(1) coefficient, also represented as λ. ? is the magnitude of the recovery rate to equilibrium in the system and Δ? is the change between t-1 and t, which is 1. Thus, as recovery rates move toward zero, i.e. the system is not recovering as fast to equilibrium, the AR(1) coefficient moves toward 1, because ? raised to a number approaching 0, approaches 1.1\n\nIncreased Variance\n\nRising variance is also indicative of CSD.\n\n“Again, this can be…intuitively understood: as the eigenvalue [the recovery rate of the system, ?−?Δ? or λ] approaches zero, the impacts of shocks do not decay, and their accumulating effect increases the variance of the state variable”\n\nScheffer, M. Critical Transitions in Nature and Society (Princeton Univ. Press, 2009).\n\nScheffer offers a mathematical proof of this, explained briefly below:\n\nThe expectation of an AR(1) process (the equation above) is that it will move toward the mean values of ?t and ?t-1, which means you can transform the equation above as follows (with E standing for expected value and c is a constant):\n\nE(?t) = c + λE(?t-1) + E(??) >> mean (?) = c + λ*mean(?) + 0 >>\n\nmean(y) = c / 1 – λ\n\nWith λ (the recovery rate of the system) approaching zero and c = 0, the mean(y) equals zero and Scheffer proves that variance progresses toward infinity when autocorrelation tends to one:\n\n“Close to the critical point, the return speed to equilibrium decreases, implying that [?] approaches zero and the autocorrelation tends to one. Thus, the variance tends to infinity.”\n\nScheffer, M. Critical Transitions in Nature and Society (Princeton Univ. Press, 2009).\n\nCSD in the S&P 500\n\nLeveraging the open source ewstools Python library, I calculated core Early Warnings Signals (including some that are not covered in this post) for the S&P 500 over the last two years. I used daily closing price data for this analysis which I adjusted to achieve mean-stationarity of the data." ]

[ null, "https://complexityeverywhere.com/wp-content/uploads/2019/06/Screen-Shot-2019-06-01-at-9.05.26-AM-1.png", null, "https://complexityeverywhere.com/wp-content/uploads/2019/06/Screen-Shot-2019-06-02-at-6.49.09-AM.png", null ]

[ "MIMO Frequency Response Data Models\n\nThis example shows how to create a MIMO frequency-response model using frd.\n\nFrequency response data for a MIMO system includes a vector of complex response data for each of the input/output (I/O) pair of the system. Thus, if you measure the frequency response of each I/O pair of your system at a set of test frequencies, you can use the data to create a frequency response model:\n\n1. Load frequency response data in AnalyzerDataMIMO.mat.\n\nload AnalyzerDataMIMO H11 H12 H21 H22 freq\n\nThis command loads the data into the MATLAB® workspace as five column vectors H11, H12, H21, H22, and freq. The vector freq contains 100 test frequencies. The other four vectors contain the corresponding complex-valued frequency response of each I/O pair of a two-input, two-output system.\n\nTip\n\nTo inspect these variables, enter:\n\nwhos H11 H12 H21 H22 freq\n2. Organize the data into a three-dimensional array.\n\nHresp = zeros(2,2,length(freq));\nHresp(1,1,:) = H11;\nHresp(1,2,:) = H12;\nHresp(2,1,:) = H21;\nHresp(2,2,:) = H22;\n\nThe dimensions of Hresp are the number of outputs, number of inputs, and the number of frequencies for which there is response data. Hresp(i,j,:) contains the frequency response from input j to output i.\n\n3. Create a frequency-response model.\n\nH = frd(Hresp,freq);\n\nH is an frd model object, which is a data container for representing frequency response data.\n\nYou can use frd models with many frequency-domain analysis commands. For example, visualize the response of this two-input, two-output system using bode.\n\nTip\n\nBy default, the frd command assumes that the frequencies are in radians/second. To specify different frequency units, use the TimeUnit and FrequencyUnit properties of the frd model object. For example:\n\nsets the frequency units to in radians/minute." ]

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[ "# Words with LCU\n\nA list of all LCU words with their Scrabble and Words with Friends points. You can also find a list of all words that start with LCU. Also commonly searched for are words that end in LCU. Try our five letter words with LCU page if you’re playing Wordle-like games or use the New York Times Wordle Solver for finding the NYT Wordle daily answer.\n\n15 Letter Words\nincalculability31miscalculations28\n14 Letter Words\nmiscalculating29miscalculation27calculatedness25recalculations24\n13 Letter Words\ncalculability28calculatingly28uncalculating27miscalculated26precalculuses26miscalculates25recalculating25calculational24recalculation23\n12 Letter Words\nincalculably28incalculable26calculatedly25miscalculate24uncalculated24calculations22recalculated22recalculates21\n11 Letter Words\nanimalculum25calculative24precalculus24calculating23dyscalculia23animalcular21animalcules21calculation21calculators20recalculate20\n10 Letter Words\ncalculably25bellcurves23calculable23orichalcum23precalculi22animalcula20animalcule20calculated20calculuses20calculates19calculator19\n9 Letter Words\nbellcurve22wellcurbs21talcuming20calculous19acalculia18calculate18\n8 Letter Words\nwellcurb20calculus18talcumed17\n7 Letter Words\ncalculi16oilcups15talcums15\n6 Letter Words\noilcup14talcum14sulcus12\nStarts\nEnds\nContains\nLength\nWord Finder\nPlay Games\nDictionaries\nLists of Words\nWords by Length\nPopular Searches\nFrequent Searches\nWord Lists" ]

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[ "", null, "Linear Momentum and Collisions\n\nPresentation on theme: \"Linear Momentum and Collisions\"— Presentation transcript:\n\nLinear Momentum and Collisions\nChapter 9 Linear Momentum and Collisions\n\n9-1 Linear Momentum Momentum is a vector; its direction is the same as the direction of the velocity.\n\n9-1 Linear Momentum Change in momentum: mv 2mv\n\n9-2 Momentum and Newton’s Second Law\nNewton’s second law, as we wrote it before: is only valid for objects that have constant mass. Here is a more general form, also useful when the mass is changing:\n\n9-3 Impulse Impulse is a vector, in the same direction as the average force.\n\n9-3 Impulse We can rewrite as So we see that\nThe impulse is equal to the change in momentum.\n\n9-3 Impulse Therefore, the same change in momentum may be produced by a large force acting for a short time, or by a smaller force acting for a longer time.\n\n9-4 Conservation of Linear Momentum\nThe net force acting on an object is the rate of change of its momentum: If the net force is zero, the momentum does not change:\n\n9-4 Conservation of Linear Momentum\nInternal Versus External Forces: Internal forces act between objects within the system. As with all forces, they occur in action-reaction pairs. As all pairs act between objects in the system, the internal forces always sum to zero: Therefore, the net force acting on a system is the sum of the external forces acting on it.\n\n9-4 Conservation of Linear Momentum\nFurthermore, internal forces cannot change the momentum of a system. However, the momenta of components of the system may change.\n\nConservation of momentum\nThe total momentum of a group of interacting objects remains the same in the absence of external forces Applications: Collisions, analyzing action/reaction interactions\n\n9-4 Conservation of Linear Momentum\nAn example of internal forces moving components of a system:\n\n9-5 Inelastic Collisions\nCollision: two objects striking one another Time of collision is short enough that external forces may be ignored Inelastic collision: momentum is conserved but kinetic energy is not Completely inelastic collision: objects stick together afterwards\n\n9-5 Inelastic Collisions\nA completely inelastic collision:\n\n9-5 Inelastic Collisions\nSolving for the final momentum in terms of the initial momenta and masses:\n\n9-5 Inelastic Collisions\nBallistic pendulum: the height h can be found using conservation of mechanical energy after the object is embedded in the block.\n\n9-5 Inelastic Collisions\nFor collisions in two dimensions, conservation of momentum is applied separately along each axis:\n\n9-6 Elastic Collisions In elastic collisions, both kinetic energy and momentum are conserved. One-dimensional elastic collision:\n\n9-6 Elastic Collisions We have two equations (conservation of momentum and conservation of kinetic energy) and two unknowns (the final speeds). Solving for the final speeds:\n\n9-6 Elastic Collisions Two-dimensional collisions can only be solved if some of the final information is known, such as the final velocity of one object.\n\n9-7 Center of Mass The center of mass of a system is the point where the system can be balanced in a uniform gravitational field.\n\n9-7 Center of Mass For two objects:\nThe center of mass is closer to the more massive object.\n\n9-7 Center of Mass The center of mass need not be within the object:\n\n9-7 Center of Mass Motion of the center of mass:\n\n9-7 Center of Mass The total mass multiplied by the acceleration of the center of mass is equal to the net external force: The center of mass accelerates just as though it were a point particle of mass M acted on by\n\n9-8 Systems with Changing Mass: Rocket Propulsion\nIf a mass of fuel Δm is ejected from a rocket with speed v, the change in momentum of the rocket is: The force, or thrust, is\n\nSummary of Chapter 9 Linear momentum: Momentum is a vector\nNewton’s second law: Impulse: Impulse is a vector The impulse is equal to the change in momentum If the time is short, the force can be quite large\n\nSummary of Chapter 9 Momentum is conserved if the net external force is zero Internal forces within a system always sum to zero In collision, assume external forces can be ignored Inelastic collision: kinetic energy is not conserved Completely inelastic collision: the objects stick together afterward\n\nSummary of Chapter 9 A one-dimensional collision takes place along a line In two dimensions, conservation of momentum is applied separately to each Elastic collision: kinetic energy is conserved Center of mass:\n\nSummary of Chapter 9 Center of mass:\n\nSummary of Chapter 9 Motion of center of mass: Rocket propulsion:" ]

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[ "# The Mole & the Molar Mass\n\n## The Mole\n\n### Definition\n\nIt is the amount of matter that contains as many objects as the number of atoms in exactly 12 grams of isotopically pure 12C, which contains 6.02 × 1023 carbon atoms.\n1 Mole = 6.02 × 1023 called \"Avogadro's Number\"\n\n### Examples\n\n1 mol 12C atoms = 6.02 × 1023 atoms of 12C\n1 mol H2O molecules = 6.02 × 1023 molecules of H2O\n1 mol NO3 ions = 6.02 × 1023 nitrate ions\n\nRemarks:\n• The reason why 12 grams, it is because the atomic mass of a C atom.\n• isotopically-pure means there are 4 different isotopes of C, of which 12C is the most stable and abundant isotope.\n2 mole of N2 molecules, should contain 2 × 6.02 × 1023 of N2 molecules.\n½ mole of N2 molecules, should contain ½ 6.02 × 1023 of N2 molecules.\n\n### Exercise: Mole & Molar Mass\n\nExercise on Mole & Molar Mass\n\nCheck your answers here: Solution to the Exercise on Mole & Molar Mass\n\nExercise on Number of Atoms and Moles\n\nCheck your answers here: Solution to the Exercise on Number of Atoms and Moles\n\n## The Molar Mass\n\n### Definition:\n\nIt is the mass in grams of 1 mol of a substance. It has the unit of g/mol . i.e.\nIt numerically equals the formula weight (in amu) of the substance.\n\n### Examples\n\n1 atom of 12C has a mass of 12 amu ⇒ 1 mol of 12C has a mass of 12 g\n1 atom of 35.5Cl has a mass of 35.5 amu ⇒ 1 mol of 35.5Cl has a mass of 35.5 g\n1 atom of H2O has a mass of 18 amu ⇒ 1 mol of H2O has a mass of 18 g\nHence, 1 mole of H2O contains 6.02 × 1023 of H2O\nA mass of all these molecules is 18 g\n\nExercise on Finding Mass and Number of Moles\n\nCheck your answers here: Solution to the Exercise on Finding Mass and Number of Moles" ]

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[ "B. Two Fairs\ntime limit per test\n3 seconds\nmemory limit per test\n256 megabytes\ninput\nstandard input\noutput\nstandard output\n\nThere are $n$ cities in Berland and some pairs of them are connected by two-way roads. It is guaranteed that you can pass from any city to any other, moving along the roads. Cities are numerated from $1$ to $n$.\n\nTwo fairs are currently taking place in Berland — they are held in two different cities $a$ and $b$ ($1 \\le a, b \\le n$; $a \\ne b$).\n\nFind the number of pairs of cities $x$ and $y$ ($x \\ne a, x \\ne b, y \\ne a, y \\ne b$) such that if you go from $x$ to $y$ you will have to go through both fairs (the order of visits doesn't matter). Formally, you need to find the number of pairs of cities $x,y$ such that any path from $x$ to $y$ goes through $a$ and $b$ (in any order).\n\nPrint the required number of pairs. The order of two cities in a pair does not matter, that is, the pairs $(x,y)$ and $(y,x)$ must be taken into account only once.\n\nInput\n\nThe first line of the input contains an integer $t$ ($1 \\le t \\le 4\\cdot10^4$) — the number of test cases in the input. Next, $t$ test cases are specified.\n\nThe first line of each test case contains four integers $n$, $m$, $a$ and $b$ ($4 \\le n \\le 2\\cdot10^5$, $n - 1 \\le m \\le 5\\cdot10^5$, $1 \\le a,b \\le n$, $a \\ne b$) — numbers of cities and roads in Berland and numbers of two cities where fairs are held, respectively.\n\nThe following $m$ lines contain descriptions of roads between cities. Each of road description contains a pair of integers $u_i, v_i$ ($1 \\le u_i, v_i \\le n$, $u_i \\ne v_i$) — numbers of cities connected by the road.\n\nEach road is bi-directional and connects two different cities. It is guaranteed that from any city you can pass to any other by roads. There can be more than one road between a pair of cities.\n\nThe sum of the values of $n$ for all sets of input data in the test does not exceed $2\\cdot10^5$. The sum of the values of $m$ for all sets of input data in the test does not exceed $5\\cdot10^5$.\n\nOutput\n\nPrint $t$ integers — the answers to the given test cases in the order they are written in the input.\n\nExample\nInput\n3\n7 7 3 5\n1 2\n2 3\n3 4\n4 5\n5 6\n6 7\n7 5\n4 5 2 3\n1 2\n2 3\n3 4\n4 1\n4 2\n4 3 2 1\n1 2\n2 3\n4 1\n\nOutput\n4\n0\n1" ]

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[ "## Probabilistic Graphical Models\n\n### Principles and Techniques\n\nby Koller, Friedman\n\n### Instructor Requests\n\nMost tasks require a person or an automated system to reason—to reach conclusions based on available information. The framework of probabilistic graphical models, presented in this book, provides a general approach for this task. The approach is model-based, allowing interpretable models to be constructed and then manipulated by reasoning algorithms. These models can also be learned automatically from data, allowing the approach to be used in cases where manually constructing a model is difficult or even impossible. Because uncertainty is an inescapable aspect of most real-world applications, the book focuses on probabilistic models, which make the uncertainty explicit and provide models that are more faithful to reality.\n\nProbabilistic Graphical Models discusses a variety of models, spanning Bayesian networks, undirected Markov networks, discrete and continuous models, and extensions to deal with dynamical systems and relational data. For each class of models, the text describes the three fundamental cornerstones: representation, inference, and learning, presenting both basic concepts and advanced techniques. Finally, the book considers the use of the proposed framework for causal reasoning and decision making under uncertainty. The main text in each chapter provides the detailed technical development of the key ideas. Most chapters also include boxes with additional material: skill boxes, which describe techniques; case study boxes, which discuss empirical cases related to the approach described in the text, including applications in computer vision, robotics, natural language understanding, and computational biology; and concept boxes, which present significant concepts drawn from the material in the chapter. Instructors (and readers) can group chapters in various combinations, from core topics to more technically advanced material, to suit their particular needs.\n\nThis landmark book provides a very extensive coverage of the field, ranging from basic representational issues to the latest techniques for approximate inference and learning. As such, it is likely to become a definitive reference for all those who work in this area. Detailed worked examples and case studies also make the book accessible to students.\n\nKevin Murphy Department of Computer Science, University of British Columbia\nExpand/Collapse All\nContents (pg. ix)\nAcknowledgments (pg. xxiii)\nList of Figures (pg. xxv)\nList of Algorithms (pg. xxxi)\nList of Boxes (pg. xxxiii)\n1 Introduction (pg. 1)\n1.1 Motivation (pg. 1)\n1.2 Structured Probabilistic Models (pg. 2)\n1.3 Overview and Roadmap (pg. 6)\n1.4 Historical Notes (pg. 12)\n2 Foundations (pg. 15)\n2.1 Probability Theory (pg. 15)\n2.2 Graphs (pg. 34)\n2.3 Relevant Literature (pg. 39)\n2.4 Exercises (pg. 39)\nPart I: Representation (pg. 43)\n3 The Bayesian Network Representation (pg. 45)\n3.1 Exploiting Independence Properties (pg. 45)\n3.2 Bayesian Networks (pg. 51)\n3.3 Independencies in Graphs (pg. 68)\n3.4 From Distributions to Graphs (pg. 78)\n3.5 Summary (pg. 92)\n3.6 Relevant Literature (pg. 93)\n3.7 Exercises (pg. 96)\n4 Undirected Graphical Models (pg. 103)\n4.1 The Misconception Example (pg. 103)\n4.2 Parameterization (pg. 106)\n4.3 Markov Network Independencies (pg. 114)\n4.4 Parameterization Revisited (pg. 122)\n4.5 Bayesian Networks and Markov Networks (pg. 134)\n4.6 Partially Directed Models (pg. 142)\n4.7 Summary and Discussion (pg. 151)\n4.8 Relevant Literature (pg. 152)\n4.9 Exercises (pg. 153)\n5 Local Probabilistic Models (pg. 157)\n5.1 Tabular CPDs (pg. 157)\n5.2 Deterministic CPDs (pg. 158)\n5.3 Context-Specific CPDs (pg. 162)\n5.4 Independence of Causal Influence (pg. 175)\n5.5 Continuous Variables (pg. 185)\n5.6 Conditional Bayesian Networks (pg. 191)\n5.7 Summary (pg. 193)\n5.8 Relevant Literature (pg. 194)\n6 Template-Based Representations (pg. 199)\n6.1 Introduction (pg. 199)\n6.2 Temporal Models (pg. 200)\n6.3 Template Variables and Template Factors (pg. 212)\n6.4 Directed Probabilistic Models for Object-Relational Domains (pg. 216)\n6.5 Undirected Representation (pg. 228)\n6.6 Structural Uncertainty (pg. 232)\n6.7 Summary (pg. 240)\n6.8 Relevant Literature (pg. 242)\n7 Gaussian Network Models (pg. 247)\n7.1 Multivariate Gaussians (pg. 247)\n7.2 Gaussian Bayesian Networks (pg. 251)\n7.3 Gaussian Markov Random Fields (pg. 254)\n7.4 Summary (pg. 257)\n7.5 Relevant Literature (pg. 258)\n8 The Exponential Family (pg. 261)\n8.1 Introduction (pg. 261)\n8.2 Exponential Families (pg. 261)\n8.3 Factored Exponential Families (pg. 266)\n8.4 Entropy and Relative Entropy (pg. 269)\n8.5 Projections (pg. 273)\n8.6 Summary (pg. 282)\n8.7 Relevant Literature (pg. 283)\nPart II: Inference (pg. 285)\n9 Exact Inference: Variable Elimination (pg. 287)\n9.1 Analysis of Complexity (pg. 288)\n9.2 Variable Elimination: The Basic Ideas (pg. 292)\n9.3 Variable Elimination (pg. 296)\n9.4 Complexity and Graph Structure: Variable Elimination (pg. 305)\n9.5 Conditioning (pg. 315)\n9.6 Inference with Structured CPDs (pg. 325)\n9.7 Summary and Discussion (pg. 336)\n9.8 Relevant Literature (pg. 337)\n9.9 Exercises (pg. 338)\n10 Exact Inference: Clique Trees (pg. 345)\n10.1 Variable Elimination and Clique Trees (pg. 345)\n10.2 Message Passing: Sum Product (pg. 348)\n10.3 Message Passing: Belief Update (pg. 364)\n10.4 Constructing a Clique Tree (pg. 372)\n10.5 Summary (pg. 376)\n10.6 Relevant Literature (pg. 377)\n10.7 Exercises (pg. 378)\n11 Inference as Optimization (pg. 381)\n11.1 Introduction (pg. 381)\n11.2 Exact Inference as Optimization (pg. 386)\n11.3 Propagation-Based Approximation (pg. 391)\n11.4 Propagation with Approximate Messages (pg. 430)\n11.5 Structured Variational Approximations (pg. 448)\n11.6 Summary and Discussion (pg. 473)\n11.7 Relevant Literature (pg. 475)\n11.8 Exercises (pg. 477)\n12 Particle-Based Approximate Inference (pg. 487)\n12.1 Forward Sampling (pg. 488)\n12.2 Likelihood Weighting and Importance Sampling (pg. 492)\n12.3 Markov Chain Monte Carlo Methods (pg. 505)\n12.4 Collapsed Particles (pg. 526)\n12.5 Deterministic Search Methods (pg. 536)\n12.6 Summary (pg. 540)\n12.7 Relevant Literature (pg. 541)\n12.8 Exercises (pg. 544)\n13 MAP Inference (pg. 551)\n13.1 Overview (pg. 551)\n13.2 Variable Elimination for (Marginal) MAP (pg. 554)\n13.3 Max-Product in Clique Trees (pg. 562)\n13.4 Max-Product Belief Propagation in Loopy Cluster Graphs (pg. 567)\n13.5 MAP as a Linear Optimization Problem (pg. 577)\n13.6 Using Graph Cuts for MAP (pg. 588)\n13.7 Local Search Algorithms (pg. 595)\n13.8 Summary (pg. 597)\n13.9 Relevant Literature (pg. 598)\n13.10 Exercises (pg. 601)\n14 Inference in Hybrid Networks (pg. 605)\n14.1 Introduction (pg. 605)\n14.2 Variable Elimination in Gaussian Networks (pg. 608)\n14.3 Hybrid Networks (pg. 615)\n14.4 Nonlinear Dependencies (pg. 630)\n14.5 Particle-Based Approximation Methods (pg. 642)\n14.6 Summary and Discussion (pg. 646)\n14.7 Relevant Literature (pg. 647)\n14.8 Exercises (pg. 649)\n15 Inference in Temporal Models (pg. 651)\n15.2 Exact Inference (pg. 653)\n15.3 Approximate Inference (pg. 661)\n15.4 Hybrid DBNs (pg. 675)\n15.5 Summary (pg. 688)\n15.6 Relevant Literature (pg. 690)\n15.7 Exercises (pg. 692)\nPart III: Learning (pg. 695)\n16 Learning Graphical Models: Overview (pg. 697)\n16.1 Motivation (pg. 697)\n16.2 Goals of Learning (pg. 698)\n16.3 Learning as Optimization (pg. 702)\n16.5 Relevant Literature (pg. 715)\n17 Parameter Estimation (pg. 717)\n17.1 Maximum Likelihood Estimation (pg. 717)\n17.2 MLE for Bayesian Networks (pg. 722)\n17.3 Bayesian Parameter Estimation (pg. 733)\n17.4 Bayesian Parameter Estimation in Bayesian Networks (pg. 741)\n17.5 Learning Models with Shared Parameters (pg. 754)\n17.6 Generalization Analysis (pg. 769)\n17.7 Summary (pg. 776)\n17.8 Relevant Literature (pg. 777)\n17.9 Exercises (pg. 778)\n18 Structure Learning in Bayesian Networks (pg. 783)\n18.1 Introduction (pg. 783)\n18.2 Constraint-Based Approaches (pg. 786)\n18.3 Structure Scores (pg. 790)\n18.4 Structure Search (pg. 807)\n18.5 Bayesian Model Averaging (pg. 824)\n18.6 Learning Models with Additional Structure (pg. 832)\n18.7 Summary and Discussion (pg. 838)\n18.8 Relevant Literature (pg. 840)\n19 Partially Observed Data (pg. 849)\n19.1 Foundations (pg. 849)\n19.2 Parameter Estimation (pg. 862)\n19.3 Bayesian Learning with Incomplete Data (pg. 897)\n19.4 Structure Learning (pg. 908)\n19.5 Learning Models with Hidden Variables (pg. 925)\n19.6 Summary (pg. 933)\n19.7 Relevant Literature (pg. 934)\n19.8 Exercises (pg. 935)\n20 Learning Undirected Models (pg. 943)\n20.1 Overview (pg. 943)\n20.2 The Likelihood Function (pg. 944)\n20.3 Maximum (Conditional) Likelihood Parameter Estimation (pg. 949)\n20.4 Parameter Priors and Regularization (pg. 958)\n20.5 Learning with Approximate Inference (pg. 961)\n20.6 Alternative Objectives (pg. 969)\n20.7 Structure Learning (pg. 978)\n20.8 Summary (pg. 996)\n20.9 Relevant Literature (pg. 998)\n20.10 Exercises (pg. 1001)\nPart IV: Actions and Decisions (pg. 1007)\n21 Causality (pg. 1009)\n21.1 Motivation and Overview (pg. 1009)\n21.2 Causal Models (pg. 1014)\n21.3 Structural Causal Identifiability (pg. 1017)\n21.4 Mechanisms and Response Variables (pg. 1026)\n21.5 Partial Identifiability in Functional Causal Models (pg. 1031)\n21.6 Counterfactual Queries (pg. 1034)\n21.7 Learning Causal Models (pg. 1040)\n21.8 Summary (pg. 1053)\n21.9 Relevant Literature (pg. 1054)\n21.10 Exercises (pg. 1055)\n22 Utilities and Decisions (pg. 1059)\n22.1 Foundations: Maximizing Expected Utility (pg. 1059)\n22.2 Utility Curves (pg. 1064)\n22.3 Utility Elicitation (pg. 1068)\n22.4 Utilities of Complex Outcomes (pg. 1071)\n22.5 Summary (pg. 1081)\n22.6 Relevant Literature (pg. 1082)\n22.7 Exercises (pg. 1084)\n23 Structured Decision Problems (pg. 1085)\n23.1 Decision Trees (pg. 1085)\n23.2 Influence Diagrams (pg. 1088)\n23.3 Backward Induction in Influence Diagrams (pg. 1095)\n23.4 Computing Expected Utilities (pg. 1100)\n23.5 Optimization in Influence Diagrams (pg. 1107)\n23.6 Ignoring Irrelevant Information (pg. 1119)\n23.7 Value of Information (pg. 1121)\n23.8 Summary (pg. 1126)\n23.9 Relevant Literature (pg. 1128)\n23.10 Exercises (pg. 1130)\n24 Epilogue (pg. 1133)\nA Background Material (pg. 1137)\nA.1 Information Theory (pg. 1137)\nA.2 Convergence Bounds (pg. 1143)\nA.3 Algorithms and Algorithmic Complexity (pg. 1146)\nA.4 Combinatorial Optimization and Search (pg. 1154)\nA.5 Continuous Optimization (pg. 1161)\nBibliography (pg. 1173)\nNotation Index (pg. 1211)\nSubject Index (pg. 1215)\n\n#### Daphne Koller\n\nDaphne Koller is Professor in the Department of Computer Science at Stanford University.\n\n#### Nir Friedman\n\nNir Friedman is Professor in the Department of Computer Science and Engineering at Hebrew University.\n\nInstructors\nYou must have an instructor account and submit a request to access instructor materials for this book.\neTextbook" ]

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[ "If the Universe is Flat, why does it look 3D from our View?\n\nSo I know that the universe is flat in a 4D point of view, but what about a 3D point of view? The universe does not SEEM flat from our view, so how can it really be so? Please answer more simple, im not an astronomy pro.\n\nSorry if this is a dumb question, I don't have a degree in astrophysics or anything, i'm just a kid who is really interested in the universe, so i'm sorry (in advance) if this is a simply answered bad question.\n\nSo I know that the universe is flat in a 4D point of view, but what about a 3D point of view?\n\nYou seem to be misunderstanding the term \"flatness\". In this context, it does not mean the universe looks like a piece of paper in any sense. It instead refers to whether space is warped.\n\ni'm just a kid who is really interested in the universe\n\nI'll try to simplify as much as possible. I'm assuming you've learned that triangles have 180°. For the mathematics you'll be dealing with in middle school, high school and even most of college, you'll only have to deal with that.\n\nThe more accurate statement is that triangles have 180° when our geometry is not warped. When triangles have 180°, we say we are dealing with Euclidean geometry (named after Euclid, an ancient Greek mathematician). There are in fact cases in higher mathematics when this does not hold, which we call Non-Euclidean geometries. Just as a side note: Euclidean geometry was developed thousands of years ago. On the other hand, Non-Euclidean geometries only began to develop in the 1800s, so they're pretty recent.\n\nAnyway, Euclidean geometry deals with flat spaces. For example, if you draw a triangle on a piece of paper, it will indeed have 180°. However, Non-Euclidean geometry deals with curved (warped) spaces. For example, if you draw a triangle on the surface of a ball (which is not flat), the triangle will actually have more than 180°. If you instead draw the triangle on a potato chip or saddle-shaped surface (also not flat), it will have less than 180°.\n\nSo with regards to the Universe: let's say we draw a triangle connecting three stars in outer space (we can actually use any object or simply points, but I'm using stars as an example). Will that triangle be less than, equal to or more than 180°?\n\nAccording to our measurements, it will actually be equal to 180°; the Universe is indeed flat. This has major implications for physics and astronomy." ]

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[ "# How to show that $B$ embeds to $G/U^-$ densely?\n\nLet $G=GL_n$ and $B$ its Borel subgroup consisting of all upper triangular matrices. Let $U^-$ be the unipotent subgroup of $G$ consisting of all unipotent lower triangular matrices. How to show that $B$ embeds to $G/U^-$ and this embedding is dense? Thank you very much.\n\nThe embedding of $B$ is simply given by mapping an element of $B$ to its coset in $G/U^-$. Looking at the Lie algebras, you see immediately that this embedding is a local diffemorphism around $e$. I am not sure what's the simplest way of proving that the image is dense, it should be the \"big cell\" in the Bruhat decompositon." ]

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[ "Choice and sizing\n\nIn order to calculate the capacity of a pressure tank with membrane, it is necessary to know the plant specifications. For example, water consumption required.\n\nThe sizing of the tank can be calculated using the following formula:\n\nTaking into account:\n\nK\nWorking coefficient of the pump (see table)\nAmax\nAverage flow (liters per minute)\nPmax\nMaximum working pressure of the pump(bar)\nPmin\nMinimum working pressure of the pump(bar)\nPprec\nPre-charge pressure of the tank(bar)\n\nWarning!Always set the pre-charge of the tank 0,2 bar less than the pump power pressure\n\nExample\n\nSystem data:\n\n• Pump power4 HP\n• K= 0,375\n• Amax= 120liters per minute\n• Pmax= 7 bar\n• Pmin= 2,2 bar\n• Pprec= 2 bar\n\nPump power\n(HP)\nCoefficient\n(K)\n1–20,25\n2,5–40,375\n5–80,625\n9–120,875\n\nIn any case we will adopt the closest measure to the calculated value" ]

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[ "", null, "Previous Article in Journal\nBioH2 Production Using Microalgae: Highlights on Recent Advancements from a Bibliometric Analysis\n\nFont Type:\nArial Georgia Verdana\nFont Size:\nAa Aa Aa\nLine Spacing:\nColumn Width:\nBackground:\nArticle\n\n# Capacity Optimization of Rainwater Harvesting Systems Based on a Cost–Benefit Analysis: A Financial Support Program Review and Parametric Sensitivity Analysis\n\n1\nIndustry-University Cooperation Foundation, Pukyong National University, Busan 48513, Republic of Korea\n2\nCivil Engineering, Department of Sustainable Engineering, Pukyong National University, Busan 48513, Republic of Korea\n3\nDisaster Prevention Research Institute, Pukyong National University, Busan 48513, Republic of Korea\n4\nHydrology Engineering & Consulting Center, Korea Inc., 1304-1306, 233, Gasan digital 1-ro, Geumcheon-gu, Seoul 08501, Republic of Korea\n*\nAuthor to whom correspondence should be addressed.\nWater 2023, 15(1), 186; https://doi.org/10.3390/w15010186\nReceived: 20 November 2022 / Revised: 16 December 2022 / Accepted: 28 December 2022 / Published: 2 January 2023\n\n## Abstract\n\n:\nWater risk has been continuously rising due to climate change and ownership disputes of water resources. Dam construction to secure water resources may lead to environmental problems and upstream immersion. On the other hand, rainwater harvesting systems can effectively supply water at a low cost, although economic efficiency of these systems is still debatable. This study evaluates financial support programs to promote installation of rainwater harvesting systems, increasing economic feasibility. Based on a cost–benefit analysis, capacity optimization methods are further suggested. A sensitivity analysis is performed to determine the relative importance among uncertain parameters such as inflation and discount rates. In doing so, priority factors to consider in the design of rainwater harvesting systems are ultimately identified. A net present value, although it is sensitive to the inflation rate, is shown to be more appropriate to estimate the economic efficiency of rainwater harvesting system, compared to the typical cost–benefit ratio. Because the high future value overestimates the economic feasibility of rainwater harvesting systems, proper inflation rates should be applied. All in all, a funding program to promote rainwater harvesting systems significantly increases the benefits. Thus, national financial support policies are recommended to ensure economic feasibility of rainwater harvesting systems.\n\n## 1. Introduction\n\nMany countries across the world have focused on improving water resource management to cope with the water stress caused by climate change, rapid urbanization, and ownership disputes over water resources. The easiest way to mitigate water stress is to construct dams. Dams have been constructed for centuries for various purposes, such as water security, flood control, and power generation. However, as the negative impacts of dams on the environment and hydrological cycles have been discovered, new strategies have been sought to secure water resources. Rainwater harvesting (RWH) systems offer a practical way to mitigate water stress. RWH systems provide direct runoff containment while simultaneously storing water, which can then be used for irrigation, household water supplies, etc. . This favorable function of RWH systems can reduce public water demand concentrated around water sources and mitigate water stress .\nVarious factors influence the performance of RWH systems, including catchment area, rainfall patterns, and demand. However, as shown in previous studies that have focused on determining optimal size, the performance of RWH systems significantly depends on their capacity [3,4,5,6,7]. In general, the capacity of an RWH system is determined by the expected demand. However, when designed with the capacity to supply all of the expected demand, the installation and operation costs of RWH systems can be irrecoverable.\nThe capacity of RWH systems can be constrained by the reliability or runoff reduction rate, which is a representative index for evaluating the performance of RWH systems, to avoid over-design. Reliability refers the ratio of the number of periods that satisfy the demand to a given total period. The appropriate capacity for RWH systems can be determined by restricting the reliability. On the other hand, various other conditions also need to be considered to determine the appropriate capacity for RWH systems, which means that the optimization problem to determine the optimal capacity becomes more complicated. The following optimization methods have been used to solve the optimization problem to determine the optimal capacity of RWH systems: linear programming [8,9,10], non-linear programming [11,12], and the heuristic method [13,14,15,16,17,18].\nThe goal of avoiding over-designing the capacity of RWH systems is to maximize the revenue that they generate, i.e., the capacity can be determined using optimization models with objective functions that maximize the benefits. Typical methods for analyzing the economic feasibility of RWH systems include the net present value (NPV) and the benefit–cost ratio (BCR). The NPV is an indicator used in cost–benefit analyses, which represents the difference between the present value of benefits and the present value of costs over a period of time. The BCR refers to the ratio between the present value of benefits and the present value of costs over a period of time. In this regard, the authors of found that the BCR has the limitation of not reflecting the size of a project’s funds. In contrast, the NPV has the limitation of missing investment-efficient opportunities.\nThe social discount rate and the inflation rate are key parameters in cost–benefit analysis because they represent the potential costs of resource use over time and they compare projected revenues and costs over different time periods . The higher the social discount rate, the smaller the present value of any future benefit or cost. Accordingly, many previous studies have defined the social discount rate as the subject of sensitivity analysis [19,20,21,22,23,24,25,26,27]. At the same time, for the purpose of sustainable water resource management, Korean local governments have implemented financial support programs to promote the installation and operation of RWH systems. For example, in Incheon City, Korea, some of the installation costs for RWH systems are supported, and water utility billing relief is also provided. Therefore, these national or social support policies can affect the cost–benefit analysis of RWH systems.\nOver the past decade, RWH systems have been recognized as inefficient facilities because of their low benefits relative to costs [24,28]. Accordingly, recent studies considered social benefits such as constant production cost reduction, dam-related cost reduction, excellent outflow reduction, environmental pollution reduction, heat wave reduction, and fine dust reduction as factors of benefits [29,30,31,32,33]. However, the social benefits are ambiguous to evaluate and are not real economic benefits that users who use RWH systems can actually feel.\nThe purpose of this study is to determine the capacity of the economic RWH systems, considering the real economic benefits that users who use rainwater utilization facilities can actually feel. At the same time, it is difficult to find cases in previous studies on the cost benefits of RWH systems [8,13,27,34,35,36,37,38,39,40,41] where BCR and NPV have been compared and evaluated or the effects of social discount rate and inflation on cost–benefit analysis have been examined. Thus, in this study, we explored which indicator would be the most appropriate for the cost–benefit analysis of RWH systems based on the results from a comparative evaluation of the BCR and NPV. In addition, via sensitivity analysis, we identified the priority factors to be considered when determining the capacity of RWH systems. We also analyzed the economic feasibility of RWH systems while considering financial support programs to review the necessity of national financial support policies. Because the sensitivity analysis needed a lot of computational time, an optimization model that could determine the economically feasible capacity of RWH systems using a particle swarm algorithm was further developed.\n\n## 2. Methods\n\n#### 2.1. Financial Support Policies for Rainwater Harvesting Systems in South Korea\n\nThe annual precipitation in South Korea is 1.2 times (1207 mm) higher than the world’s average (about 1000 mm), but most precipitation is concentrated in the monsoon season (from 25 June to 25 September), and more than 60% of the annual precipitation occurs over the sea. Thus, the actual available water resources per person do not amount to much. Additionally, the authors found that Korea is among the few OECD countries that experience medium-high water stress .\nThe Korean Ministry of Environment enacted the Act for the Promotion and Support of Water Reuse (APSWR) to utilize water resources more efficiently and promote the sustainable use of water resources. In the APSWR, water reuse is defined as the recycling of treated wastewater (rainwater, sewage, etc.) for beneficial purposes, such as household water supplies and stream or lake maintenance. In addition, the APSWR states that local governments should make efforts to provide technical and financial support for the installation and operation of RWH systems to promote water reuse. The local governments in South Korea provide various financial support programs to promote the installation of RWH systems in accordance with the APSWR. Table 1 shows representative local governments that implement financial support programs for the installation and operation of RWH systems. As shown in Table 1, local governments in South Korea cover most of the costs of installing RWH systems, although the scale of support varies depending on the financial situation. In South Korea, water utility bills consist of water supply charges, sewage charges, and water usage charges. Table 2 presents a water bill for 1000 $m 3$ of water over 1 month in Incheon, Korea. In the case of Incheon, 10% of rainwater usage is deducted from the total water consumption as a benefit for the operation of RWH systems. For example, if rainwater usage is 1000 $m 3$ and the total water consumption is 1000 $m 3$, the water utility bill would only charge for 900 $m 3$.\n\n#### 2.2. Study Area\n\nIn 2007, Cheongna International City, Incheon, South Korea, planned to supply rainwater to public facilities using RWH systems to enhance its image as an eco-friendly city. The Korea Land and Housing Corporation divided Cheongna-dong into five drainage areas and designed a large-scale RWH system to supply household water for schools and parks, irrigation water for public parks, and water for road cleaning. In this study, a cost–benefit analysis was conducted on the RWH system that was planned for District 1 (Cheongna 1-dong, CN1-D), and the analysis results were compared and reviewed using design data .\nThe watershed area of the study area (CN1-D) is 73.92 ha and the catchment area of the RWH system is 19.24 ha (Figure 1). The Korea Land Corporation simulated the daily inflow to the RWH system over 10 years (1995–2004) via the long-term simulation of watershed runoff using the storm water management model from the EPA (EPA SWMM) and daily precipitation data from Incheon Weather Station (Figure 2) .\nFigure 3 presents the average daily demand by month, with the calculation method as follows: the daily inflow of waterways was averaged with the daily evaporation over 30 years (1971–2000). The household water for parks was calculated by multiplying the area of the park, the green area per capita, and the water usage per capita per day. The household water for parks was designed to supply 10% of the target draft during winter (January, February, and December) and the monsoon season (from 16 June to 15 July). The household water for schools was calculated by considering the number of schools, grades, classes, students per class, teachers, and the water usage per capita per day. During the winter and summer vacations, the water usage of schools only corresponded to the number of teachers. The water for road cleaning was calculated by considering the road lengths and the water usage per 1 km.\n\n#### 2.3.1. Cost–Benefit Analysis\n\nCost–benefit analyses are conducted to establish the economic feasibility of public projects by comparing the investment costs to the expected benefits. In terms of the cost–benefit analysis results, a project is economically feasible when the expected benefits are greater than the investment costs. Several criteria are used in cost–benefit analyses to determine the economic efficiency of particular investment projects and whether they should be undertaken. Representative criteria for cost–benefit analysis include net present value (NPV) and the benefit–cost ratio (BCR).\nThe NPV measures the present value of the net benefits of the considered project. The NPV is formulated as follows:\n$NPV = ∑ t = 0 T B t ( 1 + r ) t − ∑ t = 0 T C t ( 1 + r ) t$\nwhere $B t$ is the benefit at time t, r is the discount rate at time t, $C t$ represents the costs at time t, and T is the planning horizon year. A positive NPV means that the considered project could be implemented based on economic feasibility.\nThe BCR is the ratio of the present value of benefits to the present value of costs. The BCR is formulated as follows:\n$BCR = ∑ t = 0 T B t ( 1 + r ) t / ∑ t = 0 T C t ( 1 + r ) t$\nWhen the BCR of the considered project exceeds 1, it means that the project is acceptable in terms of economic efficiency, i.e., the present value of the benefits of the project is greater than the present value of the costs.\nThe BCR should not be used to rank mutually exclusive options, as it can lead to rankings that are inconsistent with those obtained using the NPV . For example, when the initial investment costs of project A are 50 and the present value of the total benefits is 100, the NPV and BCR of project A are calculated as 50 and 2, respectively. Conversely, when the initial investment costs of project B are 100 and the present value of the total benefits is 200, the NPV and BCR of project B are 100 (which is twice that of project A) and 2 (which is the same as that of project A), respectively. Therefore, the BCR has a limitation in terms of not reflecting the size of a project’s funds, whereas the NPV has a limitation in terms of missing investment-efficient opportunities. Thus, the appropriate criterion for each individual project should be selected after a comparative evaluation of these two criteria.\nIn order to economically determine the capacity of rainwater harvesting systems, the costs and benefits must be quantitatively determined. Costs are inevitably incurred throughout the operation of rainwater harvesting systems, including installation costs, labor expenses for the operation and maintenance of facilities, and electricity charges for the use of pumps. Benefits can include reduced water utility charges due to replacing water usage with rainwater usage.\nTable 3 shows the costs and benefits that were considered in the cost–benefit analysis of the RWH system in this study. In the cost–benefit analysis, the costs included the installation and maintenance costs of the RWH system. It was not easy to estimate the installation and maintenance costs of the RWH system because the installation and maintenance costs of rainwater facilities vary greatly depending on the area, local environment, operation method, and materials and structure. In accordance with the APSWR, which recommends the installation of RWH systems in densely populated areas and facilities with many users, large-scale rainwater facilities with concrete tanks are mostly installed underground in urban areas. Based on previous studies [44,45,46], construction of rainwater harvesting systems with concrete tanks costs 450,000 KRW/$m 3$ (346 USD/$m 3$), and the maintenance/operation costs are 2% of the construction costs (Table 3). In addition, Article 19 (1) 1 of the Enforcement Rules of the Local Public Enterprises Act of South Korea stipulates that the life span of water supply facilities (water intake, water supply, water purification, and drainage facilities) is 30 years; accordingly, the cost–benefit analysis period in this study was set as 30 years.\nThe benefits of the operation of the RWH system in this study included reductions in water utility bills by replacing water usage with rainwater usage, financial support for installation costs, and water utility billing relief (Table 3). The study area (CH1-D) is located in Incheon (Figure 1), and the considered benefits of the installation/operation of the RWH system were determined based on Incheon’s water billing standards (Table 2) and financial support programs (Table 1). The benefits of replacing water usage with rainwater usage were calculated as monthly benefits by applying rainwater usage to the water utility billing regulations. Incheon fully or partially supported the installation costs for the RWH system. Because the designed RWH system in CH1-D is a large-scale facility that supplies up to 282 $m 3$/day, the local government was not able to support all of the installation costs. Thus, the financial support for the installation costs was determined to be KRW 10 million (USD 7692) by referring to the financial support from other local governments (e.g., Busan and Suwon offer up to KRW 10 million). Incheon reduces water supply charges, sewage charges, and water usage charges by 10% each when rainwater is utilized. In order to determine the reduced charges, the total water usage had to be determined. As it was not easy to quantify the total water usage in the design stage, we determined the total water usage as the difference between the rainwater supply and the target supply.\n\n#### 2.3.2. Discount Rate and Inflation Rate\n\nThe discount rate is the amount of returns used to discount future cash flows back to their present value . The discount rate used in the cost–benefit analysis of public projects is the social discount rate. Table A1 shows estimates for the social discount rate in a number of countries across the world. In addition, the cost–benefit analysis results depend on the adopted social discount rate (i.e., r in Equations (1) and (2)). The social discount rate is a key parameter in the cost–benefit analysis because it represents the potential costs of resource use over time and compares projected revenues and costs (net cash flows) over different periods of time . The higher the social discount rate, the smaller the present value of any future benefits or costs. Accordingly, many studies in the literature have defined the social discount rate as the subject of sensitivity analysis [19,20,21].\nIn the cost–benefit analysis of the RWH system, the inflation rate was also used as one of the uncertain parameters because the RWH system constantly incurred operating and maintenance costs after installation. Considering the inflation rate, the benefits $B n$ and the costs $C n$ in Equations (1) and (2) were determined as follows:\n$B t = B i n s t a l l + ( B r e p l a c e + B r e l i e f ) × ( 1 + I R ) t$\n$C t = C i n s t a l l + C o p e r × ( 1 + I R ) t$\nwhere $I R$ is the inflation rate, $C i n s t a l l$ is the initial installation costs, $B i n s t a l l$ is the subsidies for $C i n s t a l l$. As B and C are incurred at the present point, they do not reflect the inflation rate. The expression $B r e p l a c e$ is the benefit of replacing water usage with rainwater usage. It is not easy to determine water usage in designing RWH systems. The rainwater supply should be replaced with a water supply to meet demand when rainwater is insufficient. Thus, this study assumed that the benefit of replacing water usage with rainwater usage is the value obtained by subtracting the utility for water usage from the utility when the water supply meets all demand. The expression $B r e l i e f$ is the water utility billing relief, and $C o p e r$ represents the operation and maintenance costs. Hence, Equations (1) and (2) can be modified as follows:\n$NPV = B i n s t a l l + ∑ t = 0 T ( B r e p l a c e + B r e l i e f ) ( 1 + I R ) t ( 1 + r ) t − C i n s t a l l + ∑ t = 0 T C o p e r ( 1 + I R ) t ( 1 + r ) t$\n$BCR = B i n s t a l l + ∑ t = 0 T ( B r e p l a c e + B r e l i e f ) ( 1 + I R ) t ( 1 + r ) t / C i n s t a l l + ∑ t = 0 T C o p e r ( 1 + I R ) t ( 1 + r ) t$\nThus, we analyzed the sensitivity of the social discount rate and the inflation rate to review the effects of changes in those rates on the cost–benefit analysis results for the RWH system.\n\n#### 2.4.1. Simulation Model for Rainwater Harvesting Systems\n\nThe economically feasible capacity of rainwater harvesting systems can be determined using cost–benefit analysis and mass balance analysis. Mass balance analysis considers the target draft, inflow (caught rainwater), and capacity of rainwater harvesting systems. We used the following reservoir mass balance equation for the mass balance analysis of the rainwater harvesting system :\n$S T t = S T t − 1 + P P t + Q F t − R t − E V t$\nwhere $S t$ is the reservoir storage at the end of time period t, $P P t$ is the precipitation amount on the reservoir surface, $Q F t$ is the reservoir inflow over period t, $R t$ is the reservoir release, and $E V t$ is the evaporation.\nUnlike reservoirs, underground rainwater harvesting systems are sealed, so there is little precipitation or evaporation on the surface. Thus, $P P t$ and $E V t$ could be ignored in Equation (7). However, because rainwater harvesting systems involve rainwater usage (yield, $Y t$), it was necessary to divide the release ($R t$) in Equation (7) into yield ($Y t$) and spill ($E X R t$). Table 4 presents the RWH system mass balance equation that was derived by modifying the reservoir mass balance equation, as well as the conditions and equations for determining the yield and spill of the RWH system. In Table 4, $T D t$ is the target draft in period t.\n\n#### 2.4.2. Optimization Model to Determine the Capacity of Rainwater Harvesting Systems Considering Benefit–Cost Analysis\n\nIn this study, we aimed to analyze the sensitivity of the parameters (discount rate and inflation rate) that were determined in the cost–benefit analysis and evaluate the Korean policies that promote the installation/operation of rainwater harvesting systems. In the cost–benefit analysis of RWH systems, the NPV and BCR are calculated differently depending on the capacity of the selected RWH system, even when constant discount rates and inflation rates are applied. As it was inefficient to determine the optimum capacity to maximize the NPV or BCR by generating various capacity scenarios for the considered RWH system, we designed a simple optimization model that could determine the optimum economically feasible capacity of the RWH system using a PSO.\nPSO algorithms are meta-heuristic methods that are based on a stochastic optimization technique [49,50]. Particle swarm algorithms are developed by mimicking the social behavioral modalities of the migration of bird populations. The concept of PSO algorithms involves accelerating each particle toward its $P b e s t$ and $G b e s t$ locations in each iteration, as shown in Figure 4 .\nIn Figure 4, $P b e s t$ means that particle i memorized the best position that it has ever found, $G b e s t$ is the best position of the swarm (particle group), and $x i k$ is the current position of the the ith particle x in iteration k. The next position of the $x i k + 1$ particle is formulated as follows:\n$x i k + 1 = ω x i k + c 1 ( P b e s t − x i k ) + c 2 ( G b e s t − x i k )$\nwhere $ω$, $c 1$, and $c 2$ are weighting factors. The best static parameters are recommended as $ω = 0.72984$ and $c 1 = c 2 = 2.05$ .\nFigure 5 shows a schematic diagram of a PSO searching for the economically feasible capacity of an RWH system. The objective function of the optimization problem is to maximize the NPV or BCR and is formulated as follows:\n$M a x i m i z e z = NPV or BCR$\nwhere the NPV/BCR is calculated from the simulation results of an RWH system using the capacity determined by the PSO. Thus, in the optimization problem, the decision variable is the capacity of the selected RWH system. In this study, the PSO evaluated the NPV/BCR that was calculated using the simulation model to search for the optimum capacity of the RWH system to maximize the NPV/BCR.\nIn our RWH system simulation model, the NPV and BCR were determined as follows. The RWH system was simulated on a daily basis from 1995 to 2004 (10 years). The water utility bills were calculated on a monthly basis for those 10 years using the simulation results of the RWH system to determine the benefits. The NPV and BCR were calculated for 30 years, i.e., the durable life span of RWH systems with concrete tanks, using the previously determined average monthly water utility bills.\n\n## 3. Results\n\n#### 3.1. Water Balance Analysis of the Rainwater Harvesting System\n\nIn this study, we performed a water balance analysis for the capacity of the considered RWH system using the equations in Table 4. The capacity of the RWH system used in the water balance analysis was 1961 cases, comprising 5 $m 3$ units from 200 $m 3$ to 10,000 $m 3$.\nFigure 6 shows the results for the temporal and volumetric reliability of the capacity of the RWH system. Reliability has been applied in numerous previous studies because it is easy to use and shows the interrelationship between yield and target draft. Reliability is divided into temporal reliability ($R e l t e m p$) and volumetric reliability ($R e l v o l$). Temporal reliability ($R e l t e m p$) represents the ratio of the period when the yield is satisfied with the target draft to the total period T. Temporal reliability is formulated as follows:\n$R e l t e m p = 1 T ∑ t = 1 T B V t$\nwhere $B V t$ is a binary variable, which is 1 or 0. If the yield ($Y t$) satisfies the target draft ($T D t$) for period t, $B V t$ is 1; otherwise, it is 0. Volumetric reliability is expressed as:\n$R e l v o l = ∑ t = 1 T Y t T D t$\nAs shown in Figure 6, the temporal and volumetric reliability increased as the size of the RWH system increased, whereas the uplift rate for the reliability decreased as the capacity increased. These results were representative of the meteorological characteristics of South Korea, where most of the annual precipitation is concentrated in the summer. Further research on the water rationing strategies of RWH systems is needed to increase reliability.\n\n#### 3.2. Comparative Evaluation of Two Cost–Benefit Analysis Methods\n\nTo compare the cost–benefit analysis results using the NPV and BCR, we assumed that the RWH system operated for 30 years, which is the durable life span of water resource facilities. The NPV and BCR were determined under the following conditions: an inflation rate of 4.5% and a discount rate of 3.4%.\nFigure 7 shows the NPV and BCR of the capacity of the considered RWH system. The BCR in Figure 7a peaked at a capacity of 285 $m 3$ and continued to decrease as the capacity increased. As shown in Figure 7b, the net present value increased as the capacity increased, peaked at a capacity of 1105 $m 3$, and then gradually decreased as the capacity continued to increase. According to the cost–benefit analysis of the RWH system using BCR, the return on investment was 2.89 times the initial investment when the capacity of the RWH system was 285 $m 3$. However, as shown in Figure 7, when the capacity was 285 $m 3$, the temporal reliability was 16.7% and the volumetric reliability was 7.3%, indicating that the operational efficiency of the RWH system was significantly unreliable. On the other hand, in the case of NPV, the maximum profit of KRW 581 million (USD 446,923) was generated when the capacity was 1105 $m 3$. The capacity of 285 $m 3$, which was found to generate the maximum return in the cost–benefit analysis using BCR, generated KRW 366 million (USD 281,538) in the cost–benefit analysis using NPV, which was a much lower amount of profit.\nThese results showed that the BCR had significant drawbacks. The BCR can be misleading when two compared projects incur different costs (i.e., when the costs vary depending on the capacity of the selected RWH system). The BCR having a value greater than 1 indicates that a project is worthwhile in the absolute but does not provide a basis for comparison to other projects , i.e., the BCR has the disadvantage of not reflecting the scale of investment. For example, a project that costs USD 100 could generate a more significant increase in real wealth than a project that costs USD 10 but the benefit–cost ratio may be lower. In general, in the case of public works, because the scale of the projects is relatively large, it would be more appropriate to conduct cost–benefit analysis using NPV rather than BCR.\n\n#### 3.3. Analysis of the Effectiveness of Financial Support Programs\n\nUnlike existing long-distance water supply systems, which purify and supply water from rivers or reservoirs, RWH systems have the advantage of reducing energy consumption from water transportation. In addition, RWH systems can reduce water stress by reducing water intake from existing sources of water, such as rivers, and can be used as tools to cope with drought. Thus, in countries with high water stress, it is necessary to promote the installation of RWH systems. This section presents the effects of governmental financial support policies that promote the installation and operation of RWH systems.\nFigure 7 shows the results from the cost–benefit analysis reflecting Incheon’s financial support programs, as presented in Table 1. In contrast, Figure 8 shows the results of our cost–benefit analysis excluding governmental financial support programs. As shown in Figure 8b, the maximum profit generated without financial support programs was KRW 279 million (USD 214,615), which was about 52% smaller than the profit generated with help from financial support programs, as shown in Figure 7b. It can be seen in Figure 7b that the increase in the NPV with the increase in capacity was greater than the increase in the NPV with the increase in capacity (Figure 8b). This result meant that the benefit of reducing water utility bills by 10% of rainwater usage generated a greater profit than was gained from governmental financial support for the installation costs of the RWH system.\nIn Figure 7b, the maximum profit was generated when temporal reliability was 32.0% and volumetric reliability was 25.1%. This result may not be enough to indicate that it perfectly corresponds with the purpose of RWH systems. The fact that NPV is larger than zero means that a project has economic feasibility. In other words, RWH systems may be designed as the capacity analyzed that NPV exceeds 0. In excluding financial support, the marginal capacity, which is the largest capacity that can be indicated as having economic feasibility, is 2770 $m 3$, reliability was 48.6%, and volumetric reliability was 46.0% (Figure 8b). In the subsidized case, the marginal capacity is 3385 $m 3$, reliability was 52.9%, and volumetric reliability was 51.4% (Figure 7b). Therefore, governmental financial support for the installation and billing relief system may contribute to the economic feasibility and operational efficiency of RWH systems.\n\n#### 3.4. Sensitivity Analysis of the Discount Rate and Inflation Rate\n\nThe NPV evaluates the economic feasibility of a project by converting future value into present value. As shown in Equation (1), the general NPV formula includes the discount rate as the parameter with uncertainty. In the case of the NPV formula for RWH, not only the discount rate but also the inflation rate for water and electric utility bills should be considered (Equation (5)). Thus, this study conducted scenario-based uncertainty modeling to evaluate how the inflation and discount rates uncertainties affect the optimization model output, i.e., the maximum net present value and corresponding capacity.\nIn order to assess the sensitivity and uncertainty of the model to inflation and discount rates using scenario-based uncertainty modeling, the probability distribution to be sampled should be determined. This study determined the probability distribution of the inflation and discount rate based on the collected data from the literature and the Korean Statistical Information Service . The probability distribution of the inflation rate was determined and fitted using the historical records for the inflation rate of utility prices in South Korea from 1989 to 2020. Figure 9 presents the inflation rates of consumer and utility prices in South Korea from 1989 to 2020 . A set of distributions is tested to find the best one fitted to the historical inflation rate by year. This study considered the following probability distributions: normal, lognormal, gamma, t, and beta distributions. To find the best distribution fitted to the inflation rate, the Kolmogorov–Smirnov (K–S) test is applied. Figure 10a shows a histogram and five probability distribution curves calculated from the parameters estimated by the maximum likelihood estimation from the historical inflation rate. In Figure 10a, the p-value shown in the legend is a significant probability. The higher the p-value, the stronger the evidence that the null hypothesis should be adopted. The t distribution was found to be the best fit for the inflation rate of the water, electric, and gas utility prices. The light red region represents the 90% confidence interval to the fitted t distribution. This study set the range of the inflation rate for our sensitivity analysis as −1.1% to 7.7%, with a 90% confidence interval.\nThe appropriate social discount rate for public environmental projects in Korea is suggested based on the results of a survey among 114 experts in the economic field . The survey period was from 26 October to 3 November 2015, and according to the survey, the appropriate social discount rate for environmental public projects in Korea was 3.26% on average (standard deviation = 1.06%). As the available data were limited to the mean and standard deviation , this study assumed that the discount rate followed a normal distribution and determined the range for the sensitivity analysis. Figure 10b presents the normal distribution with a mean of 3.26 and a variance of 1.06, and the light blue region represents the 90% confidence interval. Thus, this study set the range of the discount rate for the sensitivity analysis as 1.5% to 5.0%, with a 90% confidence interval.\nFigure 11 illustrates the effect of the maximum NPV and capacity due to uncertainty in the inflation rate. The maximum NPV and corresponding capacity were derived by the optimization model presented in Section 2.4.2. In Figure 11a, the 90% confidence interval (light blue shaded region) represents the maximum NPV derived by inputting the inflation rate in the range from −1.1% to 7.7%. The 50% confidence interval (deep blue shaded region) was derived from the inflation rate in the range from 1.7% to 4.9%.\nAs shown in Figure 11a, it can be seen that the uncertainty of the derived maximum NPV is more affected by the uncertainty of the inflation rate as the discount rate decreases. In particular, at an inflation rate of more than 4.9%, which corresponds to the 75th–90th percentile, the volatility of maximum NPV is more significant than the less than 75th percentile. These results can be figured out via Equation (5). The higher inflation rate and lower discount rate contribute to the benefit growth in maximizing NPV. In other words, as the present value of the future cash flow increases, RWH can be evaluated as a profitable facility.\nFigure 12 presents the relationship between the two variables with uncertainty and the dependent variable. Figure 12a presents the maximum NPV figures determined by the optimization model according to changes in the social discount rate and inflation rate. Figure 12b shows the different capacities of the RWH system that correspond to the maximum NPV figures in Figure 12a. In order to present the results in Figure 12 more clearly, they are also presented numerically in Table A2 (Figure 12a) and Table A3 (Figure 12b).\nIn Figure 12, it can be seen that the maximum NPV and capacity increased as the inflation rate increased. Conversely, the maximum NPV and capacity increased as the discount rate decreased. As the future value increased, the maximum NPV occurred at larger capacities. As an example, as shown in Figure 7b, the inflection point of the NPV curve occurred because the operation and maintenance costs of the RWH system were greater than the benefits. Installation costs are fixed costs that do not increase or decrease at the present point, and maintenance and operation costs depend on installation costs. As the benefits of using the RWH may be increased by the future value increases, the maximum NPV may appear at a larger capacity. On the other hand, as shown in Figure 6b, the uplift rate for reliability decreased as the capacity increased. That is, the maximum NPV may not be increased indefinitely because the efficiency of rainwater utilization facilities decreases as capacity increases.\n\n## 4. Discussion\n\nIn this study, we suggested a method to determine the economically feasible capacity of an RWH system based on a cost–benefit analysis. As the social and environmental benefits are ambiguous to evaluate and are not real economic benefits, this study set the benefits as the real profits: the governmental financial support for the installation of RWH, profits from replacing water usage with rainwater usage, and water utility billing relief. The costs were defined as installation and operation costs of the considered RWH systems.\nFirst, the cost–benefit analysis was performed using the NPV and BCR; then, NPV was indicated as an appropriate method to analyze the economic feasibility of RWH systems. BCR indicates economics as a ratio of costs to benefits. BCR tends to derive a low-investment alternative because it presents the most efficient way to invest as the best alternative. Under the same conditions in Section 3.2, BCR and NPV present 285 $m 3$ and 1105 $m 3$ as the most economic capacity of the RWH system. When the capacity is 285 $m 3$, temporal reliability is 16.7%, and net profit is KRW 366 million (USD 281,538). The 16.7% of reliability means that rainwater supply is available about 61 days a year, indicating that the operational efficiency is significantly unreasonable in the study area. In the case of NPV, the maximum net profit is derived as KRW 581 million (USD 446,923), which is a larger net profit than the case of BCR. Therefore, the BCR may be inappropriate for applying cost–benefit analysis for a large-scale RWH, because BCR is not reflecting the investment scale.\nSecond, we reviewed the effectiveness of Korean governmental financial support to encourage the installation of RWH via cost–benefit analysis. Incheon in South Korea provides financial support, such as subsidies for installation costs and water utility billing relief. As a result of comparing the maximum NPV according to the presence or absence of financial support, the maximum net profit increases by 52%. In addition, the marginal capacity, which is the maximum capacity of RWH to generate revenue, increases from 2770 $m 3$ to 3385 $m 3$ with financial support. That is, governmental financial support can contribute to the economic feasibility of RWH but also the installation and operation of a more effective RWH. The above results show as an example how much actual worth a support program presented only as a percentage can actually generate.\nFinally, we present the results of the uncertainty modeling and sensitivity analysis for the inflation and discount rates that are uncertain parameters of NPV. In addition, as it takes time to determine the maximum NPV for the scenarios of inflation and discount rates, the simple optimization model using the PSO is presented to reduce the computation time in Section 2.4.2. As a result, as the present value of the future value increases, the volatility of the derived maximum NPV and capacity of RWH significantly increases. Giving high future value can lead to an increase in the capacity of the RWH corresponding to the maximum NPV, which may lead to concern about over-design of the RWH. Thus, in the case of cost–benefit analysis for RWH, caution is needed when determining the inflation and discount rates.\nThe appropriate discount rate for long-term environmental projects, such as projects to cope with climate change, and the appropriate rate for other countries, are presented in Table A1. Most developed countries recommend an appropriate discount rate of less than 3% for long-term projects. The conservative tendency to apply low discount rates to long-term projects reflects uncertainty about the future. In contrast, in the case of developing countries, a high discount rate of more than 10% is announced for economic growth and infrastructure improvement.\n\n## 5. Conclusions\n\nThis study suggested a method to evaluate the economically feasible capacity of an RWH system using cost–benefit analysis. The cost–benefit analysis was performed using the NPV and BCR; then, an appropriate method was suggested to analyze the economic feasibility of a selected RWH system. Because the potential rainwater usage varies depending on the capacity of the RWH system in question, a simulation model was developed based on the reservoir mass balance equation, which could determine the yield of the considered RWH system, and the costs and benefits were defined. We defined the costs as the installation and operation costs of the considered RWH system. The benefits were the profits from replacing water usage with rainwater usage, the governmental financial support for the installation and operation costs of the RWH system, and water utility billing relief. South Korean policies regarding financial support for the installation and operation costs of RWH systems were also presented in this study, and the effects of those policies were reviewed using cost–benefit analysis. In addition, the key parameters to consider in cost–benefit analysis were identified via a sensitivity analysis.\nAs a result, the NPV, which could derive the threshold as the capacity increased, was determined to be appropriate for analyzing the economic feasibility of RWH systems. Because the BCR presents the economic feasibility of a project as a ratio, it was challenging to derive an absolute value for economic feasibility using this criterion. Thus, it was determined to be inappropriate to conduct cost–benefit analysis using BCR for relatively large-scale RWH systems. When comparing the maximum NPV for cases with and without help from governmental financial support programs, we found that the maximum NPV with financial support programs was nearly twice as large as that without. This study could be meaningful as an economic analysis case that considered the benefits for the users of RWH systems from the current financial support programs in South Korea. The results from the sensitivity analysis, in the case of cost–benefit analysis for RWH, showed that caution is needed when determining the discount and inflation rates because the future value may change significantly depending on the discount and inflation rates used.\nIn this study, the results were derived through the simulation of a designed RWH system in CN1-D, Incheon, South Korea. However, different results could also be derived, because RWH systems are designed in various ways depending on the purpose of their use, the installation location, and the support policies in the relevant country. Thus, the method proposed in this study needs to be verified through application to various meteorological characteristics.\n\n## Author Contributions\n\nConceptualization, Y.J. and T.K.; methodology, Y.J. and S.L.; software, Y.J.; investigation, J.P.; writing—original draft preparation, Y.J. and Y.K.; writing—review and editing, S.L. and Y.K. All authors have read and agreed to the submitted version of the manuscript.\n\n## Funding\n\nThis work was supported by the Korea Environment Industry & Technology Institute (KEITI) through the Smart Water City Research Program, funded by the Korean Ministry of Environment (MOE) (2019002950004).\n\nNot applicable.\n\nNot applicable.\n\n## Conflicts of Interest\n\nThe authors declare no conflict of interest.\n\n## Abbreviations\n\nThe following abbreviations are used in this manuscript:\n BCR Benefit–cost ratio NPV Net present value RWH Rainwater harvesting APSWR Act for the Promotion and Support of Water Reuse PSO Particle swarm optimization\n\n## Appendix A\n\nTable A1. Estimates for social discount rates in countries across the world.\nTable A1. Estimates for social discount rates in countries across the world.\nInstitution/CountrySocial Discount RateRemark\nWorld BankProjects for developing countries: 10–12%\nUnited States Environmental\nProtection Agency\nIntergenerational discount rate: 2–3%\n(subject to sensitivity analysis)\n\nEuropean UnionLong-term projects/policies: 3%\nUnited KingdomStandard: 3.5%\nLong-term projects of 30–125 years: 3%; 125–200 years: 2%; 200+ years: 1.5%\n\nFranceStandard: 4%\nLong-term projects/policies: 2%\n\nNetherlandsStandard: 5.5%\nProjects/policies for climate change: 4%\n\nGermanyLong-term projects/policies: 1%\nJapanProjects within 50 years: 4%\nAustraliaStandard: 7%\nSubject to sensitivity analysis: 3% and 10%\n\nChinaShort- and mid-term projects: 8%\nLong-term projects: less than 8%\n\nIndia12%\nRepublic of the Philippines15%\nSouth KoreaStandard: 4.5%\nWater resources projects of 0–30 years: 4.5%; 30+ years: 3.5%\n\nTable A2. The results of our sensitivity analysis of inflation and discount rates: the maximum net present value.\nTable A2. The results of our sensitivity analysis of inflation and discount rates: the maximum net present value.\nClassificationInflation Rate\n−1.1%−0.2%0.8%1.7%2.6%3.5%4.5%5.4%6.3%7.2%\nDiscount Rate1.5%447550677837105113181648205325453144\n1.9%41250862376995812041504187723282878\n2.2%38046857570787610991374171521312636\n2.6%35243253165180310021256156619502414\n3.0%3263984906017399151148143217842211\n3.3%3023694535556808391048131016312025\n3.7%280342418513627771957119814911854\n4.1%260317386474580711877109613651697\n4.4%242294358438536655805100112501552\n4.8%22527233240449660574291511441421\nTable A3. The results of the sensitivity analysis of inflation and discount rates: capacity.\nTable A3. The results of the sensitivity analysis of inflation and discount rates: capacity.\nClassificationInflation Rate\n−1.1%−0.2%0.8%1.7%2.6%3.5%4.5%5.4%6.3%7.2%\nDiscount Rate1.5%80083710151272160216811895199220932235\n1.9%7718409131054147416111801194220542222\n2.2%6698258411032128015971682191719972096\n2.6%641804836957112215781638187119512090\n3.0%633720823843105113441632173219351997\n3.3%617674823844101412671607166918971986\n3.7%62463577382689410571470163317861929\n4.1%53762866882384010451280160516921916\n4.4%5026226808208589871102157916471864\n4.8%4906116317748328471043134616121704\n\n## References\n\n1. Ghisi, E.; Ferreira, D.F. Potential for potable water savings by using rainwater and greywater in a multi-storey residential building in southern Brazil. Build. Environ. 2007, 42, 2512–2522. [Google Scholar] [CrossRef]\n2. Guo, R.; Guo, Y. Stochastic modelling of the hydrologic operation of rainwater harvesting systems. J. Hydrol. 2018, 562, 30–39. [Google Scholar] [CrossRef]\n3. 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Available online: https://www.law.go.kr/LSW/admRulLsInfoP.do?admRulSeq=2100000207680 (accessed on 1 November 2022).\nFigure 1. The location of the study area in Cheongna-dong, Incheon, South Korea.\nFigure 1. The location of the study area in Cheongna-dong, Incheon, South Korea.\nFigure 2. Rainfall–runoff time series data in the study area (1995–2004): (a) box and whisker plot with monthly distributions of rainfall; (b) result of rainfall–runoff simulation.\nFigure 2. Rainfall–runoff time series data in the study area (1995–2004): (a) box and whisker plot with monthly distributions of rainfall; (b) result of rainfall–runoff simulation.\nFigure 3. Monthly target draft of the rainwater harvesting system in the study area.\nFigure 3. Monthly target draft of the rainwater harvesting system in the study area.\nFigure 4. A conceptual searching scheme of particles in a particle swarm algorithm.\nFigure 4. A conceptual searching scheme of particles in a particle swarm algorithm.\nFigure 5. A schematic diagram of the connection between a rainwater harvesting system simulation model and a particle swarm algorithm.\nFigure 5. A schematic diagram of the connection between a rainwater harvesting system simulation model and a particle swarm algorithm.\nFigure 6. The capacity–reliability curves for the considered rainwater harvesting system: (a) the capacity–temporal reliability curve; (b) the capacity–volumetric reliability curve.\nFigure 6. The capacity–reliability curves for the considered rainwater harvesting system: (a) the capacity–temporal reliability curve; (b) the capacity–volumetric reliability curve.\nFigure 7. The results of the cost–benefit analysis for the capacity of the considered rainwater harvesting system: (a) the capacity–BCR curve; (b) the capacity–NPV curve.\nFigure 7. The results of the cost–benefit analysis for the capacity of the considered rainwater harvesting system: (a) the capacity–BCR curve; (b) the capacity–NPV curve.\nFigure 8. The results of the cost–benefit analysis of the considered rainwater harvesting system without financial support programs: (a) the capacity–BCR curve; (b) the capacity–NPV curve.\nFigure 8. The results of the cost–benefit analysis of the considered rainwater harvesting system without financial support programs: (a) the capacity–BCR curve; (b) the capacity–NPV curve.\nFigure 9. Historical records for the inflation rate in South Korea.\nFigure 9. Historical records for the inflation rate in South Korea.\nFigure 10. The curves of fitted probability distributions and their 90% confidence intervals: (a) histogram against the five distributions of the inflation rate; (b) the normal distribution with a mean of 3.26 and a variance of 1.06 for the discount rate.\nFigure 10. The curves of fitted probability distributions and their 90% confidence intervals: (a) histogram against the five distributions of the inflation rate; (b) the normal distribution with a mean of 3.26 and a variance of 1.06 for the discount rate.\nFigure 11. The results of the uncertainty analysis for the inflation rate: (a) the maximum net present value; (b) the corresponding capacity.\nFigure 11. The results of the uncertainty analysis for the inflation rate: (a) the maximum net present value; (b) the corresponding capacity.\nFigure 12. The results of our sensitivity analysis for the inflation and discount rates: (a) the maximum net present value; (b) the corresponding capacity.\nFigure 12. The results of our sensitivity analysis for the inflation and discount rates: (a) the maximum net present value; (b) the corresponding capacity.\nTable 1. The costs and benefits of the installation and operation of rainwater harvesting systems (USD 1 = KRW 1300).\nTable 1. The costs and benefits of the installation and operation of rainwater harvesting systems (USD 1 = KRW 1300).\nCityFinancial Support and Billing Relief\nInstallation CostsWater Utility Bill\nWater Supply ChargeSewage ChargeWater Usage Charge\nSeoul90% of installation costs\nup to KRW 20 million (USD 15,384)\nIncheonFull or partial support10% of RWU 110% of RWU10% of RWU\nSuwon90% of installation costs\nup to KRW 10 million (USD 7692)\nSome RWUSome RWUSome RWU\nSejongFull or partial support10% of RWU30% of RWU\nBusan90% of installation costs\nup to KRW 10 million (USD 7692)\n10% of RWU10% of RWU\nNote(s): 1 RWU, rainwater usage.\nTable 2. A water utility bill for 1000 $m 3$ of water in Incheon, South Korea (USD 1 = KRW 1300).\nTable 2. A water utility bill for 1000 $m 3$ of water in Incheon, South Korea (USD 1 = KRW 1300).\nClassificationPricing BracketUnit ChargeCalculation Details\n($m 3$)(KRW/$m 3$)\nWater Supply Charge1~300870 (USD 0.67)300 $m 3 ×$ 870 KRW/$m 3$ = KRW 261,000 (USD 201)\nMore than 3001120 (USD 0.86)700 $m 3 ×$ 1120 KRW/$m 3$ = KRW 784,000 (USD 603)\nSewage Charge1~50490 (USD 0.38)50 $m 3 ×$ 490 KRW/$m 3$ = KRW 24,500 (USD 18.8)\n51~100510 (USD 0.39)50 $m 3 ×$ 510 KRW/$m 3$ = KRW 25,500 (USD 19.6)\n101~3001010 (USD 0.78)200 $m 3 ×$ 1010 KRW/$m 3$ = KRW 202,000 (USD 155)\n301~5001100 (USD 0.85)200 $m 3 ×$ 1100 KRW/$m 3$ = KRW 220,000 (USD 169)\n501~10001130 (USD 0.87)500 $m 3 ×$ 1130 KRW/$m 3$ = KRW 565,000 (USD 435)\nMore than 10001160 (USD 0.89)-\nWater Usage ChargeWhole range170 (USD 0.13)1000 $m 3 ×$ 170 KRW/$m 3$ = KRW 170,000 (USD 131)\nTotal Water Utility BillKRW 2,432,000 (USD 1871)\nTable 3. The costs and benefits of the installation and operation of a rainwater harvesting system (USD 1 = KRW 1300).\nTable 3. The costs and benefits of the installation and operation of a rainwater harvesting system (USD 1 = KRW 1300).\nClassificationCategoryContentRemark\nCostsInstallation, construction, equipment, etc.350,000–450,000 KRW/$m 3$\n(269–346 USD/$m 3$)\n[44,45]\nMaintenance expenses (labor, electricity, etc.)2% of installation costs[44,45,46]\nBenefitsSavings on water utility bills by replacing\nwater usage with rainwater usage\nEquivalent to the amount\nof rainwater usage\nSee\nTable 1\nSubsidies for installation costsUp to\nKRW 10 million\n(UDS 7692)\nFull or partial support in Incheon, South Korea\nWater Utility Bill\nConcessions\nWater supply charge concessions10% of water supply chargesIncheon,\nSouth Korea\nSewage charge concessions10% of sewage charges\nWater usage charge concessions10% of water usage charges\nTable 4. The equations for the simulation model for rainwater harvesting systems.\nTable 4. The equations for the simulation model for rainwater harvesting systems.\nClassificationSimulation Model\nConditionEquation\nMass Balance Equation$S T t = S T t − 1 + Q F t − Y t − E X R t$\nYield Determination$S T t − 1 + Q F t ≥ T D t$$Y t = T D t$\n$S T t − 1 + Q F t ≤ T D t$, and $S T t − 1 ≠ 0$$Y t = S t − 1$\n$S T t − 1 + Q F t ≤ T D t$, and $S T t − 1 = 0$$Y t = 0$\nSpill Determination$S t − 1 + Q F t − Y t ≥ C a p$$E X R t = S T t − 1 + Q F t − Y t − C a p$\n$S t − 1 + Q F t − Y t < C a p$$E X R t = 0$\n Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.\n\n## Share and Cite\n\nMDPI and ACS Style\n\nJin, Y.; Lee, S.; Kang, T.; Park, J.; Kim, Y. Capacity Optimization of Rainwater Harvesting Systems Based on a Cost–Benefit Analysis: A Financial Support Program Review and Parametric Sensitivity Analysis. Water 2023, 15, 186. https://doi.org/10.3390/w15010186\n\nAMA Style\n\nJin Y, Lee S, Kang T, Park J, Kim Y. Capacity Optimization of Rainwater Harvesting Systems Based on a Cost–Benefit Analysis: A Financial Support Program Review and Parametric Sensitivity Analysis. Water. 2023; 15(1):186. https://doi.org/10.3390/w15010186\n\nChicago/Turabian Style\n\nJin, Youngkyu, Sangho Lee, Taeuk Kang, Jongpyo Park, and Yeulwoo Kim. 2023. \"Capacity Optimization of Rainwater Harvesting Systems Based on a Cost–Benefit Analysis: A Financial Support Program Review and Parametric Sensitivity Analysis\" Water 15, no. 1: 186. https://doi.org/10.3390/w15010186\n\nNote that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here." ]

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[ "# How do you apply boundary conditions in a time-stepping problem?\n\nIt looks to me like a very common problem, yet I haven't been able to find any practical guide on the subject despite many hours searching.\n\nHere is a clearer statement of my question:\n\nI have a system of (linear) PDEs in the variables $x$ and $t$. I can use the method of lines to discretise the space coordinate $x$ which yields a system of ODEs in time: $$\\frac{d\\vec{u}}{dt}=A\\vec{u}+\\vec{b}$$ $A$ is a square matrix.\n\nNow, to me, the straightforward way to solve the above on a computer is to use an explicit time-stepping method of some sort. So I write:\n\n$$\\vec{u}_{n+1}=\\vec{u}_{n}+\\Delta t(A\\vec{u}_{n}+\\vec{b}_n)$$\n\nNow, how do I enforce boundary conditions in all this ?\n\nIn their most general form, these are $$B\\vec{u}=0$$ $B$ is a rectangular matrix.\n\nI don't really see how this can fit into the above. Any help would be super appreciated.\n\n• Boundary conditions are implicitly defined in the discrete spatial derivative operator $A$ – HBR Dec 20 '17 at 22:36\n• How does it work in practice ? – jrekier Dec 20 '17 at 22:49\n• This is discussed in every book on numerical methods for PDEs. – David Ketcheson Dec 21 '17 at 4:37\n• David, your comment is really not helping. I have browsed through a fair number of such books and haven't found an explanation I could understand. Hence the question. For your comment to be of any value, you should at least provide a reference for one such book. – jrekier Dec 21 '17 at 8:39\n• @jerkier a good introductory reference to numerical PDEs is this book by Randall Leveque. you could start by reading chapter 2.4 that deals with Dirichlet boundary conditions for a simple steady-state problem (in your notation, that'd be $Ax = b$), then work your way through the more advanced examples. – GoHokies Dec 21 '17 at 12:52\n\nAs HBR mentioned, the boundary conditions can often be immediately incorporated into $A$ and $b$. For example, suppose we wish to solve the 1D heat equation with Dirchilet boundary conditions\n\n$$u_t = \\sigma u_{xx} + h(x,t), \\quad u(a,t) = f(t), \\quad u(b,t) = g(t).$$\n\nWe then discretize $u_j^n \\approx u(x_j, t_n)$ where $x_j = a+j \\Delta x$ and $t_n = n\\Delta t$. Here $j = 0,1,2,\\ldots,N+1$ and $\\Delta x = (b-a)/(N+1)$. Note that by our boundary conditions $u_0^n = f(t_n)$ and $u_{N+1}^n = g(t_n)$. As these values are completely determined for all time, we don't even include them in our linear system. For interior gridpoints ($2 \\le j \\le N-1$), we take a second-order centered difference\n\n$$u_{xx}(x_j,t_n) \\approx \\frac{u_{j-1}^n - 2u_j^n + u_{j+1}^n}{\\Delta x^2}. \\tag{1}$$\n\nFor the boundary-adjacent gridpoints, we get\n\n\\begin{align} u_{xx}(x_1,t_n) &\\approx \\frac{u_{0}^n - 2u_1^n + u_{2}^n}{\\Delta x^2}=\\frac{f(t_n) - 2u_1^n + u_{2}^n}{\\Delta x^2}, \\tag{2} \\\\ u_{xx}(x_N,t_n) &\\approx \\frac{u_{N-1}^n - 2u_{N}^n + u_{N+1}^n}{\\Delta x^2}=\\frac{u_{N-1}^n - 2u_{N}^n + g(t_n)}{\\Delta x^2}. \\tag{3} \\end{align}\n\nCollecting our discretization, we get\n\n$$\\frac{d}{dt} \\begin{pmatrix} u_1 \\\\ u_2 \\\\ u_3 \\\\ \\vdots \\\\ u_{n-1} \\\\ u_n \\end{pmatrix} = \\frac{1}{\\Delta x^2}\\begin{pmatrix} -2 & 1 & 0 & 0 & \\cdots & 0 & 0 & 0 \\\\ 1 & -2 & 1 & 0 & \\cdots & 0 & 0& 0 \\\\ 0 & 1 & -2 & 1 & \\cdots & 0& 0 & 0 \\\\ 0 & 0 & 1 & -2 & \\cdots & 0 & 0& 0 \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots\\\\ 0 & 0 & 0 & 0 & \\cdots & 1 & -2 & 1 \\\\ 0 & 0 & 0 & 0 & \\cdots & 0 & 1 & -2 \\end{pmatrix}\\begin{pmatrix} u_1 \\\\ u_2 \\\\ u_3 \\\\ u_4 \\\\ \\vdots \\\\ u_{n-1} \\\\ u_n \\end{pmatrix} + \\begin{pmatrix} f(t)/\\Delta x^2 + h(x_1,t) \\\\ h(x_2,t) \\\\ h(x_3,t) \\\\ h(x_4,t) \\\\ \\vdots \\\\ h(x_{n-1},t) \\\\ g(t)/\\Delta x^2 + h(x_n, t) \\end{pmatrix}. \\tag{4}$$\n\nAs you can see, we have baked our boundary conditions into the matrix $A$ and the vector $b$. (Make sure you can see how we got from the (1)-(3) to (4).) Similar approaches work for other PDEs and other discretization. Handling boundary conditions for hyperbolic problems is more tricky though, and you should always try to be very careful when treating the boundary in your discretization.\n\n• Hi @eepperly, and thanks for your detailed answer. Much appreciated. It seems to me like this wouldn't be directly applicable with methods based on spectral differentiation (spectral methods). Am I right ? I will try your suggestion using collocation points though. Cheers ! – jrekier Dec 21 '17 at 8:49\n• @jerkier a good reference for spectral methods is Jan Hesthaven's book. I can't point you right now to the specific section dealing with BCs (don't have the book at hand), but could do so later today if you're interested. – GoHokies Dec 21 '17 at 13:00\n\nA simple way which should work well for an implicit time-integrator that can handle stiff equations is to write your BC's as relaxation ODEs. For example, if we want to enforce $u_j=u_{j0}$ where $u_j$ is the value of the unknown at a boundary grid point $j$, then use $\\frac{d}{dt} u_j = - \\lambda_{rlx} (u_j - u_{j0})$, where $\\lambda_{rlx}$ is a \"relaxation rate\" chosen to be fast compared to time scales in the operator A.\n\nThe advantage of this method is it is very easy to implement. This way your BC is maintained not exactly (depending on the chosen $\\lambda_{rlx}$), but the error can be made small, and sometimes it is actually desired to have some freedom in the boundary values, e.g., if your main goal is achieving a steady state.\n\n• I will give it a try right away. Thx! – jrekier Dec 21 '17 at 8:50" ]

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[ "Documentation\n\n# cdf\n\nCumulative distribution function\n\n## Syntax\n\n``y = cdf('name',x,A)``\n``y = cdf('name',x,A,B)``\n``y = cdf('name',x,A,B,C)``\n``y = cdf('name',x,A,B,C,D)``\n``y = cdf(pd,x)``\n``y = cdf(___,'upper')``\n\n## Description\n\nexample\n\n````y = cdf('name',x,A)` returns the cumulative distribution function (cdf) for the one-parameter distribution family specified by `'name'` and the distribution parameter `A`, evaluated at the values in `x`.```\n\nexample\n\n````y = cdf('name',x,A,B)` returns the cdf for the two-parameter distribution family specified by `'name'` and the distribution parameters `A` and `B`, evaluated at the values in `x`.```\n````y = cdf('name',x,A,B,C)` returns the cdf for the three-parameter distribution family specified by `'name'` and the distribution parameters `A`, `B`, and `C`, evaluated at the values in `x`.```\n````y = cdf('name',x,A,B,C,D)` returns the cdf for the four-parameter distribution family specified by `'name'` and the distribution parameters `A`, `B`, `C`, and `D`, evaluated at the values in `x`.```\n\nexample\n\n````y = cdf(pd,x)` returns the cdf of the probability distribution object `pd`, evaluated at the values in `x`.```\n````y = cdf(___,'upper')` returns the complement of the cdf using an algorithm that more accurately computes the extreme upper-tail probabilities. `'upper'` can follow any of the input arguments in the previous syntaxes.```\n\n## Examples\n\ncollapse all\n\nCreate a standard normal distribution object with the mean, $\\mu$, equal to 0 and the standard deviation, $\\sigma$, equal to 1.\n\n```mu = 0; sigma = 1; pd = makedist('Normal','mu',mu,'sigma',sigma);```\n\nDefine the input vector x to contain the values at which to calculate the cdf.\n\n`x = [-2,-1,0,1,2];`\n\nCompute the cdf values for the standard normal distribution at the values in x.\n\n`y = cdf(pd,x)`\n```y = 1×5 0.0228 0.1587 0.5000 0.8413 0.9772 ```\n\nEach value in y corresponds to a value in the input vector x. For example, at the value x equal to 1, the corresponding cdf value y is equal to 0.8413.\n\nAlternatively, you can compute the same cdf values without creating a probability distribution object. Use the `cdf` function, and specify a standard normal distribution using the same parameter values for $\\mu$ and $\\sigma$.\n\n`y2 = cdf('Normal',x,mu,sigma)`\n```y2 = 1×5 0.0228 0.1587 0.5000 0.8413 0.9772 ```\n\nThe cdf values are the same as those computed using the probability distribution object.\n\nCreate a Poisson distribution object with the rate parameter, $\\lambda$, equal to 2.\n\n```lambda = 2; pd = makedist('Poisson','lambda',lambda);```\n\nDefine the input vector x to contain the values at which to calculate the cdf.\n\n`x = [0,1,2,3,4];`\n\nCompute the cdf values for the Poisson distribution at the values in x.\n\n`y = cdf(pd,x)`\n```y = 1×5 0.1353 0.4060 0.6767 0.8571 0.9473 ```\n\nEach value in y corresponds to a value in the input vector x. For example, at the value x equal to 3, the corresponding cdf value y is equal to 0.8571.\n\nAlternatively, you can compute the same cdf values without creating a probability distribution object. Use the `cdf` function, and specify a Poisson distribution using the same value for the rate parameter, $\\lambda$.\n\n`y2 = cdf('Poisson',x,lambda)`\n```y2 = 1×5 0.1353 0.4060 0.6767 0.8571 0.9473 ```\n\nThe cdf values are the same as those computed using the probability distribution object.\n\nCreate a standard normal distribution object.\n\n`pd = makedist('Normal')`\n```pd = NormalDistribution Normal distribution mu = 0 sigma = 1 ```\n\nSpecify the `x` values and compute the cdf.\n\n```x = -3:.1:3; p = cdf(pd,x);```\n\nPlot the cdf of the standard normal distribution.\n\n`plot(x,p)`", null, "Create three gamma distribution objects. The first uses the default parameter values. The second specifies `a = 1` and `b = 2`. The third specifies `a = 2` and `b = 1`.\n\n`pd_gamma = makedist('Gamma')`\n```pd_gamma = GammaDistribution Gamma distribution a = 1 b = 1 ```\n`pd_12 = makedist('Gamma','a',1,'b',2)`\n```pd_12 = GammaDistribution Gamma distribution a = 1 b = 2 ```\n`pd_21 = makedist('Gamma','a',2,'b',1)`\n```pd_21 = GammaDistribution Gamma distribution a = 2 b = 1 ```\n\nSpecify the `x` values and compute the cdf for each distribution.\n\n```x = 0:.1:5; cdf_gamma = cdf(pd_gamma,x); cdf_12 = cdf(pd_12,x); cdf_21 = cdf(pd_21,x);```\n\nCreate a plot to visualize how the cdf of the gamma distribution changes when you specify different values for the shape parameters `a` and `b`.\n\n```figure; J = plot(x,cdf_gamma); hold on; K = plot(x,cdf_12,'r--'); L = plot(x,cdf_21,'k-.'); set(J,'LineWidth',2); set(K,'LineWidth',2); legend([J K L],'a = 1, b = 1','a = 1, b = 2','a = 2, b = 1','Location','southeast'); hold off;```", null, "Fit Pareto tails to a $t$ distribution at cumulative probabilities 0.1 and 0.9.\n\n```t = trnd(3,100,1); obj = paretotails(t,0.1,0.9); [p,q] = boundary(obj)```\n```p = 2×1 0.1000 0.9000 ```\n```q = 2×1 -1.8487 2.0766 ```\n\nCompute the cdf at the values in `q`.\n\n`cdf(obj,q)`\n```ans = 2×1 0.1000 0.9000 ```\n\n## Input Arguments\n\ncollapse all\n\nProbability distribution name, specified as one of the probability distribution names in this table.\n\n`'name'`DistributionInput Parameter `A`Input Parameter `B`Input Parameter `C`Input Parameter `D`\n`'Beta'`Beta Distributiona first shape parameterb second shape parameter\n`'Binomial'`Binomial Distributionn number of trialsp probability of success for each trial\n`'BirnbaumSaunders'`Birnbaum-Saunders Distributionβ scale parameterγ shape parameter\n`'Burr'`Burr Type XII Distributionα scale parameterc first shape parameterk second shape parameter\n`'Chisquare'`Chi-Square Distributionν degrees of freedom\n`'Exponential'`Exponential Distributionμ mean\n`'Extreme Value'`Extreme Value Distributionμ location parameterσ scale parameter\n`'F'`F Distributionν1 numerator degrees of freedomν2 denominator degrees of freedom\n`'Gamma'`Gamma Distributiona shape parameterb scale parameter\n`'Generalized Extreme Value'`Generalized Extreme Value Distributionk shape parameterσ scale parameterμ location parameter\n`'Generalized Pareto'`Generalized Pareto Distributionk tail index (shape) parameterσ scale parameterμ threshold (location) parameter\n`'Geometric'`Geometric Distributionp probability parameter\n`'HalfNormal'`Half-Normal Distributionμ location parameterσ scale parameter\n`'Hypergeometric'`Hypergeometric Distributionm size of the populationk number of items with the desired characteristic in the populationn number of samples drawn\n`'InverseGaussian'`Inverse Gaussian Distributionμ scale parameterλ shape parameter\n`'Logistic'`Logistic Distributionμ meanσ scale parameter\n`'LogLogistic'`Loglogistic Distributionμ mean of logarithmic valuesσ scale parameter of logarithmic values\n`'Lognormal'`Lognormal Distributionμ mean of logarithmic valuesσ standard deviation of logarithmic values\n`'Nakagami'`Nakagami Distributionμ shape parameterω scale parameter\n`'Negative Binomial'`Negative Binomial Distributionr number of successesp probability of success in a single trial\n`'Noncentral F'`Noncentral F Distributionν1 numerator degrees of freedomν2 denominator degrees of freedomδ noncentrality parameter\n`'Noncentral t'`Noncentral t Distributionν degrees of freedomδ noncentrality parameter\n`'Noncentral Chi-square'`Noncentral Chi-Square Distributionν degrees of freedomδ noncentrality parameter\n`'Normal'`Normal Distributionμ mean σ standard deviation\n`'Poisson'`Poisson Distributionλ mean\n`'Rayleigh'`Rayleigh Distributionb scale parameter\n`'Rician'`Rician Distributions noncentrality parameterσ scale parameter\n`'Stable'`Stable Distributionα first shape parameterβ second shape parameterγ scale parameterδ location parameter\n`'T'`Student's t Distributionν degrees of freedom\n`'tLocationScale'`t Location-Scale Distributionμ location parameterσ scale parameterν shape parameter\n`'Uniform'`Uniform Distribution (Continuous)a lower endpoint (minimum)b upper endpoint (maximum)\n`'Discrete Uniform'`Uniform Distribution (Discrete)n maximum observable value\n`'Weibull'`Weibull Distributiona scale parameterb shape parameter\n\nExample: `'Normal'`\n\nValues at which to evaluate the cdf, specified as a scalar value or an array of scalar values.\n\nIf one or more of the input arguments `x`, `A`, `B`, `C`, and `D` are arrays, then the array sizes must be the same. In this case, `cdf` expands each scalar input into a constant array of the same size as the array inputs. See `'name'` for the definitions of `A`, `B`, `C`, and `D` for each distribution.\n\nExample: `[0.1,0.25,0.5,0.75,0.9]`\n\nData Types: `single` | `double`\n\nFirst probability distribution parameter, specified as a scalar value or an array of scalar values.\n\nIf one or more of the input arguments `x`, `A`, `B`, `C`, and `D` are arrays, then the array sizes must be the same. In this case, `cdf` expands each scalar input into a constant array of the same size as the array inputs. See `'name'` for the definitions of `A`, `B`, `C`, and `D` for each distribution.\n\nData Types: `single` | `double`\n\nSecond probability distribution parameter, specified as a scalar value or an array of scalar values.\n\nIf one or more of the input arguments `x`, `A`, `B`, `C`, and `D` are arrays, then the array sizes must be the same. In this case, `cdf` expands each scalar input into a constant array of the same size as the array inputs. See `'name'` for the definitions of `A`, `B`, `C`, and `D` for each distribution.\n\nData Types: `single` | `double`\n\nThird probability distribution parameter, specified as a scalar value or an array of scalar values.\n\nIf one or more of the input arguments `x`, `A`, `B`, `C`, and `D` are arrays, then the array sizes must be the same. In this case, `cdf` expands each scalar input into a constant array of the same size as the array inputs. See `'name'` for the definitions of `A`, `B`, `C`, and `D` for each distribution.\n\nData Types: `single` | `double`\n\nFourth probability distribution parameter, specified as a scalar value or an array of scalar values.\n\nIf one or more of the input arguments `x`, `A`, `B`, `C`, and `D` are arrays, then the array sizes must be the same. In this case, `cdf` expands each scalar input into a constant array of the same size as the array inputs. See `'name'` for the definitions of `A`, `B`, `C`, and `D` for each distribution.\n\nData Types: `single` | `double`\n\nProbability distribution, specified as a probability distribution object created with a function or app in this table.\n\nFunction or AppDescription\n`makedist`Create a probability distribution object using specified parameter values.\n`fitdist`Fit a probability distribution object to sample data.\nDistribution FitterFit a probability distribution to sample data using the interactive Distribution Fitter app and export the fitted object to the workspace.\n`paretotails`Create a piecewise distribution object that has generalized Pareto distributions in the tails.\n\n## Output Arguments\n\ncollapse all\n\ncdf values, returned as a scalar value or an array of scalar values. `y` is the same size as `x` after any necessary scalar expansion. Each element in `y` is the cdf value of the distribution, specified by the corresponding elements in the distribution parameters (`A`, `B`, `C`, and `D`) or the probability distribution object (`pd`), evaluated at the corresponding element in `x`.\n\n## Alternative Functionality\n\n• `cdf` is a generic function that accepts either a distribution by its name `'name'` or a probability distribution object `pd`. It is faster to use a distribution-specific function, such as `normcdf` for the normal distribution and `binocdf` for the binomial distribution. For a list of distribution-specific functions, see Supported Distributions.\n\n• Use the Probability Distribution Function app to create an interactive plot of the cumulative distribution function (cdf) or probability density function (pdf) for a probability distribution." ]

[ null, "https://www.mathworks.com/help/examples/stats/win64/PlotStandardNormalDistributioncdfExample_01.png", null, "https://www.mathworks.com/help/examples/stats/win64/PlotGammaDistributioncdfExample_01.png", null ]

[ "# optuna.visualization.plot_optimization_history¶\n\n`optuna.visualization.``plot_optimization_history`(study, *, target=None, target_name='Objective Value')[source]\n\nPlot optimization history of all trials in a study.\n\nExample\n\nThe following code snippet shows how to plot optimization history.\n\n```import optuna\n\ndef objective(trial):\nx = trial.suggest_float(\"x\", -100, 100)\ny = trial.suggest_categorical(\"y\", [-1, 0, 1])\nreturn x ** 2 + y\n\nsampler = optuna.samplers.TPESampler(seed=10)\nstudy = optuna.create_study(sampler=sampler)\nstudy.optimize(objective, n_trials=10)\n\nfig = optuna.visualization.plot_optimization_history(study)\nfig.show()\n```\nParameters\n• study (optuna.study.Study) – A `Study` object whose trials are plotted for their target values.\n\n• target (Optional[Callable[[optuna.trial._frozen.FrozenTrial], float]]) –\n\nA function to specify the value to display. If it is `None` and `study` is being used for single-objective optimization, the objective values are plotted.\n\nNote\n\nSpecify this argument if `study` is being used for multi-objective optimization.\n\n• target_name (str) – Target’s name to display on the axis label and the legend.\n\nReturns\n\nA `plotly.graph_objs.Figure` object.\n\nRaises\n\nValueError – If `target` is `None` and `study` is being used for multi-objective optimization.\n\nReturn type\n\nplotly.graph_objs._figure.Figure" ]

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[ "Cody\n\n# Problem 44385. Extra safe primes\n\nSolution 1656196\n\nSubmitted on 22 Oct 2018 by Sharon Spelt\nThis solution is locked. To view this solution, you need to provide a solution of the same size or smaller.\n\n### Test Suite\n\nTest Status Code Input and Output\n1   Pass\nx = 0; assert(isequal(isextrasafe(x),false))\n\ntf = 0\n\n2   Pass\nx = 5; assert(isequal(isextrasafe(x),false))\n\np = 2 tf = 0\n\n3   Pass\nx = 7; assert(isequal(isextrasafe(x),false))\n\np = 3 p = 1 tf = 0\n\n4   Pass\nx = 11; assert(isequal(isextrasafe(x),true))\n\np = 5 p = 2 tf = 1\n\n5   Pass\nx = 15; assert(isequal(isextrasafe(x),false))\n\ntf = 0\n\n6   Pass\nx = 23; assert(isequal(isextrasafe(x),true))\n\np = 11 p = 5 tf = 1\n\n7   Pass\nx = 71; assert(isequal(isextrasafe(x),false))\n\np = 35 tf = 0\n\n8   Pass\nx = 719; assert(isequal(isextrasafe(x),true))\n\np = 359 p = 179 tf = 1\n\n9   Pass\nx = 2039; assert(isequal(isextrasafe(x),true))\n\np = 1019 p = 509 tf = 1\n\n10   Pass\nx = 2040; assert(isequal(isextrasafe(x),false))\n\ntf = 0\n\n11   Pass\nx = 5807; assert(isequal(isextrasafe(x),true))\n\np = 2903 p = 1451 tf = 1" ]

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[ "# K & R C Programs Exercise 5-6.\n\nK and R C, Solution to Exercise 5-6:\nK and R C Programs Exercises provides the solution to all the exercises in the C Programming Language (2nd Edition)", null, ". You can learn and solve K&R C Programs Exercise.\nC Program to Rewrite appropriate programs from earlier chapters and exercises with pointers instead of array indexing. Good possibilities include getline (Chapters 1 and 4), atoi , itoa , and their variants (Chapters 2, 3, and 4), reverse (Chapter 3), and strindex and getop (Chapter 4). Read more about C Programming Language .\n\n`/************************************************************ You can use all the programs on www.c-program-example.com* for personal and learning purposes. For permissions to use the* programs for commercial purposes,* contact [email protected]* To find more C programs, do visit www.c-program-example.com* and browse!* * Happy Coding***********************************************************/ #include <ctype.h>#include <stdio.h>#include <stdlib.h> /* getline: get line into s, return length */ int getline(char *s, int lim){ char *p; int c; p = s; while (--lim > 0 && (c = getchar()) != EOF && c != 'n') *p++ = c; if (c == 'n') *p++ = c; *p = ''; return (int)(p - s);}int atoi(char *s){ int n, sign; while (isspace(*s)) s++; sign = (*s == '+' || *s == '-') ? ((*s++ == '+') ? 1 : -1) : 1; for (n = 0; isdigit(*s); s++) n = (n * 10) + (*s - '0'); return sign * n;}/*The itoa() function converts an integer value into anASCII string of digits.*/char *utoa(unsigned value, char *digits, int base){ char *s, *p; s = \"0123456789abcdefghijklmnopqrstuvwxyz\"; if (base == 0) base = 10; if (digits == NULL || base < 2 || base > 36) return NULL; if (value < (unsigned) base) { digits = s[value]; digits = ''; } else { for (p = utoa(value / ((unsigned)base), digits, base); *p; p++); utoa( value % ((unsigned)base), p, base); } return digits;}char *itoa(int value, char *digits, int base){ char *d; unsigned u; d = digits; if (base == 0) base = 10; if (digits == NULL || base < 2 || base > 36) return NULL; if (value < 0) { *d++ = '-'; u = -((unsigned)value); } else u = value; utoa(u, d, base); return digits;}static void swap(char *a, char *b, size_t n){ while (n--) { *a ^= *b; *b ^= *a; *a ^= *b; a++; b++; }}void my_memrev(char *s, size_t n){ switch (n) { case 0: case 1: break; case 2: case 3: swap(s, s + n - 1, 1); break; default: my_memrev(s, n / 2); my_memrev(s + ((n + 1) / 2), n / 2); swap(s, s + ((n + 1) / 2), n / 2); break; }}void reverse(char *s){ char *p; for (p = s; *p; p++) ; my_memrev(s, (size_t)(p - s));}static char *strchr(char *s, int c){ char ch = c; for ( ; *s != ch; ++s) if (*s == '') return NULL; return s;}int strindex(char *s, char *t){ char *u, *v, *w; if (*t == '') return 0; for (u = s; (u = strchr(u, *t)) != NULL; ++u) { for (v = u, w = t; ; ) if (*++w == '') return (int)(u - s); else if (*++v != *w) break; } return -1;}#define NUMBER '0' int getop(char *s){ int c; while ((*s = c = getch()) == ' ' || c == 't') ; *(s + 1) = ''; if (!isdigit(c) && c != '.') return c; /* not a number */ if (isdigit(c)) /* collect integer part */ while (isdigit(*++s = c = getch())) ; if (c == '.') /* collect fraction part */ while (isdigit(*++s = c = getch())) ; *++s = ''; if (c != EOF) ungetch(c); return NUMBER;}`\n`Read more c programs `\n`C BasicC StringsK and R C Programs Exercise`\n\nYou can easily select the code by double clicking on the code area above." ]

[ null, "http://www.assoc-amazon.com/e/ir", null ]

[ "# Adding and Subtracting with Negative Numbers\n\nAdding and subtracting with negative numbers is taught through the use of a number line.  As learning progresses students begin to visualise the calculation and use written methods for larger and smaller values.  Students later apply their learning to complete number pyramids involving a mixture of positive and negative integers.\n##### Differentiated Learning Objectives\n• All students should be able to add and subtract with negative numbers using a number line.\n• Most students should be able to add and subtract with negative numbers using written methods.\n• Some students should be able to recognise that a 3 - -2 becomes 3 + 2 through a change of direction on the number line.\n\n### Mr Mathematics Blog\n\n#### Multiplication and Division Rule of Indices\n\nUsing the rules of indices students learn how to evaluate expressions in index notation.\n\n#### Problem Solving Maths Lessons\n\nFour new problem solving lessons to develop student’s mathematical reasoning and communication skills.\n\n#### Enlarging Shapes by a Negative Scale Factor\n\nHow to teach performing and describing negative scale factor enlargements." ]

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[ "", null, "Can I force a number of digits on a number?\n\n#1\n\nIs it possible to force a number to have a number of digits?\n\nExample force \"8\" to render as \"08\"\n\nI'm trying to use this because I want to render a specific date format after having added/subtracted a number of days. Note that I'm not talking about the CURRENT date/time. I want to specify the starting date/\n\ne.g. I specify a start date of 08/15/19 (MM/DD/YY) and I want to subtract 6 days. If I just use a regular equation, it will return 08/9/19. But I want it to return 08/09/19, because that's the format I need to paste into the specific field I want to fill.\n\n#2\n\nYou can use the format attribute for the equation command to do this:\n\n{=4+5}\nPad number with 0's to a length of two:\n{=4+5; format=02}\nPad number with 0's to a length of three:\n{=4+5; format=03}\n\nAnother example:\n{=4+7}\n{=4+7; format=02}\n{=4+7; format=03}\n\nMore examples are available here:\n\n#3\n\nNeat! Thanks", null, "" ]

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[ "petsc-3.11.2 2019-05-18\nReport Typos and Errors\n\n# MatCreateAIJ\n\nCreates a sparse parallel matrix in AIJ format (the default parallel PETSc format). For good matrix assembly performance the user should preallocate the matrix storage by setting the parameters d_nz (or d_nnz) and o_nz (or o_nnz). By setting these parameters accurately, performance can be increased by more than a factor of 50.\n\n### Synopsis\n\n```#include \"petscmat.h\"\nPetscErrorCode MatCreateAIJ(MPI_Comm comm,PetscInt m,PetscInt n,PetscInt M,PetscInt N,PetscInt d_nz,const PetscInt d_nnz[],PetscInt o_nz,const PetscInt o_nnz[],Mat *A)\n```\nCollective on MPI_Comm\n\n### Input Parameters\n\n comm - MPI communicator m - number of local rows (or PETSC_DECIDE to have calculated if M is given) This value should be the same as the local size used in creating the y vector for the matrix-vector product y = Ax. n - This value should be the same as the local size used in creating the x vector for the matrix-vector product y = Ax. (or PETSC_DECIDE to have calculated if N is given) For square matrices n is almost always m. M - number of global rows (or PETSC_DETERMINE to have calculated if m is given) N - number of global columns (or PETSC_DETERMINE to have calculated if n is given) d_nz - number of nonzeros per row in DIAGONAL portion of local submatrix (same value is used for all local rows) d_nnz - array containing the number of nonzeros in the various rows of the DIAGONAL portion of the local submatrix (possibly different for each row) or NULL, if d_nz is used to specify the nonzero structure. The size of this array is equal to the number of local rows, i.e 'm'. o_nz - number of nonzeros per row in the OFF-DIAGONAL portion of local submatrix (same value is used for all local rows). o_nnz - array containing the number of nonzeros in the various rows of the OFF-DIAGONAL portion of the local submatrix (possibly different for each row) or NULL, if o_nz is used to specify the nonzero structure. The size of this array is equal to the number of local rows, i.e 'm'.\n\n### Output Parameter\n\nA -the matrix\n\nIt is recommended that one use the MatCreate(), MatSetType() and/or MatSetFromOptions(), MatXXXXSetPreallocation() paradgm instead of this routine directly. [MatXXXXSetPreallocation() is, for example, MatSeqAIJSetPreallocation]\n\n### Notes\n\nIf the *_nnz parameter is given then the *_nz parameter is ignored\n\nm,n,M,N parameters specify the size of the matrix, and its partitioning across processors, while d_nz,d_nnz,o_nz,o_nnz parameters specify the approximate storage requirements for this matrix.\n\nIf PETSC_DECIDE or PETSC_DETERMINE is used for a particular argument on one processor than it must be used on all processors that share the object for that argument.\n\nThe user MUST specify either the local or global matrix dimensions (possibly both).\n\nThe parallel matrix is partitioned across processors such that the first m0 rows belong to process 0, the next m1 rows belong to process 1, the next m2 rows belong to process 2 etc.. where m0,m1,m2,.. are the input parameter 'm'. i.e each processor stores values corresponding to [m x N] submatrix.\n\nThe columns are logically partitioned with the n0 columns belonging to 0th partition, the next n1 columns belonging to the next partition etc.. where n0,n1,n2... are the input parameter 'n'.\n\nThe DIAGONAL portion of the local submatrix on any given processor is the submatrix corresponding to the rows and columns m,n corresponding to the given processor. i.e diagonal matrix on process 0 is [m0 x n0], diagonal matrix on process 1 is [m1 x n1] etc. The remaining portion of the local submatrix [m x (N-n)] constitute the OFF-DIAGONAL portion. The example below better illustrates this concept.\n\nFor a square global matrix we define each processor's diagonal portion to be its local rows and the corresponding columns (a square submatrix); each processor's off-diagonal portion encompasses the remainder of the local matrix (a rectangular submatrix).\n\nIf o_nnz, d_nnz are specified, then o_nz, and d_nz are ignored.\n\nWhen calling this routine with a single process communicator, a matrix of type SEQAIJ is returned. If a matrix of type MPIAIJ is desired for this type of communicator, use the construction mechanism\n\n``` MatCreate(...,&A); MatSetType(A,MATMPIAIJ); MatSetSizes(A, m,n,M,N); MatMPIAIJSetPreallocation(A,...);\n```\n\n``` MatCreate(...,&A);\n```\n``` MatSetType(A,MATMPIAIJ);\n```\n``` MatSetSizes(A, m,n,M,N);\n```\n``` MatMPIAIJSetPreallocation(A,...);\n```\n\nBy default, this format uses inodes (identical nodes) when possible. We search for consecutive rows with the same nonzero structure, thereby reusing matrix information to achieve increased efficiency.\n\n### Options Database Keys\n\n -mat_no_inode - Do not use inodes -mat_inode_limit - Sets inode limit (max limit=5)\n\n### Example usage\n\nConsider the following 8x8 matrix with 34 non-zero values, that is assembled across 3 processors. Lets assume that proc0 owns 3 rows, proc1 owns 3 rows, proc2 owns 2 rows. This division can be shown as follows\n\n``` 1 2 0 | 0 3 0 | 0 4\nProc0 0 5 6 | 7 0 0 | 8 0\n9 0 10 | 11 0 0 | 12 0\n-------------------------------------\n13 0 14 | 15 16 17 | 0 0\nProc1 0 18 0 | 19 20 21 | 0 0\n0 0 0 | 22 23 0 | 24 0\n-------------------------------------\nProc2 25 26 27 | 0 0 28 | 29 0\n30 0 0 | 31 32 33 | 0 34\n```\n\nThis can be represented as a collection of submatrices as\n\n``` A B C\nD E F\nG H I\n```\n\nWhere the submatrices A,B,C are owned by proc0, D,E,F are owned by proc1, G,H,I are owned by proc2.\n\nThe 'm' parameters for proc0,proc1,proc2 are 3,3,2 respectively. The 'n' parameters for proc0,proc1,proc2 are 3,3,2 respectively. The 'M','N' parameters are 8,8, and have the same values on all procs.\n\nThe DIAGONAL submatrices corresponding to proc0,proc1,proc2 are submatrices [A], [E], [I] respectively. The OFF-DIAGONAL submatrices corresponding to proc0,proc1,proc2 are [BC], [DF], [GH] respectively. Internally, each processor stores the DIAGONAL part, and the OFF-DIAGONAL part as SeqAIJ matrices. for eg: proc1 will store [E] as a SeqAIJ matrix, ans [DF] as another SeqAIJ matrix.\n\nWhen d_nz, o_nz parameters are specified, d_nz storage elements are allocated for every row of the local diagonal submatrix, and o_nz storage locations are allocated for every row of the OFF-DIAGONAL submat. One way to choose d_nz and o_nz is to use the max nonzerors per local rows for each of the local DIAGONAL, and the OFF-DIAGONAL submatrices. In this case, the values of d_nz,o_nz are\n\n``` proc0 : dnz = 2, o_nz = 2\nproc1 : dnz = 3, o_nz = 2\nproc2 : dnz = 1, o_nz = 4\n```\nWe are allocating m*(d_nz+o_nz) storage locations for every proc. This translates to 3*(2+2)=12 for proc0, 3*(3+2)=15 for proc1, 2*(1+4)=10 for proc3. i.e we are using 12+15+10=37 storage locations to store 34 values.\n\nWhen d_nnz, o_nnz parameters are specified, the storage is specified for every row, coresponding to both DIAGONAL and OFF-DIAGONAL submatrices. In the above case the values for d_nnz,o_nnz are\n\n``` proc0: d_nnz = [2,2,2] and o_nnz = [2,2,2]\nproc1: d_nnz = [3,3,2] and o_nnz = [2,1,1]\nproc2: d_nnz = [1,1] and o_nnz = [4,4]\n```\nHere the space allocated is sum of all the above values i.e 34, and hence pre-allocation is perfect.\n\n### Keywords\n\nmatrix, aij, compressed row, sparse, parallel\n\n### See Also\n\nMatCreate(), MatCreateSeqAIJ(), MatSetValues(), MatMPIAIJSetPreallocation(), MatMPIAIJSetPreallocationCSR(),\nMATMPIAIJ, MatCreateMPIAIJWithArrays()\n\nintermediate\n\n### Location\n\nsrc/mat/impls/aij/mpi/mpiaij.c\n\n### Examples\n\nsrc/mat/examples/tutorials/ex4.c.html\nsrc/mat/examples/tutorials/ex4f.F90.html\nsrc/ksp/ksp/examples/tutorials/ex14f.F90.html\nsrc/tao/bound/examples/tutorials/plate2.c.html\nsrc/tao/bound/examples/tutorials/plate2f.F90.html\n\nIndex of all Mat routines\nTable of Contents for all manual pages\nIndex of all manual pages" ]

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[ "# DAVIDSTUTZ\n\n## A Formal Definition of Watertight Meshes\n\nIn computer graphics, watertight meshes usually describe meshes consisting of one closed surface. In this sense, watertight meshes do not contain holes and have a clearly defined inside. Therefore, they are commonly required by many applications in computer graphics as well as in computer vision — for example, when voxelizing meshes into occupancy grids or signed distance functions. However, I found it very difficult to find a proper formal definition of watertightness. In this article, I want to discuss the definition I used for my master thesis.\n\nAs part of my master thesis, I worked with CAD models — specifically, triangular meshes — from ShapeNet [] to learn shape completion of cars. One of the problems with the models in ShapeNet is their complexity. As illustrated below, these models may be very detailed — comprising several hundred thousand faces and vertices — and not watertight (e.g. due to missing windows). In order to \"learn\" shape completion using deep neural networks, for example, I needed to voxelize car models — which required watertight meshes, and simplified ones for efficiency. Figure 1 shows some examples of complex models.", null, "Figure 1: Examples of very complex car models (left) and examples of the semi-convex hull algorithm for simplification (right).\n\nFor the discussion of watertight meshes, we first need to formally define what makes up a proper triangular mesh (e.g. following []):\n\nDefinition. A triangular mesh $\\mathcal{M} = (V, F)$ is defined by a set of vertices $V \\subseteq \\mathbb{R}^3$ and a set of triangular faces $F \\subseteq \\{1,\\ldots,|V|\\}^3$ such that $f = (f_1, f_2, f_3) \\in F$ defines a triangular face enclosed by the corresponding vertices $v_{f_1}$, $v_{f_2}$ and $v_{f_3}$. Faces implicitly also define the edges $E(F)$ between the vertices.\n\nThe concepts of adjacency and incidency are naturally extended to triangular meshes. We note that a triangular mesh only defines the surface of an object. Without additional constraints it is generally hard to reason about the interior and exterior of the surface — and whether the surface is closed. This question naturally leads to the concept of watertight meshes. In the literature, for examples in [], watertight meshes are usually defined as 2-manifold meshes without boundary edges.\n\nHowever, an exact definition — especially one that I could use in my master thesis — was not very easy to find. In the end, I additionally read parts of [] and [] to make the following definitions:\n\nDefinition.\n\n1. A self intersection is an intersection of two faces of the same mesh.\n2. A non-manifold edge has more than two incident faces.\n3. The star of a vertex is the union of all its incident faces.\n4. A non-manifold vertex is a vertex where the corresponding star is not connected when removing the vertex.\n5. A mesh is 2-manifold if it does contain neither self intersections, nor non-manifold edges, nor non-manifold vertices.", null, "Figure 2: Non-manifold vertex (left) and non-manifold edge (right); both from [].\n\nIllustrations of these somewhat abstract definitions can be found in Figure 2 (originally by []). In general, 2-manifold meshes are preferable to arbitrary meshes as many algorithms and applications are not applicable to non-manifold meshes. In our case, however, the definition of 2-manifold meshes is only motivated by the need to formally define watertight meshes (which are sometimes also referred to as closed meshes). Intuitively, the only constraint missing from 2-manifold meshes is a notion of \"closedness\" — meaning a clear interior and exterior. This becomes apparent when considering the definition of a non-manifold edge which also allows edges with only one incident faces, so-called boundary edges.\n\nDefinition. A 2-manifold mesh is called watertight if each edge has exactly two incident faces, i.e. no boundary edges exist.\n\nThe above definitions, while appearing abstract, are also useful in practice. In software, e.g. in MeshLab, it is easy to identify and label non-manifold vertices, edges as well as boundary edges to help design and work with triangular meshes.\n\nFrom the above definition, we can also deduce that \"closedness\" is essentially a design choice. This means that we cannot extract watertight meshes from non-watertight meshes algorithmically. For my application, however, I assumed that all meshes are supposed to be \"closed\", however, may contain complex details within the model — for example, cars that are modeled as \"closed\" exterior but also include interior. In this case, I decided to tackle simplification and the problem of watertightness together by using a semi-convex hull approximation. Results of this approximation can be found in Figure 1.\n\n• [] Mario Botsch, Leif Kobbelt, Mark Pauly, Pierre Alliez, and Bruno Levy. Polygon Mesh Processing. AK Peters, 2010.\n• [] H. Edelsbrunner. Surface Reconstruction by Wrapping Finite Sets in Space. Springer Verlag, 2003.\n• [] P. Giblin. Graphs, Surfaces and Homology. Cambridge University Press, 2010.\n• [] Angel X. Chang, Thomas A. Funkhouser, Leonidas J. Guibas, Pat Hanrahan, Qi-Xing Huang, Zimo Li, Silvio Savarese, Manolis Savva, Shuran Song, Hao Su, Jianxiong Xiao, Li Yi, Fisher Yu: ShapeNet: An Information-Rich 3D Model Repository. CoRR abs/1512.03012 (2015)" ]

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[ "# Hope to get some help on this Grade 6 question.. Hopefully not a need to use algebra to solve, since just learned algebra?\n\nUpdate:\n\nnephew learnt basic algebra only. so i'm trying to find an easier way to explain to him... creating two unknowns to solve is not part of the syllabus at the moment", null, "Relevance\n\nTotal number of pairs of shoes produced = 850. SP = 'sellng price of'.\n\nSP female pair = \\$45. SP male pair = \\$39. Let m = number of male pairs sold.\n\nThen amount collected from sale of m male pairs = \\$(39*m)...(1).\n\nAlso, amount collected from sale of (850-m) female pairs =\\$45(850-m) =\\$(39*m) +\n\n\\$(12210). Then 45(850-m) = 39m + 12210, ie., m(39+45) = 45*850 -12210, ie.,\n\n84m = 38250 -12210 = 26040 and m = 310. Then (1) above --->\\$(39*m) = \\$(39*310)\n\n= \\$12090 = \\$12,100 (rounded to nearest \\$100.) was collected from the sale of the\n\nmale shoes.\n\n•", null, "Login to reply the answers\n• There were 850 pairs of shoes produced in a factory. Each pair of female shoes was sold at \\$45. Each pair of male shoes was sold at \\$39. When all the shoes were sold , the amount of money collected from the sale of female shoes was \\$12210 more than the amount of money collected from the sale of male shoes. How much money was collected from the sale of male shoes?\n\nx + y = 850\n\n45y - 39x = 12210\n\n45x + 45y = 38250\n\n84x = 26040\n\nx = 310\n\n39x = 12090\n\n\\$12100 was collected from the sale of male shoes.\n\n(answer rounded to the nearest hundred dollars).\n\n•", null, "Hirq1 month agoReport\n\nnephew learnt basic algebra only. so i'm trying to find an easier way to explain to him... creating two unknowns to solve is not part of the syllabus at the moment\n\n•", null, "Login to reply the answers\n• That's why you have algebra homework because this is practice for what you should have learned in class.\n\nLet f = pairs of female shoes sold\n\nLet m = pairs of male shoes sold\n\nWe know there are 850 pairs of shoes sold so we can create this equation:\n\nf + m = 850\n\nWe are then told that female shoes are sold for \\$45 each and male shoes are sold for \\$39 each.  So when \"f\" shoes are sold, (45f) dollars have been collected.  When \"m\" shoes are sold, (39m) dollars have been collected.  We are told that the money collected from the female shoes is \\$12,210 more than the amount collected for the male shoes.  We can use this information to create this equation:\n\n45f = 12210 + 39m\n\nWe now have a system of two equations and two unknowns so we can find out how many pairs of each type of shoe was sold.\n\nIf we solve the first equation for f in terms of m we can then substitute that into the other equation and solve for the remaining unknown:\n\nf + m = 850\n\nf = 850 - m\n\n45f = 12210 + 39m\n\n45(850 - m) = 12210 + 39m\n\n38250 - 45m = 12210 + 39m\n\n26040 = 84m\n\n310 = m\n\nWe can find \"f\" if we wanted, but we are asked to find how much money was collected from the men's shoe sales rounded to the nearest hundred.  We know \"m\", so the total amount collected from above is (39m):\n\n39m\n\n39 * 310\n\n12090\n\nRounded to the nearest hundred:\n\n\\$12,100\n\n•", null, "Hirq1 month agoReport\n\nnephew learnt basic algebra only. so i'm trying to find an easier way to explain to him... creating two unknowns to solve is not part of the syllabus at the moment\n\n•", null, "Login to reply the answers\n• Anonymous\n1 month ago\n\nYes, you need algebra.  Unless you prefer to guess.\n\n•", null, "Login to reply the answers" ]

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