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[ "## Cookies on the SKF website\n\nSKF uses cookies on our websites to align the information shown as closely as possible to our visitors’ preferences, such as country and language choices.\n\n# Requisite bearing size\n\nTo determine the requisite size of a spherical plain bearing or rod end, it is necessary to know the requisite rating life for the application. This depends on the type of machine, the operating conditions and the demands regarding operational reliability. The following steps can be used to determine requisite bearing size:\n1. Use the guideline values of the load ratio C/P provided in table 1 to obtain a requisite basic dynamic load rating C. Compare this value with the basic dynamic load rating of the bearings listed in the product tables.\n2. To check whether the sliding contact surface combination of the selected bearing or rod end can be used under the actual load p and sliding velocity v conditions use the pv diagrams:\n• for steel/steel sliding contact surface combination (diagram 1)\n• for SKF Explorer steel/steel bearings (diagram 2)\n• for steel/bronze sliding contact surface combination (diagram 3)\n• for steel/PTFE sintered bronze sliding contact surface combination (diagram 4)\n• for steel/PTFE fabric sliding contact surface combination, metric bearings (diagram 5)\n• for steel/PTFE fabric sliding contact surface combination, inch bearings (diagram 6)\n• for steel/PTFE FRP sliding contact surface combination, FS and FBAS designs (diagram 7)\n• for steel/PTFE FRP sliding contact surface combination, F design (diagram 8)\n3. The specific bearing load p and the sliding velocity v needed to perform this check can be calculated as explained in the following sections.\n• If the pv diagram indicates that the basic rating life equation can be used, proceed to step 4.\n• If the pv diagram shows that the pv range is exceeded, select a bearing with a higher load carrying capacity.\n4. Calculate the Basic rating life and proceed as follows:\n• If the calculated rating life is shorter than the requisite rating life, a larger bearing or rod end should be selected and the calculation repeated.\n• If the calculated rating life is larger than the requisite rating life, the bearing or rod end can be selected for the application.\nThe bearing or rod end size is often determined by the dimensions of the associated components. In these cases, check the pv diagram to determine if the product is suitable.\n\n## Specific bearing load\n\nThe magnitude of the specific bearing load can be calculated using\n\np = K P/C\n\nwhere\n p = specific bearing load [N/mm2] K = specific load factor depending on the bearing design and sliding contact surface combination (table 2) [N/mm2] P = equivalent dynamic bearing load [kN] C = basic dynamic load rating [kN]\n\n## Mean sliding velocity\n\nThe mean sliding velocity for constant movement can be calculated using\n\nv = 5,82 × 10-7 dm β f\n\nwhere\n v = mean sliding velocity [m/s] When the operation is intermittent (not continuous), the mean sliding velocity should be calculated for a cycle of operation. dm = inner ring mean diameter [mm] for radial spherical plain bearings and rod ends dm = dk for angular contact spherical plain bearings dm = 0,9 dk for spherical plain thrust bearings dm = 0,7 dk dk = inner ring sphere diameter [mm], refer to the relevant product table β = half the angle of oscillation (fig. 1) [degrees, °], for rotation β = 90° f = frequency of oscillation [min-1], or rotational speed [r/min]\nFor intermittent movement, the angle of oscillation is usually given per unit time. In this case the mean sliding velocity can be calculated using\n\nv = 8,73 × 10-6 dm 4β / t\n\nwhere\n β = half angle of oscillation (fig. 1) [degrees, °] t = time taken to pass through complete oscillation [s]", null, "" ]

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[ "## 2013/05/23\n\nIn this post, I introduce the widely used stochastic optimization technique, namely the stochastic gradient descent. I also implement the algorithm for the linear-regression problem and provide the Matlab code.\n\nWe consider solving the following type of problems, \\begin{align*} \\min_{\\theta \\in \\bR^d} f(\\theta) := \\frac{1}{n}\\sum_{k=1}^n f_k(\\theta) \\end{align*} where $n$ is very large. To make it a bit concrete, you can write linear/nonlinear regression problems in this form. You can also write neural network training and lots of machine learning problems in this form, too. Starting from an initial parameter estimate $\\theta_0$, gradient descent aims to minimise the cost by implementing the following iteration, \\begin{align*} \\theta_t = \\theta_{t-1} - \\eta \\nabla_\\theta f(\\theta_{t-1}). \\end{align*} Since $f$ is defined as the sum of very large number of functions (each function is parameterised by an individual sample from our dataset), you can easily see the problem: For each $\\theta_{t-1}$, you need to sum over a great number of functions. SGD type methods implement the following type of iteration in order to get rid of this problem, \\begin{align*} \\theta_t = \\theta_{t-1} - \\eta_t g_t(\\theta_{t-1}). \\end{align*} where $\\bE[g_t(\\theta)] = \\nabla_\\theta f(\\theta)$ and $\\eta_t$ is the step-size satisfying $\\sum_t \\eta_t = \\infty$ and $\\sum_t \\eta_t^2 < \\infty$. This is the general way to write the algorithm. Of course the simplest thing to do is, \\begin{align*} \\theta_t = \\theta_{t-1} - \\eta_t \\nabla_\\theta f_{i_t}(\\theta) \\end{align*} where $i_t$ is drawn uniformly random from the index set $[n] = \\{1,\\ldots,n\\}$. Since we draw the index uniformly random, the expectation of the gradient we used here is the same with the exact gradient. Using a single sample would result in a high-variance estimate of the gradient so people usually use a minibatch of samples in order to reduce the variance. There is a huge literature on variance reduction algorithms, I will not even try to address that.\n\nIn my previous post, I introduced the algorithm for maximum-likelihood (ML) estimation. It is fairly well known that you can use SGD for ML estimation by minimising the averaged negative log-likelihood. This time I will not go into any probabilistic interpretation but just sketch how you can apply SGD to linear regression.\n\nLinear Regression.\n\nLet's try to solve the large-scale linear regression problem with SGD. Assume we have lots of unidimensional observations (outputs) $(y_k)_{k=1}^n$ and coefficients $(x_k)_{k=1}^n$ (inputs) where $x_k \\in \\bR^d$.\n\nWe would like to fit a linear model to model input-output relationship: $Y \\approx X^\\top \\theta$ where $X$ is the input and $Y$ is the output. In general, the usual approach in machine learning is to formulate the problem as an empirical risk minimisation problem. In this framework, you assume $(X,Y) \\sim \\bP$ where $\\bP$ is an unknown probability measure. Ideally, you would like to minimise (the risk), \\begin{align*} \\min_\\theta \\bE_\\bP\\left[(Y - X^\\top \\theta)^2\\right] \\end{align*} where the expectation is taken with respect to the unknown probability measure and $(X,Y) \\sim \\bP$ are random variables. Since we don't know $\\bP$ but instead have samples $(X_{1:n} = x_{1:n},Y_{1:n} = y_{1:n})$ from it, we can instead minimise the empirical risk (the Monte Carlo estimation of the expectation, if you like): \\begin{align*} \\min_\\theta \\frac{1}{n}\\sum_{k = 1}^n (y_k - x_k^\\top \\theta)^2. \\end{align*} To be clear, you can arrive to the same optimisation problem by assuming $y_k$ is generated via $x_k^\\top \\theta$ plus Gaussian noise and deriving the maximum likelihood estimate of $\\theta$ (being Bayesian but without priors). The factor $\\frac{1}{n}$ will be missing but it can be put into the step size of the algorithm. For finite $n$, it is a constant so it doesn't change the argument. Here $f_k(\\theta) = (y_k - x_k^\\top \\theta)^2$, so if you implement stochastic gradient descent, you will have, \\begin{align*} \\theta_k &= \\theta_{k-1} - \\eta_k \\nabla_\\theta f_k(\\theta_{k-1}) \\\\ &= \\theta_{k-1} + \\eta_k x_k (y_k - x_k^\\top \\theta_{k-1}). \\end{align*} That's it. Actually there was a 2-factor but it can be embedded into the step-size. Of course, there is a usual assumption on the step size, you need, \\begin{align*} \\sum_k \\eta_k = \\infty \\,\\,\\,\\,\\,\\,\\, \\text{and} \\,\\,\\,\\,\\,\\,\\, \\sum_k \\eta_k^2 < \\infty \\end{align*} What's done in practice is to parameterise the entire $(\\eta_k)_{k=1}^\\infty$ sequence by two parameters. Basically, you choose, \\begin{align*} \\eta_k = \\frac{\\alpha}{k^\\beta} \\end{align*} with $\\alpha > 0$ and $0.5 < \\beta < 1$. A typical choice is $\\beta = 0.51$ and then you can tune only $\\alpha$. Here is a simple Matlab code to generate data and perform regression:\n\n% DATA GENERATION\nN = 1000; % small dataset\ndx = 3; % dimension of the parameter and inputs\nth_tr = [1,2,3]'; % the real parameter\nX = rand(dx,N); % arbitrary input data generation\nsig = 0.5; % noise variance\nY = X' * th_tr + sqrt(sig) * randn(N,1); % output data generation\n%\n\n% ALGORITHM INITIALISATION\nthinit = [0,0,0]';\na = 0.7;\nb = 0.51;\n%\n\n% The first iteration\nt = 1;\nind = randi(N);\ngam(t) = a ./(t^b);\nth(:,t) = thinit + gam(t) * X(:,ind) * (Y(ind) - X(:,ind)' * thinit);\nerr(t) = sqrt((1/N) * norm(Y - X' * th(:,t))^2);\n% Note that computing the error is not necessary and it is not done in\n% practice since datasets are massive (Look at the code line and see that\n% all data is required to compute the error). For this example, you can\n% plot it.\n%\n\n% For loop.\nfor t = 2:2*N\nind = randi(N);\ngam(t) = a ./(t^b);\nth(:,t) = th(:,t-1) + gam(t) * X(:,ind) * (Y(ind) - X(:,ind)' * th(:,t-1));\nerr(t) = sqrt((1/N) * norm(Y - X' * th(:,t))^2);\n\nend\n\nLast updated: 17/01/2017" ]

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[ "# NMath User's Guide\n\n18.1 Creating Matrices (.NET, C#, CSharp, VB, Visual Basic, F#)\n\nThis section describes how to create instances of the structured sparse matrix classes.\n\nCreating Default Matrices\n\nYou can construct default structured sparse matrices by supplying the necessary parameters to describe the matrix shape, as shown in Table 12. All stored values are initialized to zero.\n\n Matrix Type Shape Parameters Lower Triangular Order Upper Triangular Order Symmetric Order Hermitian Order Banded Rows, Columns, Lower Bandwidth, Upper Bandwidth TriDiagonal Rows, Columns Symmetric Banded Order, Half Bandwidth Hermitian Banded Order, Half Bandwidth\n\nSquare matrix types are characterized by their order--that is, the number of rows and columns. For example, a matrix of order 3 is a 3 x 3 matrix. Thus, this code creates a default 5 x 5 Hermitian matrix of double-precision complex numbers:\n\nCode Example – C# matrix\n\n```var A = new DoubleHermitianMatrix( 5 );\n```\n\nCode Example – VB matrix\n\n```Dim A As New DoubleHermitianMatrix(5)\n```\n\nConstructors for rectangular matrix types accept separate row and column shape parameters. For example:\n\nCode Example – C# matrix\n\n```var A = new DoubleTriDiagMatrix ( 3, 5 );\n```\n\nCode Example – VB matrix\n\n```Dim A As New DoubleTriDiagMatrix(3, 5)\n```\n\nConstructors for banded matrix types also accept bandwidth parameters that describe the width of the banded region. Thus, the following code creates a 4 x 5 FloatComplexBandMatrix with a lower bandwidth of 1 and an upper bandwidth of 2:\n\nCode Example – C# matrix\n\n```var A = new FloatComplexMatrix( 4, 5, 1, 2 );\n```\n\nCode Example – VB matrix\n\n```Dim A As New FloatComplexMatrix(4, 5, 1.0F, 2.0F)\n```\n\nThis creates an 8 x 8 FloatSymBandMatrix with a half bandwidth of 2:\n\nCode Example – C# matrix\n\n```var A = new FloatSymBandMatrix( 8, 2 );\n```\n\nCode Example – VB matrix\n\n```Dim A As New FloatSymBandMatrix(8, 2)\n```\n\nOnce you've constructed a default matrix, you can set individual values using the provided indexers (Section 18.2). In some case, methods are also provided that return vector views of the underlying data, which can also be used to set matrix values (Section 18.5).\n\nCreating Sparse Matrices from General Matrices\n\nYou can construct all NMath structured sparse matrix types from general matrix types. Such constructors extract the appropriate values from the general matrix. Data is copied.\n\nFor example, this code constructs a FloatUpperTriMatrix instance by extracting the upper triangular region of a square general matrix:\n\nCode Example – C# matrix\n\n```var genMat = new FloatMatrix( 5, 5, 0, 1 );\n```\n```var A = new FloatUpperTriMatrix( genMat );\n```\n\nCode Example – VB matrix\n\n```Dim GenMat As New FloatMatrix(5, 5, 0.0F, 1.0F)\n```\n```Dim A As New FloatUpperTriMatrix(GenMat)\n```\n\nConstructors for square matrix types, such as upper triangular matrices, throw a MatrixNotSquareException if the given general matrix is not square. Alternatively, you can pass in a non-square general matrix and specify the order of the square submatrix to extract. Thus, this code creates a 3 x 3 DoubleSymmetricMatrix by extracting the upper triangular region of the 3 x 3 leading submatrix from the given 4 x 6 general matrix:\n\nCode Example – C# matrix\n\n```var genMat = new DoubleMatrix( 4, 6, 0, 0.25 );\n```\n```var A = new DoubleSymmetricMatrix( A, 3 );\n```\n\nCode Example – VB matrix\n\n```Dim GenMat As New DoubleMatrix(4, 6, 0.0, 0.25)\n```\n```Dim A As New DoubleSymmetricMatrix(A, 3)\n```\n\nAn IndexOutOfRangeException is raised if the given order specifies a submatrix that is out of bounds.\n\nBanded matrix types can also be constructed from general matrices by specifying the desired bandwidth. For instance, the following code extracts the values required to construct a Hermitian banded matrix with a half bandwidth of 3 from the given general matrix:\n\nCode Example – C# matrix\n\n```var incr = new DoubleComplex( 1, 0.25 );\n```\n```var genMat = new DoubleComplexMatrix( 12, 12, 0, incr );\n```\n```var A = new DoubleHermitianBandMatrix( A, 3 );\n```\n\nCode Example – VB matrix\n\n```Dim Incr As New DoubleComplex(1.0, 0.25)\n```\n```Dim GenMat As New DoubleComplexMatrix(12, 12, 0.0, Incr)\n```\n```Dim A As New DoubleHermitianBandMatrix(A, 3)\n```\n\nCreating Sparse Matrices from Other Sparse Matrices\n\nSome structured sparse matrix types can be constructed from other structured sparse matrices. For example, a tridiagonal matrix is really a special case of a banded matrix with lower and upper bandwidth equal to 1. Therefore, banded matrices can be constructed from tridiagonal matrices, and vice versa. For example:\n\nCode Example – C# matrix\n\n```int rows = 8, cols = 8, ub = 0, lb = 2;\n```\n```var data = new FloatVector( (ub+lb+1)*cols, 1, 1 );\n```\n```var A = new FloatBandMatrix( data, rows, cols, lb, ub );\n```\n```var B = new FloatTriDiagMatrix( A );\n```\n\nCode Example – VB matrix\n\n```Dim Rows As Integer = 8\n```\n```Dim Cols As Integer = 8\n```\n```Dim UB As Integer = 0\n```\n```Dim LB As Integer = 2\n```\n```Dim Data As New FloatVector((UB + LB + 1) * Cols, 1.0F, 1.0F)\n```\n```Dim A As New FloatBandMatrix(Data, Rows, Cols, LB, UB)\n```\n```Dim B As New FloatTriDiagMatrix(A)\n```\n\nSimilarly, you can construct banded matrices from symmetric or Hermitian banded matrices, or triangular matrices from symmetric or Hermitian matrices, and vice versa.\n\nCreating Sparse Matrices from a Data Vector\n\nYou can construct all NMath structured sparse matrix types from an appropriate data vector and shape parameters. The vector storage scheme used by each structured sparse matrix type is described in Chapter 17. For example, you could create this 4 x 4 symmetric matrix:", null, "like this:\n\nCode Example – C# matrix\n\n```var data = new DoubleVector( 10, 0, 1 );\n```\n```var A = new DoubleSymmetricMatrix( data, 4 );\n```\n\nCode Example – VB matrix\n\n```Dim Data As New DoubleVector(10, 0.0, 1.0)\n```\n```Dim A As New DoubleSymmetricMatrix(Data, 4)\n```\n\nSimilarly, you could create this 5 x 7 banded matrix with an upper bandwidth of 1 and a lower bandwidth of 0:", null, "using this code:\n\nCode Example – C# matrix\n\n```var data = new FloatVector( 14, 1 );\n```\n```var A = new FloatBandMatrix( data, 5, 7, 0, 1 );\n```\n\nCode Example – VB matrix\n\n```Dim Data As New FloatVector(14, 1.0F)\n```\n```Dim A As New FloatBandMatrix(Data, 5, 7, 0, 1)\n```\n\nImplicit Conversion\n\nNMath provides implicit conversion operators for the structured sparse matrix classes. Single-precision types are implicitly promoted to double-precision types, and real types are implicitly promoted to complex types, as shown in Figure 4. An arrow indicates implicit promotion.\n\nFigure 4 – Implicit conversion for matrix data types", null, "For example, Figure 5 shows the pattern for implicit conversion among the tridiagonal types.\n\nFigure 5 – Implicit conversion for tridiagonal matrices", null, "Copying Matrices\n\nThe NMath structured sparse matrix classes provide three copy methods:\n\nClone() returns a deep copy of a matrix. Data is copied; each matrix references different data.\n\nShallowCopy() returns a shallow copy of a matrix. Data is not copied; both matrices reference the same data.\n\nDeepenThisCopy() copies the data viewed by a matrix to new data block. This guarantees that there is only one reference to the underlying data, and that this data is in contiguous storage.\n\nFor instance:\n\nCode Example – C# matrix\n\n```var A = new FloatUpperTriMatrix( 5 );\n```\n```FloatUpperTriMatrix B = A.ShallowCopy();\n\n```\n\n```B[0,0] = 1; // A[0,0] == B[0,0]\n```\n```B.DeepenThisCopy();\n```\n```B[0,0] = 2; // A[0,0] != B[0,0]\n```\n\nCode Example – VB matrix\n\n```Dim A As New FloatUpperTriMatrix(5)\n```\n```Dim B As FloatUpperTriMatrix = A.ShallowCopy()\n\n```\n\n```B(0, 0) = 1 ' A[0,0] == B[0,0]\n```\n```B.DeepenThisCopy()\n```\n```B(0, 0) = 2 ' A[0,0] != B[0,0]\n```\n\nTop\n\nTop" ]

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[ "# Equivalent narcotic depth\n\nEquivalent narcotic depth (END) is used in technical diving as a way of estimating the narcotic effect of a breathing gas mixture, such as heliox and trimix. The method is, for a given mix and depth, to calculate the depth which would produce the same narcotic effect when breathing air.\n\nThe equivalent narcotic depth of a breathing gas mix at a particular depth is calculated by finding the depth of a dive when breathing air that would have the same total partial pressure of nitrogen and oxygen as the breathing gas in question. For example, a trimix containing 20% oxygen, 40% helium, 40% nitrogen (trimix 20/40) being used at 60 metres (200 ft) has an END of 32 metres (105 ft).\n\nSince air is composed of approximately 21% oxygen and 79% nitrogen, the narcotic gases make up 100% of the mix, or equivalently the fraction of the total gases which are narcotic is 1.0. Oxygen is assumed equivalent in narcotic effect to nitrogen for this purpose. In contrast, the oxygen and nitrogen component in a trimix containing, for example, 40% helium accounts for only 60% of the mix, i.e. a fraction of 0.6. In a trimix, the fraction of narcotic gases (oxygen and nitrogen) is equal to 1.0 minus the fraction of non-narcotic gas (helium).\n\n## Calculations\n\nIn diving calculations it is assumed unless otherwise stipulated that the atmospheric pressure is 1 bar or 1 atm. and that the diving medium is water. The ambient pressure at depth is the sum of the hydrostatic pressure due to depth and the atmospheric pressure on the surface.\n\n### Metres\n\nThe partial pressure of a gas in a mixture at a particular depth in metres is given by:\n\nfraction of gas × (depth/10 + 1)\n\nSo the equivalent narcotic depth can be calculated as follows:\n\npartial pressure of narcotic gases in air at END = partial pressure of narcotic gases in trimix at a given depth.\n\nor\n\n(fraction of O2 + fraction of N2) in air × (END/10 + 1) = (fraction of O2 + fraction of N2) in trimix × (depth/10 +1)\n\nwhich gives:\n\n1.0 × (END/10 + 1) = (fraction of O2 + fraction of N2) in trimix × (depth/10 +1)\n\nresulting in:\n\nEND = (depth + 10) × (fraction of O2 + fraction of N2) in trimix − 10\n\nSince (fraction of O2 + fraction of N2) in a trimix = (1 − fraction of helium), the following formula is equivalent:\n\nEND = (depth + 10) × (1 − fraction of helium) − 10\n\n#### Example in metres\n\nWorking the earlier example, for a gas mix containing 40% helium being used at 60 metres, the END is:\n\nEND = (60 + 10) × (1 − 0.4) − 10\nEND = 70 × 0.6 − 10\nEND = 42 − 10\nEND = 32 metres\n\nSo at 60 metres on this mix, the diver would feel the same narcotic effect as a dive on air to 32 metres.\n\n### Feet\n\nThe partial pressure of a gas in a mixture at a particular depth in feet is given by:\n\nfraction of gas × (depth/33 + 1)\n\nSo the equivalent narcotic depth can be calculated as follows:\n\npartial pressure of narcotic gases in air at END = partial pressure of narcotic gases in trimix at a given depth.\n\nor\n\n(fraction of O2 + fraction of N2) in air × (END/33 + 1) = (fraction of O2 + fraction of N2) in trimix × (depth/33 +1)\n\nwhich gives:\n\n1.0 × (END/33 + 1) = (fraction of O2 + fraction of N2) in trimix × (depth/33 +1)\n\nresulting in:\n\nEND = (depth + 33) × (fraction of O2 + fraction of N2) in trimix − 33\n\nSince (fraction of O2 + fraction of N2) in a trimix = (1 − fraction of helium), the following formula is equivalent:\n\nEND = (depth + 33) × (1 − fraction of helium) − 33\n\n#### Example in feet\n\nAs an example, for a gas mix containing 40% helium being used at 200 feet, the END is:\n\nEND = (200 + 33) × (1 − 0.4) − 33\nEND = 233 × 0.6 − 33\nEND = 140 − 33\nEND = 107 feet\n\nSo at 200 feet on this mix, the diver would feel the same narcotic effect as a dive on air to 107 feet.\n\n## Oxygen narcosis\n\nSince there is evidence that oxygen plays a part in the narcotic effects of a gas mixture, the NOAA diving manual recommends treating oxygen and nitrogen as equally narcotic. This is now preferred to the previous method of considering only nitrogen as narcotic, since it is more conservative. In this analysis, it is assumed that the narcotic potentials of nitrogen and oxygen are similar. Although oxygen has greater lipid solubility than nitrogen and therefore should be more narcotic (Meyer-Overton correlation), it is likely that some of the oxygen is metabolised, thus reducing its effect to a level similar to that of nitrogen." ]

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[ "# EP1998435B1 - Method and system for estimating rotor angular position and rotor angular velocity at low speeds or standstill - Google Patents\n\nMethod and system for estimating rotor angular position and rotor angular velocity at low speeds or standstill Download PDF\n\n## Info\n\nPublication number\nEP1998435B1\nEP1998435B1 EP08251107.2A EP08251107A EP1998435B1 EP 1998435 B1 EP1998435 B1 EP 1998435B1 EP 08251107 A EP08251107 A EP 08251107A EP 1998435 B1 EP1998435 B1 EP 1998435B1\nAuthority\nEP\nEuropean Patent Office\nPrior art keywords\nβ\nα\nω\nproduce\nsignal\nPrior art date\nLegal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)\nActive\nApplication number\nEP08251107.2A\nOther languages\nGerman (de)\nFrench (fr)\nOther versions\nEP1998435A3 (en\nEP1998435A2 (en\nInventor\nJun Hu\nCurrent Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)\nHamilton Sundstrand Corp\nOriginal Assignee\nHamilton Sundstrand Corp\nPriority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)\nFiling date\nPublication date\nPriority to US11/754,396 priority Critical patent/US7577545B2/en\nApplication filed by Hamilton Sundstrand Corp filed Critical Hamilton Sundstrand Corp\nPublication of EP1998435A2 publication Critical patent/EP1998435A2/en\nPublication of EP1998435A3 publication Critical patent/EP1998435A3/en\nApplication granted granted Critical\nPublication of EP1998435B1 publication Critical patent/EP1998435B1/en\nActive legal-status Critical Current\nAnticipated expiration legal-status Critical\n\n• 230000000875 corresponding Effects 0.000 claims description 5\n• 238000001914 filtration Methods 0.000 claims description 2\n• 230000001131 transforming Effects 0.000 claims 3\n• 238000010168 coupling process Methods 0.000 claims 2\n• 238000005859 coupling reaction Methods 0.000 claims 2\n• 238000000034 methods Methods 0.000 description 4\n\n## Images\n\n•", null, "•", null, "•", null, "•", null, "•", null, "•", null, "## Classifications\n\n• HELECTRICITY\n• H02GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER\n• H02PCONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS\n• H02P21/00Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation\n• H02P21/06Rotor flux based control involving the use of rotor position or rotor speed sensors\n• HELECTRICITY\n• H02GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER\n• H02PCONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS\n• H02P21/00Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation\n• H02P21/14Estimation or adaptation of machine parameters, e.g. flux, current or voltage\n• H02P21/18Estimation of position or speed\n\n## Description\n\nBACKGROUND OF THE INVENTION\n• This invention relates to rotor angular position and velocity sensing systems for mechanical shaft sensorless control of dynamoelectric machines, and more particularly to an improved system for resolving the position and velocity of a rotor for a dynamoelectric machine using an estimate of extended rotor flux.\n• In some vehicles, including some aircraft, a motor may be utilized both as a motor and as a generator. Because of this dual function, the motor may be called a dynamoelectric machine. A typical motor comprises a stationary stator, and a rotating rotor. In some motors, it is necessary to detect a position of a rotor in order to sustain operation of the motor. Determining a rotor position typically requires a shaft position sensor. It is desirable to eliminate a mechanical shaft sensor to reduce cost and improve reliability.\n• Some methods of sensorless rotor position detection include the back EMF method, which determines rotor position based on voltage, the signal injection method, which injects high frequencies into a system, and the method discussed in U.S. Patent No. 7,072,790 which uses flux to determine rotor position. It is desirable to improve the method U.S. Patent No. 7,072,790 for applications operating at low speeds or at a standstill.\n• SUMMARY OF THE INVENTION\n• The present invention relates to a method of estimating rotor angular position and rotor angular velocity for a dynamoelectric machine as claimed in claim 1, and to a control for estimating an initial rotor angular position and a rotor angular velocity for a dynamoelectric machine from a standstill as claimed in claim 8.\n• A method and system for estimating an angular position and an angular velocity of a rotor in a dynamoelectric machine measures an AC current and a potential for each of a plurality of windings coupled to a stator of the dynamoelectric machine, transforms the measured currents and potentials to a stationary frame to produce transformed currents and transformed potentials, and processes the transformed currents and transformed potentials to produce a first intermediate signal and a second intermediate signal. The first intermediate signal and the second intermediate signal are cross-coupled and processed to obtain a first extended rotor flux value and a second extended rotor flux value. The first extended rotor flux value and the second extended rotor flux value are applied to a phase lock loop to derive an estimated rotor angular position and an estimated rotor angular velocity for the dynamoelectric machine.\n• These and other features of the present invention can be best understood from the following specification and drawings, the following of which is a brief description, and which describe certain embodiments of the invention by way of example only.\n• BRIEF DESCRIPTION OF THE DRAWINGS\n• Figure 1 illustrates a block diagram of a mechanical sensorless rotor angular position and velocity sensing system.\n• Figure 2 illustrates a phasor diagram of electrical parameters related to extended rotor flux.\n• Figure 3 illustrates a block diagram of operations performed within the system of Figure 1 to calculate a first extended rotor flux value and a second extended rotor flux value.\n• Figure 4 illustrates a block diagram of how a microprocessor of Figure 1 uses a phase lock loop (PLL) to obtain an estimated rotor angular position and an estimated rotor angular velocity.\n• Figure 5 illustrates how a stationary frame of the system of Figure 1 aligns with multiple phases of AC.\n• Figure 6 illustrates an initial stator voltage and an initial rotor position as a function of time.\nDETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT\n• As shown in Figure 1, the rotor angular position and velocity sensing system 10 comprises a motor 12 that is able to operate as a starter to start an engine 14, or as a generator to power a load (not shown). Because of this dual function, the motor 12 may be called a dynamoelectric machine. In one example, the motor 12 is a brushless motor that requires a controller to know a position of its rotor to operate.\n• To start the motor 12, an AC power supply 16 provides an AC voltage along supply lines 18 to a rotating exciter 19. In the example of Figure 1, the AC power supply 16 provides three phases of AC power, however it is understood that other numbers of phases of AC power could be provided. The rotating exciter is connected to a shaft 21 that is also connected to the motor 12 and the engine 14.\n• The AC voltage from the supply lines 18 induces an AC voltage along motor terminals 20a, 20b, and 20c. The induced voltage causes a current to flow through an output filter 22. A microprocessor 24 measures a voltage 26 and a current 28 from each of the terminals 20a, 20b, and 20c. A position and speed estimator 30 uses the voltage and current measurements to estimate a flux of the motor 12 and to estimate a rotor position 32 and a rotor angular velocity 34.\n• Once the estimated rotor position 32 and estimated rotor angular velocity 34 have been calculated, an inverter 38 is turned ON. The microprocessor 24 processes the estimated rotor position 32 and estimated rotor angular velocity 34 to control a pulse width modulated (PWM) generator 36. Inverter 38 is coupled to the PWM generator 36 and converts a DC voltage from DC voltage supply lines 40 to AC. This voltage enables AC to flow through the output filter 22, which improves power quality by filtering out harmonics and reducing electromagnetic interference (EMI). The AC from the terminals 20a, 20b, and 20c then flows to a stator of the motor 12 to sustain operation of the motor 12.\n• Figure 1 illustrates how the microprocessor 24 processes the estimated rotor position 32 and estimated rotor angular velocity 34 to control the inverter 38. An abc to d-q frame transformer 42 uses the estimated rotor position 32 to transform the current measurements 28 to a rotating d-q frame to obtain current values Id and Iq. A torque current profile generator 44 uses the estimated rotor angular velocity 34 to lookup reference current values Id* and Iq*. Comparators 45 and 46 compare the transformed Id and Iq values to reference current values Id* and Iq* to determine differences ΔId and ΔIq between the transformed values and the reference values.\n• Proportional and integral (PI) regulators 47 and 48 process the differences ΔId and ΔIq using proportional and integral gains, and transmit an output signal to d-q to alpha-beta frame transformer 50, which converts the output into a stationary α -β frame to produce Valpha* and Vbeta* signals which are transmitted to the PWM generator 36. The PWM generator then controls the inverter 38 accordingly to produce a desired AC voltage.\n• The output filter 22 comprises an inductor and a capacitor (not shown) in each phase. An input current Iinvt flows from the inverter 38 along the windings 23 a, 23b, and 23c to the output filter 22, and an output current Is flows from the output filter 22 along the terminals 20a, 20b, and 20c to the motor 12. The current flowing through the capacitor can be calculated by the following equation: $I ^ c = C ⁢ ⅆ V s ⅆ t$\n• where Îc is an estimated capacitor current; and\n• Vs is one of the voltage measurements 26.\nA motor current can then be calculated using the following equation: $I s = I invt - I ^ c$\n• where Is is the calculated motor current; and\n• Iinvt is the inverter output current.\nEquations 1 and 2 apply to all three phases A, B, and C corresponding to the three windings 20a, 20b, and 20c.\n• The voltage measurements 26 and current measurements 28 are measured from each of the three terminals (20a, 20b, 20c) and each of the three windings (23a, 23b; 23c) in an a-b-c frame. The current measurement 28 is a measurement of the inverter output current Iinvt. A flux estimation is implemented in an alpha-beta (α-β) stationary frame. The relationship between the α-β frame and the a-b-c frame is described in the following equation: $f α f β = 2 3 ⁢ 1 - 1 2 - 1 2 0 3 2 - 3 2 ⁢ f a f b f c$\n• where f can be replaced with voltage, current, or flux;\n• α, b, and c represent the phases of current on the terminals 20a, 20b, and 20c in the a-b-c frame; and\n• α and β represent axes of the α-β frame.\n• The stationary α-β frame is a two phase frame and is a necessary step in calculating flux. Equation #3 is used to determine an α-axis voltage Vα, a β-axis voltage Vβ, an α-axis current Iα, and a β-axis current Iβ.\n• The following equation can then be used to determine an extended rotor flux in the α-β stationary frame: $λ ext_α λ ext_β = 1 s ⁢ V α V β - R s 0 0 R s ⁢ I α I β - L q 0 0 L q ⁢ I α I β$\n• where λ ext is an alpha extended rotor flux;\n• λ ext is a beta extended rotor flux;\n• Rs is a stator resistance;\n• Lq is a q-axis inductance; and\n• 1/s is an integrator.\nEquation #4 can be used to determine flux in both salience and non-salience motors. As shown in equation #4, an integrator $1 s$ is required to calculate extended rotor flux. The integrator $1 s$ is an operator, not a variable, s being a Laplace operator. One problem that may arise when using a pure integrator, such as $\" 1 s \" ,$ is a DC drift problem, in which a small DC component in an AC signal can cause a substantial error in a flux determination. To avoid the DC drift problem associated with a pure integrator, lag functions, such as $1 s + ω i$ and $ω i s + ω i ,$ may be used, as shown in the following equation: $λ ext_α λ ext_β = 1 ω i s + ω i - ω i s + ω i 1 ⁢ ω i s + ω i ⁢ V α V β - R s 0 0 R s ⁢ I α I β - L q 0 0 L q ⁢ I α I β$\n• where ωi is a selected corner frequency.\nSelected corner frequency ωi may be fixed or it may be adjustable.\n• Figure 2 illustrates a phasor diagram in the rotating d-q frame. Figure 2 illustrates the relationship between extended rotor flux and back EMF.\n• A flux λs in the stator of the motor 12 is represented by a phasor 60. A stator current Is is represented by a phasor 62. A stator potential Vs is represented by a phasor 64. A phasor 66 represents Is*Lq where Lq is a q-axis rotor inductance. A vector sum of the phasor 60, representing λs, and the phasor 66, representing Is*Lq, is an extended rotor flux λext, which aligns with the d-axis of the d-q frame, and is represented by a phasor 67.\n• A back electromotive force (EMF) Es is represented by a phasor 68. As shown in Figure 2, the back EMF Es is perpendicular to the stator flux λs . The back EMF Es , represented by the phasor 68, is a vector sum of the stator potential Vs represented by phasor 64 and stator resistance potential drop Is*Rs represented by a phasor 70, where Rs is the stator resistance.\n• An extended back electromotive force (EEMF), Eext , in the stator is represented by a phasor 72, and aligns with the q-axis of the d-q frame. Is*Xq , where Xq is a q-axis stator reactance, is represented by a phasor 74. The extended back EMF represented by phasor 72 is a vector sum of Es represented by phasor 68 and Is*Xq represented by a phasor 74.\n• Figure 3 illustrates a block diagram of the flux estimation algorithm shown in equation #5. As shown in Figure 3, a transformed measured current Iα for the α -axis on a signal path 80 is multiplied by the stator resistance Rs 82 to produce Iα *Rs on a signal path 84. A summer 86 subtracts Iα * Rs on the signal path 84 from the transformed potential V α on a signal path 88 to produce V α-(Iα*Rs ) on a signal path 90. Vα- (Iα *Rs ) on the signal path 90 is multiplied by a $1 s + ω i$ first lag function 92 to produce first intermediate signal $1 s + ω i ⁢ V α - I α * R s$ on a signal path 93. $1 s + ω i ⁢ V α - I α * R s$ on the signal path 93 is multiplied by a $ω i s + ω i$ second lag function 94 to produce $ω i s + ω i ⁢ V α - I α * R s$ on a signal path 95.\n• Additionally, a transformed measured current I β for the β-axis on a signal path 96 is multiplied by the stator resistance Rs 98 to produce Iβ *Rs on a signal path 100. A summer 102 subtracts Iβ *Rs on the signal path 100 from the transformed potential Vβ on a signal path 104 to produce Vβ-(Iβ*Rs ) on a signal path 106. Vβ -(Iβ*Rs ) on the signal path 106 is multiplied by the $1 s + ω i$ first lag function 108 to produce second intermediate signal $1 s + ω i ⁢ V β - I β * R s$ on a signal path 110. $1 s + ω i ⁢ V β - I β * R s$ on the signal path 110 is multiplied by the $ω i s + ω i$ second lag function 112 to produce $ω i s + ω i ⁢ V β - I β * R s$ on a signal path 114.\n• The transformed measured current I α for the α-axis on the signal path 80 is also multiplied by a q-axis inductance Lq 116 to produce I α *Lq on the signal path 118. A summer 120 subtracts Iα*Lq on the signal path 118 from $1 s + ω i ⁢ V α - I α * R s$ on the signal path 93 and adds $ω i s + ω i ⁢ V β - I β * R s$ from the signal path 114 to produce $1 s + ω i ⁢ V α - I α * R s + ω i s + ω i 2 ⁢ V β - I β * R s - I α * L q$ which corresponds to the extended rotor flux on the α-axis λ ext_α on the signal path 122. The \"^\"notation indicates that the extended rotor flux is an estimate based on measured values.\n• Additionally, the transformed measured current Iβ for the β-axis on the signal path 96 is also multiplied by a q-axis inductance Lq 124 to produce Iβ*Lq on the signal path 126. A summer 128 subtracts Iβ*Lq on the signal path 126 from $1 s + ω i ⁢ V β - I β * R s$ on the signal path 110 and subtracts $ω i s + ω i ⁢ V α - I α * R s$ on the signal path 95 to produce $1 s + ω i ⁢ V β - I β * R s - ω i s + ω i 2 ⁢ V α - I α * R s - I β * L q$ which corresponds to the extended rotor flux on the β-axis λ est_β on the signal path 130. Once again, the \"^\"notation indicates that the extended rotor flux is an estimate based on measured values.\n• As shown in Figure 3, the signal paths 95 and 114 cross-couple the signal paths 93 and 110.\n• The following equation can be used to describe the relationship between the extended rotor flux and the rotor position: $λ ext_α λ ext_β = λ ⁢ cos θ sin θ$\n• where θ is the rotor position; and\n• λ is a flux amplitude.\n• Using equation #6, it would be possible to use an arctangent function to calculate a rotor position. Another option is to used a phase-locked loop (PLL) to derive position and angular velocity information.\n• Figure 4 is a block diagram illustrating how the microprocessor 24 uses a phase lock loop (PLL) to improve an estimate of rotor angular position and rotor angular velocity. The estimated α-axis extended rotor flux λ ext_α and the estimated β-axis extended rotor flux λ ext_β are applied to the signal paths 122 and 130. A multiplier 132 multiplies the estimated α-axis extended rotor flux λ ext_α with a feedback signal on a signal path 134 from a sine function 136 to produce an α-axis multiplier output signal on a signal path 138. Likewise, a multiplier 140 multiplies the estimated β-axis extended rotor flux λ ext_β with a feedback signal on a signal path 142 from a cosine function 144 to produce a β-axis multiplier output signal on a signal path 146.\n• A summer 148 subtracts the α-axis multiplier output signal on the signal path 138 from the β-axis multiplier output signal on the signal path 146 to produce a difference signal on a signal path 150. A proportional and integral (PI) regulator function 152 multiplies the difference signal on the signal path 150 by the function $K p + K i s$ to produce a PI output signal on a signal path 154. Ki is an integral gain of the PI function 152, and Kp is a proportional gain of the PI function 152. Both Ki and Kp are constants based on a design of the system 10 as shown in Figure 1.\n• An integral function 156 multiplies the PI output signal on the signal path 154 by the function $1 s$ to produce an integration output signal on a signal path 158. The integration output signal on the signal path 158 is also fed into the inputs of the sine function 136 and the cosine function 144 to provide the PLL.\n• A low pass filter (LPF) function 160 multiplies the PI output signal on the signal path 154 by a third lag function $ω c s + ω c$ to produce an estimated rotor angular velocity ω̂ on a signal line 162, where ωc is a corner or cutoff frequency of the LPF function 160. A low pass filter associated with the LPF function 160 is used to smooth out the signal on the signal line 154.\n• The integration output signal on the signal path 158 is compensated by an offset Δθ to obtain a final estimated rotor angular position θ̂ . The offset Δθ can be a lump-sum error of miscellaneous delays, including delays introduced by the lag functions 92, 108 of Figure 3, digital sampling delays introduced in measured voltage and current signals, and computation delays in the microprocessor 24 as shown in Figure 1. A lookup table 164 may be used to compensate for this phase delay Δθ. The lookup table 164 generates a suitable phase delay Δθ on a signal path 166, and a summer 168 subtracts the phase delay Δθ from the integration output signal on the signal path 158 to produce the estimated rotor angular position θ̂ on a signal path 170.\n• Figure 5 illustrates how the α-β frame 173 comprises an α-axis 174 and a β -axis 175 that are perpendicular to each other. The α-β frame 173 aligns with a first phase 178, a second phase 180 and a third phase 182 of the system 10. A rotor 172 rotates, and its displacement from the α-axis is shown by the angle θ 184, which is the rotor angular position to be estimated.\n• When the motor 12 is at a standstill, as magnetic flux in the motor 12 changes in magnitude, a voltage is induced on the motor terminals 20a, 20b, and 20c, which can be sensed by the microprocessor 24. The induced stator voltages in the α-β frame can be described by the following equation: $V α = ⅆ λ s ⅆ t ⁢ cos θ 0$ $V β = ⅆ λ s ⅆ t ⁢ sin θ 0$\n• where λs is a magnitude of stator flux; and\n• θ0 is an initial rotor position angle at standstill.\n• The measured voltage 26 can be transformed to an alpha-beta frame. The transformed measured voltages Vα and Vβ may be fed into the PLL as shown in Figure 4 to obtain the initial position angle θ0 . In that case, the voltage V α replaces the flux λ̂ ext_α on the signal path 122, and the voltage V β replaces the flux λ̂ est_β on the signal path 130 in Figure 4.\n• Figure 6 illustrates an initial stator voltage and an initial rotor position as a function of time. During startup the rotating exciter 19 is powered on by the ac power supply 16 in Figure 1. Graph 186 illustrates a line-to-line voltage for each phase of the motor 12 as a function of time, and graph 188 illustrates an estimated rotor position as a function of time in the motor 12. The inverter 38 is OFF in the time periods shown in graphs 186 and 188. A voltage 190 corresponds to a VBC line-to-line voltage, a voltage 192 corresponds to a VCA line-to-line voltage, and a voltage 194 corresponds to a VAB line-to-line voltage. An estimated rotor position θ 0 200 corresponds to an angle of the rotor 172.\n• During an initial time period 196, the AC power supply 16 is OFF, and the three voltages 190, 192, and 194 are close to zero and the estimated rotor position θ0 200 cannot be used to determine actual rotor position. Specifically, if each of the voltages does not exceed a certain threshold, the extended rotor flux values are not applied to the phase lock loop to derive an estimated rotor angular position and an estimated rotor angular velocity, and the estimated rotor angular position and estimated rotor angular velocity are designated as invalid. At time 198, the AC power supply 16 turns ON and current flows to the rotating exciter 19 through the supply line 18 in the system 10. During this period, an excitation magnetic field of the motor 12 is arising. The rising magnetic flux induces voltage at the terminals 20a, 20b, and 20c of the system 10. The magnitude of voltages 190, 192 and 194 is sufficient for the microprocessor 24 to be able to estimate rotor position θ0 200. In graph 188, from time 198 to approximately time 202 the value of θ0 remains stable, and after time 202 the value starts to fluctuate due to a decaying voltage signal as shown in graph 186. This stable period demonstrates that a rotor position can be estimated from the voltages 190, 192, and 194 during the stable time period. If the rotor angular position cannot be determined within a predetermined period of time after the rotating exciter is turned on, the system indicates a fault condition. If the rotor angular position can be determined within a predetermined period of time after the rotating exciter is turned on, the inverter is turned on.\n\n## Claims (13)\n\n1. A method of estimating rotor angular position and rotor angular velocity for a dynamoelectric machine (12) comprising the steps of:\n(i) measuring an AC current (28) and a potential (26) for each of a plurality of windings coupled to a stator of the dynamoelectric machine;\n(ii) transforming the plurality AC currents and potentials to a stationary frame (α-β) to produce a first transformed current, a second transformed current, a first transformed potential, and a second transformed potential;\n(iii) processing the first transformed current and the first transformed potential, and processing the second transformed current and the second transformed potential to obtain a first intermediate signal (93) and a second intermediate signal (110);\n(iv) processing the first intermediate signal and the second intermediate signal to obtain a first extended rotor flux value (122) and a second extended rotor flux value (130); and\n(v) applying the first extended rotor flux value and the second extended rotor flux value to a phase lock loop (132-158) to derive an estimated rotor angular position and an estimated rotor angular velocity for the dynamoelectric machine;\nwherein the stationary frame is a two-phase α-β frame having an α-axis and a β-taxis, and the first transformed current is Iα, the second transformed current is Iβ , the first transformed potential is Vα , the second transformed potential is Vβ , the first extended rotor flux value corresponds to the α-axis, and the second extended rotor flux value corresponds to the β-axis; and\nthe step of processing the first transformed current and the first transformed potential, and processing the second transformed current and the second transformed potential to obtain a first intermediate signal and a second intermediate signal comprises the steps of:\nmultiplying the first transformed current Iα and the second transformed current Iβ by a resistance Rs of the stator to produce signals Iα *Rs ,Iβ *Rs ;\nsubtracting the signals I α*Rs ,I β*Rs from the first transformed potential Vα and the second transformed potential Vβ to produce signals V α-I α*Rs , V β -I β*Rs ; and\nmultiplying the signals V α-I α*Rs , V β-I β*Rs by a first lag function $1 s + ω i 92 108 ,$ wherein ω i is a selected corner frequency for the lag function and s is a Laplace operator, to produce first intermediate signal $1 s + ω i ⁢ V α - I α * R s$ and second intermediate signal $1 s + ω i ⁢ V β - I β * R s ⁢ 110 ;$ multiplying the transformed currents Iα,Iβ by a q-axis inductance Lq of the stator to produce signals Iα*Lq,Iβ*Lq ;\ncharacterised in that step (iv) comprises cross-coupling the first intermediate signal and the second intermediate signal to obtain the first extended rotor flux value (122) and the second extended rotor flux value (130); and\nin that the step of cross-coupling and processing the first intermediate signal and the second intermediate signal comprises the steps of:\nmultiplying the first intermediate signal $1 s + ω i ⁢ V α - I α * R s$ and the second intermediate signal $1 s + ω i ⁢ V β - I β * R s ⁢ 110$ by a second lag function $ω i s + ω i$ to produce signals $ω i s + ω i 2 ⁢ V α - I α * R s ⁢ 95 , ω i s + ω i 2 ⁢ V β - I β * R s ⁢ 114 ;$\nadding the signal $ω i s + ω i 2 ⁢ V β - I β * R s ⁢ 114$ to the first intermediate signal (93) and subtracting the signal I α*Lq (118) to produce a signal $1 s + ω i ⁢ V α - I α * R s + ω i s + ω i 2 ⁢ V β - I β * R s - I α * L q$ that corresponds to the first extended rotor flux value (λext_α; 122) and\nsubtracting the signal $ω i s + ω i 2 ⁢ V α - I α * R s$ from the second intermediate signal (110) and subtracting the signal I β*Lq (126) to produce a signal $1 s + ω i ⁢ V β - I β * R s - ω i s + ω i 2 ⁢ V α - I α * R s - I β * L q$ that corresponds to the second extended rotor flux value (λext-β; 130).\n2. The method of claim 1, wherein the selected corner frequency ωi is fixed.\n3. The method of claim 1 or 2, wherein the selected corner frequency ωi is adjustable.\n4. The method of any preceding claim, wherein the step of applying the first extended rotor flux value and the second extended rotor flux value to a phase lock loop to derive an estimated rotor angular position and an estimated rotor angular velocity for the dynamoelectric machine comprises the steps of:\nmultiplying the first extended rotor flux value with a sine function feedback signal to produce an α-axis multiplier output signal;\nmultiplying the second extended rotor flux value with a cosine function feedback signal to produce an β-axis multiplier output signal;\nsubtracting the α-axis multiplier output signal from the β-axis multiplier output signal to produce a difference signal;\nmultiplying the difference signal by a proportional and integral regulator function $K p + K i s$ to produce a PI output signal, wherein Kp is a constant value corresponding to a proportional gain of the proportional and integral regulator function, Ki is a constant value corresponding to an integral gain of the proportional and integral regulator function, and s is a Laplace operator;\nmultiplying the PI output signal by an integral function $1 s$ to produce an integration output signal;\nmultiplying the integration output signal by a sine function to produce the sine feedback signal; and\nmultiplying the integration output signal by a cosine function to produce the cosine feedback signal, wherein the estimated rotor angular position and the estimated rotor angular velocity for the dynamoelectric machine are derived from the proportional and integral regulator function output signal and the integration output signal.\n5. The method of claim 4, further comprising the steps of:\nmultiplying the proportional and integral regulator function output signal by a low pass filter function $ω c s + ω c$ to produce the estimated rotor angular velocity ω̂, where ωc is a corner frequency of the low pass filter function;\ngenerating a delay compensation based on a value of the estimated rotor angular velocity; and\nsubtracting the delay compensation from the integration output signal to produce the estimated rotor angular position.\n6. The method as recited in any preceding claim, further comprising the step of monitoring the potentials of the plurality of windings coupled to the stator of the dynamoelectric machine, wherein if the potentials do not exceed a certain threshold, the step of applying the first extended rotor flux value and the second extended rotor flux value to a phase lock loop to derive an estimated rotor angular position and an estimated rotor angular velocity for the dynamoelectric machine is not performed.\n7. The method of any preceding claim, further comprising the step of indicating a fault condition if the rotor angular position for the dynamoelectric machine cannot be determined within a predetermined period of time.\n8. A control for estimating an initial rotor angular position and a rotor angular velocity for a dynamoelectric machine (12) from a standstill comprising:\na reference frame transformation function for transforming an AC potential (26) for each of a plurality of windings coupled to a stator of the dynamoelectric machine to a stationary frame (α-β) to produce a first transformed potential (V α) and a second transformed potential (Vβ), and\na phase lock loop (132-158) to derive an estimated rotor angular position (Θ) and an estimated rotor angular velocity for the dynamoelectric machine from the first transformed potential and the second transformed potential;\nwherein the control is arranged to estimate the initial rotor angular position and the rotor angular velocity according to the method of claim 1.\n9. The control of claim 8, wherein if the potential of each of the plurality of windings does not exceed a certain threshold, the control designates the estimated rotor angular position and estimated rotor angular velocity as invalid.\n10. The control of claim 8 or 9, wherein if the rotor angular position for the dynamoelectric machine cannot be determined within a predetermined period of time after a rotating exciter (19) is powered on, the system indicates a fault condition.\n11. The control of claim 8, 9 or 10, wherein if the rotor angular position for the dynamoelectric machine can be determined within a predetermined period of time after a rotating exciter (19) is powered on, an inverter is turned on.\n12. The control of any of claims 8 to 11, wherein the phase lock loop to derive an estimated rotor angular position value and an estimated rotor velocity value for the dynamoelectric machine from the first transformed potential and the second transformed potential comprises:\na first multiplier (132) for multiplying the first transformed potential with a sine function feedback signal to produce an α-axis multiplier output signal;\na second multiplier (140) for multiplying the second transformed potential with a cosine function feedback signal to produce an β-axis multiplier output signal;\na first summer (148) for subtracting the α-axis multiplier output signal from the β-axis multiplier output signal to produce a difference signal;\na proportional and integral regulator function (152) for multiplying the difference signal by a function $K p + K i s$ to produce a PI output signal, wherein Kp is a constant value corresponding to a proportional gain of the proportional and integral regulator function, Ki is a constant value corresponding to an integral gain of the proportional and integral regulator function, and s is a Laplace operator;\nan integral function (156) for multiplying the PI output signal by an integral function $1 s$ to produce an integration output signal; s\na sine function (136) for multiplying the integration output signal by a sine function to produce the sine feedback signal; and\na cosine function (144) for multiplying the integration output signal by a cosine function to produce the cosine feedback signal, wherein the estimated rotor angular position and the estimated rotor angular velocity for the dynamoelectric machine are derived from the proportional and integral regulator function output signal and the integration output signal.\n13. The control of claim 12, further comprising:\na low pass filter (160) for filtering the PI output signal by a low pass filter function $ω c s + ω c$ to produce the estimated rotor angular velocity, where ω c is a corner frequency of the low pass filter function;\na lookup table (164) for generating a delay compensation based on a value of the estimated rotor angular velocity; and\na second summer (168) for subtracting the delay compensation from the integration output signal to produce the estimated rotor angular position.\nEP08251107.2A 2007-05-29 2008-03-27 Method and system for estimating rotor angular position and rotor angular velocity at low speeds or standstill Active EP1998435B1 (en)\n\n## Priority Applications (1)\n\nApplication Number Priority Date Filing Date Title\nUS11/754,396 US7577545B2 (en) 2007-05-29 2007-05-29 Method and system for estimating rotor angular position and rotor angular velocity at low speeds or standstill\n\n## Publications (3)\n\nPublication Number Publication Date\nEP1998435A2 EP1998435A2 (en) 2008-12-03\nEP1998435A3 EP1998435A3 (en) 2011-12-07\nEP1998435B1 true EP1998435B1 (en) 2013-07-24\n\n# Family\n\n## Family Applications (1)\n\nApplication Number Title Priority Date Filing Date\nEP08251107.2A Active EP1998435B1 (en) 2007-05-29 2008-03-27 Method and system for estimating rotor angular position and rotor angular velocity at low speeds or standstill\n\n## Country Status (2)\n\nUS (1) US7577545B2 (en)\nEP (1) EP1998435B1 (en)\n\n## Families Citing this family (30)\n\n* Cited by examiner, † Cited by third party\nPublication number Priority date Publication date Assignee Title\nEP2023479B1 (en) * 2007-08-06 2014-04-16 Baumüller Nürnberg Gmbh System for seamless velocity and/or location determination including standstill for a permanent magnet rotor of an electric machine\nDK176958B1 (en) * 2007-12-19 2010-07-26 Vestas Wind Sys As Generator System with intelligent processing of position signal\nWO2009088908A1 (en) * 2008-01-04 2009-07-16 Delphi Technologies, Inc. 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Method and apparatus for determining rotor position in a fluid pressure pulse generator\nJP6614825B2 (en) * 2015-06-30 2019-12-04 日立ジョンソンコントロールズ空調株式会社 Power conversion device, motor drive device, refrigeration device\nUS9705438B2 (en) 2015-07-14 2017-07-11 Infineon Technologies Austria Ag Controller for a free-running motor\nITUB20155550A1 (en) * 2015-11-13 2017-05-13 Piaggio & C Spa of a combined propulsion control method of assisted bicycles and relative pedal assisted bicycles\nCN105515486B (en) * 2015-12-03 2017-12-29 北京机械设备研究所 A kind of magnetic pole position of permanent magnet synchronous motor rotor real-Time Compensation bearing calibration\nCN105322846B (en) * 2015-12-23 2017-12-29 北京机械设备研究所 A kind of magnetic pole position of permanent magnet synchronous motor rotor detection means based on phaselocked loop\nUS9985565B2 (en) 2016-04-18 2018-05-29 Rockwell Automation Technologies, Inc. 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International Rectifier Corporation Self tuning method and apparatus for permanent magnet sensorless control\n\n## Also Published As\n\nPublication number Publication date\nEP1998435A2 (en) 2008-12-03\nEP1998435A3 (en) 2011-12-07\nUS20080300820A1 (en) 2008-12-04\nUS7577545B2 (en) 2009-08-18\n\n## Similar Documents\n\nPublication Publication Date Title\nTWI499198B (en) Motor control apparatus and motor control method\nUS9124209B2 (en) Method and apparatus for controlling power converter with inverter output filter\nUS7245104B2 (en) Position-sensorless motor control device\nJP3979561B2 (en) AC motor drive system\nDE102005054897B4 (en) Method and device for controlling the start of an electric motor\nJP4716118B2 (en) Motor control device\nEP2566045B1 (en) Sensorless control apparatus for a synchronous motor\nJP4665360B2 (en) Electric motor control device\nUS6831439B2 (en) Apparatus for measuring magnetic flux of synchronous reluctance motor and sensorless control system for the same 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Method of estimating magnetic pole position in synchronous motor\nUS6724168B2 (en) Apparatus and method for controlling rotation speed of synchronous reluctance motor\nJP4022630B2 (en) Power conversion control device, power conversion control method, and program for power conversion control\n\n## Legal Events\n\nDate Code Title Description\nAK Designated contracting states\n\nKind code of ref document: A2\n\nDesignated state(s): AT BE BG CH CY CZ DE DK EE ES FI FR GB GR HR HU IE IS IT LI LT LU LV MC MT NL NO PL PT RO SE SI SK TR\n\nAX Request for extension of the european patent to:\n\nExtension state: AL BA MK RS\n\nREG Reference to a national code\n\nRef country code: DE\n\nRef legal event code: R079\n\nFree format text: PREVIOUS MAIN CLASS: H02P0021000000\n\nIpc: H02P0021060000\n\nRIC1 Information provided on ipc code assigned before grant\n\nIpc: H02P 21/14 20060101ALI20111031BHEP\n\nIpc: H02P 21/06 20060101AFI20111031BHEP\n\nAX Request for extension of the european patent 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[ "# Slalom Corporation retains and reinvests all its earnings. So, Slalom does not pay any dividends, and...\n\n##### Part a, b, c & d are correct Week 4, Question 2: Use the force method...\npart a, b, c & d are correct Week 4, Question 2: Use the force method to determine the force in each member of the truss. Take a = 7 kN, b = 8 kN, c = 11 kN and EA = 20000 kN (for all the members). B) 2 m 3 m— 3 m Please fill in your solutions here: a) FAB = -11 kN Your last answer was int...\n##### A container is filled t0 the brim with water: solid object of mass 5.00 kg is lowered into the container; s0 that the object is fully submerged in the water (see Figure 7.1). During - this proce 3.00 kg of water flows over the rim and out Of the container: Considering only the weight and buoyant force; what is the net force acting On the submerged objecl; and, when released. does object rise. sink: or remain in place?ObjectGroup of answer choices294N. The object sinks196 N. The object sinks:29AN\nA container is filled t0 the brim with water: solid object of mass 5.00 kg is lowered into the container; s0 that the object is fully submerged in the water (see Figure 7.1). During - this proce 3.00 kg of water flows over the rim and out Of the container: Considering only the weight and buoyant for...\n##### Multiply: (n - S)(n - 1) n(n - 1) - S(n - 1) n? _ Sn + 5 (c) n2 _ 6n + 5 (d) n? + 5 (9) Divide: (8x8 8r2 ~8x) + (4x) (a)2x8 ~8r? 28x (b) 2x8 _ 2r2 _ 2x 8x\" _ 8x - 28 (d) 2x\" _ 2r - 2 (10) Divide: (x? + 2x - 63) + (x + 9) (a) * -7 (b) x? _ 7 (c) x +7 (d) x2 + 3* - 54 (11) Factor: 10x? 100x ()5(2x2 20x) (6) 1Ox(x + 10) (c) Sx(2x + 20) (d) x(10x + 100) (12) Factor: x? + 41 - 77 (x+46 77) (x - 7)6 + 11) (x +7)6 - 11) (x - 7)6 - 11) (13) Graph the Function f(x) =x' + 3r7 _ 6x - 8\nMultiply: (n - S)(n - 1) n(n - 1) - S(n - 1) n? _ Sn + 5 (c) n2 _ 6n + 5 (d) n? + 5 (9) Divide: (8x8 8r2 ~8x) + (4x) (a)2x8 ~8r? 28x (b) 2x8 _ 2r2 _ 2x 8x\" _ 8x - 28 (d) 2x\" _ 2r - 2 (10) Divide: (x? + 2x - 63) + (x + 9) (a) * -7 (b) x? _ 7 (c) x +7 (d) x2 + 3* - 54 (11) Factor: 10x? 100x ...\n##### 7FF beluy Euluale Fp EZds 0 + Le poibl 41 bXLZLlZ(z\n7FF beluy Euluale Fp EZds 0 + Le poibl 41 bXLZLl Z(z...\n##### Use tne Lyapunov equation method t0 determine whether the following system is asymptotically stable:1 ;]\nUse tne Lyapunov equation method t0 determine whether the following system is asymptotically stable: 1 ;]...\n##### Perform the addition or subtraction and simplify: Su u + 9 + u + 9\nPerform the addition or subtraction and simplify: Su u + 9 + u + 9...\n##### Buffer\nWhat mass of ammonium chloride should you add to 2.66 L of a 0.168 M NH3 to obtain a buffer with a pH of 9.57?Kb of NH3 = 1.76 x 10^-5_____ g...\n##### Given the force field find the work required to move an object on the given oriented curveF= (y;on the path consisting of the line egment from (1,4) to (0,0) followed by the line segment from 0,0) to (0,9)The amount ofwork required is (Simplify your answer)\nGiven the force field find the work required to move an object on the given oriented curve F= (y; on the path consisting of the line egment from (1,4) to (0,0) followed by the line segment from 0,0) to (0,9) The amount ofwork required is (Simplify your answer)...\n##### Cyclopropane decomposes to propene when heated to 500 \"C. Using the data shown below_ calculate the rate constant for this first order reaction;ConcentrationTimecyclopropane(mnutCS)(mmolL)1.48 1.27 0.98 0.84\nCyclopropane decomposes to propene when heated to 500 \"C. Using the data shown below_ calculate the rate constant for this first order reaction; Concentration Time cyclopropane (mnutCS) (mmolL) 1.48 1.27 0.98 0.84...\n##### Brothers Harry and Herman Hausyerday began operations of their machine shop (H&H Tool, Inc.) on January...\nBrothers Harry and Herman Hausyerday began operations of their machine shop (H&H Tool, Inc.) on January 1, 2016. The annual reporting period ends December 31. The trial balance on January 1, 2018, follows the amounts are rounded to thousands of dollars to simplify: Account Titles Cash Accounts R...\n##### Problem 19. Prove that there exists a Jordan measurable set contained in IR which is not...\nProblem 19. Prove that there exists a Jordan measurable set contained in IR which is not a Borel set....\n##### 6.4.9 An elevator has a placard starting that the maximum capacity is 1384 lb- passengers. So,...\n6.4.9 An elevator has a placard starting that the maximum capacity is 1384 lb- passengers. So, 8 adult male passengers can have a mean weight of up to 1384/8 - 173 pounds. If the elevator is loaded with Badult male passengers find the probability that it is overloaded because they have a man weight ...\n##### (three parts): Thinking of R2 as the plane, consider the following three functions (R2)2- R: ....\n(three parts): Thinking of R2 as the plane, consider the following three functions (R2)2- R: . de((a, b), (с, d))-V (a-c)2 + (b-d)2. The first is the usual distance formula (which we will take for granted to be a metric), the second is the \"grid\" distance (imagine you can only move hor...\n##### Evaluate the integral by making an appropriate change of variables. Slo 3 cos (5(X+3) dA where...\nEvaluate the integral by making an appropriate change of variables. Slo 3 cos (5(X+3) dA where R is the trapezoidal region with vertices (8,0), (9, 0), (0, 9), and (0,8) 17 sin(5) 2 x...\n##### Identify X in each of the following nuclear equations: (a) 33 78As X + –1 0β...\nIdentify X in each of the following nuclear equations: (a) 33 78As X + –1 0β (b) 1 1H + 2 4He X (c) 10 25 0 8Fm 10 25 0 7Fm + X...\n##### Problem 2.Consider the data you used in Lab 4: {125-c(2.7,4.6,2.6,3.0,3.2,3.81 {200-c(4.6,3.4,2,9,3.5,4.1,5.1)You can obtain the relevant sums using R to answer the questions below:(i) Consider the means model:Yij = +6.f = 1,2;6=1_nThe least squares normal equations [Refer to page 4 of Chap 2 Part 2 notes] are given by: np, 2; Y1fnjzYziFor the given data (ie, substitute the actual values for the relevant sums). solve the normal equations to get estimate of /4, and /2\nProblem 2. Consider the data you used in Lab 4: {125-c(2.7,4.6,2.6,3.0,3.2,3.81 {200-c(4.6,3.4,2,9,3.5,4.1,5.1) You can obtain the relevant sums using R to answer the questions below: (i) Consider the means model: Yij = +6.f = 1,2;6=1_n The least squares normal equations [Refer to page 4 of Chap 2 ...\n##### Situation: The town of Waterville has the following past population data projected beyond 1990 Year Population...\nSituation: The town of Waterville has the following past population data projected beyond 1990 Year Population 10,240 12,150 18,430 26210 22,480 32.410 45,050 57,200 64,030 77,320 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 The average water consumption (including every possible sector) is 160...\n##### Enter your answer in the provided box:How many total moles of ions are released when the following sample dissolves completely in water? 5.26 g of Ba(OH)z ' 8H,O: mol\nEnter your answer in the provided box: How many total moles of ions are released when the following sample dissolves completely in water? 5.26 g of Ba(OH)z ' 8H,O: mol..." ]

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[ "# (9, 12)\n\nA = 9\n\nB = 12\n\n Sum 21 Difference 3 A × B 108 A ÷ B 0.75 B ÷ A 1.33333 GCD 3 LCM 36 Average 10.5\n\n## Combination\n\n220\n\n${}_{ 12 } C_{ 9 } = 220$\n\n## Sum of squares\n\n225\n\n$9^2 + 12^2 = 225$\n\n## Norm\n\n$\\sqrt{ 9^2 + 12^2 } = 15.0$\n\n## Geometric mean\n\n10.392304845413264\n\nNo\n\n## Stirling number of the first kind\n\n0\n\n$S(9,\\ 12) = 0$\n\n## Stirling number of the second kind\n\n0\n\n$S(9,\\ 12) = 0$\n\n## Properties of 9 and 12\n\nProperties of 9 and 12: The GCD and LCM of 9 and 12 are 3 and 36 respectively. 9 + 12 = 21, |9 - 12| = 3, 9 * 12 = 108, 9 / 12 = 0.75, 12 / 9 = 1.3333333333333333. The average of 9 and 12 is 10.5 and the norm of them is 15.0. The geometric mean of two numbers is 10.392304845413264." ]

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[ "By Topic", null, "• Abstract\n• Authors\n• Figures\n• Multimedia\n• References\n• Cited By\n• Keywords\nSECTION I\n\n## INTRODUCTION\n\nDynamic spectrum allocation is becoming more prevalent in practice, requiring frequency-agile transmitters with amplifiers capable of high-efficiency operation at multiple operating frequencies and power levels based on system needs and limitations. This focus session paper presents a simplex algorithm that is expected to be useful in the real-time fast reconfiguration of power amplifier circuits for wireless communications and radar as they reconfigure between frequencies.\n\nCircuit optimization algorithms originally intended for computer-aided design (CAD) applications can be used for real-time optimization. Bandler, Charalambous, and Steer describe multiple algorithms useful for generic CAD circuit optimization, including simplex (used in this paper), pattern, and gradient searches . In real-time reconfigurable amplifiers, adaptive amplifier output matching networks are needed. Tunable matching networks using micro-electrical mechanical systems (MEMS) devices are discussed by Lu and Vaka-Heikkila , . Real-time impedance matching is discussed by Deve and Sun . Qiao demonstrates a reconfigurable low-power communication amplifier using a genetic tuning algorithm , and Baylis, Martin, and Fellows demonstrate unconstrained and constrained optimizations using a gradient-based approach . This paper demonstrates the use of simplex in amplifier load-impedance optimizations. To our knowledge, this paper represents the first direct application of simplex optimization to optimize in the load-reflection coefficient", null, "$(\\Gamma_{L})$ plane.\n\nSECTION II\n\n## SIMPLEX CIRCUIT OPTIMIZATION APPROACH\n\nThe simplex method originally presented by Nelder and Mead and discussed by Bandler has been applied to our work presented in this paper. In our algorithm, the search is performed in the Smith Chart, which is the complex plane of the load reflection coefficient", null, "$\\Gamma_{L}$. Figure 1 shows how the search unfolds. The first point of the simplex is established at the designated starting point, shown as point A. An additional point is then measured in each coordinate direction of the search space, separated from the initial point by", null, "$D_{s}$ (points B and C). The “simplex” is the polygon connecting each of the evaluation points. In two dimensions, the initial simplex is a right triangle.\n\nFig. 1: Simplex construction for the search. The initial simplex consists of points A, B, and C. The optimization plane is the smith chart (the complex plane of", null, "$\\Gamma_{L}$).\n\nPower-added efficiency (PAE) is used as the optimization criterion for maximization, and is first evaluated at each point of the initial simplex (points A, B, and C). The value of", null, "$\\Gamma_{L}$ in the simplex possessing the lowest PAE (the worst value of the criterion) is then replaced by a new point .\n\nFigure 1 also illustrates how the replacement simplex point is selected in the case where point A has the lowest (worst) PAE of the original simplex. In this case, PAE is next measured at point 1, the reflection of point A about the centroid of the line connecting the two remaining simplex points B and C . If point 1 has a greater (better) PAE value than both B and C, an additional point is measured at a factor", null, "$k$ times the initial distance from the line between the other two points (point 2). The PAE at point 2 is then compared with the PAE at point 1. If the PAE at point 2 is greater than the PAE at point 1, then the next simplex consists of points B, C, and 2. This is known as expansion of the simplex. If the PAE at point 2 is lower than at point 1, but point 1 has higher PAE than both B and C, the simplex operations resume using the simplex consisting of points B, C, and 1. If the PAE at point 1 is greater than at point A, but less than the PAE at both B and C, then the new simplex consists of points B, C, and 4 (point 4 is", null, "$1/k$ times the distance between the centroid and point 1). If point 1 has lower (worse) PAE than the original worst simplex point A, the new simplex point is placed on the original side of the mirroring line at a distance", null, "$1/k$ between this line and the original worst point A (this is point 3 in Fig. 2). Simplex operations are re-performed using the simplex consisting of points B, C, and 3. The search ends when the distance from the centroid to the new simplex point falls below a prespecified resolution distance", null, "$D_{r}$. The highest-PAE point of the final simplex is chosen as the optimum.\n\nSECTION III\n\n## SIMULATION RESULTS\n\nSimulations were performed using a Modelithics model for a Qorvo TGF2960 high electron mobility transistor (HEMT). For the simulation tests, the algorithms were coded in MATLAB, and MATLAB controls simulations of Keysight Technologies' Advanced Design System software. A bias setting of", null, "$V_{GS}=-1\\ \\text{V}, V_{DS}=5\\ \\text{V}$ and input power of 10 dBm were used, with a CDMA2K waveform as the excitation.", null, "$D_{s}=0.2, D_{r}=0.05$, and", null, "$k=2$ were used. Figure 2 shows the simplex search results in the Smith Chart for a starting point", null, "$\\Gamma_{L}=0.25\\underline{/-90^{\\circ}}$, An optimum PAE value of 47.64% is obtained at the optimum", null, "$\\Gamma_{L}=0.39\\underline{/56.8^{\\circ}}$ with only 16 measured points. Figure 3 shows that a similar optimum (PAE = 47.64% at", null, "$\\Gamma_{L}=0.33\\underline{/57.7^{\\circ}})$ was reached for a starting condition", null, "$\\Gamma_{L}=0.9\\underline{/180^{\\circ}}$ using 27 measured points. Figure 4 shows a traditional load-pull for PAE, and it can be seen that the efficiency and endpoint values of the searches shown in Figs. 2 and 3 match the optimum from traditional load pull. (PAE = 47.49% at", null, "$\\Gamma_{L}=0.32\\underline{/60.0^{\\circ}})$.\n\nFig. 2: Smith chart view of simulated search results starting from", null, "$\\Gamma_{L}=0.25\\underline{/-90^{\\circ}}$. 18 measured points were required.\n\nTable I shows the simulation results taken from multiple starting values of", null, "$\\Gamma_{L}$. The mean end value of PAE is 47.66% with standard deviation of only 0.021 %. The mean number of measurements required for the search for this sample is 18.60 measurements with a sample standard deviation of 5.22.\n\nFig. 3: Smith chart view of simulated search results starting from", null, "$\\Gamma_{L}=0.9\\underline{/180^{\\circ}}.\\ 27$ measured points were required.\nFig. 4: Traditional tgf2960 PAE load-pull simulation results. The optimum PAE = 47.49% at", null, "$\\Gamma_{L}=0.32\\underline{/60.0^{\\circ}}$\nSECTION IV\n\n## MEASUREMENT RESULTS\n\nThe algorithm was then measurement tested on a Microwave Technology MWT-173 GaAs MESFET, using a chirp waveform as the excitation. The device is different than the simulation device to provide diversity of situations for algorithm demonstration.", null, "$D_{s}=0.2, D_{r}=0.05$, and", null, "$k=2$ were used. Figure 5 shows the results for the starting point", null, "$\\Gamma_{L}= 0.5\\underline{/0^{\\circ}}$ (PAE = 32.63% at", null, "$\\Gamma_{L}=0.26\\underline{/-92.6^{\\circ}}$). Figure 6 shows the results for the starting point", null, "$\\Gamma_{L}=0.75\\underline{/90^{\\circ}}$ (PAE = 32.52% at", null, "$\\Gamma_{L}=0.25\\underline{/-90.0^{\\circ}})$. Figure 7 shows the traditional load-pull results, again showing good correspondence between the endpoints of the measurement search algorithm and the optimum as chosen by a traditional load-pull (PAE = 32.76% at", null, "$\\Gamma_{L}=0.21\\underline{/-96.3^{\\circ}})$. Table II shows measurement results for multiple starting values of", null, "$\\Gamma_{L}$.\n\nTable II: Measurement results\n\nThe mean end PAE value is 32.63% with a very small sample standard deviation of 0.083%. This shows excellent convergence between the different starting points. In addition, the mean number of measurements for this sample is 19.6, with sample standard deviation of 6.66 measurements.\n\nFig. 5: Smith chart view of measured search results starting from", null, "$\\Gamma_{L}=0.5\\underline{/0^{\\circ}}.\\ 22$ measured points were required.\nFig. 6: Smith chart view of measured search results starting from", null, "$\\Gamma_{L}=0.75\\underline{/90^{\\circ}}$. 25 measured points were required.\nFig. 7: Traditional MWT-173 PAE load-pull measurement results. The optimum PAE = 32.76% at", null, "$\\Gamma_{L}=0.21\\underline{/-96.3^{\\circ}}$\nSECTION V\n\n## CONCLUSIONS\n\nA simplex algorithm for fast load-pull search in the Smith Chart has demonstrated in simulation and measurement. The algorithm has been designed for intended application in a fast, reconfigurable amplifier for use in a frequency-agile transmitter. Ongoing work that cannot be included in this paper due to space limitations, includes constrained optimization, which allows the optimum PAE meeting adjacent-channel power ratio requirements to be quickly found. In addition, ongoing work shows that the simplex method performs very well in comparison to gradient and pattern load-pull searches.\n\n### Acknowldegments\n\nThis work has been funded by the National Science Foundation (Grant Number ECCS-1343316). The authors wish to thank Lawrence Cohen of the Naval Research Laboratory for helpful collaboration in this work. In addition, the authors gratefully acknowledge the support of Keysight Technologies in providing cost-free loan of the Advanced Design System software, and Modelithics for donation of simulation model libraries through the Modelithics University Program.\n\n## Footnotes\n\nNo Data Available\n\nNone\n\n## Corrections\n\nNone\n\nThis paper appears in:\n2016 IEEE MTT-S International Microwave Symposium (IMS)\nConference Date(s):\n22-27 May 2016\nConference Location:\nSan Francisco, CA\nOn page(s):\n1 - 4\nE-ISBN:\nNo Data Available\nPrint ISBN:\nNew-2005_Electronic_978-1-5090-0698-4\nINSPEC Accession Number:\n16213811\nDigital Object Identifier:\n10.1109/MWSYM.2016.7540376\nDate of Current Version:\nNo Data Available\nDate of Original Publication:\nNo Data Available" ]

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[ "# Tips on using Google Sheets for an old Excel and 1-2-3 deal", null, "# Tips on using Google Sheets for an old Excel and 1-2-3 deal\n\nGoogle Sheets is wonderful for cooperative spreadsheet development, but really using it there a number of formulas tips and tricks:\n\n1. Blank cells. If you want to detect if a cell has a number, then BLANK() and then if you want to see if a cell is non-blank, then NOT(BLANK())\n2. Indexing in a range. If you have a table then you can use VLOOKUP() if the numbers are sorted, but if you want to figure out what is a particular row and column, the and that makes it easy.\n3. Vlookup. This is perfect for looking numbers up in a table. Note that normally these won't be sorts, so make sure the third parameter is FALSE, so vlookup(\"hello\",a1:b23,2, false) means look for the word hello in the lookup table at a1:b23 and return the value in the 2nd column and assume the list isn't sorted.\n4. Regex. If you want to check is a string is one of several and then spit out a number, then regexmatch is your friend. So this would look like `Regexmatch(a1, \"(office|work)*\") which would match either of these strings that are in between the (parenthesis)\n5. Roundup. If you normally just to INT(A1+0.5), then you will appreciate that there is actually a function that rounds up to any arbitrary digit, so ROUNDUP(23.234, -1) works and you get 23.2.\n6. SUMIFS. This is a super useful function, you basically give it a range and it will pick out elements and sum them if a criterion is met. So you can, for instance, add up inventory based on a bunch of different conventions. It's a poor man's SQL Select statement. There is a simpler version called SUMIF that inverts the range and criteria so be careful.\n7. TRIM. This is super useful if you have a bunch of white space around a header, it just cuts it all." ]

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[ "# 2.9 MC Answers and Review\n\n269 resources\nSee Units\n\n## Answers and Review for Multiple Choice Practice on Molecular and Ionic Compound Structures and Properties\n\nSTOP ! ⛔ Before you look at the answers, make sure you gave this practice quiz a try so you can assess your understanding of the concepts covered in Unit 2. Click here for the practice questions: AP Chemistry Unit 2 Multiple Choice Questions.", null, "Image courtesy of Pixabay\n\nFacts about the test: The AP Chemistry exam has 60 multiple choice questions and you will be given 1 hour 30 minutes to complete the section. That means it should take you around 15 minutes to complete 10 questions.\n*The following questions were not written by College Board and although they cover information outlined in the AP Chemistry Course and Exam Description the formatting on the exam may be different.", null, "1. Lewis dot diagrams for CF_4 and SF_4 are provided in the image above. Which of the following statements are true?\nA. The molecular geometry for both substances is different because S has a greater number of electron domains than C.\nB. Both molecules have the same shape because the molecular formulas are the same.\nC. Both molecules have similar shapes and are nonpolar because the electronegativities for C and S are similar.\nD. The lone pairs on S make is the negative end of the dipole.\nAnswer: An electron domain is a lone pair or a bonding group. The two molecules have different shapes because the total number of electron domains are different.\n📄 Study AP Chem, Unit 2.5: Lewis Dot Diagrams\n\n2. Which of the following ionic compounds will have the highest melting point? Predict the relative lattice energies of the salts.\nA. NaCl\nB. KF\nC. K_2O\nD. CaO\nAnswer: The lattice energy for CaO is greatest because the magnitude of charge difference is greatest between the Ca^2+ ions and the O^2- ions. As lattice energy increases, the melting point of the ionic compound increases.\n📄 Study AP Chem, Unit 2.3: Ionic Bonding and Ionic Solids\n\n3. In the molecule COCl_2, how many sigma and pi bonds are present?\nA. 2 sigma and 1 pi bond\nB. 3 sigma and 1 pi bond\nC. 3 sigma bonds only\nD.3 pi bonds only\nAnswer: In the Lewis structure, there are 2 single bonds and 1 double bond. For every single bond there is 1 sigma bond. For every double bond, there is 1 sigma bond and 1 pi bond.", null, "4.", null, "A. Gold, Au(s)\nB. Ammonium Nitrate, NH_4NO_3(s)\nC. Ethanol, CH_20H(I)\nD. Potassium Chloride, KCl(s)\nAnswer: Covalent compounds have relatively low melting and boiling points compared to ionic compounds and metallic solids. The only covalent compound is ethanol.\n📄 Study AP Chem, Unit 2.4: Metallic Bonding and Alloys", null, "5. A potential energy graph of O_2 is represented in the graph above. Which of the following is the potential energy on the y-axis representing?\nA. Bond Length\nB. Electronegativity\nC. Bond Energy\nD. Type of Bond\nAnswer: The potential energy on the y-axis measures the bond energy, the amount of energy required to separate the two oxygen atoms.\n📄 Study AP Chem, Unit 2.2: Intramolecular Force and Potential Energy\n\n6. PCl_3 is polar and PF_5 is nonpolar. Which of the following statements correctly justifies the difference in polarity?\nA. The P--F bonds are nonpolar while the P--Cl bonds are polar\nB. PF_5 has a symmetrical geometry, whereas PCl_3 has a nonsymmetrical geometry.\nC. P--F is a single bond while P--Cl is a double bond.\nD. P--Cl is an ionic bond while P--F is a covalent bond.\nAnswer: Symmetrical molecules are nonpolar regardless of the type of bonds. Unsymmetrical molecules, like PCl_3, are polar if the bonds have a significant electronegativity difference.\n\n7. Which of the following descriptions seems most appropriate for the substance CBr_4?\nA. CBr_4 is an electrolyte.\nB. CBr_ 4 is a solid at room temperature and is hard, but brittle.\nC. CBr_4 is malleable, ductile, and lustrous.\nD. CBr_4 has a boiling point of 97C and cannot conduct electricity.\nAnswer: CBr_4 is a covalent molecule; therefore, it has a relatively low boiling point and cannot conduct electricity.\n📄 Study AP Chem, Unit 2.1: Types of Chemical Bonds\n\n8. What is the molecular geometry of formaldehyde, CH_2O?\nA.  Trigonal Pyramidal\nB. Trigonal Planar\nC. Tetrahedral\nD. Linear\nAnswer: Formaldehyde's Lewis structure contains three bonding groups and no lone pairs. This is consistent with trigonal planar molecular geometry.\n\n9. Which of the following descriptions best fits the substance molybdenum, Mo?\nA. Mo is malleable, ductile, and lustrous.\nB. Mo is a gas at room temperature.\nC. Mo is a solid at room temperature and is hard, but brittle.\nD. Mo has a boiling point of 97C and cannot conduct electricity.\nAnswer: Molybdenum is a metal. Metals are located on the left side of the periodic table and tend to be malleable, ductile, and lustrous.\n📄 Study AP Chem, Unit 2.4: Metallic Bonding and Alloys\n\n10. Which of the following molecules has a resonance structure?\nA. NH_3\nB.CCI_4\nC.SF_2\nD. SO_2\nAnswer: SO_2 is the only molecule in the list that has a resonance structure. Remember resonance structures are when there is more than one appropriate Lewis structure.\n📄 Study AP Chem, Unit 2.6: Resonance and Formal Charge\n\n11. What is the hybridization for the SO_3\nA. sp\nB. sp^2\nC. sp^3\nD. SO_3 does not have hybridization because it has a resonance structure.\nAnswer: The hybridization for SO_3 is sp^2 because the molecule has three electron domains. The completed Lewis structure contains three bonds to oxygen and no lone pairs.\n\n12. Which of the following is a rigid alloy?\nA. Steel (Fe & C)\nB. Brass (Cu & Zn)\nC. Sodium chloride (NaCl)\nD. Dry Ice (CO_2)\nAnswer: Steel is an interstitial alloy. Interstitial alloys are more rigid than pure metals and substitutional alloys because the smaller atoms fit in the holes created by the parent atoms. This makes it difficult for the host atom to move around.\n📄 Study AP Chem, Unit 2.4: Metallic Bonding and Alloys\n\n13. Which of the following is the formula for the compound formed by aluminum and chlorine atoms?\nA. Al_3Cl\nB. AlCl\nC. AlCl_2\nD. AlCl_3\nAnswer: When aluminum is in an ionic compound, it has a charge of 3+. When chlorine is in an ionic compound, it has a charge of 1-. Therefore the ratio that elements come in to make a neutral compound is 1 Al: to 3 Cl ions.\n📄 Study AP Chem, Unit 2.3: Ionic Bonding and Ionic Solids\n\n14. Which of the following substances conducts electricity in the solid phase?\nA. Diamond (C)\nB. Ethanol (CH_3OH)\nC. Zinc (Zn)\nD. Potassium bromide (KBr)\nAnswer: Zinc is a metal. Metals are the only substance that conduct electricity in the solid phase.\n\n15. A substance that is composed of only nonmetals but has a very high melting point is called....\nA. Network covalent solid\nB. Ionic Solid\nC. Molecular Solid\nD. Metallic Solid\nAnswer: Network covalent solids are giant molecules. In order to melt them, the bonds between the atoms must be broken, which requires more energy than overcoming intermolecular forces.\n📄 Study AP Chem, Unit 2.4: Metallic Bonding and Alloys\n\n🔍Check out all of the resources for AP Chem Unit 3\n🤝Connect with other students studying AP Chem with Hours\nBrowse Study Guides By Unit\n⚛️Unit 1 – Atomic Structure & Properties\n🌀Unit 3 – Intermolecular Forces & Properties\n🧪Unit 4 – Chemical Reactions\n👟Unit 5 – Kinetics\n🔥Unit 6 – Thermodynamics\n⚖️Unit 7 – Equilibrium\n🍊Unit 8 – Acids & Bases\n🔋Unit 9 – Applications of Thermodynamics\n✍️Free Response Questions\n🧐Multiple Choice Questions\n📆Big Reviews: Finals & Exam Prep", null, "" ]

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[ "## Friday, November 5, 2010\n\n### Price Inflation - fractal\n\nInflation - a fractal explanation\n\nHere I hypothesis price inflation is a fractal phenomena and is equivalent to the coastline paradox (the measurement of a coastline is related to the length of a the 'measuring stick') where the unit of 'money' is the coastline.\n\nUpdate: 29th 11, 2012\nIf the fractal demonstrates the likes of price and quantity, supply and demand - as I have shown in my previous entries - then it should also be able to demonstrate inflation.\nThis entry has been a real challenge for me, but has been very existing : my thoughts maybe different, but they at least can't be over looked.\n\nJust what is inflation?\nOf course we have the textbook answer, a general increase in the average price level - over time, but to be fractal, it must be a universal definition, and go beyond prices of goods.\nA fractal explanation will have to demonstrate the following - I am convinced that the general fractal does.\n1. the increase in the descriptive value placed upon an 'unchanging' object\n2. there must be devaluation of a kind of the measuring stick.\n3. there must be a notion of equilibrium, where one gets used to the new nominal change or size\n\nA fractal explanation for inflation\nWhen we look at the  Koch Snowflake development/production (below), what is it that inflates - the area,, the number of 'triangles', or the perimeter - or nothing?", null, "Inflation and perimeter\n\nI have become convinced that inflation is represented by the perimeter of the fractal, the reason being that the area of the Koch is finite, while the perimeter is infinite - as the fractal iterates (grows). Also note that area and perimeter are related through the base length of the triangle side too.\n\nThis came to me after thinking about what the perimeter length of an island, or the length of a piece of string. Fractal theory says both are infinite, and that the result depends on the actual length of the measuring stick. The shorter the stick, the larger the perimeter length: all while, at the same time the shape – the object – remains the same, or at equilibrium. This measuring problem is well described from around 2 minutes into the following: youtube clip: BBC, How long is a piece of string?\n\nP= s.l\nThe perimeter (P) of the Koch snowflake fractal is calculated by multiplying number of sides (s) by length of a side (l).  The below diagram shows as the fractal iterates - and the shape develops towards equilibrium, at iteration 7 (plus or minus 2) - the perimeter of the snowflake rises exponentially to infinity. This is due to relationship between the diminishing length of each side, and the exponentially rising quantity of sides.", null, "This to me is also analogise to price inflation and the concept of value in economies: (Domestic) Production (GDP), where the measuring stick is units of currency or money.\nMeasuring an economies GDP should be the same principle as measuring an island, or the Koch snowflake's perimeter, only the measuring tool is money, or currency in this context.\n\nTo go any further we need to clear what money is? By definition, money should be - with the help of the circular flow of income model: Y=P=E - a liquid means of exchange, derived from income (Y), which comes from production (P), and will in turn be what exspended or spent (E).\nIt follows that if the Product (the production) of the Koch snowflake is the triangle bits; if so, then these bits represent 'money' as a means of exchange .\nAs the fractal grows (as domonstrated in the fractal zoom below) the sides and bits get smaller and smaller, while the perimeter gets larger and larger.", null, "the Koch fractal zoom - growth\nIf we were to stop the zoom at any time, we see the same 'snowflake' ahead, just like shape as earlier in the zoom: that is to say, in our analogy, the money looks the same, only the the numbers have changed. We still see an equilibrium shape.\n\nWith that, I conclude that inflation is natural, and that growth, development and now inflation are inextricably linked through the mechanics of the fractal.\n\nHowever, it begs the question, what about printed or fiat money? Does it distort the reality? Is there such a thing as unnatural inflation?\nIt is my thinking that fiat money is another fractal of its own. If we print (more) money - than what our resources call for - it follows that we will shorten measuring tool more than what is natural, distorting the reality with an even longer perimeter, or larger (misleading) GDP.\n\nAlso, for inflation to be fractal, inflation has to be a universal concept: I am interested to learn that hotels now come with 6, 7, and possibly 8 stars to rate their quality.  Have hotels really changed so much to deserve more numbers than the ‘5 star’. Of course they have, but maybe the system is open to some distortion - 5 should be the ultimate. As the 'stars' increase or inflate above the original ‘5 star’, that 5 star loses value.  Other examples exist:  academic grade inflation - where the school leaving qualification creeps up and up, from a Degree to a Masters to where you need a PhD to work behind a bar.\n\nWe know, from experience, and from neo-classical economic's, that price inflation fits the above 'rules': (nominal) price's rise for the same object;  consumers get used to the new price's - there is new equilibrium; and finally there is a devaluation of money itself, when relating to its past value.\n\nI also consider changes in models of car (for example) without any inovation or evolution, just the colour of style, as inflation - design inflation. The new car devalues the old just because it is has something different (no practicle) - even if they are basically the same car.\n\nI am still of minds that there is more to this: there could be something in the quantity of 'triangles' to explore.\n\n" ]

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[ "# tunablePID\n\nTunable PID controller\n\n## Syntax\n\n```blk = tunablePID(name,type) blk = tunablePID(name,type,Ts) blk = tunablePID(name,sys) ```\n\n## Description\n\nModel object for creating tunable one-degree-of-freedom PID controllers. `tunablePID` lets you parametrize a tunable SISO PID controller for parameter studies or for automatic tuning with tuning commands such as `systune`, `looptune`, or the Robust Control Toolbox™ command, `hinfstruct`.\n\n`tunablePID` is part of the family of parametric Control Design Blocks. Other parametric Control Design Blocks include `tunableGain`, `tunableSS`, and `tunableTF`.\n\n## Construction\n\n`blk = tunablePID(name,type)` creates the one-degree-of-freedom continuous-time PID controller:\n\n`$blk={K}_{p}+\\frac{{K}_{i}}{s}+\\frac{{K}_{d}s}{1+{T}_{f}s},$`\n\nwith tunable parameters `Kp`, `Ki`, `Kd`, and `Tf`. The `type` argument sets the controller type by fixing some of these values to zero (see Input Arguments).\n\n`blk = tunablePID(name,type,Ts)` creates a discrete-time PID controller with sample time `Ts`:\n\n`$blk={K}_{p}+{K}_{i}IF\\left(z\\right)+\\frac{{K}_{d}}{{T}_{f}+DF\\left(z\\right)},$`\n\nwhere IF(z) and DF(z) are the discrete integrator formulas for the integral and derivative terms, respectively. The values of the `IFormula` and `DFormula` properties set the discrete integrator formulas (see Properties).\n\n`blk = tunablePID(name,sys)` uses the dynamic system model, `sys`, to set the sample time, `Ts`, and the initial values of the parameters `Kp`, `Ki`, `Kd`, and `Tf`.\n\n### Input Arguments\n\n`name`\n\nPID controller `Name`, specified as a character vector such as `'C'` or `'PI1'`. (See Properties.)\n\n`type`\n\nController type, specified as one of the values in the following table. Specifying a controller type fixes up to three of the PID controller parameters.\n\nValue for `type`Controller TypeEffect on PID Parameters\n`'P'`Proportional only`Ki` and `Kd` are fixed to zero; `Tf` is fixed to 1; `Kp` is free\n`'PI'`Proportional-integral`Kd` is fixed to zero; `Tf` is fixed to 1; `Kp` and `Ki` are free\n`'PD'`Proportional-derivative with first-order filter on derivative action`Ki` is fixed to zero; `Kp`, `Kd`, and `Tf` are free\n`'PID'`Proportional-integral-derivative with first-order filter on derivative action`Kp`, `Ki`, `Kd`, and `Tf` are free\n\n`Ts`\n\nSample time, specified as a scalar.\n\n`sys`\n\nDynamic system model representing a PID controller.\n\n## Properties\n\n`Kp, Ki, Kd, Tf`\n\nParametrization of the PID gains `Kp`, `Ki`, `Kd`, and filter time constant `Tf` of the tunable PID controller `blk`.\n\nThe following fields of `blk.Kp`, `blk.Ki`, `blk.Kd`, and `blk.Tf` are used when you tune `blk` using a tuning command such as `systune`:\n\nFieldDescription\n`Value`Current value of the parameter.\n`Free`\n\nLogical value determining whether the parameter is fixed or tunable. For example,\n\n• If `blk.Kp.Free = 1`, then `blk.Kp.Value` is tunable.\n\n• If `blk.Kp.Free = 0`, then `blk.Kp.Value` is fixed.\n\n`Minimum`\n\nMinimum value of the parameter. This property places a lower bound on the tuned value of the parameter. For example, setting ```blk.Kp.Minimum = 0``` ensures that `Kp` remains positive.\n\n`blk.Tf.Minimum` must always be positive.\n\n`Maximum`Maximum value of the parameter. This property places an upper bound on the tuned value of the parameter. For example, setting ```blk.Tf.Maximum = 100``` ensures that the filter time constant does not exceed 100.\n\n`blk.Kp`, `blk.Ki`, `blk.Kd`, and `blk.Tf` are `param.Continuous` objects. For general information about the properties of these `param.Continuous` objects, see the `param.Continuous` object reference page.\n\n`IFormula, DFormula`\n\nDiscrete integrator formulas IF(z) and DF(z) for the integral and derivative terms, respectively, specified as one of the values in the following table.\n\nValueIF(z) or DF(z) Formula\n`'ForwardEuler'`\n\n`$\\frac{{T}_{s}}{z-1}$`\n\n`'BackwardEuler'`\n\n`$\\frac{{T}_{s}z}{z-1}$`\n\n`'Trapezoidal'`\n\n`$\\frac{{T}_{s}}{2}\\frac{z+1}{z-1}$`\n\nDefault: `'ForwardEuler'`\n\n`Ts`\n\nSample time. For continuous-time models, `Ts = 0`. For discrete-time models, `Ts` is a positive scalar representing the sampling period. This value is expressed in the unit specified by the `TimeUnit` property of the model. Unspecified sample time (`Ts = -1`) is not supported for PID blocks.\n\nChanging this property does not discretize or resample the model.\n\nDefault: `0` (continuous time)\n\n`TimeUnit`\n\nUnits for the time variable, the sample time `Ts`, and any time delays in the model, specified as one of the following values:\n\n• `'nanoseconds'`\n\n• `'microseconds'`\n\n• `'milliseconds'`\n\n• `'seconds'`\n\n• `'minutes'`\n\n• `'hours'`\n\n• `'days'`\n\n• `'weeks'`\n\n• `'months'`\n\n• `'years'`\n\nChanging this property has no effect on other properties, and therefore changes the overall system behavior. Use `chgTimeUnit` to convert between time units without modifying system behavior.\n\nDefault: `'seconds'`\n\n`InputName`\n\nInput channel name, specified as a character vector. Use this property to name the input channel of the controller model. For example, assign the name `error` to the input of a controller model `C` as follows.\n\n`C.InputName = 'error';`\n\nYou can use the shorthand notation `u` to refer to the `InputName` property. For example, `C.u` is equivalent to `C.InputName`.\n\nInput channel names have several uses, including:\n\n• Identifying channels on model display and plots\n\n• Specifying connection points when interconnecting models\n\nDefault: Empty character vector, `''`\n\n`InputUnit`\n\nInput channel units, specified as a character vector. Use this property to track input signal units. For example, assign the concentration units `mol/m^3` to the input of a controller model `C` as follows.\n\n`C.InputUnit = 'mol/m^3';`\n\n`InputUnit` has no effect on system behavior.\n\nDefault: Empty character vector, `''`\n\n`InputGroup`\n\nInput channel groups. This property is not needed for PID controller models.\n\nDefault: `struct` with no fields\n\n`OutputName`\n\nOutput channel name, specified as a character vector. Use this property to name the output channel of the controller model. For example, assign the name `control` to the output of a controller model `C` as follows.\n\n`C.OutputName = 'control';`\n\nYou can use the shorthand notation `y` to refer to the `OutputName` property. For example, `C.y` is equivalent to `C.OutputName`.\n\nInput channel names have several uses, including:\n\n• Identifying channels on model display and plots\n\n• Specifying connection points when interconnecting models\n\nDefault: Empty character vector, `''`\n\n`OutputUnit`\n\nOutput channel units, specified as a character vector. Use this property to track output signal units. For example, assign the unit `Volts` to the output of a controller model `C` as follows.\n\n`C.OutputUnit = 'Volts';`\n\n`OutputUnit` has no effect on system behavior.\n\nDefault: Empty character vector, `''`\n\n`OutputGroup`\n\nOutput channel groups. This property is not needed for PID controller models.\n\nDefault: `struct` with no fields\n\n`Name`\n\nSystem name, specified as a character vector. For example, `'system_1'`.\n\nDefault: `''`\n\n`Notes`\n\nAny text that you want to associate with the system, stored as a string or a cell array of character vectors. The property stores whichever data type you provide. For instance, if `sys1` and `sys2` are dynamic system models, you can set their `Notes` properties as follows:\n\n```sys1.Notes = \"sys1 has a string.\"; sys2.Notes = 'sys2 has a character vector.'; sys1.Notes sys2.Notes```\n```ans = \"sys1 has a string.\" ans = 'sys2 has a character vector.' ```\n\nDefault: `[0×1 string]`\n\n`UserData`\n\nAny type of data you want to associate with system, specified as any MATLAB® data type.\n\nDefault: `[]`\n\n## Examples\n\nTunable Controller with a Fixed Parameter\n\nCreate a tunable PD controller. Then, initialize the parameter values, and fix the filter time constant.\n\n```blk = tunablePID('pdblock','PD'); blk.Kp.Value = 4; % initialize Kp to 4 blk.Kd.Value = 0.7; % initialize Kd to 0.7 blk.Tf.Value = 0.01; % set parameter Tf to 0.01 blk.Tf.Free = false; % fix parameter Tf to this value blk```\n```blk = Parametric continuous-time PID controller \"pdblock\" with formula: s Kp + Kd * -------- Tf*s+1 and tunable parameters Kp, Kd. Type \"pid(blk)\" to see the current value and \"get(blk)\" to see all properties. ```\n\nController Initialized by Dynamic System Model\n\nCreate a tunable discrete-time PI controller. Use a `pid` object to initialize the parameters and other properties.\n\n```C = pid(5,2.2,'Ts',0.1,'IFormula','BackwardEuler'); blk = tunablePID('piblock',C)```\n```blk = Parametric discrete-time PID controller \"piblock\" with formula: Ts*z Kp + Ki * ------ z-1 and tunable parameters Kp, Ki. Type \"pid(blk)\" to see the current value and \"get(blk)\" to see all properties. ```\n\n`blk` takes the value of properties, such as `Ts` and `IFormula`, from `C`.\n\nController with Named Input and Output\n\nCreate a tunable PID controller, and assign names to the input and output.\n\n```blk = tunablePID('pidblock','pid') blk.InputName = {'error'} % assign input name blk.OutputName = {'control'} % assign output name```\n\n## Tips\n\n• You can modify the PID structure by fixing or freeing any of the parameters `Kp`, `Ki`, `Kd`, and `Tf`. For example, `blk.Tf.Free = false` fixes `Tf` to its current value.\n\n• To convert a `tunablePID` parametric model to a numeric (nontunable) model object, use model commands such as `pid`, `pidstd`, `tf`, or `ss`. You can also use `getValue` to obtain the current value of a tunable model.\n\n## Compatibility Considerations\n\nexpand all\n\nBehavior changed in R2016a" ]

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[ "8M: 2-4 QC #1 (variables on both sides)\n 1.  Solve the equation below.  Just type the solution     in the box, not x = the number.-14x + 7 = -6x - 89 2.  Solve the equation below.  Just type the solution     in the box, not x = the number.19x - 9 = 8x + 57 3.  Solve the equation below.  Just type the solution     in the box, not x = the number.17x + 4 = 10x - 45 4.  Solve the equation below.  Just type the solution     in the box, not x = the number.-13x - 6 = -4x + 75 5.  Solve the equation below.  Just type the solution     in the box, not x = the number.-3x - 10 = -7(x + 6) 6.  Solve the equation below.  Just type the solution     in the box, not x = the number.-9(x + 1) = -5x + 11 7.  Solve the equation below.  Just type the solution     in the box, not x = the number.-13x - 6x + 7 = -9x + 87 8.  Solve the equation below.  Just type the solution     in the box, not x = the number.3x - 4 = 50 -2x + 1 9.  Solve the equation below.  Just type the solution     in the box, not x = the number.10x + 7 + 5x = 7x + 31 10.  Solve the equation below.  Just type the solution     in the box, not x = the number.x - 2 = 4x - 5 11.  Solve the equation below.  Just type the solution       in the box, not x = the number.-2(x - 8) = 4(x + 1) 12.  Solve the equation below.  Just type the solution       in the box, not x = the number.9x + 10 - 6x = 6x + 40\nStudents who took this test also took :\n\nCreated with That Quiz — the math test generation site with resources for other subject areas." ]

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[ "# Non-Moment Mechanics\n\nAutor:\nVerfügbarkeit:\nAuf Lager.\nArtikelnummer:\n1172402\n49,30\n• ### Produktbeschreibung\n\n#### Non-Moment Mechanics\n\nLaplace transform has the advantage of directly giving the solution of differential equation with the given boundary values without the necessity of first finding the general solution and then transform any real or complex variable bearing function into the integral form. In case of more then two and three variables so as for thermo dynamical phenomenae some variables are kept constant and partial differentiation comes into play. Wave equation, heat equation etc. are such equation which may not be solved with the help of Laplace operator L. So these types of problem on which Laplace operator or transform fails will be non-moment problem function .A Non-Moment function is directly proportional to any exponential function. for the purpose of generating a new ideology and corollary on Non-Moment problem, we take into account equation (1).as the standard form of an equation representing it self as the generator of NON-MOMENT PROBLEMB function. To make our result more accurate and widely considerable, we would like to introduce some other facts and finding coherent to the convergence of result and making our formula as a very common men discussion.\n• ### Zusatzinformation\n\nAutor\nBindung\nTaschenbuch\nVerlag\nScholar's Press\nISBN / EAN\n9783639719789\n• ### 0 Kundenmeinungen Vielen Dank für das Interesse an unsere Kundenmeinungen. Wir können die Authentizität der Bewertungen nicht prüfen und sicherstellen.\n\nSchreiben Sie selbst eine Rezension\n\nIhre Meinung interessiert uns – und hilft anderen Kunden bei der Auswahl." ]

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[ "Algorithms\nTopics:\nMin-cut\n• Searching\n• Sorting\n• Greedy Algorithms\n• Graphs\n• String Algorithms\n• Dynamic Programming\n\n# Min-cut\n\nMin-Cut of a weighted graph is defined as the minimum sum of weights of (at least one)edges that when removed from the graph divides the graph into two groups. Mechthild Stoer and Frank Wagner proposed an algorithm in 1995 to find minimum cut in an undirected weighted graphs.\n\nThe algorithm works on a method of shrinking the graph by merging the most tightly connected vertex until only one node is left in the graph and for each step performed, the weight of the merged cut is stored in a list. Minimum value in the list would be the minimum cut value of the graph.\n\nPseudocode for the algorithm is given below,\n\nfunction: MinCutPhase(Graph G, Weights W, Vertex a):\nA <- {a}\nwhile A != V:\nadd tightly connected vertex to A\nstore cut_of_the_phase and shrink G by merging the two vertices added last\n\nminimum = INF\nfunction: MinCut(Graph G, Weights W, Vertex a):\nwhile |V| > 1:\nMinCutPhase(G,W,a)\nif cut_of_the_phase < minimum:\nminimum = cut_of_the_phase\nreturn minimum\n\n\nVertex $a$ can be arbitrary and remains same throughout the whole algorithm. $A$ in MinCutPhase is a collection of vertices which starts with the arbitrary vertex $a$ and starts adding other vertices of graph $G$ one at a time until it is equal to $V$ which is collection of all vertices. Tightly connected vertex to $A$ implies a vertex whose edge weight to any of the vertex in $A$ is maximum. It can mathematically represented as $z \\not\\in A\\ such\\ that\\ W(A,z) = max\\{W(A,y) | y \\not\\in A\\}$, $z$ will be the new tightly connected vertex. The cut_of_the_phase is the sum of edges connecting last added vertex with rest of the graph. Minimum value of $cut\\_of\\_the\\_phase$ is the required result.\n\nLast two vertices added to the set $A$ are merged by creating a single node for both of them and edges joining them are deleted. Edges connecting these vertices to other vertices are replaced with edges weighing sum of edges to both the vertex. Ex: Let vertices $P$ and $Q$ are the two vertices added last to the set $A$. Let there are three edges connecting $P$ and $Q$ to the set $A$, $E(X,P), W(X,P) = 10; E(X,Q), W(X,Q) = 20; E(Y,Q), W(Y,Q) = 15$. Now as they are added last, vertices $P$ and $Q$ are merged to a single node, say $R$. Now, edges connecting to $R$ will be, $E(X,R), W(X,R) = 30; E(Y,R), W(Y,R) = 15$.\n\nFor a flow network with $a$ as in the algo mentioned above, is either $S$ or $T$, will be the maximum flow of the network from $S$ to $T$. This is because, minimum cut will be the bottleneck capacity of the flow network. And always, the amount of flow from source side to sink side has to pass through these set of edges that are to be cut. Working on a paper, source should be on left and destination on right. The cut should always start from top of the graph, move to bottom of the graph(should not stop in between). The edges that are to be considered in min-cut should move from left of the cut to right of the cut. Sum of capacity of all these edges will be the min-cut which also is equal to max-flow of the network.\n\nWorking on a directed graph to calculate max flow of the graph using min-cut concept is shown in image below.", null, "Few possible cuts in the graph are shown in the graph and weights of each cut are as follows: $Cut 1: 25, Cut 2: 12, Cut 3: 16, Cut 4: 10, Cut 5: 15$. As mentioned these are only few possible cuts but considering any valid cut would not have weight less than $Cut 4$. And it is the min-cut of this graph which is also the max-flow of the graph as explained above.\n\nContributed by: Vinay Kumar\nNotifications\n\n?" ]

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[ "# Remove space in empty node so that draw line is connected\n\nI'm trying to remove the white space that appears between two lines as a result of going to an empty node. Specifically, I have a node drawing to the left of the empty node and then the empty node drawing to the a new full node.\n\nI've already used inner sep=0 and outer sep=0 on the node to reduce the empty gap. However,a small sliver still remains.\n\nI need to have the empty node there for space reasons.\n\nImage:", null, "MWE:\n\n\\documentclass{article}\n\\usepackage{tikz}%pictures\n\\usetikzlibrary{shapes,arrows}\n\\begin{document}\n\\begin{tikzpicture}[auto,>=latex']\n\\tikzstyle{block} = [draw, shape=rectangle, minimum height=3em, minimum width=3em, node distance=2cm, line width=1pt]\n\\tikzstyle{sum} = [draw, shape=circle, node distance=1.5cm, line width=1pt, minimum width=1.25em]\n\\tikzstyle{connection}=[inner sep=0,outer sep=0]\n%Creating Blocks and Connection Nodes\n\\node [block, right of=input] (h1) {$0.99$};\n\\node [right of=h1] (hsum) {};\n\\node [connection, right of=hsum] (cont) {};\n\\path (h1) -- coordinate (hmed) (h1);\n%Connecting Blocks\n\\begin{scope}[line width=1pt]\n\\draw (input) -- (h1);\\draw (h1) -- (cont);\n\\end{scope}\n%Creating Blocks and Connection Nodes\n\\node [block, right of=cont] (m1) {$0.18$};\\node [block, right of=m1] (m2) {$0.16$};\n\\node [right of=m2] (msum) {};\n\\node [connection, right of=msum] (output) {};\n\\path (m1) -- coordinate (mmed) (m2);\n%Connecting Blocks\n\\begin{scope}[line width=1pt]\n\\draw (cont) -- (m1);\\draw (m1) -- (m2);\\draw (m2) -- (output);\n\\end{scope}\n\\end{tikzpicture}\n\\end{document}\n\n\nYou can use a coordinate like\n\n\\coordinate[right of=hsum] (cont);\n\n\nCode:\n\n\\documentclass{article}\n\\usepackage{tikz}%pictures\n\\usetikzlibrary{shapes,arrows}\n\\begin{document}\n\\begin{tikzpicture}[auto,>=latex']\n\\tikzstyle{block} = [draw, shape=rectangle, minimum height=3em, minimum width=3em, node distance=2cm, line width=1pt]\n\\tikzstyle{sum} = [draw, shape=circle, node distance=1.5cm, line width=1pt, minimum width=1.25em]\n\\tikzstyle{connection}=[inner sep=0,outer sep=0]\n%Creating Blocks and Connection Nodes\n\\coordinate (input);\n\\node [block, right of=input] (h1) {$0.99$};\n\\node [right of=h1] (hsum) {};\n\\coordinate[right of=hsum] (cont);\n\\path (h1) -- coordinate (hmed) (h1);\n%Connecting Blocks\n\\begin{scope}[line width=1pt]\n\\draw (input) -- (h1);\\draw (h1) -- (cont);\n\\end{scope}\n%Creating Blocks and Connection Nodes\n\\node [block, right of=cont] (m1) {$0.18$};\\node [block, right of=m1] (m2) {$0.16$};\n\\node [right of=m2] (msum) {};\n\\node [connection, right of=msum] (output) {};\n\\path (m1) -- coordinate (mmed) (m2);\n%Connecting Blocks\n\\begin{scope}[line width=1pt]\n\\draw (cont) -- (m1);\\draw (m1) -- (m2);\\draw (m2) -- (output);\n\\end{scope}\n\\end{tikzpicture}\n\\end{document}", null, "If you want to stick to the node, use cont.center for the connections instead of cont like:\n\n\\draw (input) -- (h1);\\draw (h1) -- (cont.center);\n\n\nCode again:\n\n\\documentclass{article}\n\\usepackage{tikz}%pictures\n\\usetikzlibrary{shapes,arrows}\n\\begin{document}\n\\begin{tikzpicture}[auto,>=latex']\n\\tikzset{block/.style = {draw, shape=rectangle, minimum height=3em, minimum width=3em, node\ndistance=2cm, line width=1pt},\nsum/.style = {draw, shape=circle, node distance=1.5cm, line width=1pt, minimum\nwidth=1.25em},\nconnection/.style={inner sep=0,outer sep=0}\n}\n%Creating Blocks and Connection Nodes\n\\coordinate (input);\n\\node [block, right of=input] (h1) {$0.99$};\n\\node [right of=h1] (hsum) {};\n\\node [connection, right of=hsum] (cont) {};\n\\path (h1) -- coordinate (hmed) (h1);\n%Connecting Blocks\n\\begin{scope}[line width=1pt]\n\\draw (input) -- (h1);\\draw (h1) -- (cont.center);\n\\end{scope}\n%Creating Blocks and Connection Nodes\n\\node [block, right of=cont] (m1) {$0.18$};\\node [block, right of=m1] (m2) {$0.16$};\n\\node [right of=m2] (msum) {};\n\\node [connection, right of=msum] (output) {};\n\\path (m1) -- coordinate (mmed) (m2);\n%Connecting Blocks\n\\begin{scope}[line width=1pt]\n\\draw (cont.center) -- (m1);\\draw (m1) -- (m2);\\draw (m2) -- (output);\n\\end{scope}\n\\end{tikzpicture}\n\\end{document}\n\n\nAlso, use tikzset instead of tikzstyle (deprecated) as I did in the second code.\n\nYou can do this by adding some property to tikzpicture as follows:\n\n\\begin{tikzpicture}[every node/.style={inner sep=0,outer sep=0}]\n\n• This applies to all nodes inside the picture and needs to be overwritten for nodes, where the space is needed as for the three rectangular nodes with the numbers. Using coordinate instead of node is the better choice for denoting a point without node contents. – Heiko Oberdiek Jul 2 '17 at 7:25" ]

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

[ "# Finitely generated abelian groups with GAP.¶\n\nThis module provides a python wrapper for abelian groups in GAP.\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: AbelianGroupGap([3,5])\nAbelian group with gap, generator orders (3, 5)\n\n\nFor infinite abelian groups we use the GAP package Polycyclic:\n\nsage: AbelianGroupGap([3,0]) # optional - gap_packages\nAbelian group with gap, generator orders (3, 0)\n\n\nAUTHORS:\n\n• Simon Brandhorst (2018-01-17): initial version\n\nclass sage.groups.abelian_gps.abelian_group_gap.AbelianGroupElement_gap(parent, x, check=True)\n\nAn element of an abelian group via libgap.\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: G = AbelianGroupGap([3,6])\nsage: G.gens()\n(f1, f2)\n\nexponents()\n\nReturn the tuple of exponents of this element.\n\nOUTPUT:\n\n• a tuple of integers\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: G = AbelianGroupGap([4,7,9])\nsage: gens = G.gens()\nsage: g = gens^2 * gens^4 * gens^8\nsage: g.exponents()\n(2, 4, 8)\nsage: S = G.subgroup(G.gens()[:1])\nsage: s = S.gens()\nsage: s\nf1\nsage: s.exponents()\n(1,)\n\n\nIt can handle quite large groups too:\n\nsage: G = AbelianGroupGap([2^10, 5^10])\nsage: f1, f2 = G.gens()\nsage: g = f1^123*f2^789\nsage: g.exponents()\n(123, 789)\n\n\nWarning\n\nCrashes for very large groups.\n\nTodo\n\nMake exponents work for very large groups. This could be done by using Pcgs in gap.\n\norder()\n\nReturn the order of this element.\n\nOUTPUT:\n\n• an integer or infinity\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: G = AbelianGroupGap()\nsage: g = G.gens()\nsage: g.order()\n4\nsage: G = AbelianGroupGap() # optional - gap_packages\nsage: g = G.gens() # optional - gap_packages\nsage: g.order() # optional - gap_packages\n+Infinity\n\nclass sage.groups.abelian_gps.abelian_group_gap.AbelianGroupElement_polycyclic(parent, x, check=True)\n\nAn element of an abelian group using the GAP package Polycyclic.\n\nexponents()\n\nReturn the tuple of exponents of self.\n\nOUTPUT:\n\n• a tuple of integers\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: G = AbelianGroupGap([4,7,0]) # optional - gap_packages\nsage: gens = G.gens() # optional - gap_packages\nsage: g = gens^2 * gens^4 * gens^8 # optional - gap_packages\nsage: g.exponents() # optional - gap_packages\n(2, 4, 8)\n\n\nEfficiently handles very large groups:\n\nsage: G = AbelianGroupGap([2^30,5^30,0]) # optional - gap_packages\nsage: f1, f2, f3 = G.gens() # optional - gap_packages\nsage: (f1^12345*f2^123456789).exponents() # optional - gap_packages\n(12345, 123456789, 0)\n\nclass sage.groups.abelian_gps.abelian_group_gap.AbelianGroupGap(generator_orders)\n\nAbelian groups implemented using GAP.\n\nINPUT:\n\n• generator_orders – a list of nonnegative integers where $$0$$ gives a factor isomorphic to $$\\ZZ$$\n\nOUTPUT:\n\n• an abelian group\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: AbelianGroupGap([3,6])\nAbelian group with gap, generator orders (3, 6)\nsage: AbelianGroupGap([3,6,5])\nAbelian group with gap, generator orders (3, 6, 5)\nsage: AbelianGroupGap([3,6,0]) # optional - gap_packages\nAbelian group with gap, generator orders (3, 6, 0)\n\n\nWarning\n\nNeeds the GAP package Polycyclic in case the group is infinite.\n\nclass sage.groups.abelian_gps.abelian_group_gap.AbelianGroupQuotient_gap(G, N)\n\nQuotients of abelian groups by a subgroup.\n\nNote\n\nDo not call this directly. Instead use quotient().\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: A = AbelianGroupGap([4,3])\nsage: N = A.subgroup([A.gen(0)^2])\nsage: Q1 = A.quotient(N)\nsage: Q1\nQuotient abelian group with generator orders (2, 3)\nsage: Q2 = Q1.quotient(Q1.subgroup(Q1.gens()[:1]))\nsage: Q2\nQuotient abelian group with generator orders (1, 3)\n\ncover()\n\nReturn the covering group of this quotient group.\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: G = AbelianGroupGap([2,3,4,5])\nsage: gen = G.gens()[:2]\nsage: S = G.subgroup(gen)\nsage: Q = G.quotient(S)\nsage: Q.cover() is G\nTrue\n\nlift(x)\n\nLift an element to the cover.\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: A = AbelianGroupGap()\nsage: N = A.subgroup([A.gen(0)^2])\nsage: Q = A.quotient(N)\nsage: Q.lift(Q.0)\nf1\n\nnatural_homomorphism()\n\nReturn the defining homomorphism into self.\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: A = AbelianGroupGap()\nsage: N = A.subgroup([A.gen(0)^2])\nsage: Q = A.quotient(N)\nsage: Q.natural_homomorphism()\nGroup morphism:\nFrom: Abelian group with gap, generator orders (4,)\nTo: Quotient abelian group with generator orders (2,)\n\nrelations()\n\nReturn the relations of this quotient group.\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: G = AbelianGroupGap([2,3,4,5])\nsage: gen = G.gens()[:2]\nsage: S = G.subgroup(gen)\nsage: Q = G.quotient(S)\nsage: Q.relations() is S\nTrue\n\nclass sage.groups.abelian_gps.abelian_group_gap.AbelianGroupSubgroup_gap(ambient, gens)\n\nSubgroups of abelian groups with GAP.\n\nINPUT:\n\n• ambient – the ambient group\n\n• gens – generators of the subgroup\n\nNote\n\nDo not construct this class directly. Instead use subgroup().\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: G = AbelianGroupGap([2,3,4,5])\nsage: gen = G.gens()[:2]\nsage: S = G.subgroup(gen)\n\nlift(x)\n\nCoerce to the ambient group.\n\nThe terminology comes from the category framework and the more general notion of a subquotient.\n\nINPUT:\n\n• x – an element of this subgroup\n\nOUTPUT:\n\nThe corresponding element of the ambient group\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: G = AbelianGroupGap()\nsage: g = G.gen(0)\nsage: H = G.subgroup([g^2])\nsage: h = H.gen(0); h\nf2\nsage: h.parent()\nSubgroup of Abelian group with gap, generator orders (4,) generated by (f2,)\nsage: H.lift(h)\nf2\nsage: H.lift(h).parent()\nAbelian group with gap, generator orders (4,)\n\nretract(x)\n\nConvert an element of the ambient group into this subgroup.\n\nThe terminology comes from the category framework and the more general notion of a subquotient.\n\nINPUT:\n\n• x – an element of the ambient group that actually lies in this subgroup.\n\nOUTPUT:\n\nThe corresponding element of this subgroup\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: G = AbelianGroupGap()\nsage: g = G.gen(0)\nsage: H = G.subgroup([g^2])\nsage: H.retract(g^2)\nf2\nsage: H.retract(g^2).parent()\nSubgroup of Abelian group with gap, generator orders (4,) generated by (f2,)\n\nclass sage.groups.abelian_gps.abelian_group_gap.AbelianGroup_gap(G, category, ambient=None)\n\nFinitely generated abelian groups implemented in GAP.\n\nNeeds the gap package Polycyclic in case the group is infinite.\n\nINPUT:\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: G = AbelianGroupGap([3, 2, 5])\nsage: G\nAbelian group with gap, generator orders (3, 2, 5)\n\nElement\nall_subgroups()\n\nReturn the list of all subgroups of this group.\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: G = AbelianGroupGap([2, 3])\nsage: G.all_subgroups()\n[Subgroup of Abelian group with gap, generator orders (2, 3) generated by (),\nSubgroup of Abelian group with gap, generator orders (2, 3) generated by (f1,),\nSubgroup of Abelian group with gap, generator orders (2, 3) generated by (f2,),\nSubgroup of Abelian group with gap, generator orders (2, 3) generated by (f2, f1)]\n\naut()\n\nReturn the group of automorphisms of self.\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: G = AbelianGroupGap([2, 3])\nsage: G.aut()\nFull group of automorphisms of Abelian group with gap, generator orders (2, 3)\n\nautomorphism_group()\n\nReturn the group of automorphisms of self.\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: G = AbelianGroupGap([2, 3])\nsage: G.aut()\nFull group of automorphisms of Abelian group with gap, generator orders (2, 3)\n\nelementary_divisors()\n\nReturn the elementary divisors of this group.\n\nSee sage.groups.abelian_gps.abelian_group_gap.elementary_divisors().\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: G = AbelianGroupGap([2,3,4,5])\nsage: G.elementary_divisors()\n(2, 60)\n\nexponent()\n\nReturn the exponent of this abelian group.\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: G = AbelianGroupGap([2,3,7])\nsage: G\nAbelian group with gap, generator orders (2, 3, 7)\nsage: G = AbelianGroupGap([2,4,6])\nsage: G\nAbelian group with gap, generator orders (2, 4, 6)\nsage: G.exponent()\n12\n\ngens_orders()\n\nReturn the orders of the generators.\n\nUse elementary_divisors() if you are looking for an invariant of the group.\n\nOUTPUT:\n\n• a tuple of integers\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: Z2xZ3 = AbelianGroupGap([2,3])\nsage: Z2xZ3.gens_orders()\n(2, 3)\nsage: Z2xZ3.elementary_divisors()\n(6,)\nsage: Z6 = AbelianGroupGap()\nsage: Z6.gens_orders()\n(6,)\nsage: Z6.elementary_divisors()\n(6,)\nsage: Z2xZ3.is_isomorphic(Z6)\nTrue\nsage: Z2xZ3 is Z6\nFalse\n\nidentity()\n\nReturn the identity element of this group.\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: G = AbelianGroupGap([4,10])\nsage: G.identity()\n1\n\nis_subgroup_of(G)\n\nReturn if self is a subgroup of G considered in the same ambient group.\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: G = AbelianGroupGap([2,3,4,5])\nsage: gen = G.gens()[:2]\nsage: S1 = G.subgroup(gen)\nsage: S1.is_subgroup_of(G)\nTrue\nsage: S2 = G.subgroup(G.gens()[1:])\nsage: S2.is_subgroup_of(S1)\nFalse\n\nis_trivial()\n\nReturn True if this group is the trivial group.\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: G = AbelianGroupGap([])\nsage: G\nAbelian group with gap, generator orders ()\nsage: G.is_trivial()\nTrue\nsage: AbelianGroupGap().is_trivial()\nTrue\nsage: AbelianGroupGap([1,1,1]).is_trivial()\nTrue\nsage: AbelianGroupGap().is_trivial()\nFalse\nsage: AbelianGroupGap([2,1]).is_trivial()\nFalse\n\nquotient(N)\n\nReturn the quotient of this group by the normal subgroup $$N$$.\n\nINPUT:\n\n• N – a subgroup\n\n• check – bool (default: True) check if $$N$$ is normal\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: A = AbelianGroupGap([2,3,4,5])\nsage: S = A.subgroup(A.gens()[:1])\nsage: A.quotient(S)\nQuotient abelian group with generator orders (1, 3, 4, 5)\n\nsubgroup(gens)\n\nReturn the subgroup of this group generated by gens.\n\nINPUT:\n\n• gens – a list of elements coercible into this group\n\nOUTPUT:\n\n• a subgroup\n\nEXAMPLES:\n\nsage: from sage.groups.abelian_gps.abelian_group_gap import AbelianGroupGap\nsage: G = AbelianGroupGap([2,3,4,5])\nsage: gen = G.gens()[:2]\nsage: S = G.subgroup(gen)\nsage: S\nSubgroup of Abelian group with gap, generator orders (2, 3, 4, 5)\ngenerated by (f1, f2)\nsage: g = G.an_element()\nsage: s = S.an_element()\nsage: g * s\nf2^2*f3*f5\nsage: G = AbelianGroupGap([3,4,0,2]) # optional - gap_packages\nsage: gen = G.gens()[:2] # optional - gap_packages\nsage: S = G.subgroup(gen) # optional - gap_packages\nsage: g = G.an_element() # optional - gap_packages\nsage: s = S.an_element() # optional - gap_packages\nsage: g * s # optional - gap_packages\ng1^2*g2^2*g3*g4" ]

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[ "Feet To Cm\n\n# 1810 ft to cm1810 Feet to Centimeters\n\nft\n=\ncm\n\n## How to convert 1810 feet to centimeters?\n\n 1810 ft * 30.48 cm = 55168.8 cm 1 ft\nA common question is How many foot in 1810 centimeter? And the answer is 59.3832020997 ft in 1810 cm. Likewise the question how many centimeter in 1810 foot has the answer of 55168.8 cm in 1810 ft.\n\n## How much are 1810 feet in centimeters?\n\n1810 feet equal 55168.8 centimeters (1810ft = 55168.8cm). Converting 1810 ft to cm is easy. Simply use our calculator above, or apply the formula to change the length 1810 ft to cm.\n\n## Convert 1810 ft to common lengths\n\nUnitLengths\nNanometer5.51688e+11 nm\nMicrometer551688000.0 µm\nMillimeter551688.0 mm\nCentimeter55168.8 cm\nInch21720.0 in\nFoot1810.0 ft\nYard603.333333333 yd\nMeter551.688 m\nKilometer0.551688 km\nMile0.3428030303 mi\nNautical mile0.297887689 nmi\n\n## What is 1810 feet in cm?\n\nTo convert 1810 ft to cm multiply the length in feet by 30.48. The 1810 ft in cm formula is [cm] = 1810 * 30.48. Thus, for 1810 feet in centimeter we get 55168.8 cm.\n\n## 1810 Foot Conversion Table", null, "## Alternative spelling\n\n1810 Feet to Centimeters, 1810 Feet in Centimeters, 1810 ft to Centimeter, 1810 ft in Centimeter, 1810 Foot to Centimeter, 1810 Foot in Centimeter, 1810 ft to Centimeters, 1810 ft in Centimeters, 1810 Feet to cm, 1810 Feet in cm, 1810 Foot to cm, 1810 Foot in cm, 1810 Feet to Centimeter, 1810 Feet in Centimeter" ]

[ null, "https://feet-to-cm.appspot.com/image/1810.png", null ]

[ "## Friday, December 31, 2021\n\n### About TGD counterparts of twistor amplitudes\n\nThe twistor lift of TGD, in which H=M4 × CP2 is replaced with the product of twistor spaces T(M4) and T(CP2), and space-time surface X4⊂ H with its 6-D twistor space as 6-surface X6 ⊂ T(M4)× T(CP2), is now a rather well-established notion and M8-H duality predicts it at the level of M8.\n\nNumber theoretical vision involves M8-H duality. At the level of ⊂H⊂, the quark mass spectrum is determined by the Dirac equation in ⊂H⊂. In M8 mass squared spectrum is determined by the roots of the polynomial P determining space-time surface and are in general complex. By Galois confinement the momenta are integer valued when p-adic mass is used as a unit and mass squared spectrum is also integer valued. This raises hope about a generalization of the twistorial construction of scattering amplitudes to TGD context.\n\nIt is always best to start from a problem and the basic problem of the twistor approach is that physical particles are not massless.\n\n1. The intuitive TGD based proposal has been that since quark spinors are massless in H, the masslessness in the 8-D sense could somehow solve the problems caused by the massivation in  the construction of twistor scattering amplitudes. However, no obvious mechanism has been identified. One step in this direction was the realization that in H quarks propagate with well-defined chiralities and only the square of Dirac equation is satisfied. For a quark of given helicity the  spinor can be identified as helicity spinor.\n2.  M8 quark momenta are in general complex as algebraic integers. They are the counterparts of the area momenta xi of momentum twistor space whereas H momenta are identified as ordinary momenta. Total momenta  of Galois confined states  have as components ordinary integers.\n3.   The  M8 counterpart of  the 8-D massless condition in H is the restriction of momenta to mass shells m2= rn determined as roots  of P. The M8 counterpart of Dirac equation in H is octonionic Dirac equation, which is algebraic as everything in M8 and analogous to massless Dirac equation. The solution is a helicity spinor λ associated with the massive momentum.\nThe outcome is an extremely simple proposal for the scattering amplitudes.\n1. Vertices correspond to trilinears of Galois confined many-quark states as states  of super symplectic algebra acting as isometries of the \"world of classical worlds\" (WCW). Quarks are on-shell with H  momentum p and M8 momenta xi,xi+1, pi=xi+1-xi. Dirac operator xkiγk restricted to fixed helicity L,R appears as a vertex factor and has an  interpretation as a residue of a pole from an on-mass-shell propagator D so that a correspondence with twistorial construction becomes obvious. D  is expressible in terms of the helicity spinors of given chirality and gives two independent holomorphic factors as in case of massless theories.\n2. MHV construction utilizing k=2 MHV amplitudes as building bricks does not seem to be needed at the level of a single space-time surface. One can of course ask, whether the M8 quark lines could be regarded as analogs of lines connecting different MHV diagrams replaced with Galois singlets. The scattering amplitudes would be rational functions in accordance with the number theoretic vision. The absence of logarithmic radiative corrections is not a problem: the coupling constant evolution would be discrete and defined by the hierarchy of extensions of rationals.\n3. The scattering amplitudes for a single 4-surface X4 are determined by a polynomial. The integration over WCW is replaced with a summation of polynomials characterized by rational coefficients. Monic polynomials are highly suggestive. A connection with p-adicization emerges via the identification of the p-adic prime as one of the ramified primes of P. Only (monic) polynomials having a common p-adic prime are allowed in the sum. The counterpart of the vacuum functional exp(-K) is naturally identified as the discriminant D of the extension associated with P and p-adic coupling constant evolution emerges from the identification of exp(-K) with D.\nSee the article About TGD counterparts of twistor amplitudes or the chapter with the same title.\n\nFor a summary of earlier postings see Latest progress in TGD." ]

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[ "## Research Journal of Optics and Photonics.\n\nAll submissions of the EM system will be redirected to Online Manuscript Submission System. Authors are requested to submit articles directly to Online Manuscript Submission System of respective journal.\n\nResearch Article, Res J Opt Photonics Vol: 2 Issue: 1\n\n# Supercontinuum Generation Using Microstructured Optical Fibers\n\nFerreira MFS*\n\nInstitute of Nanostructures, Nanomodelling and Nanofabrication, Department of Physics, University of Aveiro, 3810-193 Aveiro, Portugal\n\n*Corresponding Author : Ferreira MFS\nInstitute of Nanostructures, Nanomodelling and Nanofabrication, Department of Physics, University of Aveiro, 3810-193 Aveiro, Portugal\nTel: +351 23 4370356\nE-mail: [email protected]\n\nReceived: November 23, 2017 Accepted: January 01, 2018 Published: January 08, 2018\n\nCitation: Ferreira MFS (2018) Supercontinuum Generation Using Microstructured Optical Fibers. Res J Opt Photonics 2:1.\n\n## Abstract\n\nThe supercontinuum generation results generally from the synergy between several fundamental nonlinear processes, such as selfphase modulation, cross-phase modulation, stimulated Raman scattering, and four-wave mixing. The relative importance of these processes depends on the spectral location and power of the pump, as well as the nonlinear and dispersive characteristics of the medium. Several types of microstructured optical fibers with optimized designs, have been developed during the recent years in order to enhance the supercontinuum generation. This paper provides an overview of the peculiar dispersive and nonlinear properties exhibited by these fibers, including the supercontinuum generation by femtosecond pumping.\n\n## Introduction\n\nA number of third order nonlinear processes can occur when high-power light is launched into the optical fibers . These can grow to appreciable magnitudes over the long lengths available in fibers, even though the nonlinear index of the silica glass is very small (n2 = 2.35×10-20 m2/W) at 1550 nm ).\n\nFiber nonlinearities fall into two general categories . The first category arises from modulation of the refractive index of silica by intensity changes in the signal (Kerr effect). This gives rise to nonlinearities such as self-phase modulation (SPM) , crossphase modulation (XPM) , four-wave mixing (FWM) , and modulation instability (MI). The second category of nonlinearities corresponds to stimulated scattering processes, such as stimulated Brillouin scattering (SBS) and stimulated Raman scattering (SRS) , which are interactions between optical signals and acoustic or molecular vibrations in the fiber, respectively.\n\nA rather interesting nonlinear phenomenon which can be observed in optical fibers is the supercontinuum generation (SCG), which corresponds to an extremely wide spectrum, achieved by an optical pulse along its propagation . It results generally from the synergy between several fundamental nonlinear processes, such as SPM, XPM, SRS, MI, and FWM . The spectral locations and powers of the pumps, as well as the nonlinear and dispersive characteristics of the fiber determine the relative importance and the interaction between these nonlinear processes. A supercontinuum source can find applications in the area of bio-medical optics, where it allows the improvement of longitudinal resolution in optical coherence tomography (OCT) by more than an order of magnitude ; in optical frequency metrology ; in all kinds of spectroscopy, and as multiwavelength source in the telecommunications area .\n\nThe fiber nonlinearity is commonly characterized by the nonlinear parameter λ (defined in the next section), which in standard single-mode silica fibers has a typical value of λ ≈ 1.3W-1/km . Such value is too small for some applications requiring highly efficient nonlinear processes, namely for optical signal processing. An improved performance has resulted from the use of microstructured optical fibres (MOFs) [13,14]. MOFs with a small core dimension and a cladding with a large air-fill fraction allow for extremely tight mode confinement, i.e., small effective mode area, and hence, a higher value of γ. Besides, the large refractive index difference between silica and air critically cause a strong waveguide contribution to the overall group-velocity dispersion (GVD) of the guided mode . As a consequence, these structures can provide anomalous group-velocity dispersion and hence conditions for soliton formation at shorter wavelengths than is possible with conventional fibres .\n\n## Fiber Non-linearity and Dispersion\n\nThe fiber nonlinearity is commonly characterized by the nonlinear parameter γ, which is given by (1)", null, "(1)\n\nwhere λ is the light wavelength, n2 is the nonlinear-index coefficient of the fiber core and Aeff is the effective mode area, given by:", null, "(2)\n\nF(x,y) representing the spatial distribution of the fiber mode.\n\nEquation (1) shows that, for a fixed wavelength, since n2 is determined by the material from which the fiber is made, the most practical way of increasing the nonlinear parameter γ is to reduce the effective mode area Aeff. However, the nonlinear parameter γ can also be enhanced using the dopant dependence of the nonlinear refractive index n2. It has been shown that n2 increases by doping the bulk glass with GeO2 . On the other hand, n2 decreases in the case of an F-doped bulk glass.\n\nFiber dispersion\n\nThe mode-propagation constant β(ω) is related with the refractive index n(ω) in the form", null, "(3)\n\nMathematically, the effects of fiber dispersion are accounted for by expanding β(ω) in a Taylor series about the carrier frequency ω0 at which the pulse spectrum is centered:", null, "(4)\n\nwhere", null, "(5)\n\nThe group velocity dispersion (GVD) is characterized by the parameter β2", null, "(6)\n\nwhere Vg=c/ng and ng=n+ω(dn/dω) represent the group velocity and the group refractive index, respectively. In practice, the GVD is often characterized by another parameter, D, given by", null, "(7)\n\nThe parameter D gives the delay of arrival time in ps unit for two wavelength components separated by 1 nm over a distance of 1 km. In the case of fused silica, D=0 at λ=λZD=1.276 μm, which is referred to as the silica zero-dispersion wavelength. The parameter D becomes negative (normal GVD) at shorter wavelengths and positive (anomalous GVD) at longer wavelengths.\n\nThe overall GVD of a guided mode in an optical fiber depends not only on the material dispersion, but also on the waveguide dispersion . The last contribution becomes especially important in the case of fibers using a large difference in refractive index between the core and cladding materials. Besides affecting the dispersion characteristics, increasing such refractive index difference provides also an enhancement of the nonlinear parameter of the fiber, as discussed previously.\n\nMicrostructured optical fibers\n\nMicrostructured optical fibers (MOFs) represent a new class of optical fibers that are characterized by the fact that the silica cladding presents an array of embedded air holes. They are also referred as photonic crystal fibers (PCFs), since they were first realized in 1996 in the form of a photonic crystal cladding with a periodic array of air holes .\n\nMicrostructured fibers can be divided in two main types. One class of fibers, first proposed in 1999 , have a central region containing air. Such fibers are usually called hollow-core MOFs and the light propagating in them is confined to the core by the photonic bandgap effect. The nonlinear effects are strongly reduced and the dispersion becomes negligible in this kind of MOFs. A nonlinear coefficient =0.023 W-1/km was reported recently for a hollow core fiber . However, the nonlinear effects in this type of MOFs can be greatly enhanced if air is replaced with a suitable gas or liquid .\n\nThe second type of microstructured fibers has a solid core, in which the light is guided due to total internal reflection, since the core has a higher refractive index than the cladding. A solid core MOF with an hexagonal pattern of holes is represented in Figure 1 has two main structural parameters: the hair-hole diameter d and the holeto- hole spacing ∧. In such fibers the periodic nature of air holes in the cladding is not important as long as they provide an effective reduction of its refractive index below that of the silica core [21,22]. This helps to concentrate the mode field in a very small area, which is particularly the case in MOFs with and large air-filling fractions. The tapered fiber considered above can be used to model the behavior of the fundamental mode of air-silica microstructured optical fibers with a high air-filling fraction in their cladding .\n\nFigure 1: Schematic of a solid-core microstructured fiber with a hexagonal geometry. The two main parameters are the hair-hole diameter d and the hole-to-hole spacing Λ.\n\nFigure 2a shows the spatial mode distribution for a microstructured fiber with a hexagonal pattern of holes of diameter d = 0.5μm, spaced by ∧ = 10 μm, considering a wavelength λ=1.55 μm. Figure 2b shows the normalized effective mode area against the normalized wavelength for different hole-pitch ratios. We observe that increasing the hole-pitch ratio the effective area is reduced. Consequently, the nonlinear parameter is increased, as given by Equation (1).\n\nFigure 2: (a) Spatial mode distribution for a microstructured fiber with d=0.5 μm and ∧ = 10 μm. (b) Normalized effective area against the normalized wavelength for different hole-pitch ratios.\n\nIn the case of microstructured fibers and to take into account of the different proportions of light in glass and air, the nonlinear coefficient λ must be redefined as follows :", null, "(8)\n\nIn Equation (11)", null, "is the nonlinear refractive index of material i (2.9×10-23 m2W-1 for air, 2.6×10-20 m2W-1 for silica),", null, "is the effective area for the light in material i, and", null, "is the effective nonlinear index for fiber, with a core area Acore. A nonlinear coefficient λ= 240 W-1 /km at 850 nm was measured for a solid-core MF with a core diameter of 1μm .\n\nA peak power of P0=5kW launched into a silica MOF with a nonlinear coefficient λ=240W-1/km provides a nonlinear length LNL=(λP0)-1<1mm. For dispersion values in the range -400<β2<400ps2/ km and pulse durations T0=300fs, we have a dispersion length", null, ". Considering also typical values of fiber loss (α~1-100dB/km), we conclude that both the effective fiber length Leff = (1-exp(-αL)) and the dispersion length LD are much longer than the nonlinear length LNL, which means that strong nonlinear effects will be readily observable in solid-core MOFs.\n\nThe microstructured cladding offers greatly enhanced design flexibility and can manipulate the dispersion characteristics by controlling structural parameters such as the hair-hole diameter d and the hole-to-hole spacing Λ . In fact, the dispersive properties of MOFs are quite sensitive to these parameters and can be tailored by changing appropriately each of them. Figure 3 shows the dispersion characteristics of a microstructured fiber with a hexagonal pattern of holes spaced by (a) ∧=2 μm, and (b) ∧=1 μm, for different hole-pitch ratios.\n\nIn solid-core MOF, as the holes get larger, the core becomes more and more isolated, until it resembles an isolated strand of silica glass. The core diameter of a MOF is given by Dcore = 2∧-d. MOFs with larger cores exhibit semi-infinite anomalous dispersion above the ZDW. By decreasing the core size, the ZDW tends to be shifted to a shorter wavelength, leading to the anomalous dispersion at near infrared and visible wavelengths. When the core size is decreased further, a second ZDW arises in the longer wavelength side, such that the GVD is anomalous in the spectral window between the two ZDWs and normal outside it. This situation can be seen in some cases of Figure 3a, 3b. Submicron-diameter MOF cores have been fabricated using a conventional tapering process .\n\nFigure 3: Dispersion properties of a microstructured fiber with a hexagonal pattern of holes spaced by by (a) ∧ = 2 μm, and (b) ∧ = 1 μm, for different hole-pitch ratios.\n\n## Modelling of the Supercontinuum\n\nModelling the supercontinuum generation can be realized considering a generalized NLSE that includes higher-order dispersion effects, as well as intrapulse Raman scattering. Such equation can be written as (1).", null, "(9)\n\nwhere U(z,t) is the electric field envelope, ω0 is the center frequency, βk are the dispersion coefficients at the center frequency, α(ω) is the frequency-dependent fiber loss, and λ is the nonlinear parameter.\n\nThe nonlinear response function R(t) in Equation (9) can be written as", null, "(10)\n\nwhere the δ function represents the instantaneous electron response (responsible for the Kerr effect), h(t) represents the delayed ionic response (responsible for the Raman scattering) and fR is the fractional contribution of the delayed Raman response to the nonlinear polarization, in which a value 0.18 fR = is often assumed . It is common to approximate h(t) in the form [28,29]", null, "(11)\n\nwhere τ1 =12.2 fs and τ2 = 32 fs. More accurate forms of the response function h(t) have also recently been investigated .\n\nEquation (9) can be used to describe the propagation of femtosecond pulses in optical fibers, in both the normal and anomalous dispersion regimes. When such pulses have enough power, their spectra undergo extreme broadening. In the anomalous dispersion regime, this process is mainly influenced by the phenomenon of soliton fission, which occurs whenever a higher-order soliton is affected by third- or higher-order dispersion. The soliton order N is given by [1,28]", null, "(12)\n\nwhere", null, "is the dispersion distance and LNL=1/λP0 is the nonlinear length.\n\nIn the presence of higher-order dispersion, an Nth-order soliton gives origin to N fundamental solitons whose widths and peak powers are given by", null, "(13)", null, "(14)\n\nwhere k = 1 to Ñ, where Ñ is the integer closest to N when N is not an integer. Soliton fission occurs generally after propagation distance Lfiss ∼ LD/N, at which the injected higher-order soliton attains its maximum bandwidth. The fission distance Lfiss is a particularly significant parameter in the context higher-order soliton effect compression [32,33]\n\nBesides higher-order dispersion, another main effect affecting the dynamics of a higher-order soliton fission is the intrapulse Raman scattering (IRS). This phenomenon leads to a continuous downshift of the soliton carrier frequency, an effect known as the soliton selffrequency shift (SSFS) . Such effect was observed for the first time by Mitschke and Mollenauer in 1986 . The origin of SSFS can be understood by noting that for ultrashort solitons the pulse spectrum becomes so broad that the high-frequency components of the pulse can transfer energy through Raman amplification to the lowfrequency components of the same pulse. Such an energy transfer appears as a red shift of the soliton spectrum, with shift increasing with distance.\n\nThe rate of frequency shift per propagation length is given by", null, "(15)\n\nwhere", null, "is the Raman parameter andis the soliton peak power. Since the SSFS effect is proportional to (λP0)2, it will be enhanced if short pulses with high peak pulses are propagated in highly nonlinear fibers.\n\nThe fission phenomenon can be observed in Figure 4, which shows the temporal and spectral evolution of an optical pulse along a highly nonlinear MOF with hole diameter d=1.4 μm and pitch =1.6 μm. In this case, the dispersion curve has only one zero dispersion wavelength, located at 735 nm. We consider a pump in the anomalous dispersion region at 790 nm, where the nonlinear parameter is λ =117W-1km-1. An input pulse", null, "is assumed, where P0=5kW and T0=28.4fs, which corresponds to an intensity full width at half maximum (FWHM) of 50 fs. For the assumed parameters, we have N=6.02.\n\nFigure 4: (a) Spectral and (b) temporal evolution of an optical pulse along a microstructured fiber obtained from Equation (9). The input sech pulse at 790 nm has a peak power of 5 kW and 50 fs FWHM.\n\nA clear signature of soliton fission is the appearance of a new spectral peak in Figure 4a at z = Lfiss. The new spectral peak observed corresponds to the so-called nonsoliton radiation (NSR) or Cherenkov radiation which is emitted by the solitons resulting from the fission process in the presence of higher-order dispersion. When the third-order dispersion coefficient (β3),the NSR is emitted at wavelengths shorter than that of the soliton, as observed in Figure 4a. Regarding the time domain, Figure 4b shows that a dispersive wave spreads with propagation and lags behind the main soliton.\n\nAnother noticeable feature in Figure 4 is the red-shift of the solitons created by the fission process. As a consequence of the SSFS induced by intrapulse Raman scattering, such solitons separate from each other. Since the SSFS is the largest for the shortest soliton, its spectrum shifts the most toward the red side in Figure 4a. The change of the soliton´s frequency determines a reduction in the soliton´s speed because of dispersion. This deceleration appears as a bending of the soliton trajectory in the time domain, as observed in Figure 4b.\n\nFigure 5 shows the initial and final pulse profiles, both in frequency and time domains. We observe that a nearly uniform supercontinuum is achieved between 500 nm and 1100nm. These features are particularly important for to achieve high performance OCT. In fact, the extreme nonlinear spectral broadening provided by SCG permits a high resolution OCT, while the smooth spectral profile avoids the appearance of side-lobes on the axial point spread function.\n\nFigure 5: Initial and final pulse profiles in (a) frequency and (b) time domains for the case of Figure 4.\n\n## Conclusion\n\nThe peculiar dispersive characteristics of both tapered and microstrutured fibers make them particularly suitable to generate the supercontinuum using commercially available pump sources around 1060 nm and 800 nm. Pumping with femtosecond pulses in the anomalous dispersion region of the fiber, the soliton fission process gives origin to multiple fundamental solitons of different widths and peak powers. Among the host of soliton related effects, there are two which become then particularly important: the SSFS induced by the Raman scattering and the emission of dispersive radiation. Solitons dominate the long-wavelength edge of the supercontinuum, while the dispersive radiation determines the short wavelength expansion. As a result, a wide and nearly uniform supercontinuum can be achieved. This is important for several applications, namely in the area of optical coherence tomography.\n\n## Acknowledgments\n\nThe authors acknowledge FCT (Fundação para a Ciência e Tecnologia), FEDER, and the program COMPETE for financial support through project Pest-C/CTM/LA0025/2011.\n\n## Recommended Conferences\n\n### International Conference on Photonics & Optoelectronics\n\nSingapore, Singapore" ]

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[ "## Review of Levy Flights\n\nIf Levy flights occur, the variance increases faster than just linearly with time. The variance denotes the typical size of a group of random walkers, and is the average of the squares of the distance moved minus square of the average of the distance moved. D is the rate at which the variance grows. Diffusion is faster in watery hemolymph (large diffusion constant) than in hemolymph of high osmolality (small diffusion constant). Volume changes in the hemocoel complicate these relationships.\n\nIf variance equals time raised to some exponent, the exponent being one for normal diffusion, this exponent is larger than one, but usually less than two, during a Levy flight. Superdiffusion is when the exponent is greater than one but less than two. Having an exponent of two would mean that all random walkers were moving apart from each other at a constant rate as might occur in an explosion. We still must learn what role, if any, Levy flight dynamics and superdiffusion play in the bee's hemocoel (Ref: Levy Flight Dynamics).\n\n0 0" ]

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[ "# 70.79 percent as a fraction\n\nHere you will see step by step solution to convert 70.79 percent into fraction. 70.79 percent as a fraction is 7079/10000. Please check the explanation that how to write 70.79% as a fraction.\n\n## Answer: 70.79% as a fraction is\n\n= 7079/10000\n\n### How to convert 70.79 percent to fraction?\n\nTo convert the 70.79% as a fraction form simply divide the number by 100 and simplify the fraction by finding GCF. If given number is in decimal form then multiply numerator and denominator by 10^n and simplify it as much as possible.\n\n#### How to write 70.79% as a fraction?\n\nFollow these easy steps to convert 70.79% into fraction-\n\nGiven number is => 70.79\n\n• Write down the 70.79 in a percentage form like this:\n• 70.79% = 70.79/100\n• Since, 70.79 is not a whole number, now we need multiply numerator and denominator by 10^n, n = decimal points in number. [n=2]\n• => 70.79 × 100/100 × 100 = 7079/10000\n• We also need to check to simplify the fraction.\n• Greatest common factor [GCF] of 7079 and 10000 is 1, so this is the simplest form is 7079/10000.\n• Conclusion: 70.79% = 7079/10000\n\nTherefore, the 70.79 percent as a fraction, final answer is 7079/10000." ]

[ null ]

[ "", null, "# Differential Geometry: A First Course in Curves and Surfaces", null, "Differential Geometry: A First Course in Curves and Surfaces\nby\n\nPublisher: University of Georgia\nNumber of pages: 127\n\nDescription:\nContents: Curves (Examples, Arclength Parametrization, Local Theory: Frenet Frame, Some Global Results), Surfaces: Local Theory (Parametrized Surfaces and the First Fundamental Form, The Gauss Map and the Second Fundamental Form, The Codazzi and Gauss Equations, Covariant Differentiation, Parallel Translation, and Geodesics) Surfaces: Further Topics (Holonomy and the Gauss-Bonnet Theorem, Hyperbolic Geometry, Surface Theory with Differential Forms, Calculus of Variations and Surfaces of Constant Mean Curvature).\n\n(1.8MB, PDF)\n\n## Similar books", null, "Differential Geometry: A Geometric Introduction\nby - Project Euclid\nThis is the only book that introduces differential geometry through a combination of an intuitive geometric foundation, a rigorous connection with the standard formalisms, computer exercises with Maple, and a problems-based approach.\n(199 views)", null, "Notes on Differential Geometry\nby - Van Nostrand\nA concise introduction to differential geometry. The ten chapters of Hicks' book contain most of the mathematics that has become the standard background for not only differential geometry, but also much of modern theoretical physics and cosmology.\n(10086 views)", null, "Differential Geometry Of Three Dimensions\nby - Cambridge University Press\nThe book is devoted to differential invariants for a surface and their applications. By the use of vector methods the presentation is both simplified and condensed, and students are encouraged to reason geometrically rather than analytically.\n(5133 views)", null, "A Course Of Differential Geometry\nby - Clarendon Press\nContents: Tensor theory; The ground form when n=2; Geodesics in two-way space; Two-way space as a locus in Euclidean space; Deformation of a surface and congruences; Curves in Euclidean space and on a surface; The ruled surface; Minimal surface; etc.\n(3207 views)" ]

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[ "# 一、认识HelloWorld\n\n``````% hello world program\n-module(helloworld).\n-export([start/0]).\n\nstart() ->\nio:fwrite(\"Hello, world!\\n\").``````\n\n`%`：注释\n\n`-module(模块名)`：定义模块，模块名必须和文件名一致（去掉后缀.erl），否则编译会出错。\n\n`-export([函数名/参数个数]) `：定义外部可以调用的函数，斜杠后面是参数个数\n\n`start()`：一个函数，名字任意取\n\n`io:fwrite`：调用io模块的fwrite函数，输出消息到控制台\n\n`.`：语句结束符\n\n# 二、数据类型\n\n## 1、标准数据类型\n\nErlang有7种标准数据类型，数字、原子、布尔、位字符串、元组、映射、列表\n\n1）数字 number\n\n【进制整数】可以用数字+“#”表示2-36的进制整数，例如emqttd_session.erl第824行中`16#FFFF`\n\n【整数范围】Erlang的则整数精度只受到内存限制，支持大数运算。\n\n``````start() ->\nA = 1234567890*99999999999999999999,\nio:fwrite(\"~B~n\",[A]).\n\n% 输出123456788999999999998765432110``````\n\n【浮点数的科学计数法】和其他语言类似，用e/E后面带+/-和指数：\n\n``````start() ->\nA = 3.468e+3,\nio:fwrite(\"~f~n\",[A]).\n\n% 输出3468.000000``````\n\n【ASCII】可以用\\$char来表达ASCII的数字，如`\\$A`值为65，`\\$\\n`值为10。\n\n2）原子 atom", null, "``````start() ->\nA = ok,\nB = error,\nC = 'Error',\nD = a@12_3Axf,\nE = 'error ans',\nio:fwrite(\"~w~n\",[A]).``````\n\n【运算】原子的意义就是文字，只支持比较和赋值运算\n\n【创建】出现即创建，除非系统重启否则不会被清除，单系统最大可用原子数1048576个。\n\nEMQ通常使用原子来定义一些字符串常量，例如emqttd.hrl第43行自定义类型时，取值只能为两个原子：\n\n``-type(pubsub() :: publish | subscribe).``\n\nemqttd_protocol.hrl第101行定义MQTT报文名称时，使用了原子：\n\n``````-define(TYPE_NAMES, [\n'CONNECT',\n……``````\n\nemqttd_trie.erl第156行删除订阅树边时，返回的原子：\n\n``````delete_path([]) ->\nok;``````\n\nemqttd_trie.erl第82行，从根节点遍历订阅树时，根节点用的原子：\n\n``````match(Topic) when is_binary(Topic) ->\nTrieNodes = match_node(root, emqttd_topic:words(Topic)),``````\n\n3）布尔 boolean\n\n4）位字符串 binary\n\n``<<E1, ..., EN>>``\n\n``````Value % 必须是整数、浮点数或另一个位串\nValue:Size % 指定数据的二进制位数\nValue/TypeSpecifierList % 指定数据的类型\nValue:Size/TypeSpecifierList % 指定数据的位数和类型``````\n\n<<\"abc\">>是<<\\$a,\\$b,\\$c>>的语法糖，在EMQ超级常用，例如emqttd_packet.erl第34行：\n\n``protocol_name(?MQTT_PROTO_V5) -> <<\"MQTT\">>.``\n\n<<1:3, 2:4>>表达式中的位串位数是3+4=7位，二进制被拼接成0b0010010，所以最终得到的值是<<18:7>>。\n\n<<1,17,42:12>>表达式中的位串位数是8+8+12=28位，二进制高位用0填充，整个串为0b 0000 0010 1010，所以最终得到的值是<<2,10:4>>\n\n``````start() ->\nA = <<1:3, 2:4>>,\nB = <<18:7>>,\nC = A == B,\nio:fwrite(\"~w ~w ~w ~n\",[A,B,C]).``````\n\nTypeSpecifierList可以控制编码细节，控制参数用横杠隔开：\n\n• 数据类型（Type）：integer、float、bytes/binary、bits/bitstring、utf8、utf16、utf32，其中bytes是binary缩写，bits是bitstring的缩写，作用相同。\n• 符号（Signedness）：signed、unsigned，默认无符号。\n• 大小端（Endianness）：big、little、native，默认大端。\n• 单位（Unit）：unit:1~256，默认整数、浮点数都是1bit为单位，位串8bit为单位。\n\n``````-define( UINT, 32/unsigned-little-integer).\n-define( INT, 32/signed-little-integer).\n-define( USHORT, 16/unsigned-little-integer).\n-define( SHORT, 16/signed-little-integer).\n-define( UBYTE, 8/unsigned-little-integer).\n-define( BYTE, 8/signed-little-integer).\n-define( CHAR, 1/binary-unit:8).``````\n\n``````validate3(<<C/utf8, _Rest/binary>>) when C == \\$#; C == \\$+; C == 0 ->\nfalse;``````\n\n5）元组 tuple\n\n``A={Term1, ..., TermN}.``\n\n``{A, B, C, _} = {apple, banana, orange, hhhhhh}``\n\n``````% emqttd_protocol.erl第493行，调用函数，发送参数：\ncase emqttd_topic:validate({name, Topic}) of\ntrue -> ok;\nend;\n% emqttd_topic.erl第91行，接受参数：\nvalidate({filter, Topic}) when is_binary(Topic) ->\nvalidate2(words(Topic));``````\n\n6）映射 map\n\nmap是k-v映射表。\n\n``````% 定义map\nmymap = #{Key1=>Value1,...,KeyN=>ValueN}\n% 获取值\nmyvalue = maps:get(Key1,mymap)\n% 设置数据\nmaps:put(Value1, Value2, ..., ValueN)\n% 更新数据\nmaps:update(Value1, Value2, ..., ValueN)``````\n\n``maps:put(Topic, NewQos, SubMap);``\n\n7）列表 list\n\n``[Term1,...,TermN]``\n\n``-export([start/0]).``\n\n``````[H|T]=[apple, banana]\n% H = apple\n% T = [banana]``````\n\n``````validate({_, <<>>}) ->\nfalse;\nvalidate({_, Topic}) when is_binary(Topic) and (size(Topic) > ?MAX_TOPIC_LEN) ->\nfalse;\nvalidate({filter, Topic}) when is_binary(Topic) ->\nvalidate2(words(Topic));\nvalidate({name, Topic}) when is_binary(Topic) ->\nWords = words(Topic),\nvalidate2(Words) and (not wildcard(Words)).\n\nvalidate2([]) ->\ntrue;\nvalidate2(['#']) -> % end with '#'\ntrue;\nvalidate2(['#'|Words]) when length(Words) > 0 ->\nfalse;\nvalidate2([''|Words]) ->\nvalidate2(Words);\nvalidate2(['+'|Words]) ->\nvalidate2(Words);\nvalidate2([W|Words]) ->\ncase validate3(W) of true -> validate2(Words); false -> false end.``````\n\n## 2、高级数据类型\n\n1）记录 record\n\n``````-record(Name, {Field1 [= Value1],\n...\nFieldN [= ValueN]}).``````\n\n``R = #Name{Field1=Expr1,...,FieldK=ExprK}``\n\n``R2 = R#Name{Field1=Expr1,...,FieldM=ExprM}``\n\n``#Name.Field``\n\n``````terminate(_Reason, #state{stats_timer = TRef}) ->\ntimer:cancel(TRef),\nekka:unmonitor(membership).``````\n\n``````% 定义：emqttd.hrl 第148行\n-record(mqtt_delivery,\n{ sender :: pid(), %% Pid of the sender/publisher\nmessage :: mqtt_message(), %% Message\nflows :: list()\n}).\n\n% 创建：emqttd_pubsub.erl第89行\ndelivery(Msg) -> #mqtt_delivery{sender = self(), message = Msg, flows = []}.\n\n% 更新：eqmttd_pubsub.erl第78行\ndispatch(To, Delivery#mqtt_delivery{flows = [{route, Node, To} | Flows]});\n\n% 读值：eqmttd_pubsub.erl第72行\nroute([], #mqtt_delivery{message = #mqtt_message{topic = Topic}}) ->\ndropped(Topic), ignore;``````\n\n2）进程ID Pid\n\n``````-record(mqtt_session,\n{ client_id :: binary(),\nsess_pid :: pid(),\nclean_sess :: boolean()\n}).``````\n\n``SubPid ! {dispatch, Topic, Msg};``\n\n`self() `可以获取自己的进程ID。\n\n3）引用 Reference\n\n``Tag = make_ref()``\n\n4）函数 function\n\n``````parse(<<\"\\$local/\", Topic1/binary>>, Options) ->\nif_not_contain(local, Options, fun() ->\nparse(Topic1, [local | Options])\nend);``````\n\n``````if_not_contain(local, Options, Fun) ->\n?IF(lists:member(local, Options), error(invalid_topic), Fun());``````\n\n# 三、运算符\n\n+ 加\n- 减\n* 乘\n/ 除\nrem 取余（模）\ndiv 整除\n\n## 2、关系运算符\n\n== 相等\n/= 不等 注意不等号\n< 小于\n=< 小于等于 注意先等后小\n> 大于\n>= 大于等于\n=:= 恒等于 重点\n=/= 恒不等于 重点\n\n1）恒等于和恒不等于\n\n``````start() ->\nA = 1 == 1.0,\nB = 1 =:= 1.0,\nC = 1 /= 1.0,\nD = 1 =/= 1.0,\nio:fwrite(\"1==1.0: ~w~n1=:=1.0: ~w~n1/=1.0: ~w~n1=/=1.0: ~w~n\",[A,B,C,D]).\n\n% 1==1.0: true\n% 1=:=1.0: false\n% 1/=1.0: false\n% 1=/=1.0: true``````\n\n``````% emqttd_pubsub.erl第77行\nNode =:= node()\n% emqttd_trie.erl第83行\nName =/= undefined``````\n\n2）不同数据类型做比较\n\nnumber < atom < reference < fun < port < pid < tuple < list < bitstring\n\nband 按位与\nbor 按位或\nbxor 按位异或\nbnot 按位非\n\n## 4、逻辑布尔运算符\n\nor 或\nand 与\nnot 非\nxor 异或\nandalso 短路与\norelse 短路或\n\n``````% eqmttd_serializer.erl第43行：\nwhen ?CONNECT =< Type andalso Type =< ?DISCONNECT ->\n% emqttd_protocol.erl第427行：\nwhen NullId =:= undefined orelse NullId =:= <<>> ->``````\n\n# 五、函数和递归\n\n``````[Name](Pattern11,...,Pattern1N) [when GuardSeq1] ->\nBody1;\n...;\n[Name](PatternK1,...,Pa<code>tternKN) [when GuardSeqK] ->\nBodyK.``````\n\n1）简单函数示例\n\n``````start() ->\nA = 1<2.``````\n\n2）多子句函数示例\n\n``````start() ->\nA = 1<2,\nio:fwrite(\"~w~n\",[A]).``````\n\n3）带保护序列函数示例\n\n``````add(X) when X>3 ->\nio:fwrite(\"~w~n\",[X]).\n\nstart() ->\n\n4）匿名函数\n\n``````StatFun = fun() ->\ncase Conn:getstat([recv_oct]) of\n{ok, [{recv_oct, RecvOct}]} -> {ok, RecvOct};\n{error, Error} -> {error, Error}\nend\nend\n\n% 作为参数传递：emqttd_keepalive:start(StatFun, Interval, {keepalive, check})\n% 直接调用：StatFun()``````\n\n5）多函数声明执行过程示例\n\n``````validate2([]) ->\ntrue;\nvalidate2(['#']) -> % end with '#'\ntrue;\nvalidate2(['#'|Words]) when length(Words) > 0 ->\nfalse;\nvalidate2([''|Words]) ->\nvalidate2(Words);\nvalidate2(['+'|Words]) ->\nvalidate2(Words);\nvalidate2([W|Words]) ->\ncase validate3(W) of true -> validate2(Words); false -> false end.``````\n\n``````validate2(['#']) -> % end with '#'\ntrue;\nvalidate2(['#']) -> % end with '#'\nfalse.``````\n\n• 如果为空，返回true，合法主题，\n• 如果为\"#\"且是最后一个，返回true，合法主题，\n• 如果为\"#\"且后面还有层级，返回false，非法主题，\n• 如果为空层，继续匹配，\n• 如果为\"+\"，继续匹配，\n• 如果为任意字符串，则调用validate3看里面是否有通配符（如果有则是非法主题，通配符必须独占层级）。\n\n6）递归\n\n# 六、宏定义\n\n1）宏定义常量示例\n\n``````% -define(Const, Replacement).\n-define(PROTOCOL_VERSION, \"MQTT/5.0\").``````\n\n2）宏定义函数示例（支持函数重载）\n\n``````% -define(Func(Var1,...,VarN), Replacement).\n-define(ALLOW_DENY(A), ((A =:= allow) orelse (A =:= deny))).``````\n\n3）使用宏定义示例\n\n``````% ?Const\n% ?Func(Arg1,...,ArgN)\ncompile({A, all}) when ?ALLOW_DENY(A) ->\n{A, all};``````\n\n4）系统预定义宏\n\n``````% 当前模块名\n?MODULE\n% 当前模块名，字符串格式\n?MODULE_STRING.\n% 当前模块的文件名\n?FILE.\n% 当前行号\n?LINE.\n% 当前机器名称\n?MACHINE.\n% 当前函数名称\n?FUNCTION_NAME\n% 当前函数有多少个参数\n?FUNCTION_ARITY\n% OTP版本号\n?OTP_RELEASE``````\n\n5）控制宏\n\n``````% 如果定义了宏，则\n-ifdef(Macro).\n% 如果未定义宏，则\n-ifndef(Macro).\n% 否则\n-else.\n% 结束控制宏流程\n-endif.\n% 如果条件为真，则\n-if(Condition).\n% 否则如果条件2为真，则\n-elif(Condition).\n% 让宏表现得像没有被定义过一样\n-undef(Macro).``````\n\n# 七、行为 Behaviour\n\nErlang/OTP有四个经典behaviour：gen_server、gen_fsm、gen_event、supervisor，在前文《EMQ源码分析（二）：Erlang相关概念》已经描述过了：\n\n• gen_server：用于实现C/S结构中的服务端\n• gen_fsm：用于实现有限状态机\n• gen_event：用于实现事件处理功能\n• supervisor：用于实现监督树中的监督进程\n\n``-behaviour(gen_server).``\n\n# 八、其他EMQ中常见Erlang语法\n\n1）定义模块名\n\n``-module(emqttd_router).``\n\n2）定义作者\n\n``-author(\"Feng Lee <[email protected]>\").``\n\n3）引入模块或头文件\n\n``-include(\"emqttd.hrl\").``\n\n4）定义函数声明\n\n``````%% @doc Match Routes.\n-spec(match(Topic:: binary()) -> [mqtt_route()]).\nmatch(Topic) when is_binary(Topic) ->\n%% Optimize: ets???\nMatched = mnesia:ets(fun emqttd_trie:match/1, [Topic]),\n%% Optimize: route table will be replicated to all nodes.\nlists:append([ets:lookup(mqtt_route, To) || To <- [Topic | Matched]]).``````\n\n5）自定义数据类型\n\n``-type(mqtt_qos() :: ?QOS0 | ?QOS1 | ?QOS2).``" ]

[ null, "http://cdn.bewindoweb.com/uploadpic/95333cd2ad4ba1417d108b30f3a041b8.png", null ]

[ "# Ask the Engineer- How do I calculate power from 3-phase LED readings?\n\nAnnie Paquette\nOctober 09, 2010\n\nQuestion from Steven G.: How do I calculate the power usage for devices attached to one of your Smart PDUs by looking at the digital readouts on them? I understand the 3-phase power calculation of P=VA*1.73, but is that done on each leg/phase (x,y,z) for the PDU? An example from a CS-24VY-L30MA is X=4.6, Y=4.6, Z=6.3\n\nAnswer: the shortcut way to find power consumption is (X+Y+Z)*120V = 1860W for your example. This is only accurate when loads are fairly well balanced. It turns out that even though one leg is 37% higher than the other two legs, your example is balanced well enough that the power calculation is only 1.6% overestimated.\n\nThe more accurate calculation of 1831W must be found by first calculating the outlet branch amperages (XY, YZ, ZX) through an iterative process using the equations:\n\nX = sqrt([XY]*[XY] + [ZX]*[ZX] + [XY]*[ZX])\n\nY = sqrt([XY]*[XY] + [YZ]*[YZ] + [XY]*[YZ])\n\nZ = sqrt([YZ]*[YZ] + [ZX]*[ZX] + [YZ]*[ZX])\n\nThese are derived from the law of cosines with the assumption that the three legs are mutually 120 degrees apart.\n\nAsk the Engineer- Is branch circuit protection on a 20 amp rated power strip required in North America?" ]

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[ "# Formula\n\nUpdated: 05/01/2023 by Computer Hope", null, "A formula is an expression telling the computer what mathematical operation to perform upon a specific value. When referring to computer software, formulas are used in spreadsheet programs, such as Microsoft Excel. Using formulas in spreadsheets can allow you to quickly make calculations and get totals of multiple cells, rows, or columns in a spreadsheet.\n\nIn the picture below is an example of a Microsoft Excel formula =SUM(A\\$1:A\\$3), which adds the total of cells A1, A2, and A3. In this formula, SUM is the function of the formula.", null, "Tip\n\nPress the keyboard shortcut key Shift+F3 to open the Excel Formula or Insert Function window.\n\nNote\n\nA formula is not be seen unless the cell is selected. If you want Excel or Google Sheets to always show you formulas instead of the values use the shortcut Ctrl+`." ]

[ null, "https://www.computerhope.com/jargon/f/function-f-of-x.png", null, "https://www.computerhope.com/jargon/f/formula.png", null ]

[ "# Solve for z square root of 5-z+3=6", null, "Move all terms not containing to the right side of the equation.\nSubtract from both sides of the equation.\nSubtract from .\nTo remove the radical on the left side of the equation, square both sides of the equation.\nSimplify each side of the equation.\nMultiply the exponents in .\nApply the power rule and multiply exponents, .\nCancel the common factor of .\nCancel the common factor.\nRewrite the expression.\nSimplify.\nRaise to the power of .\nSolve for .\nMove all terms not containing to the right side of the equation.\nSubtract from both sides of the equation.\nSubtract from .\nMultiply each term in by\nMultiply each term in by .\nMultiply .\nMultiply by .\nMultiply by .\nMultiply by .\nSolve for z square root of 5-z+3=6\n\n## Our Professionals", null, "### Robert Kristofer", null, "### Anna Frok", null, "### Magnus Flores", null, "### Lydia Fran\n\n#### We are MathExperts\n\nSolve all your Math Problems: https://elanyachtselection.com/\n\nScroll to top" ]

[ null, "https://jmservera.com/wp-content/uploads/ask60.png", null, "https://www.jmservera.com/wp-content/uploads/2022/07/Robert-Fox.jpg", null, "https://www.jmservera.com/wp-content/uploads/2022/07/Jessica-Alba.jpg", null, "https://www.jmservera.com/wp-content/uploads/2022/07/Albert-Flores.jpg", null, "https://www.jmservera.com/wp-content/uploads/2022/07/Lydia-Francii.jpg", null ]

[ "# C++ program to find the smallest of two numbers", null, "In this example, you will learn a C++ program to find the smallest of two numbers. This program takes two numbers from the user and displays the smallest value. If the user enters 4 and 7. The program finds the smallest from both which is 4.\n\n## Program: Find the Smallest of Two Numbers in C++\n\n``````// C++ program to find smallest of two numbers\n#include<iostream>\nusing namespace std;\n\nint main()\n{\nint number1, number2, smallest ;\ncout<<\"Enter first number: \";\ncin>>number1;\ncout<<\"Enter second number: \";\ncin>>number2;\n\nif(number1<number2)\n{\nsmallest=number1;\n}\nelse\n{\nsmallest=number2;\n}\ncout<<\"Smallest of the two number is: \"<<smallest;\nreturn 0;\n}``````\n\n## Description and working of this program\n\n• Take two numbers from the user and store them in “number1” and “number2”.\n• Condition if the first number is smaller than the second than smallest= number1\n• If the above condition is not true then smallest = number2\n• Print the smallest value on the screen." ]

[ null, "https://logictoprogram.com/wp-content/uploads/2022/03/Smallest-of-two-numbers-in-cpp.png", null ]

[ "PDA\n\nView Full Version : Excel help needed\n\nstep_andy\n17th March 2008, 04:29 PM\nFolks,\nI need some assistance here:\n\nI have a following problem on my hand where I need to calculate an exam score based on 3 assessments. Each assessment consist of number of tasks where a student is awarded a score for each task which then is used in calculations. The issue is, that I need to discard the lowest scoring assessment.\nSo.\nAssessment 1: consists of 5 tasks, eg score: 6 6 7 7 5 resulting in Assessment overall 6\nAssessment 2: consists of 5 tasks, eg score: 5 5 6 5 5 resulting in Assessment overall 5\nAssessment 3: consists of 5 tasks, eg score: 3 6 4 4 4 resulting in Assessment overall 4\n\nExam overall: (drop lowest, eg Assessment 3) and Assessments 1 and 2 are used to calculate the overall score.\n\nSeveral issues:\nThere are two equal low values - which one to drop?\nHow do I instruct excel to select the right task scores in calculation overall?\n\nteamshaw\n17th March 2008, 04:43 PM\nYou could possibly have the average calculated manually by using the sum of the range of scores - minimum of the range of scores (MIN function) then dividing this by 4.\n\ngeektechnu\n17th March 2008, 05:36 PM\nHere's a little example of how you could do something like this...\n\nLets say the 5 scores are in cells A1 to E5:\n\n=( SUM(A1:E5) - MIN(A1:E5) ) / ( COUNT(A1:E5) - 1 )\n\nMIN is used here to get the lowest number.\nThe equation then gets the average of 4 scores.\n\nHope you can adapt this to your needs.\n\nstep_andy\n18th March 2008, 09:47 AM\nThanks for the suggestions, but this isn't what I need.\nI will try from different angle ..\n\nI have 3 data arrays, A1; A2; A3 each consisting of collection of 5 values.\nI need to use 2 arrays with highest average values in calculating overall score.\n\nExample here\nhttp://img215.imageshack.us/img215/8025/examplebc0.th.jpg (http://img215.imageshack.us/my.php?image=examplebc0.jpg)\n\nSo depending on the values as calculated Average 1 in cell F2 and Average 2 in L2 and Average 3 in R2 I need to use 2 highest averages in calculating overall average.\nThen, I need to round the number based on values in the Arrays used on calculating the overall average; in this instance values In Array 2 and Array 3. I need to round up to 6 if the majority (85%) of the achieved scores are above 5.9 (so, 85% of 10 scores is 8 values must be above 5.9). In this instance 5.9 would be rounded down to 5.\n\nI have figured out the rounding part.\nThe problem is how do I select the right arrays to calculate my rounding!\n\nneo\n18th March 2008, 10:08 AM\nHi step_andy,\n\nHope this is more what you need. (Formulae shown in row 9 calculation cells.)\n\n(Edit: Cell T7 should have read \"Average of Top 2 Assessments\". Sorry about any possible confusion.)\n\nscokim\n18th March 2008, 10:09 AM\nDoes replacing A1:E5 in the suggested formula above with F2,L2,R2 work ?\n\nstep_andy\n18th March 2008, 01:28 PM\nHi Neo,\nThanks for that.\nWhilst this allows Excel to use 2 highest average values, how do I instruct Excel to look into respective arrays?\nIn your example, array 3 results in lowest Average, so values in the Arrays 1 and 2 should be used in determining if the average of 2 highest assessments should be rounded up or rounded down.\nNote, that I am not using arithmetic rounding, eg if above .5 it rounded up and if below, rounded down.\nI am looking for something that will say, if Average for Assessment 3 = min, use Arrays 1 & 2 in rounding calculation.\n\nneo\n18th March 2008, 04:06 PM\nHi step_andy,\n\nI am looking for something that will say, if Average for Assessment 3 = min, use Arrays 1 & 2 in rounding calculation.\nI thought that my example actually did that. I've updated the sample to include simple rounding (below).\n\nI'll explain how I've set it up, so you can perhaps help me understand the holes in my understanding of what you need...\n\nRow P: (Rounded) Average of Assessment 1\nRow Q: (Rounded) Average of Assessment 2\nRow R: (Rounded) Average of Assessment 3\n\nRow S: The lowest (Rounded) Assessment value. (If the three rounded average assessment values are 2, 3 and 2 respectively, then this value will be 2...doesn't explicitly matter if we're discarding Assessment 1 or or Assessment 3.)\n\nRow T: Adds ALL rounded average values from the three assessments, then removes the lowest rounded assessment value (as calculated in row S). This added value (here the rounded average of assessments 1 & 2) is divided by two to get the overall average assessment value (from the two 'included' assessments - 1 & 2).\n\n...I think this is what you're after.\n\nBUT - here is my \\$0.02:\n\nI would think it would be MUCH fairer to use the calculations from my first example (ie DON'T use the rounding).\n\nIf you don't round, you effectively find the average of the tasks from the 'included' assessments.\nWith rounding, you bias the results; finding the average of the rounded (up or down) averages of the 'included' assessments.I've made an example for two students, showing their non-rounded scores in yellow, and their rounded scores in green:" ]

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[ "", null, "# How to Implement JavaScript for Statistics Method\n\nOver these past few weeks, I’ve learned that JavaScript is an extremely useful programming language. In this blog, we will see how we can use JavaScript to implement statistical methods into our programs.", null, "As you can probably imagine, there exist a couple libraries for statistical methods inside JavaScript. One of them is Simple Statistics. Here’s the link for the documentation for this library:\nhttps://simplestatistics.org/\n\nI will give a basic demonstration of some of the simpler statistical methods.\n\n## min and max\n\n```\t\t\t\t```\nmin(x: Array<number>): number\n```\n```\n\n(max is written in the same way, just swap min with max)\n\nmin and max return the smallest/largest number in an array. The only parameter is that you input an array of numbers.\n\n## sum\n\n```\t\t\t\t```\nsum(x:Array<number>): number\n```\n```\n\nsum returns the sum of all the numbers in an array. Again, the only parameter is that you input an array of numbers.\n\n## mean\n\n```\t\t\t\t```\nmean(x: Array<number>): number\n```\n```\n\nMean returns the average of all the numbers in an array. The only parameter is that you input an array of numbers.\n\n## variance\n\n```\t\t\t\t```\nvariance(x: Array<number>): number\n```\n```\n\nVariance returns the sum of squared deviations from all the numbers in an array. The only parameter is that you input an array of numbers.\n\n## zScore\n\n```\t\t\t\t```\nzScore(x: number, mean: number, standardDeviation: number): number\n```\n```\n\nzScore measures the score above or below the mean. For this, you need only provide the x (observed value), the mean (mu), and the standard deviation (sigma).\n\nAs you can see from these examples, Simple Statistics is a very intuitive JavaScript library. If you have any interest in this library, you can install it with this terminal command (assuming you have node.js):\n\n```\t\t\t\t```\nnpm install --save simple-statistics\n```\n```\n\nAgain, here is the link to the documentation if you need it:\nhttps://simplestatistics.org/\n\nIn researching for this blog, I also found another library, jstat. This blog is already long enough, so I will not be delving into it, but here’s the documentation for that library if you need it:\nhttps://jstat.github.io/overview.html\n\nThis has been Warren Ball, data science intern at iSmile Technologies. Thanks for reading, I hope this helps.", null, "### Get free consultation from our tech experts", null, "", null, "## Related articles you may would like to read\n\n##### Azure DevOps for Data Science", null, "" ]

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[ "# Rational functions in real life applications\n\n### What are some real life Applications of Polynomial Functions", null, "\"Real world\" applications of rational functions. 16/02/2014 · Rational expressions in math can be used in a wide variety of ways in real life, including in areas like distance measurement. Find out how rational expressions can be used in real life with help, In What Real-Life Situations Would You Use Polynomials? Using Polynomials in Real World Real Life Situations for Polynomials Adding Polynomials Dividing Polynomials Solving Word Problems Using Polynomial Equation Polynomials and Factoring Describe a Real Life Situation that could be Modeled by a Function.\n\n### What Are Some Real-Life Examples of Polynomials\n\nReal World Examples for Rational Numbers Blogger. Rational Function Applications - Work and Rate The video explains application problems that use rational equations. Part 2 of 2. Examples: 1. One person can complete a task 8 hours sooner than another person., 21/05/2015 · When Would I Use Rational Equations in Real Life?. Part of the series: Equations. Rational equations are used in real life all the time, even if you don't realize it. Learn when you would use.\n\n10/05/2017 · SAT Math Test Prep Online Crash Course Algebra & Geometry Study Guide Review, Functions,Youtube - Duration: 2:28:48. The Organic Chemistry Tutor 1,244,023 views There are no real life applications of reciprocal functions As with any advanced math, whether you use it in your \"real life\" or not depends on whether you work in the engineering field or not.\n\nA2A. The most commonly used polynomial equation is a line. It is used all the time, as I'm sure you know. So let's go on to quadratic polynomials. These are in the form [math]y=ax^2+bx+c[/math], where a, b, and c are real constants. You'll be surp... That depends on what you mean with \"real-life\". You won't need polynomial functions to sell stuff at a supermarket, or to cut off a dead branch from your tree... but if you work in science and\n\nAbsolute Value Functions Real World Applications As you know Absolute Value is a way to eliminate negative numbers in situations in which they are not called for. For example, if your student walks to school and it is a 2 mile walk we would say he or she walked 2 miles. Polynomial Functions in Real Life To find the motion of a particle under the influence of gravity, this equation is used: The weight, w, of a sick patient can be modeled with this equation: x0 = initial position v0 = initial velocity a = acceleration due to gravity t = time\n\nAbsolute Value Functions Real World Applications As you know Absolute Value is a way to eliminate negative numbers in situations in which they are not called for. For example, if your student walks to school and it is a 2 mile walk we would say he or she walked 2 miles. Section 7.8 Applications of Rational Functions 737 Version: Fall2007 d(mi) v(mi/h) t(h) Upstream 150 32 −c? Downstream 150 32 + c? Table3. Entering the given data in a dis-tance,speed,andtimetable. Solvingd = vt forthetimet, t = d v. In the rst row of Table3, we have d = …\n\nDid you say functions? Let's focus on how rational functions can be used in the medical field Why is this important you ask? Now let's do the same for another function like logarithmic functions Let's look more into how logarithmic functions are used for pH levels But how does As a math teacher, I tend to get the class involved by finding real-life applications of the math- with functions and vertical asymptotes I am having trouble finding simple enough (rational) functions that describe real-life phenomena.\n\nA2A. The most commonly used polynomial equation is a line. It is used all the time, as I'm sure you know. So let's go on to quadratic polynomials. These are in the form [math]y=ax^2+bx+c[/math], where a, b, and c are real constants. You'll be surp... As a math teacher, I tend to get the class involved by finding real-life applications of the math- with functions and vertical asymptotes I am having trouble finding simple enough (rational) functions that describe real-life phenomena.\n\n16/02/2014 · Rational expressions in math can be used in a wide variety of ways in real life, including in areas like distance measurement. Find out how rational expressions can be used in real life with help That depends on what you mean with \"real-life\". You won't need polynomial functions to sell stuff at a supermarket, or to cut off a dead branch from your tree... but if you work in science and\n\nIt will help to understand where the numbers come from and why they can be used in real life situations, and how mathematics can be independent discipline no matter how many real life examples are there. Moreover, we should not move from integers to rational numbers without having another, perhaps deeper look, what numbers are, and their That helps learn the functions of each different method for solving. Jenni Coburn, IN Students struggling with all kinds of algebra problems find out that our software is a life-saver.\n\n06/08/2015 · Application Of Functions In Real Life Situations Rofus Dino Esquierdo. Loading... Unsubscribe from Rofus Dino Esquierdo? Cancel Unsubscribe. Working... Subscribe Subscribed Unsubscribe 72. Loading A \"real world\" application of rational functions is the \"Thin Lens Equation\" which relates focal length, object distance, and real image distance. It is as practical as it gets. Cameras, eyeballs, magnifying glasses, etc all operate using this principle in some capacity. The derivation of it is straight forward and only uses similar triangles\n\nThat helps learn the functions of each different method for solving. Jenni Coburn, IN Students struggling with all kinds of algebra problems find out that our software is a life-saver. 06/08/2015 · Application Of Functions In Real Life Situations Rofus Dino Esquierdo. Loading... Unsubscribe from Rofus Dino Esquierdo? Cancel Unsubscribe. Working... Subscribe Subscribed Unsubscribe 72. Loading\n\nFUNCTION! FUNCTIONS ALSO ALLOW US TO VISUALIZE RELATIONSHIPS IN TERMS OF GRAPHS, WHICH ARE MUCH EASIER TO READ AND INTERPRET THAN LIST OF NUMBERS Temperature as a function of various factors. Temperature is a very complicated function because it … Rational equations can be used to solve a variety of problems that involve rates, times and work. Using rational expressions and equations can help us answer questions about how to combine workers or machines to complete a job on schedule.\n\nAs a math teacher, I tend to get the class involved by finding real-life applications of the math- with functions and vertical asymptotes I am having trouble finding simple enough (rational) functions that describe real-life phenomena. A2A. The most commonly used polynomial equation is a line. It is used all the time, as I'm sure you know. So let's go on to quadratic polynomials. These are in the form [math]y=ax^2+bx+c[/math], where a, b, and c are real constants. You'll be surp...\n\nThere are no real life applications of reciprocal functions As with any advanced math, whether you use it in your \"real life\" or not depends on whether you work in the engineering field or not. In this lesson you will learn about rational functions, discontinuities and how we can use them to model real-life scenarios. Some examples will include average cost, percentage and mixture problems.\n\nHow long will a trip take if you travel at a certain speed? Use the relationship between speed and time to explore rational functions and discover asymptotes in the real world. Plan your 60-minute lesson in Math or Precalculus and Calculus with helpful tips from Hilary Yamtich In the case you actually have service with math and in particular with examples of real life rational expressions word problems or linear algebra come pay a visit to us at Rational-equations.com. We have got a huge amount of excellent reference material on topics ranging from variables to adding and subtracting fractions\n\nApplication of Rational Functions Application 1. This application is a Cost-Benefit Model.A utility company burns coal to generate electricity. The cost C (in dollars) of removing p amount (percent) of the smokestack pollutants is given by:. Is it possible for the company … Rational-equations.com contains helpful resources on real life applications of algebra, terms and subtracting polynomials and other math subject areas. Whenever you need to have guidance on syllabus for college or even common factor, Rational-equations.com is undoubtedly the perfect place to visit!\n\nWhat are the uses of rational numbers in real life? How about its your ‘birthday’ party and someone brings out a cake. There are 12 people at the party; all want some cake and, to be fair, you want to divide it so that everyone gets the same amount. So, each person gets one twelfth of the cake. How can we create functions to describe these real world situations? Students explore the behavior of rational functions that they generate. Plan your 90-minute lesson in Math or Precalculus and Calculus with helpful tips from Hilary Yamtich\n\n06/08/2015 · Application Of Functions In Real Life Situations Rofus Dino Esquierdo. Loading... Unsubscribe from Rofus Dino Esquierdo? Cancel Unsubscribe. Working... Subscribe Subscribed Unsubscribe 72. Loading Polynomial Functions in Real Life To find the motion of a particle under the influence of gravity, this equation is used: The weight, w, of a sick patient can be modeled with this equation: x0 = initial position v0 = initial velocity a = acceleration due to gravity t = time\n\nAs a math teacher, I tend to get the class involved by finding real-life applications of the math- with functions and vertical asymptotes I am having trouble finding simple enough (rational) functions that describe real-life phenomena. Continuous functions as we've introduced them here are just the tip of the iceberg. The field of point-set topology defines more general notions of open and closed sets and then defines continuity in terms of those open and closed sets. Continuous functions that take real numbers as inputs and give real numbers as outputs are just one kind of continuous function.\n\nTo apply the domain and range in real-world settings, we take a function that represents a real-world situation and then analyze what the domain and range represent in the function. This allows us Fractions are used in real life in many different ways, but they are most commonly used in the cooking, construction and science industries. Because fractions describe an object or substance that has been divided into different equal parts, fractions can be found almost anywhere.\n\nIn this lesson you will learn about rational functions, discontinuities and how we can use them to model real-life scenarios. Some examples will include average cost, percentage and mixture problems. When medicine is given overtime a certain amount is going to be absorbed in the body so we need to know the exact amount of the medicine that is existing in that body in a certain period of time. We will use the rational function in determining the concentration of the medicine\n\nA Real-life Application of Rational Functions Add Remove This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here! I am more talking visually, when you graph some rational equation, and you get a hole from the cancelled factors that can't be vertical asymptotes. I guess I am just looking for some real life example of a rational expression vs. something contrived in a textbook. Who would ever graph a rational expression?\n\nexamples of functions with vertical asymptotes in real life. Application of Rational Functions Application 1. This application is a Cost-Benefit Model.A utility company burns coal to generate electricity. The cost C (in dollars) of removing p amount (percent) of the smokestack pollutants is given by:. Is it possible for the company …, A rational exponent is an exponent in fraction form. Any expression containing the square root of a number is a radical expression. Both have real world applications in fields including architecture, carpentry, masonry, financial services, electrical engineering and sciences like biology..\n\n### Applications of Rational Functions YouTube", null, "A Real-life Application of Rational Functions. 18/03/2008 · Equations with Rational Expressions in the Real World? Hi all. Need help with the following question: Come up with a real-world example from your work or personal life of using an equation with a rational expression to solve a real-world problem. Please give the example, setup the problem, then solve it, explaining your steps., Polynomial Functions in Real Life To find the motion of a particle under the influence of gravity, this equation is used: The weight, w, of a sick patient can be modeled with this equation: x0 = initial position v0 = initial velocity a = acceleration due to gravity t = time.\n\nReal World Examples for Rational Numbers Blogger. 16/02/2014 · Rational expressions in math can be used in a wide variety of ways in real life, including in areas like distance measurement. Find out how rational expressions can be used in real life with help, A \"real world\" application of rational functions is the \"Thin Lens Equation\" which relates focal length, object distance, and real image distance. It is as practical as it gets. Cameras, eyeballs, magnifying glasses, etc all operate using this principle in some capacity. The derivation of it is straight forward and only uses similar triangles.\n\n### Function Application for the Real World Video & Lesson", null, "Real life application to the reciprocal function Answers. In this lesson, we'll recall what a function is and then look at applying functions in real life that involve both numbers and objects. We'll then learn how to solve problems using these functions. Section 7.8 Applications of Rational Functions 737 Version: Fall2007 d(mi) v(mi/h) t(h) Upstream 150 32 −c? Downstream 150 32 + c? Table3. Entering the given data in a dis-tance,speed,andtimetable. Solvingd = vt forthetimet, t = d v. In the rst row of Table3, we have d = ….", null, "• What are some real-life applications of rational algebraic\n• What are some real life Applications of Polynomial Functions\n• How Are Fractions Used in Real Life? Reference.com\n\n• FUNCTION! FUNCTIONS ALSO ALLOW US TO VISUALIZE RELATIONSHIPS IN TERMS OF GRAPHS, WHICH ARE MUCH EASIER TO READ AND INTERPRET THAN LIST OF NUMBERS Temperature as a function of various factors. Temperature is a very complicated function because it … In the Real World. Polynomials and rational expressions are used to express properties of the physical world, like how gravity works.If you need a funnier example of gravity at work (or not at work), check this out.. Polynomials can also be used to do approximation, something that's usually covered in …\n\nIt will help to understand where the numbers come from and why they can be used in real life situations, and how mathematics can be independent discipline no matter how many real life examples are there. Moreover, we should not move from integers to rational numbers without having another, perhaps deeper look, what numbers are, and their FUNCTION! FUNCTIONS ALSO ALLOW US TO VISUALIZE RELATIONSHIPS IN TERMS OF GRAPHS, WHICH ARE MUCH EASIER TO READ AND INTERPRET THAN LIST OF NUMBERS Temperature as a function of various factors. Temperature is a very complicated function because it …\n\n06/08/2015 · Application Of Functions In Real Life Situations Rofus Dino Esquierdo. Loading... Unsubscribe from Rofus Dino Esquierdo? Cancel Unsubscribe. Working... Subscribe Subscribed Unsubscribe 72. Loading That helps learn the functions of each different method for solving. Jenni Coburn, IN Students struggling with all kinds of algebra problems find out that our software is a life-saver.\n\nA2A. The most commonly used polynomial equation is a line. It is used all the time, as I'm sure you know. So let's go on to quadratic polynomials. These are in the form [math]y=ax^2+bx+c[/math], where a, b, and c are real constants. You'll be surp... How can we create functions to describe these real world situations? Students explore the behavior of rational functions that they generate. Plan your 90-minute lesson in Math or Precalculus and Calculus with helpful tips from Hilary Yamtich\n\nThe formulas of algebra are used every day in real life when distance needs to be determined, volumes in containers need to be figured out and when sale prices need to be calculated. When using a recipe and doubling it or cutting it in half, algebra is used to determine the exact amount of ingredients. Basic algebraic equations are used almost I am more talking visually, when you graph some rational equation, and you get a hole from the cancelled factors that can't be vertical asymptotes. I guess I am just looking for some real life example of a rational expression vs. something contrived in a textbook. Who would ever graph a rational expression?\n\nAbsolute Value Functions Real World Applications As you know Absolute Value is a way to eliminate negative numbers in situations in which they are not called for. For example, if your student walks to school and it is a 2 mile walk we would say he or she walked 2 miles. In the case you actually have service with math and in particular with examples of real life rational expressions word problems or linear algebra come pay a visit to us at Rational-equations.com. We have got a huge amount of excellent reference material on topics ranging from variables to adding and subtracting fractions\n\nA rational exponent is an exponent in fraction form. Any expression containing the square root of a number is a radical expression. Both have real world applications in fields including architecture, carpentry, masonry, financial services, electrical engineering and sciences like biology. A2A. The most commonly used polynomial equation is a line. It is used all the time, as I'm sure you know. So let's go on to quadratic polynomials. These are in the form [math]y=ax^2+bx+c[/math], where a, b, and c are real constants. You'll be surp...\n\n20/06/2016 · Applications of Rational Functions - Duration: 12:57. April Pena [East CTA] 2,561 Representing Real Life Situations using Functions and Piecewise Function - Duration: 7:56. Mariane Viterbo 06/08/2015 · Application Of Functions In Real Life Situations Rofus Dino Esquierdo. Loading... Unsubscribe from Rofus Dino Esquierdo? Cancel Unsubscribe. Working... Subscribe Subscribed Unsubscribe 72. Loading\n\nAs a math teacher, I tend to get the class involved by finding real-life applications of the math- with functions and vertical asymptotes I am having trouble finding simple enough (rational) functions that describe real-life phenomena. Rational Function Applications - Work and Rate The video explains application problems that use rational equations. Part 2 of 2. Examples: 1. One person can complete a task 8 hours sooner than another person.\n\nAs a math teacher, I tend to get the class involved by finding real-life applications of the math- with functions and vertical asymptotes I am having trouble finding simple enough (rational) functions that describe real-life phenomena. Rational equations can be used to solve a variety of problems that involve rates, times and work. Using rational expressions and equations can help us answer questions about how to combine workers or machines to complete a job on schedule.\n\n20/06/2016 · Applications of Rational Functions - Duration: 12:57. April Pena [East CTA] 2,561 Representing Real Life Situations using Functions and Piecewise Function - Duration: 7:56. Mariane Viterbo A Real-life Application of Rational Functions Add Remove This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!\n\n## Real life application to the reciprocal function Answers", null, "7.8 ApplicationsofRationalFunctions. Real Life Linear Model. Many everyday activities require the use of mathematical models, perhaps unconsciously. One difficulty with mathematical models lies in translating the real world application into an accurate mathematical representation. Example: Renting a Moving Van, Continuous functions as we've introduced them here are just the tip of the iceberg. The field of point-set topology defines more general notions of open and closed sets and then defines continuity in terms of those open and closed sets. Continuous functions that take real numbers as inputs and give real numbers as outputs are just one kind of continuous function..\n\n### Application of Rational Functions\n\nExamples of real life rational expressions word problems. 21/05/2015 · When Would I Use Rational Equations in Real Life?. Part of the series: Equations. Rational equations are used in real life all the time, even if you don't realize it. Learn when you would use, I am more talking visually, when you graph some rational equation, and you get a hole from the cancelled factors that can't be vertical asymptotes. I guess I am just looking for some real life example of a rational expression vs. something contrived in a textbook. Who would ever graph a rational expression?.\n\n16/02/2014 · Rational expressions in math can be used in a wide variety of ways in real life, including in areas like distance measurement. Find out how rational expressions can be used in real life with help Fractions are used in real life in many different ways, but they are most commonly used in the cooking, construction and science industries. Because fractions describe an object or substance that has been divided into different equal parts, fractions can be found almost anywhere.\n\nContinuous functions as we've introduced them here are just the tip of the iceberg. The field of point-set topology defines more general notions of open and closed sets and then defines continuity in terms of those open and closed sets. Continuous functions that take real numbers as inputs and give real numbers as outputs are just one kind of continuous function. Did you say functions? Let's focus on how rational functions can be used in the medical field Why is this important you ask? Now let's do the same for another function like logarithmic functions Let's look more into how logarithmic functions are used for pH levels But how does\n\nThe formulas of algebra are used every day in real life when distance needs to be determined, volumes in containers need to be figured out and when sale prices need to be calculated. When using a recipe and doubling it or cutting it in half, algebra is used to determine the exact amount of ingredients. Basic algebraic equations are used almost I am more talking visually, when you graph some rational equation, and you get a hole from the cancelled factors that can't be vertical asymptotes. I guess I am just looking for some real life example of a rational expression vs. something contrived in a textbook. Who would ever graph a rational expression?\n\nIn the case you actually have service with math and in particular with examples of real life rational expressions word problems or linear algebra come pay a visit to us at Rational-equations.com. We have got a huge amount of excellent reference material on topics ranging from variables to adding and subtracting fractions In the Real World. Polynomials and rational expressions are used to express properties of the physical world, like how gravity works.If you need a funnier example of gravity at work (or not at work), check this out.. Polynomials can also be used to do approximation, something that's usually covered in …\n\nWhat are the uses of rational numbers in real life? How about its your ‘birthday’ party and someone brings out a cake. There are 12 people at the party; all want some cake and, to be fair, you want to divide it so that everyone gets the same amount. So, each person gets one twelfth of the cake. As a math teacher, I tend to get the class involved by finding real-life applications of the math- with functions and vertical asymptotes I am having trouble finding simple enough (rational) functions that describe real-life phenomena.\n\n21/05/2015 · When Would I Use Rational Equations in Real Life?. Part of the series: Equations. Rational equations are used in real life all the time, even if you don't realize it. Learn when you would use Real Life Linear Model. Many everyday activities require the use of mathematical models, perhaps unconsciously. One difficulty with mathematical models lies in translating the real world application into an accurate mathematical representation. Example: Renting a Moving Van\n\nWhat Are Some Real-Life Examples of Polynomials? Credit: U.S. Army Corps of Engineers Europe District/CC-BY 2.0 The motion of an object that’s thrown 3m up at a velocity of 14 m/s can be described using the polynomial -5tsquared + 14t + 3 = 0. That helps learn the functions of each different method for solving. Jenni Coburn, IN Students struggling with all kinds of algebra problems find out that our software is a life-saver.\n\nFractions are used in real life in many different ways, but they are most commonly used in the cooking, construction and science industries. Because fractions describe an object or substance that has been divided into different equal parts, fractions can be found almost anywhere. Polynomial Functions in Real Life To find the motion of a particle under the influence of gravity, this equation is used: The weight, w, of a sick patient can be modeled with this equation: x0 = initial position v0 = initial velocity a = acceleration due to gravity t = time\n\nA Real-life Application of Rational Functions Add Remove This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here! How long will a trip take if you travel at a certain speed? Use the relationship between speed and time to explore rational functions and discover asymptotes in the real world. Plan your 60-minute lesson in Math or Precalculus and Calculus with helpful tips from Hilary Yamtich\n\nJill: 500 bulbs/2 hours = 250 bulbs/hour Think about how many bulbs each person can plant in one hour. This is their planting rate. Jack: 450 bulbs/3 hours = 150 bulbs/hour Jill & Jack together: Combine their hourly rates to determine the rate they work together. 250 + 150 When medicine is given overtime a certain amount is going to be absorbed in the body so we need to know the exact amount of the medicine that is existing in that body in a certain period of time. We will use the rational function in determining the concentration of the medicine\n\nIn What Real-Life Situations Would You Use Polynomials? Using Polynomials in Real World Real Life Situations for Polynomials Adding Polynomials Dividing Polynomials Solving Word Problems Using Polynomial Equation Polynomials and Factoring Describe a Real Life Situation that could be Modeled by a Function That depends on what you mean with \"real-life\". You won't need polynomial functions to sell stuff at a supermarket, or to cut off a dead branch from your tree... but if you work in science and\n\nGraphing Linear Equations Quilt Project I have created this project as a fun way to get my kids excited about graphing lines. I have always found that including art and coloring with math gets kids a bit more excited about learning and as an added bonus- they look great as classroom decorations whe... The formulas of algebra are used every day in real life when distance needs to be determined, volumes in containers need to be figured out and when sale prices need to be calculated. When using a recipe and doubling it or cutting it in half, algebra is used to determine the exact amount of ingredients. Basic algebraic equations are used almost\n\nA rational exponent is an exponent in fraction form. Any expression containing the square root of a number is a radical expression. Both have real world applications in fields including architecture, carpentry, masonry, financial services, electrical engineering and sciences like biology. When medicine is given overtime a certain amount is going to be absorbed in the body so we need to know the exact amount of the medicine that is existing in that body in a certain period of time. We will use the rational function in determining the concentration of the medicine\n\nIn this lesson, we'll recall what a function is and then look at applying functions in real life that involve both numbers and objects. We'll then learn how to solve problems using these functions. In What Real-Life Situations Would You Use Polynomials? Using Polynomials in Real World Real Life Situations for Polynomials Adding Polynomials Dividing Polynomials Solving Word Problems Using Polynomial Equation Polynomials and Factoring Describe a Real Life Situation that could be Modeled by a Function\n\nRational-equations.com contains helpful resources on real life applications of algebra, terms and subtracting polynomials and other math subject areas. Whenever you need to have guidance on syllabus for college or even common factor, Rational-equations.com is undoubtedly the perfect place to visit! In this lesson you will learn about rational functions, discontinuities and how we can use them to model real-life scenarios. Some examples will include average cost, percentage and mixture problems.\n\nApplication of Rational Functions Application 1. This application is a Cost-Benefit Model.A utility company burns coal to generate electricity. The cost C (in dollars) of removing p amount (percent) of the smokestack pollutants is given by:. Is it possible for the company … How long will a trip take if you travel at a certain speed? Use the relationship between speed and time to explore rational functions and discover asymptotes in the real world. Plan your 60-minute lesson in Math or Precalculus and Calculus with helpful tips from Hilary Yamtich\n\nA \"real world\" application of rational functions is the \"Thin Lens Equation\" which relates focal length, object distance, and real image distance. It is as practical as it gets. Cameras, eyeballs, magnifying glasses, etc all operate using this principle in some capacity. The derivation of it is straight forward and only uses similar triangles 06/08/2015 · Application Of Functions In Real Life Situations Rofus Dino Esquierdo. Loading... Unsubscribe from Rofus Dino Esquierdo? Cancel Unsubscribe. Working... Subscribe Subscribed Unsubscribe 72. Loading\n\nIn the case you actually have service with math and in particular with examples of real life rational expressions word problems or linear algebra come pay a visit to us at Rational-equations.com. We have got a huge amount of excellent reference material on topics ranging from variables to adding and subtracting fractions 20/06/2016 · Applications of Rational Functions - Duration: 12:57. April Pena [East CTA] 2,561 Representing Real Life Situations using Functions and Piecewise Function - Duration: 7:56. Mariane Viterbo\n\nGraphing Linear Equations Quilt Project I have created this project as a fun way to get my kids excited about graphing lines. I have always found that including art and coloring with math gets kids a bit more excited about learning and as an added bonus- they look great as classroom decorations whe... Rational equations can be used to solve a variety of problems that involve rates, times and work. Using rational expressions and equations can help us answer questions about how to combine workers or machines to complete a job on schedule.\n\nIt will help to understand where the numbers come from and why they can be used in real life situations, and how mathematics can be independent discipline no matter how many real life examples are there. Moreover, we should not move from integers to rational numbers without having another, perhaps deeper look, what numbers are, and their FUNCTION! FUNCTIONS ALSO ALLOW US TO VISUALIZE RELATIONSHIPS IN TERMS OF GRAPHS, WHICH ARE MUCH EASIER TO READ AND INTERPRET THAN LIST OF NUMBERS Temperature as a function of various factors. Temperature is a very complicated function because it …\n\n### What Can Rational Expressions in Math Be Used for in Real", null, "Real World Examples for Rational Numbers Blogger. Polynomial Functions in Real Life To find the motion of a particle under the influence of gravity, this equation is used: The weight, w, of a sick patient can be modeled with this equation: x0 = initial position v0 = initial velocity a = acceleration due to gravity t = time, 21/05/2015 · When Would I Use Rational Equations in Real Life?. Part of the series: Equations. Rational equations are used in real life all the time, even if you don't realize it. Learn when you would use.\n\nWhat are some real-life applications of rational algebraic. In this lesson you will learn about rational functions, discontinuities and how we can use them to model real-life scenarios. Some examples will include average cost, percentage and mixture problems., Rational Function Applications - Work and Rate The video explains application problems that use rational equations. Part 2 of 2. Examples: 1. One person can complete a task 8 hours sooner than another person..\n\n### Real life example of Rational Function Rational function", null, "Module 6 Applications Involving Rational Equations. There are no real life applications of reciprocal functions As with any advanced math, whether you use it in your \"real life\" or not depends on whether you work in the engineering field or not. 10/05/2017 · SAT Math Test Prep Online Crash Course Algebra & Geometry Study Guide Review, Functions,Youtube - Duration: 2:28:48. The Organic Chemistry Tutor 1,244,023 views.", null, "That depends on what you mean with \"real-life\". You won't need polynomial functions to sell stuff at a supermarket, or to cut off a dead branch from your tree... but if you work in science and 21/05/2015 · When Would I Use Rational Equations in Real Life?. Part of the series: Equations. Rational equations are used in real life all the time, even if you don't realize it. Learn when you would use\n\nRational Function Applications - Work and Rate The video explains application problems that use rational equations. Part 2 of 2. Examples: 1. One person can complete a task 8 hours sooner than another person. Did you say functions? Let's focus on how rational functions can be used in the medical field Why is this important you ask? Now let's do the same for another function like logarithmic functions Let's look more into how logarithmic functions are used for pH levels But how does\n\nIn this lesson you will learn about rational functions, discontinuities and how we can use them to model real-life scenarios. Some examples will include average cost, percentage and mixture problems. In the Real World. Polynomials and rational expressions are used to express properties of the physical world, like how gravity works.If you need a funnier example of gravity at work (or not at work), check this out.. Polynomials can also be used to do approximation, something that's usually covered in …\n\nSection III: Rational Expressions, Equations, and Functions Module 6: Applications Involving Rational Equations When solving real-world problems we occasionally come across rational equations. In this module we will study two applications that require us to solve rational equations: applications involving work and applications involving motion. That depends on what you mean with \"real-life\". You won't need polynomial functions to sell stuff at a supermarket, or to cut off a dead branch from your tree... but if you work in science and\n\nA function is merely a “machine” that generates some output in correlation to a given input. So, if [math]f(x) = 2x + 1[/math], then, [math]f(3) = 7[/math]. Understanding this behavior is essential to recognizing the variety of input\\output correl... What Are Some Real-Life Examples of Polynomials? Credit: U.S. Army Corps of Engineers Europe District/CC-BY 2.0 The motion of an object that’s thrown 3m up at a velocity of 14 m/s can be described using the polynomial -5tsquared + 14t + 3 = 0.\n\nA \"real world\" application of rational functions is the \"Thin Lens Equation\" which relates focal length, object distance, and real image distance. It is as practical as it gets. Cameras, eyeballs, magnifying glasses, etc all operate using this principle in some capacity. The derivation of it is straight forward and only uses similar triangles In the case you actually have service with math and in particular with examples of real life rational expressions word problems or linear algebra come pay a visit to us at Rational-equations.com. We have got a huge amount of excellent reference material on topics ranging from variables to adding and subtracting fractions\n\nWhen medicine is given overtime a certain amount is going to be absorbed in the body so we need to know the exact amount of the medicine that is existing in that body in a certain period of time. We will use the rational function in determining the concentration of the medicine In the case you actually have service with math and in particular with examples of real life rational expressions word problems or linear algebra come pay a visit to us at Rational-equations.com. We have got a huge amount of excellent reference material on topics ranging from variables to adding and subtracting fractions\n\nIn the case you actually have service with math and in particular with examples of real life rational expressions word problems or linear algebra come pay a visit to us at Rational-equations.com. We have got a huge amount of excellent reference material on topics ranging from variables to adding and subtracting fractions Rational-equations.com contains helpful resources on real life applications of algebra, terms and subtracting polynomials and other math subject areas. Whenever you need to have guidance on syllabus for college or even common factor, Rational-equations.com is undoubtedly the perfect place to visit!\n\nIn the Real World. Polynomials and rational expressions are used to express properties of the physical world, like how gravity works.If you need a funnier example of gravity at work (or not at work), check this out.. Polynomials can also be used to do approximation, something that's usually covered in … Rational-equations.com contains helpful resources on real life applications of algebra, terms and subtracting polynomials and other math subject areas. Whenever you need to have guidance on syllabus for college or even common factor, Rational-equations.com is undoubtedly the perfect place to visit!\n\nTo apply the domain and range in real-world settings, we take a function that represents a real-world situation and then analyze what the domain and range represent in the function. This allows us There are actually a lot of real life applications of rational algebraic expressions. You can’t solve any problems that involve work,rate and time without rational equations. That’s why there are equations like these- 1. Final velocity= Initial ve...\n\nIn What Real-Life Situations Would You Use Polynomials? Using Polynomials in Real World Real Life Situations for Polynomials Adding Polynomials Dividing Polynomials Solving Word Problems Using Polynomial Equation Polynomials and Factoring Describe a Real Life Situation that could be Modeled by a Function In this lesson, we'll recall what a function is and then look at applying functions in real life that involve both numbers and objects. We'll then learn how to solve problems using these functions." ]

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[ "## Conversion formula\n\nThe conversion factor from gallons to quarts is 4, which means that 1 gallon is equal to 4 quarts:\n\n1 gal = 4 qt\n\nTo convert 662 gallons into quarts we have to multiply 662 by the conversion factor in order to get the volume amount from gallons to quarts. We can also form a simple proportion to calculate the result:\n\n1 gal → 4 qt\n\n662 gal → V(qt)\n\nSolve the above proportion to obtain the volume V in quarts:\n\nV(qt) = 662 gal × 4 qt\n\nV(qt) = 2648 qt\n\nThe final result is:\n\n662 gal → 2648 qt\n\nWe conclude that 662 gallons is equivalent to 2648 quarts:\n\n662 gallons = 2648 quarts\n\n## Alternative conversion\n\nWe can also convert by utilizing the inverse value of the conversion factor. In this case 1 quart is equal to 0.00037764350453172 × 662 gallons.\n\nAnother way is saying that 662 gallons is equal to 1 ÷ 0.00037764350453172 quarts.\n\n## Approximate result\n\nFor practical purposes we can round our final result to an approximate numerical value. We can say that six hundred sixty-two gallons is approximately two thousand six hundred forty-eight quarts:\n\n662 gal ≅ 2648 qt\n\nAn alternative is also that one quart is approximately zero times six hundred sixty-two gallons.\n\n## Conversion table\n\n### gallons to quarts chart\n\nFor quick reference purposes, below is the conversion table you can use to convert from gallons to quarts\n\ngallons (gal) quarts (qt)\n663 gallons 2652 quarts\n664 gallons 2656 quarts\n665 gallons 2660 quarts\n666 gallons 2664 quarts\n667 gallons 2668 quarts\n668 gallons 2672 quarts\n669 gallons 2676 quarts\n670 gallons 2680 quarts\n671 gallons 2684 quarts\n672 gallons 2688 quarts" ]

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[ "# How to solve the problem of particles penetrating through boundary in triaxial test?\n\nAsked by Rong Zhao on 2019-09-21\n\nHello everybody:\nI am using the following code to conduct the triaxial test. But the particles penetrate through the boundary in the triaxial test. How to fix this problem?\n\n############################################\n### DEFINING VARIABLES AND MATERIALS ###\n############################################\n\n# The following 5 lines will be used later for batch execution\nnum_spheres = 1000, # number of spheres\ncompFricDegree = 30, # contact friction during the confining phase\nkey = '_triax_base_', # put you simulation's name here\nunknownOk = True\n)\n\nnum_spheres = table.num_spheres # number of spheres\nkey = table.key\ntargetPorosity = 0.43 #the porosity we want for the packing\ncompFricDegree = table.compFricDegree # initial contact friction during the confining phase (will be decreased during the REFD compaction process)\ndamp = 0.2 # damping coefficient\nstabilityThreshold = 0.01 # we test unbalancedForce against this value in different loops (see below)\nyoung = 5e6 # contact stiffness\nmn, mx = Vector3(0,0,0), Vector3(1,1,1) # corners of the initial packing\n\n# create materials for spheres and plates\nO.materials.append(FrictMat(young=young,poisson=0.5,frictionAngle=0,density=0,label='walls'))\n\n# create walls around the packing\nwalls = aabbWalls([mn,mx],thickness=0,material='walls')\nwallIds = O.bodies.append(walls)\n\n## use a SpherePack object to generate a random loose particles packing\nsp = pack.SpherePack()\nsp.makeCloud(mn,mx,-1,0.3333,num_spheres,False, 0.95,seed=1) #\"seed\" make the \"random\" generation always the same\n#or alternatively (higher level function doing exactly the same):\n#sp.toSimulation(material='spheres')\n\n############################\n### DEFINING ENGINES ###\n############################\n\ntriax = TriaxialStressController(\nmaxMultiplier=1.+2e4/young, # spheres growing factor (fast growth)\nfinalMaxMultiplier=1.+2e3/young, # spheres growing factor (slow growth)\nthickness=0,\ninternalCompaction=True, # If true the confining pressure is generated by growing particles\n)\n\nnewton = NewtonIntegrator(damping=damp)\n\nO.engines = [\nForceResetter(),\nInsertionSortCollider([Bo1_Sphere_Aabb(),Bo1_Box_Aabb()]),\nInteractionLoop(\n[Ig2_Sphere_Sphere_ScGeom(),Ig2_Box_Sphere_ScGeom()],\n[Ip2_FrictMat_FrictMat_FrictPhys()],\n[Law2_ScGeom_FrictPhys_CundallStrack()]\n),\n## We will use the global stiffness of each body to determine an optimal timestep (see https://yade-dem.org/w/images/1/1b/Chareyre&Villard2005_licensed.pdf)\nGlobalStiffnessTimeStepper(active=1,timeStepUpdateInterval=100,timestepSafetyCoefficient=0.5),\ntriax,\nnewton\n]\n\n# Display spheres with 2 colors for seeing rotations better\nGl1_Sphere.stripes = 0\n\n#######################################\n### APPLYING CONFINING PRESSURE ###\n#######################################\n\n# the value of (isotropic) confining stress defines the target stress to be applied in all three directions\ntriax.goal1 = triax.goal2 = triax.goal3 = -10000\n\nwhile 1:\nO.run(1000, True)\n# the global unbalanced force on dynamic bodies, thus excluding boundaries, which are not at equilibrium\nunb = unbalancedForce()\n# print(f\"unbalanced force:{unb},mean stress:{triax.meanStress}\")\nprint 'unbalanced force:',unb,' mean stress: ',triax.meanStress\nif unb < stabilityThreshold and abs(-10000-triax.meanStress)/10000 < 0.001:\nbreak\n\nprint \"### Isotropic state saved ###\"\n\n###################################################\n### REACHING A SPECIFIED POROSITY PRECISELY ###\n###################################################\n\nimport sys #this is only for the flush() below\nwhile triax.porosity > targetPorosity:\n## we decrease friction value and apply it to all the bodies and contacts\ncompFricDegree = 0.95*compFricDegree\nprint \"\\r Friction: \",compFricDegree,\" porosity:\",triax.porosity,\nsys.stdout.flush()\nO.run(500,1)\n\nprint \"### Compacted state saved ###\"\n\n##############################\n##############################\n\n# We move to deviatoric loading, let us turn internal compaction off to keep particles sizes constant\ntriax.internalCompaction = False\n\n# Change contact friction (remember that decreasing it would generate instantaneous instabilities)\n\n# set stress control on x and z, we will impose strain rate on y\n# now goal2 is the target strain rate\ntriax.goal2 = rate\n# we define the lateral stresses during the test, here the same 10kPa as for the initial confinement.\ntriax.goal1=-10000\ntriax.goal3=-10000\n\n# we can change damping here. What is the effect in your opinion?\nnewton.damping=0.1\n\nO.engines= O.engines + [PyRunner(iterPeriod=20,command='history()',label='recorder')]\n\n# Save temporary state in live memory. This state will be reloaded from the interface with the \"reload\" button.\nO.saveTmp()\n\n#####################################################\n### Example of how to record and plot data ###\n#####################################################\n\n# a function saving variables\ndef history():\ne11=-triax.strain,e22=-triax.strain,e33=-triax.strain,\ns11=-triax.stress(triax.wall_right_id),\ns22=-triax.stress(triax.wall_top_id),\ns33=-triax.stress(triax.wall_front_id),\nDeviatorStress=(-triax.stress(triax.wall_top_id)-(-triax.stress(triax.wall_right_id)-triax.stress(triax.wall_front_id))/2)*1e-6,\ni=O.iter\n)\n\nplot.plots={'e22':('DeviatorStress')}\nplot.plot()\n\nO.run(124700)\n\n## Question information\n\nLanguage:\nEnglish Edit question\nStatus:\nExpired\nFor:\nAssignee:\nNo assignee Edit question\nLast query:\n2019-09-22\n2019-10-08\n Jan Stránský (honzik) said on 2019-09-21: #1\n\nHello,\ntry to increase the stiffness of walls, something like (theoretically can be infinity):\nO.materials.append(FrictMat(young=1e3*young,...,label='walls'))\ncheers\nJan\n\n Rong Zhao (zhaorong) said on 2019-09-22: #2\n\nHello, Jan Stránský (honzik) . I try to the solution you provided for me to increase the stiffness of walls up to young=1e50*young, but it doesn't work, particles still penetrate through the boundary in triaxial test.\n\n Launchpad Janitor (janitor) said on 2019-10-08: #3\n\nThis question was expired because it remained in the 'Open' state without activity for the last 15 days." ]

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[ "# Document (#38589)\n\nAuthor\nStiller, J.\nOlensky, M.\nPetras, V.\nTitle\n¬A framework for the evaluation of automatic metadata enrichments\nSource\nMetadata and semantics research: 8th Research Conference, MTSR 2014, Karlsruhe, Germany, November 27-29, 2014, Proceedings. Eds.: S. Closs et al\nImprint\nCham : Springer\nYear\n2014\nPages\nS.238-249\nSeries\nCommunications in computer and information science; 478\nAbstract\nAutomatic enrichment of collections connects data to vocabularies, which supports the contextualization of content and adds searchable text to metadata. The paper introduces a framework of four dimensions (frequency, coverage, relevance and error rate) that measure both the suitability of the enrichment for the object and the enrichments' contribution to search success. To verify the framework, it is applied to the evaluation of automatic enrichments in the digital library Europeana. The analysis of 100 result sets and their corresponding queries (1,121 documents total) shows the framework is a valuable tool for guiding enrichments and determining the value of enrichment efforts.\nTheme\nObject\nEuropeana\n\n## Similar documents (author)\n\n1. Petras, V.: Heterogenitätsbehandlung und Terminology Mapping durch Crosskonkordanzen : eine Fallstudie (2010) 1.87\n```1.8734696 = sum of:\n1.8734696 = product of:\n3.7469392 = sum of:\n3.7469392 = weight(author_txt:petras in 731) [ClassicSimilarity], result of:\n3.7469392 = score(doc=731,freq=1.0), product of:\n0.6539319 = queryWeight, product of:\n9.167778 = idf(docFreq=11, maxDocs=42306)\n0.07132938 = queryNorm\n5.7298613 = fieldWeight in 731, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n9.167778 = idf(docFreq=11, maxDocs=42306)\n0.625 = fieldNorm(doc=731)\n0.5 = coord(1/2)\n```\n2. Gradmann, S.; Olensky, M.: Semantische Kontextualisierung von Museumsbeständen in Europeana (2013) 1.87\n```1.8650792 = sum of:\n1.8650792 = product of:\n3.7301583 = sum of:\n3.7301583 = weight(author_txt:olensky in 2940) [ClassicSimilarity], result of:\n3.7301583 = score(doc=2940,freq=1.0), product of:\n0.7565535 = queryWeight, product of:\n1.0756068 = boost\n9.860925 = idf(docFreq=5, maxDocs=42306)\n0.07132938 = queryNorm\n4.9304624 = fieldWeight in 2940, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n9.860925 = idf(docFreq=5, maxDocs=42306)\n0.5 = fieldNorm(doc=2940)\n0.5 = coord(1/2)\n```\n3. Petras, V.; Bank, M.: Vergleich der Suchmaschinen AltaVista und HotBot bezüglich Treffermengen und Aktualität (1998) 1.50\n```1.4987756 = sum of:\n1.4987756 = product of:\n2.9975512 = sum of:\n2.9975512 = weight(author_txt:petras in 3515) [ClassicSimilarity], result of:\n2.9975512 = score(doc=3515,freq=1.0), product of:\n0.6539319 = queryWeight, product of:\n9.167778 = idf(docFreq=11, maxDocs=42306)\n0.07132938 = queryNorm\n4.583889 = fieldWeight in 3515, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n9.167778 = idf(docFreq=11, maxDocs=42306)\n0.5 = fieldNorm(doc=3515)\n0.5 = coord(1/2)\n```\n4. Mayr, P.; Petras, V.: Crosskonkordanzen : Terminologie Mapping und deren Effektivität für das Information Retrieval 1.50\n```1.4987756 = sum of:\n1.4987756 = product of:\n2.9975512 = sum of:\n2.9975512 = weight(author_txt:petras in 3997) [ClassicSimilarity], result of:\n2.9975512 = score(doc=3997,freq=1.0), product of:\n0.6539319 = queryWeight, product of:\n9.167778 = idf(docFreq=11, maxDocs=42306)\n0.07132938 = queryNorm\n4.583889 = fieldWeight in 3997, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n9.167778 = idf(docFreq=11, maxDocs=42306)\n0.5 = fieldNorm(doc=3997)\n0.5 = coord(1/2)\n```\n5. Mayr, P.; Petras, V.: Cross-concordances : terminology mapping and its effectiveness for information retrieval (2008) 1.50\n```1.4987756 = sum of:\n1.4987756 = product of:\n2.9975512 = sum of:\n2.9975512 = weight(author_txt:petras in 143) [ClassicSimilarity], result of:\n2.9975512 = score(doc=143,freq=1.0), product of:\n0.6539319 = queryWeight, product of:\n9.167778 = idf(docFreq=11, maxDocs=42306)\n0.07132938 = queryNorm\n4.583889 = fieldWeight in 143, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n9.167778 = idf(docFreq=11, maxDocs=42306)\n0.5 = fieldNorm(doc=143)\n0.5 = coord(1/2)\n```\n\n## Similar documents (content)\n\n1. Nicholls, P.; Ridley, J.: ¬A context for evaluating for multimedia (1996) 0.09\n```0.092269436 = sum of:\n0.092269436 = product of:\n0.576684 = sum of:\n0.07798128 = weight(abstract_txt:introduces in 5201) [ClassicSimilarity], result of:\n0.07798128 = score(doc=5201,freq=1.0), product of:\n0.10674066 = queryWeight, product of:\n5.8445415 = idf(docFreq=332, maxDocs=42306)\n0.018263308 = queryNorm\n0.7305677 = fieldWeight in 5201, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n5.8445415 = idf(docFreq=332, maxDocs=42306)\n0.125 = fieldNorm(doc=5201)\n0.17842467 = weight(abstract_txt:adds in 5201) [ClassicSimilarity], result of:\n0.17842467 = score(doc=5201,freq=1.0), product of:\n0.18534161 = queryWeight, product of:\n1.317715 = boost\n7.7014403 = idf(docFreq=51, maxDocs=42306)\n0.018263308 = queryNorm\n0.96268004 = fieldWeight in 5201, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n7.7014403 = idf(docFreq=51, maxDocs=42306)\n0.125 = fieldNorm(doc=5201)\n0.100232154 = weight(abstract_txt:evaluation in 5201) [ClassicSimilarity], result of:\n0.100232154 = score(doc=5201,freq=2.0), product of:\n0.12618499 = queryWeight, product of:\n1.5376372 = boost\n4.4933925 = idf(docFreq=1285, maxDocs=42306)\n0.018263308 = queryNorm\n0.7943271 = fieldWeight in 5201, product of:\n1.4142135 = tf(freq=2.0), with freq of:\n2.0 = termFreq=2.0\n4.4933925 = idf(docFreq=1285, maxDocs=42306)\n0.125 = fieldNorm(doc=5201)\n0.2200459 = weight(abstract_txt:framework in 5201) [ClassicSimilarity], result of:\n0.2200459 = score(doc=5201,freq=2.0), product of:\n0.26854795 = queryWeight, product of:\n3.1723125 = boost\n4.635178 = idf(docFreq=1115, maxDocs=42306)\n0.018263308 = queryNorm\n0.8193914 = fieldWeight in 5201, product of:\n1.4142135 = tf(freq=2.0), with freq of:\n2.0 = termFreq=2.0\n4.635178 = idf(docFreq=1115, maxDocs=42306)\n0.125 = fieldNorm(doc=5201)\n0.16 = coord(4/25)\n```\n2. 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Rindflesch, T.C.; Fizsman, M.: The interaction of domain knowledge and linguistic structure in natural language processing : interpreting hypernymic propositions in biomedical text (2003) 0.07\n```0.07426988 = sum of:\n0.07426988 = product of:\n0.30945787 = sum of:\n0.039111335 = weight(abstract_txt:contribution in 4098) [ClassicSimilarity], result of:\n0.039111335 = score(doc=4098,freq=1.0), product of:\n0.106960826 = queryWeight, product of:\n1.0010308 = boost\n5.850566 = idf(docFreq=330, maxDocs=42306)\n0.018263308 = queryNorm\n0.36566037 = fieldWeight in 4098, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n5.850566 = idf(docFreq=330, maxDocs=42306)\n0.0625 = fieldNorm(doc=4098)\n0.041679796 = weight(abstract_txt:valuable in 4098) [ClassicSimilarity], result of:\n0.041679796 = score(doc=4098,freq=1.0), product of:\n0.1115938 = queryWeight, product of:\n1.0224806 = boost\n5.97593 = idf(docFreq=291, maxDocs=42306)\n0.018263308 = queryNorm\n0.37349564 = fieldWeight in 4098, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n5.97593 = idf(docFreq=291, maxDocs=42306)\n0.0625 = fieldNorm(doc=4098)\n0.048526496 = weight(abstract_txt:supports in 4098) [ClassicSimilarity], result of:\n0.048526496 = score(doc=4098,freq=1.0), product of:\n0.123502456 = queryWeight, product of:\n1.0756546 = boost\n6.2867084 = idf(docFreq=213, maxDocs=42306)\n0.018263308 = queryNorm\n0.39291927 = fieldWeight in 4098, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n6.2867084 = idf(docFreq=213, maxDocs=42306)\n0.0625 = fieldNorm(doc=4098)\n0.06251332 = weight(abstract_txt:error in 4098) [ClassicSimilarity], result of:\n0.06251332 = score(doc=4098,freq=1.0), product of:\n0.1462193 = queryWeight, product of:\n1.1704082 = boost\n6.8405 = idf(docFreq=122, maxDocs=42306)\n0.018263308 = queryNorm\n0.42753124 = fieldWeight in 4098, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n6.8405 = idf(docFreq=122, maxDocs=42306)\n0.0625 = fieldNorm(doc=4098)\n0.035437416 = weight(abstract_txt:evaluation in 4098) [ClassicSimilarity], result of:\n0.035437416 = score(doc=4098,freq=1.0), product of:\n0.12618499 = queryWeight, product of:\n1.5376372 = boost\n4.4933925 = idf(docFreq=1285, maxDocs=42306)\n0.018263308 = queryNorm\n0.28083703 = fieldWeight in 4098, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n4.4933925 = idf(docFreq=1285, maxDocs=42306)\n0.0625 = fieldNorm(doc=4098)\n0.08218949 = weight(abstract_txt:automatic in 4098) [ClassicSimilarity], result of:\n0.08218949 = score(doc=4098,freq=1.0), product of:\n0.25308958 = queryWeight, product of:\n2.6670601 = boost\n5.1959147 = idf(docFreq=636, maxDocs=42306)\n0.018263308 = queryNorm\n0.32474467 = fieldWeight in 4098, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n5.1959147 = idf(docFreq=636, maxDocs=42306)\n0.0625 = fieldNorm(doc=4098)\n0.24 = coord(6/25)\n```\n4. Hobson, S.P.; Dorr, B.J.; Monz, C.; Schwartz, R.: Task-based evaluation of text summarization using Relevance Prediction (2007) 0.07\n```0.074103326 = sum of:\n0.074103326 = product of:\n0.37051663 = sum of:\n0.03899064 = weight(abstract_txt:introduces in 2939) [ClassicSimilarity], result of:\n0.03899064 = score(doc=2939,freq=1.0), product of:\n0.10674066 = queryWeight, product of:\n5.8445415 = idf(docFreq=332, maxDocs=42306)\n0.018263308 = queryNorm\n0.36528385 = fieldWeight in 2939, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n5.8445415 = idf(docFreq=332, maxDocs=42306)\n0.0625 = fieldNorm(doc=2939)\n0.04999226 = weight(abstract_txt:corresponding in 2939) [ClassicSimilarity], result of:\n0.04999226 = score(doc=2939,freq=1.0), product of:\n0.12597707 = queryWeight, product of:\n1.0863776 = boost\n6.349379 = idf(docFreq=200, maxDocs=42306)\n0.018263308 = queryNorm\n0.3968362 = fieldWeight in 2939, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n6.349379 = idf(docFreq=200, maxDocs=42306)\n0.0625 = fieldNorm(doc=2939)\n0.061379407 = weight(abstract_txt:evaluation in 2939) [ClassicSimilarity], result of:\n0.061379407 = score(doc=2939,freq=3.0), product of:\n0.12618499 = queryWeight, product of:\n1.5376372 = boost\n4.4933925 = idf(docFreq=1285, maxDocs=42306)\n0.018263308 = queryNorm\n0.486424 = fieldWeight in 2939, product of:\n1.7320508 = tf(freq=3.0), with freq of:\n3.0 = termFreq=3.0\n4.4933925 = idf(docFreq=1285, maxDocs=42306)\n0.0625 = fieldNorm(doc=2939)\n0.14235637 = weight(abstract_txt:automatic in 2939) [ClassicSimilarity], result of:\n0.14235637 = score(doc=2939,freq=3.0), product of:\n0.25308958 = queryWeight, product of:\n2.6670601 = boost\n5.1959147 = idf(docFreq=636, maxDocs=42306)\n0.018263308 = queryNorm\n0.56247425 = fieldWeight in 2939, product of:\n1.7320508 = tf(freq=3.0), with freq of:\n3.0 = termFreq=3.0\n5.1959147 = idf(docFreq=636, maxDocs=42306)\n0.0625 = fieldNorm(doc=2939)\n0.07779797 = weight(abstract_txt:framework in 2939) [ClassicSimilarity], result of:\n0.07779797 = score(doc=2939,freq=1.0), product of:\n0.26854795 = queryWeight, product of:\n3.1723125 = boost\n4.635178 = idf(docFreq=1115, maxDocs=42306)\n0.018263308 = queryNorm\n0.28969863 = fieldWeight in 2939, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n4.635178 = idf(docFreq=1115, maxDocs=42306)\n0.0625 = fieldNorm(doc=2939)\n0.2 = coord(5/25)\n```\n5. Parinov, S.: Semantic enrichment of research outputs metadat : new CRIS facilities for authors (2014) 0.07\n```0.07319458 = sum of:\n0.07319458 = product of:\n0.60995483 = sum of:\n0.035437416 = weight(abstract_txt:evaluation in 3585) [ClassicSimilarity], result of:\n0.035437416 = score(doc=3585,freq=1.0), product of:\n0.12618499 = queryWeight, product of:\n1.5376372 = boost\n4.4933925 = idf(docFreq=1285, maxDocs=42306)\n0.018263308 = queryNorm\n0.28083703 = fieldWeight in 3585, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n4.4933925 = idf(docFreq=1285, maxDocs=42306)\n0.0625 = fieldNorm(doc=3585)\n0.08013183 = weight(abstract_txt:metadata in 3585) [ClassicSimilarity], result of:\n0.08013183 = score(doc=3585,freq=3.0), product of:\n0.15072869 = queryWeight, product of:\n1.6805367 = boost\n4.9109836 = idf(docFreq=846, maxDocs=42306)\n0.018263308 = queryNorm\n0.53162956 = fieldWeight in 3585, product of:\n1.7320508 = tf(freq=3.0), with freq of:\n3.0 = termFreq=3.0\n4.9109836 = idf(docFreq=846, maxDocs=42306)\n0.0625 = fieldNorm(doc=3585)\n0.4943856 = weight(abstract_txt:enrichment in 3585) [ClassicSimilarity], result of:\n0.4943856 = score(doc=3585,freq=3.0), product of:\n0.5804082 = queryWeight, product of:\n4.038894 = boost\n7.8684945 = idf(docFreq=43, maxDocs=42306)\n0.018263308 = queryNorm\n0.8517895 = fieldWeight in 3585, product of:\n1.7320508 = tf(freq=3.0), with freq of:\n3.0 = termFreq=3.0\n7.8684945 = idf(docFreq=43, maxDocs=42306)\n0.0625 = fieldNorm(doc=3585)\n0.12 = coord(3/25)\n```" ]

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[ "", null, "# University Physics Vol. 2 The comprehensive contents from this book, combined with Odigia’s Teaching and Learning Tools have everything you need to engage, collaborate, track and assess your students.\n\n## This course includes:", null, "# 342\n\n### practice questions", null, "# Helping Teachers Do What They Do Best: Teach", null, "# Customize\n\nUse our courses as is or easily customize them to fit your teaching style and the needs of your students. You can add your favorite resources, hide and show our existing content and pre-built assessments, or make them your own. Everything your students need, in one place!", null, "# Engage and Collaborate\n\nOdigia combines learning materials, discussions, and tools to create a familiar social experience for students allowing you to easily connect and redirect students attention.", null, "# Track\n\nSee how much time students are spending on different areas of the course, which areas are creating the most amount of engagement and identify topics the students are struggling with. Flag and provide feedback on assignments to proactively meet individual students' needs.", null, "# Assess\n\nGame theory allows students to monitor their progress visually and motivates them to stay on track. Students can see exactly what activities they need to complete, which ones have been flagged and compare their progress against the overall class.\n\n### What is the difference between temperature and heat?\n\nConcepts Covered:\n\n• How is thermal equilibrium related to temperature?\n• How is temperature measured and converted between different scales?\n• What are the effects of thermal expansion?\n• How are problems involving heat transfer solved?\n• What are the different phase transitions of heat flow?\n• What are the different mechanisms of heat transfer?\n\n### How is the Kinetic Theory of Gases explained?\n\nConcepts Covered:\n\n• What does a molecular model of an ideal gas look like?\n• What is the relationship between pressure, temperature, and RMS speed?\n• How is the heat capacity and equipartition of energy determined?\n• How are molecular speeds distributed in an ideal gas?\n\n### What is the First Law of Thermodynamics?\n\nConcepts Covered:\n\n• How are thermodynamic systems defined?\n• What is the relationship between work, heat, and internal energy?\n• How is the first law of thermodynamics applied?\n• What are the different types of thermodynamic processes?\n• How is the heat capacity of an ideal gas determined?\n• What are the adiabatic processes for an ideal gas?\n\n### What is the Second law of Thermodynamics?\n\nConcepts Covered:\n\n• How are reversible and irreversible processes defined?\n• What are the functions and components of a heat engine?\n• What are the differences between refrigerators and heat pumps?\n• What are the different statements of the second law of thermodynamics?\n• What is the Carnot Cycle and its implications?\n• How is the entropy of simple processess calculated?\n• What is the meaning of entropy at a microscopic scale?\n\n### What is the significance of electric charges and fields?\n\nConcepts Covered:\n\n• How does an electric charge create force?\n• What are the similarities and differences between conductors and insulators?\n• What is Coulomb’s Law?\n• What is the purpose and properties of the electric field?\n• How are the electric fields of charge distributions calculated?\n• How are electric field lines diagramed?\n• What are electric dipoles?\n\n### How is Gauss's Law used?\n\nConcepts Covered:\n\n• How is electric flux calculated?\n• How is Gauss’s Law explained?\n• How is Gauss’s Law applied?\n• What is the electric field within and immediately outside a conductor at equilibrium like?\n\n### How is electric potential related to voltage and electrical energy?\n\nConcepts Covered:\n\n• What is electric potential energy?\n• How are electric potential, voltage, and potential difference related?\n• How is electric potential calculated?\n• How is the electric field calculated from a given potential?\n• What are equipotential surfaces and conductors?\n• What are the different application of electrostatics?\n\n### Why is capacitance important?\n\nConcepts Covered:\n\n• How is the capacitance of a capacitor evaluated?\n• How is the equivalent capacitance of capacitors in series and in parallel combinations determined?\n• How is energy stored in a capacitor?\n• What effects does a dielectric in a capacitor have on capacitance?\n• What does the molecular model of a dielectric look like?\n\n### How does resistance affect the flow of electrical current?\n\nConcepts Covered:\n\n• What is an electrical current?\n• What does the model of conduction in metals look like?\n• How do you differentiate between resistance and resistivity?\n• When does Ohm’s law apply to a situation and when does it not?\n• What is the relationship between electrical energy and power?\n• Why are superconductors important?\n\n### How do direct-current circuits work?\n\nConcepts Covered:\n\n• What is an electromotive force?\n• How do resistors connected in series and in parallel behave?\n• What are Kirchoff’s Rules, and what are their purpose?\n• How are different types of electrical measuring instruments used?\n• What are the applications of RC circuits?\n• What are the basics of wiring and electrical safety?\n\n### How do magnetic forces and field form and behave?\n\nConcepts Covered:\n\n• What are the different historical and contemporary applications of magnetism?\n• How do magnetic fields and lines behave?\n• What is the motion of a charged particle like in a magnetic field?\n• How does a magnetic force affect a current-carrying conductor?\n• How is the force and torque on a current loop calculated?\n• What is the Hall Effect?\n• What are the different applications of magnetic forces and fields?\n\n### What are the sources of magnetic fields?\n\nConcepts Covered:\n\n• How is the Biot-Savart Law defined?\n• What is the magnetic field created from a thin straight wire like?\n• How does the magnetic force between two parallel current act?\n• What is the magnetic field of a current loop like?\n• What is Ampère’s Law?\n• What is the relationship between magnetic fields, solenoids, and toroids?\n• Why are certain materials magnetic and others not?\n\n### Why is electromagnetic induction important in everyday life?\n\nConcepts Covered:\n\n• For what reasons is Lenz’s law used?\n• What common processess and applications use motional emf?\n• What is the relationship between an induced emf and an electric field?\n• Where are eddy currents beneficial, and where are they not helpful?\n• How do electric generators and back emfs work?\n• What are the different applications of electromagnetic induction?\n\n### What are the applications of inductance in electronic devices?\n\nConcepts Covered:\n\n• When is mutual inductance desirable, and when is it not?\n• How do self-inductance and inductors work?\n• How can energy be stored in a magnetic field?\n• What are RL circuits?\n• How do oscillations in an LC circuit behave?\n• How do RLC series circuits operate?\n\n### How do alternating-current circuits work?\n\nConcepts Covered:\n\n• What are the different types of AC sources?\n• What processes power simple AC circuits?\n• How do RLC series with AC work?\n• How is power created in, and how does it move through, an AC circuit?\n• What is the process to determine resonance in an AC circuit?\n• What processes make transformers work?\n\n### How are electromagnetic waves created, and how do they transport energy and momentum?\n\nConcepts Covered:\n\n• What do Maxwell’s equations explain about electromagnetic waves?\n• How are plane electromagnetic waves produced and detected?\n• How is energy carried by electromagnetic waves?\n• What is the relationship between the momentum and radiation pressure of an electromagnetic wave?\n• How is the electromagnetic spectrum arranged?\n\n# University Physics Vol. 2\n\nUniversity Physics is a three-volume collection that meets the scope and sequence requirements for two- and three-semester calculus-based physics courses.\n\nThis textbook emphasizes connections between theory and application, making physics concepts interesting and accessible to students while maintaining the mathematical rigor inherent in the subject. Frequent, strong examples focus on how to approach a problem, how to work with the equations, and how to check and generalize the result.\n\n## Senior Contributing Authors\n\nSamuel J. Ling, Truman State University\nWilliam Moebs, Formerly of Loyola Marymount University\nJeff Sanny, Loyola Marymount University\n\n## Contributing Authors\n\nGerald Friedman, Santa Fe Community College\nStephen D. Druger\nAlice Kolakowska, University of Memphis\nDavid Anderson, Albion College\nDaniel Bowman, Ferrum College\nLev Gasparov, University of North Florida\nLee LaRue, Paris Junior College\nMark Lattery, University of Wisconsin\nRichard Ludlow, Daniel Webster College\nPatrick Motl, Indiana University Kokomo\nDedra Demaree, Georgetown University\nEdw. S. Ginsberg, University of Massachusetts\nDavid Smith, University of the Virgin Islands\nJoseph Trout, Richard Stockton College\nKevin Wheelock, Bellevue College\nTao Pang, University of Nevada, Las Vegas\nKenneth Podolak, Plattsburgh State University\nTakashi Sato, Kwantlen Polytechnic University" ]

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[ "# 括号三分之一加四分之一减六分之一减八分之一括号乘二十四等于多少\n\n(1/3十1/4一1/6)*24=1/3*24十1/4*24一1/6*24=8十6一4=14一4=10\n\n-2\n\n10\n\n24*(1/3+1/4-1/6)=24*1/3+24*1/4-24*1/6=8+6-4=10\n\n24乘(三分之一加四分之一减六分之一件八分之一)=24X(1/3+1/4-1/6-1/8)=8+6-4-3=7\n\n(1/3+1/4-1/6)*12=4+3-2=5\n\n(1/4+1/6+1/8+1/2)24=(6/24+4/24+3/24+14/24)24=27/24x24=27\n\nAll rights reserved Powered by www.mwfd.net\ncopyright ©right 2010-2021。" ]

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[ "# Finding the Compression of a Spring Using the Work-Energy Theorem\n\n## Homework Statement\n\nA 6.0kg box moving at 3.0m/s on a horizontal, frictionless surface runs into a light spring of force constant 75 N/cm. Use the work-energy theorem to find the maximum compression in the spring.\n\n## Homework Equations\n\nW = K2 - K1\n\nW = (1/2)*m*v^2 - (1/2)*m*vo^2\n\nW = - (1/2)*k*x^2\n\n## The Attempt at a Solution\n\n- (1/2)*k*x^2 = 0 - (1/2)*m*vo^2, where k = 75000N/m\n\nx = ((m*v^2)/k)^(1/2) = ((6.0kg*(3m/s)^2)/75000N/m)^(1/2)\n\nx = 0.027m\n\nBut the answer in the book is x = 0.085m. I just can't see what I am doing wrong unless my textbook is wrong." ]

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[ "# Some questions about Hartshorne chapter 2 proposition 2.6\n\nIn Hartshorne chapter 2 proposition 2.6,Hartshorne shows that there is a fully faithful functor $t:\\mathcal{Var}\\rightarrow \\mathcal{Sch}(k)$ from the category of varieties over $k$ to the category of schemes over $k$.He proceed as follows:\n\nFirst,for any topological space $X$,define $t(X)$ to be the set of irreducible closed subsets of $X$.We can define a topology on $t(X)$ by taking as closed sets the subsets of the form $t(Y)$,where $Y$ is closed subset of $X$.If $f:X_1\\rightarrow X_2$ is continuous ,then define a map $$t(f):t(X_1)\\rightarrow t(X_2)\\\\ Y\\mapsto \\overline{f(Y)},$$where $Y$ is a irreducible closed subset of $X_1$. And for any topological space $X$,define $\\alpha :X\\rightarrow t(X)$ by $\\alpha(p)=\\overline{\\{p\\}}$.\n\nQuestion 1:why is $\\overline{f(Y)}$ a irreducible closed subset of $X_2$?\n\nNext,let $V$ be an affine variety over $k$ with coordinate ring $A$,and $O_V$ its sheaf of regular functions,he then shows that $(t(V),\\alpha_*(O_V))$ is isomorphic to the affine scheme $(X,O_X)$,where $X=\\operatorname{Spec}A$.\n\nQuestion 2:why is $(t(V),\\alpha_*(O_V))$ a locally ringed spaces,I mean why is $(\\alpha_*(O_V))_Y$ a local ring for any $Y\\in t(V)$?\n\nNow define a morphism of locally ringed spaces $\\beta:(V,O_V)\\rightarrow X=\\operatorname{Spec}A,$by $\\beta(p)=m_p$.And for any open set $U\\subset X$,define a homomorphism of rings $O_X(U)\\rightarrow \\beta_*(O_V)(U)$:given $s\\in O_X(U)$,$p\\in \\beta^{-1}(U)$,we get a regular function $g$ on $\\beta^{-1}(U)$ by $g(p):=\\overline{s_{\\beta(p)}}\\in A_{m_p}/m_p=k$,where $s_{\\beta(p)}\\in O_{X,\\beta(p)}$ and we identify the stalk $O_{X,\\beta(p)}$ with the local ring $A_{m_p}$.\n\nQuestion 3:why is $g$ a regular function?It seems to me that $g$ is locally constant,is this ture?\n\nThen he claim that the above homomorphism $O_X(U)\\rightarrow \\beta_*(O_V)(U)$ is a isomorphism and use the fact that there is a 1-1 correspondence $t(V)\\leftrightarrow \\operatorname{Spec}A=X$,then $(X,O_X)\\cong (t(V),\\alpha_*(O_V))$ as locally ringed spaces.I can't follow this part of proof.\n\nQuestion 4:what is this $\\beta$ used for?Shouldn't we construct a morphism $(t(V),\\alpha_*(O_V))\\rightarrow (X,O_X)$?And why $O_X(U)\\rightarrow \\beta_*(O_V)(U)$ is a isomorphism?\n\n• question 1: because continuous maps preserve irreducibility and closure also preserves irreducibility. – Seth Sep 7 '14 at 16:00\n• @Seth:You are right.This is easier than I thought.Thanks! – Wei Xia Sep 8 '14 at 15:08\n\nQuestion 2. We know that $$(\\alpha_*(O_V))_Y=\\varinjlim_{Y\\in U}[\\alpha_*(O_V)](U)=\\varinjlim_{Y\\in U}O_V(\\alpha^{-1}(U)),$$ now,each open subset of $t(V)$ is of the form $t(V)-t(W)$ for some irreducible closed subset $W$ of $V$,let $U=t(V)-t(W)$ for some $W$ ,then $Y\\in U \\Leftrightarrow Y\\notin t(W)\\Leftrightarrow \\exists p\\in Y,s.t. p\\notin W$$\\Leftrightarrow Y\\cap (V-W)\\neq \\emptyset,but V-W=\\alpha^{-1}(U),hence Y\\in U \\Leftrightarrow Y\\cap \\alpha^{-1}(U) \\neq \\emptyset.And \\alpha induce a 1-1 correspondence between open subsets of V and t(V),we have$$\\varinjlim_{Y\\in U}O_V(\\alpha^{-1}(U))=\\varinjlim_{Y\\cap U\\neq \\emptyset}O_V(U),$$where U runs over all open subsets of V s.t. Y\\cap U\\neq \\emptyset. Note that \\varinjlim_{Y\\cap U\\neq \\emptyset}O_V(U) is just the local ring of a subvariety which has been shown to be a local ring(excercize 3.13 chapter 1). Question 3.In fact,the function g is defined by the following map:let p\\in \\beta^{-1}(U),$$O_X(U)\\rightarrow O_{X,\\beta(p)}\\rightarrow A_{m_p}\\rightarrow k\\\\s\\mapsto s(\\beta(p))\\mapsto \\frac{a}{b}\\mapsto \\frac{a(p)}{b(p)}=g(p),$$where$O_{X,\\beta(p)}\\cong A_{m_p}$and$a,b\\in A,b\\notin m_p$.Note that$a,b$are polynomial functions on$V$since$A$is the coordinate ring of$V$,and by the definition of$s\\in O_X(U)$,for any$p\\in \\beta^{-1}(U)$,$s(\\beta(p))$is locally quotient of elements in$A$,so$g$is a regular function on$\\beta^{-1}(U)$.Clearly,$g$is a regular function in the usual sense and need not to be locally constant. Question 4.Our original plan is to show$(t(V),\\alpha_*(O_V))\\cong(X,O_X)$.There is a natural homeomorphism$\\gamma:t(V)\\rightarrow X$,so it's left to show$\\gamma':O_X\\rightarrow \\gamma_*(\\alpha_*(O_V))$is an isomorphism.But$\\beta=\\gamma\\circ\\alpha$,so it's enough to show$O_X\\cong \\beta_*(O_V)$. We next show that for any open set$U\\subset X$,$\\varphi:O_X(U)\\rightarrow \\beta_*(O_V)(U)$is an isomorphism.$\\varphi$is injective: suppose$g(p)=0$for all$p\\in\\beta^{-1}(U)$.For any$p\\in\\beta^{-1}(U)$,there exist an open set$p\\in W\\subset U$and$a,b\\in A,b\\notin m_p$s.t. for any$q\\in W$we have$s(q)=\\frac{a}{b}$and$b\\notin q$. For any$q\\in\\beta^{-1}(W),\\beta(q)\\in W$,we have$s(\\beta(q))=\\frac{a}{b}$and$g(q)=\\frac{a(q)}{b(q)}=0$. It follows that$a(q)=0$for all$q\\in\\beta^{-1}(W)$,since$\\beta^{-1}(W)$is open and dense in$V$,we see that$a=0$in$A$.So$s(\\beta(q))=0$for all$\\beta(q)\\in W$.Since$p$is a arbitrary piont in$U$,these W can cover U,we see that$s=0$in$O_X(U)$.$\\varphi$is surjective:let$g\\in O_V(\\beta^{-1}(U))$,then for any$p\\in\\beta^{-1}(U)$,there exist$W\\subset \\beta^{-1}(U)$,$p\\in W$and$a,b\\in A,b\\notin \\beta(p)=m_p$such that$g=\\frac{a}{b}$on$W$.Let$s(\\beta(q)):=\\frac{a}{b}$for all$q\\in W\\subset\\beta^{-1}(U)$,since$p$is a arbitrary piont in$\\beta^{-1}(U)$,these open sets$\\beta(W)$cover$U$,in this way we get$s$as an element of$O_X(U)$.Then one can check that$s$is well- defined and$\\varphi(s)=g$. Alternatively,one can show$O_X\\cong \\beta_*(O_V)\\$ by the following diagram: \\begin{matrix} O_X(U) &\\longrightarrow & \\beta_*(O_V)(U)=O_V(\\beta^{-1}(U)) \\\\\\ \\downarrow & & \\downarrow \\\\\\ O_{X,p}& \\longrightarrow& [\\beta_*(O_V)]_p=O_{V,\\beta^{-1}(p)} \\\\\\ \\downarrow & & \\downarrow \\\\\\ A_p & \\longrightarrow & A_p \\\\\\ \\end{matrix}" ]

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[ "# Rotating Multi-Line Text with Multi-Rows\n\nI am trying to build a basic table and have come up with the following code. My problem is that the rotated boxes are weirdly aligned:", null, "Optimally, all the small rotated boxes should be centered left-right, and the row-spanning, rotated ones should be centered in both ways. I tried using origin=c with rotatebox, but it didn't change anything.\n\n\\begin{table}\n\\begin{tabularx}{\\textwidth}{X|X|X|X|X|X|X}\n\\multirow{2}{*}{\\rotatebox[]{90}{Tool}} & \\multicolumn{2}{c|}{\\makecell{Usability,\\\\ Consistency,\\\\Reuse}} & \\multicolumn{3}{c|}{Support for \\ac{UCSD}} & \\multirow{2}{*}{\\rotatebox[]{90}{\\makecell{Structured\\\\Storage}}} \\\\\n& \\rotatebox[origin=c]{90}{\\makecell{Shape\\\\Recognition}} & \\rotatebox[origin=c]{90}{\\makecell{Similarity\\\\Search}} & \\rotatebox[origin=c]{90}{\\makecell{Real-Time\\\\Collaboration}} & \\rotatebox[origin=c]{90}{\\makecell{Low- \\& High\\\\Fidelity\\\\Prototypes}} & \\rotatebox[origin=c]{90}{\\makecell{Interactive\\\\Testing}} & \\\\ \\hline\n\\end{tabularx}\n\n\\caption{Comparison}\n\\label{tab:existing_tools}\n\\end{table}", null, "\\documentclass{article}\n\\usepackage{rotating}\n\\usepackage{makecell, tabularx}\n\\newcolumntype{C}{>{\\centering\\arraybackslash}X}\n\n\\begin{document}\n\\begin{table}\n\\begin{tabularx}{\\textwidth}{C|C|C|C|C|C|C}\n& \\multicolumn{2}{c|}{\\makecell[b]{Usability,\\\\ Consistency,\\\\Reuse}}\n& \\multicolumn{3}{c|}{Support for UCSD}\n& \\\\" ]

[ null, "https://i.stack.imgur.com/NWj6D.png", null, "https://i.stack.imgur.com/GEmVJ.png", null ]

[ "# Printable Math Worksheets For Kindergarten Addition And Subtraction, Mixed Addition And Subtraction Free Math Worksheets Adding Subtracting More Arithmetic V1 Simple Math Worksheets Addition And Subtraction Worksheets Basic College Mathematics Practice Tests Math Games For High School Printable Wh Questions\n\nPrintable Math Worksheets For Kindergarten Addition And Subtraction Indeed recently is being hunted by consumers around us, perhaps one of you personally. Individuals now are accustomed to using the internet in gadgets to view video and image data for inspiration, and according to the name of the article I will talk about about Printable Math Worksheets For Kindergarten Addition And Subtraction.\n\n• Mixed Operations Math Worksheets – Below You Will Find Well Over 100 Free Addition Worksheets Specifically For This Skill.\n• Kindergarten Addition Worksheets Free Printables Education Com . 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Make Custom Quick Math Facts Printable.\n• 2Nd Grade Math Word Problem Worksheets Free And Printable K5 Learning , Featuring Worksheets That Include Fun Themes, And Engaging Pictures, We Know That Your Child Will Have A Blast While Strengthening Critical Early Math Skills.\n• Number Practice Set Math Worksheets Kindergarten Adding And Subtracting Money 4Th Grade Number Practice Worksheets Worksheets Quiz Generator Printable Math Project On Geometry 3Rd Grade Money Worksheets Kumon Schedule Of Classes 2016 , Looking For Free Printable Kindergarten Math Worksheets Or Preschool Math Worksheets For Your Child?\n\nFind, Read, And Discover Printable Math Worksheets For Kindergarten Addition And Subtraction, Such Us:\n\n• Free Printable Number Addition Worksheets 1 10 For Kindergarten And Grade 1 Addition On Number Line Addition With Pictures Objects Megaworkbook . Addition Is A Process Which Children Learn Quite Naturally As Soon As They Learn To Start Counting.\n• Worksheetfun Free Printable Worksheets Kindergarten Subtraction 1St Grade Math Addition For Kg2 Free Printable Math Worksheets For Kg2 Worksheets Game Faction 2Nd Grade Homework Mathematics Workbook Teacher Sites For Worksheets 10Th Mathematics , Learning Addition By Adding One More To The Group Is An Excellent Way To Introduce This Math Concept To Your Child.\n• Kindergarten Subtraction Worksheets Free Printables Education Com – Download, Print, Or Use The Kindergarten Worksheets Online.\n• Addition And Subtraction Worksheets For Kindergarten , Worksheets And No Prep Teaching Resources Addition Subtraction Math Worksheets.\n• Worksheetfun Free Printable Worksheets Kindergarten Subtraction 1St Grade Math Addition For Kg2 Free Printable Math Worksheets For Kg2 Worksheets Game Faction 2Nd Grade Homework Mathematics Workbook Teacher Sites For Worksheets 10Th Mathematics : Free, Printable Subtraction Worksheets To Help Build Math Skills.\n• Free Fact Family Worksheets . 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Math Worksheet Phenomenal Free Printable Kindergarten Math Worksheets Free Printable Kindergarten Math Sheets Free Printable Kindergarten Math Worksheets With Pictures Book Free Kindergarten Math Worksheets Also Math Worksheets\n\nKindergarten math worksheets in pdf printable format. End of year math assessment addition and subtraction word problems within 10. Free preschool and kindergarten math worksheets, including patterns, more than / less than, addition, subtraction, measurement, money and graphing. Addition and subtraction worksheets are worksheets designed for teachers to help their students free math addition and subtraction worksheets. Add & subt within 1000 (7) grade/level: These worksheets are of the finest quality. Just click on the math worksheet title and click on the download link under the." ]

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[ "Localization with a Nonlinear, Noninvertible Sensor Mapping\n\nLocalization with a\nNonlinear, Noninvertible\nSensor Mapping\n\nAaron Becker\n\n237a Everitt Lab\n\n11/4/2008\n\nOverview\n\nProblem statement\n\nSystem model\n\nSensor model\n\nState estimator\n\nResults\n\nRelated Work\n\nLocalization via triangulation\n\n(GPS)\n\nP. Enge and P. Misra, “Special issue on GPS: The global positioning system,”, 1999.\n\n(IR signals)\n\nR. Want, A. Hopper, V. Falcao, and J. Gibbons,\n\n“The active badge location system,” 1992.\n\n(Ultrasonic & RF)\n\nN. Prijantha, A. Chakraborty, and H. Balakrishan,\n\n“The cricket location\n-\nsupport system,” 2000.\n\n(RF signals)\n\nT. W. Christ, P. Godwin, and R. Lavigne,\n\n“A prison guard duress alarm location system,” 1993.\n\nMap\n-\nbased localization\n\n(LIDAR & ceiling image)\n\nS. Thrun, M. Bennewitz, W. Burgard, et. al,\n\n“Minerva: A second generation museum tour\n-\nguide robot,” 1999.\n\n(Light intensity)\n\nN. Ravi and L. Iftode, “Fiatlux: Fingerprinting rooms using light intensity,” 2007.\n\n(GSM and K\n-\nNNSS fingerprinting)\n\nV. Otsason, A. Varshavsky, A. LaMarca, and E. de Lara,\n\n”Accurate GSM indoor localization,” 2005.\n\nWiFi localization\n\n(NNSS)\n\nP. Bahl, V. Padmanabhan, and A. Balachandran,\n\n“A software system for locating mobile users,” 2000.\n\n(Blueprints and transmitter locations)\n\nM. Robinson and I. N. Psaromiligkos,\n\n“Received signal strength based location estimation of a wireless LAN client,” 2005.\n\n(Gaussian Processes)\n\nB. Ferris, D. Hahnel, and D. Fox,\n\n“Gaussian processes for signal strength\n-\nbased location estimation,” 2006.\n\nSystem Model\n\n2\n(2),\n,\nk k\nk k\nx\nSE y\nv\n\n \n \n \n \n \n \n \n \n \n \nx x\nu u\nSystem State (pose)\nControl Inputs\nDerivation of control inputs\n\nv\nSystem Evolution\n\nState update equation:\n\nt\nv\nt\n\nt\nt\n\nt\nt\n\nx\nt\nx\nSystem\nH\nx\nz\nu\n+\n+\nF\nG\nZ\n-\n1\n,\n2 2\n,,,,\n,\n,~ (0,),~ (0,),...\nx k\nk y k x k x k x k\nk\nn\nn assume n n N n N i i d\nn\n \n\n \n \n \n\n \n \n \nn\n\n1,\n1\n1\n1\n1\n1\n1\n1\n1\ng\nsin sin\ncos co\nf\ns\nk k k\nk\nk k k\nk\nk\nk\nk k k k\nk\nk\nxk\nk\nk\nk\nk\nk\nk\nk\nk\ny\nk\nn\nn\nx\ny\nv\nt\nx\nt\nn\nv\ny\nt\n\n  \n\n  \n\n  \n \n  \n \n \n \n \n \n      \n \n \n \n \n \n\n \n\n \n \n \n\n \n \n\n \n \n \n \n\n \nx x u\nx\nn\nx\nLocalization: Problem Statement\n\n0\n0 0 0 0\n0 0\n0 0\n0 0\n0\n0\n,,[0,2 )\n~ (,)\nT T\nk k k k\nk k k k\nX Y X Y\nX x\nY y\nf\nelse\nx y\nX Unif a b\n\n \n    \n\n \n\n \n\n \n\n \n \n \n\n   \nX\nX X\n\nLocalization\nfrom a known\nstarting point\n\nLocalization\nfrom an unknown\nstarting point\n0\n0\n,\n~ (,),\n~ (0,2 ),\n,\nY Unif c d\nUnif\na b c d\n\n  \n\nObjective\n\n2\nˆ\nmin [ ]\nk\nk k k\nk\nE\n \nx\nx x x\nx\n\nMinimize the current expected\nsquared error\n\nAt all times, have a state\nestimate\nclose\n\nto the true state\n\nReckoning\n\nTrue Position\n\nCauses of Drift\n\nModel errors\n\nProcess noise\n\nExpected Cost\n\n2 2\n2\n[ ]\nˆ\nˆ\n[( ) ] [( ) ]\n[ ] [ ]\n0\n( )\nk k\nk k k\nk k k\nk\nE\nE E\nE E\nVar\n \n \n\nX X\nX X X\nX X X\nX\nThe optimal estimate of is\nThough the expected error is zero, th\ne expected\nvariance is nonzero and increases wi\nth time\nSensor Model\n\n[,]\n- -\nT\ni\nP\n\nEvery four seconds WiFi dongle\ncan scan for access points\nReceives for each access po\nint detected\nIdeally, a range to landmark sen\nsor\n\n2 2\n1 1\n2 2\nAccessPoint\n2 2\n2\n2 2\n4\nk landmark k landmark\n,i\nk landmark k landmark\nk\ni\nk landmark i k landmark i\nx x y y\nx x y y\nR\nx x y y\n \n \n  \n \n \n \n  \n \n \n \n \n \n  \n \nz\nMaximum Received Signal Strength\nin Everitt Lab\n\nCommercial scan, January 2008\n\nActual WiFi Signal\n\nSensor Model\n\n[,,] [,]\nj j j i i\ni i AP\n  \nx x\nx x\nGather data , separate by access point i\nnto\nFor each access point\n1.) Fit a linear model to the da\nta\n2.) Subtract linear model from t\nhe data, leav\n\n1\n2\n2\n1\n2\n-\nf( ),~ (0,)\n,,\n(\ni i i AP\nT T\nk k\ni\nf\ni n\nn\nz RSS m b\nz z\nz N\nk\n \n \n\n   \n \n \nx x\nX x x Z\nx\nx\ning a zero mean data set\n\n3.) Fit a covariance function to\nthis data set\n\n2\n2\n2\n,\n2 2\n1\nexp\n2\n,)\n(,)\ni j\ni j\ni j\nf\nf n\ni j\ni j\ni j\nk\n\n \n\n \n\n \n\n \n\n \n\n \n\n \nx x\nx\nK x x\n4.) Save the covariance for all d\nata points\n\nPrediction algorithm\n\nSensor Model\nf(z|x)\n\n*\n2\n* *\n* *\n1\n* *\n1\n2 2 2\n* * *\n(,)\nT\nT\nf n\nk\n \n\n  \n\n  \nx\nk x X\nk K Z\nk K k\nGiven a new location , generate the expe\ncted\nvalue and expected variance\n1.)\n2.)\n3.)\nThen the probability of a certain se\n\n2\n* *\n2\n*\n* *\n* * * * *\n2\n* * *\n*\n|;,\n1\n;,\n2\nz\nf RSS f z\nf z e\n\n\n\n \n\nx\nx\nnsor\nmeasurement at location is:\n\nSensor Model\n\nData Collection\n\n[,,]\nj j j\nx\n32 access points met the minimum detecti\non threshold\n(recorded 4596 data points )\nState Estimation\n\nObservers without Probability\n\nOften we have fewer sensors than states or\nsensors that do not return our state directly\n\nObserver\n\nCorrection Factor\nTrue System\nH\nx\nz\nu\n+\n+\nF\nG\nH\n+\n+\nF\nG\nx\n^\n\n+\nH\nT\n(\nHH\nT\n)\n-\n1\n\n+\n+\nz\n^\nZ\n-\n1\nZ\n-\n1\ne\nState Observer Example\n\nWith feedback we can ‘estimate’ our state\n\nWe can design our observer to converge to the\ncorrect state at any rate if system is:\n\nStable\n\nObservable\n\nLinear\n\n1\n1:1:0\n| 1 1:1 1:1 0\n| 1:1:0\n( |,,)\n:\n( |,,)\nk\nk k k\nk\nk k\nk\nk\nk k\nk\nf\nf\n\n  \nX\nX U Z X\nx u z x\nu z x\nx\nu\nz\nGiven\nPrevious location probability\nNew control input\nCalculate\nPropagating the probability distribution\n\n1\n1:1:0 1\n| 1 1:1 1:1 0\n| 1:1:0 | 1\n( |,,)\n:\n( |,,) ( |,,)\nk\nk k k k k k k\nk\nk k\nk\nk\nk\nk k\nk\nk\nk\nk\nf\nf f\n\n  \n\nX\nX U Z X X X U Z\nu\nz\nu z\nx u z x\nu z x x\nx x\nGiven\nPrevious location probability\nNew control input\nCalculate with Markov assumption\n0\nx\n1\nx\n1\nu\n1\nz\n2\nx\n2\nu\n2\nz\n1\nk\n\nu\nk\nu\n1\nk\n\nx\n1\nk\n\nz\nk\nx\nk\nz\n0\nx\n1\nx\n1\nu\n1\nz\n2\nx\n2\nu\n2\nz\n1\nk\n\nu\n1\nk\n\nz\nBayesian Propagation\n\n1\n1:1:1\n1 1:1 1:1\n1:1:\n1:1:\n1:1:\n1\n1 1:1 1\n1\n:\n1\n1\n( |,)\n(\n( |\n( |,)\n( |,\n( |,)\n( |\n,)\n( |,)\n|,)\n)\n)\nk\nk k\nk k k\nk k k\nk\nk k\nk k k k\nk k\nk k\nk\nk k k\nk\nf\nnew posterior f\nprior\nf\nf\nf\nf\nf\nd\nf\n\n \n\n \n \n\nx X\nx u x\nx u z\nx\nx u\nx\nz\nx u z\nx u\nu z\nz\nz\nx u z\nx\n\nKalman Filter\n\n1\n1|\n1 1 1\n1 1 1 1\nk\nk k k k\nT\nk k k\nk k\nk k k k\nk k k k k k\n\n  \n   \n \n \n \n \nx Fx Gu\nP FP F V\nx x Re\nP P RHP\nTwo Steps:\n1. Predict the state and covariance mat\nrix using motion model\n2. Correct the state and covariance bas\ned on sensor data\n\n1\n1\n1\n1\n1\n1\nk\nk k\nT\nk\nk k\nT\nk k\n\n \n \n\ne z Hx\nS HP H W\nR P H S\nwhere\n1\n1 1:1 1:\n1:1:1\n1:1:\n1\n1\n1 1\n1\n1::\n( |,)\n( |,\n( |,)\n(\n(\n| )\n)\n| )\n|\n,\n(,)\nk\nk k\nk k k k\nk k\nk k k\nk k\nk\nk\nk\nk k\nf\nf\nf\nf\nf\nd\nf\n\n  \n\n \n\nx X\nx u x x\nx u z\nx u z\nx u\nx u z\nz\nz x\nKalman Filter, Block Diagram\n\nUpdate\nError Term\nKalman State Estimate\nTrue System\nH\nx\nz\nF\nG\nH\n+\n+\nF\nG\nx\n^\n\n+\ne\n+\n+\nz\n^\n+\n+\nw\nv\n+\n+\n+\nZ\n-\n1\nR\n+\n+\nV\nH\n(\ninput\n)\nH\nT\n(\nH\n(\ninput\n)\nH\nT\n+\nW\n)\n-\n1\nF\n(\ninput\n)\nF\nT\nP\n(\nk\n|\nk\n)\nR\n×\nZ\n-\n1\n+\n\nZ\n-\n1\n×\nu\nP\n(\nk\n+\n1\n|\nk\n)\n1\n1 1 1\n1 1 1 1\n1|\nk\nk k k k\nk k k k\nT\nk k k k k k\nk k k\nk k\n\n  \n   \n\n \n \n \n \nx Fx Gu\nx x Re\nP P RHP\nP FP F V\n1. Predict 2. Correct\nKalman Filter Performance\n\nExtended Kalman Filter\n\nDealing\nWith Nonlinearities\n\nThe Kalman filter is the\noptimal linear estimator\n\nThe robotic system is nonlinear\n\nSystem can be linearized\n\nWe will still have the best\nlinear\nestimator at the estimated\noperating point\n\nEKF algorithm\n\nParticle Filter\n\nWe can approximate any\nprobability distribution with\nsamples (particles)\n\n6\n4\n2\n2\n4\n6\n4\n2\n2\n4\nParticle filters can\nrepresent multimodal\ndistributions:\n\n1\n1 1:1 1:\n1:1:1\n1:1:\n1\n1\n1 1\n1\n1::\n( |,)\n( |,\n( |,)\n(\n(\n| )\n)\n| )\n|\n,\n(,)\nk\nk k\nk k k k\nk k\nk k k\nk k\nk\nk\nk\nk k\nf\nf\nf\nf\nf\nd\nf\n\n  \n\n \n\nx X\nx u x x\nx u z\nx u z\nx u\nx u z\nz\nz x\nParticle Filter Benefits\n\n| { }\n:\n( | )\nk free\nC_space k Q\nf C\n\n \nX x\nx\nIncorporating a pdf for\nobstacles in the workspace\nis a binary evaluation\nParticle Filter\n\n,\n1\n,\n,\n,\n,\n,\n:\n1\n_ _,\n,\ni k\nk\ni k i k\ni k\ni k\ni k\ni k\ni k\ni k\ni k\nM\nfor i to M\nx\nsample from motion_model\ny\nx\ny\n\n \n\n \n\n \n \n \n \n \n\n \n \n \np p u\np\np\nTwo Steps:\nUsing two sets\n1. Predict using motion model\nof particles\n2. Correct based on sensor data\n\n1\n,,\n1\nk\ni k i k\nj k\ni k j k\nfor i to M\nw measurement_model C_space\nfor i to M\ndraw j with probability w\n\np z\np p\nParticle Filters in Action\n\nHere, a uniform\ndistribution is\napproximated (blue)\n\nGlobal sensors are\nused to weigh the\nparticles\n\nAnd the distribution\nis resampled (red)\n\nInitialization\n\nParticle Filters in Action\n\nStep\n\nHere, a distribution\nis propagated\n(blue)\n\nGlobal sensors are\nused to weigh the\nparticles, along\nwith the\nC_space\n\nAnd then the\ndistribution is\nresampled (red)\n\nResults\n\nThe SegMonster can localize and track itself in the hallways of Everitt Lab\n\nLiterature Results, Wireless Track\ning Using WiFi Signal Strength\n1.09 m SegMonster\n2.12 m Gaussian Process, Particle Fil\nter\n2.65 m Nearest Neighbor in Signal Sp\nace\n3.88 m AP Locatio\nns and blueprints (no data-gathering sta\nge)\nVideo Of Particle Filter\n\nReckoning\n\nTrue Position\n\nParticle Filter\n\n95% Confidence\nEllipse\n(+/\n-\n2 std)\n\nVideo of Particles\n\nPropagate\n\nUpdate\n\nIncorporating\nC\n-\nspace\n\ndata\n\nQuestions\n\nDead reckoning: velocities from encoders\n\nNonlinear observer example\n\nBayesian methods:\n\nContinuous\n\nDiscrete\n\nParticle filtering:\n\nBlock diagram\n\nMotion model\n\nSegMonster\n\nDynamics\n\nImprovements\n\nPotential fields\n\nMultimodal distributions\n\nWireless mapping (outdoors)\n\nAdaptive Control: parameter estimation/RWP\n\n1,2,1\n2 2 1\n2,,\n1 1 1\nmean cov\n\n \n   \n\n \n   \n   \n \nx\nDimension Parameters Example Representation\nObserving With Probability Distributions\n\nAssumption:\n\nour probability is normally distributed\n\nImplication:\n\nposition can be represented with just the mean and\ncovariance\n\n4\n2\n2\n4\n6\n0.1\n0.2\n0.3\n0.4\n1\n1 1:1 1:\n1:1:1\n1:1:\n1\n1\n1 1\n1\n1::\n( |,)\n( |,\n( |,)\n(\n(\n| )\n)\n| )\n|\n,\n(,)\nk\nk k\nk k k k\nk k\nk k k\nk k\nk\nk\nk\nk k\nf\nf\nf\nf\nf\nd\nf\n\n  \n\n \n\nx X\nx u x x\nx u z\nx u z\nx u\nx u z\nz\nz x\n\nHas high dependence on initial\nconditions (open loop)\n\nOne of Igor's former masters had\n\na tick\n-\ntock man, all\nlevers and gearwheels and cranks and clockwork. Instead of a\nbrain, it had a long tape punched with holes. Instead of a heart,\nit had a big spring. Provided everything in the kitchen was very\ncarefully positioned, the thing could sweep the floor and make a\npassable cup of tea. If it\nwasn't\n\ncarefully positioned, or if the\nticking, clicking thing hit an unexpected bump, then it'd strip\nthe plaster off the walls and make a furious cup of cat.\n\nThief of Time\n, Terry Pratchet\n\nBayesian Filters\n\n1 1:1 1:1\n1\n1:1:\n1 1\n1:1:\n1:\n1\n1:\n1 1:1\n1:1\n( |,)\n(\nBayesi\n(\nan F\n|\niltering(,,)\n,)\n( |,\n1:0\n2\n|\n:for do\n3:( |,)\n4:\n_\n,\n( | )\n5:\n( )\n)\n|,\n)\nk k k\nk k\nk\nk k\nk\nk k k\nk k\nk k\nk\nk k k\nk k\nk\nk\nf\nf\nunNormalized f\nunNormal\nal\ni\nl\nzed\nf\nf\nf\nd\nf\n\n \n  \n  \n\n \n\n \n\nu z\nx X\nx u x\nx u\nx u\nz\nx u z\nz\nx u z\nx\nx\nu z\nz x\n1:\n1\n1:\n1:\n:1:\n1:1:\n1:\n(\n6:endfor\n7:for do\n1\n8:\n9:endfor\n10:retur\n|\n_ ( |,)\n_\n,\n( |,)\n)\n( |,)\nn\nk k k\nk k k\nk k\nk k k\nk\nk\nf\nunNormalized f\nf\nall\nf\n\n \nx\nx\nx u z\nx\nu z\nx z\nu z\nu\nX\ncorrection using\neffect of mo\nsensor\nnew po\ntion\nster\nmode\nior\ni r\nl\npr o\n\nDiscrete Bayesian Filter\n\n1\n1\n1:\n1 1:1 1\n1:1\n1\n:\n1:1\n1\n:1:1\n1:1\n1\n:\n1\n:1\n|,\n|,\n|,\n|\n|,\n|\n|\n|,\n,\nk\nk\nk k\nk\nk k k\nk\nk\nk k\nk k\nx X\nk k k\nk k k\nx\nk\nk\nk k\nX\nk\nk\nP x u z\nP x u z\nP x u x\nP x u z\nP x u\nP z x\nP z x\nP x u z\nz\n\n  \n\n \n\n \n\nPrediction\n\nCorrection\n\nTwo Steps to propagate belief\n\nsensor\ne\n\nff\nmo\nect\ndel\nn\nof motio\new poste\nn mod\nrior\ni\ne\npr or\nl\n\nParticle Filter Block Diagram\n\nWeigh\n(\nfor each particle\n)\nPredict\n(\nfor each particle\n)\nError Term\nParticle State Estimate\nExtract mean\n,\nor Maximize\nExpectation\nTrue System\nH\nx\nz\nF\nG\nH\nF\nG\nx\n^\n\n+\nz\n^\n+\n+\nw\nv\n+\n+\n+\nZ\n-\n1\nZ\n-\n1\nu\nResample\n(\nuntil we have n particles\n)\nK\n()\nDraw\n\ni\n\nwith probability\nw\ni\np\n[\nj\n]\nt\n\n=\np\n[\ni\n]\nt\n-\n1\n\n+\n+\n+\nV\nrand\nMean\n\nState Estimation Cookbook\n\nPropagates one sample (do this to each sample):\n\n1\n1 1\n1 1 1\n1 1\nsample_motion_model(,)\n.: ,\n.\nk k\nk k k\nk k k k k\nk k k\nx x x\ny y y\n  \n\n \n  \n \n   \n     \n   \n     \n \n   \n     \n     \n   \n     \n   \nu x\nu x\nthis is the code for\n1 ) Inputs\n2 ) Calculate\nchange in\n\n1 1 1\n2 2\n1 1\n2 1 1\n1 1 1 2 2\n3 4 1\natan2,\n:\nˆ\n.:\nˆ\nrot k k k k k\ntrans k k k k\nrot k k rot\nrot rot rot trans\ntrans trans trans rot rot\ny y x x\nx x y y\nsample\nsample\n \n\n   \n   \n     \n \n \n\n   \n   \n  \n  \n   \nposition\n3 ) Add noise\n\n2\n2 2 1 2 2\n1 1\n1 1\n1 1 2\nˆ\nˆ ˆ\nˆ\ncos\n.\nˆ ˆ\n:\nˆ\nsin\nˆ ˆ ˆ\nrot rot rot trans\nk trans rot\nk trans rot\nk rot rot\nsample\nx x\ny y\n   \n  \n  \n   \n\n  \n  \n  \n  \n4 ) Calculate\nestimate to return\nEffect of Propagation Parameters\n\nTuning Parameters\n\nα\n1\n,\n\nα\n2\n,\n\nα\n3\n,\n\nα\n4\n\nProportional to\n\nEffects\n\nα\n1\n\nrotation\n\nrotation\n\nα\n2\n\ntranslation\n\nrotation\n\nα\n3\n\ntranslation\n\ntranslation\n\nα\n4\n\nrotation\n\ntranslation\n\nLeft:\n\n[0.005, 0.001, 0.46, 0.02]\n\nRight:\n\n[0.01, 0.005, 0.0016, 0.02]\n\nCalculating Errors\n\nm\ni\nm\ni\ni\ni\nm\n1\n1\n2\n1\n2\n2\n2\n1\n1\n2\n1\ncos\n,\nsin\n2\natan\n,\n,\n,\n,\n0\ncos\ncos\n,\n\n:\nangles\n\nof\n\nset\n\na\n\nof\n\nmean\n\n:\ngiven\n\nHow to calculate Covariance from\nparticles\n\nAssumes\nw\n[\ni\n]k\n\nis normalized (sums to 1) and that you\nhave calculated the means for x and y,\n\ny\ny\nx\nx\nw\nM\nto\ni\nx\nfor\nk\ni\nk\ni\nk\ni\n\np\np\ncov\ncov\n0\ncov\n\ny\nx\n,\n\nHow to deal with a Multimodal\nprobability distribution\n\nThe particle filter can represent multimodal\ndistributions, but the covariance and mean\nare now meaningless statistics.\n\nk\n-\nmeans algorithm to find k means, and\nchoose the mean with the highest\nprobability (or highest average probability)\n\nBetter:\n\nuse Expectation Maximization\nalgorithm to identify most probable point.\n\nGenerating Velocities from Encoders\n\nEquations of motion, differential drive robot\n\n[Steve M LaValle, Planning Algorithms pg 728 http://planning.cs.uiuc.edu/, 13.1]\n\ncos\n2\n2\nsin\n2\n2\nl r\nl l\nk k t\nl\nl r\nr r\nk k t\nr\nr l\nr\nx u u\nu t\nr\ny u u\nu t\nr\nu u\nL\n\n \n\n \n\n\n\n \n\n \n \n\n \n \nVelocity Pose Update\n\nSegMonster Dynamics\n\nThe objective for the design of the SegMonster was to produce a low cost robotic platform built\nto handle both indoor and outdoor environments that could move at the speed of a jogging\nhuman.\n\nThese design objectives resulted in a non\n-\ndestructive addition to a standard Segway that\nexhibited a certain level of autonomy. Specifically, the SegMonster could follow a sidewalk by\nusing a color camera to segment between grass and concrete, and follow another SegWay by\ntracking a red LED array attached to the lead Segway. The SegMonster was equipped with a\nsingle IR sensor to detect obstacles immediately in front of it and programmed to stop until the\nobstacle was removed. This level of autonomy can be called 'behavior level', since the\nSegMonster relied entirely on local data (wheel encoders, and the IR sensor) or calculated a\nlocal control law based on the immediate inputs (using the camera to determine whether the\nrobot was close or far from the sidewalk edge, or visual servo\n-\ning based on the size and\nlocation of a red color blob segmented from the camera).\n\nThe SegMonster did calculate a position estimate directly from wheel encoder data, which was\nreasonably accurate for paths of length less than 20 feet. We were able to use this data to move\nthe SegMonster in a 5’x5’ box pattern. Though the SegMonster had one global sensor (the\ncamera), it was used entirely for local based control.\n\nThe first controller for the Segmonster used classical techniques\n\ninner PID control loops for\neach motor, and outer loop velocity based PI loops for the reference body angle and turning\nangle.\n\nEnabling the SegMonster to carryout complex tasks, a global position estimate is necessary\n\nThe SegMonster\n\nMoves by tilting forwards/backwards\n\nComparison to inverted pendulum\n\nθ\nis angle from vertical\n\nlength to Center Of Mass\n\nM\nC\n\nis mass of cart\n\nM\np\n\nmass of pendulum\n\nI inertia of pendulum\n\nx horizontal displacement\n\nof cart\n\nM\nC\nM\np\n,\nI\n?\nx\nl\nInverted Pendulum\nat rest\nInverted Pendulum perturbed from\nequilibrium by\n?\nThe cart cannot move in vertical\ndirection, so\n\nSum forces on FBD of cart in\nhorizontal direction:\n\nSum forces on pendulum in horizontal\ndirection\n\nSub second equation in for N:\n\nCart and Pendulum forces can be\ncalculated separately\n\nM\np\n,\nI\n?\nl\nP\nN\nmg\nx\n..\nI\n?\n.\nI\n?\n2\nPendulum\n,\nand forces upon it\nM\nC\nx\nP\nN\nx\n..\nbx\n.\nF\nCart\n,\nand forces upon it\nF\nN\nx\nb\nx\nM\nc\n\nsin\ncos\n2\n\nc\np\np\nM\nM\nx\nM\nN\n\nF\nM\nM\nx\nb\nx\nM\nM\nc\np\np\nc\n\nsin\ncos\n2\n\nForces perpendicular to pendulum\n\n1.\nForces perp to pendulum\n\n2.\nTo remove Pan N, sum\nthe moments about the\nCOM of the pendulum\n\n3.\nSub (2) into (1)\n\n4.\nGroup angular\naccelerations\n\n5.\nDivide by mass of\npendulum\n\n6.\nIsolate horizontal\nacceleration\n\n7.\nSimplify\n\n8.\nNear the equilibrium\nposition, with small\nangular accelerations,\nthe horizontal\nacceleration is fixed\n\nM\np\n,\nI\n?\nl\nP\nN\nmg\nx\n..\nI\n?\n.\nI\n?\n2\nPendulum\n,\nand forces upon it\n\nx\ng\nx\ng\nM\nI\nx\ng\nM\nI\nx\ng\nM\nI\nx\nM\ng\nM\nM\nI\nx\nM\nM\ng\nM\nI\nI\nN\nP\nx\nM\nM\ng\nM\nN\nP\np\np\np\np\np\np\np\np\np\np\np\np\n\ntan\n)\n8\n(\ntan\ncos\n)\n7\n(\ncos\nsin\n)\n6\n(\ncos\nsin\n)\n5\n(\ncos\nsin\n)\n4\n(\ncos\nsin\n)\n3\n(\ncos\nsin\n)\n2\n(\ncos\nsin\ncos\nsin\n)\n1\n(\nEquations not entirely accurate\n\nOffset of pivot force\n\nConstant gravity on handle\n\nOffset\nx\n\n= 2.25”\n\nOffset\ny\n\n= 8.25”\n\nM\np\n,\nI\n?\nl\nP\nN\nmg\nx\n..\nI\n?\n.\nI\n?\n2\nOffset\nx\nOffset\ny\nt\nhandle\nThe Segmonster has a slightly different structure\nthan the basic inverted pendulum\n.\nThere is a\nconstant torque due to gravity on the handle\n,\nand\nthe pivot point is offset from the center by\n(\nOffset\nx\n,\n\nOffset\ny\n)\nEquivalent System\n\nLet theta\n2\n\nbe the angle to the COM from the\ncenter of the wheel, and l\n2\n\nbe the distance\nfrom the COM to the center of the wheel\n\n?\nl\nmg\nx\n..\nOffset\nx\nOffset\ny\nt\nhandle\nl\n2\n?\n2\n25\n.\n8\ncos\n25\n.\n2\nsin\n25\n.\n8\ncos\n25\n.\n2\nsin\n\ntan\nsin\n5\n.\n4\ncos\n5\n.\n16\n125\n.\n73\n25\n.\n8\ncos\n25\n.\n2\nsin\n2\n2\n2\n2\n2\n2\n2\n\ng\nx\ny\nx\ny\nx\nConversion from motor angle\n\nto body position.\n\n?\na\n1\na\n2\nb\ns\nL\na\n\nPotential Fields\n\n*\n*\n*\n*\n*\n,\n,,\n,,\n,\n1 1 1\ngoal\ngoal goal\ngoal\natt\ngoal goal\ngoal goal\ngoal\nrep\nq q\nd q q d\nd\nU q\nd q q\nd q q d\nd q q\nU q\nD q Q\n\n \n\n \n\n \n \n \n\nwhere is the threshold distance to\nswitch from a conic to a parabolic well\n\n*\n*\n2\n*\n,\n,\n,\n0\nQ\nD q\nD q Q\nD q\nD q\nD q Q\n\nwhere is the threshold distance\nfor detecting an obstacle, is\ndistance to nearest obstacle\nWireless Networks on Campus\n\nDuring our data gathering sessions we\nidentified 1819 unique wireless networks on\ncampus.\n\nMRAC Controller\n*\n\nu\nB\nAx\nx\n\nr\nm\nm\nm\nB\nx\nA\nx\n\nPlant:\n\nReference\nmodel:\n\nIdeal Control:\n\nm\nm\nr\nu\nB\nBL\nA\nBK\nA\nL\nx\nK\n\n*\n*\n*\n*\nbu\nax\nx\n\nr\nb\nx\na\nx\nm\nm\nm\n\nm\n*\nm\n*\n*\n*\nb\nbl\na\nbk\na\nr\nl\nx\nk\nu\n\nscalar\n\nVector/matrix\n\n*\nP. A. Ioannou, J. Sun,\n\nPrentice\n-\nHall, 1996, Ch 6.2\n\nMRAC Controller\n*\n\nFeedback\ngain\n\nReference\ngain:\n\nConstraint:\n\nb\nex\nk\nsgn\n1\n\nb\ner\nl\nsgn\n2\n\nl\nT\nT\nm\nsgn\nPex\nB\nK\n\nl\nr\nT\nm\nsgn\nPe\nB\nL\n\nMust know\nsign of\nb\nsince\n\n:\nsgn\nl\nIs 1 if\nL\n*\n\nis positive\ndefinite,\n-\n1 if\nnegative definite\nsince\n\n*\nP. A. Ioannou, J. Sun,\n\nPrentice\n-\nHall, 1996, Ch 6.2\n\nm\nB\nBL\n\n*\nscalar\n\nVector/matrix\n\nm\n*\nb\nbl\n\n]\n~\n~\n~\n~\n[\n)\n~\n~\n(\nL\nΓ\nL\nK\nΓ\nK\nPe\ne\nL\n,\nK\ne,\nT\nT\nT\n\ntr\nV\nb\nl\nb\nk\ne\n)\nl\n,\nk\nV(e,\n2\n2\n1\n2\n2\n2\n~\n2\n~\n2\n~\n~\n\nMRAC Simulation Model\n\nSegMonster Improvements\n\nRemade hand attachment to\nallow full range of turn commands\n\nAdded wireless sensor and\ngumstix processor\n\nAdded limit switches to robot\nbody\n\nImproved precision for encoder\n\nAdded two IR distance sensors\n\nDead zone compensation for turn\ncommands\n\nForgetting factor on Integral\nterms\n\nSimple Pendulum\n\nRotating wheel on end\n\nWheel actuated by DC\nmotor\n\nMotor exerts torque on\nwheel\n\nWheel produces\nreaction torque on\npendulum\n\np\n\nr\n\nPENDULUM\nREACTION\nWHEEL\nState Estimation\n\nExample: RWP\n\nState Estimation Example: RWP\n\nr\nr\np\np\n\nx\n10\n\nu\nu\nk\nki\nu\n\nr\np\ny\n\nState Estimation Example: RWP\n\nWith feedback we can ‘estimate’ our\nstate\n\nWe can design our estimator to\nconverge to the correct state at any rate\nif system is:\n\nStable\n\nObservable\n\nLinear\n\nState Estimation Example: RWP\n\nWhy can’t we do the same\nthing with our robots?\n\nThree Problems:\n\n1.\nNot stable\n\n2.\nNot observable\n\n3.\nNot linear\n\nGP_predict Algorithm\n\n1\n*\n* *\n*\n1\n*\n*\n*\n2 2\n*\n*\nGP_predict(,,,)\n1:,\n2:\n3:\n4:\n5:return,\nT\nT\nf n\nk\n\n  \nx X L\nα\nk x X\nk\nα\nv L k\nv v\n\n1\n1\nchol\n\\\\\nT\n\nL K\nα L L z\n\nEKF\n\n1\n1 1\n1\n1\n1\n1 1 1\n1 1 1\n,\nk k\nk\nk k k k\nk k k k\nk\nk\nk k\nT\nk k k\nk k\nf\n\n  \n \n\n \n\n \n \n\nx\nP P V\nx\nx\nx Re\nP P\nu\nF F\nTwo Steps:\n1. Predict the state and covariance mat\nrix using motion model\n2. Correct the state and covariance bas\ned on sensor data\n\n1\n1\n1\n1\n1\n1\n1\n1\n1\n,\n1\n1 1\n1\n1\n1 1\nk\nk k k k\nk\nk\nk\nT\nk k\nk\nk\nk k\nk k\nk\nT\nk\nk k\nk\nk\nf h\n\n \n\n \n \n\n \n \n \nx u x\nH\nH\ne z x\nS P W\nR P\nR P S\nx x\nH\nH\nH\nF H\nwhere\nThe linearized state and observation\nmatrices are the following Jacobians:" ]

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[ "# Pizza theorem", null, "Pizza theorem Example of application of the theorem with eight sectors: by cutting the pizza along the blue lines and, alternately taking one slice each, proceeding clockwise or counterclockwise, two diners eat the same amount (measured in area) of pizza. Proof without words for 8 sectors by Carter & Wagon (1994a).\n\nIn elementary geometry, the pizza theorem states the equality of two areas that arise when one partitions a disk in a certain way.\n\nContents 1 Statement 2 History 3 Generalizations 4 Related results 5 See also 6 References 7 External links Statement Let p be an interior point of the disk, and let n be a multiple of 4 that is greater than or equal to 8. Form n sectors of the disk with equal angles by choosing an arbitrary line through p, rotating the line n / 2  − 1 times by an angle of 2π / n radians, and slicing the disk on each of the resulting n / 2 lines. Number the sectors consecutively in a clockwise or anti-clockwise fashion. Then the pizza theorem states that: The sum of the areas of the odd-numbered sectors equals the sum of the areas of the even-numbered sectors (Upton 1968).\n\nThe pizza theorem is so called because it mimics a traditional pizza slicing technique. It shows that, if two people share a pizza sliced in this way by taking alternating slices, then they each get an equal amount of pizza.\n\nHistory The pizza theorem was originally proposed as a challenge problem by Upton (1967). The published solution to this problem, by Michael Goldberg, involved direct manipulation of the algebraic expressions for the areas of the sectors. Carter & Wagon (1994a) provide an alternative proof by dissection. They show how to partition the sectors into smaller pieces so that each piece in an odd-numbered sector has a congruent piece in an even-numbered sector, and vice versa. Frederickson (2012) gave a family of dissection proofs for all cases (in which the number of sectors is 8, 12, 16, ...).\n\nGeneralizations 12 sectors: green area = orange area The requirement that the number of sectors be a multiple of four is necessary: as Don Coppersmith showed, dividing a disk into four sectors, or a number of sectors that is not divisible by four, does not in general produce equal areas. Mabry & Deiermann (2009) answered a problem of Carter & Wagon (1994b) by providing a more precise version of the theorem that determines which of the two sets of sectors has greater area in the cases that the areas are unequal. Specifically, if the number of sectors is 2 (mod 8) and no slice passes through the center of the disk, then the subset of slices containing the center has smaller area than the other subset, while if the number of sectors is 6 (mod 8) and no slice passes through the center, then the subset of slices containing the center has larger area. An odd number of sectors is not possible with straight-line cuts, and a slice through the center causes the two subsets to be equal regardless of the number of sectors.\n\nMabry & Deiermann (2009) also observe that, when the pizza is divided evenly, then so is its crust (the crust may be interpreted as either the perimeter of the disk or the area between the boundary of the disk and a smaller circle having the same center, with the cut-point lying in the latter's interior), and since the disks bounded by both circles are partitioned evenly so is their difference. However, when the pizza is divided unevenly, the diner who gets the most pizza area actually gets the least crust.\n\nAs Hirschhorn et al. (1999) note, an equal division of the pizza also leads to an equal division of its toppings, as long as each topping is distributed in a disk (not necessarily concentric with the whole pizza) that contains the central point p of the division into sectors.\n\nRelated results Hirschhorn et al. (1999) show that a pizza sliced in the same way as the pizza theorem, into a number n of sectors with equal angles where n is divisible by four, can also be shared equally among n/4 people. For instance, a pizza divided into 12 sectors can be shared equally by three people as well as by two; however, to accommodate all five of the Hirschhorns, a pizza would need to be divided into 20 sectors.\n\nCibulka et al. (2010) and Knauer, Micek & Ueckerdt (2011) study the game theory of choosing free slices of pizza in order to guarantee a large share, a problem posed by Dan Brown and Peter Winkler. In the version of the problem they study, a pizza is sliced radially (without the guarantee of equal-angled sectors) and two diners alternately choose pieces of pizza that are adjacent to an already-eaten sector. If the two diners both try to maximize the amount of pizza they eat, the diner who takes the first slice can guarantee a 4/9 share of the total pizza, and there exists a slicing of the pizza such that he cannot take more. The fair division or cake cutting problem considers similar games in which different players have different criteria for how they measure the size of their share; for instance, one diner may prefer to get the most pepperoni while another diner may prefer to get the most cheese.\n\nSee also Other mathematical results related to pizza slicing involve the lazy caterer's sequence, a sequence of integers that counts the maximum number of pieces of pizza that one can obtain by a given number of straight slices, and the ham sandwich theorem, a result about slicing three-dimensional objects whose two-dimensional version implies that any pizza, no matter how misshapen, can have its area and its crust length simultaneously bisected by a single carefully chosen straight-line cut, and whose three-dimensional version implies that a plane cut exists that equally shares base, tomato and cheese.\n\nReferences Carter, Larry; Wagon, Stan (1994a), \"Proof without Words: Fair Allocation of a Pizza\", Mathematics Magazine, 67 (4): 267, doi:10.1080/0025570X.1994.11996228, JSTOR 2690845. Carter, Larry; Wagon, Stan (1994b), \"Problem 1457\", Mathematics Magazine, 67 (4): 303–310, JSTOR 2690855. Cibulka, Josef; Kynčl, Jan; Mészáros, Viola; Stolař, Rudolf; Valtr, Pavel (2010), \"Solution of Peter Winkler's pizza problem\", Fete of Combinatorics and Computer Science, Bolyai Society Mathematical Studies, vol. 20, János Bolyai Mathematical Society and Springer-Verlag, pp. 63–93, arXiv:0812.4322, doi:10.1007/978-3-642-13580-4_4, ISBN 978-3-642-13579-8, S2CID 18272355. Hirschhorn, J.; Hirschhorn, M. D.; Hirschhorn, J. K.; Hirschhorn, A. D.; Hirschhorn, P. M. Hirschhorn (1999), \"The pizza theorem\" (PDF), Austral. Math. Soc. Gaz., 26: 120–121. Frederickson, Greg (2012), \"The Proof Is in the Pizza\", Mathematics Magazine, 85 (1): 26–33, doi:10.4169/math.mag.85.1.26, JSTOR 10.4169/math.mag.85.1.26, S2CID 116636161. Knauer, Kolja; Micek, Piotr; Ueckerdt, Torsten (2011), \"How to eat 4/9 of a pizza\", Discrete Mathematics, 311 (16): 1635–1645, arXiv:0812.2870, doi:10.1016/j.disc.2011.03.015, S2CID 15566728. Mabry, Rick; Deiermann, Paul (2009), \"Of Cheese and Crust: A Proof of the Pizza Conjecture and Other Tasty Results\", American Mathematical Monthly, 116 (5): 423–438, doi:10.4169/193009709x470317, JSTOR 40391118. Ornes, Stephen (December 11, 2009), \"The perfect way to slice a pizza\", New Scientist. Upton, L. J. (1967), \"Problem 660\", Mathematics Magazine, 40 (3): 163, JSTOR 2688484. Problem statement. Upton, L. J. (1968), \"Problem 660\", Mathematics Magazine, 41 (1): 42, JSTOR 2687962. Solution by Michael Goldberg. Berzsenyi, George (1994), \"The Pizza Theorem - Part I\" (PDF), Quantum Magazine: 29 Berzsenyi, George (1994), \"The Pizza Theorem - Part II\" (PDF), Quantum Magazine: 29 External links Weisstein, Eric W. \"Pizza Theorem\". MathWorld. Sillke, Torsten, Pizza Theorem, retrieved 2009-11-24 Categories: AreaTheorems about circlesPizzaProof without wordsMetaphors referring to food and drink\n\nSi quieres conocer otros artículos parecidos a Pizza theorem puedes visitar la categoría Area.\n\nSubir\n\nUtilizamos cookies propias y de terceros para mejorar la experiencia de usuario Más información" ]

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[ "# N-distances and Their Applications\n\nLev B. Klebanov", null, "", null, "### Distributed for Karolinum Press, Charles University\n\n96 pages | 8 x 12\nPaper \\$15.00 ISBN: 9788024611525 Published April 2006 Not for sale in the Czech Republic and the Slovak Republic\n\nThe book focuses on probability metrics suitable for the characterization of random variables in Hilbert or Banach space. It provides details of various stochastic processes, such as testing non-deterministic statistical hypotheses, characterization of probability distribution or constructing multidimensional test for two selections. The book is published in the English language.\n\nTable of Contents", null, "• Contents\nContents\n\nPreface\n\nChapter 1. Positive and Negative Definite Kernels and Their Properties\n\n1. Definitions of positive and negative definite kernels\n\n2. Examples of positive definite kernels\n\n3. Positive definite functions\n\n4. Negative definite kernels\n\n5. Coarse embeddings of metric spaces into Hilbert space\n\n6. Strictly and strongly positive and negative definite kernels\n\nChapter 2. N-Metrics in the Set of Probability Measures\n\n1. A class of positive definite kernels in the set of probabilities and N-distances\n\nChapter 3. m-negative Definite Kernels and Metrics\n\nChapter 4. N-metrics and the Problem of Recovering Measures From Potential\n\n1. Recovering Measures From Potential\n\n2. Stability in the Problem of Recovering a Measure from Potential\n\nChapter 5. N-metrics in the Study of Certain Problems of Characterization of Distributions\n\n1. Some characterization of Gaussian and related distributions\n\n2. Characterization of distributions symmetric to a group of transformations\n\nChapter 6. Commutative Semigroups with Positive Definite Kernel\n\n1. General considerations\n\n2. Distances in X\n\n3. Special representations\n\n4. Properties of x(t)\n\n5. Infinitely divisible elements\n\n6. Accompanying infinitely divisible elements\n\n7. Examples\n\nChapter 7. Statistical Estimates obtained by the Minimal Distances Method\n\n1. Estimating a location parameter, I\n\n2. Estimating a location parameter, II\n\n3. Estimating a general parameter\n\n4. Estimating a location parameter, III\n\n5. Semiparametric estimation\n\nChapter 8. Some Statistical Tests Based on N-Distances\n\n1. Multivariate two-sample test\n\n2. Test for two distributions to belong to the same additive type\n\n3. Some Tests for Observations to be Gaussian\n\n4. A Test for Closeness of Probability Distributions\n\nChapter 9. A Permutation Test Motivated by Microarray Data Analysis 1. Introduction\n\n2. Theoretical framework: the test statistic\n\n3. Computational framework: accuracy of permutation quantiles and p-values\n\n4. Power of the test: a simulation study\n\n5. Examples of data analysis\n\n6. Conclusions of the Chapter\n\nBibliography\n\nAuthor Index\n\nIndex", null, "" ]

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[ "Intel® Fortran Compiler\nBuild applications that can scale for the future with optimized code designed for Intel® Xeon® and compatible processors.\n27530 Discussions\n\n## do concurrent: Bug in code or in compiler?", null, "Novice\n115 Views\n\nHi,\n\nI've started to substitute \"forall\" statements with \"do concurrent\" loops and found an unexpected behavior when compiling with -Ox, x>1 and openMP and autoparallelization, i.e.\n\n` ifort test.f90 -O2 -parallel -qopenmp`\n\nfor the Intel compiler in versions 18.0.1 and 18.0.3. I seems like that the \"do concurrent\" loop is not executed at all. Is my code invalid or does the compiler makes a mistake when doing the optimization?\n\n```program test\nimplicit none\nreal, dimension(4) :: c\n\nc = [1.0,2.0,5.0,3.3]\nwrite(6,*) math_qToR(c)\nwrite(6,*) math_qToR2(c)\n\ncontains\n\nfunction math_qToR(q)\nimplicit none\nreal, dimension(4), intent(in) :: q\nreal, dimension(3,3) :: math_qToR\ninteger :: i, j\n\nforall (i = 1:3, j = 1:3)\nmath_qToR(i,j) = q(i+1) * q(j+1)\nend forall\n\nend function math_qToR\n\nfunction math_qToR2(q)\nimplicit none\nreal, dimension(4), intent(in) :: q\nreal, dimension(3,3) :: math_qToR2\ninteger :: i, j\n\ndo concurrent (i = 1:3, j = 1:3)\n!write(6,*) i,j ! cures the behavior\nmath_qToR2(i,j) = q(i+1) * q(j+1)\nenddo\n\nend function math_qToR2\n\nend program```\n\n0 Replies", null, "" ]

[ null, "https://community.intel.com/t5/image/serverpage/avatar-name/intelyellow/avatar-theme/candy/avatar-collection/Intel_Customer/avatar-display-size/message/version/2", null, "https://community.intel.com/skins/images/F43E08D03F0846F5FE99C58CE2E475A0/responsive_peak/images/icon_anonymous_message.png", null ]

[ "Solutions by everydaycalculation.com\n\n## 33 is what percent of 85?\n\n33 of 85 is 38.82%\n\n#### Steps to solve \"what percent is 33 of 85?\"\n\n1. 33 of 85 can be written as:\n33/85\n2. To find percentage, we need to find an equivalent fraction with denominator 100. Multiply both numerator & denominator by 100\n\n33/85 × 100/100\n3. = (33 × 100/85) × 1/100 = 38.82/100\n4. Therefore, the answer is 38.82%\n\nIf you are using a calculator, simply enter 33÷85×100 which will give you 38.82 as the answer.\n\nMathStep (Works offline)", null, "Download our mobile app and learn how to work with percentages in your own time:" ]

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

[ "# [Solved]More qwt plot questions\n\n• In the documentation of QWT plot there is a code like this\n\n@#include \"../include/qwt_plot.h>\n\nQwtPlot *myPlot;\nlong curve1, curve2; // keys\ndouble x, y1, y2; // x and y values\n\nmyPlot = new QwtPlot(\"Two Graphs\", parent, name);\n\ncurve1 = myPlot->insertCurve(\"Graph 1\");\ncurve2 = myPlot->insertCurve(\"Graph 2\");\n\ngetSomeValues(x, y1, y2);\n\n// copy the data into the curves\nmyPlot->setCurveData(curve1, x, y1, 100);\nmyPlot->setCurveData(curve2, x, y2, 100);\n\n// finally, refresh the plot\nmyPlot->replot();\n@\n\nBut when I try to use this methods setCurveData and insertCurve the compiler says\nerror: ‘class QwtPlot’ has no member named ‘insertCurve’ etc.\n\nAnd the documentation is upto date I think.\n\nThe other QWT widgets works and got there members but not this.\n\nis it a old doc on the net or;\n\n• I have downloaded qwt-6.0.1 as a zip file less than a week ago. It is fairly small (<4MB). It contains in the doc directory an html documentation. The files have been generated on August 1, 2011. I assume that this the most recent documentation available. At least it refers to the compatibility of qwt 6.x on the main page.\n\n• This seems to be the same documentation on the \"web \":http://qwt.sourceforge.net/index.html\n\n• Did you look in Qwt's headers files for setCurveData and insertCurve?\n\n• Well, QwtPlot neither has a setCurveData or a insertCurve member nor does its documentation have an example like the one you've listed above. You're most probably mixing 5.x libraries with 6.x documentation and vice versa. They are not compatible.\n\n• There was never such methods. Try something like this:\n@\nQwtPlot *myPlot = new QwtPlot(\"Two Graphs\");\nQwtPlotCurve *curve1 = new QwtPlotCurve(\"Graph 1\");\nQwtPlotCurve *curve2 = new QwtPlotCurve(\"Graph 2\");\n\n//set curve color\ncurve1->setPen(QPen(Qt::green, 2));\ncurve1->setPen(QPen(Qt::red, 2));\n\ncurve1->attach(myPlot);\ncurve2->attach(myPlot);\n\nconst int MAX_VALUES = 100;\ndouble x[MAX_VALUES], y1[MAX_VALUES], y2[MAX_VALUES];// x and y values\ngetSomeValues(x, y1, y2);\n\n// copy the data into the curves\ncurve1->setRawSamples(x, y1, MAX_VALUES);\ncurve1->setRawSamples(x, y2, MAX_VALUES);\n\n// finally, refresh the plot\nmyPlot->replot();\n@\nit's untestet...\nAnd of course take a look at very well examples in the qwt source.\n\n• Yes this looks like my old function.\nThank you" ]

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[ "", null, "y = 27*[1/2*(x/27)^(1+2/3) / (1+2/3) -1/2*(x/27)^(1-2/3)/(1-2/3)+(2/3)/(1-(2/3)^2)]" ]

[ null, "https://www.wolframalpha.com/_next/static/images/Logo_1tof6SJC.svg", null ]

[ "# Efficient way to scan an image with a window\n\nHi, I need to develop, in an efficient way, a blur filter that considers only the pixels with value different from 0, which are provided by a mask image.\n\nI have developed the following function, but it requires too much time, in case of a large blur kernel size (example: kernel size = 49).\n\nvoid adaptiveMeanFilter(cv::Mat& source_p, cv::Mat& destination_p, cv::Mat& mask_p, int filterSize_p){\n\nif ((filterSize_p % 2) == 0){\n\nfilterSize_p += 1;\n}\n\nfloat xWeightedMean = 0;\nfloat yWeightedMean = 0;\n\nint count = 0;\n\ncv::Mat temp = cv::Mat::zeros(source_p.rows, source_p.cols, source_p.type());\n\nfor (int yIter = (filterSize_p / 2); yIter < source_p.rows - (filterSize_p / 2); yIter++){\nfor (int xIter = (filterSize_p / 2); xIter < source_p.cols - (filterSize_p / 2); xIter++){\n\nxWeightedMean = 0;\nyWeightedMean = 0;\ncount = 0;\n\nfor (int yIterFilter = -(filterSize_p / 2); yIterFilter <= (filterSize_p / 2); yIterFilter++){\nfor (int xIterFilter = -(filterSize_p / 2); xIterFilter <= (filterSize_p / 2); xIterFilter++){\n\nif (mask_p.at<uchar>(yIter + yIterFilter, xIter + xIterFilter) != 0){\n\ncount++;\nxWeightedMean += source_p.at<cv::Vec2f>(yIter + yIterFilter, xIter + xIterFilter);\nyWeightedMean += source_p.at<cv::Vec2f>(yIter + yIterFilter, xIter + xIterFilter);\n}\n}\n}\n\ntemp.at<cv::Vec2f>(yIter, xIter) = (xWeightedMean / count);\ntemp.at<cv::Vec2f>(yIter, xIter) = (yWeightedMean / count);\n\n}\n}\n\ntemp.copyTo(destination_p);\n\n\n}\n\nCan someone help me?\n\nThanks\n\nedit retag close merge delete\n\n## Comments\n\nBlur filters (like yours, too) are separable. So you should compute first the mean of the pixels in the X direction to a temporary buffer, then you make a vertical blur of the temporary buffer to the final image.\n\nThe process can be further optimized by adding the next pixel and subtracting the last one during the x blur instead of making the sum of all the elements." ]

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[ "## Table of Content", null, "Open Access\n\nARTICLE\n\n# Performance Analysis of RIS Assisted NOMA Networks over Rician Fading Channels\n\nXianli Gong1, Chongwen Huang2,3,4, Xinwei Yue5, Zhaohui Yang2,4,6, Feng Liu1,*\n\n1 School of Electronic and Information Engineering, Beihang University, Beijing, 100191, China\n2 College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou, 310027, China\n3 International Joint Innovation Center, Zhejiang University, Haining, 314400, China\n4 Zhejiang Provincial Key Laboratory of Information Processing, Communication and Networking (IPCAN), Hangzhou, 310027, China\n5 School of Information and Communication Engineering, Beijing Information Science and Technology University, Beijing, 100101, China\n6 Zhejiang Laboratory, Hangzhou, 311121, China\n\n* Corresponding Author: Feng Liu. Email:", null, "(This article belongs to this Special Issue: Recent Advances in Backscatter and Intelligent Reflecting Surface Communications for 6G-enabled Internet of Things Networks)\n\nComputer Modeling in Engineering & Sciences 2023, 135(3), 2531-2555. https://doi.org/10.32604/cmes.2023.024940\n\n## Abstract\n\nIn this paper, we consider a downlink non-orthogonal multiple access (NOMA) network assisted by two reconfigurable intelligent surfaces (RISs) over Rician fading channels, in which each user communicates with the base station by the virtue of a RIS to enhance the reliability of the received signal. To evaluate the system performance of our proposed RIS-NOMA network, we first derive the exact and asymptotic expressions for the outage probability and ergodic rate of two users. Then, we derive the exact and asymptotic upper bound expressions for the ergodic rate of the nearby user. Based on asymptotic analytical results, the diversity orders for the outage probability and the high signal-to-noise ratio (SNR) slopes for the ergodic rate of the two users are obtained in the high SNR regime. Moreover, we derive the system throughputs of the proposed RIS-NOMA network in delay-limited and delay-tolerant transmission modes. Numerical results confirm our analysis and demonstrate that: 1) The outage probability and ergodic rate of RIS-NOMA networks are superior to that of RIS-assisted orthogonal multiple access (OMA) networks; 2) The RIS-NOMA networks have ability to achieve a larger system throughput compared to RIS-OMA networks; and 3) The system performance of RIS-NOMA networks can be significantly improved as the number of reflecting elements and Rician factor increases.\n\n## Keywords\n\n1  Introduction\n\nAs a promising multiple access technology in the next generation wireless communication network, non-orthogonal multiple access (NOMA) has the ability to tremendously enhance the spectral and energy efficiency [1,2]. The distinguishing feature of NOMA is that multiple users with different quality-of-service requirements are served by exploiting the superposition coding scheme in the same time/frequency/code resource block . In [6,7], the authors have proved that the outage probability and ergodic rate of NOMA can be better than that of conventional orthogonal multiple access (OMA). Zhu et al.  have proposed an optimal power allocation scheme learned by adopting the deep deterministic policy gradient algorithm in multiple-input multiple-output NOMA vehicular edge computing.\n\nRecently, reconfigurable intelligent surface (RIS) has been regarded as a promising technology for the next generation wireless communication, which can improve the spectral and energy efficiency of future wireless communication networks in a cost-effective way . Specifically, RIS is a low-cost planar array composed of a large number of reconfigurable passive elements, in which the amplitude and phase of the incident signal for each element can be changed independently [12,13]. By adjusting the amplitude and phase coefficients of RIS elements through programmable controller, the electromagnetic signal can be reconfigured to significantly enhance the signal quality and the coverage of wireless communication . Compared with the conventional relaying technology, the advantage of RIS is that the passive reflection does not consume energy and it does not generate self-interference in full duplex (FD) mode . The RIS-empowered multiuser uplink communication system has been investigated in , in which a channel estimation framework based on the parallel factor decomposition has been proposed to unfold the resulting cascaded channel model. In , the authors have studied the downlink wireless communication system, in which the transmit beamforming and the RIS have been jointly optimized to maximize the energy efficiency of the system. A new joint design of the transmit beamforming and the phase shifting of the RIS has been proposed in , in which the sum rate of the multiuser system has been maximized by using deep reinforcement learning (DRL). In , the multiuser communication system assisted by RIS has been considered, in which the power allocation and the phase shifting of the RIS have been jointly optimized to minimize the sum transmit power. Considering RIS-empowered terahertz band communications, a DRL-based hybrid beamforming scheme has been proposed in to improve the communication coverage.\n\n1.1 Related Work\n\nSince both RIS and NOMA can greatly improve the spectrum and energy efficiency, it is critical to study their combined advantages to further enhance the spectrum and energy efficiency of future wireless communication networks. In , a simple RIS-NOMA transmission scheme has been designed, in which the outage performance of the cell-edge user has been improved with the assistance of RIS in a NOMA system. In , the authors have investigated the performance of RIS-NOMA networks in terms of the outage probability, ergodic rate, and energy efficiency by exploiting 1-bit coding scheme. The effects of coherent phase shifting and random phase shifting on outage performance for RIS-NOMA networks have been investigated in . By designing the passive beamforming weights, the authors in have investigated the outage probability and ergodic rate of prioritized user for RIS-NOMA. The RIS reflection with discrete phase shifting for RIS-NOMA and RIS-OMA has been analyzed in . In , a novel NOMA solution with RIS partitioning has been proposed to reduce the transmission interference. Considering the impact of the direct link from the base station (BS) to users on the network performance, the outage behaviors of multiple RISs-assisted NOMA networks have been studied by using discrete phase shifting in . The authors in have examined the outage performance of RIS-NOMA networks under presence of hardware impairment. In , the outage probability and ergodic rate of RIS-aided uplink and downlink NOMA networks have been investigated by employing continuous phase shifting. The authors in have studied RIS assisted downlink NOMA networks, in which the performance of the networks has been analyzed by using the coherent phase shifting.\n\n1.2 Motivation and Contributions\n\nAs mentioned in the above works, compared with the conventional relaying technology, the RIS is capable of reflecting the incident signal transmitted from the transmitter to the receiver in FD mode without self-interference and energy consumption, which not only enhances the spectrum and energy efficiency, but also has a very low cost. Therefore, the integration of RIS and NOMA can further improve the spectrum and energy efficiency of the next generation wireless communication networks. Furthermore, the current literature on RIS-NOMA mainly concentrates on the optimization problem, but the research on performance analysis is not enough. Compared with conventional NOMA systems, the performance analysis of RIS-NOMA systems is more complex and challenging owing to the influence of the RIS. These motivate us to study the performance of RIS assisted NOMA systems, which provides an important theoretical basis for deployment of RIS in the future.\n\nIn this paper, we consider a RIS assisted NOMA network, in which a pair of NOMA users in a cell cannot directly receive signals from the BS due to the obstruction of obstacles, and each user needs the help of a RIS. Moreover, we comprehensively analyze the performance of RIS-NOMA networks in terms of the outage probability, ergodic rate, and system throughput. The primary contributions of this paper are summarized as follows:\n\n•   We derive the closed-form and asymptotic expressions for the outage probability of a pair of NOMA users over Rician fading channels for the proposed RIS-assisted NOMA network. To gain further insight, we obtain the diversity orders of the pair of NOMA users for RIS-NOMA network according to the asymptotic analyses at high signal-to-noise ratio (SNR). We observe that the diversity orders of the pair of NOMA users depend on the number of reflecting elements and Rician factor.\n\n•   We compare the outage performance of the pair of users for RIS-NOMA and RIS-OMA networks. We demonstrate that the outage performance of RIS-NOMA outperforms that of RIS-OMA. We prove that the RIS-NOMA network is capable of improving outage performance with increasing the number of reflecting elements and Rician factor.\n\n•   We derive the exact and asymptotic expressions for the ergodic rate of the pair of NOMA users over Rician fading channels for our proposed RIS-NOMA network. To approximate the exact ergodic rate, we also derive the exact and asymptotic upper bound expressions for the ergodic rate of the nearby user. Based on theoretical analyses, we obtain the high SNR slopes for the ergodic rate of the pair of NOMA users. We observe that the throughput ceiling appears in the ergodic rate of the distant user at high SNR. We demonstrate that the superiority of RIS-NOMA over RIS-OMA in terms of the ergodic rate. We evidence that the ergodic rate of RIS-NOMA becomes larger with the increase of the number of reflecting elements.\n\n•   We analyze the system throughput of our proposed RIS-NOMA network in two transmission modes. In delay-limited transmission mode, we observe that the system throughput of RIS-NOMA converges to a constant at high SNR. Furthermore, we prove that the proposed RIS-NOMA is capable of enhancing the system throughput compared to RIS-OMA. In delay-tolerant transmission mode, we evidence that the proposed RIS-NOMA can achieve a larger system throughput than RIS-OMA at high SNR. Moreover, we confirm that as the number of reflecting elements increases, the system throughput of RIS-NOMA can be significantly improved in the two transmission modes.\n\nThe main notations used in this paper are explained as follows. Cx×y denotes the space of x×y complex-valued matrices. VH represents the conjugate-transpose of V. fX() and FX() mean the probability density function (PDF) and cumulative distribution function (CDF) of random variable X, respectively. diag() denotes a diagonal matrix. Pr() denotes the probability. E() and D() stand for the expectation and variance, respectively.\n\n2  System Model\n\nConsider a two-RIS assisted NOMA network with one single-antenna BS and two single-antenna users as shown in Fig. 1, in which the BS sends the superposed signals to the pair of NOMA users by means of two RISs. Assume that there is no direct link between the BS and each NOMA user due to strong attenuation, and each NOMA user needs a RIS to assist in communicating with the BS. D1 and D2 are the nearby user and the distant user, respectively. R1 and R2 are RISs corresponding to D1 and D2, respectively. Each RIS has K reflecting elements. The reflecting phase shifting matrix of Rn is denoted by ϑn=diag(βejθn1,,βejθnk,,βejθnK), n{1,2}, where β[0,1] and θnk[0,2π) represent the amplitude coefficient and the phase shift of the k-th reflecting element of the RIS, respectively.", null, "Figure 1: System model of RIS assisted downlink NOMA networks\n\n2.1 Channel Model\n\nIt is assumed that the channel state information of all channels is completely known at the BS. The complex channel coefficients from the BS to Rn and from Rn to Dn are represented by GnCK×1 and HnCK×1, respectively. Assume that the wireless links for RIS-NOMA networks are modeled as Rician fading channels. The cascade channel gains from the BS to Rn and from Rn to Dn can be expressed as HnHϑnGn. The channel coefficient matrixes can be written as\n\nGn=[Gn1,,Gnk,,GnK]H,(1)\n\nand\n\nHn=[Hn1,,Hnk,,HnK]H,(2)\n\nwhere the channel coefficients from the BS to the k-th reflecting element of Rn and from the k-th reflecting element of Rn to Dn are respectively given by\n\nGnk=αn,1k(κκ+1+1κ+1G~nk),(3)\n\nand\n\nHnk=αn,2k(κκ+1+1κ+1H~nk).(4)\n\nDefining αn,1k=ηdn,1α, αn,2k=ηdn,2α, dn,1 and dn,2 are the distances from the BS to Rn and from Rn to Dn, respectively. κ, α, and η are the Rician factor, path loss exponent, and frequency dependent factor, respectively. The fading gains G~nk and H~nk follow complex Gaussian distribution with zero mean and unit variance, i.e., G~nkCN(0,1) and H~nkCN(0,1).\n\n2.2 Signal Model\n\nBased on the superposition coding scheme, the BS broadcasts superposed signal s=a1Pbs1+a2Pbs2 to D1 and D2, where Pb means the normalized transmit power of the BS, sn represents the unit power signal transmitted to Dn with an denoting the power allocation factor of Dn. For simplicity, we assume a fixed power allocation between two users. The optimized power sharing between non-orthogonal users can further enhance the system performance of RIS-NOMA networks, which will be studied in future work. For user fairness, the power allocation factor satisfies a2>a1 with a1+a2=1. The received signals at D1 and D2 are respectively given by\n\nyD1=H1Hϑ1G1(a1Pbs1+a2Pbs2)+nD1,(5)\n\nand\n\nyD2=H2Hϑ2G2(a1Pbs1+a2Pbs2)+nD2,(6)\n\nwhere nDn represents the additive white Gaussian noise with the mean power N0 at Dn. On the basis of the successive interference cancellation (SIC) principle, the signal of D2 is detected first at D1. The received signal-to-interference-plus-noise ratio (SINR) at D1 to detect signal s2 of D2 is given by\n\nγD1s2=ρ|H1Hϑ1G1|2a2ρ|H1Hϑ1G1|2a1+1,(7)\n\nwhere ρ=PbN0 is the transmit SNR. After performing SIC, the signal of D1 is detected at D1. The received SINR of detecting signal s1 of D1 at D1 can be given by\n\nγD1s1=ρ|H1Hϑ1G1|2a1.(8)\n\nThe signal of D2 is directly detected by treating the signal of D1 as interference at D2. The received SINR of D2 to detect its own signal can be given by\n\nγD2s2=ρ|H2Hϑ2G2|2a2ρ|H2Hϑ2G2|2a1+1.(9)\n\n2.3 RIS-OMA\n\nFor comparison, we regard the RIS-OMA scheme as the benchmark, in which the BS sends information to two OMA users through two RISs in two time slots. In each time slot, signal sn is transmitted from the BS to Dn by utilizing Rn. The observation at Dn is given by\n\nyn,DnOMA=HnHϑnGnPbsn+nDn.(10)\n\nThe received SINR at Dn for detecting its own signal can be given by\n\nγn,DnsnOMA=ρ|HnHϑnGn|2.(11)\n\n2.4 Channel Statistics\n\nIn order to achieve optimal performance, we use the coherent phase shift design. For the wireless communication links of RIS-NOMA networks, our goal is to optimize the channel by adjusting the amplitude and phase of the RIS to achieve the optimal channel gain for each user. For the BS-RIS-Dn link, we maximize |HnHϑnGn|=|k=1KβGnkHnkejθnk|, where Gnk and Hnk are the k-th element of Gn and Hn, respectively. In order to achieve maximization, the sum of the phase of GnkHnk and the phase shift θnk of the k-th reflecting element of the RIS is equal to zero. Therefore, we have\n\n|HnHϑnGn|=k=1K|GnkHnk|,(12)\n\nwhere β=1 without loss of generality.\n\nThe channel coefficients Gnk from the BS to the k-th reflecting element and Hnk from the k-th reflecting element to Dn follow Rician distribution, hence the probability density function (PDF) of cascade Rician channel Xnk=|GnkHnk| in is expressed as\n\nfXnk(x)=1αn,1αn,2i=0j=04xi+j+1(κ+1)i+j+2κi+j(i!)2(j!)2e2κ(αn,1αn,2)i+j+12Kij[2x(κ+1)αn,1αn,2],(13)\n\nwhere Kν() denotes the modified Bessel function of the second kind with order ν . The mean and variance of Xnk are respectively written as\n\nμn=E(Xnk)=παn,1αn,24(1+κ)[L12(κ)]2,(14)\n\nand\n\nΩn=D(Xnk)=αn,1αn,2{1π216(1+κ)2[L12(κ)]4},(15)\n\nwhere L12() represents the Laguerre polynomial and L12(κ)=e12κ[(1κ)K0(κ2)κK1(κ2)].\n\n3  Outage Probability\n\nIn communication systems, the outage probability can be defined as the probability that the outage occurs for a user when the information rate of the user is lower than its target rate. The outage probability is a widely used system performance metric in fixed-rate transmission systems. In this section, the outage performance of downlink RIS-NOMA networks is investigated.\n\n3.1 Outage Probability of D1\n\nBy using the SIC scheme at nearby user D1, the signal of distant user D2 is detected first, and then its own signal is detected. Therefore, when both signals s1 and s2 cannot be successfully detected by D1, an outage occurs for D1. Based on Eqs. (7) and (8), the outage probability of D1 for RIS-NOMA networks can be expressed as\n\nPD1=1Pr(γD1s1>γth1,γD1s2>γth2),(16)\n\nwhere γth1=2r11 and γth2=2r21 represent the target SNRs with r1 and r2 denoting the target rates of D1 and D2 to detect signals s1 and s2, respectively. The outage probability of D1 is derived in the following theorem.\n\nTheorem 3.1. Under Rician fading channels, the closed-form expression for the outage probability of D1 for RIS-NOMA networks can be approximated as\n\nPD11Γ(Kμ12Ω1)γ(Kμ12Ω1,μ1λΩ1),(17)\n\nwhere λmax(λ1,λ2), λ1=γth1ρa1, λ2=γth2ρ(a2a1γth2) with a2>a1γth2, μ1=πα1,1α1,24(1+κ)[L12(κ)]2, Ω1=α1,1α1,2{1π216(1+κ)2[L12(κ)]4}, γ(α,x)=0xettα1dt is the lower incomplete Gamma function , and Γ(z)=0ettz1dt is the gamma function .\n\nProof: See Appendix A.\n\nRemark 3.1. From Eq. (17), it is observed that the approximate outage probability of D1 for RIS-NOMA is subject to ρ, K, κ, and α when the target SNR and the power allocation coefficient are fixed. Specifically, increasing ρ, K, κ, and α, the approximate outage probability of D1 for RIS-NOMA decreases. As a result, the outage performance of D1 for RIS-NOMA can be enhanced.\n\n3.2 Outage Probability of D2\n\nOn account of the NOMA principle, the signal of D2 is directly detected at distant user D2. Thus the outage occurs for D2 when signal s2 cannot be successfully detected by D2. In view of Eq. (9), the outage probability of D2 for RIS-NOMA networks can be expressed as\n\nPD2=Pr(γD2s2<γth2).(18)\n\nThe outage probability of D2 is derived in the following theorem.\n\nTheorem 3.2. Under Rician fading channels, the closed-form expression for the outage probability of D2 for RIS-NOMA networks can be approximated as\n\nPD21Γ(Kμ22Ω2)γ(Kμ22Ω2,μ2λ2Ω2),(19)\n\nwhere μ2=πα2,1α2,24(1+κ)[L12(κ)]2 and Ω2=α2,1α2,2{1π216(1+κ)2[L12(κ)]4}.\n\nProof: See Appendix B.\n\nRemark 3.2. From Eq. (19), it can be seen that the approximate outage probability of D2 for RIS-NOMA depends on ρ, K, κ, and α. Similarly, as ρ, K, κ, and α increase, the approximate outage probability of D2 for RIS-NOMA decreases. Consequently, the outage performance of D2 for RIS-NOMA can be improved.\n\n3.3 Outage Probability of Dn for RIS-OMA\n\nFor RIS-OMA networks, Dn detects its own signal sn in each time slot. Hence the outage occurs for Dn when its own signal cannot be successfully detected. According to Eq. (11), the outage probability of Dn for RIS-OMA networks can be expressed as\n\nPDnOMA=Pr(γn,DnsnOMA<γthnOMA),(20)\n\nwhere γthnOMA=22rnOMA1 is the target SNR with rnOMA being the target rate of Dn to detect signal sn. The derivation process of the outage probability for RIS-OMA networks is similar to that of RIS-NOMA networks. The outage probability of Dn for RIS-OMA networks can be provided in the following corollary.\n\nCorollary 3.1. Under Rician fading channels, the closed-form expression for the outage probability of Dn for RIS-OMA networks can be approximated as\n\nPDnOMA1Γ(Kμn2Ωn)γ(Kμn2Ωn,μnτnΩn),(21)\n\nwhere τn=γthnOMAρ.\n\n3.4 Diversity Analysis\n\nIn communication systems, the diversity order can be defined as the slope of the outage probability at high SNR. The diversity order describes the speed at which the outage probability decreases with the transmit SNR, which is usually used to evaluate the outage performance of wireless communication networks. The diversity order can be expressed as\n\nd=limρlog(PD(ρ))log(ρ).(22)\n\nIn order to obtain further insights, the approximate results of the outage probability are analyzed at high SNR, which can be used to determine the diversity order. The asymptotic outage probabilities of D1 and D2 at high SNR for RIS-NOMA networks can be provided in the following corollary.\n\nCorollary 3.2. Based on Eqs. (17) and (19), when ρ, the asymptotic expressions for outage probability of D1 and D2 for RIS-NOMA networks can be respectively given by\n\nPD1Ω1Kμ121Γ(Kμ12Ω1)(μ1λΩ1)Kμ12Ω1,(23)\n\nand\n\nPD2Ω2Kμ221Γ(Kμ22Ω2)(μ2λ2Ω2)Kμ22Ω2.(24)\n\nProof: Applying the series representation , when ρ, we have λ0 and λ20, by extracting the first term (n=0) of series representation, the high SNR approximate expressions can be given by\n\nγ(Kμ12Ω1,μ1λΩ1)Ω1Kμ12(μ1λΩ1)Kμ12Ω1,(25)\n\nand\n\nγ(Kμ22Ω2,μ2λ2Ω2)Ω2Kμ22(μ2λ2Ω2)Kμ22Ω2.(26)\n\nSubstituting Eqs. (25) and (26) into Eqs. (17) and (19), respectively, Eqs. (23) and (24) can be obtained. The proof is completed.\n\nRemark 3.3. From Eqs. (23) and (24), it can be observed that the asymptotic outage probabilities of D1 and D2 for RIS-NOMA are related to ρ, K, κ, and α. Hence the asymptotic outage probabilities of D1 and D2 for RIS-NOMA increase with the increase of ρ, K, κ, and α at high SNR.\n\nSubstituting Eqs. (23) and (24) into Eq. (22), the diversity orders of D1 and D2 for RIS-NOMA networks can be respectively calculated as\n\ndD1=limρlog(PD1(ρ))log(ρ)=Kμ122Ω1,(27)\n\nand\n\ndD2=limρlog(PD2(ρ))log(ρ)=Kμ222Ω2.(28)\n\nRemark 3.4. From Eqs. (27) and (28), we can observe that the diversity orders of D1 and D2 for RIS-NOMA are influenced by the number of reflecting elements of RIS, the Rician factor, and the path loss exponent.\n\nRemark 3.5. It can be observed that the diversity orders increase as K, κ, and α increase. The outage probability curve becomes steeper with the increase of the diversity order, resulting in the faster convergence of the outage probability. Therefore, increasing the diversity order can improve the outage performance of RIS-NOMA at high SNR.\n\nRemark 3.6. Similarly, the diversity order of Dn for RIS-OMA is Kμn22Ωn, which is also in connection with K, κ, and α.\n\n3.5 Delay-Limited System Throughput\n\nThe system throughput is an important performance metric in wireless communication systems. The research on system throughput is of great significance for practical implementation. In the delay-limited transmission mode, the information is transmitted from the BS to users at a constant rate, which leads to outage due to the influence of the wireless fading channels. Therefore, the system throughput depends on the outage probability in the delay-limited transmission mode. The delay-limited system throughput of RIS-NOMA is expressed as\n\nRDL=(1PD1)r1+(1PD2)r2,(29)\n\nwhere PD1 and PD2 are the outage probabilities from Eqs. (17) and (19), respectively.\n\n4  Ergodic Rate\n\nThe ergodic rate is a commonly used system performance metric in adaptive-rate transmission systems. In this section, the ergodic rates of two users for RIS-NOMA networks are analyzed.\n\n4.1 Ergodic Rate of D1\n\nBased on SIC principle, if D1 can successfully detect the signal of D2, the achievable rate of D1 can be given by RD1=log(1+γD1s1). On account of Eq. (8), the ergodic rate of D1 for RIS-NOMA networks is expressed as\n\nRD1erg=E[log(1+γD1s1)].(30)\n\nThe ergodic rate of D1 is obtained in the following theorem.\n\nTheorem 4.1. Under Rician fading channels, the exact expression for the ergodic rate of D1 for RIS-NOMA is calculated as\n\nRD1erg=ρa1ln20Γ(ω1+1)γ(ω1+1,yυ1)(1+ρa1y)Γ(ω1+1)dy,(31)\n\nwhere ω1=Kμ12Ω11 and υ1=Ω1μ1.\n\nProof: See Appendix C.\n\nRemark 4.1. From Eq. (31), we can observe that the ergodic rate of D1 for RIS-NOMA is dependent on ρ, K, κ, and α when the target SNR and the power allocation coefficient are constants. More specifically, with the increase of ρ, K, κ, and α, the ergodic rate of D1 for RIS-NOMA increases.\n\n4.2 Ergodic Rate of D2\n\nAssuming that D2 can detect its own signal s2, the achievable rate of D2 can be given by RD2=log(1+γD2s2). In virtue of Eq. (9), the ergodic rate of D2 for RIS-NOMA networks is expressed as\n\nRD2erg=E[log(1+γD2s2)].(32)\n\nThe ergodic rate of D2 is obtained in the following theorem.\n\nTheorem 4.2. Under Rician fading channels, the approximate expression for the ergodic rate of D2 for RIS-NOMA is calculated as\n\nRD2ergπLln2l=1La21tl22a1+a2(tl+1)[11Γ(ω2+1)γ(ω2+1,1υ2tl+1ρa1(1tl))],(33)\n\nwhere tl=cos(2l12Lπ), ω2=Kμ22Ω21, and υ2=Ω2μ2.\n\nProof: See Appendix D.\n\nRemark 4.2. From Eq. (33), it is observed that the ergodic rate of D2 for RIS-NOMA is related to ρ, K, κ, and α. Similarly, the ergodic rate of D2 for RIS-NOMA increases with ρ, K, κ, and α.\n\n4.3 Ergodic Rate of Dn for RIS-OMA\n\nFor RIS-OMA networks, since Dn can detect its own signal sn, the achievable rate of Dn can be given by RDn,OMA=12log(1+γn,DnsnOMA). According to Eq. (11), the ergodic rate of Dn for RIS-OMA networks can be expressed as\n\nRDn,OMAerg=E[12log(1+γn,DnsnOMA)].(34)\n\nSimilar to the derivation process of Eq. (31), the ergodic rate of Dn can be presented in the following corollary.\n\nCorollary 4.1. Under Rician fading channels, the exact expression for the ergodic rate of Dn for RIS-OMA is given by\n\nRDn,OMAerg=ρ2ln20Γ(ωn+1)γ(ωn+1,yυn)(1+ρy)Γ(ωn+1)dy.(35)\n\n4.4 Slope Analysis\n\nSimilar to the diversity order, the high SNR slope can be defined as the slope of the ergodic rate at high SNR, which describes how fast the ergodic rate increases with the transmit SNR. The high SNR slope can be expressed as\n\nS=limρRD,erg(ρ)log(ρ).(36)\n\nTo gain deep insights into the system performance, the approximate ergodic rate at high SNR is studied, which can be used to calculate the high SNR slope. However, the approximate ergodic rate of D1 at high SNR cannot be obtained from Eq. (31). For the convenience of analyses, the upper bound for the ergodic rate of D1 can be derived in the following theorem.\n\nTheorem 4.3. Under Rician fading channels, the upper bound for the ergodic rate of D1 for RIS-NOMA can be calculated as\n\nRD1,UBerg=log(1+ρa1((Kμ1)2+KΩ1)).(37)\n\nProof: Based on Eq. (30), by adopting the Jensen’s inequality, the upper bound for the ergodic rate of D1 can be given by\n\nRD1erg=E[log(1+|H1Hϑ1G1|2ρa1)]log(1+ρa1E(|H1Hϑ1G1|2)).(38)\n\nOn the basis of Eq. (12), the upper bound for the ergodic rate of D1 can be expressed as\n\nRD1,UBerg=log(1+ρa1E(|X1|2)).(39)\n\nAccording to Eqs. (14) and (15), the upper bound for the ergodic rate of D1 can be calculated as\n\nRD1,UBerg=log(1+ρa1(E2(X1)+D(X1)))=log(1+ρa1((Kμ1)2+KΩ1)).(40)\n\nHence Eq. (37) can be obtained. The proof is completed.\n\nRemark 4.3. From Eq. (37), we can observe that the upper bound for the ergodic rate of D1 for RIS-NOMA is related to ρ, K, κ, and α. Specifically, the upper bound for the ergodic rate of D1 for RIS-NOMA increases with ρ, K, κ, and α.\n\nThe asymptotic upper bound for the ergodic rate of D1 at high SNR for RIS-NOMA networks can be provided in the following corollary.\n\nCorollary 4.2. Based on Eq. (37), when ρ, the asymptotic upper bound for the ergodic rate of D1 for RIS-NOMA can be given by\n\nRD1,UBerg,=log(ρa1((Kμ1)2+KΩ1)).(41)\n\nRemark 4.4. From Eq. (41), we can observe that the asymptotic upper bound for the ergodic rate of D1 for RIS-NOMA depends on ρ, K, κ, and α. Similarly, increasing ρ, K, κ, and α, the asymptotic upper bound for the ergodic rate of D1 for RIS-NOMA increases at high SNR.\n\nSubstituting Eq. (41) into Eq. (36), the high SNR slope of D1 for RIS-NOMA networks can be calculated as\n\nSD1=limρRD1,UBerg,(ρ)log(ρ)=1.(42)\n\nRemark 4.5. From Eq. (42), we can observe that the high SNR slope of D1 for RIS-NOMA is a non-zero value. This implies that the ergodic rate of D1 for RIS-NOMA tends to infinity at high SNR. The higher the high SNR slope, the steeper the ergodic rate curve, which results in the faster convergence of the ergodic rate. Hence the ergodic rate increases with the increase of the high SNR slope.\n\nThe asymptotic ergodic rate of D2 at high SNR for RIS-NOMA networks can be presented in the following corollary.\n\nCorollary 4.3. Based on Eq. (32), when ρ, the asymptotic ergodic rate of D2 for RIS-NOMA can be calculated as\n\nRD2erg,=log(1+a2a1).(43)\n\nProof: With the aid of Eqs. (9) and (12), we can obtain\n\nlimργD2s2=limρρ|H2Hϑ2G2|2a2ρ|H2Hϑ2G2|2a1+1=limρρ|X2|2a2ρ|X2|2a1+1=a2a1.(44)\n\nIn view of Eq. (32), when ρ, the asymptotic ergodic rate of D2 can be calculated as\n\nRD2erg,=limρE[log(1+γD2s2)]=E[log(1+limργD2s2)]=log(1+a2a1).(45)\n\nHence Eq. (43) can be obtained. The proof is completed.\n\nRemark 4.6. From Eq. (43), it can be observed that the asymptotic ergodic rate of D2 for RIS-NOMA is a constant when the power allocation coefficient is fixed. Hence the asymptotic ergodic rate of D2 for RIS-NOMA remains unchanged at high SNR.\n\nSubstituting Eq. (43) into Eq. (36), the high SNR slope of D2 for RIS-NOMA networks can be calculated as\n\nSD2=limρRD2erg,(ρ)log(ρ)=0.(46)\n\nRemark 4.7. From Eq. (46), it can be observed that the high SNR slope of D2 for RIS-NOMA is zero. This means that the ergodic rate of D2 for RIS-NOMA is a constant. Hence the ergodic rate of D2 for RIS-NOMA converges to a throughput ceiling at high SNR.\n\nSimilar to the derivation process of Eq. (37), the upper bound for the ergodic rate of Dn for RIS-OMA networks can be obtained in the following corollary.\n\nCorollary 4.4. Under Rician fading channels, the upper bound for the ergodic rate of Dn for RIS-OMA is given by\n\nRDn,OMAerg,UB=12log(1+ρ((Kμn)2+KΩn)).(47)\n\nThe asymptotic upper bound for the ergodic rate of Dn at high SNR for RIS-OMA networks can be provided in the following corollary.\n\nCorollary 4.5. Based on Eq. (47), when ρ, the asymptotic upper bound for the ergodic rate of Dn for RIS-OMA can be given by\n\nRDn,OMAerg,UB,=12log(ρ((Kμn)2+KΩn)).(48)\n\nRemark 4.8. Substituting Eq. (48) into Eq. (36), the high SNR slope of Dn for RIS-OMA networks is equal to 12.\n\n4.5 Delay-Tolerant System Throughput\n\nIn the delay-tolerant transmission mode, the information is transmitted from the BS to users at any constant rate, which is subject to the channel conditions of the users. Hence the delay-tolerant system throughput of RIS-NOMA is expressed as\n\nRDT=RD1erg+RD2erg,(49)\n\nwhere RD1erg and RD2erg are the ergodic rates from Eqs. (31) and (33), respectively.\n\n5  Numerical Results\n\nIn this section, numerical results are presented to evaluate the performance of RIS-NOMA networks over Rician fading channels. The effects of the Rician factor, the number of reflecting elements of the RIS, and the pass loss exponent on the performance of RIS-NOMA networks are discussed. The accuracy of the derived theoretical results can be verified by Monte Carlo simulations. In order to compare the performance with the RIS-NOMA networks, the conventional RIS-OMA networks are also provided. Without loss of generality, the power allocation factors of two users are set to a1=0.2 and a2=0.8, the target rates of two users are set to r1=0.0005 bits per channel use (BPCU) and r2=0.002 BPCU. In order to simplify the calculation, we adopt the normalized distance for dimensionless physical quantity. It is assumed that the distances from the BS to R2 and from R2 to D2 are normalized to unity. The normalized distances from BS to R1 and R2, from R1 to D1, and from R2 to D2 are set to d1,1=0.4, d2,1=0.5, d1,2=0.4, and d2,2=0.5, respectively. For clarity, the main parameters used for numerical results are summarized in Table 1.", null, "5.1 Outage Probability\n\nFig. 2 plots the outage probabilities of two users vs. the transmit SNR for RIS-NOMA networks. The parameters are set to K=8, κ=10 dB, and α=2. We can observe that the accurate outage probability curves given by Monte Carlo simulations are in good agreement with the derived theoretical outage probability results for RIS-NOMA networks, which verifies the accuracy of the derived theoretical outage probability results. It can be observed that the outage performance of RIS-NOMA networks is better than that of RIS-OMA networks. The reason is that RIS-NOMA can provide services for multiple users in the same time slot by adopting the superposition coding scheme, while RIS-OMA can only provide services for one user in each time slot. One can observe that the asymptotic outage probability curves converge to the derived theoretical results in the high SNR regime, which also demonstrates the accuracy of the theoretical analysis. As can be observed that the outage probability decreases with the increase of the transmit SNR. One can make the following observation from figure that the outage probabilities of two users converge to zeros in the high SNR regime and thus the non-zero diversity order can be obtained.", null, "Figure 2: Outage probability vs. the transmit SNR when K = 8, κ = −10 dB, and α = 2\n\nFig. 3 plots the outage probabilities of two users vs. the transmit SNR with different K for RIS-NOMA networks. The parameters are set to κ=10 dB and α=2. It is observed that the outage probabilities of two users for RIS-NOMA decrease with increasing the number of reflecting elements K, thus the outage performance for RIS-NOMA can be improved. The reason is that the diversity orders of two users for RIS-NOMA depend on the number of reflecting elements.", null, "Figure 3: Outage probability vs. the transmit SNR with different K when κ = −10 dB and α = 2\n\nIn order to explain the influence of the Rician factor on the outage performance of RIS-NOMA networks, Fig. 4 plots the outage probabilities of two users vs. the transmit SNR with different κ for RIS-NOMA networks. The parameters are set to K=8 and α=2. As can be seen from the figure that as the value of the Rician factor κ increases, the outage probabilities of two users for RIS-NOMA reduce, hence the outage performance of RIS-NOMA becomes better. This is due to the fact that the diversity orders of two users for RIS-NOMA are related to the value of the Rician factor. The reason behind this phenomenon is that there are a large number of LoS components of Rician fading channels, which greatly improves the outage performance of RIS-NOMA networks.", null, "Figure 4: Outage probability vs. the transmit SNR with different κ when K = 8 and α = 2\n\nTo further illustrate the impact of the path loss exponent on the outage performance of RIS-NOMA networks, Fig. 5 plots the outage probabilities of two users vs. the transmit SNR with different α for RIS-NOMA networks. The parameters are set to K=8 and κ=10 dB. It can be seen that the outage probabilities of two users for RIS-NOMA increase significantly with the decrease of the path loss exponent α, therefore the outage performance for RIS-NOMA deteriorates sharply. This is because that the path loss exponent reflects the number of obstacles in the propagation environment.", null, "Figure 5: Outage probability vs. the transmit SNR with different α when K = 8 and κ = −10 dB\n\n5.2 Ergodic Rate\n\nFig. 6 plots the ergodic rates of two users vs. the transmit SNR for RIS-NOMA networks. The parameters are set to K=20, κ=10 dB, and α=2. We can see from the figure that the Monte Carlo simulation ergodic rate curves are completely consistent with the derived theoretical ergodic rate results for RIS-NOMA, which demonstrates the accuracy of the derived theoretical ergodic rate results. Another observation is that the upper bound curve for ergodic rate of D1 is very close to the derived theoretical ergodic rate. As can be observed from this figure that the ergodic rate of D2 converges to a throughput ceiling at high SNR and therefore leads to a zero high SNR slope. The important observation is that the ergodic rate of D1 for RIS-NOMA is higher than that of D1 for RIS-OMA at high SNR. The reason is that the high SNR slope of D1 for RIS-NOMA exceeds that of D1 for RIS-OMA. On the other hand, the ergodic rate of D2 for RIS-NOMA is lower than that of D2 for RIS-OMA at high SNR. This is because that D2 for RIS-NOMA has a smaller the high SNR slope compared with D2 for RIS-OMA.", null, "Figure 6: Ergodic rate vs. the transmit SNR when K = 20, κ = −10 dB, and α = 2\n\nConsidering the influence of the number of reflecting elements on the ergodic performance of RIS-NOMA networks, Fig. 7 plots the ergodic rates of two users vs. the transmit SNR with different K for RIS-NOMA networks. The parameters are set to κ=10 dB and α=2. We can observe from the figure that the ergodic rates of two users for RIS-NOMA increase as the number of reflecting elements K increases. Another observation is that the ergodic rate of D1 is extremely large with respect to that of D2 for RIS-NOMA with the increase of the number of reflecting elements. This is due to the fact that there is a throughput ceiling at high SNR for the ergodic rate of D2 for RIS-NOMA.", null, "Figure 7: Ergodic rate vs. the transmit SNR with different K when κ = −10 dB and α = 2\n\n5.3 System Throughput\n\nFig. 8 plots the system throughput vs. the transmit SNR with different K for RIS-NOMA networks in delay-limited transmission mode. The parameters are set to κ=10 dB and α=2. It can be seen that RIS-NOMA is capable of enhancing the system throughput compared to RIS-OMA. The reason is that the system throughput depends on the outage probability in delay-limited transmission mode. One can make the following observation from figure that the system throughput of RIS-NOMA converges to a constant at high SNR. This phenomenon indicates that the system throughput no longer increases with the increase of SNR when it reaches a fixed value at high SNR. Another observation is that as the number of reflecting elements increases, the system throughput of RIS-NOMA can be significantly improved.", null, "Figure 8: System throughput in delay-limited transmission mode vs. the transmit SNR with different K when κ = −10 dB and α = 2\n\nAs a further advance, Fig. 9 plots the system throughput vs. the transmit SNR with different K for RIS-NOMA networks in delay-tolerant transmission mode. The parameters are set to κ=10 dB and α=2. It can be observed that RIS-NOMA can achieve a larger system throughput than RIS-OMA at high SNR. The main reason behind this is that the system throughput is related to the ergodic rate in delay-tolerant transmission mode. It is observed from the figure that the system throughput of RIS-NOMA becomes larger with increasing the number of reflecting elements.", null, "Figure 9: System throughput in delay-tolerant transmission mode vs. the transmit SNR with different K when κ = −10 dB and α = 2\n\n6  Conclusion\n\nIn this paper, the system performance of RIS-assisted downlink NOMA communication networks has been investigated in terms of the outage probability, ergodic rate, and system throughput over Rician fading channels. The exact and asymptotic expressions for the outage probability and ergodic rate of two users for RIS-NOMA networks have been derived. Based on the theoretical analyses, the diversity orders and high SNR slopes of the two users have been obtained at high SNR. In addition, the system throughputs of RIS-NOMA networks have been derived in delay-limited and delay-tolerant transmission modes. Simulation results have indicated that the outage probability and ergodic rate of RIS-NOMA outperform that of RIS-OMA. It has been shown that the system throughput of RIS-NOMA is superior to that of RIS-OMA. Moreover, it has been demonstrated that the system performance of RIS-NOMA networks becomes better with increasing the number of reflecting elements and Rician factor. Based on this work, we will consider extending the research to multiple users in the future. In addition, optimized power allocation and multiple antennas of device nodes are also promising research directions.\n\nFunding Statement: This work was supported in part by the Key R&D Program of Zhejiang Province under Grant No. 2020C05005, in part by the National Natural Science Foundation of China under Grants U2033215, 91738301, 91538202, and 91638301, in part by the Program for New Century Excellent Talents in University under Grant NCET-09-0025, and in part by the Fundamental Research Funds for the Central Universities. The work of C. Huang was partially supported by the China National Key R&D Program under Grant 2021YFA1000500, National Natural Science Foundation of China under Grant 62101492, Zhejiang Provincial Natural Science Foundation of China under Grant LR22F010002, Distinguished Young Scholars of the National Natural Science Foundation of China, Ng Teng Fong Charitable Foundation in the form of ZJU-SUTD IDEA Grant, Zhejiang University Education Foundation Qizhen Scholar Foundation, and Fundamental Research Funds for the Central Universities under Grant 2021FZZX001-21. The work of X. Yue was supported by the National Natural Science Foundation of China under Grant 62071052 and the R&D Program of Beijing Municipal Education Commission under Grant KM202011232003.\n\nConflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.\n\n## References\n\n1. Ding, Z., Liu, Y., Choi, J., Sun, Q., & Elkashlan, M. (2017). 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Non-orthogonal multiple access (NOMA) with multiple intelligent reflecting surfaces. IEEE Transactions on Wireless Communications, 20(11), 7184-7195. [Google Scholar] [CrossRef]\n28. Hemanth, A., Umamaheswari, K., Pogaku, A. C., Do, D. T., & Lee, B. M. (2020). Outage performance analysis of reconfigurable intelligent surfaces-aided NOMA under presence of hardware impairment. IEEE Access, 8, 212156-212165. [Google Scholar] [CrossRef]\n29. Cheng, Y., Li, K. H., Liu, Y., Teh, K. C., & Vincent Poor, H. (2021). Downlink and uplink intelligent reflecting surface aided networks: NOMA and OMA. IEEE Transactions on Wireless Communications, 20(6), 3988-4000. [Google Scholar] [CrossRef]\n30. Gong, X., Huang, C., Yue, X., Liu, F., & Yang, Z. (2022). Performance analysis for reconfigurable intelligent surface assisted downlink NOMA networks. IET Communications, 16(13), 1593-1605. [Google Scholar] [CrossRef]\n31. Simon, M. K. (2006). Probability distributions involving gaussian random variables. USA: Springer.\n32. Gradshteyn, I. S., Ryzhik, I. M. (2000). Table of integrals, series and products. 6th edition. New York, NY, USA: Academic Press.\n33. Primak, S., Kontorovitch, V., Lyandres, V. (2004). Stochastic methods and their applications to communications: Stochastic differential equations approach. West Sussex, UK: Wiley.\n34. Hildebrand, B. F. (1987). Introduction to numerical analysis. New York, NY, USA: Dover.\n\nAppendix A: Proof of Theorem 1\n\nSubstituting Eqs. (7) and (8) into Eq. (16), based on Eq. (12), the outage probability of D1 for RIS-NOMA networks is expressed as\n\nPD1=1Pr(|H1Hϑ1G1|2>λ1,|H1Hϑ1G1|2>λ2)=1Pr(|H1Hϑ1G1|2>max(λ1,λ2))=1Pr(|H1Hϑ1G1|2>λ)=Pr(k=1K|G1kH1k|<λ),(A.1)\n\nwhere λmax(λ1,λ2), λ1=γth1ρa1, and λ2=γth2ρ(a2a1γth2) with a2>a1γth2.\n\nApplying the series of Laguerre polynomials , the PDF of X1=k=1K|G1kH1k| is approximated as\n\nfX1(x)xω1(υ1)ω1+1Γ(ω1+1)exp(xυ1),(A.2)\n\nwhere ω1=Kμ12Ω11 and υ1=Ω1μ1.\n\nSubstituting Eq. (A.2) into Eq. (A.1), the outage probability of D1 can be approximated as\n\nPD1=Pr(X1<λ)0λxω1(υ1)ω1+1Γ(ω1+1)exp(xυ1)dx=1(υ1)ω1+1Γ(ω1+1)0λxω1exυ1dx.(A.3)\n\nUsing t=xυ1, the outage probability of D1 is expressed as\n\nPD11Γ(ω1+1)0λυ1tω1etdt.(A.4)\n\nApplying the lower incomplete Gamma function , the outage probability of D1 can be calculated as\n\nPD11Γ(ω1+1)0λυ1t(ω1+1)1etdt=1Γ(ω1+1)γ(ω1+1,λυ1)=1Γ(Kμ12Ω1)γ(Kμ12Ω1,μ1λΩ1),(A.5)\n\nwhere Ω1=α1,1α1,2{1π216(1+κ)2[L12(κ)]4} and μ1=πα1,1α1,24(1+κ)[L12(κ)]2. Thus, Eq. (17) can be obtained. The proof is completed.\n\nAppendix B: Proof of Theorem 2\n\nSubstituting Eq. (9) into Eq. (18), the outage probability of D2 for RIS-NOMA networks can be expressed as\n\nPD2=Pr(ρ|H2Hϑ2G2|2a2ρ|H2Hϑ2G2|2a1+1<γth2)=Pr(|H2Hϑ2G2|2<λ2)=Pr(k=1K|G2kH2k|<λ2),(B.1)\n\nwhere the last equality holds because of Eq. (12).\n\nSimilarly, the PDF of X2=k=1K|G2kH2k| by utilizing the series of Laguerre polynomials can be approximated as\n\nfX2(x)xω2(υ2)ω2+1Γ(ω2+1)exp(xυ2),(B.2)\n\nwhere ω2=Kμ22Ω21 and υ2=Ω2μ2.\n\nSubstituting Eq. (B.2) into Eq. (B.1), using u=xυ2, the outage probability of D2 can be approximated as\n\nPD2=Pr(X2<λ2)0λ2xω2(υ2)ω2+1Γ(ω2+1)exp(xυ2)dx=1Γ(ω2+1)0λ2υ2uω2eudu.(B.3)\n\nBy referring to the lower incomplete Gamma function, the outage probability of D2 can be calculated as\n\nPD21Γ(ω2+1)γ(ω2+1,λ2υ2)=1Γ(Kμ22Ω2)γ(Kμ22Ω2,μ2λ2Ω2),(B.4)\n\nwhere μ2=πα2,1α2,24(1+κ)[L12(κ)]2 and Ω2=α2,1α2,2{1π216(1+κ)2[L12(κ)]4}. Hence Eq. (19) can be obtained. The proof is completed.\n\nAppendix C: Proof of Theorem 3\n\nSubstituting Eq. (8) into Eq. (30), according to Eq. (12), the ergodic rate of D1 for RIS-NOMA networks can be expressed as\n\nRD1erg=E[log(1+|H1Hϑ1G1|2ρa1)]=E[log(1+|k=1K|G1kH1k||2Y1ρa1)]=ρa1ln201FY1(y)1+ρa1ydy.(C.1)\n\nThe CDF of Y1 can be written as\n\nFY1(y)=Pr(|k=1K|G1kH1k||2<y)=Pr(X1<y).(C.2)\n\nBased on Eq. (17), the CDF of Y1 is given by\n\nFY1(y)1Γ(ω1+1)γ(ω1+1,yυ1).(C.3)\n\nSubstituting Eq. (C.3) into Eq. (C.1), the ergodic rate of D1 can be approximated as\n\nRD1ergρa1ln20Γ(ω1+1)γ(ω1+1,yυ1)(1+ρa1y)Γ(ω1+1)dy.(C.4)\n\nHence Eq. (31) can be obtained. The proof is completed.\n\nAppendix D: Proof of Theorem 4\n\nSubstituting Eq. (9) into Eq. (32), in view of Eq. (12), the ergodic rate of D2 for RIS-NOMA networks can be expressed as\n\nRD2erg=E[log(1+ρ|H2Hϑ2G2|2a2ρ|H2Hϑ2G2|2a1+1)]=E[log(1+ρ|k=1K|G2kH2k||2a2ρ|k=1K|G2kH2k||2a1+1Y2)]=1ln201FY2(y)1+ydy.(D.1)\n\nThe CDF of Y2 can be written as\n\nFY2(y)=Pr(ρ|k=1K|G2kH2k||2a2ρ|k=1K|G2kH2k||2a1+1<y)=Pr(|k=1K|G2kH2k||2<yρ(a2a1y))=Pr(X2<yρ(a2a1y)),(D.2)\n\nwhere a2>a1y. According to Eq. (19), the CDF of Y2 can be given by\n\nFY2(y)1Γ(ω2+1)γ(ω2+1,1υ2yρ(a2a1y)).(D.3)\n\nSubstituting Eq. (D.3) into Eq. (D.1), the ergodic rate of D2 can be approximated as\n\nRD2erg1ln20a2a111+y[11Γ(ω2+1)γ(ω2+1,1υ2yρ(a2a1y))]dy.(D.4)\n\nUsing t=2(a1a2)y1, the ergodic rate of D2 is expressed as\n\nRD2erg1ln211a22a1+a2(t+1)[11Γ(ω2+1)γ(ω2+1,1υ2t+1ρa1(1t))]dt.(D.5)\n\nApplying Chebyshev-Gauss quadrature , the ergodic rate of D2 can be calculated as\n\nRD2ergπLln2l=1La21tl22a1+a2(tl+1)[11Γ(ω2+1)γ(ω2+1,1υ2tl+1ρa1(1tl))],(D.6)\n\nwhere tl=cos(2l12Lπ). Hence Eq. (33) can be obtained. The proof is completed.\n\nGong, X., Huang, C., Yue, X., Yang, Z., Liu, F. (2023). Performance Analysis of RIS Assisted NOMA Networks over Rician Fading Channels. CMES-Computer Modeling in Engineering & Sciences, 135(3), 2531–2555.", null, "This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.\n\nView\n\nLike\n\n## Related articles\n\n• Shanshan Yu, Ju Liu, Xiaoqing...\n• Yinghua Zhang, Yanfang Dong, Lei...\n• Lei Wang, Jian Liu, Alan Yang\n• Ruoyu Su, Xiaojun Sun, Fei Ding,...\n• Jian Liu, Lei Wang, Changming...", null, "", null, "", null, "" ]

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[ "# Control Statements for Printing Variables\n\nThere are two control statements that provide the ability to select data output by SIMPLIS. These are .PRINT and .KEEP. These are detailed in the following sections.\n\nIn this topic:\n\n## .PRINT\n\nThe .PRINT statement is used to specify the output variables to be recorded for printing/plotting. Note that with the SIMetrix/SIMPLIS, .PRINT only specifies data to be saved in the binary file and does not create ASCII tabular output. .PRINT instructs SIMPLIS to send the specified data to the SIMetrix front end. SIMetrix saves the data as a vector in its binary file. The actual vector name used is described in the following paragraphs.\n\nThe format of .PRINT is:\n\n`.PRINT var1 var2 ...`\nwhere var1, var2, and ... are legal print variables. Forms of legal print variables and their meanings are listed below:\n V(DName) Branch voltage across a two-terminal device in the current circuit. DName is the device name of a device whose element keyword is one of the following: R, L, C, V, I, E, G, H, F, Q, S, !R, !L, and !C SIMetrix vector name will be DName. V(#NodeName) Voltage on mapped node NodeName. Mapped nodes are created using the .NODE_MAP statement. SIMetrix vector name will be #NodeName. I(DName) Branch current through a two-terminal device in the current circuit. DName is the device name of a device whose element keyword is one of the following: R, L, C, V, I, E, G, H, F, Q, S, !R, !L, and !C SIMetrix vector name will be of the form: DName#pinname where pinname will be p for the first pin and n for the second pin. I(DName#pinname) Current through a device pin. DName is the device name while pinname is either a pin number or mapped pin name. If Dname is a subcircuit device and the subcircuit definition includes .NODE_MAP statements to map its external nodes to names, then those mapped names may be used for pinname. For example: ```X1 1 2 3 SUB1 .SUBCKT SUB1 100 200 300 .NODE_MAP INP 100 .NODE_MAP INN 200 .NODE_MAP OUT 300 ... ... .ENDS SUB1 .PRINT I(X1#INP)``` The above .PRINT will instruct SIMPLIS to output the current into pin connected to node 1 of X1. The resulting SIMetrix vector will be called \"X1#INP\". V(Node1,Node2) Differential voltage from node Node1 to node Node2 where Node1 and Node2 are node names in the current circuit. V(Node1) Voltage from Node1 to node 0, the ground node. This form is allowable only in the main circuit and only if node 0 is present in the main circuit. SIMetrix vector name will be: Node1 V(Xname1.Xname2.DName) Branch voltage across the device named DName in the subcircuit referred to as Xname2 in the subcircuit referred to as Xname1 in the current circuit. The allowable device is same as those listed for the form of V(DName). SIMetrix vector name will be: Xname1.Xname2.DName I(Xname1.Xname2.DName) Branch current through the device named DName in the subcircuit referred to as Xname2 in the subcircuit referred to as Xname1 in the current circuit. SIMetrix vector name will be of the form: Xname1.Xname2.DName#pinname where pinname will be p for the first pin and n for the second pin. V(Xname1.Xname2.Node1,Node2) Differential voltage from node Node1 to node Node2 where Node1 and Node2 are node numbers in the subcircuit referred to as Xname2 in the subcircuit referred to as Xname1 in the current circuit. NODE_V Print all node voltages in the main circuit with respect to the ground node, node 0, in the main circuit. ALL Print all node voltages in the main circuit with respect to the ground node, node 0, in the main circuit and print all branch currents of the two-terminal devices in the main circuit. The two-terminal devices refer to those whose branch current can be printed through the form of I(DName).\nIn the normal case, no more than 200 output variables can be created for printing/plotting in one simulation. If more than 200 output variables are needed, the output variables can be generated through more than one pass of the simulation.\n\nIf the print variable NODE_V or ALL is used, then node 0 must be present in the main circuit. Although the number of output variables that are associated with NODE_V or ALL are obviously large, NODE_V or ALL is counted as one print variable in counting towards the maximum number of 200 print variables allowed in one simulation. Simulation can be substantially slower with NODE_V or ALL as a printing variable. In addition, if ALL is used as the print variable, other print variables defined in the main circuit, or in any of its descendant subcircuits, are ignored.\n\nThere are two subtle points that need to be understood when figuring out the SIMetrix vector names. If DName or Xname as appeared in the .PRINT statement is a device/subcircuit name containing a dollar sign '\\$', then all characters ahead of the first dollar sign and the first dollar sign are stripped in the corresponding DName or Xname in the SIMetrix vector name. For example, the following .PRINT statement\n\n`.PRINT I(X\\$U12.XABC.R234)`\n\nwill result in a SIMetrix vector name of:\n\n`U12.XABC.R234#p`\n\nIn addition, the SIMetrix vector names shown for each type of print variables above assume the .PRINT statement is defined in the top-level circuit/schematic. If the .PRINT statement is defined in a subcircuit/component schematic, then the actual SIMetrix vector name will be prepended by a path prefix. Let us take the following two .PRINT statements as examples:\n\n```.PRINT V(#VOUT)\n.PRINT I(X\\$U12.XABC.R234)```\n\nIf these .PRINT statements were defined in the top-level circuit/schematic, the SIMetrix vector names would be #VOUT, and U12.XABC.R234#p, respectively. Suppose the subcircuit/component schematic containing these two .PRINT statements is two-level down from the top-level circuit/schematic. In addition, suppose that the subcircuit/component schematic containing these two .PRINT statements can be reached from the top-level schematic by first descending into a component schematic named \"Component A\" with a reference designator of U76 and then descending into a component schematic named \"Component B\" with a reference designator of U34 as shown in the figure below.", null, "The SIMetrix vector names associated with the same two .PRINT statements would then be U76.U34.#VOUT and U76.U34.U12.XABC.R234#p, respectively.\n\n## .KEEP\n\nThe purpose of the .KEEP statement is to instruct SIMPLIS in a very generic way to produce and save voltages/currents for printing/plotting in the subcircuits/child components, thus enabling the debugging of the sucbircuits/child component schematics through random probing without knowing in advance what specific voltages/currents needed to be produced and kept for printing/plotting. Similar to the .PRINT statement, the .KEEP statement in the SIMetrix/SIMPLIS environment only specifies data to be saved in the aforementioned binary file and does not create the ASCII tabular output.\n\nThe format of .KEEP is:\n\n`.KEEP keep_var1, keep_var2, ...`\nwhere keep_var1, keep_var2, and ... are legal keep variables.\n\nMultiple .KEEP statements may be defined in a subcircuit/schematic. Forms of legal keep variables and their meanings are listed below:\n\n *V All node voltages in the current circuit/schematic. With this keep variable specified, all node voltages in the current/schematic are produced and saved for printing/plotting. This keep variable is allowed only if node 0 is present in the top-level circuit/schematic because the node voltages are measured with respect to the voltage of node 0 in the top-level circuit/schematic. The SIMetrix vector names for the node voltages generated by the *V keep variable will be same as the SIMetrix vector names for the node voltages generated through .PRINT statements. So a node with a node number of NodeNum without any node mapping will result in a node voltage whose SIMetrix vector name is NodeNum, plus a path prefix if this node is not in the top-level schematic. Similarly, a node mapped to a name of NodeName will result in a node voltage whose SIMetrix vector name is #NodeName, plus a path prefix if this node is not in the top-level schematic. *I All printable/plottable currents for the current circuit/schematic. With this keep variable specified, all branch currents through two-terminal devices in the current circuit/schematic and all pin currents for subcircuit devices in the current circuit/schematic are produced and saved for printing/plotting. The SIMetrix vector names for the currents generated by the *I keep variable will be same as the SIMetrix vector names for the currents generated through .PRINT statements. So the SIMetrix vector name for a branch current will have the form of DName#pinname, where DName is the device name, and pinanme will be p for the first pin and n for the second pin. If the device is not in the top-level schematic, the SIMetrix vector name will have a path prefix. Similarly, the SIMetrix vector name for a subcircuit pin current will have the form of DName#pinname where DName is the device name, and pinname is either a pin number or a mapped pin name. Again, the SIMetrix vector name will have a path prefix if the device DName is not placed in the top-level schematic. **V This is similar to the *V keep variable used for producing node voltages. Specifying **V as a keep variable in the current circuit/schematic is equivalent to specifying *V in the current circuit/schematic and in all its descendant subcircuits/schematics. Hence, it will produce node voltages for the current circuit/schematic and all its descendant subcircuits/schematics for printing/plotting. Similar to the *V keep variable, this keep variable is allowed only if node 0 is present in the top-level circuit/schematic. For a very large hierarchical design, having the **V keep variable defined in the top-level schematic could result in a large number of print/plot variables and can lead to substantially slower simulations. **I This is similar to the *I keep variable used for producing branch and device pin currents. Specifying **I as a keep variable in the current circuit/schematic is equivalent to specifying *I in the current circuit/schematic and in all its descendant subcircuits/schematics. Hence, it will produce branch currents through two-terminal devices and subcircuit pin currents for the current circuit/schematic and all its descendant subcircuits/schematics for printing/plotting. For a very large hierarchical design, having the **I keep variable defined in the top-level schematic could result in a large number of print/plot variables and can lead to substantially slower simulations." ]

[ null, "https://help.simetrix.co.uk/8.4/simplis/library/images/simplis_reference/SIMPLISReference-39.png", null ]

[ "# An intrinsically curved surface with no extrinsic curvature\n\nAn intrinsic curvature is the property of a manifold itself. It shows it's deviation from euclidean geometry. The extrinsic curvature, on the other hand, depends on how the surface is embedded in a higher dimensional manifold. Now as a beginner student wherever I've seen examples of difference between these two I'm always led to the example of a cylinder. It has no intrinsic curvature but looking at it from R3 it does have an extrinsic curvature.\n\nMy question is : Is the opposite true? Can I have a surface with no extrinsic curvature(due to my \"clever\" embedding) but is intrinsically curved? If yes, what will be a simple example?\n\n• I am by no means an expert in this field, but may I ask what exterior manifolds you consider? Only embeddings in Euclidean spaces? Is a manifold embedded into itself considered to be extrinsically curved? – M. Winter Sep 6 '18 at 13:39\n• To answer the first part, No. I expect if the answer to my question is positive it's going to be embedded in a non euclidean space. For the second part, I don't know the concept of \"embedded into itself\". – Ari Sep 6 '18 at 14:48\n• @Ali So you have a concept of \"relative curvature\" w.r.t. to the embedding manifold? I mean if you have a concept of embedding a manifold into other manifolds, then the identity map $\\mathrm{id}:M\\to M$ can be seen as an embedding of $M$ into itself. And what would be the \"relative curvature\" here? – M. Winter Sep 6 '18 at 14:50\n• Let me formulate it that way: I would consider a manifold as embedded in an extrinsically flat way, already when all its geodesics are mapped to geodesiscs of the embedding manifold. This is true for self-embeddings and I would assume there are surfaces in non-flat 3-manifolds that only contain geodesics. So there would give flat embeddings of intrisically curved 2-manifolds. – M. Winter Sep 6 '18 at 15:08\n\nLet's first clarify the terminology and setting. Let's assume that we have an embedded submanifold $S$ (e.g. a surface) in a Riemannian manifold $M$ (e.g. Euclidean 3-space).\n\nI assume that by intrinsic curvature of $S$, you mean the Gaussian curvature of a surface. For higher dimensional manifolds, this generalizes to the sectional curvature, but this is a little more complicated: it assigns a number to each 2-dimensional subspace of the tangent space, namely the Gaussian curvature of the submanifold (surface) tangent to that plane.\n\n\"Extrinsic curvature\" could mean several things, but for a hypersurface (e.g. an embedded surface $S$ in Euclidean 3-space), we could summarize by saying that an extrinsic curvature is a quantity defined by the second fundamental form, or equivalently its associated shape operator $B$. If you don't know what these things are, it's ok, you can keep reading. Think of the shape operator as a symmetric matrix depending on $p \\in S$. The main \"extrinsic curvatures\" that are worth considering are:\n\n• The eigenvalues of $B$, called principal curvatures. Note that they are equal to the curvature of curves lying in $S$, seen as curves in $M$.\n• The trace of $B$ (maybe divided by the dimension), called the mean curvature, equal to the sum (or average) of the principal curvatures.\n\nFor a surface in a 3-dimensional manifold, the Gauss equation says that: $$\\det B = K_S - K_M$$ where $K_S$ is the Gauss curvature of $S$ and $K_M$ is the sectional curvature in $M$ of the plane tangent to $S$. This equation is probably one of things you're looking for answering your question: it tells you the relation between the second fundamental form (defining the \"extrinsic curvatures\"), the intrinsic curvature of $S$ and the intrinsic curvature of $M$.\n\nNow let's answer your question more precisely. As you can tell from Gauss equation, if all the extrinsic curvatures are zero, i.e. $B$ vanishes (FYI, in this case one says that $S$ is a totally geodesic submanifold), then the intrinsic curvature of $S$ is equal to the intrinsic curvature of $M$. In particular, if $M$ has zero curvature (i.e. Euclidean 3-space), then a submanifold for which all the extrinsic curvatures are zero also has zero intrinsic curvature.\n\nThat being said, if by \"the extrinsic curvature\" we just signify the mean curvature $\\mathrm{tr}(B)$, we are just looking for surfaces with zero mean curvature, these things are called minimal surfaces. Now the question is: are there minimal surfaces that are not totally geodesic (basically, minimal surfaces that are not planes in Euclidean 3-space)? You can guess that the answer is probably \"yes, there are plenty\", because that's pretty much asking if there are some symmetric matrices $B$ whose trace is zero, but are not the zero matrix. In fact, the \"fundamental theorem of surface theory\" basically guarantees that there are many examples. You'll see examples by looking for images of \"minimal surface\" on the web.\n\nI tried to give a complete and detailed answer, I hope it was useful, but if you're looking for the short answer: yes, any minimal surface in Euclidean 3-space that is not a plane has zero mean curvature but nonzero Gaussian curvature. For example, the catenoid:", null, "• Perhaps this was in there somewhere, but the affirmative answer to the OP comes from taking a totally geodesic surface (so extrinsically non-curving) in a curved space. The simplest example will be a great $2$-sphere in the $3$-sphere $S^3$. – Ted Shifrin Sep 6 '18 at 21:56\n• Right, I wrote: \"If all the extrinsic curvatures are zero, i.e. B vanishes (FYI, in this case one says that S is a totally geodesic submanifold), then the intrinsic curvature of S is equal to the intrinsic curvature of M\". But as I also wrote, I find that a better answer is a minimal surface in Euclidean space. – Seub Sep 6 '18 at 22:34" ]

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[ "Mirandas financial aide stipulates that fer tuition not exceed 1,200 if her college charges $55 regestration fee for the term plus$450 per course what is the greatest number of courses miranda can register for_ answer should be whole number.\n\nCall the number of courses ’ C ’.\n\nWhatever it is, it will cost her [ 55 + 450C ].\n\nHow many courses can she get for $1,200? 55 + 450C = 1200 Subtract 55 from each side: 450C = 1145 Divide each side by 450 : C = 1145 / 450 = 2.544. She can only register for a whole number of courses, so 2 is the limit. That will cost her (55 + 2x450) = (55 + 900) =$955\nand she’ll have \\$245 left over. not enough for another course.\n\nMirandas can sign up for 2 courses\n\nRELATED:" ]

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[ "# How To Make Logic Circuit From Truth Table\n\nBy | February 25, 2023\n\nIf you are looking to make a logic circuit from a truth table—a diagram that shows what output each combination of inputs will give—then it’s important to know the steps involved. This process can be daunting for many due to its complexity, but with the right guidance and knowledge, it can be relatively straightforward.\n\nThe first step in this process is to set up the truth table with all the input combinations and the corresponding results written out. Make sure to include all the possibilities and confirm that the results are correct.\n\nNext, use a logic diagram, which consists of various elements—AND, OR, NOT, NAND, and NOR—that represent the conditions of the truth table. It's important to have the correct symbols and understand what they mean in order to make sure the circuit works as expected.\n\nThe third step is to convert the truth table into an equivalent logic diagram using the elements mentioned above. To do this, start by drawing arrows between the inputs and outputs, and then creating rectangles around the elements of the logic diagram that satisfy the table. Finally, add the elements inside the rectangles to get the final logic circuit.\n\nOnce the logic circuit is completed, it's time to test and check the results. The simplest way to check whether the circuit is working correctly is by using a logic simulation tool. With this tool, you can feed in different input combinations and compare the results to the truth table’s output. If everything matches up, then the circuit is working properly.\n\nCreating a logic circuit from a truth table may take some time and effort, but it can be done with the right knowledge and guidance. With plenty of practice and understanding, anyone can learn how to make this type of circuit, and potentially save time in the long run.", null, "Xor Gate Xnor Truth Table Symbol Boolean Expression Electrical4u", null, "Basic Logic Gates Truth Table", null, "Logic And Gate Tutorial 2 3 Input Truth Table Electronics Area", null, "Schematic Representation Of Logic Gate For Probe 1b And 2b A Not B Scientific Diagram", null, "Converting Truth Tables Into Boolean Expressions Algebra Electronics Textbook", null, "Circuitmix On Twitter Logic Gates Symbols And Truth Table Retweet If You Liked It Electronics Raspberrypi Arduino Ai Robotics Automation Iot Https T Co Wl9fombkei", null, "Truth Table Boolean Expression And Logic Gates Notes S Qa Tests Grade 11 Computer Science Function Algebra Kullabs", null, "Teaching Digital Logic Fundamentals Theory Simulation And Deployment Ni", null, "Logic Gates And Truth Tables Inst Tools", null, "Digital Electronics Logic Gates Basics Tutorial Circuit Symbols Truth Tables", null, "Logicblocks Digital Logic Introduction Learn Sparkfun Com", null, "Free Truth Table To Logic Circuit Converter Software For Windows", null, "Digital Electronics Logic Gates Basics Tutorial Circuit Symbols Truth Tables", null, "What Is The Easiest Way To Learn Truth Table Of Nor Gate And Xor Class 12th Cbse Quora", null, "Truth Tables Circuit Diagrams Of Logic Gates Your Electrical Guide", null, "8 Best Free Truth Table Calculator Software For Windows", null, "Digital Logic Mb5fdb0a87e2fa1的技术博客 51cto博客", null, "Breadboard 2 Logic Multiple Gates Joe S Hobby Electronics", null, "Basic Logic Gates", null, "Question Using Truth Tables To Find The Output Of Logic Circuits Nagwa" ]

[ null, "https://i2.wp.com/www.electrical4u.com/wp-content/uploads/XOR-and-XNOR-Gate-Truth-Table-Symbol.png", null, "https://i1.wp.com/theorycircuit.com/wp-content/uploads/2017/05/or-gate-truth-table.png", null, "https://i1.wp.com/electronicsarea.com/wp-content/uploads/2-input-3-input-logic-and-gate-truth-table.png", null, "https://i3.wp.com/www.researchgate.net/profile/Sivaraman-Gandhi/publication/325015897/figure/fig5/AS:657932110811138@1533874647937/Schematic-representation-of-logic-gate-for-probe-1b-and-2b-a-NOT-b-YES-and-c-AND.png", null, "https://i1.wp.com/www.allaboutcircuits.com/uploads/articles/converting-truth-tables-into-boolean-expressions9.jpg", null, "https://i0.wp.com/pbs.twimg.com/media/D5Q_RIoWkAAHHFT.jpg:large", null, "https://i3.wp.com/kullabs.com/uploads/not3.JPG", null, "https://i0.wp.com/ni.scene7.com/is/image/ni/image5250904457128845092", null, "https://i2.wp.com/instrumentationtools.com/wp-content/uploads/2018/10/Universal-Logic-Gates-and-Truth-tables.png", null, "https://i3.wp.com/www.circuitstoday.com/wp-content/uploads/2010/04/2-Input-AND-Gate-Truth-Table.jpg", null, "https://i3.wp.com/cdn.sparkfun.com/assets/learn_tutorials/2/1/5/example_circuit.png", null, "https://i1.wp.com/cdn.ilovefreesoftware.com/wp-content/uploads/2018/01/Truth-Table-to-Logic-Circuit-Converter-Software-for-Windows.png", null, "https://i2.wp.com/www.circuitstoday.com/wp-content/uploads/2010/04/2-Input-NOR-Gate-Truth-Table.jpg", null, "https://i2.wp.com/qph.cf2.quoracdn.net/main-qimg-3fc999bd79d2a129e0f9932d3e941273-lq", null, "https://i2.wp.com/www.yourelectricalguide.com/wp-content/uploads/2019/10/OR-gate.png", null, "https://i2.wp.com/static.listoffreeware.com/wp-content/uploads/logisim_truth-table-calculator_2018-01-02_15-19-12.png", null, "https://i2.wp.com/cdn.sparkfun.com/assets/9/e/4/5/b/524f44a3757b7fc74a8b4567.png", null, "https://i2.wp.com/www.hobbyelectronics.net/uploads/6/5/7/2/6572022/6753887.jpg", null, "https://i0.wp.com/grace.bluegrass.kctcs.edu/~kdunn0001/files/3_Basic_Logic_Gates/paste_image18.png", null, "https://i1.wp.com/media.nagwa.com/898129090264/en/thumbnail_l.jpeg", null ]

[ "Factors of 439\n\nHere we have calculated and created a list of all the factors of four hundred thirty-nine.\nThe factors of 439 are:\n\n1, 439\n\nFactors of 439 are integers that you can multiply together to get 439. Here are all the combinations of factors of 439 that you can multiply together to get 439:\n\n1 x 439 = 439\n439 x 1 = 439\n\nIf you divide 439 by a factor of 439, you will get one of the other factors of 439. Here are all combinations of dividing 439 by its factors:\n\n439 / 1 = 439\n439 / 439 = 1\n\nIn summary, we have not only listed all the factors of 439, but also illustrated that our solution is correct by multiplying and dividing the factors of four hundred thirty-nine.\n\nFactoring Calculator\nDo you need the factors for another number? No problem - just enter it below and press \"Factor\".\n\nFactors of 440\nHere is the next number on our list for which we have calculated the factors." ]

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[ "# Mathematical Treasures - Johann Faulhaber's Academia Algebrae\n\nAuthor(s):\nFrank J. Swetz and Victor J. Katz", null, "This is the title page of the Academia Algebrae by Johann Faulhaber (1580 - 1635). Faulhaber was a German cossist, who evidently had influence on both Johann Kepler and René Descartes. In this work, he exhibited formulas (in the German cossist notation) for sums from 1 to n of the kth powers of the positive integers for k = 13, 14, 15, 16, and 17. He had earlier exhibited the formulas for smaller values of k. Unfortunately, he left little indication as to how he had developed these formulas.", null, "On this double page, signature B, f. i (verso) and ii (recto) of the Academia Algebrae, Faulhaber gave the formula for the sum of the 13th powers of the integers, with some steps leading to this formula. The final formula appears in the third from the last line of the paragraph with all the algebraic notation. In translation to modern notation, it reads that the sum of the 13th powers from 1 to n is equal to (30n14 + 210n13 + 455n12 – 1001n10 + 2145n8 – 3003n6 + 2275n4 691n2)/420. For more information on Faulhaber and sums of powers in general, consult the article in this magazine by Janet Beery, \"Sums of Powers of Positive Integers.\"" ]

[ null, "https://www.maa.org/sites/default/files/images/cms_upload/080080145-155926.jpg", null, "https://www.maa.org/sites/default/files/images/cms_upload/080080145-259487.jpg", null ]

[ "# What is 410 Feet/Second in Knots?\n\n## Convert 410 Feet/Second to Knots\n\nTo calculate 410 Feet/Second to the corresponding value in Knots, multiply the quantity in Feet/Second by 0.59248380129641 (conversion factor). In this case we should multiply 410 Feet/Second by 0.59248380129641 to get the equivalent result in Knots:\n\n410 Feet/Second x 0.59248380129641 = 242.91835853153 Knots\n\n410 Feet/Second is equivalent to 242.91835853153 Knots.\n\n## How to convert from Feet/Second to Knots\n\nThe conversion factor from Feet/Second to Knots is 0.59248380129641. To find out how many Feet/Second in Knots, multiply by the conversion factor or use the Velocity converter above. Four hundred ten Feet/Second is equivalent to two hundred forty-two point nine one eight Knots.\n\n## Definition of Foot/Second\n\nThe foot per second (plural feet per second) is a unit of both speed (scalar) and velocity (vector quantity, which includes direction). It expresses the distance in feet (ft) traveled or displaced, divided by the time in seconds (s, or sec). The corresponding unit in the International System of Units (SI) is the metre per second. Abbreviations include ft/s, ft/sec and fps, and the rarely used scientific notation ft s−1.\n\n## Definition of Knot\n\nThe knot is a unit of speed equal to one nautical mile (1.852 km) per hour, approximately 1.151 mph. The ISO Standard symbol for the knot is kn. The same symbol is preferred by the IEEE; kt is also common. The knot is a non-SI unit that is \"accepted for use with the SI\". Worldwide, the knot is used in meteorology, and in maritime and air navigation—for example, a vessel travelling at 1 knot along a meridian travels approximately one minute of geographic latitude in one hour. Etymologically, the term derives from counting the number of knots in the line that unspooled from the reel of a chip log in a specific time.\n\n## Using the Feet/Second to Knots converter you can get answers to questions like the following:\n\n• How many Knots are in 410 Feet/Second?\n• 410 Feet/Second is equal to how many Knots?\n• How to convert 410 Feet/Second to Knots?\n• How many is 410 Feet/Second in Knots?\n• What is 410 Feet/Second in Knots?\n• How much is 410 Feet/Second in Knots?\n• How many kt are in 410 ft/s?\n• 410 ft/s is equal to how many kt?\n• How to convert 410 ft/s to kt?\n• How many is 410 ft/s in kt?\n• What is 410 ft/s in kt?\n• How much is 410 ft/s in kt?" ]

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[ "# R/marginal_assoc_tests.r In CCpop: One and two locus GWAS of binary phenotype with case-control-population design\n\n#### Documented in marginal.assoc.test.hwe.in.controlsmarginal.assoc.test.kpx.kpymarginal.assoc.test.pop.hwe.kpymarginal.assoc.test.unconstrained.chisqmarginal.assoc.test.unconstrained.gsq\n\n```# Tests for association (or independence) between a SNP locus and a binary\n# disease phenotype.\n\nmarginal.assoc.test.unconstrained.chisq = function(t0, t1) {\n# The marginal Pearson chi-squared test on the 2x3 (or less) contingency table.\n\n# NOTE: the tables must be pruned so that they do not contain columns with all\n# zeros.\n\nn0 = sum(t0)\nn1 = sum(t1)\nnx = t0 + t1\nn = n0 + n1\ne0 = (n0 / n) * nx\ne1 = (n1 / n) * nx\n\nstatistic = sum((t0 - e0) ^ 2 / e0 + (t1 - e1) ^ 2 / e1)\np.value = pchisq(statistic, df = length(t0), lower.tail = F)\n\nreturn (list(pen = t1 / (t0 + t1), statistic = statistic, p.value = p.value))\n}\n\nmarginal.assoc.test.unconstrained.gsq = function(t0, t1) {\n# The marginal G^2 test (GLRT conditioned on the observed marginals of the 2x3\n# contingency table)\n\npen.full = t1 / (t0 + t1)\npen.null = sum(t1) / sum(t1 + t0)\n\nstatistic = 2 * (sum(t0 * log(1 - pen.full) + t1 * log(pen.full), na.rm = T) -\nsum(t0 * log(1 - pen.null) + t1 * log(pen.null), na.rm = T))\np.value = pchisq(statistic, df = length(t0), lower.tail = F)\n\nreturn (list(pen = pen.full, statistic = statistic, p.value = p.value))\n}\n\nmarginal.assoc.test.hwe.in.controls = function(t0, t1) {\n# Assuming HWE among the controls (justifiable if the disease is rare). This is\n# a Wald test for marginal association assuming HWE among the controls (Chen and\n# Chatterjee 2007)\n\nt1[t1 == 0] = 1 # is this a good solution to the case where I have zero t1 cells?\nn0 = sum(t0)\n\nef = (t0 + 2 * t0) / (2 * n0)\net0 = n0 * c((1 - ef) ^ 2, 2 * ef * (1 - ef), ef ^ 2)\neor.hom = (t1 * et0) / (et0 * t1)\neor.het = (t1 * et0) / (et0 * t1)\n\nbeta = log(c(eor.hom, eor.het))\n\nbeta.cov = matrix(c(\n1/t1 + 1/t1 + 1/(2*et0 + et0) + 1/(2*et0 + et0),\n1/t1 + 1/n0 * 1/(ef * (1 - ef)),\n1/t1 + 1/n0 * 1/(ef * (1 - ef)),\n1/t1 + 1/t1 + 4/(2*et0 + et0) + 4/(2*et0 + et0)), nrow = 2)\n\nstatistic = t(beta) %*% solve(beta.cov) %*% beta\np.value = pchisq(statistic, df = 2, lower.tail = F)\n\nreturn (list(statistic = statistic, p.value = p.value))\n}\n\nmarginal.assoc.test.pop.hwe.kpy = function(t0, t1, tp, prevalence, pen.initial = NULL, f.initial = NULL) {\n# Assuming HWE in the general population, and that disease prevalence is\n# known. This test uses an extraneous population sample to improve estimation\n# of population parameters such as SNP distributions.\n\ntx = t0 + t1 + tp\n\nf.null.est = min(0.5, (0.5 * tx + tx) / sum(tx))\npx.null.est = c((1 - f.null.est) ^ 2, 2 * f.null.est * (1 - f.null.est), f.null.est ^ 2)\nnull.loglik = sum(t0 * log(1 - prevalence) + t1 * log(prevalence) + tx * log(px.null.est))\n\nif (is.null(pen.initial)) {\npars.initial = c(rep(prevalence, 3), f.null.est)\n} else {\npars.initial = c(pen.initial, f.initial)\n}\n\neval_f = function(pars) {\npen = pars[1:3]\nf = pars\n\npx = c((1 - f) ^ 2, 2 * f * (1 - f), f ^ 2)\ndpx.df = c(2 * f - 2, 2 - 4 * f, 2 * f)\n\nobjective = null.loglik - sum(t0 * log(1 - pen) + t1 * log(pen)) - sum(tx * log(px))\ngradient = -c(t1 / pen - t0 / (1 - pen), sum(tx / px * dpx.df))\n\n}\n\nlb = rep(.Machine\\$double.eps, 4)\nub = c(rep(1, 3) - .Machine\\$double.neg.eps, 0.5)\n\neval_g_eq = function(pars) {\npen = pars[1:3]\nf = pars\n\npx = c((1 - f) ^ 2, 2 * f * (1 - f), f ^ 2)\ndpx.df = c(2 * f - 2, 2 - 4 * f, 2 * f)\n\nconstraints = sum(pen * px) - prevalence\njacobian = matrix(c(px, sum(pen * dpx.df)), nrow = 1)\n\nreturn (list(constraints = constraints, jacobian = jacobian))\n}\n\nsol = nloptr(x0 = pars.initial, eval_f = eval_f, eval_g_eq = eval_g_eq, lb = lb, ub = ub,\nopts = list(algorithm = 'NLOPT_LD_SLSQP', ftol_abs = 1e-8, xtol_rel = 0, maxeval = 1000))\n\npen = t0 * 0 + sol\\$solution[1:3]\nf = sol\\$solution\npx = c((1 - f) ^ 2, 2 * f * (1 - f), f ^ 2)\n\nstatistic = -2 * sol\\$objective\np.value = pchisq(statistic, df = 2, lower.tail = F)\n\nreturn (list(pen = pen, statistic = statistic, p.value = p.value))\n}\n\nmarginal.assoc.test.kpx.kpy = function(t0, t1, prevalence, px, pen.initial = NULL) {\n# Assuming the SNP distribution is known, and that disease prevalence is known\n\n# The GLRT for testing a marginal association given a known SNP distribution\n# (specified as genotypic, but can come from a HWE assumption and known MAF)\n\n# NOTE: this is exactly the same algorithm as that for a the pairwise test with\n# known pxx.\n\nm = length(px)\ndf = m - 1\n\nif (is.null(pen.initial)) {\npars.initial = rep(prevalence, df)\n} else {\npen.initial = pen.initial[2:m]\n}\n\nfn = function(x) {\npen = c((prevalence - sum(x * px[2:m])) / px, x)\nreturn (-sum(t0 * log(1 - pen) + t1 * log(pen)))\n}\n\ngr = function(x) {\nx0 = (prevalence - sum(x * px[2:m])) / px\ng = t1[2:m] / x - t0[2:m] / (1 - x) - t1 / x0 / px * px[2:m] + t0 / (1 - x0) / px * px[2:m]\nreturn (-g)\n}\n\nui = rbind(diag(df), -diag(df), -px[2:m], px[2:m])\nci = c(rep(.Machine\\$double.eps, df), rep(-(1 - .Machine\\$double.neg.eps), df), .Machine\\$double.eps - prevalence, prevalence - px * (1 - .Machine\\$double.neg.eps))\nsol = constrOptim(pars.initial, fn, gr, ui, ci)\n\npen = t0 * 0 + c((prevalence - sum(sol\\$par * px[2:m])) / px, sol\\$par)\n\nstatistic = 2 * sum(t0 * log((1 - pen) / (1 - prevalence)) + t1 * log(pen / prevalence))\np.value = pchisq(statistic, df = df, lower.tail = F)\n\nreturn (list(pen = pen, statistic = statistic, p.value = p.value))\n}\n```\n\n## Try the CCpop package in your browser\n\nAny scripts or data that you put into this service are public.\n\nCCpop documentation built on May 2, 2019, 3:25 a.m." ]

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[ "# dplyr: How to Summarise Data But Keep All Columns\n\nWhen using the summarise() function in dplyr, all variables not included in the summarise() or group_by() functions will automatically be dropped.\n\nHowever, you can use the mutate() function to summarize data while keeping all of the columns in the data frame.\n\nThe following example shows how to use this function in practice.\n\n## Example: Summarise Data But Keep All Columns Using dplyr\n\nSuppose we have the following data frame that contains information about various basketball players:\n\n```#create data frame\ndf <- data.frame(team=rep(c('A', 'B', 'C'), each=3),\npoints=c(4, 9, 8, 12, 15, 14, 29, 30, 22),\nassists=c(3, 3, 2, 5, 8, 10, 4, 5, 12))\n\n#view data frame\ndf\n\nteam points assists\n1 A 4 3\n2 A 9 3\n3 A 8 2\n4 B 12 5\n5 B 15 8\n6 B 14 10\n7 C 29 4\n8 C 30 5\n9 C 22 12```\n\nWe can use the following syntax to summarize the mean points scored by team:\n\n```library(dplyr)\n\n#summarize mean points values by team\ndf %>%\ngroup_by(team) %>%\nsummarise(mean_pts = mean(points))\n\n# A tibble: 3 x 2\nteam mean_pts\n\n1 A 7\n2 B 13.7\n3 C 27\n```\n\nThe column called mean_pts displays the mean points scored by each team.\n\nFrom the output we can see:\n\n• The mean points scored by players on team A is 7.\n• The mean points scored by players on team B is 13.7.\n• The mean points scored by players on team C is 27.\n\nHowever, suppose we would like to keep all other columns from the original data frame.\n\nWe can use the following syntax with the mutate() function to do so:\n\n```library(dplyr)\n\n#summarize mean points values by team and keep all columns\ndf %>%\ngroup_by(team) %>%\nmutate(mean_pts = mean(points)) %>%\nungroup()\n\n# A tibble: 9 x 4\nteam points assists mean_pts\n\n1 A 4 3 7\n2 A 9 3 7\n3 A 8 2 7\n4 B 12 5 13.7\n5 B 15 8 13.7\n6 B 14 10 13.7\n7 C 29 4 27\n8 C 30 5 27\n9 C 22 12 27\n```\n\nBy using the mutate() function, we’re able to create a new column called mean_pts that summarizes the mean points scored by team while also keeping all other columns from the original data frame." ]

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[ "# Current Transformer\n\nHello,\n\nGood day, trust we are doing well.\n\nI am handling a project to design a digital clamp meter, so, I chose a LEM HO 100-S current transducer, it uses 5V as supply volt and out voltage of 0-5V. To see my results, I interfaced the Arduino with an LCD\n\nBut, on powering the CT with 5V, the analog output Vout was reading 2.8V constantly, with and without current through it, I have tried several CTs, same result, please, what can I do.\n\nAttached is the datasheet for the CT.\n\nThis is the code below.\n\n<\n#include <Wire.h>\n#include <LiquidCrystal_I2C.h>\n\nLiquidCrystal_I2C lcd(0x27, 20, 4);\n\nint current = A1;\nfloat val = 0;\nfloat readCurrent = 0;\nString par = “Amps”;\n\nunsigned long i = 0;\n\nvoid setup() {\n\nlcd.init();\nlcd.backlight();\nlcd.clear();\n\n// pinMode(A0, INPUT);\n\n}\n\nvoid loop() {\n\nval = (readCurrent * 50) / 1023;\n\nlcd.setCursor(0, 0);\nlcd.print(val);\nlcd.setCursor(6, 0);\nlcd.print(par);\n\ndelay(500);\n\n}/>\n\nho_50_250-s-0100_series.pdf (1.78 MB)\n\nSearching on the Internet I found this:\nhttps://www.digikey.com/en/datasheets/lem-usa/lem-usa-inc-ho_50_250-s-0100_series\nIt looks like the output is balanced at some 2.8 vots for 0 current. 2-8 - 5 volt for positive current, 2.8 to 0 for negative current is my guess.\n\nThe output is biased at 1/2 vcc. How else do you measure an ac waveform?\n\nStart over using the library for the acs712 sensors. They work the same way, you, just need to figure out the scaling value for the transducer rating you have. You can do that, right? Right?\n\nthe sensor is for 125 A. with how much current did you test it?\n\nAC oscillates. you want a RMS value, not individual values.\n\n``````  unsigned long long sum = 0;\nint n = 0;\nunsigned long start_time = millis();\nwhile (millis() - start_time < 200) { // in 200 ms measures 10 50Hz AC oscillations\nint v = analogRead(A1);\nsum += v * v;\nn++;\n}\nint rms = sqrt((double) sum / n);\n``````\n\nThat's not \"just\" a current transformer... There's some active circuitry inside.\n\nI chose a LEM HO 100-S current transducer,\n\nThat's a 100 Amp device. How much current are you trying to measure?\n\nBut, on powering the CT with 5V, the analog output Vout was reading 2.8V constantly,\n\nIf I'm reading the datasheet correctly, the output is biased. (That's good for negative/reversed current or AC since the Arduino can't read negative voltages.) Positive/forward current should read above the bias, negative/reversed current should read below the bias, and AC should read above & below.\n\nHello wattsThat, I haven't used the library before, but I'll see what I can do\n\nDVDdoug, I did not know that\n\nI tested it on an Air conditioner, using my clamp meter, I got a reading of 13 amps, though I noticed some changes in the readings, but was around +2\n\nMaybe the air conditioner is reulating its doings and the reading then changes. What about using an oldfashioned heater that burns the same power al the time?\n\nBig thanks to everyone, I used the ACS712 library and I got positive results, I used the examples in the library\n\n``````#include \"ACS712.h\"\n\n/*\nThis example shows how to measure the power consumption\nof devices in 230V electrical system\nor any other system with alternative current\n*/\n\n// We have 30 amps version sensor connected to A1 pin of arduino\n// Replace with your version if necessary\nACS712 sensor(ACS712_30A, A1);\n\nvoid setup() {\nSerial.begin(9600);\n\n// This method calibrates zero point of sensor,\n// It is not necessary, but may positively affect the accuracy\n// Ensure that no current flows through the sensor at this moment\nsensor.calibrate();\n}\n\nvoid loop() {\n// We use 230V because it is the common standard in European countries\n// Change to your local, if necessary\nfloat U = 230;\n\n// To measure current we need to know the frequency of current\n// By default 50Hz is used, but you can specify own, if necessary\nfloat I = sensor.getCurrentAC();\nSerial.println(I*4.165);\n\n// To calculate the power we need voltage multiplied by current\nfloat P = U * I;\n\nSerial.println(String(\"P = \") + P + \" Watts\");\n\ndelay(1000);\n}\n``````" ]

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[ "identify\n\n0th\n\nPercentile\n\nIdentify a Time Series Model\n\nChecks the white noise and stationarity of a univariate time series, and also identifies an appropriate ARIMA model using AICC criterion.\n\nUsage\nidentify(x, p = NULL, q = NULL, nlag = 6, intercept = TRUE,\nstat.test = FALSE, method = c(\"adf\", \"pp\", \"kpss\"), output = TRUE)\nArguments\nx\na numeric vector or univariate time series.\np\nthe maximum lag order for AR process. The default is NULL.\nq\nthe maximum lag order for MA process. The default is NULL.\nnlag\nthe lag parameter to calculate the Ljung-Box test statistic. The default is 6.\nintercept\nan intercept to be included in ARIMA model, only valid for p > 0 or q > 0. The default is TRUE.\nstat.test\nthe stationary test for time series, see stationary.test for more details. The default is FALSE.\nmethod\nthe method of stationary test, only valid for stat.test = TRUE.\noutput\na logical value indicating to print the results in R console. The default is TRUE.\nDetails\n\nThis function is similar to IDENTIFY statement in ARIMA procedure of SAS software, which is to check the white noise and stationarity for a univariate time series. The white noise check is accomplished by using Box.test in stats package, with the default method type = \"Ljung-Box\". The stationary check uses the stationary.test implemented in this package.\n\nThe AICC criterion (Burnham and Anderson (2002)) is used to identify an optimal model which has the minimum AICC value. The AICC is defined as $$AICC = AIC + 2k(k+1)/(n-k-1),$$ where $AIC = 2k - 2log(Loglik)$ which is called Akaike information criterion (Akaike (1974)). Here, $k,n$ are the number of estimated parameters and observations, respectively. $Loglik$ is the maximized value of the likelihood function for the model.\n\nFour plots are made: plot of original data, ACF plot, PACF plot and p.value of white noise check.\n\nValue\n\n• A list with class \"identify\" containing the following components:\n• WNchecka matrix with three columns for results of white noise check for each lag.\n• aiccthe minimum AICC value, only available for p > 0 or q > 0.\n• min.pthe optimal order p for AR process, only available for p > 0 or q > 0.\n• min.qthe optimal order q for AR process, only available for p > 0 or q > 0.\n• stnt.testa list of stationary test results with three components. See stationary.test for more details.\n\nNote\n\nMissing values are removed before the analysis.\n\nReferences\n\nAkaike, H. (1974), \"A new look at the statistical model identification\", IEEE Transactions on Automatic Control, 19 (6): 716-723.\n\nBox, G. E. P. and Pierce, D. A. (1970), Distribution of residual correlations in autoregressive-integrated moving average time series models. Journal of the American Statistical Association, 65, 1509-1526.\n\nBurnham, K. P.; Anderson, D. R. (2002), Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach (2nd ed.), Springer-Verlag\n\nLjung, G. M. and Box, G. E. P. (1978). On a measure of lack of fit in time series models. Biometrika 65, 297-303.\n\nHarvey, A. C. (1993) Time Series Models. 2nd Edition, Harvester Wheatsheaf, NY, pp. 44, 45.\n\n• identify\nExamples\nx <- arima.sim(list(order = c(2,0,0),ar = c(0.2,0.4)),n = 100)\nidentify(x) # white noise check\nidentify(x,stat.test = TRUE) # white noise and stationarity check\nidentify(x,p = 3,q = 2) # white noise check and optimal model identification.\nDocumentation reproduced from package aTSA, version 3.1.2, License: GPL-2 | GPL-3\n\nCommunity examples\n\nLooks like there are no examples yet." ]

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[ "# A global meshless collocation particular solution method (integrated Radial Basis Function) for two-dimensional Stokes flow problems\n\nC. A. Bustamante, H. Power, Y. H. Sua, W. F. Florez\n\nResultado de la investigación: Contribución a una revistaArtículorevisión exhaustiva\n\n29 Citas (Scopus)\n\n## Resumen\n\nA global version of the Method of Approximate Particular Solutions (MAPS) is developed to solve two-dimensional Stokes flow problems in bounded domains. The velocity components and the pressure are approximated by a linear superposition of particular solutions of the non-homogeneous Stokes system of equations with a Multiquadric Radial Basis Function as forcing term. Although, the continuity equation is not explicitly imposed in the resulting formulation, the scheme is mass conservative since the particular solutions exactly satisfy the mass conservation equation. The present scheme is validated by comparing the obtained numerical result with the analytical solution of two boundary value problems constructed from the Stokeson exterior fundamental solution, i.e. regular everywhere except at infinity. For these two cases, convergence of the method and the influence of the value of the Multiquadric's shape parameter on the numerical results are studied by computing the relative Root Mean Square (RMS) error for several homogeneous distributions of collocation points and values of the shape parameter. From this analysis is observed that the proposed MAPS results are stable and accurate for a wide range of shape parameter values. In addition, the lid-driven cavity and backward-facing step flow problems are solved and the obtained results compared with the solutions found with more conventional numerical schemes, showing good agreement between them.\n\nIdioma original Inglés 4538-4547 10 Applied Mathematical Modelling 37 6 https://doi.org/10.1016/j.apm.2012.09.069 Publicada - 15 mar. 2013 Sí\n\n## Huella\n\nProfundice en los temas de investigación de 'A global meshless collocation particular solution method (integrated Radial Basis Function) for two-dimensional Stokes flow problems'. En conjunto forman una huella única." ]

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[ "# How to Combine Complicated Models with Tricky Effects\n\nNeed to dummy code in a Cox regression model?", null, "Interpret interactions in a logistic regression?", null, "This is where statistical analysis starts to feel really hard. You’re combining two difficult issues into one.\n\nYou’re dealing with both a complicated modeling technique at Stage 3 (survival analysis, logistic regression, multilevel modeling) and tricky effects in the model (dummy coding, interactions, and quadratic terms).\n\nThe only way to figure it all out in a situation like that is to break it down into parts.  Trying to understand all those complicated parts together is a recipe for disaster.\n\nBut if you can do linear regression, each part is just one step up in complexity.  Take one step at a time.\n\n### The Type of Model Reflects the Distribution of the Response Variable and the Design\n\nMost statistical models are extensions of linear models (aka. the General Linear Model).  The extensions deal with linear model assumptions that your data don’t meet.\n\nSince most assumptions are about the residuals, whose distribution reflects the measurement of the response variable, these extensions are all for response variables that don’t fit the general linear model.  Our examples above are necessary for censored time-to-event, categorical, and clustered response variables.\n\nThe other regression models mostly use maximum likelihood estimation, which requires different measures of model fit.\n\nThey also often use link functions, making interpretation of coefficients harder. (Remember, coefficients measure the relationship between the predictor and response, so the link function on the response always affects the scaling of the coefficients).\n\nSo the first thing you need to work on is how to implement, interpret coefficients, and understand model fit in the model you’re using.  Don’t worry yet about the tricky effects until you’ve mastered the model you’re using with simple, continuous predictors.\n\n### The Tricky Effects are About the Predictor Variables\n\nThe nice thing about all these models is that the fixed part of the right hand side of the equation are all the same. This is the part with the regression coefficients and X variables. This part of the model measures and tests the effects you’re interested in.\n\nA dummy variable is a dummy variable in every kind of model.\n\nThe random parts will differ, but they’re not directly involved with the effects of predictor variables. (In most models, the only random part is the residuals).\n\nAll those tricky effects occur when the predictors aren’t the textbook-perfect numerical variables you used in regression class.  Examples include dummy variables for categorical predictors, interactions and quadratic terms that are the products of two predictors, centered or rescaled predictor variables.\n\nSo once you learn how and when to use, and how to interpret any tricky effect in a linear model, you can use it in any type of model.\n\nI generally recommend mastering these effects in the context of linear models. Even if that’s not your immediate need.  Because learning how to work with and interpret the effects of dummy variables is a lot harder if you’re doing it in terms of odd ratios than in terms of comparing means.\n\nSo the next time you need to figure out a combination of a tricky effect and a complicated model, break it down first.  Simplify your model as much as you can as you master each part.\n\nThis might mean you run some simple preliminary models that you never could publish. That’s okay. Taking the time to understand each part will be worth it.", null, "Interpreting Linear Regression Coefficients: A Walk Through Output\nLearn the approach for understanding coefficients in that regression as we walk through output of a model that includes numerical and categorical predictors and an interaction.\n\n### Related Posts\n\n1.", null, "Claudia says\n\nThat was great!\nThank you very much!\n\n2.", null, "Ingo Peters says\n\nThis short article on tricky effects was very useful for me and clearified my interpretation of a cox regression with a quadratic term. Thanks!\n\n3.", null, "HdS says\n\nHello,\nPart of the problem is that there are not many publications regarding “tricky” problems. I wanted to do a interaction in a cox regression and where not able to find any good adivce for their interpretation.\n\n•", null, "Karen says\n\nAgreed.\n\nWhen you can find an example of exactly what you need to do, that’s the clearest and best way to learn.\n\nSo in the absence of that, you just have to break down the problem into different pieces.\n\nKaren\n\nPlease note that, due to the large number of comments submitted, any questions on problems related to a personal study/project will not be answered. We suggest joining Statistically Speaking, where you have access to a private forum and more resources 24/7." ]

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[ "", null, "", null, "# 原型链\n\n### 含义\n\n• 构造函数,原型,实例的关系,每个构造函数的原型都会有一个内部指针指向该构造函数,每个构造函数的实例都会有一个内部指针指向该构造函数的原型;如果把该构造函数的原型当做另外一个构造函数的实例,那么该实例就会有一个内部指针指向新的构造函数的原型对象,新的构造函数的原型对象会有一个内部指针指向新的构造函数,这样层层嵌套下去就形成了原型链\n• 特别的一支原型链: 每个继承者的原型都是被继承者的实例,所以每个函数的原型都是Object的实例\n\n### 图示\n\n• 原型对象.constructor = 构造函数\n\n• 实例.proto = 原型对象\n\n• Function 是顶层构造器\n\n• Object 是顶层原型", null, "", null, "### 代码\n\n```function A() { this.a = true } A.prototype.getAProperty = function () { return this.a } function B() { this.b = false } B.prototype = new A() //此处的B的原型指向了A的构造函数 //可以手动把B的原型指向重新改回来 // B.prototype = { // constructor: B // } B.prototype.getBProperty = function(){ return this.b } let instance = new B() let res = instance.getAProperty() console.log(res) //true```" ]

[ null, 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[ "# What is the matlab code syntax to create a temporary array in MATLAB ?\n\n1 view (last 30 days)\nSiddharth Kamila on 9 Jun 2022\nAnswered: Ganesh Gudipati on 13 Jun 2022\nI am working on the fault detection and fault classification in power system using wavelet and ANN in matlab/simulink. I need to sense the fault current from the power system model drawn in simulink, and the waveform of the fault current displayed in the scope will be converted into data and has to be stored temporarily in the form of array in workspace (as a memory) and this data samples stored in the array shall be fed to wavelet code (along with 0.05 sec time step, so that window can be made) and the wavelet code can calculate the fault coefficients and these fault coefficients can be fed to the ANN controller to detect the correct type of fault. Once the data samples is fed to the wavelet code after 0.05 sec, that data samples stored in array will be cleared and new data samples will be stored in the form of array in workspace. Again this data samples is fed to wavelet code and wavelet code will calculate the fault coefficients and after next 0.05 sec the data samples stored in the array will be cleared and new data samples will be stored in the array in workspace. In this way at every 0.05 sec time step, the data samples will be stored in the array and will be given to wavelet code and the array will be cleared and new data samples will be stored in the array. This array is called \"temporary array\".\nThe question is that how to create this temporary array in MATLAB ? What is the matlab code syntax for creating this temporary array ? I just want only suggestion. I am not asking for an exact answer.\ndpb on 9 Jun 2022\ntmp=zeros(N,1);\ncreates a column vector of N elements you can write into...it will have to be in scope of whatever it is that is doing the calculation or be global.\n\nGanesh Gudipati on 13 Jun 2022\nHi Siddharth,\nAs per my understanding you want to create an array and at each timestep the new samples should replace the old samples.\nCreate a array of initial size ‘n’ using the command\ntempArray = zeros(1,n);\n% no of rows =1, now of columns = n;\nLet us consider your model as a function. Now this function takes the tempArray and it will modify the array at each time step.", null, "Once this whole execution is done, and the tempArray is no more required\nclear(tempArray)\n%This removes the array from the workspace and frees the memory.\n\nR2021b\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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[ "", null, "23rd International Symposium on\nGraph Drawing & Network Visualization\nSeptember 24-26, 2015, Los Angeles, CA, USA\n\n# Accepted papers\n\nMarcus Schaefer and Daniel Stefankovic. The Degenerate Crossing Number and Higher-Genus Embeddings [+]\nAbstract: If a graph embeds in a surface with k crosscaps, does it always have an embedding in the same surface in which every edge passes through each crosscap at most once? This well-known open problem can be restated using crossing numbers: the degenerate crossing number, dcr(G), of G equals the smallest number k so that G has an embedding in a surface with k crosscaps in which every edge passes through each crosscap at most once. The genus crossing number, gcr(G), of G equals the smallest number k so that G has an embedding in a surface with k crosscaps. The question then becomes whether dcr(G) = gcr(G), and it is in this form that it was first asked Mohar.\nWe show that dcr(G) <= 6 gcr(G), and dcr(G) = gcr(G) as long as dcr(G) <= 3. We can separate dcr and gcr for (single-vertex) graphs with embedding schemes, but it is not clear whether the separating example can be extended into separations on simple graphs. Finally, we show that if a graph can be embedded in a surface with crosscaps, then it has an embedding in that surface in which every edge passes through each\ncrosscap at most twice.\nTherese Biedl and Jens M. Schmidt. Small-Area Orthogonal Drawings of 3-Connected Graphs [+]\nAbstract: It is well-known that every graph with maximum degree 4 has an orthogonal drawing with area at most $\\frac{49}{64}n^2+O(n)\\approx 0.76n^2$. In this paper, we show that if the graph is 3-connected, then the area can be reduced even further to $\\frac{25}{36}n^2+O(n) \\approx 0.56n^2$. The drawing uses the 3-canonical order for (not necessarily planar) 3-connected graphs, which can be computed in linear time from the Mondshein-sequence. To our knowledge, this is the first application of the 3-canonical order on non-planar graphs in graph-drawing.\nClinton Bowen, Stephane Durocher, Maarten Löffler, Anika Rounds, André Schulz and Csaba D. Tóth. Realization of simply connected polygonal linkages and recognition of unit disk contact trees [+]\nAbstract: We consider two variants of the fundamental question of determining whether a simply connected flexible combinatorial structure can be realized in Euclidean space. Two models are considered: body-and-joint frameworks and contact graphs of unit disks in the plane. (1) We show that it is strongly NP-hard to decide whether a given polygonal linkage (body-and-bar framework) is realizable in the plane when the bodies are convex polygons and their contact graph is a tree; the problem is weakly NP-hard already for a chain of rectangles; but efficiently decidable for a chain of triangles hinged at distinct vertices. (2) We also show that it is strongly NP-hard to decide whether a given tree is the contact graph of interior-disjoint unit disks in the plane.\nUlf Rüegg, Christoph Daniel Schulze, John Julian Carstens and Reinhard Von Hanxleden. Size- and Port-Aware Horizontal Node Coordinate Assignment [+]\nAbstract: The approach by Sugiyama et al. is widely used to automatically draw directed graphs. One of its step is to assign horizontal coordinates to nodes. Brandes and Köpf presented a method that proved to work well in practice. We extend this method to make it possible to draw diagrams with nodes that have considerably different sizes and edges that have fixed attachment points on a node's perimeter (ports). Our extensions integrate seamlessly with the original method and preserve the linear execution time.\nLinear-size Universal Point Sets for One-bend Drawings [+]\nAbstract: For every integer n ≥ 4, we construct a planar point set S_n of size 6n-10 such that every n-vertex planar graph G admits a plane embedding in which the vertices are mapped to points in S_n, and every edge is either a line segment or a polyline with one bend, where the bend point is also in S_n.\nBoris Klemz, Martin Nöllenburg and Roman Prutkin. Recognizing Weighted Disk Contact Graphs [+]\nAbstract: Disk contact representations realize graphs by mapping vertices bijectively to interior-disjoint disks in the plane such that two disks touch each other if and only if the corresponding vertices are adjacent in the graph. Deciding whether a vertex-weighted planar graph can be realized such that the disks' radii coincide with the vertex weights is known to be NP-hard. In this work, we reduce the gap between hardness and tractability by analyzing the problem for special graph classes. We show that it remains NP-hard for outerplanar graphs with unit weights and for stars with arbitrary weights, strengthening the previous hardness results. On the positive side, we present constructive linear-time recognition algorithms for caterpillars with unit weights and for embedded stars with arbitrary weights.\nDisplaying User Behavior in the Collaborative Graph Visualization System OnGraX [+]\nAbstract: The visual analysis of complex networks is a challenging task in many fields, such as systems biology or social sciences. Often, various domain experts work together to improve the analysis time or the quality of the analysis results. Collaborative visualization tools can facilitate the analysis process in such situations. We propose a new web-based visualization environment which supports distributed, synchronous and asynchronous collaboration. In addition to standard collaboration features like event tracking or synchronizing, our client/server-based system provides a rich set of visualization and interaction techniques for better navigation and overview of the input network. Changes made by specific analysts or even just visited network elements are highlighted on demand by heat maps. They enable us to visualize user behavior data without affecting the original graph visualization. We evaluate the usability of the heat map approach against two alternatives in a user experiment.\nWill Evans, Giuseppe Liotta, Henk Meijer and Steve Wismath. Alternating Paths and Cycles of Minimum Length [+]\nAbstract: Let R be a set of n red points and B be a set of n blue points in the Euclidean plane.\nWe study the problem of computing a planar drawing of a cycle of minimum length that contains vertices at points R ∪ B and alternates colors.\nWhen these points are collinear, we describe a Θ(n log n)-time algorithm to find such a shortest alternating cycle where every edge has at most two bends. We extend our approach to compute shortest alternating paths in O(n2) time with two bends per edge and to compute shortest alternating cycles on 3-colored point-sets in O(n2) time with O(n) bends per edge. We also prove that for arbitrary k-colored point-sets, the problem of computing an alternating shortest cycle is NP-hard, where k is any positive integer constant.\nAlexander Igamberdiev, Wouter Meulemans and André Schulz. Drawing Planar Cubic 3-Connected Graphs with Few Segments: Algorithms & Experiments [+]\nAbstract: A drawing of a graph can be understood as an arrangement of geometric objects.\nIn the most natural setting the arrangement is formed by straight-line segments.\nEvery cubic planar 3-connected graph with n vertices has such a drawing with only n/2+3 segments, matching the lower bound.\nThis result is due to Mondal et al. [J. of Comb. Opt., 25], who gave an algorithm for constructing such drawings.\n\nWe introduce two new algorithms that also produce drawings with n/2+3 segments.\nOne algorithm is based on a sequence of dual edge contractions, the other\nis based on a sequence of removed cycles. We also show a flaw in the algorithm\nof Mondal et al. and present a fix for it. We then compare the performance of these three\nalgorithms by measuring angular resolution, edge length and face aspect ratio of the constructed drawings.\nWe observe that the corrected algorithm of Mondal et al. mostly outperforms the other algorithms, especially in terms of angular resolution.\nHowever, the new algorithms perform better in terms of edge length and minimal face aspect ratio.\nDrago Bokal, Mojca Bracic, Marek Dernár and Petr Hlinený. On Degree Properties of Crossing-critical Families of Graphs [+]\nAbstract: Answering an open question from 2007, we construct infinite k-crossing-critical families of graphs which contain arbitrarily often vertices of any prescribed odd degree, for sufficiently large k. From this we derive that, for any set of integers D such that min(D)=3 and 3,4 ? D, and for all sufficiently large k there exists an infinite k-crossing-critical family such that the numbers in D are precisely the vertex degrees which occur arbitrarily often in this family. We also investigate what are the possible average degrees of such crossing-critical families.\nNabil Mustafa and Janos Pach. On the Zarankiewicz Problem for Intersection Hypergraphs [+]\nAbstract: Let d and t be fixed positive integers, and let Kdt,...,t denote the complete d-partite hypergraph with t vertices in each of its parts, whose hyperedges are the d-tuples of the vertex set with precisely one element from each part. According to a fundamental theorem of extremal hypergraph theory, due to Erdős, the number of hyperedges of a d-uniform hypergraph on n vertices that does not contain Kdt,...,t as a subhypergraph, is nd-1/td-1. This bound is not far from being optimal.\n\nWe address the same problem restricted to intersection hypergraphs of (d-1)-dimensional simplices in Rd. Given an n-element set S of such simplices, let Hd(S) denote the d-uniform hypergraph whose vertices are the elements of S, and a d-tuple is a hyperedge if and only if the corresponding simplices have a point in common. We prove that if Hd(S) does not contain Kdt,...,t as a subhypergraph, then its number of edges is O(n) if d=2, and O(nd-1+ε for any ε> 0 if d ≥3. This is almost a factor of n better than Erdős's above bound. Our result is tight, apart from the error term ε in the exponent.\n\nIn particular, for d=2, we obtain a theorem of Fox and Pach, which states that every Kt,t-free intersection graph of n segments in the plane has O(n) edges. The original proof was based on a separator theorem that does not generalize to higher dimensions. The new proof works in any dimension and is simpler: it uses size-sensitive cuttings, a variant of random sampling. We demonstrate the flexibility of this technique by extending the proof of the planar version of the theorem to intersection graphs of x-monotone curves.\nFranz J. Brandenburg, Walter Didimo, William S. Evans, Philipp Kindermann, Giuseppe Liotta and Fabrizio Montecchiani. Recognizing and Drawing IC-planar Graphs [+]\nAbstract: IC-planar graphs are those graphs that admit a drawing where no two crossed edges share an end-vertex and each edge is crossed at most once. They are a proper subfamily of the 1-planar graphs. Given an IC-planar graph G with n vertices, we present an O(n)-time algorithm that computes a straight-line drawing of G in quadratic area, and an O(n3)-time algorithm that computes a straight-line drawing of G with right-angle crossings in exponential area. Both these area requirements are worst-case optimal. We also show that it is NP-complete to test IC-planarity both in the general case and in the case in which a rotation system is fixed for the input graph. Furthermore, we describe a polynomial-time algorithm to test whether a set of matching edges can be added to a triangulated planar graph such that the resulting graph is IC-planar.\nWouter Meulemans and André Schulz. A Tale of Two Communities: Assessing Homophily in Node-Link Diagrams [+]\nAbstract: Homophily is a concept in social network analysis that states that in a network a link is more probable, if the two individuals have a common characteristic. We study the question if an observer can assess homophily by looking at the node-link diagram of the network. We design an experiment that investigates three different layout algorithms and asks the users to estimate the degree of homophily in the displayed network. One of the layout algorithms is a classical force-directed method, the other two are designed to improve node distinction based on the common characteristic. We study how each of the three layout algorithms helps to get a fair estimate, and whether there is a tendency to over or underestimate the degree of homophily. The stimuli in our experiments use different network sizes and different proportions of the cluster sizes.\nTowards Characterizing Graphs with a Sliceable Rectangular Dual [+]\nAbstract: Let G be a plane triangulated graph. A rectangular dual of G is a partition of a rectangle into a set of interior-disjoint rectangles, one for each vertex, such that two regions are adjacent if and only if the corresponding vertices are connected by an edge. A rectangular dual is sliceable if it can be recursively subdivided along horizontal or vertical lines. A graph is rectangular if it has a rectangular dual and sliceable if it has a sliceable rectangular dual. There is a clear characterization of rectangular graphs. However, a full characterization of sliceable graphs is still lacking. The currently best result (Yeap and Sarrafzadeh, 1995) proves that all rectangular graphs without a separating 4-cycle are sliceable. In this paper we introduce a recursively defined class of graphs, so-called rotating pyramids, which contain exactly one separating 4-cycle in their core''. We conjecture that configurations of rotating pyramids determine if a graph is sliceable. We verify this conjecture for the class of rectangular graphs that contain exactly one separating 4-cycle. The nonsliceable graphs in this class are exactly the graphs that reduce to rotating windmills: rotating pyramids with a specific corner assignment.\nVida Dujmovic, David Eppstein and David R. Wood. Genus, Treewidth, and Local Crossing Number [+]\nAbstract: We consider relations between the size, treewidth, and local crossing number (maximum crossings per edge) of graphs embedded on topological surfaces. We show that an n-vertex graph embedded on a surface of genus g with at most k crossings per edge has treewidth O((gkn)1/2) and layered treewidth O(gk), and that these bounds are tight up to a constant factor. As a special case, the k-planar graphs with n vertices have treewidth O((kn)) and layered treewidth O(k), which are tight bounds that improve a previously known O(k3/4n1/2) treewidth bound. Additionally, we show that for g<m, every m-edge graph can be embedded on a surface of genus g with O((m/g)log2g) crossings per edge, which is tight to within a polylogarithmic factor.\nOswin Aichholzer, Therese Biedl, Thomas Hackl, Martin Held, Stefan Huber, Peter Palfrader and Birgit Vogtenhuber. Representing Directed Trees as Straight Skeletons [+]\nAbstract: The straight skeleton of a polygon is the geometric graph obtained by tracing the vertices during a mitered offsetting process. It is known that the straight skeleton of a simple polygon is a tree, and one can naturally derive directions on the edges of the tree from the propagation of the shrinking process. In this paper, we ask the reverse question: Given a tree with directed edges, can it be the straight skeleton of a polygon? And if so, can we find a suitable simple polygon? We answer these questions for all directed trees where the order of edges around each node is fixed.\nLev Nachmanson, Roman Prutkin, Bongshin Lee, Nathalie Henry Riche, Alexander Holroyd and Xiaoji Chen. GraphMaps: Browsing Large Graphs as Interactive Maps [+]\nAbstract: Algorithms for laying out large graphs have seen significant progress in the past decade. However, browsing large graphs remains a challenge. Rendering thousands of graphical elements at once often results in a cluttered image, and navigating these elements naively can cause disorientation. To address this challenge we propose a method called GraphMaps, mimicking the browsing experience of online geographic maps.\nGraphMaps creates a sequence of layers, where each layer refines the previous one. During graph browsing, GraphMaps chooses the layer corresponding to the zoom level, and renders only those entities of the layer that intersect the current viewport. The result is that, regardless of the graph size, the number of entities rendered at each view does not exceed a predefined threshold, yet all graph elements can be explored by the standard zoom and pan operations.\nGraphMaps preprocesses a graph in such a way that during browsing, the geometry of the entities is stable, and the viewer is responsive. Our case studies indicate that GraphMaps is useful in gaining an overview of a large graph, and also in exploring a graph on a finer level of detail.\nDrawing Large Graphs by Multilevel Maxent-Stress Optimization [+]\nAbstract: Drawing large graphs appropriately is an important step for the visual analysis of data from real-world networks. Here we present a novel multilevel algorithm to compute a graph layout with respect to a recently proposed metric that combines layout stress and entropy. As opposed to previous work, we do not solve the linear systems of the maxent-stress metric with a typical numerical solver. Instead we use a simple local iterative scheme within a multilevel approach. To accelerate local optimization, we approximate long-range forces and use shared-memory parallelism. Our experiments validate the high potential of our approach, which is particularly appealing for dynamic graphs. In comparison to the previously best maxent-stress optimizer, which is sequential, our parallel implementation is on average 30 times faster already for static graphs (and still faster if executed on one thread) while producing a comparable solution quality.\nPeter Eades, Seok-Hee Hong, Karsten Klein and An Nguyen. Shape-based Quality Metrics for Large Graph Visualization [+]\nAbstract: We propose a new family of quality metrics for graph drawing; in particular, we concentrate on larger graphs. We illustrate these metrics with examples and apply the metrics to data from previous experiments, leading to the suggestion that the new metrics are effective.\nCarla Binucci, Markus Chimani, Walter Didimo, Martin Gronemann, Karsten Klein, Jan Kratochvil, Fabrizio Montecchiani and Ioannis Tollis. 2-Layer Fan-planarity: From Caterpillar to Stegosaurus [+]\nAbstract: In a fan-planar drawing of a graph there is no edge that crosses two other independent edges. We study 2-layer fan-planar drawings, i.e., fan-planar drawings such that the vertices are assigned to two distinct horizontal layers and edges are straight-line segments that connect vertices of different layers. We fully characterize 2-layer fan-planar drawable graphs and describe a linear-time testing and embedding algorithm for biconnected graphs. We also study the relationship between 2-layer fan-planar graphs and 2-layer right-angle crossing graphs.\nWalter Didimo, Francesco Giacchè and Fabrizio Montecchiani. Kojaph: Visual Definition and Exploration of Patterns in Graph Databases [+]\nAbstract: We present Kojaph, a new system for the visual definition and exploration of patterns in graph databases. It offers a powerful visual language integrated in a simple user interface, to define complex patterns as a combination of topological properties and node/edge attribute properties. Users can also interact with the query results and visually explore the graph incrementally, starting from such results. From the application perspective, Kojaph has been designed to run on top of every desired graph database management system (GDBMS). As a proof of concept, we integrated it with Neo4J, the most popular GDBMS.\nIgnacio Garcia Marco and Kolja Knauer. Drawing graphs with vertices and edges in convex position [+]\nAbstract: A graph has strong convex dimension 2, if it admits a straight-line drawing in the plane such that its vertices are in convex position and the midpoints of its edges are in convex position. Halman, Onn, and Rothblum conjectured that graphs of strong convex dimension 2 are planar and therefore have at most 3n-6 edges. We prove that all such graphs have at most 2n-3 edges while on the other hand we present a class of non-planar graphs of strong convex dimension 2. We also give lower bounds on the maximum number of edges a graph of strong convex dimension 2 can have and discuss variants of this graph class. We apply our results to questions about large convexly independent sets in Minkowski sums of planar point sets, that have been of interest in recent years.\nDavid Auber, Nicolas Bonichon, Paul Dorbec and Claire Pennarun. Rook-drawing for Plane Graphs [+]\nAbstract: Motivated by visualization of large graphs, we introduce a new type of graph drawing called rook-drawing\". A rook-drawing of a graph G is obtained by placing the n nodes of G on the intersections of a regular grid, such that each line and column of the grid supports exactly one node. This paper focuses on rook-drawings of planar graphs. We first give a linear algorithm to compute a planar straight-line rook-drawing for outerplanar graphs. We then show that there exist planar graphs which have no planar straight-line rook-drawing. Finally, we give a linear time algorithm to compute a polyline planar rook-drawing for plane graphs with at most n-3 bent edges.\nFabrizio Frati, Michael Hoffmann and Vincent Kusters. Simultaneous Embeddings with Few Bends and Crossings [+]\nAbstract: A simultaneous embedding with fixed edges (SEFE) of two planar graphs R and B is a pair of plane drawings of R and B that coincide when restricted to R\\cap B. We show that whenever R and B admit a SEFE, they also admit a SEFE in which every edge is a polygonal curve with few bends and every pair of edges has few crossings. Specifically: (1) if R and B are trees then one bend per edge and four crossings per edge pair suffice (and one bend per edge is sometimes necessary), (2) if R is a planar graph and B is a tree then six bends per edge and eight crossings per edge pair suffice, and (3) if R and B are planar graphs then six bends per edge and sixteen crossings per edge pair suffice. Our results improve on a paper by Grilli et al. (GD'14), which proves that nine bends per edge suffice, and on a paper by Chan et al. (GD'14), which proves that twenty-four crossings per edge pair suffice.\nIrina Kostitsyna, Martin Nöllenburg, Valentin Polishchuk, André Schulz and Darren Strash. On Minimizing Crossings in Storyline Visualizations [+]\nAbstract: In a storyline visualization, we visualize a collection of interacting characters (e.g., in a movie, play, etc.) by x-monotone curves that converge for each interaction, and diverge otherwise. Given a storyline with n character interactions, we show tight lower and upper bounds on the number of crossings required in any storyline visualization for a restricted case. In particular, we show that if (1) each meeting consists of exactly two characters and (2) the meetings can be modeled as a tree, then we can always find a storyline visualization with O(n log n) crossings. Furthermore, we show that there exist storylines in this restricted case that require Ω(n log n) crossings. Lastly, we show that, in the general case, computing a storyline visualization with a minimum number of crossings is fixed-parameter tractable, when parameterized on the number of characters k. Our algorithm runs in time O(k!^2k log k + k!^2n)$, where n is the number of meetings. Till Bruckdorfer, Michael Kaufmann, Patrizio Angelini and Tamara Mchedlidze. A Universal Point Set for 2-Outerplanar Graphs [+] Abstract: A point set S ⊂ R2 is universal for a class G if every graph of G has a planar straight-line embedding on S. It is well-known that the integer grid is a quadratic-size universal point set for planar graphs, while the existence of a sub-quadratic universal point set for them is one of the most fascinating open problems in Graph Drawing. Motivated by the fact that outerplanarity is a key property for the existence of small universal point sets, we study 2-outerplanar graphs and provide for them a universal point set of size O(n log n). The Utility of Untangling [+] Abstract: In this paper we show how techniques developed for untangling planar graphs by Bose et. al. [Discrete & Computational Geometry 42(4): 570-585 (2009)] and Goaoc et. al [Discrete & Computational Geometry 42(4): 542-569 (2009)] imply new results about some recent graph drawing models. These include column planarity, universal point subsets, and partial simultaneous geometric embeddings (with or without mappings). Some of these results answer open problems posed in previous papers. Martin Balko, Josef Cibulka and Pavel Valtr. Drawing graphs using a small number of obstacles [+] Abstract: An obstacle representation of a graph G is a set of points in the plane representing the vertices of G, together with a set of polygonal obstacles such that two vertices of G are connected by an edge in G if and only if the line segment between the corresponding points avoids all the obstacles. The obstacle number obs(G) of G is the minimum number of obstacles in an obstacle representation of G. We provide the first non-trivial general upper bound on the obstacle number of graphs by showing that every n-vertex graph satisfies obs(G) <= 2n log(n). This refutes a conjecture of Mukkamala, Pach, and Pálvölgyi. For bipartite n-vertex graphs, we improve this bound to n-1. Both bounds apply even when the obstacles are required to be convex. We also prove a lower bound 2Ω(hn) on the number of n-vertex graphs with obstacle number at most h for h < n and an asymptotically matching lower bound Ω(n4/3M2/3) for the complexity of a collection of M ≥ Ω(n) faces in an arrangement of n2 line segments with 2n endpoints. Muhammad Jawaherul Alam, Thomas Bläsius, Ignaz Rutter, Torsten Ueckerdt and Alexander Wolff. Pixel and Voxel Representations of Graphs [+] Abstract: We study contact representations for graphs, which we call pixel representations in 2D and voxel representations in 3D. Our representations are based on the unit square grid whose cells we call pixels in 2D and voxels in 3D. Two pixels are adjacent if they share an edge, two voxels if they share a face. We call a connected set of pixels or voxels a blob. Given a graph, we represent its vertices by disjoint blobs such that two blobs contain adjacent pixels or voxels if and only if the corresponding vertices are adjacent. We are interested in the size of a representation, which is the number of pixels or voxels it consists of. We first show that finding minimum-size representations is NP-complete. Then, we bound representation sizes needed for certain graph classes. In 2D, we show that, for k-outerplanar graphs with n vertices, Θ(kn)$ pixels are always sufficient and sometimes necessary. In particular, outerplanar graphs can be represented with a linear number of pixels, whereas general planar graphs sometimes need a quadratic number. In 3D, Θ(n2) voxels are always sufficient and sometimes necessary for any n-vertex graph. We improve this bound to Θ(n τ) for graphs of treewidth τ and to O((g+1)2 n log2n) for graphs of genus g. In particular, planar graphs admit representations with O(n log2n) voxels.\nTill Bruckdorfer, Michael Kaufmann, Stephen Kobourov and Sergey Pupyrev. On Embeddability of Buses in Point Sets [+]\nAbstract: Set membership of points in the plane can be visualized by connecting corresponding points via graphical features, like paths, trees, polygons, ellipses. In this paper we study the bus embeddability problem (BEP): given a set of colored points we ask whether there exists a planar realization with one horizontal straight-line segment per color, called bus, such that all points with the same color are connected with vertical line segments to their bus. We present an ILP and an FPT-approach for the general problem. For restricted versions of this problem, such as when the relative order of buses is predefined, or when a bus must be placed above all its points, we provide efficient algorithms. We show that another restricted version of the problem can be solved using 2-stack pushall sorting. On the negative side we prove the NP-completeness of a special case of BEP.\nLukas Barth, Andreas Gemsa, Benjamin Niedermann and Martin Nöllenburg. On the Readability of Boundary Labeling [+]\nAbstract: Boundary labeling deals with annotating features in images such that labels are placed outside of the image and are connected by curves (so-called leaders) to the corresponding features. While boundary labeling has been extensively investigated from an algorithmic perspective, the research on its readability has been neglected. In this paper we present the first formal user study on the readability of boundary labeling. We consider the four most studied leader types with respect to their performance, i.e., whether and how fast a viewer can assign a feature to its label and vice versa. We give a detailed analysis of the results regarding the readability of the four models and discuss their aesthetic qualities based on the users’ preference judgments and interviews.\nRadoslav Fulek and Rados Radoicic. Vertical Visibility among Parallel Polygons in Three Dimensions [+]\nAbstract: Motivated by a paper of Babilon et al. entitled Visibility Representations of Complete Graphs'' we study the following problem. Let C={C_1,..., C_n} denote a collection of translates of a regular convex k-gon in the plane with the stacking order. The collection C forms a visibility clique if for every i<j the intersection of C_i and C_j is not covered by the elements that are stacked between them, i.e., (C_i∩ C_j) ∖ i<k<j C_k ≠ ∅.\nWe show that if C forms a visibility clique its size is bounded from above by O(k4) thereby improving the upper bound of 22k from the aforementioned paper. We also obtain an upper bound of 22(k choose 2)+2 on the size of a visibility clique for homothetes of a convex (not necessarily regular) k-gons.\nJan Kyncl. Simple Realizability of Complete Abstract Topological Graphs Simplified [+]\nAbstract: An abstract topological graph (briefly an AT-graph) is a pair A=(G,X) where G=(V,E) is a graph and X⊂ (E choose 2) is a set of pairs of its edges. The AT-graph A is simply realizable if G can be drawn in the plane so that each pair of edges from X crosses exactly once and no other pair crosses. We characterize simply realizable complete AT-graphs by a finite set of forbidden AT-subgraphs, each with at most six vertices. This implies a straightforward polynomial algorithm for testing simple realizability of complete AT-graphs, which simplifies a previous algorithm by the author.\nMichael J. Bannister, David A. Brown and David Eppstein. Confluent Orthogonal Drawings of Syntax Diagrams [+]\nAbstract: We provide a pipeline for generating syntax diagrams (also called railroad diagrams) from context free grammars. Syntax diagrams are a graphical representation of a context free language, which we formalize abstractly as a set of mutually recursive nondeterministic finite automata and draw by combining elements from the confluent drawing, layered drawing, and smooth orthogonal drawing styles. Within our pipeline we introduce several heuristics that modify the grammar but preserve the language, improving the aesthetics of the final drawing.\nTarik Crnovrsanin, Jacqueline Chu and Kwan-Liu Ma. An Incremental Layout Method for Visualizing Online Dynamic Graphs [+]\nAbstract: Graphs provide a visual means for examining relation data and force-directed methods are often used to lay out graphs for viewing. Making sense of a dynamic graph as it evolves over time is challenging, and previous force-directed methods were designed for static graphs. In this paper, we present an incremental version of a multilevel multi-pole layout method with a refinement scheme incorporated, which enables us to visualize online dynamic networks while maintaining a mental map of the graph structure. We demonstrate the effectiveness of our method and compare it to previous methods using several dynamic network data sets.\nMichael Bekos, Michael Kaufmann and Christian Zielke. The Book Embedding Problem from a SAT-Solving Perspective [+]\nAbstract: In a book embedding, the vertices of a graph are placed on the spine of a book and the edges are assigned to pages, so that edges of the same page do not cross. In this paper, we approach the problem of determining whether a graph can be embedded in a book of a certain number of pages from a different perspective: We propose a simple and quite intuitive SAT formulation, which is robust enough to solve non-trivial instances of the problem in reasonable amount of time. In addition, we demonstrate how our approach helps to elaborate some hypotheses on the book embedding problem and develop new ones.\nRadoslav Fulek, Michael Pelsmajer and Marcus Schaefer. Hanani-Tutte for Radial Planarity [+]\nAbstract: A drawing of a graph G is radial if the vertices of G are placed on concentric circles C1, ... , Ck with common center c and edges are drawn radially: every edge crosses every circle centered at c at most once. G is radial planar if it has a radial embedding, that is, a crossing-free radial drawing. If the vertices of G are ordered or partitioned into ordered levels (as they are for leveled graphs), we require that the assignment of vertices to circles corresponds to the given ordering or leveling.\n\nWe show that a graph G is radial planar if G has a radial drawing in which every\ntwo edges cross an even number of times; the radial embedding has the same leveling as the radial drawing. In other words, we establish the weak variant of the Hanani-Tutte theorem for radial planarity. This generalizes a result by Pach and Tóth.\nAlessio Arleo, Walter Didimo, Giuseppe Liotta and Fabrizio Montecchiani. A Million Edge Drawing for a Fistful of Dollars [+]\nAbstract: In this paper we study the problem of designing a graph drawing algorithm for large graphs. The algorithm must be simple to implement and the computing infrastructure should not require major hardware or software investments. We describe a simple implementation of a spring embedder in Giraph, a vertex-centric open source framework for distributed computing. The algorithm is tested on real small-world, scale-free graphs of up to 1 million edges by using a cheap PaaS (Platform as a Service) infrastructure of Amazon. We can afford drawing graphs with about one million edges in about 20 minutes, by spending less than 2 USD per drawing for the cloud computing infrastructure.\nJonathan Klawitter, Martin Nöllenburg and Torsten Ueckerdt. Combinatorial Properties of Triangle-Free Rectangle Arrangements and the Squarability Problem [+]\nAbstract: We consider arrangements of axis-aligned rectangles in the plane. A geometric arrangement specifies the coordinates of all rectangles, while a combinatorial arrangement specifies only the respective intersection type in which any two rectangles intersect, i.e., which corners of one rectangle are contained in the other rectangle and vice versa. First, we exclude two particular intersection types and investigate combinatorial rectangle arrangements with a triangle-free intersection graph. We show that such rectangle arrangements are in bijection with the 4-orientations of an underlying planar multigraph and prove that there is a corresponding geometric arrangement with only touching rectangles.\nMoreover, we prove that every triangle-free planar graph is the contact graph of such an arrangement. Secondly, we study whether a given rectangle arrangement has a combinatorially equivalent square arrangement. In addition to some necessary conditions and counterexamples, we show that rectangle arrangements pierced by a horizontal line are squarable under certain sufficient conditions.\nIntersection-Link Representations of Graphs [+]\nAbstract: We consider graphs which are partitioned into dense subgraphs. We introduce intersection-link representations for such graphs, in which each vertex u is represented by a geometric object R(u) and in which each edge (u,v) is represented by the intersection between R(u) and R(v) if it belongs to a dense subgraph or by a curve connecting the boundaries of R(u) and R(v) otherwise. We study a notion of planarity, called Clique Planarity, for intersection-link representations of graphs in which the dense subgraphs are cliques.\nFabian Lipp, Alexander Wolff and Johannes Zink. Faster Force-Directed Graph Drawing with the Well-Separated Pair Decomposition [+]\nAbstract: The force-directed paradigm is one of the few generic approaches to drawing graphs. Since force-directed algorithms can be extended easily, they are used frequently. Most of these algorithms are, however, quite slow on large graphs as they compute a quadratic number of forces in each iteration. We speed up this computation by using an approximation based on the well-separated pair decomposition.\nWe perform experiments on a large number of graphs and show that we can strongly reduce the runtime—even on graphs with less then a hundred vertices—without a significant influence on the quality of the drawings (in terms of number of crossings and deviation in edge lengths).\nChristian Duncan. Maximizing the Degree of (Geometric) Thickness-t Regular Graphs [+]\nAbstract: In this paper, we show that there exist (6t-1)-regular graphs with thickness t, by constructing such an example graph. Since all graphs of thickness t must have at least one node with degree less than 6t, this construction is optimal. We also show, by construction, that there exist 5t-regular graphs with geometric thickness at most t. Our construction for the latter builds off of a relationship between geometric thickness and the Cartesian product of two graphs." ]

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[ "# logistic回归\n\n## 深入解读Logistic回归结果（一）：回归系数，OR\n\n(2016-03-08 06:40:50)\n\nLogistic回归虽然名字叫”回归” ，但却是一种分类学习方法。使用场景大概有两个：第一用来预测，第二寻找因变量的影响因素。\n\ny =β0 +β1*x1 +β2*x2 +β3*x3 +...+βn*xn\n\ny = 1 / (1+e-x)\n\ny = 1 / (1+e-z),\n\nLn(y/(1-y)) =β0 +β1*x1 +β2*x2 +β3*x3 +...+βn*xn\n\nLn(y/(1-y))称为Logit变换。我们再将y视为y取值为1的概率p(y=1)，因此，1-y就是y取值为0的概率p(y=0)，所以上式改写为：\n\np(y=1) = ez/(1+ez),\n\np(y=0) = 1/(1+ez),\n\nodds: 称为几率、比值、比数，是指某事件发生的可能性(概率)与不发生的可能性（概率）之比。用p表示事件发生的概率，则：odds = p/(1-p)。\n\nOR：比值比，为实验组的事件发生几率(odds1)/对照组的事件发生几率(odds2)。\n\n我们用一个例子来说明，这个例子中包含200名学生数据，包括1个自变量和4个自变量：\n\n因变量:  hon，表示学生是否在荣誉班(honors class)，1表示是，0表示否；\n\n自变量：\n\nfemale ：性别，分类变量，1=女，0=男\n\nwrite: 写作成绩，为连续变量\n\nmath：数学成绩，为连续变量\n\n1、不包含任何变量的Logistic回归\n\n首先拟合一个不包含任何变量的Logistic回归，\n\n模型为 ln(p/(1-p) =β0\n\n回归结果如下（结果经过编辑）：\n\nhon\n\nP\n\n-1.12546\n\n0.164\n\n0.000\n\n这里的系数β就是模型中的β0 = -1.12546，\n\n我们用p表示学生在荣誉班的概率，所以有ln(p/(1-p) =β0 = -1.12546，\n\n解方程得：p = 0.245。\n\nodds = p/1-p = 0.3245\n\n这里的p是什么意思呢？p就是所有数据中hon=1的概率。\n\n我们来统计一下整个hon的数据:\n\nhon\n\n0\n\n151\n\n75.5%\n\n1\n\n49\n\n24.5%\n\nhon取值为1的概率p为49/(151+49) = 24.5% = 0.245，我们可以手动计算出ln(p/(1-p) = -1.12546，等于系数β0。可以得出关系：\n\nβ0=ln(odds)。\n\n2、包含一个二分类因变量的模型\n\n拟合一个包含二分类因变量female的Logistic回归，\n\n模型为 ln(p/(1-p)  =β0 +β1* female.\n\n回归结果如下（结果经过编辑）：\n\nhon\n\nP\n\nfemale\n\n0.593\n\n.3414294\n\n0.083\n\n-1.47\n\n.2689555\n\n0.000\n\n在解读这个结果之前，先看一下hon和female的交叉表：\n\nhon\n\nfemale\n\nTotal\n\nMale\n\nFemale\n\n0\n\n74\n\n77\n\n151\n\n1\n\n17\n\n32\n\n49\n\nTotal\n\n91\n\n109\n\n3、包含一个连续变量的模型\n\n拟合一个包含连续变量math的Logistic回归，\n\n模型为 ln(p/(1-p)  =β0 +β1* math.\n\n回归结果如下（结果经过编辑）：\n\nhon\n\nP\n\nmath\n\n.1563404\n\n.0256095\n\n0.000\n\n-9.793942\n\n1.481745\n\n0.000\n\n这里截距系数的含义是在荣誉班中math成绩为0的odds的对数。我们计算出odds = exp(-9.793942) = .00005579，是非常小的。因为在我们的数据中，没有math成绩为0的学生，所以这是一个外推出来的假想值。\n\n怎么解释math的系数呢？根据拟合的模型，有：\n\nln(p/(1-p)) =  - 9.793942  + .1563404*math\n\n我们先假设math=54，有：\n\nln(p/(1-p))(math=54) = - 9.793942 + .1563404 *54\n\n然后我们把math提高提高一个单位，令math=55，有：\n\nln(p/(1-p))(math=55) = - 9.793942 + .1563404 *55\n\n两者之差：\n\nln(p/(1-p))(math=55) - ln(p/1-p))(math = 54) = 0.1563404.\n\n正好是变量math的系数。\n\n由此我们可以说，math每提高1个单位，odds（即p/(1-p)，也即处于荣誉班的几率）的对数增加0.1563404。\n\n那么odds增加多少呢？根据对数公式：\n\nln(p/(1-p))(math=55) - ln(p/1-p))(math = 54) = ln((p/(1-p)(math=55)/ (p/(1-p)(math=54))) = ln(odds(math=55)/ odds(math=54)) = 0.1563404.\n\n所以：\n\nodds(math=55)/ odds(math=54)  =  exp(0.1563404) = 1.169.\n\n因此我们可以说，math每升高一个单位，odds增加16.9%。且与math的所处的绝对值无关。\n\n聪明的读者肯定发现，odds(math=55)/ odds(math=54)不就是OR嘛！\n\n4、包含多个变量的模型（无交互效应）\n\n模型为 ln(p/(1-p) = β0 +β1* math+β2* female+β3* read.\n\n回归结果如下（结果经过编辑）：\n\nhon\n\nP\n\nmath\n\n.1229589\n\n0.000\n\nfemale\n\n0.979948\n\n0.020\n\n.0590632\n\n0.026\n\n-11.77025\n\n0.000\n\n该结果说明:\n\n5、包含交互相应的模型\n\n拟合一个包含female、math和两者交互相应的Logistic回归，\n\n模型为 ln(p/(1-p)  =β0 +β1* female+β2* math+β3* female *math.\n\n所谓交互效应，是指一个变量对结果的影响因另一个变量取值的不同而不同。\n\n回归结果如下（结果经过编辑）：\n\nhon\n\nP\n\nfemale\n\n-2.899863\n\n0.349\n\nmath\n\n.1293781\n\n0.000\n\nfemale*math\n\n.0669951\n\n0.210\n\n-8.745841\n\n0.000\n\n注意：female*math项的P为0.21，可以认为没有交互相应。但这里我们为了讲解交互效应，暂时忽略P值，姑且认为他们是存在交互效应的。\n\n由于交互效应的存在，我们就不能说在保持math和female*math不变的情况下，female的影响如何如何，因为math和female*math是不可能保持不变的!\n\n对于这种简单的情况，我们可以分别拟合两个方程，\n\n对于男性（female=0）：\n\nlog(p/(1-p))= β0 + β2*math.\n\n对于女性（female=1）：\n\nlog(p/(1-p))= (β0 + β1) + (β2 + β3 )*math.\n\n然后分别解释。\n\n0 条评论\n\n• ### 安卓手机免root实现对其他软件最高管理（sandbox思想）\n\nroot之后的安卓系统并不稳定，root后有时候会出现一些系统的错误，如果实在忍受不了的话，这时候只能恢复出厂设置了。因此不root是最优的选择，但是不ro...\n\n• ### 方便快捷的求导求积分解方程在线工具sage介绍\n\n有时候我们需要进行一些复杂的数学计算，比如求导， 求积分，解方程，还是用abcd字母代表变量的方程等，这就需要进行复杂的数学运算还需要具备良好的数学基础。不...\n\n• ### R语言进行机器学习方法及实例（一）\n\n机器学习的研究领域是发明计算机算法，把数据转变为智能行为。机器学习和数据挖掘的区别可能是机器学习侧重于执行一个已知的任务，而数据发掘是在大数据中寻找有价值的...\n\n• ### JSON Web加密中的高危漏洞\n\nJSON Web加密容易受到经典的Invalid Curve攻击，这篇文章可以帮助您了解这会对您产生什么影响以及您应该如何应对。\n\n• ### Postman使用完全指南\n\n可以看到我的请求url中有个{{domain}}，这是什么东西呢？这其实是环境变量，我们可以在如下设置环境变量\n\n• ### Havex：以工控设备为狩猎目标的恶意软件\n\n在过去一年，我们对Havex恶意程序家族及其背后的组织保持了高度的关注。Havex被认为以不同工业领域为目标进行攻击的恶意软件，并且在最初的报告中，该恶意软件对...\n\n• ### 二十年编程语言风云，哪款是你的爱豆？\n\n过去一年，编程语言也要决出这一年的最佳语言，会是谁呢，从 TIOBE 上来看，Java、C 和 Python 基本锁定了前三的位置，Java 江湖老大的地位，还...\n\n• ### 二十年编程语言风云，哪款是你的爱豆？\n\n年终岁尾了，编程语言也要决出这一年的最佳语言了，会是谁呢，从 TIOBE 上来看，Java、C 和 Python 基本锁定了前三的位置，Java 江湖老大的地位...\n\n### 活动推荐", null, "" ]

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[ "# Mathematical libraries\n\nThe ROOT mathematical libraries consist of the following components:\n\n## MathCore library\n\nThe MathCore library provides a collection of functions, C++ classes and ROOT classes for HEP numerical computing.\nThe MathCore is a self-consistent minimal set of tools required for the basic numerical computing. More advanced mathematical functionalities is provided by the MathMore library. The following is included in the MathCore library:\n\n• Special functions: Functions like the gamma, beta and error function that are used in HEP.\n\n• Statistical functions: Functions used in statistics, such as the probability density functions and the cumulative distributions functions for continuous and discrete distributions.\n\n• Function classes and interfaces: Interfaces (abstract classes) and base classes, including helper classes to wrap free (static) and non-static member functions.\n\n• Numerical algorithms: User classes with basic implementations for:\n• Fitting and parameter estimation: ROOT classes for fitting and parameter estimation from a given data set.\n\nIn addition, the MathCore library contains the following ROOT classes that were originally part of libCore:\n\n### TMath\n\nThe TMath namespace provides a collection of free functions:\n\nElementary functions\n\nSome of elementary mathematical functions refer to basic mathematical functions like the square root, the power to a number of the calculus of a logarithm, while others are used for number treatment, like rounding.\n\nAlthough there are some functions that are not in the standard C math library (like Factorial), most of the functionality offered here is just a wrapper of the first ones. Nevertheless, some of them also offer some security checks or a better precision, like the trigonometrical functions ASin(x), ACos(x) or ATan(x).\n\nExamples\n\nStatistic functions operating on arrays\n\nTMath provides functions that process arrays for calculation:\n\n• mean\n• median\n• geometrical mean\n• sample standard deviation (RMS)\n• the kth smallest element\n\nExample\n\nSpecial and statistical functions\n\nTMath provides special functions like Bessel, error functions, Gamma or similar plus statistical mathematical functions, including probability density functions, cumulative distribution and their inverse.\n\nThe majority of the special functions and the statistical distributions are provided also as free functions in the ROOT::Math namespace.\n\nFunctions not present in the ROOT::Math name that are provided only by TMath are:\n\n• Special functions:\n• DiLogarithm\n• Struve\n• Statistical functions:\n• KolmogorovProb\n• Voigt function\n• LaplaceDist\n• Vavilov\n\n### ROOT::Math\n\nThe ROOT::Math namespace provides a set of function interfaces to define the basic behaviour of a mathematical function:\n\nIn addition, helper classes, wrapping the user interfaces in the ROOT::Math function interfaces are provided. With wrapper functions you can insert your own type of function in the needed function interface.\n\nTo use the self-defined functions, they must have inherited from one of the following classes:\n\nFigure: ROOT::Math function interface structure.\n\nOne-dimensional function interfaces\n\nThis interface is used for numerical algorithms operating only on one-dimensional functions. It cannot applied to multi-dimensional functions.\n\nROOT::Math::IBaseFunctionOneDim\nThis interface provides a method to evaluate the function given a value (simple double) by implementing double operator() (const double). The user class defined only needs to reimplement the pure abstract method double DoEval(double x) that will do the work of evaluating the function at point x.\n\nExample\n\nExample for the implementation of a class that represents a mathematical function.\n\nThis interface is needed by some numerical algorithms to calculate the derivatives of the function. It introduces the method double Derivative(double x) that will return the derivative of the function at the point x. The class inherit by the user will have to implement the abstract method double DoDerivative(double x), leaving the rest of the class untouched.\n\nExample\n\nExample for the implementation of a gradient one-dimensional function.\n\nMulti-dimensional function interfaces\n\nThis interface is used for numerical algorithms operating on multi-dimensional functions.\n\nROOT::Math::IBaseFunctionMultiDim\nThis interface provides the double operator() (const double*) that takes an array of doubles with all the values for the different dimensions. In this case, the user has to provide the functionality for two different functions: double DoEval(const double*) and unsigned int NDim(). The first ones evaluates the function given the array that represents the multiple variables. The second returns the number of dimensions of the function.\n\nExample\n\nExample for the implementation of a basic multi-dimensional function.\n\nThis interface offers the same functionality as the base function and additionally the calculation of the derivative. It only adds the double Derivative(double* x, uint ivar) method for the user to implement. This method must implement the derivative of the function with respect to the variable indicated with the second parameter.\n\nExample\n\nExample for the implementation of a multi-dimensional gradient function.\n\nParametric function interfaces\n\nThis interface is used for fitting after evaluating multi-dimensional functions.\n\nROOT::Math::IParametricFunctionMultiDim\nThis interface describes a multi-dimensional parametric function. Similarly to the one dimensional version, the user needs to provide the void SetParameters(double* p) method as well as the getter methods const double * Parameters() and uint NPar().\n\nExample\n\nExample for the implementation of a parametric function.\n\nThis interface provides an interface for parametric gradient multi-dimensional functions. In addition to function evaluation, it provides the gradient with respect to the parameters, via the ParameterGradient() method. This interface is only used in case of some dedicated fitting algorithms, when is required or more efficient to provide derivatives with respect to the parameters.\n\nExample\n\nExample for the implementation of a parametric gradient function.\n\nWrapper functions\n\nTo insert your own type of function in the needed function interface, helper classes, wrapping the user interface in the ROOT::Math function interfaces are provided.\n\nThere is one possible wrapper for every interface.\n\nInterface Wrapper Description\nROOT::Math::IBaseFunctionOneDim ROOT::Math::Functor1D See → Wrapping one-dimensional functions\nROOT::Math::IBaseFunctionMultiDim ROOT::Math::Functor See → Wrapping multi-dimensional functions\n\nNote the special case when wrapping TF1 objects in parametric function interfaces.\n\nWrapping one-dimensional functions\n\nUse ROOT::Math::Functor1D to wrap one-dimensional functions.\n\nROOT::Math::Functor1D can wrap the following types:\n\n• A free C function of type double ()(double ).\n• Any C++ callable object implementation double operator()( double).\n• A class member function with the correct signature like double Foo::Eval(double ). In this case one pass the object pointer and a pointer to the member function (&Foo::Eval).\n\nExample\n\nIt can be constructed in three different ways:\n\n• Any object implementing both double operator()( double) for the function evaluation and double Derivative(double) for the function derivative.\n• Any object implementing any member function like Foo::XXX(double ) for the function evaluation and any other member function like Foo::YYY(double) for the derivative.\n• Any two function objects implementing double operator()( double). One object provides the function evaluation, the other the derivative. One or both function object can be a free C function of type double ()(double).\n\nWrapping multi-dimensional functions\n\nUse the ROOT::Math::Functor to wrap multi-dimensional function objects.\n\nIt can wrap all the following types:\n\n• Any C++ callable object implementing double operator()( const double * ).\n• A free C function of type double ()(const double *).\n• A member function with the correct signature like Foo::Eval(const double *). In this case one pass the object pointer and a pointer to the member function (&Foo::Eval).\n\nExample\n\nIt can be constructed in three different way:\n\n• From an object implementing both double operator()( const double*) for the function evaluation and double Derivative(const double *, int icoord) for the partial derivatives.\n• From an object implementing any member function like Foo::XXX(const double *) for the function evaluation and any member function like Foo::XXX(const double *, int icoord) for the partial derivatives.\n• From an function object implementing double operator()( const double *) for the function evaluation and another function object implementing double operator() (const double *, int icoord) for the partial derivatives.\n\nThe function dimension is required when constructing the functor.\n\nWrapping TF1 objects in parametric function interfaces\n\nOften the TF1 class is used.\nUse the ROOT::Math::WrappedTF1 class, if the interface to be wrapped is one-dimensional.\n\nThe default constructor takes a TF1 reference as argument, that will be wrapped with the interfaces of a ROOT::Math::IParametricGradFunctionOneDim.\n\nExample\n\nUse the ROOT::Math::WrappedMultiTF1 class, if the interface to be wrapped is multi-dimensional.\n\nFollowing the usual procedure, setting the TF1 though the constructor, will wrap it into a ROOT::Math::IParametricGradFunctionMultiDim.\n\nExample\n\n### Random numbers\n\nThe MathCore library provides the following classes for generating pseudo-random numbers:\n\n• TRandom: Using a linear congruential random generator.\n• TRandom1: Random number generator based on the Ranlux engine.\n• TRandom2: Based on the maximally equi-distributed combined Tausworthe generator by L’Ecuyer.\n• TRandom3: Based on the Mersenne and Twister pseudo-random number generator.\n\nNote\n\nFor generating non-uniform random numbers, the UNU.RAN package (see → UNU.RAN) is available.\n\nYou can work with the random number generators as follows:\n\nSeeding the random number generators\n\nWhen no value is given, the generator default seed is used. In this case an identical sequence will be generated every time the application is run.\nWhen the 0 value is used as seed, then a unique seed is generated using a TUUID, for TRandom , TRandom1 and TRandom3 .\nFor TRandom the seed is generated using only the machine clock, which has a resolution of about 1 s. Therefore, identical sequences will be generated if the elapsed time is less than a second.\n\nUsing the random number generators\n\n• Use the Rndm() method for generating a pseudo-random number distributed between 0 and 1.\n\nExample\n\nRandom number distributions\n\nThe TRandom class provides functions that can be used by all other derived classes to generate random variables according to predefined distributions. In the simplest cases, as in the exponential distribution, the non-uniform random number is obtained by suitable transformations. In the more complicated cases, the random variables are obtained by acceptance-rejection methods that require several random numbers.\n\nExample\n\nThe following table shows the various distributions that can be generated using methods of the TRandom classes.\nIn addition, you can use TF1::GetRandom() or TH1::GetRandom() to generate random numbers distributed according to a user defined function, in a limited interval, or to a user defined histogram.\n\nDistributions Description\nDouble_t Uniform(Double_t x1,Double_t x2) Uniform random numbers between x1,x2.\nDouble_t Gaus(Double_t mu,Double_t sigma) Gaussian random numbers. Default values: mu=0, sigma=1.\nDouble_t Exp(Double_t tau) Exponential random numbers with mean tau.\nDouble_t Landau(Double_t mean,Double_t sigma) Landau distributed random numbers. Default values: mean=0, sigma=1.\nDouble_t BreitWigner(Double_t mean,Double_t gamma) Breit-Wigner distributed random numbers. Default values mean=0, gamma=1.\nInt_t Poisson(Double_t mean) Poisson random numbers.\nDouble_t PoissonD(Double_t mean) Poisson random numbers.\nInt_t Binomial(Int_t ntot,Double_t prob) Binomial Random numbers\nCircle(Double_t &x,Double_t &y,Double_t r) Generate a random 2D point (x,y) in a circle of radius r.\nSphere(Double_t &x,Double_t &y,Double_t &z,Double_t r) Generate a random 3D point (x,y,z) in a sphere of radius r.\nRannor(Double_t &a,Double_t &b) Generate a pair of Gaussian random numbers with mu=0 and sigma=1.\n\n### Complex numbers\n\nThe MathCore library provides with TComplex a class for complex numbers.\n\n## MathMore library\n\nThe MathMore library provides an advanced collection of functions and C++ classes for numerical computing. This is an extension of the functionality provided by the MathCore library. The MathMore library is implemented wrapping in C++ the GNU Scientific Library (GSL). The mathematical functions are implemented as a set of free functions in the namespace ROOT::Math.\n\nThe MathMore library includes classes and functions for:\n\n## Linear algebra packages\n\nThe linear algebra packages provide a complete environment in ROOT to perform calculations like equation solving and eigenvalue decompositions.\n\nThere are the following linear algebra packages available:\n\n### Matrix package\n\nThe following topics are covered for the matrix package:\n\nMatrix classes\n\nROOT provides the following matrix classes, among others:\n\n• TMatrixDBase: Base class for matrices.\n\n• TMatrixF: Matrix with single precision (float).\n\n• TMatrixFSym: Symmetrical matrix with single precision (float).\n\n• TVectorF: Vector with single precision (float).\n\n• TMatrixD: Matrix with double precision (double).\n\n• TMatrixDSym: Symmetrical matrix with double precision (double).\n\n• TMatrixDSparse: Sparse matrix with double precision (double).\n\n• TDecompBase: Decomposition base class.\n\n• TDecompChol: Cholesky decomposition class.\n\nMatrix properties\n\nA matrix has five properties, which are all set in the constructor:\n\n• precision\nIf the precision is float (this is single precision), use the TMatrixF class family. If the precision is double, use the TMatrixD class family.\n\n• type\nPossible values are: general (TMatrixD), symmetric (TMatrixDSym) or sparse (TMatrixDSparse).\n\n• size\nNumber of rows and columns.\n\n• index\nRange start of row and column index. By default these start at 0.\n\n• sparse map\nOnly relevant for a sparse matrix. It indicates where elements are unequal 0.\n\nYou can:\n\nAccessing matrix properties\n\nUse one of the following methods to access the information about the relevant matrix property:\n\n• Int_t GetRowLwb(): Row lower-bound index.\n\n• Int_t GetRowUpb(): Row upper-bound index.\n\n• Int_t GetNrows(): Number of rows.\n\n• Int_t GetColLwb(): Column lower-bound index.\n\n• Int_t GetColUpb(): Column upper-bound index.\n\n• Int_t GetNcols: Number of columns.\n\n• Int_t GetNoElements(): Number of elements, for a dense matrix this equals: fNrows x fNcols.\n\n• Double_t GetTol(): Tolerance number that is used in decomposition operations.\n\n• Int_t *GetRowIndexArray(): For sparse matrices, access to the row index of fNrows+1 entries.\n\n• Int_t *GetColIndexArray(): For sparse matrices, access to the column index of fNelems entries.\n\n*GetRowIndexArray() and *GetColIndexArray() are specific to the sparse matrix, which is implemented according to the Harwell- Boeing format. Here, besides the usual shape/size descriptors of the matrix like fNrows, fRowLwb, fNcols and fColLwb, also a row index fRowIndex and a column index fColIndex are stored:\n\n• fRowIndex[0,..,fNrows]: Stores for each row the index range of the elements in the data and column array.\n• fColIndex[0,..,fNelems-1]: Stores the column number for each data element != 0.\n\nSetting matrix properties\n\nUse one of the following methods to set a matrix property:\n\n• SetTol (Double_t tol)\nSets the tolerance number.\n\n• ResizeTo (Int_t nrows,Int_t ncols, Int_t nr_nonzeros=-1)\nChanges the matrix shape to nrows x ncols. Index will start at 0.\n\n• ResizeTo(Int_t row_lwb,Int_t row_upb, Int_t col_lwb,Int_t col_upb, Int_t nr_nonzeros=-1)\nChanges the matrix shape to row_lwb:row_upb x col_lwb:col_upb.\n\n• SetRowIndexArray (Int_t *data)\nFor sparse matrices, it sets the row index. The array data should contain at least fNrows+1 entries column lower-bound index.\n\n• SetColIndexArray (Int_t *data)\nFor sparse matrices, it sets the column index. The array data should contain at least fNelems entries.\n\n• SetSparseIndex (Int_t nelems new)\nAllocates memory for a sparse map of nelems_new elements and copies (if exists) at most nelems_new matrix elements over to the new structure.\n\n• SetSparseIndex (const TMatrixDBase &a)\nCopies the sparse map from matrix a.\n\n• SetSparseIndexAB (const TMatrixDSparse &a, const TMatrixDSparse &b)\nSets the sparse map to the same map of matrix a and b.\n\nCreating and filling a matrix\n\nUse one of the following constructors to create a matrix:\n\n• TMatrixD(Int_t nrows,Int_t ncols)\n• TMatrixD(Int_t row_lwb,Int_t row_upb,Int_t col_lwb,Int_t col_upb)\n• TMatrixD(Int_t nrows,Int_t ncols,const Double_t *data, Option_t option= \"\")\n• TMatrixD(Int_t row_lwb,Int_t row_upb,Int_t col_lwb,Int_t col_upb, const Double_t *data,Option_t *option=\"\")\n• TMatrixDSym(Int_t nrows)\n• TMatrixDSym(Int_t row_lwb,Int_t row_upb)\n• TMatrixDSym(Int_t nrows,const Double_t *data,Option_t *option=\"\")\n• TMatrixDSym(Int_t row_lwb,Int_t row_upb, const Double_t *data, Option_t *opt=\"\")\n• TMatrixDSparse(Int_t nrows,Int_t ncols)\n• TMatrixDSparse(Int_t row_lwb,Int_t row_upb,Int_t col_lwb, Int_t col_upb)\n• TMatrixDSparse(Int_t row_lwb,Int_t row_upb,Int_t col_lwb,Int_t col_upb, Int_t nr_nonzeros,Int_t *row,Int_t *col,Double_t *data)\n\nUse one of the following methods to fill a matrix:\n\n• SetMatrixArray(const Double_t*data,Option_t*option=\"\")\nCopies array data. If option=\"F\", the array fills the matrix column-wise else row-wise. This option is implemented for TMatrixD and TMatrixDSym. It is expected that the array data contains at least fNelems entries.\n\n• SetMatrixArray(Int_t nr,Int_t *irow,Int_t *icol,Double_t *data)\nOnly available for sparse matrices. The three arrays should each contain nr entries with row index, column index and data entry. Only the entries with non-zero data value are inserted.\n\n• operator(), operator[]\nThese operators provide the easiest way to fill a matrix but are in particular for a sparse matrix expensive. If no entry for slot (i,j) is found in the sparse index table it will be entered, which involves some memory management. Therefore, before invoking this method in a loop set the index table first through a call to the SetSparseIndex() method.\n\n• SetSub(Int_t row_lwb,Int_t col_lwb,const TMatrixDBase &source)\nThe matrix to be inserted at position (row_lwb,col_lwb) can be both, dense or sparse.\n\n• Use()\nAllows inserting another matrix or data array without actually copying the data.\nThe following list shows the application of the Use() method:\n\n• Use(TMatrixD &a)\n• Use(Int_t row_lwb,Int_t row_upb,Int_t col_lwb,Int_t col_upb,Double_t *data)\n• Use(Int_t nrows,Int_t ncols,Double_t *data)\n• Use(TMatrixDSym &a)\n• Use(Int_t nrows,Double_t *data)\n• Use(Int_t row_lwb,Int_t row_upb,Double_t *data)\n• Use(TMatrixDSparse &a)\n• Use(Int_t row_lwb,Int_t row_upb,Int_t col_lwb,Int_t col_upb,Int_t nr_no nzeros, Int_t *pRowIndex,Int_t *pColIndex,Double_t *pData)\n• Use(Int_t nrows,Int_t ncols,Int_t nr_nonzeros,Int_t *pRowIndex,Int_t *pColIndex,Double_t *pData)\n\nExample\n\nA Hilbert matrix is created by copying an array.\n\nYou can also assign the data array to the matrix without actually copying it.\n\nThe array data now contains the inverted matrix.\n\nNow a unit matrix in sparse format is created.\n\nInverting a matrix\n\n• Use the Invert(Double_t &det=0) function to invert a matrix:\n\n– or –\n\n• Use the appropriate constructor to invert a matrix:\n\nBoth methods are available for general and symmetric matrices.\n\nFor matrices whose size is less than or equal to 6x6, the InvertFast(Double_t &det=0) function is available. Here the Cramer algorithm will be applied, which is faster but less accurate.\n\nUsing decomposition classes for inverting\n\nYou can also use the following decomposition classes (see → Matrix decompositions) for inverting a matrix:\n\nName Matrix type Comment\nTDecompLU General\nTDecompQRH General\nTDecompSVD General Can manipulate singular matrix.\nTDecompBK symmetric\nTDecompChol Symmetric Matrix should also be positive definite.\nTDecompSparse Sparse\n\nIf the required matrix type is general, you also can handle symmetric matrices.\n\nExample\n\nThis example shows how to check whether the matrix is singular before attempting to invert it.\n\nMatrix operators and methods\n\nThe matrix/vector operations are classified according to BLAS (basic linear algebra subroutines) levels.\n\nThe following operations and methods are available:\n\nArithmetic operations between matrices\n\nDescription Format Comment\nElement C=A+B Overwrites A\nWise sum A+=B\nTMatrixD(A,TMatrixD::kPlus,B)\nA = A + α B constructor\nElement wise subtraction C=A-B A-=B\nTMatrixD(A,TMatrixD::kMinus,B)\nOverwrites A\nConstructor\nMatrix multiplication C=A*B\nA*=B\nC.Mult(A,B)\nTMatrixD(A,TMatrixD::kMult,B)\nTMatrixD(A, TMatrixD(A, TMatrixD::kTransposeMult,B)\nTMatrixD(A, TMatrixD::kMultTranspose,B)\nOverwrites A\n\nConstructor of A.B\nConstructor of AT .B\nConstructor of A.BT\nElement wise multiplication ElementMult(A,B) A(i,j)*= B(i,j)\nElement wise division ElementDiv(A,B) A(i,j)/= B(i,j)\n\nArithmetic operations between matrices and real numbers\n\nDescription Format Comment\nElement wise sum C=r+A C=A+r A+=r overwrites A\nElement wise subtraction C=r-A C=A-r A-=r overwrites A\nMatrix multiplication C=r*A C=A*r A*=r overwrites A\n\nComparison between two matrices\n\nFormat Output Description\nA == B Bool_t Equal to\nA != B matrix Not equal\nA > B matrix Greater than\nA >= B matrix Greater than or equal to\nA < B matrix Smaller than\nA <= B matrix Smaller than or equal to\nAreCompatible(A,B) Bool_t Compare matrix properties\nCompare(A,B) Bool_t Return summary of comparison\nVerifyMatrixIdentity(A,B,verb, maxDev)   Check matrix identity within maxDev tolerance\n\nComparison between matrix and real number\n\nFormat Output Description\nA == r Bool_t Equal to\nA != r Bool_t Not equal\nA > r Bool_t Greater than\nA >= r Bool_t Greater than or equal to\nA < r Bool_t Smaller than\nA <= r Bool_t Smaller than or equal to\nVerifyMatrixValue(A,r,verb, maxDev) Bool_t Compare matrix value with r within maxDev tolerance\nA.RowNorm() Double_t Norm induced by the infinity vector norm\nA.NormInf() Double_t\nA.ColNorm() Double_t Norm induced by the 1 vector norm\nA.Norm1() Double_t\nA.E2Norm() Double_t Square of the Euclidean norm\nA.NonZeros() Int_t\nA.Sum() Double_t Number of elements unequal zero\nA.Min() Double_t\nA.Max() Double_t\nA.NormByColumn (v,\"D\") TMatrixD\nA.NormByRow (v,\"D\") TMatrixD\n\nMatrix views\n\nWith the following matrix view classes, you can access the matrix elements:\n\n• TMatrixDRow\n• TMatrixDColumn\n• TMatrixDDiag\n• TMatrixDSub\n\nMatrix view operators\n\nFor the matrix view classes TMatrixDRow, TMatrixDColumn and TMatrixDDiag, the necessary assignment operators are available to interact with the vector class TVectorD.\nThe sub matrix view classes TMatrixDSub has links to the matrix classes TMatrixD and TMatrixDSym.\n\nThe next table summarizes how to access the individual matrix elements in the matrix view classes.\n\nFormat Comment\nTMatrixDRow(A,i)(j) TMatrixDRow(A,i)[j] Element Aij\nTMatrixDColumn(A,j)(i) TMatrixDColumn(A,j)[i] Element Aij\nTMatrixDDiag(A(i) TMatrixDDiag(A[i] Element Aij\nTMatrixDSub(A(i) TMatrixDSub(A,rl,rh,cl,ch)(i,j) Element Aij\nElement Arl+i,cl+j\n\nMatrix decompositions\n\nThere are the following classes available for matrix decompositions:\n\n• TDecompLU: Decomposes a general n x n matrix A into P A = L U.\n• TDecompBK: The Bunch-Kaufman diagonal pivoting method decomposes a real symmetric matrix A.\n• TDecompChol : The Cholesky decomposition class, which decomposes a symmetric, positive definite matrix A = U^T * U where U is a upper triangular matrix.\n• TDecompQRH: QR decomposition class.\n• TDecompSVD: Single value decomposition class.\n• TDecompSparse: Sparse symmetric decomposition class.\n\nMatrix Eigen analysis\n\nWith the TMatrixDEigen and TMatrixDSymEigen classes, you can compute eigenvalues and eigenvectors for general dense and symmetric real matrices.\n\nThe following table lists the methods of the TMatrixDEigen and the TMatrixDSymEigen to obtain the eigenvalues and eigenvectors. TMatrixDSymEigen constructors can only be called with TMatrixDSym:\n\nFormat Output Description\neig.GetEigenVectors() TMatrixD Eigenvectors for both TMatrixDEigen and TMatrixDSymEigen.\neig.GetEigenValues() TVectorD Eigenvalues vector for TMatrixDSymEigen.\neig.GetEigenValues() TMatrixD Eigenvalues matrix for TMatrixDEigen.\neig.GetEigenValuesRe() TVectorD Real part of eigenvalues for TMatrixDEigen.\neig.GetEigenValuesIm() TVectorD Imaginary part of eigenvalues for TMatrixDEigen.\n\nExample\n\nThe usage of the eigenvalue class is shown in this example where it is checked that the square of the singular values of a matrix c are identical to the eigenvalues of c^T.c:\n\n### SMatrix\n\nSMatrix is a C++ package for high performance vector and matrix computations. It can be used only in problems when the size of the matrices is known at compile time, like in the tracking reconstruction of HEP experiments. It is based on a C++ technique, called expression templates, to achieve an high level optimization. The C++ templates can be used to implement vector and matrix expressions such that these expressions can be transformed at compile time to code which is equivalent to hand optimized code in a low-level language like FORTRAN or C.\n\nThe SMatrix has been developed initially by T. Glebe of the Max-Planck-Institut, Heidelberg, as part of the HeraB analysis framework. A subset of the original package has been now incorporated in the ROOT distribution, with the aim to provide to the LHC experiments a stand-alone and high performance matrix package for reconstruction. The API of the current package differs from the original one, in order to be compliant to the ROOT coding conventions.\n\nSMatrix contains the following generic classes for describing matrices and vectors of arbitrary dimensions and of arbitrary type:\n\n#### SVector\n\nThe template class ROOT::Math::SVector represents n-dimensional vectors for objects of arbitrary type. The class has two template parameters that define their properties at compile time:\n\n1. Type of the contained elements (for example float or double).\n2. Size of the vector.\n\nCreating a vector\n\nUse one of the following constructors to create a vector:\n\n• Default constructor for a zero vector (all elements equal to zero).\n• Constructor (and assignment) from a vector expression, like v=p*q+w. Due to the expression template technique, no temporary objects are created in this operation.\n• Constructor by passing directly the elements. This is possible only for vectors up to size 10.\n• Constructor from an iterator copying the data referred by the iterator. It is possible to specify the begin and end of the iterator or the begin and the size. Note that for\n\nExample The namespace ROOT::Math is used.\n\n#### SMatrix\n\nThe template class ROOT::Math::SMatrix represents a matrix of arbitrary type with nrows x ncoldimension. The class has four template parameters that define their properties at compile time:\n\n• type of the contained elements (for example float or double)\n• number of rows\n• number of columns\n• representation type\n\nCreating a matrix\n\nUse one of the following constructors to create a matrix:\n\n• Default constructor for a zero matrix (all elements equal to zero).\n• Constructor of an identity matrix.\n• Copy constructor (and assignment) for a matrix with the same representation, or from a different one when possible, for example from a symmetric to a general matrix.\n• Constructor (and assignment) from a matrix expression, like D=A*B+C. Due to the expression template technique, no temporary objects are created in this operation. In the case of an operation like A=A*B+C, a temporary object is needed and it is created automatically to store the intermediary result in order to preserve the validity of this operation.\n• Constructor from a generic STL-like iterator copying the data referred by the iterator, following its order. It is both possible, to specify the begin and end of the iterator or the begin and the size. In case of a symmetric matrix, it is required only the triangular block and the user can specify whether giving a block representing the lower (default case) or the upper diagonal part.\n\nExample Typedef’s are used in this example to avoid the full C++ names for the matrix classes. For a general matrix the representation has the default value ROOT::Math::MatRepStd. For a general square matrix, the number of columns can be omitted.\n\nExample\n\nA symmetric matrix is filled from a std::vector.\n\n## Minimization libraries and classes\n\nROOT provides several minimization libraries and classes:\n\n### TMinuit\n\nThe Minuit minimization package was originally written in Fortran by Fred James and part of PACKLIB (patch D506). It has been converted to the C++ class TMinuit , by R.Brun.\n\nTopical manual\n\nFor TMinuit, a topical manual it available at Topical Manual - TMinuit.\nIt contains in-depth information about TMinuit.\n\n### Minuit2 library\n\nThe Minuit2 library is a new object-oriented implementation, written in C++, of the popular MINUIT minimization package. These new version provides basically all the functionality present in the old Fortran version, with almost equivalent numerical accuracy and computational performances.\n\nFurthermore, it contains new functionality, like the possibility to set single side parameter limits or the FUMILI algorithm (see → FUMILI minimization package), which is an optimized method for least square and log likelihood minimizations. The package has been originally developed by M. Winkler and F. James.\n\nTopical manuals\n\nFor Minuit2, topical manuals are available at Topical Manuals - Minuit2.\nThey contain in-depth information about Minuit2.\n\n### FUMILI minimization package\n\nFUMILI is used to minimize Chi-square function or to search maximum of likelihood function.\n\nFUMILI is based on ideas, proposed by I.N. Silin. It was converted from FORTRAN to C by Sergey Yaschenko [email protected].\n\nFor detailed information on the FUMILI minimization package, see → TFumili class reference.\n\n## UNU.RAN\n\nUNU.RAN (Universal Non Uniform RAndom Number generator for generating non-uniform pseudo-random numbers) contains universal (also called automatic or black-box) algorithms that can generate random numbers from large classes of continuous (in one or multi-dimensions), discrete distributions, empirical distributions (like histograms) and also from practically all standard distributions.\n\nUNU.RAN is an ANSI C library licensed under GPL.\n\nThe TUnuran class is used to interface the UNURAN package.\n\nTutorials", null, "→ Unuran tutorials\n\n### Initializing TUnuran with string API\n\nYou can initialize UNU.RAN with the string API via TUnuran::Init(), passing the distribution type and the method.\n\nExample\n\n### Using TUnuranContDist for a one-dimensional distribution\n\n• Use TUnuranContDist for creating a continuous 1-D distribution object (for example from a TF1 object providing the PDF (probability density function).\nYou can provide additional information via TUnuranContDist::SetDomain(min,max) like the domain() for generating numbers in a restricted region.\n\nExample\n\n### Using TUnuranMultiContDist for a multi-dimensional distribution\n\n• Use TUnuranMultiContDist to create a multi-dimensional distribution, which can be created from a multi-dimensional PDF (probability density function).\n\nExample\n\n### Using TUnuranDiscrDist for a discrete one-dimensional distribution\n\n• Use TUnuranDiscrDist to create a discrete one-dimensional distribution, which can be initialized from a TF1 object or from a vector of probabilities.\n\nExample\n\n### Using TUnuranEmpDist an empirical distribution\n\nUse TUnuranEmpDist for creating an empirical distribution, which can be initialized from a TH1 object (using the bins or from its buffer for un-binned data) or from a vector of data.\n\nExample" ]

[ null, "https://root.cern/assets/images/tutorials.png", null ]

[ "", null, "", null, "", null, "", null, "• Previous Article\nInfinitely many homoclinic solutions for damped vibration problems with subquadratic potentials\n• CPAA Home\n• This Issue\n• Next Article\nBoundedness of solutions for a class of impact oscillators with time-denpendent polynomial potentials\nMarch  2014, 13(2): 635-644. doi: 10.3934/cpaa.2014.13.635\n\n## A note on the existence of global solutions for reaction-diffusion equations with almost-monotonic nonlinearities\n\nReceived  January 2013 Revised  September 2013 Published  October 2013\n\nWe show existence and uniqueness of global solutions for reaction-diffusion equations with almost-monotonic nonlinear terms in $L^q(\\Omega)$ for each $1\\leq q < \\infty$. In particular, we do not assume restriction on the growth of the nonlinearites required by the standar local existence theory.\nCitation: Aníbal Rodríguez-Bernal, Alejandro Vidal-López. A note on the existence of global solutions for reaction-diffusion equations with almost-monotonic nonlinearities. Communications on Pure & Applied Analysis, 2014, 13 (2) : 635-644. doi: 10.3934/cpaa.2014.13.635\n##### References:\n J. M. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodríguez-Bernal, Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains,, Nonlinear Anal., 56 (2004), 515. doi: 10.1016/j.na.2003.09.023.", null, "Google Scholar J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula,, Proc. Amer. Math. Soc., 63 (1977), 370. doi: 10.2307/2041821.", null, "Google Scholar H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data,, J. Anal. Math., 68 (1996), 277. doi: 10.1007/BF02790212.", null, "Google Scholar A. Carvalho, J. A. Langa and J. Robinson., \"Attractors for Infinite-dimensional Non-autonomous Dynamical Systems,'' volume 182 of Applied Mathematical Sciences,, Springer, (2012). doi: 10.1007/978-1-4614-4581-4_1.", null, "Google Scholar J. W. Cholewa and A. Rodríguez-Bernal, Extremal equilibria for dissipative parabolic equations in locally uniform spaces,, Math. Models Methods Appl. Sci., 19 (2009), 1995. doi: 10.1142/S0218202509004029.", null, "Google Scholar D. Henry, \"Geometric Theory of Semilinear Parabolic Equations,'' Volume 840 of Lecture Notes in Mathematics,, Springer-Verlag, (1981). Google Scholar M. Marcus and L. Véron, Initial trace of positive solutions of some nonlinear parabolic equations,, Comm. Partial Differential Equations, 24 (1999), 1445. doi: 10.1080/03605309908821471.", null, "Google Scholar J. C. Robinson, A. Rodríguez-Bernal and A. Vidal-López, Pullback attractors and extremal complete trajectories for non-autonomous reaction-diffusion problems,, J. Differ. Equations, 238 (2007), 289. doi: 10.1016/j.jde.2007.03.028.", null, "Google Scholar A. Rodríguez-Bernal, Attractors for parabolic equations with nonlinear boundary conditions, critical exponents, and singular initial data,, J. Differ. Equations, 181 (2002), 165. doi: 10.1006/jdeq.2001.4072.", null, "Google Scholar A. Rodríguez-Bernal and A. Vidal-López, Extremal equilibria for reaction-diffusion equations in bounded domains and applications,, Journal of Differential Equations, 244 (2008), 2983. doi: 10.1016/j.jde.2008.02.046.", null, "Google Scholar\n\nshow all references\n\n##### References:\n J. M. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodríguez-Bernal, Asymptotic behavior and attractors for reaction diffusion equations in unbounded domains,, Nonlinear Anal., 56 (2004), 515. doi: 10.1016/j.na.2003.09.023.", null, "Google Scholar J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula,, Proc. Amer. Math. Soc., 63 (1977), 370. doi: 10.2307/2041821.", null, "Google Scholar H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data,, J. Anal. Math., 68 (1996), 277. doi: 10.1007/BF02790212.", null, "Google Scholar A. Carvalho, J. A. Langa and J. Robinson., \"Attractors for Infinite-dimensional Non-autonomous Dynamical Systems,'' volume 182 of Applied Mathematical Sciences,, Springer, (2012). doi: 10.1007/978-1-4614-4581-4_1.", null, "Google Scholar J. W. Cholewa and A. Rodríguez-Bernal, Extremal equilibria for dissipative parabolic equations in locally uniform spaces,, Math. Models Methods Appl. Sci., 19 (2009), 1995. doi: 10.1142/S0218202509004029.", null, "Google Scholar D. Henry, \"Geometric Theory of Semilinear Parabolic Equations,'' Volume 840 of Lecture Notes in Mathematics,, Springer-Verlag, (1981). Google Scholar M. Marcus and L. Véron, Initial trace of positive solutions of some nonlinear parabolic equations,, Comm. Partial Differential Equations, 24 (1999), 1445. doi: 10.1080/03605309908821471.", null, "Google Scholar J. C. Robinson, A. Rodríguez-Bernal and A. Vidal-López, Pullback attractors and extremal complete trajectories for non-autonomous reaction-diffusion problems,, J. Differ. Equations, 238 (2007), 289. doi: 10.1016/j.jde.2007.03.028.", null, "Google Scholar A. Rodríguez-Bernal, Attractors for parabolic equations with nonlinear boundary conditions, critical exponents, and singular initial data,, J. Differ. Equations, 181 (2002), 165. doi: 10.1006/jdeq.2001.4072.", null, "Google Scholar A. Rodríguez-Bernal and A. Vidal-López, Extremal equilibria for reaction-diffusion equations in bounded domains and applications,, Journal of Differential Equations, 244 (2008), 2983. doi: 10.1016/j.jde.2008.02.046.", null, "Google Scholar\n Honglv Ma, Jin Zhang, Chengkui Zhong. Global existence and asymptotic behavior of global smooth solutions to the Kirchhoff equations with strong nonlinear damping. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4721-4737. doi: 10.3934/dcdsb.2019027 Kazuo Yamazaki, Xueying Wang. Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1297-1316. doi: 10.3934/dcdsb.2016.21.1297 Guangyu Xu, Jun Zhou. Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1805-1820. doi: 10.3934/cpaa.2018086 Zhoude Shao. Existence and continuity of strong solutions of partly dissipative reaction diffusion systems. Conference Publications, 2011, 2011 (Special) : 1319-1328. doi: 10.3934/proc.2011.2011.1319 Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258 Jishan Fan, Shuxiang Huang, Fucai Li. Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vacuum. Kinetic & Related Models, 2017, 10 (4) : 1035-1053. doi: 10.3934/krm.2017041 Kosuke Ono. Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 651-662. doi: 10.3934/dcds.2003.9.651 P. Blue, J. Colliander. Global well-posedness in Sobolev space implies global existence for weighted $L^2$ initial data for $L^2$-critical NLS. Communications on Pure & Applied Analysis, 2006, 5 (4) : 691-708. doi: 10.3934/cpaa.2006.5.691 Bingkang Huang, Lan Zhang. A global existence of classical solutions to the two-dimensional Vlasov-Fokker-Planck and magnetohydrodynamics equations with large initial data. Kinetic & Related Models, 2019, 12 (2) : 357-396. doi: 10.3934/krm.2019016 Michele Campiti, Giovanni P. Galdi, Matthias Hieber. Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1613-1627. doi: 10.3934/cpaa.2014.13.1613 Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. Determination of initial data for a reaction-diffusion system with variable coefficients. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 771-801. doi: 10.3934/dcds.2019032 Joana Terra, Noemi Wolanski. Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 581-605. doi: 10.3934/dcds.2011.31.581 Zongming Guo, Juncheng Wei. Asymptotic behavior of touch-down solutions and global bifurcations for an elliptic problem with a singular nonlinearity. Communications on Pure & Applied Analysis, 2008, 7 (4) : 765-786. doi: 10.3934/cpaa.2008.7.765 Shota Sato, Eiji Yanagida. Asymptotic behavior of singular solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 4027-4043. doi: 10.3934/dcds.2012.32.4027 Marco Di Francesco, Alexander Lorz, Peter A. Markowich. Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1437-1453. doi: 10.3934/dcds.2010.28.1437 Congming Li, Eric S. Wright. Global existence of solutions to a reaction diffusion system based upon carbonate reaction kinetics. Communications on Pure & Applied Analysis, 2002, 1 (1) : 77-84. doi: 10.3934/cpaa.2002.1.77 Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1553-1561. doi: 10.3934/cpaa.2014.13.1553 Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1337-1345. doi: 10.3934/cpaa.2014.13.1337 Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-21. doi: 10.3934/dcdss.2020083 Young-Pil Choi, Seung-Yeal Ha, Seok-Bae Yun. Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto--Daido model with inertia. Networks & Heterogeneous Media, 2013, 8 (4) : 943-968. doi: 10.3934/nhm.2013.8.943\n\n2018 Impact Factor: 0.925" ]

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[ "# Does the mole fraction of a solution change with Vant Hoff factor?\n\nI reason that as mole fraction is a measure of colligative properties, increase in the Vant Hoff factor ( and hence increase in the value of colligative properties) should also leave a mark on the mole fraction of the solute components. However, my instructor tells differently.\n\n• Equal doesn't always means causation. Vant Hoff factor does not change for very dilute solutions. It does change when mole fraction is high enough to cause non ideal behaviour, but it can decrease, for instance when x is high enough that the solute does not fully dissociate. – Alchimista Mar 18 at 15:04\n• That is not what I asked for. My instructor told me that if a solute component dissociates in solution to yield ions, the mole fraction of the solute increases, and this is so because the no. of solute particles increases,but the total no. of moles remain constant (why?) If the no of moles of solute increases, shouldn't the total no of moles also increase? I know that will still lead to a greater mole fraction, but doesn't the total no of moles (solute + solvent) also increase? – Aabesh Ghosh Mar 18 at 17:30\n• It depends on how you define the mole fraction, in NaCl:water or sodium ion : water chloride ion . I don't see much sense on the latter here because is account for dissociation and solvent molecules are in excess. What do you call the solute? Is NaCl, and you account for dissociation. I am afraid you mean in case you add practically more solute. Then of course you change the overall number of moles, not only x. – Alchimista Mar 18 at 18:06\n• I will explain with an example. Suppose 'x' moles of glucose are dissolved in 'y' moles of a solvent (assume water, also assume x<y). The mole fraction of glucose (solute) is x/(x+y). Now if in a similar situation, 'x' moles of NaCl are dissolved in 'y' moles of water and 100% dissociation is assumed, then is the mole fraction of solute components 2x/(2x+y) or 2x/y ?(In either case, the mole fraction increases) – Aabesh Ghosh Mar 20 at 3:31\n• In your example, for each ion it will be x/(2x+y). So the latter. Consider that in dilute solutions x + y is about y, too. – Alchimista Mar 20 at 7:29" ]

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[ "# Intuition behind the t-distributions density function\n\nI'm studying about Student's t-distribution and I started to wonder, how would one derive the t-distributions density function (from wikipedia, http://en.wikipedia.org/wiki/Student%27s_t-distribution):\n\n$$f(t) = \\frac{\\Gamma(\\frac{v+1}{2})}{\\sqrt{v\\pi}\\:\\Gamma(\\frac{v}{2})}\\left(1+\\frac{t^2}{v} \\right)^{-\\frac{v+1}{2}}$$\n\nwhere $v$ is the degrees of freedom and $\\Gamma$ is the gamma function. What is the intuition of this function? I mean, If I look at the binomial distribution's probability mass function, it makes sense to me. But t-distributions density function makes no sense at all to me...it is not intuitive at all at first sight. Or is the intuition just that it has a bell-shaped curve and it serves our needs?\n\nThnx for any help :)\n\n• This distribution has a simple (and pretty) geometric interpretation. Indeed, although Student (1908) first derived this form of the PDF through an intelligent guess (supported by Monte-Carlo simulation), Fisher (c. 1920) first obtained it with a geometric argument. The essence is that $f$ describes the distribution of the ratio of the height of a (uniformly distributed point) on the $\\nu+1$-sphere and its radius (distance from the axis): in other words, the tangent of its latitude. One account of this is provided at evolvedmicrobe.com/Literature/GeometricTDistribution.pdf.\n– whuber\nDec 3 '13 at 14:28\n\nIf you have a standard normal random variable, $Z$, and an independent chi-square random variable $Q$ with $\\nu$ df, then\n\n$T = Z/\\sqrt{Q/\\nu}$\n\nhas a $t$ distribution with $\\nu$ df. (I'm not sure what $Z/Q$ is distributed as, but it isn't $t$.)\n\nThe actual derivation is a fairly standard result. Alecos does it a couple of ways here.\n\nAs far as intuition goes, I don't have particular intuition for the specific functional form, but some general sense of the shape can be obtained by considering that the (scaled by $\\sqrt \\nu$) independent chi-distribution on the denominator is right skew:", null, "The mode is slightly below 1 (but gets closer to 1 as the df increases), with some chance of values substantially above and below 1. The variation in $\\sqrt{Q/\\nu}$ means that the variance of $t$ will be larger than that of $Z$. The values of $\\sqrt{Q/\\nu}$ substantially above 1 will lead to a $t$-value that's closer to 0 than $Z$ is, while the ones substantially below 1 will result in a $t$-value that's further from 0 than $Z$ is.\n\nAll this means that $t$ values will be (i) more variable, (ii) relatively more peaked and (iii) heavier tailed than a normal. As the df increases, $\\sqrt{Q/\\nu}$ becomes concentrated around 1, and then $t$ will be closer to the normal.", null, "(the 'relatively more peaked' results in a slightly sharper peak relative to the spread, but the larger variance pulls the center down, which means that the peak is slightly lower with lower d.f.)\n\nSo that's some intuition about why the $t$ looks as it does.\n\n• I was bit sloppy in my explanation. Of course it was square root of Chi-square distributed random variable divided by its degrees of freedom. Dec 3 '13 at 8:25\n• @Analyst I've done the same myself, more than once. Dec 3 '13 at 8:31\n\nThe answer by Glen is correct one, but from a Bayesian viewpoint it is also helpful to think of the t-distribution as a continuous mixture of normal distributions with different variances. You can find the derivation here:\n\nStudent t as mixture of gaussian\n\nI feel that this approach helps your intuition because it clarifies how the t-distribution arises when you don't know the exact variability of your population." ]

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[ "# How does 'void*' differ in C and C++?\n\nBefore learning the differentiating factor of void functions in C and C++ programming language, let us see a few examples and deeply understand the void function where it is used, use cases we can draw etc.\n\nVoid funThe void, as the name suggests, there is nothing it owes to any entity in programming. The void function, when called, will not return anything to the primary position; the control jumps to where the process is called after finishing the operation allowed in the process, which can either be calculation, recursion or printing anything to the display screens and then we will look at the play of pointers and examine How does 'void*' differ in C and C++?\n\n### C++ void function-2\n\nOutput:\n\n```//output_is_for_both_C++_codes_written_above\nAs we expect it to perform, the void function has returned the void to the central control.\n```\n\n### C void Function\n\nOutput:\n\n```C void function print statement 1C void function print statement 1 perfection in programming in C.\"\n```\n\n## How does 'void*' differ in C and C++?\n\nA c programming language allows a void* pointer to be assigned to any\n\npointer type without a cast, whereas C++ does not. We have to typecast the void* pointer in C++ explicitly.\n\nLet us look at the example below; the following is valid in C but not C++:\n\nSimilarly,\n\nTo make the above code compile in C++ as well, we have to use explicit casting, as shown below,\n\n### Feedback", null, "", null, "", null, "" ]

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[ "# Hall–Higman theorem explained\n\nIn mathematical group theory, the Hall - Higman theorem, due to, describes the possibilities for the minimal polynomial of an element of prime power order for a representation of a p-solvable group.\n\n## Statement\n\nSuppose that G is a p-solvable group with no normal p-subgroups, acting faithfully on a vector space over a field of characteristic p. If x is an element of order pn of G then the minimal polynomial is of the form (X - 1)r for some r ≤ pn. The Hall - Higman theorem states that one of the following 3 possibilities holds:\n\n• r = pn\n• p is a Fermat prime and the Sylow 2-subgroups of G are non-abelian and r ≥ pn -pn-1\n• p = 2 and the Sylow q-subgroups of G are non-abelian for some Mersenne prime q = 2m - 1 less than 2n and r ≥ 2n - 2n-m.\n\n## Examples\n\nThe group SL2(F3) is 3-solvable (in fact solvable) and has an obvious 2-dimensional representation over a field of characteristic p=3, in which the elements of order 3 have minimal polynomial (X-1)2 with r=3-1." ]

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[ "# Excel scatter chart with multiple date ranges\n\nI have multiple blocks of time series data on an Excel sheet, with each block having its own set of dates. For example, I might have dates in column A, values in column B, and then dates in column D and values in column E. The values in B go with the dates in A, and the values in E go with the dates in D. The dates in A and D may not be the same. I would like to create a scatter chart with a time category axis that is the union of my two input date ranges in columns A and D.\n\nIf I select all the data and then go insert chart (in Excel 2010), Excel treats only column A as the X axis, and looks at D as just another set of values.\n\nI can get Excel to do what I want by first just charting columns A and B, then selecting D and E and copy-pasting onto the chart. However, I would like to avoid this two-step procedure if possible.\n\n## 1 Answer\n\nIf you select a data range in Excel and generate a chart, Excel only uses the data on the far left for its X values. If you have multiple series with distinct X values, you have to add each set of X-Y values in a separate operation." ]

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[ "Additional functionality related to this tutorial has been introduced in subsequent versions of the Wolfram Language. For the latest information, see Matrices and Linear Algebra.\n\n# Matrix Types\n\nMatrices in Mathematica can be constructed from all the different types of objects that Mathematica holds. They can contain machine-precision real and complex floating-point numbers, arbitrary-precision real and complex floating-point numbers, integers, rational numbers, and general symbolic quantities. This tutorial considers the different types of matrices that Mathematica supports.\n\n## Numbers in Mathematica\n\nIn order to understand the different types of matrices that Mathematica can work with, it will help to get an overview of the different types of numbers that are supported. This section only gives an overview of numbers in Mathematica; further information is found in \"Numbers\".\n\n machine-sizeInteger integers represented by machine hardware Integer arbitrary‐length exact integer Rational integer/integer (fractions) in lowest terms machine-sizeReal approximate real number represented by machine hardware Real approximate real number, with any specified precision Complex complex number of the form number+numberI AlgebraicNumber root of an algebraic equation exact numerical expression composition of numbers and mathematical functions infinities and exceptions various special numbers\n\nIntrinsic types of numbers in Mathematica.\n\nMathematica provides an arithmetic system that unifies all these different types of numbers. For many purposes it is not necessary to know that there are two different types of integer because both types flow together. For example, the integer", null, "is represented by machine hardware. If this is raised to the tenth power, the result is represented by software:\nAnother example shows how a computation of the exponential of a machine-precision floating-point number returns an arbitrary-precision number:\n\nThese examples demonstrate the general point that Mathematica arithmetic traps overflows and underflows so that it can switch from machine to software numbers. Despite this it tries to use machine hardware functionality whenever possible, so that Mathematica computations can be fast.\n\n### Exact versus Approximate Numbers\n\nOne important issue for Mathematica is that it makes a distinction between exact and approximate numbers. This is done to support the symbolic side of Mathematica. This example compares the computation of the Sin function for an integer and a real argument:\n\nThis demonstrates that mathematical functions, such as Sin, maintain the nature of their input. If they get an exact argument, the result will be exact. If they get an approximate argument the result is approximate. This feature of Mathematica is essential to maintain the system as a symbolic system, but it stands in contrast to the operation of a purely numerical environment. For example, in many languages such as C or Java the equivalent computation would coerce the integer to a double and the result would be an approximate number.\n\nThis distinction means that when you work with numbers you need to be more aware of their types than is the case in many other languages.\n\nYou can use N to compute a numerical approximation for an exact number:\n\n### Mixed Mode Arithmetic\n\nAnother important issue concerns the operation of functions that take more than one argument. In this case Mathematica takes the lowest precision of the input. For example, the following mixed computations all return approximate results:\nThe same principle applies if the input includes an exact integer represented by software:\n\n## Matrices in Mathematica\n\nMatrices in Mathematica can contain many different types of elements.\n\nFor example, here is a matrix where each element is a machine-precision approximate real number:\nThis matrix contains complex numbers with real and imaginary parts that are machine-precision approximate real numbers:\nThis matrix contains approximate numbers with 20 digits of precision:\nThis matrix contains integers:\nThis matrix contains rational numbers:\nThis matrix contains complex numbers with real and imaginary parts that are integers:\nThis matrix contains polynomials:\n\nThese different types of matrices can be put into categories according to the types of technique used for computing with them.\n\n machine-precisionReal standard numerical techniques machine-precisionComplex arbitrary-precisionReal arbitrary‐precision numerical techniques arbitrary-precisionComplex Integer symbolic algebra techniques Rational exactComplex symbolic\n\nDifferent types of matrices in Mathematica.\n\nThese three different categories are briefly reviewed.\n\n#### Standard Numerical Techniques\n\nIn Mathematica, matrix computations involving machine-precision Real and machine-precision approximate Complex numbers are carried out with standard numerical techniques. In many cases computations involve optimized libraries, many of which are described in \"Software References\".\n\nAn important goal for many of these computations is to match and surpass the performance of any software package that is dedicated to machine-precision numbers. This is in keeping with the design goals of Mathematica, as described under \"Design Principles of Mathematica\". In the case of linear algebra computations, Mathematica makes use of a considerable amount of sophisticated technology, some of which is described under \"Performance of Linear Algebra Computation\".\n\n#### Arbitrary-Precision Numerical Techniques\n\nIn Mathematica, matrix computations involving arbitrary-precision Real and arbitrary-precision approximate Complex numbers are carried out with special numerical libraries. These libraries are adapted from standard libraries so they can work for arbitrary-precision computations.\n\n#### Symbolic Algebra Techniques\n\nMatrix computations involving exact numbers and general symbolic techniques are carried out with computer algebra techniques.\n\nAll computations provided for numerical matrices are also available for symbolic matrices. This meets the design goal of integrating symbolic and numerical computation.\n\n### Mixed Mode Matrices\n\nThis section describes what happens if the input matrix contains a mixture of different types of entries.\n\nIf the matrix contains anything that is not a number (or something that cannot be converted into a number), the matrix will be treated as a symbolic matrix and the computation will proceed with symbolic techniques. In the following example, the presence of the x makes the matrix symbolic and the result is a symbolic matrix:\nIf the matrix only contains numbers (or things that can be converted into numbers), the computation will be done according to the lowest precision. Thus in the following example, the presence of the machine-precision approximate number, 1., means that the input matrix will be converted to machine-precision numbers and the computation is done with machine-precision techniques:\nIn this example, the lowest precision is a number with 20 digits of precision. Thus, the input matrix will be converted to precision 20 numbers and the computation done with arbitrary-precision techniques:\nThe resulting matrix has entries with 20 digits of precision:\nNote that machine-precision counts as lower precision than any arbitrary-precision numbers. In this example, because there is a machine-precision number, the input matrix will be converted to machine-precision numbers and the computation done with machine-precision techniques:\nThe result is a machine-precision matrix:\nIt is important to be aware of these distinctions. For example, if the input matrix consists of integers, the computation will use symbolic algebra techniques and return an exact precision result:\n\nIn some other computation systems an input matrix of integers would be converted to a floating-point matrix. Mathematica does not do this, so that it can support symbolic computation. However, it means that users need to be aware of the differences. This is because the time required for a symbolic technique can be quite different from a machine-precision technique; in many cases the symbolic computation will take much longer.\n\n## Complex Matrices\n\nComplex matrices in Mathematica are defined in a way that is completely consistent with other types of matrices such as those that involve real numbers. All the functions that work for real matrices work for complex matrices.\n\nOne special detail about complex matrices is that certain linear algebra computations that involve a transpose for real matrices involve a conjugate transpose for complex matrices. An example of this is the Cholesky decomposition. For a real symmetric positive definite matrix it is defined as shown:\nFor a complex Hermitian positive definite matrix the definition is shown as follows:\n\n## Arbitrary-Precision Matrices\n\nMathematica arbitrary-precision computations support a form of arithmetic known as significance arithmetic. The fundamental idea of significance arithmetic is that a number is seen as an approximation with an error specified by its precision. When a number is used in a computation the precision of the result may be different according to the properties of the function. For example, here is a number with 30 digits of precision:\nNow compute the value of the Sin function for this number:\nThe result is seen to be less than 30 digits:\nIn Mathematica, linear algebra computations generally do not use significance arithmetic. They are carried out in what is known as fixed precision, where computations are done with fixed numbers of digits with no attempt to track the error in individual operations. Error tracking is done on an algorithmic level, issuing warnings, for example, when the solution of a linear system may not be correct:\nThe inverse matrix that is computed consists entirely of numbers with precision 20:" ]

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[ "## Conversion formula\n\nThe conversion factor from feet per second to miles per hour is 0.68181818181818, which means that 1 foot per second is equal to 0.68181818181818 miles per hour:\n\n1 ft/s = 0.68181818181818 mph\n\nTo convert 35 feet per second into miles per hour we have to multiply 35 by the conversion factor in order to get the velocity amount from feet per second to miles per hour. We can also form a simple proportion to calculate the result:\n\n1 ft/s → 0.68181818181818 mph\n\n35 ft/s → V(mph)\n\nSolve the above proportion to obtain the velocity V in miles per hour:\n\nV(mph) = 35 ft/s × 0.68181818181818 mph\n\nV(mph) = 23.863636363636 mph\n\nThe final result is:\n\n35 ft/s → 23.863636363636 mph\n\nWe conclude that 35 feet per second is equivalent to 23.863636363636 miles per hour:\n\n35 feet per second = 23.863636363636 miles per hour\n\n## Alternative conversion\n\nWe can also convert by utilizing the inverse value of the conversion factor. In this case 1 mile per hour is equal to 0.041904761904762 × 35 feet per second.\n\nAnother way is saying that 35 feet per second is equal to 1 ÷ 0.041904761904762 miles per hour.\n\n## Approximate result\n\nFor practical purposes we can round our final result to an approximate numerical value. We can say that thirty-five feet per second is approximately twenty-three point eight six four miles per hour:\n\n35 ft/s ≅ 23.864 mph\n\nAn alternative is also that one mile per hour is approximately zero point zero four two times thirty-five feet per second.\n\n## Conversion table\n\n### feet per second to miles per hour chart\n\nFor quick reference purposes, below is the conversion table you can use to convert from feet per second to miles per hour\n\nfeet per second (ft/s) miles per hour (mph)\n36 feet per second 24.545 miles per hour\n37 feet per second 25.227 miles per hour\n38 feet per second 25.909 miles per hour\n39 feet per second 26.591 miles per hour\n40 feet per second 27.273 miles per hour\n41 feet per second 27.955 miles per hour\n42 feet per second 28.636 miles per hour\n43 feet per second 29.318 miles per hour\n44 feet per second 30 miles per hour\n45 feet per second 30.682 miles per hour" ]

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[ "# Python Convert bytearray to String\n\n« Python Tutorial Python Examples »\n\nThis article is created to cover a program in Python that converts bytearray object to a string object. The program is created in similar was as done in bytes to string.\n\nThe question is, write a Python program to convert bytearray to string. The program given below is its answer:\n\n```data = b'Python Programming'\nprint(\"bytearray =\", data)\ndata = data.decode()\nprint(\"string =\", data)```\n\nThe snapshot given below shows the sample output of above Python program, demonstrating the bytearray to string:\n\n## bytearray to String in Python\n\nThis program allows user to enter the string. The string gets converted into bytearray with utf-8 encoding, then the bytearray gets converted back to the string using the decode() method, with utf-8 decoding.\n\n```print(\"Enter the String: \", end=\"\")\nx = input()\n\nprint(\"\\nThe string:\")\nprint(x)\n\nx = bytearray(x, \"utf-8\")\nprint(\"\\nThe bytearray:\")\nprint(x)\n\nx = x.decode()\nprint(\"\\nAgain the String:\")\nprint(x)```\n\nHere is its sample run with user input codescracker dot com", null, "If the bytearray object's data is encoded using other encoding like utf-16, then we need to use the same decoding while converting the bytearray object to string object using the decode() method.\n\nPython Online Test\n\n« Python Tutorial Python Examples »" ]

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[ "# Recognizing special cases of a defined function for permuted arguments\n\nI have a number of ugly ways to implement the following pattern recognition task, but I'm looking for something elegant to keep my notebook tidy.\n\nI need to define a function of six variables $f(a,b,c;A,B,C)$ which is known to be unchanged under the simultaneous interchange of any two of $a,b,c$ and of the corresponding $A,B,C$:\n\n\\begin{align}&f(a,b,c;A,B,C) = f(a,c,b;A,C,B) = f(b,a,c;B,A,C) \\\\ =&f(b,c,a;B,C,A) = f(c,a,b;C,A,B) = f(c,b,a;C,B,A)\\end{align}\n\nEDIT for clarity: Given one definition, how do I get Mathematica to try all possibilities within the restricted set of permutations shown above for pattern matching? I need to mimic the effect of SetAttributes[f,Orderless].\n\nMore concretely, if I define the special case:\n\nf[a_, 0, b_, A_, 0, C_] := (a+b)/(A-C)\n\n\na function call f[x, y, 0, m, n, 0] should match, and return (x+y)/(m-n). But f[x, y, 0, m, 0, n] should not match, and thus return it must be returned unevaluated.\n\nAdded question: Instead of finding a way to get Mathematica to try all the possibilities when pattern matching, would it be easier to write code such that when a representative definition for f is made, the kernel automatically adds further definitions of f for the remaining permutations of the arguments?\n\nI think Bob Hanlon had the right idea in using Orderless but his suggestion is overly naive.\n\nEdit: my answer was also wrong, but I am updating it with the correction from your comment below.\n\nSetAttributes[f1, Orderless]\n\nf[a_, b_, c_, A_, B_, C_] :=\nWith[{body = f1[{a, A}, {b, B}, {c, C}]}, body /; Head[body] =!= f1]\n\nf1[{a_, A_}, {0, 0}, {c_, C_}] := (a + c)/(A - C)\n\n\nNow:\n\nf[x, y, 0, m, n, 0]\n\nf[x, y, 0, m, 0, n]\n\n(x + y)/(m - n)\n\nf[x, y, 0, m, 0, n]\n\n• This is better, but the invariance of the function $f$ isn't for permutations of first three arguments $a,b,c$ independent of the second three $x,y,z$. They are tied together. But no problem; your answer and @BobHanlon's answer pointed me in the right direction: I could declare SetAttributes[f1, Orderless] but with the arguments arranged f1[{a_,x_},{b_,y_},{c_,z_}], and make the library of special cases for f1. Then finally, I put f[a_,b_,c_,d_,e_,f_] = f1[{a,x},{b,y},{c,z}]. I tried a few cases and it seems to work. What do you think? Jul 24, 2014 at 8:31\n• Sorry, you're right, I wasn't thinking clearly. Hopefully thinking better now, and your solution sounds just fine. Why don't you self-answer so I can vote for it? Jul 24, 2014 at 9:25\n• @QuantumDot This question just got bumped by Community♦ because it has no positively voted answer. Please post the method in your comment above as a self-answer. If you don't I shall edit my answer to include it. Aug 23, 2014 at 8:19\n• Thanks for the notification; you can go ahead and include the solution your answer. Then I'll mark it as accepted. Aug 23, 2014 at 14:39\n• Might this be a good way to enable definitions through f instead of f1?: f /: SetDelayed[f[a_, b_, c_, d_, e_, f_], expr_] := (f1[{a, d}, {b, e}, {c, f}] := expr). Ditto for set. Sometimes I overlook pitfalls with such workarounds. Sep 1, 2014 at 21:22\n\nI think you need to use a group-theoretical construction. In this way you will have full freedom in specifying any group of permutations you need. In your case the group is\n\nG = PermutationGroup[{Cycles[{{1, 2}, {4, 5}}], Cycles[{{1, 2, 3}, {4, 5, 6}}]}];\n\n\nThis generates a symmetric group on {1, 2, 3}, which also forces the same permutations on {4, 5, 6}. These are the group elements, as permutation lists:\n\nPermutationList[#, 6] & /@ GroupElements[G]\n\n{{1, 2, 3, 4, 5, 6}, {1, 3, 2, 4, 6, 5}, {2, 1, 3, 5, 4, 6}, {2, 3, 1, 5, 6, 4}, {3, 1, 2, 6, 4, 5}, {3, 2, 1, 6, 5, 4}}\n\n\nThere are six permutations, corresponding to those of the symmetric group on {1, 2, 3}.\n\nNow construct the following function:\n\nSetAttributes[SetDelayedPermuted, {HoldAll, SequenceHold}];\n\nSetDelayedPermuted[f_[args___], rhs_, group_] := ((f[##] := rhs) & @@@ Permute[{args}, group];)\n\n\nLet us try your definition, though I'm changing the - sign in the denominator to a + sign, because otherwise I think there is some inconsistency:\n\nSetDelayedPermuted[f[a_, 0, b_, A_, 0, C_], (a + b)/(A + C), G]\n\n\nNow we check your two cases (remember I changed a sign):\n\nf[x, y, 0, m, n, 0]\n\n(x + y) / (m + n)\n\nf[x, y, 0, m, 0, n]\n\nf[x, y, 0, m, 0, n]\n\n\nWe can see how many definitions were actually needed:\n\n??f\n\nGlobalf\n\nf[b_,0,a_,C_,0,A_]:=(a+b)/(A+C)\nf[b_,a_,0,C_,A_,0]:=(a+b)/(A+C)\nf[0,b_,a_,0,C_,A_]:=(a+b)/(A+C)\n\n\nNote how symmetry in both a<->b and A<->C was implemented simultaneously (hence we ended up with 3 definitions, instead of 6). That's why I think a consistent definition was having (a+b)/(A+C) or (a-b)/(A-C) but not (a+b)/(A-C) on the right hand side.\n\n• Nice answer! and kudos to you for making the RHS consistent. I like especially how it generates the necessary definitions. Sep 3, 2014 at 18:57\n• A really nice solution! Congratulations, bounty hunter. :-) Sep 5, 2014 at 10:47\n• I edited your code to make it a bit shorter. I also changed the formatting with hope of making it easier to copy and paste the code sections. I hope you don't mind. Sep 5, 2014 at 10:51\n• Thank you QuantumDot! And thank you Mr.Wizard for improving the answer.\n– jose\nSep 5, 2014 at 14:43\n• Thanks for this! I actually have an application where a function I'm interested in is only partly symmetric. I think I'll use this instead of what I have now. Aug 2, 2016 at 6:27\n\nI would do this by defining a preferred ordering to sort the first three arguments. In this example I will use canonical Mathematica Ordering but in principle you could use anything.\n\nI would define\n\nf[a_, b_, c_, A_, B_, C_] := With[\n{order = Ordering[{a, b, c}]}, (* Get ordering of first three arguments *)\nf @@ {\nSequence @@ ({a, b, c}[[order]]), (* Sort first three arguments *)\nSequence @@ ({A, B, C}[[order]]) (* Reorder second three similarly *)\n}\n] /; Not[OrderedQ[{a, b, c}]] (* Avoid infinite loop. *)\n\n\nThen\n\nf[c, b, a, C, B, A]\n(* = f[a, b, c, A, B, C] *)\nf[3, 0, -3, foo, bar, baz]\n(* = f[-3, 0, 3, baz, bar, foo] *)\n\n\netc.\n\n• How I'd have done it: f[a_, b_, c_, A_, B_, C_] /; ! OrderedQ[{a, b, c}] := f @@ Flatten[{{a, b, c}, {A, B, C}}[[All, Ordering[{a, b, c}]]]] Aug 2, 2016 at 6:37\n\nGive f the attribute Orderless\n\nSetAttributes[f, Orderless];\n\nf[0, 0, 0, 0, z_, z_] = -(1/z);\nf[0, 0, 0, 0, y_, z_] = -(1/(y - z)) Log[y/z];\n\nf @@@ Permutations[{0, 0, 0, 0, z, z}] // Union\n\n\n{-(1/z)}\n\nf @@@ Permutations[{0, 0, 0, 0, y, z}] // Union\n`\n\n{-(Log[y/z]/(y - z))}\n\n• This won't work because $f$ isn't completely orderless among all six variables. Jul 24, 2014 at 7:04" ]

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[ "# Difference between revisions of \"PythonBasics\"\n\nBack to the PythonOverview.\n\n## Preface\n\nThis guide is intended as a practical introduction to python for astronomers using CASA. Along with the accompanying pages, we hope that this introduction gets you comfortable enough to take advantage of CASA's python shell to script your data reduction or integrate your analysis, observation planning, and reduction. Despite the CASA focus, we do aim to make this guide useful to any astronomer looking for a first dive into python. Given that similar material exists across the web we will not attempt a comprehensive introduction, just a tour of most of the basic astronomy-relevant python functionality.\n\nAfter completing this guide, you can find more detailed discussion of other topics and some links to external material here: PythonOverview\n\n### Environment\n\nWe could write a lot on setting up python, a nice shell, installing key packages, etc. Because this is a CASA guide, we can short circuit some of this. CASA installs with it's own nice iPython shell and a core set of third-party packages (numpy, scipy, pylab, matplotlib). We will assume that you have downloaded and installed CASA from here and return to the issue of additional third-party software down the road.\n\nOnce you have CASA installed you can start a new session from your shell prompt by just typing\n\ncasapy\n\n\nDepending on your computing environment you likely also have python and perhaps iPython installed, but the versions of these (especially the pre-installed version on the Mac) can vary widely. If you are interested in a non-CASA distribution that folds in a similar (in fact more extensive) suite of packages you might look at the free academic version here (no promises). We focus our discussion on the version of python that comes with CASA (2.6) and the examples will assume that you are working inside of a casapy shell.\n\n### Pasting Code\n\nWe will fairly rapidly reach the point where we want to cut and paste somewhat complex blocks of code. Python's use of indentation to implement code structure can make for awkward interactions with the shell at times. Even when it goes well after cutting and pasting a 'for' loop you often have to press return twice to execute. In very complex cases, you may end up needing to paste a line at a time.\n\nFortunately, CASA's iPython shell offers a very nice way around these challenges. You can type\n\ncpaste\n\n\nto initiate a 'code paste'. You will then be able to paste code directly into the shell and the code will appear exactly as you copied it. When you have pasted all of your code got to a new line and type \"--\". This ends the code paste and the pasted code should then execute. Use this (really!), it will save you a large amount of pasting-frustration.\n\n### Getting Help, Exploring Objects, the Shell\n\nCASA's shell (and iPython shells in general) has a lot of useful functionality. For example, CASA knows some basic operating system commands like \"ls\", \"cat\", etc. CASA only knows a few of these but you can issue any commands to your Unix/Linux shell directly by prefacing those commands with an exclamation point. So this works:\n\nls\n\n\nand so does this:\n\n!ls -lh\n\n\nthis does not (because CASA does not know \"df\"):\n\ndf -h\n\n\nbut this does:\n\n!df -h\n\n\nThe shell has many other nice features. For example, if you type the name of a variable, it will simply print the variable value.\n\nYou can readily get help on most objects via \"help thisorthat,\" for example\n\nhelp list\n\n\nAlong with \"help,\" the other major exploration capability in the shell is tab-completion. This will let you explore what variables are defined in your shell. Even better, you can take any object, append a \".\", hit tab, and the shell will show you list of associated methods.\n\nTry this now by typing \"list.\" and hitting tab. The shell will print all the things that a list can do. In addition to a bunch of system-defined names that begin and end with \"__\" (e.g., see the equality operator \"__eq__\") you can see that list has a set of methods \"append\", \"count\", \"extend\", \"index\", ... \"sort\".\n\nIf you saw that list.append exists and wanted to learn a bit about it you could type\n\nhelp list.append\n\n\nWe're getting a bit ahead of ourselves but try to bear these exploratory capabilities as you look through the guide. They are a great way to poke around and find new functionality.\n\nOne very important thing to bear in mind as you begin more advanced programming is that the shell specific capabilities are not available outside the shell. In practice this means that you cannot, for example, print the name of a variable by just listing the variable inside a program.\n\n## Simple Variables\n\nLet's begin by creating an manipulating some basic variables. At the python shell prompt create your first variable by typing\n\nmy_first_var = 1\n\n\nThat's it, you've created your first variable. Note that you didn't have to specify the data type, python works this out from context. To see the value of your variable use the print function.\n\nprint (my_first_var)\n\n\nor (if and only if you are at the shell) you can see the value by just typing the variable name\n\nmy_first_var\n\n\nYou can see the type of the the variable by typing\n\nprint( type(my_first_var) )\n\n\nNow try the same thing with some other data types.\n\n# Make and print a string\na_string = \"A String\"\nprint( a_string, type(a_string) ) # a string\n\n\nNote that this first line does nothing. Python's comment string is \"#\" and it ignores everything in a line that comes after this character.\n\nSome similar examples for floating point numbers:\n\ntype(3.14159) # a float\na_float = 1e-10\nprint( type(a_float) ) # scientific notation\n\n\nNotice the in-line comment and the on-the-fly creation of a variable in the first line.\n\nFinally, python has a 'boolean' type that is either 'True' or 'False' (shorthand 'T' or 'F' in CASA, though not iPython in general).\n\ntype(True) # a boolean\nboo = False\nprint( boo, type(boo) )\n\n\nYou can chain variable creation:\n\nx = y = 1\nprint( x, y )\nx = y = v = u = t = 0\nprint( u, t )\n\n\n### Basic Math\n\nWe have seen how to create variables. Now we will manipulate them with basic math operations.\n\nBegin by creating a new variable with value 1.\n\nx=1\n\n\nand print the value after we add 2.\n\nprint ( x+2 )\n\n\nWe can assign the output of the operation to a new variable, which now exists in addition to x\n\ny = x + 2\nprint (y)\n\nz = x - 3\nprint (z)\n\n\nImplicit operations work, e.g., \"x += 3\" will add three to the value of x\n\nx = 1\nprint (x)\nx += 3\nprint (x)\nx *= 10\nprint (x)\nx /= 20\nprint (x)\n\n\nExponents are done with \"**\", so for example\n\nx = 2\ny = x**2\nprint (x, y)\n\n\nor to take the square root\n\nx = 4\nprint (x**0.5)\n\n\n### Ints and Floats\n\nWe saw that python assigns a new variable the appropriate type based on context. Overall this is a nice feature, it lets us quickly develop scripts and programs without a lot of traditional programming overhead. However, there are a few rules that we need to keep in mind.\n\nThe first point to see is that dividing two integers returns an integer that is the \"floor\" (value rounded to the nearest integer closer to zero) of the result. For example, declare a new integer x and divide it by 20:\n\nx = 1\nprint( x / 20 )\n\n\nor try these\n\nprint( 19 / 20 )\nprint( -1 / 20 )\nprint( -19 / 20 )\n\n\nThat's the division of two integers. If we divide a float and an integer, on the other hand, we get a float:\n\n(In Python 3 this changes, and the new default is to return a float. If you really want to return a truncated integer, use the int function explicitly, so your code will be forward compatible.)\n\nx = 1\nprint ( type(x) )\nprint ( x / 1.5 )\nprint ( type(x/1.5) )\n\n\nthe operation implicitly casts the result as a float.\n\nExplicit casts are okay and a reasonable way around this. Again, int() floors. For example:\n\nx = 0.6\nint_version = int(x)\nint_version # remember this prints in the shell\n\n\nSimilarly, float() casts as a float:\n\nfloat_version = float(int_version)\nfloat_version\n\n\nThe order does matter. Notice the difference between these two cases:\n\nprint ( float(19)/20 )\nprint ( float(19/20) )\n\n\nThe takeaway for the average CASA user here is: be careful tossing integers around in math because things may unexpectedly round on you. When in doubt, you should probably make things floats.\n\n### Booleans\n\nPython includes booleans as a basic data type. These are variables that have two possible values: True or False (for CASA the shorthands \"T\" and \"F\" work as well). You can declare booleans efficiently:\n\nb = True # note this is case sensitive TRUE or true does not work.\nprint(b)\nb = False\nprint(b)\n\n\nBooleans are particularly interesting as the output of many operators. For example:\n\nb = 1 == 1\nprint(b)\n\n\nor\n\nc = 1 != 1\nprint(c)\n\n\n### Deleting Variables\n\nPython variables at the shell level will hang around forever until you declare a variable with the same name or actively get rid of them (by contrast variables declared inside of a procedure will go away when the procedure ends). You can get rid of a variable (module, object, whatever) like so:\n\nx = 1\ndel x\nprint (x)\n\n\nand just like that x is gone. Be careful with this.\n\n### Checking Whether a Variable Exists\n\nWe just gave a practical example of using \"print\" to check variable existence. But this sort of catastrophic approach isn't ideal. A more robust way to check if a variable exists is:\n\ntry:\nexcept NameError:\nprint( \"Does not exist!\" )\n\n\nThe try statement accepts an \"else\":\n\nthe_answer = 42.\ntry:\nexcept NameError:\nprint( \"Does not exist!\" )\nelse:\nprint( \"We're cool.\" )\n\n\nNow try the same thing after deleting your variable:\n\ndel the_answer\ntry:\nexcept NameError:\nprint( \"Does not exist!\" )\nelse:\nprint( \"We're cool.\" )\n\n\nThis kind of approach gives you the ability to write non-explosive scripts. There are other ways to approach this. If you are interested you may want to read up further on the locals and globals python dictionaries.\n\n## Data Collections: Lists\n\nPython has several built in types of data collections. Here we'll look at \"lists\", which are loose collections of potentially mixed types of data. You can declare a list by placing a comma-delimited list of variables inside a set of square brackets. For example, let's create a list called \"bands\":\n\nbands = ['L', 'S', 'C', 'X', 12]\n\n\nOne potentially surprising feature of python lists is that they freely mix and match data types. Notice that \"bands\" mixes strings and integers. You can print the type and number of elements of the list via\n\nprint( type(bands) )\nprint( len(bands) )\n\n\n### Simple Slicing\n\nYou \"slice\" a list, accessing parts of the data using square brackets:\n\nbands = ['L', 'S', 'C', 'X', 12]\nprint( bands )\nprint( bands )\n\n\nNotice that python is \"zero-indexed\" which means that for a 5-element list like \"bands\" the indices run from 0 (first) to 4 (last). As a result this will not work\n\nprint( bands )\n\n\nbecause there is no element 5.\n\nAlso notice that python allows you to use negative indices to index \"backwards\" from the last element of the array\n\nprint( bands[-1] )\nprint( bands[-2] )\n\n\nYou can pick out a range of indices using a \":\" and the following syntax\n\nprint( bands[1:3] )\n\n\nNotice that it prints elements 1 and 2 but not element 3 of the list. So \"n:m\" retrieves elements n through m-1.\n\nSimilarly \"0:3\" will retrieve elements 0, 1, and 2. For example\n\nprint( bands[1:3] )\n\n\nA free \":\" retrieves everything along the axis:\n\nprint( bands[:] )\n\n\nwhile \":n\" implicitly takes \"0\" as the initial index and \"n:\" runs n through the last index. For example:\n\nprint( bands[2:] )\n\n\n### Assignment\n\nThese same tricks can be used for assignment. We'll recreate bands and then replace the last element:\n\nbands = ['L', 'S', 'C', 'X', 12]\nbands\nbands = \"Ku\"\nprint( bands )\n\n\nor use the more complex slicing to replace the first three elements:\n\nbands[0:3] = [1.4,3.0,6.0]\n\n\nNotice that the sizes of the left and right side of the assignment need to match. So this will not work\n\nbands[0:3] = 3.0\n\n\nIn addition to replacing elements you can add to the list like so:\n\nbands.append(\"Ka\")\n\n\n### Exploring List Functionality (Back to the Shell)\n\nLists have several neat abilities in addition to \"append.\" If you don't remember them offhand you can easily find them using the shell functionality discussed earlier. This is great opportunity to apply these techniques in practice. Do so now, making sure that \"bands\" is defined as above.\n\nType in \"bands.\" and hit <tab>. the shell will offer you the the available auto-completions. Leave aside all those system defined __ items and look at the rest. We see a list from \"append,\" \"count,\" through \"sort.\" These are methods of list and so will be called as \"append\" above.\n\nLet's say that you want to find out more about the \"count\" function. Use the \"help\" function like so:\n\nhelp(bands.count)\n\n\nso bands.count() checks the number of occurrences of a value. Try it out:\n\nbands = ['L', 'S', 'C', 'X', 'Ka']\nbands.count(\"Ka\")\n\n\nLet's play a bit with the other functions. \"Pop\" returns the most recent addition (last element) in the list and simultaneously removes it:\n\nbands = ['L', 'S', 'C', 'X', 'Ka']\nprint( bands )\nprint( bands.pop() )\nprint( bands )\n\n\nAnother function, \"index\" returns the index at which the specified element occurs for the first time in the list:\n\nbands = ['L', 'S', 'C', 'X', 'Ka']\nbands.index(\"Ka\")\n\n\nthough it gets unhappy when asked to find an item not in the list\n\nbands.index(\"Q\")\n\n\n\"insert\" places an item before the specified location. So we can for example combine \"insert\" and \"index\" to place \"K\" in the list before \"Ka\":\n\nbands = ['L', 'S', 'C', 'X', 'Ka']\nbands.insert(bands.index(\"Ka\"),\"K\")\nprint(bands)\n\n\nNow is a good time to explore the other functionality found here: removal, sorting, reversing, and so on.\n\n### A First Look at Iteration\n\nWe'll see much more on iteration in a moment, but one of the most important aspects of a list is that you can easily iterate over it. This simple syntax will get each element in bands, in order of their syntax, print its value, and tell you what type it is.\n\nfor element in bands:\nprint( element, type(element) )\n\n\none of python's quirks is that looping occurs via iteration across collections like lists.\n\n### Unpacking\n\nLists and other collections like tuples or sets have the ability to \"unpack.\" This means that they can be compactly assigned to a series of variables like so:\n\nbands = [\"X\",\"Y\",\"Z\"]\na, b, c = bands\nprint(a)\nprint(b)\nprint(c)\n\n\n## Data Collections: Dictionaries\n\nDictionaries are another basic python data collection. Dictionaries store data in pairs, with a set of unique keys each mapped to some value (the values do not have to be unique). Dictionaries are extremely useful in scripting. For example, imagine that we are reducing a complex set of observations and want to be able to look up a the calibrator associated with a given source.\n\nYou can define a dictionary using the curly braces {} and \"key\"-\"value\" pairs separated by \":\". For example\n\nmy_dict = {\"a key\":10,\n\"b key\":20}\n\n\nYou can then extract the value for a given key in a way similar to how we accessed lists. Index the dictionary with the key in square braces [].\n\nprint( my_dict[\"a key\"] )\n\n\nDictionary keys can be numbers, strings, or tuples of either (this is a place where tuples are useful compared to lists or sets - you cannot have a list as a key for a dictionary). So you could, for example, define a dictionary holding source coordinates and fluxes using tuples (defined in parentheses, e.g., as \"(1,2)\").\n\ncatalog = {(1,2):10,\n(4,5):50,\n(10,10):10}\nprint( catalog[(1,2)] )\n\n\nYou can mix and match key types and use any data type for the value. For example:\n\nmy_dict = {1:5,\n\"b\":[1,2,3],\n(1,2):\"string\"}\nmy_dict\nmy_dict[\"b\"]\nmy_dict[(1,2)]\n\n\nFor scripting inteferometry reductions one useful application might be to pair sources and calibrators like so:\n\nphasecal = {\"source_1\":\"phasecal_1\",\n\"source_2\":\"phasecal_2\",\n\"source_3\":\"phasecal_1\"}\n\n\nThen during the reduction, one can simply call:\n\nsource_name = \"source_1\"\nphasecal[source_name]\n\n\n### Manipulating Dictionaries\n\nAs with lists, dictionaries have significant associated functionality. Use tab completion and \"help\" to explore the dictionary functionality now.\n\nAmong other tricks, you can append new elements to a dictionary like so:\n\nmy_dict = {\"a\":1,\n\"b\":2,\n\"c\":3}\nmy_dict[\"d\"] = 4\n\n\nYou can check if it has a given key like so (very useful for making sure you don't index with an invalid key):\n\nprint( my_dict.has_key(\"a\"), my_dict.has_key(\"e\") )\n\n\nOr list the keys like so:\n\nprint( my_dict.keys() )\n\n\nYou can iterate over dictionaries like so:\n\nfor key in my_dict.keys():\nprint( key )\n\n\nOr iterate over pairs of keys and values using the \"iteritems\" functionality:\n\nfor key, value in my_dict.iteritems():\nprint( key, value )\n\n\n### Dictionaries of Dictionaries\n\nDictionaries can also be values in dictionaries, making 2 dimensional dictionaries. For example, we could store the white pieces on a chess board in a 2D dictionary of files and ranks as:\n\nfiles = ['a','b','c','d','e','f','g','h']\nback_row = {'a':'Rook','b':'Knight','c':'Bishop','d':'Queen', \\\n'e':'King','f':'Bishop','g':'Knight','h':'Rook'}\nwhite = {} # sets to a 1D dictionary\nfor file in files:\nwhite[file] = {} # sets to a dictionary of dictionaries\nwhite[file] = 'Pawn'\nwhite[file] = back_row[file]\n\n\nNow you can see what white piece is on a particular square simply by:\n\nprint( white['d'] )\n\n\nThere are more aspects to explore, but this gives a good introduction to the large functionality of dictionaries.\n\n## Other Types of Data Collections\n\nYou may also want to read up on sets (collections of unique elements) and tuples (immutable collections).\n\n### Sets\n\nWe won't discuss these at length, but \"sets\" are another useful python data collection. They contain unique collections of data:\n\nmy_set = set([1,2,3,3,3,3,4,4,4,4])\nprint( my_set )\n\n\nsets have a large set of interesting functionality. Explore and read away.\n\n### Tuples\n\nTuples are collections of data defined using parentheses (). They have the key characteristic of being immutable, which means that they can be used in many places that other data collections don't work (e.g., to index numpy arrays).\n\ntest = (1,2)\nprint( test )\nprint( test )\n\n\nbut notice that this doesn't work\n\ntest = 1\n\n\n## Control Flow\n\nControl flow via if, for, and while-type statements is key to any programming. Python allows these approaches and offers some slick syntax to iterate in clever ways. It also allows very ready access to this functionality inside the interpreter, which is fantastic for rapid data analysis. Here we'll just look at the basics.\n\n### If\n\nIf is our most basic control statement. You can fork on a truth statement. Maybe the biggest thing to notice here is python's unconventional lack of curly braces, begin/ends, or other delimiters. Everything is done through the : and indentation. And it works on the command line, too. For example,\n\nx = 55\nif x < 100:\nprint( \"Not enough beer bottles.\" )\nelif x > 100:\nprint( \"Too many beer bottles.\" )\nelse:\nprint( \"Just enough beer bottles.\" )\n\n\nTry setting x to different values. If you aren't using \"cpaste\" notice that you may need an extra <enter> at the end of the paste to the shell.\n\n### While\n\nWhile checks the truth statement and runs the commands inside the loop as long as it is true. This means that it can be pretty easy to get runaway loops, so get your CTRL+C fingers ready.\n\nx = 55\nwhile x < 100:\nprint( \"Too few bottles of beer. Filling one up.\" )\nx += 1\nprint(x)\n\nprint(x)\n\n\nNow let's write a loop that could get us in trouble. This would normally run away and we would need to hit CTRL+C to break out.\n\nx = 55\nwhile x < 100:\nprint(\"Too few bottles of beer. Pouring out a bottle of beer.\")\nx -= 1\nif x <= 0:\nprint(\"Out of beer. Stopping.\")\nbreak\nprint(\"Still have beer left\")\n\n\nBut we've built in a safeguard. The break statement breaks us out of our current loop. It's less destructive twin continue moves us to the next iteration of the innermost current loop.\n\nThe other key thing to notice above is the nested indentation. See how the if statement is set off by the : and indentation. But the print function below is back in line with the while and so will run every time.\n\n### For\n\nFor in python differs from its implementation in other languages in that its always iterating over some other sequence of data. This allows for some slick tricks but also takes a bit of getting used to.\n\nYour basic for statement involves creating a list and looping over it (later you may want to replace this with a numpy array).\n\nYou can replicate the basic looping over sequential values using the \"range\" function. The following syntax will loop from 0 to 9 stepping by 1:\n\nloop_over = range(10)\nfor i in loop_over:\nprint(i)\n\n\nThe \"range\" function takes several parameters. For instance, try:\n\nprint( range(2,5) )\nprint( range(100,0,-30) )\n\n\nThere's no need to focus on integers when looping. Python can loop over any collection:\n\nbeatles = ['paul','john','george','ringo']\nfor musician in beatles:\nprint musician\nif musician == 'ringo':\nprint( \"... oooh \"+musician+\" was the best.\" )\n\n\nWe could have obtained similar functionality by using \"range\" and the length of the list:\n\nbeatles = ['paul','john','george','ringo']\nfor i in range(len(beatles)):\nprint( beatles[i] )\n\n\nIt's also possible to iterate in-line, for example to create lists. For example:\n\na = [x**2 for x in range(10)]\nprint(a)\n\n\nor\n\nb = [[x,x**2,x**3] for x in range(10)]\nprint(b)\n\n\nDon't get too hung up on this syntax.\n\nThere is not a proper case statement in python though there are plenty of workarounds.\n\n### Continue, Break, and Pass\n\nThere are several additional bits of functionality that are useful for steering control flow in a program. Calling continue during an iteration will end the current iteration and move to the next one. For example\n\nfor i in range(11):\nif (i == 5):\ncontinue\nprint(i)\n\n\nbreak completely ends the innermost ongoing iteration. For example breaking the same loop yields\n\nfor i in range(11):\nif (i == 5):\nbreak\nprint(i)\n\n\nFinally, pass puts a do-nothing line of python in place, which for example lets you build an empty loop.\n\nfor i in range(10):\npass\n\n\n## More Complex Programs\n\n### Executing Scripts\n\nThe most basic way to execute a set of python commands (aside from just copying and pasting to the shell) is to the execfile command. Calling execfile('myscript.py') from inside a python shell will execute 'myscript.py' one line at a time. You can use this to run a series of reduction commands or other simple scripts. In fact, calling execfile on one or more scripts will almost certainly be sufficient to script most basic CASA data reductions.\n\nand then in the same directory where you put the script type\n\nexecfile('intro_script.py')\n\n\nThe script should execute as through you had typed each line at the shell. Alternatively, you can get exactly the same results in the Ipython shell with:\n\nrun -i intro_script.py\n\n\nThe execfile function is nice if you want to run scripts from scripts. On the other hand, run gives you control of the namespace, which is helpful when working interactively.\n\nYou can combine the control flow that we learned above (if, for, while) with execfile to refine your scripts. For example, you might have a sophisticated reduction path that requires a few user inputs, which could be collected at the top of the script as variables. The reduction might have several discrete parts, which you could turn on or off using booleans and if statements.\n\nAs you edit the variables and booleans inside the script then rerun it with execfile, various parts of the script will run tuned by the variables you set.\n\nThis simple but powerful approach can (if you desire) form the infrastructure for a lot of your CASA reduction scripting. Try playing around with these scripts now.\n\n### Functions\n\nPython allows you to define functions either from the command line (or an execfile call) or as part of modules.\n\nTo define a function use def with the following syntax.\n\ndef first_function():\n\"\"\"Prints Hello World and quits\"\"\"\nprint \"Hello World.\"\n\n\nNow it's defined. You can call it via:\n\nfirst_function()\n\n\nAnd you can find out about it:\n\nhelp(first_function)\n\n\nor\n\nfirst_function?\n\n\nA function can return values like so:\n\ndef second_function():\n\"\"\"Return the string 'Hello World'\"\"\"\nreturn \"Hello World.\"\n\nvalue = second_function()\nvalue\n\n\nIt can also accept inputs:\n\ndef third_function(inp):\n\"\"\"Just like the print function\"\"\"\nprint(inp)\n\nthird_function(\"Hello World\")\n\n\nYour definition can set defaults for these inputs:\n\ndef fourth_function(optional=\"Shark Week!\",optional2=\"Exciting!\"):\n\"\"\"Prints Shark Week Exciting unless told otherwise\"\"\"\nprint optional\nprint optional2\n\n\nIf you supply input this will override the default\n\nfourth_function(\"Hello World\")\n\n\nIf not then the default takes hold\n\nfourth_function()\n\n\nYou had yet another option here too. What if we only wanted to specify the second parameter? We can do that with a keyword call. Specify the name of the input parameter in the call to the function.\n\nfourth_function(optional2=\"Is just shameful.\")\n\n\nNote, that your keyword calls must come after your unlabeled input. These are legal:\n\nfourth_function(\"Zebra week.\", optional2=\"Is just shameful.\")\nfourth_function(optional=\"Giraffe week.\", optional2=\"Is just shameful.\")\n\n\nThis is not:\n\nfourth_function(optional=\"Giraffe week.\", \"Is just awesome.\")\n\n\nFunctions of Functions\n\nYou can also pass functions to functions. For example, imagine you define a cubic function as follows:\n\ndef cubic(x, A=1.0, B=0.0, C=0.0, D=0.0):\n\"\"\"Returns A*x**3 + B*x**2 + C*x + D.\"\"\"\nreturn A*x**3 + B*x**2 + C*x + D\n\n\nAnd the derivative function as:\n\ndef slope(f, x, h=0.00001):\n\"\"\"Takes the derivative at a point.\"\"\"\nreturn (f(x+h) - f(x-h))/(2*h)\n\n\nYou can find the derivative of $\\displaystyle{ x^3 }$ at $\\displaystyle{ x=2 }$ by calling:\n\nslope(cubic,2.0)\n\n\nThere is also a one-line function assignment called the lambda function. For example, one can simply define a function on one line as:\n\nsquare = lambda x: x**2\nprint(square(5))\n\n\nThis is handy in that it can be passed to another function. For example, what is the derivative of $\\displaystyle{ 5x^3 + 3x^2 + 2x + 3 }$ evaluated at $\\displaystyle{ x=7 }$?\n\nslope(lambda x: cubic(x,A=5.0,B=3.0,C=2.0,D=3.0), 7.0)\n\n\nUnpacking again\n\nFinally, you might want to pass out several values, for example a set of coordinates. One easy way to do this is:\n\ndef location():\n\"\"\"Returns a location.\"\"\"\nx_loc = 10\ny_loc = 20\nz_loc = 30\nreturn (x_loc,y_loc,z_loc)\n\n\nand then the function can be called like so:\n\nx, y, z = location()\nprint(x)\nprint(y)\nprint(z)\n\n\nThis is another application of the \"unpacking\" mentioned as we introduced lists.\n\n### Modules\n\nModules are collections of functions, variables, and objects that can be imported into a python session to add functionality. Python has a number of native modules, for example \"os\", \"math\", and \"time\". Third parties supply many more, including \"numpy\" for robust array handling, \"scipy\" for scientific analysis, and \"matplotlib\" for plotting. You can also easily write your own new modules in python.\n\nTo access a module in a python session or program you need to \"import\" it. For example, to import the math module type\n\nimport math\n\n\nThen type \"math.\" and try tab completion to see the math functionality. For example you can take 5 to the 3rd power via:\n\nprint( math.pow(5,3) )\n\n\nWhen you import you can change the name of the module like so\n\nimport math as m\nm.sqrt(4)\n\n\nor you can import only specific functionality. For example if you wanted only the pow function you could import it like so:\n\nfrom math import pow\nprint( pow(10,2) )\n\n\nyou can import all functionality from a module via\n\nfrom math import *\n\n\nThere are a few important subtleties to know here:\n\n• First, one reason that modules natively maintain their own namespace is to avoid variable and function names stepping on each other. For example, both math and numpy have a \"sin\" function. Having these as \"numpy.sin\" and \"math.sin\" keeps them distinct and clear.\n• A CASA-specific issue: many of the tools have two-letter names so do be careful importing things as two-letter names - you can easily lose the CASA tools.\n\nModules are easy. To define them you just need to create a file in your path with a .py extension. Here we create \"intro_module.py\". We'll define a few variables and functions.\n\nFirst, you can define variables in your module:\n\na = 10.0\nb = 15.0\nc = 20.0\n\n\nThese definitions or any other executable code placed in the main body of the module will be executed the first time that the module is imported.\n\nAnd you can define functions. You can use the variables you define in the module, if you specify them with the global statement. For example:\n\ndef square(x):\n\"\"\"\nThis takes a quadratic in form a*x**2 + b*x + c, with a,b,c\ndefined in the module.\n\"\"\"\nglobal a, b, c\nreturn a*x**2 + b*x + c\n\n\nAnd finally, you import your modules into your programs using the standard import commands, such as:\n\nimport intro_module as intro\nprint(intro.a, intro.b, intro.c)\nprint( intro.square(2) )\n\n\nIf you change your module, you cannot import it again. If a module is imported that it already knows, it will ignore the command. Rather use the reload function:\n\nreload( intro )\n\n\nand note that you cannot reload material that you have imported via a \"from X import Y\" statement. Though a reasonable workaround is to delete and re-import. This means that as you develop your own modules and test them in the shell you almost certainly want to use the \"import X\" syntax.\n\nThe main use of modules will be to store reusable code or organize complex projects into discrete files. Read more on modules.\n\nFuture sections of the python basics touch on specific modules, either basic python or third party, useful for analysis and scripting." ]

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[ "# Number 1680\n\n### Properties of number 1680\n\nCross Sum:\nFactorization:\n2 * 2 * 2 * 2 * 3 * 5 * 7\nDivisors:\n1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 20, 21, 24, 28, 30, 35, 40, 42, 48, 56, 60, 70, 80, 84, 105, 112, 120, 140, 168, 210, 240, 280, 336, 420, 560, 840, 1680\nCount of divisors:\nSum of divisors:\nPrime number?\nNo\nFibonacci number?\nNo\nBell Number?\nNo\nCatalan Number?\nNo\nBase 2 (Binary):\nBase 3 (Ternary):\nBase 4 (Quaternary):\nBase 5 (Quintal):\nBase 8 (Octal):\nBase 32:\n1kg\nsin(1680)\n0.68315165392649\ncos(1680)\n-0.73027653511358\ntan(1680)\n-0.93546981325402\nln(1680)\n7.4265490723973\nlg(1680)\n3.2253092817259\nsqrt(1680)\n40.987803063838\nSquare(1680)\n\n### Number Look Up\n\nLook Up\n\n1680 which is pronounced (one thousand six hundred eighty) is a very unique number. The cross sum of 1680 is 15. If you factorisate 1680 you will get these result 2 * 2 * 2 * 2 * 3 * 5 * 7. The figure 1680 has 40 divisors ( 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 20, 21, 24, 28, 30, 35, 40, 42, 48, 56, 60, 70, 80, 84, 105, 112, 120, 140, 168, 210, 240, 280, 336, 420, 560, 840, 1680 ) whith a sum of 5952. 1680 is not a prime number. The figure 1680 is not a fibonacci number. The number 1680 is not a Bell Number. 1680 is not a Catalan Number. The convertion of 1680 to base 2 (Binary) is 11010010000. The convertion of 1680 to base 3 (Ternary) is 2022020. The convertion of 1680 to base 4 (Quaternary) is 122100. The convertion of 1680 to base 5 (Quintal) is 23210. The convertion of 1680 to base 8 (Octal) is 3220. The convertion of 1680 to base 16 (Hexadecimal) is 690. The convertion of 1680 to base 32 is 1kg. The sine of the figure 1680 is 0.68315165392649. The cosine of the figure 1680 is -0.73027653511358. The tangent of 1680 is -0.93546981325402. The root of 1680 is 40.987803063838.\nIf you square 1680 you will get the following result 2822400. The natural logarithm of 1680 is 7.4265490723973 and the decimal logarithm is 3.2253092817259. You should now know that 1680 is amazing number!" ]

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[ "Cookie Consent by FreePrivacyPolicy.com\nSearch a number\nBaseRepresentation\nbin100110110001100…\n…0000000101101010\n310100200002201100020\n41031203000011222\n510131030133114\n6333033221310\n744145332145\noct11543000552\n93320081306\n101301021034\n116084353a5\n12303861836\n13179704941\n14c4b0905c\n1579342aa9\nhex4d8c016a\n\n1301021034 has 16 divisors (see below), whose sum is σ = 2606109120. Its totient is φ = 432995840.\n\nThe previous prime is 1301021021. The next prime is 1301021039. The reversal of 1301021034 is 4301201031.\n\nIt is not an unprimeable number, because it can be changed into a prime (1301021039) by changing a digit.\n\nIt is a polite number, since it can be written in 7 ways as a sum of consecutive naturals, for example, 165294 + ... + 172985.\n\nIt is an arithmetic number, because the mean of its divisors is an integer number (162881820).\n\nAlmost surely, 21301021034 is an apocalyptic number.\n\n1301021034 is an abundant number, since it is smaller than the sum of its proper divisors (1305088086).\n\nIt is a pseudoperfect number, because it is the sum of a subset of its proper divisors.\n\n1301021034 is a wasteful number, since it uses less digits than its factorization.\n\n1301021034 is an evil number, because the sum of its binary digits is even.\n\nThe sum of its prime factors is 338925.\n\nThe product of its (nonzero) digits is 72, while the sum is 15.\n\nThe square root of 1301021034 is about 36069.6691695391. The cubic root of 1301021034 is about 1091.6785388630.\n\nAdding to 1301021034 its reverse (4301201031), we get a palindrome (5602222065).\n\nThe spelling of 1301021034 in words is \"one billion, three hundred one million, twenty-one thousand, thirty-four\"." ]

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[ "Home » Blog » Determine functions u and v satisfying given differential equations\n\n# Determine functions u and v satisfying given differential equations\n\nThe graph of a solution", null, "of the differential equation", null, "intersects the graph of a solution", null, "of the differential equation", null, "at the origin. Furthermore, the two curves have equal slopes at the origin. Determine formulas for the functions", null, "and", null, "if", null, "First, we compute the general solutions of the second-order differential equation", null, "This is an equation of the form", null, "From this we compute", null, ", so", null, "and", null, ". Using Theorem 8.7 (pages 326-327 of Apostol) we have", null, "Next, we compute the general solutions of the second-order differential equation", null, "This is again an equation of the form", null, "From this we compute", null, "so", null, "and", null, ". Again, applying Theorem 8.7 (pages 326-327 of Apostol) we have", null, "Now, let", null, "We know that the solutions we want intersect at the origin, hence we must have", null, ". So,", null, "Plugging these values for the constants", null, "and", null, "into our expressions for", null, "and", null, "we compute the first derivatives,", null, "Next, we are given that the particular solutions", null, "and", null, "we want have equal slopes at the origin. This means", null, ". Hence,", null, "Finally,", null, "Thus,", null, "" ]

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[ "", null, "TGF - Maple Help\n\nTGF (.tgf) Graph Format\n\nTGF file format", null, "Description\n\n • TGF (Trivial Graph Format) is a simple text-based file format for storing a single graph.\n • It also allows the application of labels to vertices and edges.\n • The format does not specify whether the graph is directed or undirected.\n • The GraphTheory[ImportGraph] and GraphTheory[ExportGraph] commands can read from and write to this format.\n • The general-purpose commands Import and Export also support this format.", null, "Examples\n\nImport a TGF file encoding a directed graph.\n\n > $G≔\\mathrm{Import}\\left(\"example/digraph.tgf\",\\mathrm{base}=\\mathrm{datadir}\\right)$\n ${G}{≔}{\\mathrm{Graph 1: a directed unweighted graph with 11 vertices and 12 arc\\left(s\\right)}}$ (1)\n > $\\mathrm{GraphTheory}:-\\mathrm{DrawGraph}\\left(G,\\mathrm{style}=\\mathrm{spring}\\right)$", null, "Export the Dyck graph to a TGF file in the home directory of the current user.\n\n > $\\mathrm{DG}≔\\mathrm{GraphTheory}:-\\mathrm{SpecialGraphs}:-\\mathrm{DyckGraph}\\left(\\right)$\n ${\\mathrm{DG}}{≔}{\\mathrm{Graph 2: an undirected unweighted graph with 32 vertices and 48 edge\\left(s\\right)}}$ (2)\n > $\\mathrm{Export}\\left(\"dyck.tgf\",\\mathrm{DG},\\mathrm{base}=\\mathrm{homedir}\\right)$\n ${437}$ (3)" ]

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[ "# Quadratic formula\n\n Definition: With the quadratic formula we solve the quadratic equation: { ax^2+bx+c=0} . The formula is: x_1,x_2 = {{ - b \\,\\pm\\, \\sqrt {b^2 \\,-\\, 4 \\cdot a \\cdot c} } \\over {2 \\cdot a}} Rules The formula can give 0, 1 or 2 real number solutions (only real numbers exist on a graph!). If there are two different solutions they are written x_1 and x_2 The symbol \\pm means that for x_1 we use the + sign and for x_2 we use the - sign. The discriminant is:   D=b^2-4\\cdot a\\cdot c   (the expression inside the square root). Example Solve the equation: x^2-2x-3=0 for x. Solution: Here   а=1 \\quad b=-2 \\quad c=-3     (other examples?) x_1 ,\\,x_2 =\\, {{ - ( - 2) \\,\\pm\\, \\sqrt {( - 2)^2 - 4 \\cdot 1 \\cdot ( - 3)} } \\over {2 \\cdot 1}} = {{2 \\,\\pm\\, \\sqrt {4 + 12} } \\over 2} = {{2 \\,\\pm\\, 4} \\over 2} x_1 = {{2 + 4} \\over 2} = {6 \\over 2} = 3 and x_2 = {{2 - 4} \\over 2} = {{ - 2} \\over 2} = - 1 Answer:   x_1=3 and x_2=-1 are the two solutions to the equation f(x)=x^2-2x-3=0 (see the left graph below!).\n\nRules   Explanation\n\nDepending on the function, the discriminant can be positive, zero or negative.\n\n• If the discriminant is positive, the formula gives two distinct numbers: x_1 and x_2 .\n\nIn the above example, the discriminant was 16 (positive) and so the quadratic function f(x)=x^2-2x-3 has two distinct roots. This means the graph of the function has two x-intercepts x_1=3 and x_2=-1 .  See left graph below.\n\n• If the discriminant is zero, the formula give one root and the function \"touches\" the x-axis at this one point. See middle graph below.\n• If the discrimnant is negative, the formula has no real solution and since graphs are \"real\", the graph cannot cross or touch the x-axis. See right graph below.\n The discriminant D>0 The discriminant D=0 The discriminant D<0 D>0 - the parabola crosses the x-axis in two distinct points D=0 - the parabola touches the x-axis at a unique point D<0 - the parabola doesn't cross or touch the x-axis sample function:   f(x)=x^2-2x-3 sample function: f(x)=x^2-4x+4 sample function: f(x)=2x^2+1 We solve: x^2-2x-3=0 We solve: x^2-4x+4=0 We solve: 2x^2+1=0 D= ( - 2)^2 - 4 \\cdot 1 \\cdot ( - 3)=16>0 D= ( - 4)^2 - 4 \\cdot 1 \\cdot 4=0 D= ( 0)^2 - 4 \\cdot 2 \\cdot 2=-8<0 two different roots one unique root no roots x_1=3   and   x_2=-1 x_1=x_2=2 x_1 and x_2 do not exist!\n\nRelated topics:\n\n(:commentbox:)\n\nPage last modified on June 06, 2010, at 08:00 AM" ]

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[ "# SHA1 collision event probability after n iterations\n\nThere are String1 and String2 - some variables-strings. The probability of collision for different String1 and String2 is\n\nP{SHA1(String1) == SHA1(String2)} = p\n\n\nWhat's the probability of\n\nP{SHA1(SHA1(String1))} == P{SHA1(SHA1(String2))} //two times\nP{SHA1(...(SHA1(String1)))} == P{SHA1(...(SHA1(String2)))} //10^9 times\n\n• What do you think? What have you tried? Where did you get stuck? Where did you run into this question? What is the context/motivation for your question? This is not a site where you copy-paste your exercise and we do your exercise for you. – D.W. Jun 18 '14 at 0:56\n• Did you look at crypto.stackexchange.com/q/15068/351 ? (Did you remember to use search before asking?) – D.W. Jun 18 '14 at 0:58\n• The question came into play after using many iterations for generating hash for the password (with or without using salt). I was curious if it gets less secure after using 10^9 iterations and was curious if it gets not secure at all, if we don't use salt and the push number of iterations to infinity (neither asked nor answered this question yet). Many iterations with salt was used to exclude using of rainbow tables by hackers so the server counts the hash for some (mili-)seconds. – Haradzieniec Jun 18 '14 at 9:47\n\nAssuming the inputs and outputs are random, you would expect:\n\n\\begin{align} P[\\operatorname{SHA1}(s_1) \\not= \\operatorname{SHA1}(s_2)] &= 1 - 2^{-160} \\\\ P[\\operatorname{SHA1}^2(s_1) \\not= \\operatorname{SHA1}^2(s_2)] &= (1 - 2^{-160})^2 \\\\ ... \\\\ P[\\operatorname{SHA1}^n(s_1) \\not= \\operatorname{SHA1}^n(s_2)] &= (1 - 2^{-160})^n \\approx 1 - n/2^{160}, \\\\ \\end{align}\n\nwhere $\\operatorname{SHA1}^2 = \\operatorname{SHA1} \\circ \\operatorname{SHA1}$, etc.\n\nThus, $P[\\operatorname{SHA1}^{10^9}(s_1) = \\operatorname{SHA1}^{10^9}(s_2)] \\approx 10^9 / 2^{160} \\approx 2^{-130}$.\n\nThat's for a single pair $s_1 \\not= s_2$. If you are looking for collisions, you would expect a collision with $\\operatorname{SHA1}^{10^9}$ after something like $2^{130/2} = 2^{65}$ strings.\n\nThis approximation only holds while $n$ is small enough, and breaks down when it gets close to $2^{80}$ and cycles start to become an issue. However, for $10^9$ it should do all right.\n\nHowever, there are (theoretical) attacks to find collisions in SHA1 faster, so for non-random inputs the probabilities could be higher.\n\n• @DmitryKhovratovich: actually, it's not quite correct, however it's in the other direction. It's fairly close for small $n$ (and in this context, $10^9$ is small), however for huge $n$, the cycle structure of iterated SHA1 becomes important. Iterated SHA1 will fall into a cycle eventually, that means that two different inputs have a good probability of falling two different cycles, or two different offsets in the same cycle -- this means that there is a good probability that two different inputs will never collide for any $n$ – poncho Jun 17 '14 at 15:36\n• @poncho: the cycle effect comes into play when the number of iterations approaches $2^{n/2}$. Before that, the image size of $SHA1^i$ is about $\\frac{2^n}{i}$, so for $i=10^9$ you'd have $(1-2^{-130})$ instead of $(1-2^{-160})$. See eprint.iacr.org/2014/223.pdf for the details – Dmitry Khovratovich Jun 18 '14 at 0:22\n• I think there is danger in extending analysis of SHA1 treating it as \"true random\" through so many iterations. My feeling is that its pseudo-randomness comes into play quite early on, and probabilities of collisions for large n are much lower than estimated. I'd perhaps run an experiment with a truncated SHA1 (e.g. 16 bits) to see what properties that has. – Neil Slater Jun 18 '14 at 7:38\n• @NeilSlater: I ran a quick test with 16-bit SHA1 and the results match the formula from the answer like this (inverse probability that two random strings are equal, plotted against the expected $2^{16}/n$). Python code. – otus Jun 18 '14 at 14:19\n• @otus: I must have made a mistake in my experiment (and I have deleted comment with spurious claims that probability seems limited to $2^{-15}$ irrespective of number of iterations). I've played with a few variations of your code to make it more like my originally-intended test (longer, random strings, iterating digest, not hex digest). I get same results as you. Looking for my bug now . . . found it, I had a hard-coded value for N (or what you call N in your code) over-riding the param that I set for later tests, so most of my tests only looked at two iterations. – Neil Slater Jun 18 '14 at 15:04" ]

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[ "## Minimal Sufficient Explanation\n\nMy notes on Omar Zia Khan, Pascal Poupart, and James P Black, ICAPS 2009\n\nAutomated planning problems have long been attempted using Markov Decision Processes (MDPs). MDPs are capable of handling the probabilistic and sequential nature of planning problem. They solve the problem by providing a policy which is a mapping from states to actions. However, to use this policy in the real-world, we first need users to trust the policy. The issue of trust can be ameliorated if the policy provides an explanation for its recommeded actions. This is the pretext of khan et al 2009.\n\nPolicy in MDP is usually obtained by optimizing Bellman’s equation which considers value of a state as expected discounted total reward. This paper uses an alternate formulation of the value function introduced by and extends it to the factored MDPs. In the alternate formulation, value function is expressed as a dot product of reward and the discounted occupancy frequencies of all the states. With this formulation, an optimal policy for a given state can explain the recommended action by simply showing that the recommended action has highest occupancy frequency of highly rewarded state(s). To this effect the paper propose three explanation templates:\n\n1. ActionName is the only action that is likely to take you to Var1 = Val1, Var2 = Val2, … about $\\lambda$ times, which is higher (or lower) than any other action.\n2. ActionName is likely to take you to Var1 = Val1, Var2 = Val2, … about $\\lambda$ times, which is high (or low) as any other action''\n3. ActionName is likely to take you to Var1 = Val1, Var2 = Val2, … about $\\lambda$ times.\n\nVar1 = Val1, Var2 = Val2, … in the template represents a single state (in case of MDPs) or a scenario\\footnote{In the case of factored MDPs, the reward function is also factored and the scenario used is defined as set of states which have similar rewards.} (in case of Factored MDPs). Multiple templates can be used for explaining a recommended action.\n\nPower of these templates can be realized if they provide minimal-sufficient explanations (MSE). An explanation is sufficient if it can prove that the action recommended is optimal (or no other action has better utility than the recommended action). Sufficient explanation is minimal if the number of templates used are minimum possible.\n\nWhen the policy ($\\pi^{\\ast}$) is optimal, the value of the recommended action for state $s_0$~$\\left(V^{\\pi^\\ast}\\right.$ or $\\left.Q^{\\pi^\\ast}(s_0, \\pi^\\ast(s_0))\\right)$ will be higher than value of all other actions $\\left(Q^{\\pi^\\ast}(s_0,a), , \\forall a \\neq \\pi^\\ast(s_0)\\right)$. Khan et al. 2009 define $V_{MSE}$ (eqn. below) which has two components: (a) expected utility from all the states(or scenarios) included in the explanation, and (b) worst case utility of all the states (or scenarios) not included in the explanation. By this definition $V_{MSE}$ cannot exceed the value of optimal action. If the explanation is minimal and sufficient, $V_{MSE}$ will be higher than the value of any other action.\n\n$V_{MSE} = \\sum_{i \\leq k}\\mathbb{E}[\\operatorname{utility}(s_i)] + \\sum_{i > k} \\min(\\operatorname{utility}(s_i))$\n\n$V^{\\pi^{\\star}} \\geq V_{MSE} > Q^{\\pi^\\star}(s_0,a)\\ \\ \\forall a \\neq \\pi^\\star(s_0)$\n\nWith that definition of $V_{MSE}$, Khan et al. propose an algorithm to produce MSE at any state $s_0$ for a given optimal policy in factored MDP setting. They evaluate the MSE on two experimental domains of “course-advising” and “handwashing”. The experimental results show that the number of templates in MSE is high (or low) when the ratio of the expected utility for the optimal and the second best optimal policy is high (or low). User study with advisors in the course-advising domain show that the advisors found the explanations useful and perhaps beneficial for grade-conscious or poor performing students. User study with students show that more than 50% of the students found the explanations easy to understand, accurate and informative. However, most of them needed further information to trust the action recommended.\n\n## Critique\n\nThe paper addresses a very important topic of explaining the recommendations of a Markov Decision Process. The work is especially commendable as it was done way ahead of the Explainable AI Hype. They provide a very principled and straight forward way to explain the recommendations from the perspective of how the policy is constructed. However, it might not be intuitive enough to the users. As seen in the user-study, most of the users wanted to compare the explanation of the recommended action with their preferred actions or need more information about occupancy frequencies.\n\nIn complex domains, Bellman’s optimality equation is not solved exactly and hence, the value function of each state is only discovered approximately by the RL agent. In that scenario, as I understand, the optimality of the policy is only with respect to that approximate value function and hence the explanation of optimality is also only with respect to that approximate value function. To end user, this values might not mean anything and hence the sufficiency of the explanation might be questioned.\n\nThe explanations are only useful when it has less number of templates, the user-study in course-advising domain shown in the paper has less than 3 templates for the explanations. But in complex domain, as seen in handwashing experiments the number of templates will eventually increase and I conjecture that with increase in the number of templates the explanations will be less meaningful to the users.\n\nAlso, as raised by Dr. Natarajan in the class discussion the template only explains the action with respect to the states it visits but not with respect to the states it helps avoid. For domain where we want to avoid a certain states, we encode high negative rewards for that state. Even though the optimal policy learns to avoid that negative state, the MSE framework will only look at the occupancy frequency of highly rewarding states and try to explain positive reward states. As the occupancy frequency of the highly negative state is $0$ with the recommended action, it will not come up in the explanation. Although the templates proposed does mention “or lower” in brackets, it is not quite clear how the avoided state will be added in the explanation.\n\nFinally, in RL we usually assume the policy is mapping of states to a distribution over actions. Explaining a distribution might be complex than a single action. This paper concentrates on explaining the best action out of that distribution. However, there might be two equally good actions, or there might be more than one optimal policy. One has to extend this framework for that use cases before using it in a real system.\n\n1. Omar Zia Khan, Pascal Poupart, and James P Black, Minimal Sufficient Explanations for Factored Markov Decision Processes, ICAPS 2009\n2. Pascal Poupart, Exploiting structure to efficiently solve large scale partially observable Markov decision processes, Citeseer 2005." ]

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[ "Cody\n\n# Problem 43021. How many figures currently exist?\n\nSolution 996137\n\nSubmitted on 5 Oct 2016 by goc3\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\nassert(isequal(numfig(),0))\n\n2   Pass\nclose all; figure; assert(isequal(numfig(),1))\n\n`", null, "`\n3   Pass\nclose all; figure; figure; assert(isequal(numfig(),2))\n\n`", null, "`\n`", null, "`\n4   Pass\nclose all; figure; figure; plot(1,1); figure; imagesc(magic(5)); assert(isequal(numfig(),3))\n\n`", null, "`\n`", null, "`\n`", null, "`\n5   Pass\nclose all; n = randi(20); for ii=1:n figure; end assert(isequal(numfig(),n))\n\n`", null, "`\n`", null, "`\n`", null, "`\n`", null, "`\n`", null, "`\n`", null, "`\n`", null, "`\n`", null, "`\n`", null, "`\n`", null, "`\n6   Pass\nclose all; assert(isequal(numfig(),0))\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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null, 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null, 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null, 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", null, 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null, 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null, 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null, 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null, 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null, 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null, 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null, 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null, 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null, "data:image/png;base64,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", null, "data:image/png;base64,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", 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[ "# graph: be like equations\n\nfrom Quasimodo\n\n## Related concepts\n\n Parents toolWeight: 0.59, itemWeight: 0.58, featureWeight: 0.57, structureWeight: 0.56 Siblings chartWeight: 0.66, asymptoteWeight: 0.34, calculatorWeight: 0.33, spreadsheetWeight: 0.33, mapWeight: 0.33\n\n## Related properties\n\n Similarity Property be like equations 1.00 estimate solution for system of linear equations 0.83 linear equation on graphing calculator 0.78 be used in physics 0.76 be important in physics 0.76\n\n## Clauses\n\n### Salient implies Plausible\n\n0.20\nRule weight: 0.28\nEvidence weight: 0.72\nSimilarity weight: 1.00\nEvidence: 0.64\nPlausible(graph, be like equations)\nEvidence: 0.80\n¬ Salient(graph, be like equations)\n\n### Similarity expansion\n\n0.62\nRule weight: 0.85\nEvidence weight: 0.87\nSimilarity weight: 0.83\nEvidence: 0.81\nRemarkable(graph, be like equations)\nEvidence: 0.70\n¬ Remarkable(graph, estimate solution for system of linear equations)\n0.57\nRule weight: 0.85\nEvidence weight: 0.80\nSimilarity weight: 0.83\nEvidence: 0.80\nSalient(graph, be like equations)\nEvidence: 0.97\n¬ Salient(graph, estimate solution for system of linear equations)\n0.57\nRule weight: 0.85\nEvidence weight: 0.88\nSimilarity weight: 0.76\nEvidence: 0.81\nRemarkable(graph, be like equations)\nEvidence: 0.64\n¬ Remarkable(graph, be important in physics)\n0.57\nRule weight: 0.85\nEvidence weight: 0.85\nSimilarity weight: 0.78\nEvidence: 0.80\nSalient(graph, be like equations)\nEvidence: 0.72\n¬ Salient(graph, linear equation on graphing calculator)\n0.56\nRule weight: 0.85\nEvidence weight: 0.85\nSimilarity weight: 0.78\nEvidence: 0.81\nRemarkable(graph, be like equations)\nEvidence: 0.80\n¬ Remarkable(graph, linear equation on graphing calculator)\n0.55\nRule weight: 0.85\nEvidence weight: 0.85\nSimilarity weight: 0.76\nEvidence: 0.81\nRemarkable(graph, be like equations)\nEvidence: 0.81\n¬ Remarkable(graph, be used in physics)\n0.55\nRule weight: 0.85\nEvidence weight: 0.85\nSimilarity weight: 0.76\nEvidence: 0.80\nSalient(graph, be like equations)\nEvidence: 0.75\n¬ Salient(graph, be used in physics)\n0.53\nRule weight: 0.85\nEvidence weight: 0.80\nSimilarity weight: 0.78\nEvidence: 0.64\nPlausible(graph, be like equations)\nEvidence: 0.57\n¬ Plausible(graph, linear equation on graphing calculator)\n0.53\nRule weight: 0.85\nEvidence weight: 0.82\nSimilarity weight: 0.76\nEvidence: 0.80\nSalient(graph, be like equations)\nEvidence: 0.90\n¬ Salient(graph, be important in physics)\n0.52\nRule weight: 0.85\nEvidence weight: 0.78\nSimilarity weight: 0.78\nEvidence: 0.56\nTypical(graph, be like equations)\nEvidence: 0.49\n¬ Typical(graph, linear equation on graphing calculator)\n0.51\nRule weight: 0.85\nEvidence weight: 0.79\nSimilarity weight: 0.76\nEvidence: 0.64\nPlausible(graph, be like equations)\nEvidence: 0.60\n¬ Plausible(graph, be used in physics)\n0.50\nRule weight: 0.85\nEvidence weight: 0.77\nSimilarity weight: 0.76\nEvidence: 0.56\nTypical(graph, be like equations)\nEvidence: 0.51\n¬ Typical(graph, be used in physics)\n0.48\nRule weight: 0.85\nEvidence weight: 0.67\nSimilarity weight: 0.83\nEvidence: 0.64\nPlausible(graph, be like equations)\nEvidence: 0.92\n¬ Plausible(graph, estimate solution for system of linear equations)\n0.45\nRule weight: 0.85\nEvidence weight: 0.70\nSimilarity weight: 0.76\nEvidence: 0.64\nPlausible(graph, be like equations)\nEvidence: 0.84\n¬ Plausible(graph, be important in physics)\n0.42\nRule weight: 0.85\nEvidence weight: 0.59\nSimilarity weight: 0.83\nEvidence: 0.56\nTypical(graph, be like equations)\nEvidence: 0.93\n¬ Typical(graph, estimate solution for system of linear equations)\n0.40\nRule weight: 0.85\nEvidence weight: 0.62\nSimilarity weight: 0.76\nEvidence: 0.56\nTypical(graph, be like equations)\nEvidence: 0.87\n¬ Typical(graph, be important in physics)\n\n### Typical and Remarkable implies Salient\n\n0.13\nRule weight: 0.14\nEvidence weight: 0.91\nSimilarity weight: 1.00\nEvidence: 0.80\nSalient(graph, be like equations)\nEvidence: 0.56\n¬ Typical(graph, be like equations)\nEvidence: 0.81\n¬ Remarkable(graph, be like equations)\n\n### Typical implies Plausible\n\n0.38\nRule weight: 0.48\nEvidence weight: 0.80\nSimilarity weight: 1.00\nEvidence: 0.64\nPlausible(graph, be like equations)\nEvidence: 0.56\n¬ Typical(graph, be like equations)" ]

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[ "", null, "# Question: What Are The First 20 Perfect Cubes?\n\n## IS 400 a perfect cube?\n\nSince 2 & 5 do not occur in triplets.\n\n∴ 400 is not a perfect cube..\n\n## Is 75 a perfect square?\n\nIs 75 a perfect square number? A 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 75 is about 8.660. Thus, the square root of 75 is not an integer, and therefore 75 is not a square number.\n\n## What is the next square number after 16?\n\nInformally: When you multiply an integer (a “whole” number, positive, negative or zero) times itself, the resulting product is called a square number, or a perfect square or simply “a square.” So, 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on, are all square numbers.\n\n## What is the cube of 11?\n\n1331Cubes and Cube Roots List of 1 to 15NumberCube(a3)Cube root ∛a1113312.2241217282.2891321972.3511427442.41011 more rows•Jun 22, 2020\n\n## What are some perfect cubes?\n\nA perfect cube is a number that is the cube of an integer. For example, 125 is a perfect cube since 125 = 5 × 5 × 5= 53. Some examples of perfect cubes are 1, 8, 27, 64, 125, 216, 343, ..\n\n## Is 27 a cube number?\n\nA cube number is the answer when an integer is multiplied by itself, then multiplied by itself again. The third cube number is 27 because 3 × 3 × 3 = 27 , and so on. … The first five cube numbers are: 1, 8, 27, 64 and 125.\n\n## What are the cube numbers from 1 to 100?\n\nCube Numbers from 1 to 100NumberCube621673438512972996 more rows•Jun 4, 2020\n\n## Is 5000 a perfect cube?\n\nNow we know that a perfect cube should have a pair of three prime factor. This number has 1 pair of 2, 1 pair of 5 and 1 alone factor 5. Therefore, this number has a 5 more. So, 5 must be divided in 5000 to get a perfect cube.\n\n## Is 14 a perfect square?\n\nThe first 1000 perfect squares are listed below….List of perfect Squares?Perfect SquareFactors12111 * 1114412 * 1216913 * 1319614 * 1416 more rows\n\n## What are the perfect cubes from 1 to 20?\n\nCubes of 1 to 20NumberCube(n3)17491318583219685920800016 more rows\n\n## Is 8 a perfect square?\n\n8 is not a perfect square because you cannot express it as the product of two equal integers.\n\n## What numbers are perfect squares and perfect cubes?\n\nOne way is to list out all the square numbers with two digits: 16, 25, 36, 49, 64, 81. Then list out all the cube numbers with two digits: 27, 64. The answer is the only number in both lists: 64.\n\n## Is 24 a perfect square?\n\nSince 102.01 is a rational number and the square root of 102.01 is a rational number (10.1), 102.01 is a perfect square. 24 is NOT a perfect square. 24 is a natural number, but since there is no other natural number that can be squared to result in the number 24, 24 is NOT a perfect square.\n\n## Is 27 a perfect square?\n\nIs 27 a perfect square number? A 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 27 is about 5.196. Thus, the square root of 27 is not an integer, and therefore 27 is not a square number.\n\n## What are the first 10 perfect cubes?\n\nThe first ten cube numbers are 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000." ]

[ null, "https://mc.yandex.ru/watch/66666685", null ]

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[ "Cody\n\n# Problem 1463. Pascal's Matrix\n\nGiven an integer n ≥ 0, generate the ( n+1) × ( n+1) lower triangular Pascal's Matrix.\n\nExamples:\n\n``` pascalMat(0)\nans =\n1```\n``` pascalMat(1)\nans =\n1 0\n1 1 ```\n``` pascalMat(2)\nans =\n1 0 0\n1 1 0\n1 2 1```\n\nNeither string operations nor interpolations are allowed!\n\n### Solution Stats\n\n48.36% Correct | 51.64% Incorrect\nLast Solution submitted on Jul 02, 2020" ]

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[ "# Python File isatty() Method – Tutorial with Examples\n\nThe `isatty()` method in Python is a built-in method used to check whether the file descriptor associated with a file is an interactive terminal or not. The method returns True if the file descriptor is an interactive terminal, else returns False.\n\n## Syntax\n\n```file.isatty()\n```\n\n## Return Value\n\nThe `isatty()` method returns a boolean value – True if the file descriptor is an interactive terminal, and False otherwise.\n\n## Examples\n\n### Example 1: Checking if the file descriptor is an interactive terminal\n\nIn this example, we will check if the file descriptor associated with a file is an interactive terminal or not:\n\n```file = open(\"example.txt\", \"w\")\nprint(file.isatty())\nfile.close()\n```\n\nIn this example, we opened a file named “example.txt” in write mode and then checked if the file descriptor associated with the file is an interactive terminal or not. Since it is not an interactive terminal, the `isatty()` method returns False.\n\n### Example 2: Checking if the standard input is an interactive terminal\n\nIn this example, we will check if the standard input is an interactive terminal or not:\n\n```import sys\nprint(sys.stdin.isatty())\n```\n\nIn this example, we imported the `sys` module and used the `isatty()` method on the `stdin` object, which represents the standard input. Since the standard input is an interactive terminal, the `isatty()` method returns True.\n\n### Example 3: Checking if the standard output is an interactive terminal\n\nIn this example, we will check if the standard output is an interactive terminal or not:\n\n```import sys\nprint(sys.stdout.isatty())\n```\n\nIn this example, we imported the `sys` module and used the `isatty()` method on the `stdout` object, which represents the standard output. Since the standard output is an interactive terminal, the `isatty()` method returns True.\n\n## Use Cases\n\nThe `isatty()` method can be useful in situations where we want to check if the file descriptor associated with a file is an interactive terminal or not. This can be particularly useful when we want to determine whether the input or output is coming from an interactive terminal, as opposed to a file or pipe.\n\n## Conclusion\n\nThe `isatty()` method is a useful tool for checking if the file descriptor associated with a file is an interactive terminal or not. Whether we are working with standard input/output or a file, the `isatty()` method can help us determine if the input or output is coming from an interactive terminal, which can be useful in a variety of scenarios." ]

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[ "What is a mode? Regarding electromagnetic waves in a closed cavity\n\nThis concept is very unclear to me, and the more different sources I find, the more contradicting definitions and explanations I stumble upon.\n\nLet's use this source as a basis for my question: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/rayj.html\n\nThe second paragraph 'How many modes in the cavity' is what troubles me.\n\nFirst of all: here is my understanding of what a mode is, though I am almost certain that it is still incorrect.\n\nGiven a length $L$ where a string (or a wave) has nodes at both ends, you can have a standing wave at certain wavelengths. The 'standing wave' refers to the entire thing that is between $x=0$ and $x=L$, not just a single wavelength (Is this right?). With each standing wave corresponds a wavveector $\\vec k$ that has two properties: a length and a direction. The length defines the wavelength of the standing wave, the direction defines the direction of propagation. Now what defines a mode? A mode is like a category, so that every standing wave must belong to one and only one category. Two waves with a different amplitude still have the same mode. Two waves with a different wavelength (so their vectors $\\vec k$ have different length) have a different mode. Two waves with a different direction (so their vectors $\\vec k$ have a different orientation) have a different mode. Does the direction of the amplitude matter? After all, if a wave propagates in the $+x$ direction, the amplitude $\\vec E_0$ may be in the $+e_y$ direction, in the $+e_z$ direction, in the $e_y+e_z$ direction or anything else. If a change in amplitude direction or length means that the standing wave gets a different mode, that means there are infinitely many modes even when the wavelength is given.\n\nNow there are some troubles I have with the 'How many modes in the cavity' besides the above:\n\n\"..whereas the wave equation solution uses only positive definite values.\"\n\nWhy exactly can the wave equation not have a solution $(-1,-1,-1)$ for example?\n\nAnd the following:\n\n\"..Another technical problem is that you can have waves polarized in two perpendicular planes, so we must multiply by two to account for that.\"\n\nHow does polarizing give an extra mode? And given a propagation direction, I can choose infinitely many pairs of perpendicular planes that are also perpendicular to the wave direction.\n\nIf anyone could help me out on these troubles I would be very grateful.\n\n• Practical illustration: Get a big bowl or tub full of water, and make waves on the surface. You'll find that there are a limited number of different ways in which you can get the waves to resonate. Those different patterns are the \"modes,\" and they are determined by the mathematics of wave motion and, by the geometry of the container. www2.me.rochester.edu/courses/ME201/webexamp/coffee.pdf – Solomon Slow Feb 8 '18 at 18:24\n\nThe different modes of electromagnetic waves in a cavity correspond to different eigenfunctions of the solutions of the electromagnetic wave equation given the boundary conditions of the walls of the cavity. The eigenvalues correspond to the frequencies of the modes. For example, the wave equation for the electric field given as a phasor with a given frequency $\\omega$ becomes the Helmholtz equation $$\\nabla^2 \\vec E =-k^2 \\vec E$$ where $k=\\omega/c$ and $c$ is the wave phase velocity in the cavity. Together with the boundary conditions, this determines the different eigenvalues in $k$ (and thus frequencies $\\omega$) and the pertinent eigenfunctions $\\vec E(\\vec r)$ which are the \"modes\". There is also a magnetic field $\\vec B$ associated with the electric field of these eigenfunctions. Whenever you have a distinct electric and magnetic field pattern of these eigenfunctions, even sometimes at the same frequency, you can speak of a mode of the cavity. Different amplitudes of the same oscillation pattern are not different modes." ]

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[ "", null, "", null, "", null, "", null, "0\n\n# What is the surface area of sphere with radius 13ft?\n\nUpdated: 4/28/2022", null, "Wiki User\n\n9y ago\n\nThe surface area of a sphere with a radius of 13ft is about 2,123.7ft2", null, "Wiki User\n\n9y ago", null, "", null, "Study guides\n\n14 cards\n\n## Which of the following is not a step in the inquiry process\n\n➡️\nSee all cards\n3.89\n148 Reviews", null, "Earn +20 pts\nQ: What is the surface area of sphere with radius 13ft?\nSubmit\nStill have questions?", null, "", null, "Related questions\n\n### 'What is the surface area of a sphere with a radius of 5 m?\n\nThe surface area of a sphere with a radius of 5m is 314.2m2\n\n### What is the surface area of a sphere with radius 6?\n\nThe surface area of a sphere with radius 6 is: 452 square units.\n\n### What is the surface area of a sphere that has a radius of 10?\n\nThe surface area of a sphere that has a radius of 10 is: 1,260 square units.\n\n### What is the surface are of a sphere with a radius of 4.75?\n\nThe surface area of a sphere with a radius of 4.75 is about 283.5 units2\n\n### What is the surface are of a sphere with radius 2?\n\nThe surface area of a sphere with a radius of 2 is about 50.27 units2\n\n### What is the radius of a sphere that has a surface area of 1742400 squre ft?\n\nFinding the radius of a Sphere by using the surface area is a multi-step mathematical equation. The radius (r) of a sphere with a surface area of 1742400ft is 372.365ft.\n\n### Suppose the sphere below with radius r has a surface area equal to 50 units2 Find the surface area of the sphere with twice the radius?\n\nThe surface area of the sphere with the radius doubled is 200 units2.---> Confirmed\n\n### What is the surface area of sphere with radius of 12 units?\n\nThe surface area of a sphere with a radius of 12 units is about 1,809.6 units2\n\n### What is the surface area of a sphere with a radius of 15 units?\n\nThe surface area of a sphere with a radius of 15 units is 2,827 units2\n\n### What is the area of a sphere when the radius is 7?\n\nThe surface area of a sphere with a radius of 7 is about 615.8 units2\n\n### What is the total surface area of a sphere when the radius is 4?\n\nThe total surface area of a sphere when the radius is 4 equals 201.1 units2\n\na. 2 to 5." ]

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[ "Home | WAEC | NECO | NABTEB | UTME JAMB | WAEC GCE | NECO GCE| NABTEB GCE | JUPEB | JOINT EXAM | SCHOOL NEWS\n\nWAEC: OFFICIAL WASSCE 2021 TIMETABLE\n\nWAEC: How TO GET 2021 WAEC Question & Anwer Before Exam Start\n\n# NECO GCE CHEMISTRY PRACTICAL QUESTIONS AND ANSWER EXPO – NOV/DEC 2017\n\nNECO GCE Nov/Dec 2017 Chemistry practical Questions and Answer Expo", null, "1ai) Methyl Orange\nReason:- Titration of strong acid against base\n1aii) Yellow\n1aiii)Hydrochloricacid\n1aiv)2HCL(aq)+ x2Co3 -> 2 x CLaq + H20(aq) + CO2(g)\n1b) Molar mass of HcL = 1+35.5\n=36.5gmol^-1\nCon of A is moldm^-3 =\nCon of A of gdm^3/molar mass in gdm^3\n=7.30gdm^-3/36.5gmol^-1\n=0.200moldm^-3\n(1bii)\nCaVa/CbVb=na/nb\nCa=0.200mol/dm^3\nVa=26.20cm^3\nCb=?\nVb=25.0cm^3\nna=2\nnb=1\n(0.200*26.20)/(Cb*25.0)=2/1\nCb=(0.200*26.20*1)/(25.0*1)\nCb=5.24/36.5\nCb=0.2098mol/dm^3\n(1biii)\nConc. of B in gdm-3/Conc. of B in moldm-3\n=10.6gdm-3/0.148 gdm-3\n=101.15g\n=101gmol-1\n(1biv)\nRelative atomic mass of X\n2x+12+17.8=101\n2x=101-29.8\n2x=71.2\nx=35.6\n========================\n(2ai)\nINFERENCE: Sample X is a soluble salt\n(2aii)\nOBSERVATION: A gas is evolved. The gas turns red moist litmus blue.\nINFERENCE: Alkaline gas\n(2aiii)\nINFERENCE: CO3 gas is CO3^2-\n(2aiv)\nINFERENCE: The solution is alkaline\n======================\n3b)\n– Burrette\n– Reshort Hand\n– Solution/Acid" ]

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[ " SAT MATH section | Foreign Educator Teaching System", null, "", null, "# SAT MATH section\n\nSAT MATH section\n\nThere is going to be mathematics section on the SAT exam. People think that the math in the SAT exam would be arduous, but it is vice versa. SAT math mostly deals with the concepts or topics taught in High school. You will not encounter calculus in the SAT exam.\n\nYou will have to revise the basic math concepts and then learn to apply techniques and strategies which will help you solve the math problems quickly.\n\nThere are going to be a total of 3 math sections on the SAT mathematics part. 2 of the math sections will last for 25 minutes each while the last one will third one will last for 20 minutes. Do remember that an experimental section also comes up on the SAT exam, and if the experimental section is relating to math then you will get 4 sections of math.\n\nRating 3.00 out of 5\n" ]

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[ "OA Schedules\nFebruary 2012\nSundayMondayTuesdayWednesdayThursdayFridaySaturday\n29Jan30311Feb234\n 4M=Karen\n WIYN=Jenny\n 4M=Karen\n WIYN=Jenny\n 4M=Doug\n WIYN=Jenny\n 4M=Doug\n WIYN=Jenny\n 4M=Doug\n WIYN=Jenny\n 2M=Krissy\n 4M=Doug\n WIYN=Krissy\n 4M=Doug\n WIYN=Krissy\n567891011\n 4M=Doug\n WIYN=Krissy\n 2M=Cancelled\n 4M=Krissy\n WIYN=Dave\n 2M=Cancelled\n 4M=Krissy\n WIYN=Dave\n 4M=Krissy\n WIYN=Dave\n 2M=Cancelled\n 4M=Dave\n WIYN=Karen\n 4M=Dave\n WIYN=Karen\n 2M=Jenny\n 4M=Dave\n WIYN=Karen\n12131415161718\n 4M=Karen\n WIYN=Jenny\n 4M=Karen\n WIYN=Jenny\n 2M=Doug\n 4M=Karen\n WIYN=Jenny\n 4M=Jenny\n WIYN=Doug\n 4M=Jenny\n WIYN=Doug\n 4M=Jenny\n WIYN=Doug\n 2M=Krissy\n 4M=Doug\n WIYN=Krissy\n19202122232425\n 4M=Doug\n WIYN=Krissy\n 4M=Doug\n WIYN=Krissy\n 2M=Dave\n 4M=Krissy\n WIYN=Dave\n 4M=Krissy\n WIYN=Dave\n 4M=Krissy\n WIYN=Dave\n 4M=Dave\n WIYN=Karen\n HOLIDAY\n 4M=Dave\n WIYN=Karen\n262728291Mar23\n 4M=Dave\n WIYN=Karen\n 4M=Karen\n WIYN=Jenny\n 2M=Doug\n 4M=Karen\n WIYN=Jenny\n 4M=Karen\n WIYN=Jenny\n 4M=Jenny\n WIYN=Doug\n 4M=Jenny\n WIYN=Doug\n 4M=Jenny\n WIYN=Doug\nDisplay:\n Day Week Month Quarter Year Block List Condensed Time Plan" ]

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[ "", null, ". .\n.\nNewton's Rings-Refractive index of liquid\n.\n.\n\n## Performing Real Lab\n\nAfter experimental arrangement, the glass plate is inclined at an angle 45° to the horizontal. One or two drops og the given liquid is placed between the glass plate and lens. This glass plate reflects light from the source vertically downloads and falls normally on the convex lens. Newton's rings are seen using a long focus microscope, focussed on the air fil. The cross-wire of the microscope is made tangential to the 20th ring on the left side of the centre. The readings of the main scale and vernier scale of the microscope are noted. The cross wire is adjusted to be tangential to the 18th, 16th, 14th,  etc on the left and 2nd, 4th, 6th, etc on the right and readings are taken each time. From this the diameter of the ring is found out which is the difference between the readings on the left and right sides. The square of the diameter and hence Dn2 and D2n+m are found out.\n\n## Performing the simulation:\n\nThe simulation virtualizes the Newton’s rings experiment. The user can view the effect of Newton’s rings formed when the medium changes. Select any one type of medium. Different ring pattern can be seen by changing the radius of curvature of the lens and wavelength of light source.\n\n## Components:\n\nStart button, Light source, Filter, Microscope, Lens, Medium and Glass plate.\n\n## Help:\n\n### Variable region:\n\n1. Choose Medium Combo box helps you to choose the type of medium that the simulation have to perform.\n2. Radius Slider helps to change the radius of curvature of lens.\n3. The wavelength slider helps to change the wavelength of light used.\n\n### Measurement region:\n\n1. The start button will help to play the simulation.\n2. The variation in the rings can be seen when the medium,wavelength of light or the radius of the lens changes.\n\n## Procedure:\n\n1. Click on the \"light on\" button.\n2. Select the lens of desirable radius.\n3. Adjust the microscope position to view the Newton rings.\n4. Focus the microscope to view the rings clearly.\n5. Fix the cross-wire on 20th ring either from right or left  of the centre dark ring and take the readings .\n6. Move the crosswire and take the reading of 18th,16th...........2nd ring.\n7. You have to take the reading of rings on either side of the centre dark ring.\n8. Enter the readings in the tabular column.\n9. Calculate the refractive index of the medium by using the given formula .\n\n## Observations:\n\nTo find Least Count\n\nOne main scale division                     = ............... cm\n\nNumber of divisions on Vernier          = ...............\n\nL.C                                                     = One main scale division/ Number of division on vernier=.................................\n\n Order of ring Microscopic Reading  (cm) Diameter D(cm) D2(cm2) D'2m+p - D'2m   (cm2) Left Right\n\n## Calculation:\n\nMean value of D'2m+p - D'2m      =.......cm2\nrefractive index of given liquid,µ=D2m+p-D2m/D'2m+p-D'2m\n\n=.............................\n\n## Result:\n\nThe refractive index of the given liquid is found to be  = ..........\n\nCite this Simulator:\n\n.....\n..... .....", null, "Copyright @ 2020 Under the NME ICT initiative of MHRD", null, "" ]

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[ "Solutions by everydaycalculation.com\n\n## Reduce 249/167 to lowest terms\n\n249/167 is already in the simplest form. It can be written as 1.491018 in decimal form (rounded to 6 decimal places).\n\n#### Steps to simplifying fractions\n\n1. Find the GCD (or HCF) of numerator and denominator\nGCD of 249 and 167 is 1\n2. Divide both the numerator and denominator by the GCD\n249 ÷ 1/167 ÷ 1\n3. Reduced fraction: 249/167\nTherefore, 249/167 simplified to lowest terms is 249/167.\n\nMathStep (Works offline)", 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 ]

[ "AC\n\n2237ms\n\n#### Code [DL]\n\n```IFS= read a;while IFS= read b;do\necho \"\\$a\";f=\"\\$b\";g=\"\\$f\"\ndone\nc=`sed -r 's/T.+/T/'<<<\"\\$a\"`\nd=`sed -r 's/K.+/K/'<<<\"\\$g\"`\nprintf \"\\$d\"\nprintf \"*%.s\" `eval echo {1..\\$[\\${#c}-\\${#d}]}````\n\n#### stdin\n\n``` T\n\nK\n```\n\n#### stdout\n\n``` T\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nT\nK******```" ]

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