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# number of primes of the form |$n^2 - 6n + 5$|?

How can I find the number of primes of the form $|n^2 - 6n + 5|$ where $n$ is an integer? Through trial and error, I have found $n = 6$ (this one is obvious), and $2$. Are there any more, and what is a smarter way to solve this question?

• I think $|(n-1)(n-5)|$ is prime if either $|n-1| = 1$ or $|n-5| = 1$. – GAVD Aug 19 '15 at 6:55
• I would say that the comment by Andre Nicolas is your answer. One of the factors must equal 1, and the other factor must be a prime. – DanielWainfleet Aug 19 '15 at 7:07

The first thing I did when I saw your question was compute about a dozen values for small $n$ and look them up in the OEIS, which gave me A028347 and the formula $n^2 - 4$. Clearly for $n^2 - 4$ to be prime you need $n$ to be odd. Slightly less obvious, you need for $n$ to be a multiple of 3, because otherwise $n^2 \equiv 1 \pmod 3$. These conditions are necessary but not sufficient.
Digging a little further, we find that $n^2 - 4 = (n - 2)(n + 2)$: this is clearest when the formula gives you semiprimes with prime factors 4 apart, e.g., $19 \times 23$, $37 \times 41$, $43 \times 47$.
• Do note that the equation $n^2 - 6n + 5$ has a different "offset" (to use OEIS parlance) from $n^2 - 4$. In fact, to properly make the latter exactly equivalent to the former you need to change it to $(n - 3)^2 - 4$. Very minor detail, though. – John-Luke Aug 19 '15 at 21:59


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Physics Help Forum Time of cars lapping on second/third lap

 Kinematics and Dynamics Kinematics and Dynamics Physics Help Forum

 Sep 20th 2018, 01:11 PM #1 Junior Member   Join Date: Sep 2018 Posts: 1 Time of cars lapping on second/third lap Hi guys, I've got a problem involving a track and two cars. The problem appears to be a fairly simple simultaneous equation but actually turns out to be a bit more complex. Two toy cars start from the same position and race around a track. Both cars accelerate for 3 seconds. Car A accelerates at a rate of 2m/s/s while car B accelerates at a rate of 3.2m/s/s. The radius of the track is 2.5m. At what time does car b overtake car A. The problem in encountering in the maths is that car B completes a lap before overtaking and therefor its displacement resets to 0 while car A continues about its journey. Anyone have any ideas?
 Sep 21st 2018, 02:09 PM #2 Physics Team     Join Date: Jun 2010 Location: Morristown, NJ USA Posts: 2,312 The trck is to realize that the total distance travelled by car B must be $2 \pi R = 15.71$ meters further than the distance traveled by car A. B’s position at t=3 seconds is 14.4 m, and A’s is 9 m. So B has is 5.4 m ahead of A, and must make up an additional 15.71 - 5.4 = 10.31 meters in order to lap A, while traveling at a speed that is 3.6 m/s faster. Can you take it from here? oz93666 likes this.

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Browse Questions

Arrange the following compounds in order of increasing dipole moment? (I) Toluene (II) m-dichlorobenzene (III) o-dichlorobenzene (IV) P-dichlorobenzene

$(a)\; I < IV < II < III \\ (b)\;IV < I < II < III \\(c)\;IV < I < III < II \\(d)\;IV < II < I < III$

Can you answer this question?

$O^{-},m^{-},P^{-}$ derivatives has $\alpha=60^{\circ},120^{\circ}$ and $180^{\circ}$ respectively , p-derivative has zero dipole moment.
Dipole moment of $m-$ chlorobenzene is more than foluene.
Hence b is the correct answer.
answered Apr 1, 2014 by


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## Abstract and Applied Analysis

### An Iterative Algorithm for the Least Squares Generalized Reflexive Solutions of the Matrix Equations $AXB=E,$$CXD=F$

#### Abstract

The generalized coupled Sylvester systems play a fundamental role in wide applications in several areas, such as stability theory, control theory, perturbation analysis, and some other fields of pure and applied mathematics. The iterative method is an important way to solve the generalized coupled Sylvester systems. In this paper, an iterative algorithm is constructed to solve the minimum Frobenius norm residual problem: $‖(\begin{smallmatrix}AXB\\[5pt] CXD\end{smallmatrix})-(\begin{smallmatrix}E\\[5pt] F\end{smallmatrix})‖=$ min over generalized reflexive matrix $X$. For any initial generalized reflexive matrix ${X}_{1}$, by the iterative algorithm, the generalized reflexive solution ${X}^{\ast}$ can be obtained within finite iterative steps in the absence of round-off errors, and the unique least-norm generalized reflexive solution ${X}^{\ast}$ can also be derived when an appropriate initial iterative matrix is chosen. Furthermore, the unique optimal approximate solution $\stackrel{^}{X}$ to a given matrix ${X}_{0}$ in Frobenius norm can be derived by finding the least-norm generalized reflexive solution ${\stackrel{~}{X}}^{\ast}$ of a new corresponding minimum Frobenius norm residual problem: $\text{min}‖(\begin{smallmatrix}A\stackrel{~}{X}B\\[5pt] C\stackrel{~}{X}D\end{smallmatrix})-(\begin{smallmatrix}\stackrel{~}{E}\\[5pt] \stackrel{~}{F}\end{smallmatrix})‖$ with $\stackrel{~}{E}=E-A{X}_{0}B$, $\stackrel{~}{F}=F-C{X}_{0}D$. Finally, several numerical examples are given to illustrate that our iterative algorithm is effective.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 857284, 18 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1355495658

Digital Object Identifier
doi:10.1155/2012/857284

Mathematical Reviews number (MathSciNet)
MR2898044

Zentralblatt MATH identifier
1242.65085

#### Citation

Yin, Feng; Huang, Guang-Xin. An Iterative Algorithm for the Least Squares Generalized Reflexive Solutions of the Matrix Equations $AXB=E,$ $CXD=F$. Abstr. Appl. Anal. 2012 (2012), Article ID 857284, 18 pages. doi:10.1155/2012/857284. https://projecteuclid.org/euclid.aaa/1355495658

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# Transpose Matrix 4x4

For example, consider the following 3 X 2 matrix: 1 2 3 4 5 6 Transpose of the matrix: 1 3 5 2 4 6 When we transpose a matrix, its order changes, but for a. We know that there is a difference between a bra vector and a ket vector. This becomes explicit in the matrix representation. transpose() function. 136 461 просмотр. If the 3x3 sub-matrix is not invertible, this function returns Returns this matrix, transposed about its diagonal. This calculator solves system of four equations with four unknowns. The transpose is found by exchanging the rows with columns and columns with rows. We employ the latter, here. Since a 3D point only needs three values (x, y, and z), and the transformation matrix is a 4x4 value matrix, we need to add a fourth dimension to the point. Subdividing the 4x4 matrix into four 2x2 matrices. Fastest way to transpose 4x4 byte matrix. The equation "C = A” should appear on your screen with the inverted L cursor around the A as shown in figure 6A. The transpose of a matrix is simply a flipped version of the original matrix. If most of your matrices are used as transform matrices, because of their special property, we have a fast route for calculating their inverse. Transposes the rows and columns of a matrix. The transpose of matrix A is written A T. TRS(newVector3(6, 7, 8), Quaternion. The elements of the matrix are asked to enter and stored in the matrix ‘A’. As soon as it is changed into the reduced row echelon form the use of it in linear algebra is much easier and can be really convenient for mostly mathematicians. Here you can calculate a matrix transpose with complex numbers online for free. Product of Matrix and it's Transpose. Example, = -5 and. Active 7 years, 3 months ago. Transpose matrices 4x4. Features you might already know about matrices, such as squareness and symmetry, affect the transposition results in If you can't visualize this, draw a 4x4 matrix on a piece of paper. Автор: The Organic Chemistry Tutor. For example if you transpose a 'n' x 'm' size matrix you'll get a new one of 'm' x 'n' dimension. EXAMPLE 1 Finding the Conjugate Transpose of a Complex Matrix Determine A*for the matrix A 5 3 3 1. The Rref calculator is used to transform any matrix into the reduced row echelon form. Check that this is true for and. Matrix A has three rows and two columns, so its transpose Aᵀ will have a different shape: two rows and three columns. The transpose of a matrix is a new matrix that is obtained by exchanging the rows and columns. Transpose (const vtkMatrix4x4 *in, vtkMatrix4x4 *out). To find the conjugate trans-pose of a matrix, we first calculate the complex conjugate of each entry and then take the transpose of the matrix, as shown in the following example. Free matrix transpose calculator - calculate matrix transpose step-by-step. The Matrix, Inverse. Now fold is over the main diagonal. The equation "C = A” should appear on your screen with the inverted L cursor around the A as shown in figure 6A. The original matrix and the transpose are both printed. I've been doing some 3D graphics in OpenGL lately and I needed a way to work with 4x4 matrices. Their values should be less than The program below then computes the transpose of the matrix and prints it on the screen. Interface 4x4 Matrix Keypad With Microcontroller. •Compute the cofactor matrix by alternating + and –signs. Obviously unitary matrices (), Hermitian matrices (), and skew-Hermitian matices () are all normal. You just need to know what the transpose of a matrix looks like. denoted by , is the matrix with element , ,. An easy and fast tool to find the eigenvalues of a square matrix. Works with matrix from 2X2 to 10X10. The algorithm of matrix transpose is pretty simple. For example if you transpose a 'n' x 'm' size matrix you'll get a new one of 'm' x 'n' dimension. In this post, we’re going to discuss an algorithm for Matrix multiplication along with its flowchart, that can be used to write programming code for matrix multiplication in any high level language. The transpose of matrix A is written A T. Transpose the matrix and put it into out. Transpose of a matrix is obtained by changing rows to columns and columns to rows. The normal matrix is the transpose of the inverse of the top-left 3x3 part of this 4x4 matrix. Rows become columns and vice versa!. For matrices there is no such thing as division, you can multiply but can't divide. transpose(), you can not only transpose a 2D array (matrix) but also rearrange the axes of a multidimensional array in any order. Since a 3D point only needs three values (x, y, and z), and the transformation matrix is a 4x4 value matrix, we need to add a fourth dimension to the point. The computational time for transpose of a matrix using identity matrix as reference matrix is O(m*n). To understand transpose calculation better input any example and examine the solution. This tool transposes matrices. transpose() and numpy. •Compute the cofactor matrix by alternating + and –signs. Determinant of a 4×4 matrix is a unique number that is also calculated using a particular formula. Check that this is true for and. How do i go about it?. What does multiplying by the identity matrix look like? The easiest example is to multiply a single point by the identity matrix. In other words, for bytes A-P, I'm looking for the most efficient (in Each row of my matrix can fit into a 32-bit integer, and I'm looking for answers that can perform a transpose quickly using general purpose hardware. I have a 4x4 block of bytes. Therefore, m n must be the same as n m, and so we can conclude that m = n. m[0] contains the first element of the first row and m[1] contains the second element of the first row. The transpose of a matrix is the matrix flipped over its main diagonal, switching the matrix's row and column indices. Move the mouse cursor to the transpose button ( ) on the matrix toolbar and click it. C Program to find transpose of a matrix using function /* This is a sample C program which will ask the user for a 4X4 matrix, */ /* call a function to compute it's transpose, and output the result. To find the conjugate trans-pose of a matrix, we first calculate the complex conjugate of each entry and then take the transpose of the matrix, as shown in the following example. transpose() function. Transpose an arbitrarily sized rectangular Matrix. We can transpose a matrix by switching its rows with its columns. For example, consider the following 3 X 2 matrix: 1 2 3 4 5 6 Transpose of the matrix: 1 3 5 2 4 6 When we transpose a matrix, its order changes, but for a. Similarly, the rank of a matrix A is denoted by rank(A). int m, n, c, d. Автор: The Organic Chemistry Tutor. Interface 4x4 Matrix Keypad With Microcontroller. 5 comments, last by st0ff 4. Since a 3D point only needs three values (x, y, and z), and the transformation matrix is a 4x4 value matrix, we need to add a fourth dimension to the point. I need help with transposing a matrix. Circular Matrix (Construct a matrix with numbers 1 to m*n in spiral way) Count frequency of k in a matrix of size n where matrix (i, j) = i+j. For example if you transpose a 'n' x 'm' size matrix you'll get a new one of 'm' x 'n' dimension. transpose() and numpy. Matrix transposes are a neat tool for understanding the structure of matrices. Transposes the rows and columns of a matrix. Finally, the program must print the product of the generated matrix and it's transpose. It makes the lives of people who use matrices easier. The transpose of a matrix is simply a flipped version of the original matrix. transpose() and numpy. The elements of the matrix are asked to enter and stored in the matrix ‘A’. Related Symbolab blog posts. You just need to know what the transpose of a matrix looks like. I have a 4x4 block of bytes that I'd like to transpose using general purpose hardware. Find the matrix in reduced row echelon form that is row equivalent to the given m x n matrix A. : for transposing a Matrix, there is an intrinsics-macro, producing slightly different code inside "xmmintrin. Donate: https://www. An easy and fast tool to find the eigenvalues of a square matrix. one); produce a Matrix which represent: - scale 1, rotation none Unity uses a transposed matrix, so let's say that columns are x and rows are y. A matrix is said to be singular if its determinant is zero and non-singular. Learn about determinant of 4x4 matrix topic of maths in details explained by subject experts on vedantu. Rref Calculator for the problem solvers. A matrix with elements that are the cofactors , term-by-term, of a given square matrix. to multiply matrices doesn't require that I transpose one of them first. This matrix transpose calculator help you to find transpose matrix. Free matrix transpose calculator - calculate matrix transpose step-by-step This website uses cookies to ensure you get the best experience. For a square matrix, the trace of a matrix is the sum of the elements on the main diagonal, which is equal to the sum of all its eigenvalues with multiplicities. Press the "C", ":" and, the "A" keys. It is skew-symmetric matrix because. Learn about determinant of 4x4 matrix topic of maths in details explained by subject experts on vedantu. 136 461 просмотр. •Divide all elements in the adjugate matrix by determinant of matrix. A faster way to obtain the adjoint, adjugate or adjunct of the transpose of a 4x4 (4 by 4). Note that if A is a matrix with real entries, then A*. SSE 4x4 Matrix transpose and invert General and Gameplay Programming Programming. But I can't figure out how to do it using determinant 4x4 matrix with. Trick in using Casio fx-570ES calculator. Get step-by-step solutions. Matrix transpose 4x4, 3x4 Matrix invert 3x3 For speed and to make use of the limited stack space on board, these functions are all inlined and written as specifically as possible. This tool transposes matrices. With ndarray. Of course I use the transpose in lisp to multiply matrices and my VBA code. Using this online calculator, you will receive a detailed step-by-step solution to your problem, which will help you understand the algorithm how to find the transpose matrix. 'Multiply two 4x4 matrices. A square matrix is normal if it commutes with its conjugate transpose:. As soon as it is changed into the reduced row echelon form the use of it in linear algebra is much easier and can be really convenient for mostly mathematicians. How to find the transpose of a 4x4 matrix programing? 1 See answer cloneemperor9965 is waiting for your help. This example transposes a 3 by 3 matrix. $n\times n$ Transpose Matrix calculator calculates a transpose matrix of a matrix $A$ with real elements. Cofactor Matrix. Started by Shnoutz March 16, 2012 04:16 PM. one); produce a Matrix which represent: - scale 1, rotation none Unity uses a transposed matrix, so let's say that columns are x and rows are y. Note that all the main diagonal elements in skew-symmetric matrix are zero. This tool transposes matrices. If the 3x3 sub-matrix is not invertible, this function returns Returns this matrix, transposed about its diagonal. In this program, the user is asked to enter the number of rows r and columns c. (Recall that for square matrices, the shape doesn't change, say. i want to transpose a 4x4 matrix using sse2 intrinsics. The transpose is found by exchanging the rows with columns and columns with rows. that is because pdls are indexed by (col,row), and matrices are (row,column) by convention, so a 1-D pdl corresponds to a row vector, not a column vector. The transpose of a matrix is simply a flipped version of the original matrix. For a square matrix, the trace of a matrix is the sum of the elements on the main diagonal, which is equal to the sum of all its eigenvalues with multiplicities. For example, consider the following 3 X 2 matrix: 1 2 3 4 5 6 Transpose of the matrix: 1 3 5 2 4 6 When we transpose a matrix, its order changes, but for a. I have a 4x4 block of bytes. Matrix transposes are a neat tool for understanding the structure of matrices. Features you might already know about matrices, such as squareness and symmetry, affect the transposition results in If you can't visualize this, draw a 4x4 matrix on a piece of paper. In this post, we’re going to discuss an algorithm for Matrix multiplication along with its flowchart, that can be used to write programming code for matrix multiplication in any high level language. transpose() function. The elements of the matrix are asked to enter and stored in the matrix ‘A’. By using this website, you agree to our Cookie Policy. If is real, then. I need help with transposing a matrix. But I can't figure out how to do it using determinant 4x4 matrix with. Matrix calculator. Transpose matrices 4x4. Автор: The Organic Chemistry Tutor. Dimension also changes to the opposite. The transpose of a matrix. For example if you transpose a 'n' x 'm' size matrix you'll get a new one of 'm' x 'n' dimension. The transpose of a transpose is the initial matrix. The Hermitian Conjugate matrix is the (complex) conjugate transpose. The transpose of a matrix is a new matrix that is obtained by exchanging the rows and columns. The transpose is found by exchanging the rows with columns and columns with rows. In this post, we’re going to discuss an algorithm for Matrix multiplication along with its flowchart, that can be used to write programming code for matrix multiplication in any high level language. To find the conjugate trans-pose of a matrix, we first calculate the complex conjugate of each entry and then take the transpose of the matrix, as shown in the following example. A superscript T denotes the matrix transpose operation; for example, AT denotes the transpose of A. My implementation supports the following operations: Matrix-matrix addition. 8 and a 4x4 matrix as an array with 16 elements indexed from 0. If the 3x3 sub-matrix is not invertible, this function returns Returns this matrix, transposed about its diagonal. As soon as it is changed into the reduced row echelon form the use of it in linear algebra is much easier and can be really convenient for mostly mathematicians. Transpose an arbitrarily sized rectangular Matrix. len] * matrix[0]. Works with matrix from 2X2 to 10X10. To transpose NumPy array ndarray (swap rows and columns), use the T attribute (. 3x3 System of. Specifically, I will optimize a matrix transpose to show how to use shared memory to reorder strided global memory accesses into coalesced accesses. Public Function MxM (M1 () As Double, M2 () As Double) As Double () Dim Mat (0 To 3, 0 To 3) As Double. The inverse matrix has the property that it is equal to the product of the reciprocal of the determinant and the adjugate matrix. There are mainly two ways to obtain the inverse matrix. C Program to find transpose of a matrix using function /* This is a sample C program which will ask the user for a 4X4 matrix, */ /* call a function to compute it's transpose, and output the result. void QMatrix4x4::viewport(float left, float bottom, float width, float height, float nearPlane = 0. len) L(col) (0. i want to transpose a 4x4 matrix using sse2 intrinsics. The algorithm of matrix transpose is pretty simple. The normal matrix is the transpose of the inverse of the top-left 3x3 part of this 4x4 matrix. My implementation supports the following operations: Matrix-matrix addition. An easy and fast tool to find the eigenvalues of a square matrix. How do i go about it?. For example if you transpose a 'n' x 'm' size matrix you'll get a new one of 'm' x 'n' dimension. The elements of the matrix are asked to enter and stored in the matrix ‘A’. To obtain it, we interchange rows and columns of the matrix. The user is asked to enter the number of rows and columns of the matrix. To transpose NumPy array ndarray (swap rows and columns), use the T attribute (. Of course I use the transpose in lisp to multiply matrices and my VBA code. We employ the latter, here. Fastest way to transpose 4x4 byte matrix. F transpose(&matrix) V toRet = [[0] * matrix. Автор: The Organic Chemistry Tutor. This means that A is n n, which means that A is a square matrix. Leave feedback. Its swaps columns with rows and rows with columns. The normal matrix is the transpose of the inverse of the top-left 3x3 part of this 4x4 matrix. The program must accept two integers M and N as the input. For a square matrix, the trace of a matrix is the sum of the elements on the main diagonal, which is equal to the sum of all its eigenvalues with multiplicities. SPECIFY MATRIX DIMENSIONS Please select the size of the matrix from the popup menus, then click on the "Submit" button. Matrix calculator. Now fold is over the main diagonal. A matrix is said to be singular if its determinant is zero and non-singular. i want to transpose a 4x4 matrix using sse2 intrinsics. Dimension also changes to the opposite. Specifically, I will optimize a matrix transpose to show how to use shared memory to reorder strided global memory accesses into coalesced accesses. Therefore, m n must be the same as n m, and so we can conclude that m = n. Normal matrix. Let A be an m n matrix. To understand transpose calculation better input any example and examine the solution. Ask Question Asked 7 years, 3 months ago. Matrix4x4 -> System. For a square matrix, the trace of a matrix is the sum of the elements on the main diagonal, which is equal to the sum of all its eigenvalues with multiplicities. Similarly, if A has an inverse it will be denoted by A-1. using UnityEngine. Works with matrix from 2X2 to 10X10. An easy and fast tool to find the eigenvalues of a square matrix. Determinant of a 4×4 matrix is a unique number that is also calculated using a particular formula. Transpose (const vtkMatrix4x4 *in, vtkMatrix4x4 *out). We know that there is a difference between a bra vector and a ket vector. Trick in using Casio fx-570ES calculator. Program Explanation. Started by Shnoutz March 16, 2012 04:16 PM. A square matrix is normal if it commutes with its conjugate transpose:. The program must generate a 4x4 integer matrix by using the following conditions. The transpose of a transpose is the initial matrix. A faster way to obtain the adjoint, adjugate or adjunct of the transpose of a 4x4 (4 by 4). •Divide all elements in the adjugate matrix by determinant of matrix. The user is asked to enter the number of rows and columns of the matrix. Definition at line 38 of file vtkMatrix4x4. The Matrix, Inverse. The transpose of a matrix is simply a flipped version of the original matrix. Matrix-matrix multiplication. T), the ndarray method transpose() and the numpy. For matrices there is no such thing as division, you can multiply but can't divide. By definition, transposing means exchanging rows with columns. But I can't figure out how to do it using determinant 4x4 matrix with. Solve matrix problems for free with Open Omnia. To obtain it, we interchange rows and columns of the matrix. To find the conjugate trans-pose of a matrix, we first calculate the complex conjugate of each entry and then take the transpose of the matrix, as shown in the following example. Matrix-matrix subtraction. Note that all the main diagonal elements in skew-symmetric matrix are zero. Transposition is an operation on a matrix which swaps its columns and rows across its diagonal. Therefore, m n must be the same as n m, and so we can conclude that m = n. Transpose an arbitrarily sized rectangular Matrix. SPECIFY MATRIX DIMENSIONS Please select the size of the matrix from the popup menus, then click on the "Submit" button. transpose() and numpy. The code we wish to optimize is a transpose of a matrix of single precision values that operates out-of-place, i. Matrix calculator. We know that there is a difference between a bra vector and a ket vector. Donate: https://www. 3x3 System of. An identity matrix will be denoted by I, and 0 will denote a null matrix. Similarly, the rank of a matrix A is denoted by rank(A). Inverse of a 3 x 3 matrix (using cofactor matrix) •Calculating the inverse of a 3×3matrix is: •Compute the matrix of minors for A. - The successive elements are obtained by adding N to the current element. In other words, for bytes A-P, I'm looking for the most efficient (in Each row of my matrix can fit into a 32-bit integer, and I'm looking for answers that can perform a transpose quickly using general purpose hardware. Matrix of Cofactors. The transpose of a transpose is the initial matrix. vtkMatrix4x4 is a class to represent and manipulate 4x4 matrices. This calculator solves system of four equations with four unknowns. The inverse matrix has the property that it is equal to the product of the reciprocal of the determinant and the adjugate matrix. However, matrix multiplication does not change (it is still Arow dot. If most of your matrices are used as transform matrices, because of their special property, we have a fast route for calculating their inverse. To obtain it, we interchange rows and columns of the matrix. Interface 4x4 Matrix Keypad With Microcontroller. Therefore, m n must be the same as n m, and so we can conclude that m = n. Ask Question Asked 7 years, 3 months ago. Transformation matrix creation. Here you can calculate a matrix transpose with complex numbers online for free. The equation "C = A” should appear on your screen with the inverted L cursor around the A as shown in figure 6A. The transposed matrix is the one that has the Matrix4x4's columns exchanged with its rows. Multiplication of matrices is a very popular tutorial generally included in Arrays of C Programming. where is a diagonal matrix, i. As soon as it is changed into the reduced row echelon form the use of it in linear algebra is much easier and can be really convenient for mostly mathematicians. In this program, the user is asked to enter the number of rows r and columns c. How do i go about it?. Leave feedback. Active 7 years, 3 months ago. Specifically, I will optimize a matrix transpose to show how to use shared memory to reorder strided global memory accesses into coalesced accesses. To understand transpose calculation better input any example and examine the solution. QMatrix4x4 transposed() const; 98. Since a 3D point only needs three values (x, y, and z), and the transformation matrix is a 4x4 value matrix, we need to add a fourth dimension to the point. Therefore, m n must be the same as n m, and so we can conclude that m = n. Register free for online tutoring session to clear your doubts. Free matrix transpose calculator - calculate matrix transpose step-by-step. Started by Shnoutz March 16, 2012 04:16 PM. m[0] contains the first element of the first row and m[1] contains the second element of the first row. It is easy enough to do by hand. Public Shared Function Transpose. Specifically, it is designed to work on 4x4 transformation matrices found in 3D rendering using homogeneous coordinates [x y z w]. void QMatrix4x4::viewport(float left, float bottom, float width, float height, float nearPlane = 0. For a square matrix, the trace of a matrix is the sum of the elements on the main diagonal, which is equal to the sum of all its eigenvalues with multiplicities. Create a new matrix “C” that is the transpose of matrix “A”. This calculator solves system of four equations with four unknowns. public: static System::Numerics::Matrix4x4 Transpose(System::Numerics::Matrix4x4 static member Transpose : System. In this post we will discuss logic and interface of a matrix keypad (4x4 for this post) with microcontroller to reduce the number of port pins required to read a certain number of inputs (digital). A matrix with elements that are the cofactors , term-by-term, of a given square matrix. Transpose of a matrix is obtained by changing rows to columns and columns to rows. If and then, the dot product is. 4x4 system of equations solver. Similarly, the rank of a matrix A is denoted by rank(A). , all its off diagonal elements are 0. Donate: https://www. It is skew-symmetric matrix because. vtkMatrix4x4 is a class to represent and manipulate 4x4 matrices. for all i and j. Matrix transpose 4x4, 3x4 Matrix invert 3x3 For speed and to make use of the limited stack space on board, these functions are all inlined and written as specifically as possible. Their values should be less than The program below then computes the transpose of the matrix and prints it on the screen. If you want to process a matrix, you must hand in the transpose of the matrix, and then transpose the output when you get it back. Автор: The Organic Chemistry Tutor. We can transpose a matrix by switching its rows with its columns. Matrix transposer examples Click to use. Since a 3D point only needs three values (x, y, and z), and the transformation matrix is a 4x4 value matrix, we need to add a fourth dimension to the point. len] * matrix[0]. Suppose, if the given matrix is a square matrix, the running time will be O(n2). Here you can calculate a matrix transpose with complex numbers online for free. Specifically, it is designed to work on 4x4 transformation matrices found in 3D rendering using homogeneous coordinates [x y z w]. Similarly, if A has an inverse it will be denoted by A-1. Similarly, the rank of a matrix A is denoted by rank(A). The Matrix, Inverse. Note that if A is a matrix with real entries, then A*. In this program, the user is asked to enter the number of rows r and columns c. Lets take an example of matrix. In other words, we can say that matrix A is said to be skew-symmetric if transpose of matrix A is equal to negative of Matrix A i. Definition at line 38 of file vtkMatrix4x4. The Rref calculator is used to transform any matrix into the reduced row echelon form. But I can't figure out how to do it using determinant 4x4 matrix with. int m, n, c, d. Matrix-matrix subtraction. len) L(col) (0. for all i and j. Similarly, the rank of a matrix A is denoted by rank(A). Normally a matrix is stored in row major order, i. SPECIFY MATRIX DIMENSIONS Please select the size of the matrix from the popup menus, then click on the "Submit" button. Similarly, we can describe a matrix in row-wise fashion: given vectors in , we can define the matrix with the transposed vectors as rows:. plex matrix. Move the mouse cursor to the transpose button ( ) on the matrix toolbar and click it. Learn about determinant of 4x4 matrix topic of maths in details explained by subject experts on vedantu. If and then, the dot product is. Dimension also changes to the opposite. The elements of the matrix are asked to enter and stored in the matrix ‘A’. Conceptually, the determinant may seem the most efficient way to determine if a matrix is nonsingular. If most of your matrices are used as transform matrices, because of their special property, we have a fast route for calculating their inverse. Determinant of a 4×4 matrix is a unique number that is also calculated using a particular formula. Then, AT is by de nition an n m matrix, and therefore TAT is n m as well. len] * matrix[0]. Therefore, m n must be the same as n m, and so we can conclude that m = n. vtkMatrix4x4 is a class to represent and manipulate 4x4 matrices. where is a diagonal matrix, i. 3x3 System of. Its inverse form is basically transpose the 3x3 rotation matrix, and rescale it, and change translation part by doing dot product with 3 rescaled axes. Register free for online tutoring session to clear your doubts. Features you might already know about matrices, such as squareness and symmetry, affect the transposition results in If you can't visualize this, draw a 4x4 matrix on a piece of paper. The computational time for transpose of a matrix using identity matrix as reference matrix is O(m*n). int m, n, c, d. How to find the transpose of a 4x4 matrix programing? 1 See answer cloneemperor9965 is waiting for your help. For example if you transpose a 'n' x 'm' size matrix you'll get a new one of 'm' x 'n' dimension. 4x4 system of equations solver. friend Q_GUI_EXPORT QMatrix4x4 operator/(const QMatrix4x4& matrix, float divisor); 127. - The successive elements are obtained by adding N to the current element. Transpose of a matrix is obtained by changing rows to columns and columns to rows. To find the conjugate trans-pose of a matrix, we first calculate the complex conjugate of each entry and then take the transpose of the matrix, as shown in the following example. Let's simplify the matrix by making some definitions. To obtain it, we interchange rows and columns of the matrix. Dim I As Byte, j As Byte, k As Byte. Then, AT is by de nition an n m matrix, and therefore TAT is n m as well. The elements of the matrix are asked to enter and stored in the matrix ‘A’. In other words, for bytes A-P, I'm looking for the most efficient (in Each row of my matrix can fit into a 32-bit integer, and I'm looking for answers that can perform a transpose quickly using general purpose hardware. The transpose of a matrix is simply a flipped version of the original matrix. Matrix transposer examples Click to use. The determinant of A will be denoted by either jAj or det(A). In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix A by producing another matrix. How to find the transpose of a 4x4 matrix programing? 1 See answer cloneemperor9965 is waiting for your help. where ~A A ~ is the adjugate matrix of A. where is a diagonal matrix, i. Works with matrix from 2X2 to 10X10. len) L(col) (0. Its swaps columns with rows and rows with columns. It is skew-symmetric matrix because. •Compute the cofactor matrix by alternating + and –signs. $n\times n$ Transpose Matrix calculator calculates a transpose matrix of a matrix $A$ with real elements. For a square matrix, the trace of a matrix is the sum of the elements on the main diagonal, which is equal to the sum of all its eigenvalues with multiplicities. I already did the transpose matrix using 2 arrays. , all its off diagonal elements are 0. Here you can calculate a matrix transpose with complex numbers online for free. Their values should be less than The program below then computes the transpose of the matrix and prints it on the screen. A matrix with elements that are the cofactors , term-by-term, of a given square matrix. Register free for online tutoring session to clear your doubts. The transpose of a transpose is the initial matrix. T), the ndarray method transpose() and the numpy. plex matrix. The transpose of a matrix is simply a flipped version of the original matrix. Public Shared Function Transpose. Dimension also changes to the opposite. TRS(newVector3(6, 7, 8), Quaternion. C Program to find transpose of a matrix using function /* This is a sample C program which will ask the user for a 4X4 matrix, */ /* call a function to compute it's transpose, and output the result. Need to show: A is a square matrix. Matrix calculator. using UnityEngine. where is a diagonal matrix, i. The Rref calculator is used to transform any matrix into the reduced row echelon form. If you want to process a matrix, you must hand in the transpose of the matrix, and then transpose the output when you get it back. Cramer's Rule - 2x2 Linear System. The transpose of a matrix. Started by Shnoutz March 16, 2012 04:16 PM. Public Function MxM (M1 () As Double, M2 () As Double) As Double () Dim Mat (0 To 3, 0 To 3) As Double. Check that this is true for and. Specifically, I will optimize a matrix transpose to show how to use shared memory to reorder strided global memory accesses into coalesced accesses. int m, n, c, d. Adjoint, inverse of a matrix. A new matrix is obtained the following way: each [i, j] element of the new matrix gets the value of the [j, i] element of the original one. Matrix-matrix multiplication. An easy and fast tool to find the eigenvalues of a square matrix. 4x4 system of equations solver. Trick in using Casio fx-570ES calculator. len L(row) (0. A matrix is said to be singular if its determinant is zero and non-singular. To transpose NumPy array ndarray (swap rows and columns), use the T attribute (. - The successive elements are obtained by adding N to the current element. i want to transpose a 4x4 matrix using sse2 intrinsics. Matrix calculator. Cofactor Matrix. void QMatrix4x4::viewport(float left, float bottom, float width, float height, float nearPlane = 0. the input and output are separate. 2x2 System of equations. com/donate?business=B5K9DC6DBNZNY¤cy_code=GBPA quick reminder on the matrix transpose, with a 4x4 matrix and a 2x4 matrix. •Divide all elements in the adjugate matrix by determinant of matrix. transpose() function. Register free for online tutoring session to clear your doubts. Check that this is true for and. (Recall that for square matrices, the shape doesn't change, say. The transpose of a matrix is a new matrix that is obtained by exchanging the rows and columns. len) toRet[col][row] = matrix[row][col] R toRet V m = [[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12]]print("Original")print(m)print("After Transposition". Program Explanation. I decided to go with the matrix partitioning algorithm to compute the matrix inverse. A square matrix is normal if it commutes with its conjugate transpose:. Matrix-matrix subtraction. Since a 3D point only needs three values (x, y, and z), and the transformation matrix is a 4x4 value matrix, we need to add a fourth dimension to the point. Specifically, it is designed to work on 4x4 transformation matrices found in 3D rendering using homogeneous coordinates [x y z w]. 5 comments, last by st0ff 4. This tool transposes matrices. $n\times n$ Transpose Matrix calculator calculates a transpose matrix of a matrix $A$ with real elements. What does multiplying by the identity matrix look like? The easiest example is to multiply a single point by the identity matrix. len L(row) (0. to multiply matrices doesn't require that I transpose one of them first. Matrices - Adding & Multiplying - Gaussian Elimination & Cramer's Rule - 3x3 Matrix. The equation "C = A” should appear on your screen with the inverted L cursor around the A as shown in figure 6A. It makes the lives of people who use matrices easier. Transpose of a matrix is obtained by changing rows to columns and columns to rows. Related Symbolab blog posts. Transpose (const vtkMatrix4x4 *in, vtkMatrix4x4 *out). The transposed matrix is the one that has the Matrix4x4's columns exchanged with its rows. A matrix is said to be singular if its determinant is zero and non-singular. The Matrix, Inverse. < matrix[row]. The equation "C = A” should appear on your screen with the inverted L cursor around the A as shown in figure 6A. Let A be an m n matrix. SPECIFY MATRIX DIMENSIONS Please select the size of the matrix from the popup menus, then click on the "Submit" button. 3x3 System of. Matrices - Adding & Multiplying - Gaussian Elimination & Cramer's Rule - 3x3 Matrix. SSE 4x4 Matrix transpose and invert General and Gameplay Programming Programming. Transpose the matrix and put it into out. The normal matrix is the transpose of the inverse of the top-left 3x3 part of this 4x4 matrix. The determinant of A will be denoted by either jAj or det(A). The transpose is found by exchanging the rows with columns and columns with rows. The program must accept two integers M and N as the input. to multiply matrices doesn't require that I transpose one of them first. Now we can represent the 4x4 matrix as a 2x2 matrix: Working with 2x2 matrix multiplication is much easier. Its inverse form is basically transpose the 3x3 rotation matrix, and rescale it, and change translation part by doing dot product with 3 rescaled axes. TRS(newVector3(6, 7, 8), Quaternion. Free matrix transpose calculator - calculate matrix transpose step-by-step. for all i and j. Similarly, the determinant of a square matrix is the product of all its eigenvalues with multiplicities. Using this online calculator, you will receive a detailed step-by-step solution to your problem, which will help you understand the algorithm how to find the transpose matrix. transpose() and numpy. Register free for online tutoring session to clear your doubts. One is to use Gauss-Jordan elimination and the other is to use the adjugate matrix. Cramer's Rule - 2x2 Linear System. You just need to know what the transpose of a matrix looks like. For example if you transpose a 'n' x 'm' size matrix you'll get a new one of 'm' x 'n' dimension. For a square matrix, the trace of a matrix is the sum of the elements on the main diagonal, which is equal to the sum of all its eigenvalues with multiplicities. Lets take an example of matrix. Here you can calculate a matrix transpose with complex numbers online for free. Public Function MxM (M1 () As Double, M2 () As Double) As Double () Dim Mat (0 To 3, 0 To 3) As Double. the input and output are separate. A new matrix is obtained the following way: each [i, j] element of the new matrix gets the value of the [j, i] element of the original one. This matrix transpose calculator help you to find transpose matrix. An identity matrix will be denoted by I, and 0 will denote a null matrix. Cofactor Matrix. where is a diagonal matrix, i. void QMatrix4x4::viewport(float left, float bottom, float width, float height, float nearPlane = 0. The transposed matrix is the one that has the Matrix4x4's columns exchanged with its rows. identity, Vector3. Matrix A has three rows and two columns, so its transpose Aᵀ will have a different shape: two rows and three columns. Transposition is an operation on a matrix which swaps its columns and rows across its diagonal. Learn about determinant of 4x4 matrix topic of maths in details explained by subject experts on vedantu. public: static System::Numerics::Matrix4x4 Transpose(System::Numerics::Matrix4x4 static member Transpose : System. •Divide all elements in the adjugate matrix by determinant of matrix. for all i and j. Matrix-matrix multiplication. I have a 4x4 block of bytes. , all its off diagonal elements are 0. Now we can represent the 4x4 matrix as a 2x2 matrix: Working with 2x2 matrix multiplication is much easier. The transpose of a matrix is a new matrix that is obtained by exchanging the rows and columns. Check that this is true for and. Donate: https://www. 3x3 System of. But there exist normal matrices not belonging to any of these. void QMatrix4x4::viewport(float left, float bottom, float width, float height, float nearPlane = 0. - The successive elements are obtained by adding N to the current element. Matrix4x4 test =Matrix4x4. Try Open Omnia Today. Note that if A is a matrix with real entries, then A*. How do i go about it?. In this post, we’re going to discuss an algorithm for Matrix multiplication along with its flowchart, that can be used to write programming code for matrix multiplication in any high level language. Matrix of Cofactors. The transpose of a matrix is a new matrix that is obtained by exchanging the rows and columns. I have a 4x4 block of bytes that I'd like to transpose using general purpose hardware. Works with matrix from 2X2 to 10X10. Cofactor Matrix. The computational time for transpose of a matrix using identity matrix as reference matrix is O(m*n). Learn about determinant of 4x4 matrix topic of maths in details explained by subject experts on vedantu. It is easy enough to do by hand. A square matrix is normal if it commutes with its conjugate transpose:. The original matrix and the transpose are both printed. SSE 4x4 Matrix transpose and invert General and Gameplay Programming Programming. The Matrix, Inverse. This example transposes a 3 by 3 matrix. •Compute the cofactor matrix by alternating + and –signs. Normally a matrix is stored in row major order, i. We employ the latter, here. For matrices there is no such thing as division, you can multiply but can't divide. A square matrix is normal if it commutes with its conjugate transpose:. Transpose (const vtkMatrix4x4 *in, vtkMatrix4x4 *out). In this post we will discuss logic and interface of a matrix keypad (4x4 for this post) with microcontroller to reduce the number of port pins required to read a certain number of inputs (digital). The same logic applies to any matrix keypad of order NxN. len] * matrix[0]. SPECIFY MATRIX DIMENSIONS Please select the size of the matrix from the popup menus, then click on the "Submit" button. Solve matrix problems for free with Open Omnia. Specifically, it is designed to work on 4x4 transformation matrices found in 3D rendering using homogeneous coordinates [x y z w]. Matrix transposes are a neat tool for understanding the structure of matrices. to multiply matrices doesn't require that I transpose one of them first. Where, N is the order of the matrix. How to find the transpose of a 4x4 matrix programing? 1 See answer cloneemperor9965 is waiting for your help. Transformation matrix creation. Similarly, we can describe a matrix in row-wise fashion: given vectors in , we can define the matrix with the transposed vectors as rows:. 2x2 System of equations. Create a new matrix “C” that is the transpose of matrix “A”. I have a 4x4 block of bytes that I'd like to transpose using general purpose hardware. The code we wish to optimize is a transpose of a matrix of single precision values that operates out-of-place, i. Example, = -5 and. public: static System::Numerics::Matrix4x4 Transpose(System::Numerics::Matrix4x4 static member Transpose : System. The transpose of a matrix is simply a flipped version of the original matrix. int m, n, c, d. that is because pdls are indexed by (col,row), and matrices are (row,column) by convention, so a 1-D pdl corresponds to a row vector, not a column vector. Circular Matrix (Construct a matrix with numbers 1 to m*n in spiral way) Count frequency of k in a matrix of size n where matrix (i, j) = i+j. Subdividing the 4x4 matrix into four 2x2 matrices. , all its off diagonal elements are 0. Move the mouse cursor to the transpose button ( ) on the matrix toolbar and click it. This calculator solves system of four equations with four unknowns. But I can't figure out how to do it using determinant 4x4 matrix with. If the 3x3 sub-matrix is not invertible, this function returns Returns this matrix, transposed about its diagonal. The transpose of a matrix is simply a flipped version of the original matrix. Using this online calculator, you will receive a detailed step-by-step solution to your problem, which will help you understand the algorithm how to find the transpose matrix. Cramer's Rule - 2x2 Linear System. It is skew-symmetric matrix because. Specifically, it is designed to work on 4x4 transformation matrices found in 3D rendering using homogeneous coordinates [x y z w]. Started by Shnoutz March 16, 2012 04:16 PM. I decided to go with the matrix partitioning algorithm to compute the matrix inverse. To understand transpose calculation better input any example and examine the solution. Transpose an arbitrarily sized rectangular Matrix. An identity matrix will be denoted by I, and 0 will denote a null matrix. In this article. friend Q_GUI_EXPORT QMatrix4x4 operator/(const QMatrix4x4& matrix, float divisor); 127. The transpose is found by exchanging the rows with columns and columns with rows. len) toRet[col][row] = matrix[row][col] R toRet V m = [[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12]]print("Original")print(m)print("After Transposition". With ndarray. len) L(col) (0.


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New Contributor

# Excel Formula

Using a Mac Safari 14.1.2 and Excel. Trying to input a formula

=SUM(D5:F5) and enter and all I see is the formula, not the actual sum. WTF

2 Replies

# Re: Excel Formula

1) Make sure that the cell with the formula is not formatted as Text.

2) Make sure that Show Formulas in the Formula Auditing group of the Formulas tab of the ribbon is not highlighted.

Thanks.


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The function randperm randomly permutes the integers of a range.

## Syntax

p = randperm(n) stores in the row vector p a random permutation of the integers in the range $$[1,n]$$. p = randperm(n,k) selects k unique integers from the range $$[1,n]$$, and stores them in p.

% Random permutation of the integers in the range [1,10]
resultExample1 = randperm(10);
% Select four random integers from the range [1,15]
resultExample2 = randperm(15,4);

## Function

randperm is restricted to integers greater than or equal to 1. I developed the function randpermfree to randomly permute the integers of any range. This function is properly documented and available for download here.

The file ‘randpermfree.m’ must be in the current directory or in the MATLAB search path.
% Random permutation of the integers in the range [-2,5]
resultExample3 = randpermfree(-2,5);
% Select two random integers from the range [-10,10]
resultExample4 = randpermfree(-10,10,2);

Note that your result in the preceding examples could be different due to the random nature of the operations.

I recommend the following books to learn more on random numbers in MATLAB.

1. This book was the winner of a Textbook Excellence Award. The first chapter covers random numbers, logical operators, and common elementary math functions, among other topics.

2. MATLAB for Engineers (5th Edition)

This excellent book is especially oriented to engineers and scientists who want to learn MATLAB programming. The third chapter covers random numbers, matrix sorting, and several data analysis functions, among other topics.

I also recommend the following page in the MATLAB help documentation, and the other tutorials in this series.

doc randperm;

## Source code

I hope you have learned how to randomly permute the integers of a range in MATLAB. The source code developed in this tutorial is available at this page.


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How do you find the integral of x^5*e^(x^2) ?

Apr 25, 2018

$\frac{1}{2} {x}^{4} {e}^{{x}^{2}} - {x}^{2} {e}^{{x}^{2}} + {e}^{{x}^{2}} + C$

Explanation:

$\int {x}^{5} \cdot {e}^{{x}^{2}} \mathrm{dx}$

$= \frac{1}{2} \int {x}^{4} {e}^{{x}^{2}} \cdot 2 x \mathrm{dx}$

$= \frac{1}{2} \int {x}^{4} {e}^{{x}^{2}} {\mathrm{dx}}^{2}$

Integration by substitution

${x}^{2} = u$

$d \left({x}^{2}\right) = \mathrm{du}$

$= \frac{1}{2} \int {x}^{4} {e}^{{x}^{2}} {\mathrm{dx}}^{2} = \frac{1}{2} \int {u}^{2} {e}^{u} \mathrm{du}$

$= \frac{1}{2} \int {u}^{2} d \left({e}^{u}\right)$

$= \frac{1}{2} \left({u}^{2} {e}^{u} - \int 2 u {e}^{u} \mathrm{du}\right)$

Using integration by Parts again

$= \frac{1}{2} \left({u}^{2} {e}^{u} - 2 \int u d \left({e}^{u}\right)\right)$

$= \frac{1}{2} \left({u}^{2} {e}^{u} - 2 u {e}^{u} - \left(- 2 \int {e}^{u} \mathrm{du}\right)\right)$

$= \frac{1}{2} {u}^{2} {e}^{u} - u {e}^{u} + {e}^{u} + C$

Reverse The Substitution

$= \frac{1}{2} {x}^{4} {e}^{{x}^{2}} - {x}^{2} {e}^{{x}^{2}} + {e}^{{x}^{2}} + C$


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# Thread: Linearization of nonlinear equations with approximation

1. ## Linearization of nonlinear equations with approximation

Dear friends,

Can anybody help me to make an approximation to a nonlinear equation like (x-y)^0.5 in terms of x and y?

2. Use the Taylor formula

$f(x,y)\;\approx \;f(x_0,y_0)+\dfrac{\partial f}{\partial x}(x_0,y_0)\;(x-x_0)+\dfrac{\partial f}{\partial y}(x_0,y_0)\;(y-y_0)$

3. Thanks for your comment. Indeed, I need a parametric function in terms of x and y like what is written for (x-y)^2=x^2+y^2-2xy. However, I know there is not an exact extension for my equation. I think there is maybe a simplified form with a reasonable approximation.

4. Yes, an approximation is what you asked for and what FernandoRevilla gave you. Your function is f(x,y)= (x- y)^{0.5}, so its first order partial derivatives are f_x= (0.5)(x- y)^{-0.5} and f_y= -(0.5)(x- y)^{-0.5}. Evaluate those at some (x_0, y_0) and put into FernandoRevilla's formula. Of course, any approximation is not going to be equally accurate for all x and y. You need to decide where you want the approximation to be most accurate and pick x_0 and y_0 there.

5. Thank u HallsofIvy, I got your point. I will check it in my problem. Defining x0 and y0 is the central challenge here.

6. Originally Posted by ali1980
Thank u HallsofIvy, I got your point. I will check it in my problem. Defining x0 and y0 is the central challenge here.
There is another approach which minimises some function of the error in the linear approximation across some region of the x-y plane but from what you write I guess that would be beyond you at present.

CB

7. Thanks CaptainBlack, maybe an iterative solution can minimize the errors. But if you know a better approach, please let me know.


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# Evaluating the elliptic integral $\int_{-\pi}^\pi\frac{dx}{\sqrt{(t-2\cos x)^2-4}}$

I have the following integral,

$$I(t)=\int_{-\pi}^\pi\frac{dx}{\sqrt{(t-2\cos x)^2-4}},$$

where $t>4$ is a real parameter. I know from messing around numerically and playing with Mathematica that

$$I(t)=\frac{4}{t}K\left(\frac{16}{t^2}\right),$$

where $K$ is the complete elliptic integral of the first kind with parameter $m=k^2=16/t^2$. However, I seek proof of that fact. I have tried a handful of changes of variables which didn't get the job done, and I've searched tables of integrals without finding this integrand or similar. Any suggestions or hints would be appreciated.

-
I cannot believe that this integral yields an elliptic function. Maybe you have forgot the square root in the denominator? –  Fabian Aug 13 '12 at 22:32
Err, you are right, Fabian. Thank you. –  Jonathan Aug 13 '12 at 22:34

Here is a rather circuitous route. Maybe there is a more compact way to do this:

\require{cancel}\begin{align*} \int_{-\pi}^\pi \frac{\mathrm dx}{\sqrt{(t-2\cos x)^2-4}}&=2\int_0^\pi \frac{\mathrm dx}{\sqrt{(t-2\cos x)^2-4}}\\ &=4\int_0^\infty \frac{\mathrm du}{\sqrt{((t+4)u^2+t)(tu^2+t-4)}} \qquad \small{\left(u=\tan\frac{x}{2}\right)}\\ &=\frac4{\sqrt{t(t+4)}}\int_0^\infty \frac{\mathrm du}{\sqrt{\left(u^2+\frac{t}{t+4}\right)\left(u^2+\frac{t-4}{t}\right)}}\\ &=\frac{4\cancel{\sqrt{t(t+4)}}}{\cancel{\sqrt{t(t+4)}}}\int_0^{\pi/2} \frac{\mathrm dv}{\sqrt{t^2-16+16\sin^2 v}} \quad \small{\left(u=\sqrt{\frac{t}{t+4}}\tan\,v\right)}\\ &=4\int_0^{\pi/2} \frac{\mathrm dv}{\sqrt{t^2-16+16\sin^2 v}}\\ &=4\int_{-\pi/2}^0 \frac{\mathrm dv}{\sqrt{t^2-16+16\sin^2 v}}\qquad\text{(symmetry)}\\ &=4\int_0^{\pi/2} \frac{\mathrm dv}{\sqrt{t^2-16+16\cos^2 v}}=4\int_0^{\pi/2} \frac{\mathrm dv}{\sqrt{t^2-16\sin^2 v}}\\ &=\frac4{t}K\left(\frac{16}{t^2}\right) \end{align*}

-
Excellent, thank you very much. I'll accept this soon if nobody else comes up with something slick. –  Jonathan Aug 14 '12 at 4:00
No problem, take your time. If I conjure up something shorter, I'll edit this. –  Guess who it is. Aug 14 '12 at 4:13

Since $\cos(-x) = \cos(x)$ we can reduce the integration range to $(0,\pi)$, and then do the change of variable $u = \cos(x)$: $$\int\limits_{-\pi}^\pi \frac{\mathrm{d} x}{\sqrt{(t-2 \cos(x))^2-4}} = \int_0^\pi \frac{\mathrm{d} x}{\sqrt{\left(\frac{t}{2} - \cos(x)\right)^2 -1}} = \int_{-1}^1 \frac{2 \mathrm{d} u}{\sqrt{1-u^2} \sqrt{4 u^2 + 4 u t + t^2-4}}$$ Now perform a change of variables: $$u = \frac{v-z}{1- v z}, \qquad \text{where} \quad v = \frac{t - \sqrt{t^2-16}}{4}$$ that maps $-1<u<1$ into $-1<z<1$, that leads to $$\int_{-1}^1 \frac{ v \, \mathrm{d} z}{\sqrt{1-z^2} \sqrt{1-v^4 z^2}} = 2 v \cdot \mathrm{K}\left( v^4\right)$$ This is not exactly what the OP asked for, but quite elegant nonetheless.

Here is a confirmation of the equivalence in Mathematica:

The equivalence of the above answer to the one conjectured by OP and established by @J.M. is through the quadratic transformation: $$\mathrm{K}(z) = \frac{2}{1+\sqrt{1-z}} \mathrm{K} \left( \left(\frac{1-\sqrt{1-z}}{1+\sqrt{1-z}} \right)^2 \right)$$ where $z = \frac{16}{t^2}$. Indeed: $$\frac{2}{1+\sqrt{1-\frac{16}{t^2}}} = \frac{2t}{t + \sqrt{t^2-16}} = \frac{2t}{t + \sqrt{t^2-16}} \cdot \frac{t - \sqrt{t^2-16}}{t - \sqrt{t^2-16}} = \frac{t}{4} \cdot \frac{t-\sqrt{t^2-16}}{2}$$ and similarly: $$\left(\frac{1-\sqrt{1-\frac{16}{t^2}}}{1+\sqrt{1-\frac{16}{t^2}}} \right)^2 = \left(\frac{t-\sqrt{t^2-16}}{t+\sqrt{t^2-16}} \cdot \color\green{ \frac{t-\sqrt{t^2-16}}{t-\sqrt{t^2-16}} } \right)^2 = \left(\frac{\left(t-\sqrt{t^2-16}\right)^2}{16} \right)^2$$ Combining, we arrive at the equality: $$\frac{4}{t} \mathrm{K}\left(\frac{16}{t^2}\right) = \frac{t-\sqrt{t^2-16}}{2} \mathrm{K}\left(\left( \frac{t-\sqrt{t^2-16}}{4} \right)^4\right)$$


text:
# Secant function

This article is about a particular function from a subset of the real numbers to the real numbers. Information about the function, including its domain, range, and key data relating to graphing, differentiation, and integration, is presented in the article.
View a complete list of particular functions on this wiki
For functions involving angles (trigonometric functions, inverse trigonometric functions, etc.) we follow the convention that all angles are measured in radians. Thus, for instance, the angle of $90\,^\circ$ is measured as $\pi/2$.

## Definition

The secant function, denoted $\sec$, is defined as the composite of the reciprocal function and the cosine function.

Explicitly, it is given as follows:

$\sec x := \frac{1}{\cos x}$

The domain of the function is the set of all $x \in \R$ for which $\cos x \ne 0$.

## Key data

Item Value
default domain all real numbers except odd integer multiples of $\pi/2$.
range $(-\infty,-1] \cup [1,\infty)$, i.e., $\{ y \mid |y| \ge 1 \}$
period $2\pi$, i.e., $360\,^\circ$
vertical asymptotes lines of the form $x = n\pi + \pi/2$.
For $n$ even, left hand limit is $+\infty$ and right hand limit is $-\infty$.
For $n$ odd, left hand limit is $-\infty$ and right hand limit is $+\infty$.
local minimum values and points of attainment local minimum value of 1, attained at integer multiples of $2\pi$.
local maximum values and points of attainment local maximum value of -1, attained at odd integer multiples of $\pi$
important symmetries even function
mirror symmetry about lines of the form $x = n\pi$, $n$ varying over integers.
half turn symmetry about points $(n\pi + \pi/2,0)$, $n$ varying over integers.
first derivative $x \mapsto \sec x \tan x$
second derivative $x \mapsto \sec x \tan^2x + \sec^3x$
first antiderivative $x \mapsto \ln|\sec x + \tan x| + C$.


text:
# An obscure integral

April 7, 2010
By

(This article was first published on Xi'an's Og » R, and kindly contributed to R-bloggers)

Here is an email from Thomas I received yesterday about a computation in our book Introducing Monte Carlo Methods with R:

I’m currently reading your book “Introduction to Monte Carlo Methods with R” and I quite highly appreciate your work. I’m not able to see how the integral on page 74, that describes the marginal likelihood, simplifies to the fraction on the second line. If I’m not asking too much, could you confirm to me whether the fraction is as is given in the text.

Because the transform of the integral

$m(x) = int_{{mathbb R}^2_+} f(x|alpha,beta),pi(alpha,beta),text{d}alpha text{d}beta$

into the ratio of two integrals

$dfrac{int_{{mathbb R}^2_+} left{ frac{Gamma(alpha+beta)}{Gamma(alpha)Gamma(beta)} right}^{lambda+1}, [x x_0]^{alpha}[(1-x)y_0]^{beta} ,text{d}alpha text{d}beta}{x(1-x),int_{{mathbb R}^2_+} left{ frac{Gamma(alpha+beta)}{Gamma(alpha)Gamma(beta)} right}^{lambda}, x_0^{alpha} y_0^{beta} ,text{d}alpha text{d}beta}$

may sound curious (and possibly wrong) to many readers besides Thomas, let me explain that the bottom integral is the normalisation constant of the prior

$pi(alpha,beta)propto left{ frac{Gamma(alpha+beta)}{Gamma(alpha)Gamma(beta)} right}^lambda, x_0^{alpha}y_0^{beta}$

while the top integral is the product of the observation density:

$f(x|alpha,beta) = dfrac{Gamma(alpha+beta)}{Gamma(alpha)Gamma(beta)},dfrac{x^{alpha}(1-x)^{beta}}{x(1-x)}$

and of the prior (minus the normalisation constant). Nothing wrong then with the formula at the bottom of page 74, but this is a bit short on explanations!

Filed under: Books, R, Statistics Tagged: Introducing Monte Carlo Methods with R, normalising constant, typos

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text:
Expii

# Inverse Variation: Find Constant and Graph - Expii

Inversely proportional relationships are of the form y=k/x. To find the constant of variation, just multiply x and y together.


text:
# Markov chain problem

1. May 9, 2007

### celestra

[SOLVED] Markov chain problem

Hi, all! This is not a homework thing or something like that. I'm just curious.

Suppose a Markov chain p(n+1)=Q*p(n), where Q=[q11, q12; q21, q22] is the transition probability matrix and p(n)=[p1(n);p2(n)] is the probability vector.

Given Q and p(1), we can generate a realization from it, i.e., a sequence of ones and zeros.

Now I want to do the inverse. Given a long enough sequence of ones and zeros, Q can be obtained by counting the number of changes from zero to zero(which will be q11), zero to one(q12), one to zero(q21), and one to one(q22). But, how can I get the initial probability vector p(1) from this sequence? Maximum Likelihood Estimation, blah blah? I don't know about that. Please, someone give me a clue how I can do this. Or is it impossible to do that from just only one realization sequence?

2. May 9, 2007

### EnumaElish

How is the realization r linked to p? If p1 > 0.5 then r = 1, otherwise r = 0?

3. May 9, 2007

### celestra

Oh, I forgot that. The realization is not deterministic but stochastic through a kind of random number generation according to the probability vector. If p1=.4 and p2=.6, then the probability that a zero will come out is 40 percent and the probability that a one will come out is 60 percent.

4. May 9, 2007

### EnumaElish

So how do you construct your sequence of 1's and 0's? Toss a coin, with a 1 on one side and a 0 on the other?

5. May 9, 2007

### celestra

It's a good idea. You can use any sequence if it is composed of zeros and ones.

6. May 9, 2007

### celestra

For example, you are given this sequence: 010100101101001011010010111010.
Then, you can get Q as [3/14, 11/15; 11/14, 4/15].
And, I have no idea how I can get p(1) from this.

7. May 9, 2007

### EnumaElish

Two ideas:
1. assume uniform initial probability, p = 0.5
2. assume stationary Markov, then solve p = Qp.

8. May 9, 2007

### NateTG

It's impossible to completely determine the probability matrix from samples, even if we stipulate that it's a markov process, but it is possible to use statistics to find what the probabilities are likely to be.

Any probability matrix where all entries are non-zero is can produce any sequence. For example, a process with the probability matrix:
$$$\left[ \begin{array}{cc} \frac{1}{2}& \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \\ \end{array} \right]$$$
Will produce any particular sequence of length $n$ with probability $\frac{1}{2^n}$.

So, assuming even initial distribution it would generate 010100101101001011010010111010 about
$\frac{1}{2^{30}}$ of the time.

The probability matrix you produced
$$$\left[ \begin{array}{cc} \frac{3}{14}& \frac{11}{15} \\ \frac{11}{14} & \frac{4}{15} \\ \end{array} \right]$$$
Is just the one that has the highest likelihood of producing the sequence you gave, but, because the sample is relatively small, there really isn't a whole lot confidence that it's correct.

This 'reversing probability' stuff is a bit more advanced.

9. May 9, 2007

### celestra

What about MAP(Maximum A Posteriori) esimation, $$\hat{x}$$=argmax$$p(x|y)$$, since we're interested in the random variable $$x$$, p(1) in this case, given data as the realization $$y$$, in this case the sequence? I feel like this will work. The a posteriori pdf $$p(x|y)$$ can be calculated using Bayes's theorem $$p(x|y)=\frac{p(y|x)}{p(y)}p(x)$$ where $$p(x)$$ is the a priori pdf, $$p(y)$$ is the marginal pdf, and $$p(y|x)$$ is proportional to the likelihood function. And we can put $$\left[\begin{array}{c} .5\\ .5\\ \end{array}\right]$$ in the place of the a priori pdf $$p(x)$$ using maximum entropy principle as EnumaElish did in the first idea. But, I don't know how to calculate the rest of things. Now I'm seeking for the calculation in some books, but I haven't found yet. I guess it's just MLE(Maximum Likelihood Estimation) after all. But, I don't know how to do that.

Last edited: May 9, 2007
10. May 9, 2007

### celestra

I don't like the stationarity assumption because I want to deal with a sequence even like this: 010100100100000000000000000000000110100101.

11. May 9, 2007

### EnumaElish

Isn't there is a p that solves p = Qp for any arbitrary Q (hence an arbitrary sequence)?

12. May 10, 2007

### celestra

I did a mistake. You're right EnumaElish. Any Q has it's own eigenvectors at least one. So your stationarity assumption will be okay. And it seems the best idea for my problem currently.


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# 2 Summations

Logic Level 3

$\begin{array} { c c c c c } & & \square \\ & & + \\ \square & + & {\color{red} \square} & = & \square \\ && = \\ && \square \\ \end{array}$

Use distinct digits 1 to 5 to make the above equations true. How many possibilities are there for the center square (marked in red)?

×


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# Design of Grit Chamber with Example

“Great things are done by a series of small things brought together”

Grit chambers are long narrow tanks that are designed to slow down the flow so that solids such as sand, soil, inorganic matter will settle down.

## Design Aspects of Grit Chamber

These are the design aspects of Grit chamber, which needs to be satisfied during the design of grit chamber

### Cleaning Interval of Grit Chamber

Cleaning Interval is of grit chamber depends on the the method used to clean the chamber i.e. hand cleaning, hydraulic cleaning, machine cleaning. Also the cleaning period will also be dependent on the local conditions like what kind of waste will be available there?, intensity and frequency of rainfall which will result in silt collection in the chamber.

Generally the cleaning interval of grit chamber must be one or two weeks.

### Dimensions of Grit Chamber

The minimum depth of grit chamber must be 300mm, and the ratio between the depth and length should be 1:16.

### Detention Time of Grit Chamber

It is the average time for which water remains in the Grit chamber before moving on to further treatment. Detention Time of Grit chamber is about 30 to 90seconds but for design purposes it is generally taken as 60seconds.

### Velocity of Flow of Sewage in Grit Chamber

Grit chambers are designed to settle down the heavy inorganic particles by slowing down the velocity of flow here in order to slow down the velocity the area of chamber is increased. The velocity of flow should not be too high that the particles will not get sufficient time to settle down and if the velocity of flow becomes too slow then suspended organic matter will also settle down.

So to maintain the required velocity in the grit chamber the chambers can be divided into compartments, called proportional weir. Setting velocity by Stocke’s law depending on Reynold’s number can be given as

V = {{ g(ρ_s - ρ)d^2 } \over 18 μ} \\ \\  V= { g \over 18 } * {{(S_s - 1)} \over {γ}} * d \\

Where, Ss = Specific Gravity = (ρs / ρ)

γ = Kinematic Viscosity = (μ/ρ)

When the setting particles are greater than 1mm then nature of setting particles is termed as turbulent and in such case setting velocity can be termed by newton’s equation as

V = \sqrt{ 3.3 \, g \, {{ ρ_s - ρ } \over ρ} \, d}

Generally, when grit particles lie between 0.1mm to 1mm then the zone of settling is called as Transition Zone. And where the settling velocity is given by Hazen’s equation

V = 418 (g-1)d [{{3t + 70} \over 100}]

where d is in between 0.1mm to 1mm.

## Design of Grit Chamber Example

### Question:

Design a rectangular grit chamber for a flow rate of 20mld. Specific gravity= 2.65 and size to be removed is 0.2mm and above.

• Average temperature: 20°C
• Flow-through velocity: 0.23 m/sec
• Width of grit chamber: 1.2m
• Freeboard: 0.3m
• Space for sludge accumulation: 0.25m

here we have to calculate H and L, we have given B then if we calculate Area then we can have H. So for A we have Q = AVh where we have given Q and Vh And to calculate L, we have L = Vh Vs , So we have Vh and need to calculate Vs.

#### Step 1: Settling Velocity (Vs)

V_s = d(3T + 70)

Here, d = diameter of grit particle= 0.2mm = 0.0002m, T= temperature

Vs = 0.0002(3×20+70)

Vs= 0.026m/sec

#### Step 2: Area of Grit Chamber (A)

We can calculate area of chamber from discharge

Q = AVh

Q = 20mld = 20 x 106 letres/day = 20 x 103 m3/day

Q = 0.23 m3/day

now, from Q we will calculate A

Q = AV_h  \\  A= Q/V_h \\ A= (0.23/0.23)=1m^2

As we have calculated the area and we have width of chamber given then we can calculate height

#### Step 3: Dimensions of Grit Chamber

• If one of the quantities (B or H) is given then you can find another one.
• If any quantity is not given then take height 1m.
A= B*H \\ we \, have \, B=1.2m \\ then \, H=1/1.2 \\ H= 0.83m

To calculate length we have, L = Vh x Detention Time

So we need calculate detention time first

#### Step 4: Calculation of Detention Time

Detention \, Time = { Depth \over Settling \ velocity} \\  Detention \, Time = {0.83\over 0.026 } = 32sec

But when T= 20°C take, detention time = 40sec

#### Step 5: Calculation of Length

L= V_h \ V_s = 0.23 * 40 = 9.2m

Length of grit chamber is increased by 10 to 50% to allow non-uniformity in flow and settling of particles. It is not mandatory.

### Solution:

Overall Depth = 0.83 + 0.3(Free Board) = 0.86m

Therefore dimensions of grit chamber are L x B x H = 9.2 x 1.2 x 0.86 m

## FAQs on Design of Grit Chamber with Example

What is the detention time considered while designing a grit chamber?

Detention Time of Grit chamber is about 30 to 90seconds but for design purposes it is generally taken as 60seconds. It is the average time for which water remains in the Grit chamber before moving on to further treatment.

What is meant by grit chamber?

Grit chambers are long narrow tanks that are designed to slow down the flow so that solids such as sand, soil, inorganic matter will settle down.

What is the cleaning period for manual grit chambers?

Generally the cleaning interval of grit chamber must be one or two weeks. Cleaning period of grit chamber depends on various topics such as type of grit, local conditions, rain intensity.

Why must velocity be controlled in grit chamber?

Velocity in grit chamber should be controlled otherwise too high velocity of the particles will not get sufficient time to settle down and if the velocity of flow becomes too slow then suspended organic matter will also settle down.

What is the meaning of grit?

Grit is inorganic material present in sewage. Grit includes sand, clay, glass pieces, tea leaves, egg shells, and cinders.

Tags: Design of

5 months ago

5 months ago

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8. Сэт – продюсер, рэппер, битмейкер, поэт. Активно занимается творческой деятельностью с 1998. В 2004 открыл независимый рэп-лейбл «Revol…

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# Counterxample of representation theorem

Give an example of non-degenerate scalar product $\langle,\rangle$ on a vector space $V$ such that the application $$\varphi : V\to V^* : v\mapsto L_v$$defined by $L_v(w)=\langle v,w\rangle$ is not an isomorphism.

• Sorry, it is not clear for me what you are asking. Could you be more precise? – Janik Dec 28 '16 at 17:05
• I hope the question clear – Shabeeb Al-alawi Dec 28 '16 at 17:11
• @ZevChonoles and I edited your question almost simultaneously. Feel free to roll back to his version if you prefer it (this applies to everyone). – Arnaud D. Dec 28 '16 at 17:27
• In fact, a vector space is isomorphic to its dual if and only if it is finite-dimensional. Thus you just need to find an infinite-dimensional one. – Hanul Jeon Dec 28 '16 at 18:36
• Can you give an example ? – Shabeeb Al-alawi Dec 28 '16 at 18:57

Consider the vector space $V$ of all infinite sequences $(x_i)_{i\geq0}$ of real numbers with finitely many non-zero terms, and endow it with the inner product such that $$\langle(x_i)_{i\geq0},(y_i)_{i\geq0}\rangle=\sum_{i\geq0}x_iy_i.$$ There is a linear function $\phi:V\to\mathbb R$ such that $$\phi((x_i)_{i\geq0})=\sum_{i\geq0}x_i$$ for all sequences $(x_i)_{i\geq0}\in V$, and you can easily check that it is not in the image of the canonical map $V\to V^*$. Indeed, no vector $(\xi_i)_{i\geq0}$ in $V$ exists such that $$\sum_{i\geq0}x_i\xi_i=\sum_{i\geq0}x_i$$ for all $(x_i)_{i\geq0}\in V$.


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# Python Program for nth Catalan Number

Catalan numbers are a sequence of natural numbers that are defined by the recursive formula −

$$C_{0}= 1\:and\:C_{n+1}=\displaystyle\sum\limits_{i=0}^n C_{i}C_{n-i} for \:n\geq0;$$

The first few Catalan numbers for n = 0, 1, 2, 3, … are 1, 1, 2, 5, 14, 42, 132,429,...................

Catalan numbers can be obtained both by recursion and dynamic programming. So let’s see their implementation.

## Approach 1: Recursion Method

#### Example

Live Demo

# A recursive solution
def catalan(n):
#negative value
if n <=1 :
return 1
# Catalan(n) = catalan(i)*catalan(n-i-1)
res = 0
for i in range(n):
res += catalan(i) * catalan(n-i-1)
return res
# main
for i in range(6):
print (catalan(i))

#### Output

1
1
2
5
14
42

The scope of all the variables and recursive calls are shown below.

## Approach 2: Dynamic Programming Method

#### Example

Live Demo

# using dynamic programming
def catalan(n):
if (n == 0 or n == 1):
return 1
# divide table
catalan = [0 for i in range(n + 1)]
# Initialization
catalan[0] = 1
catalan[1] = 1
# recursion
for i in range(2, n + 1):
catalan[i] = 0
for j in range(i):
catalan[i] = catalan[i] + catalan[j] * catalan[i-j-1]
return catalan[n]
# main
for i in range (6):
print (catalan(i),end=" ")

#### Output

1
1
2
5
14
42

The scope of all the variables and recursive calls are shown below.


text:
# For isomorphism $\phi:G\to H$, show that $ab=ba$ for $a,b\in G$ iff $\phi(a)\phi(b)=\phi(b)\phi(a)$.

Could someone please verify my following proof?

For isomorphism $\phi:G\to H$, show that $ab=ba$ for $a,b\in G$ iff $\phi(a)\phi(b)=\phi(b)\phi(a)$.

Proof:

Let $ab=ba$ for all $a,b\in G$. Then $G$ is abelian. Then $H$ is abelian. Since $\phi$ is bijective, $H=\phi(G)=\left \{ \phi(g) : g\in G \right \}$. Therefore, $\phi(a)\phi(b)=\phi(b)\phi(a)$ for all $\phi(a),\phi(b)\in H$.

Let $\phi(a)\phi(b)=\phi(b)\phi(a)$ for all $\phi(a),\phi(b)\in H$. Since $\phi$ is bijective, $H=\phi(G)=\left \{ \phi(g) : g\in G \right \}$. Then $H$ is abelian. Then $G$ is abelian. Then $ab=ba$ for all $a,b\in G$.

• This is the right idea, a little clumsily written. But there's a sense in which it is too much work. You shouldn't have to make any argument at all, except to practice arguments. See math.stackexchange.com/questions/2039702/… Mar 6 '18 at 21:12
• Are you sure the intent is for all $a,b \in G$ and not for any? Mar 6 '18 at 21:13
• @gt6989b I am not sure what you mean. Don't the universal quantifiers "for all" and "for any" mean the same thing?
– user482939
Mar 6 '18 at 21:39
• I mean you don't need to assume the whole group is Abelian, just that these specific $a,b \in G$ commute Mar 6 '18 at 22:35
• @gt6989b I understand what you mean now...thanks, I will rewrite my proof without the abelian part!
– user482939
Mar 7 '18 at 0:26

If $\phi:G\to H$ is any homomorphism, and $a,b\in G$ are a particular commuting pair of elements: $ab=ba$ (we don't need to assume that the whole $G$ is Abelian), then we already have $$\phi(a)\phi(b)=\phi(ab)=\phi(ba)=\phi(b)\phi(a)\,.$$ If $\phi$ is an isomorphism, the converse also holds for the particular commuting pair $\phi(a),\phi(b)$: specifically, we can apply the same argument for the homomorphism $\phi^{-1}$.


text:
In the previous post, we saw that the sparsest cut of a graph $G=(V,E)$ (which is within a factor of 2 of the edge expansion of the graph), is defined as

$\phi := \min_{S\subseteq V} \frac{edges(S,V-S)}{\frac 1n |S|\cdot |V-S|}$

and can also be written as

$(1) \ \ \ \phi = \min_{x\in \{0,1\}^V} \frac{\sum_{i,j} A(i,j) | x(i)-x(j)|}{\frac 1n \sum_{i,j} |x(i)-x(j)|}$

while the eigenvalue gap of $G$, can be written as

$(2) \ \ \ d-\lambda_2 = \min_{x\in {\mathbb R}^V} \frac{\sum_{i,j} A(i,j) | x(i)-x(j)|^2}{\frac 1n \sum_{i,j} |x(i)-x(j)|^2}$

and so it can be seen as a relaxation of $\phi$, considering that
$|x(i)-x(j)|=|x(i)-x(j)|^2$ when $x(i),x(j)$ are boolean.

Let us now take a broader look at efficiently computable continuous relaxations of sparsest cut, and see that the eigenvalue gap is a semidefinite programming relaxation of sparsest cut, that by adding the triangle inequality to it we derive the Arora-Rao-Vazirani relaxation, and that by removing the semidefinite constraint from the ARV relaxation we get the Leighton-Rao relaxation.

We first prove that (2) is equivalent to

$(3) \ \ \ d-\lambda_2 = \min_{m, x_1,\ldots ,x_n \in {\mathbb R}^m} \frac{\sum_{i,j} A(i,j) || x_i - x_j ||^2}{\frac 1n \sum_{i,j} || x_i -x_j ||^2}$

The difference is that instead of optimizing over all possible ways to assign a real number $x(i)$ to each vertex $i$, we are now optimizing over all possible ways to assign a vector $x_i$ to each vertex $i$. The absolute value $|x(i)-x(j)|$ is then replaced by the Euclidean distance $|| x_i - x_j ||$. The right-hand-side of $(3)$ is clearly a relaxation of the right-hand-side of $(2)$, so we just need to establish that, in the right-hand-side of $(3)$, it is optimal to take 1-dimensional vectors. For every vector solution $x_1,\ldots,x_n$, we see that

$\frac{\sum_{i,j} A(i,j) || x_i - x_j ||^2}{\frac 1n \sum_{i,j} || x_i -x_j ||^2} = \frac{\sum_k \sum_{i,j} A(i,j) | x_i(k) - x_j(k) |^2}{\sum_k \sum_{i,j} | x_i(k) -x_j(k) |^2} \geq \min_k \frac{\sum_{i,j} A(i,j) | x_i(k) - x_j(k) |^2}{\sum_{i,j} | x_i(k) -x_j(k) |^2}$

where the last inequality follows from

$(4) \ \ \ \frac{\sum_{i=1}^n a_i}{\sum_{i=1}^n b_i} \geq \min_i \frac {a_i}{b_i}$

which, in turn, has a one-line proof:

$\sum_i a_i = \sum_i b_i \cdot \frac {a_i}{b_i} \geq \left( min_i \frac {a_i}{b_i} \right) \cdot \sum_i b_i.$

The formulation (3) is equivalent to the following semidefinite program:

$(5) \ \ \ \begin{array}{l} \min \sum_{i,j} A(i,j) ||x_i -x_j ||^2\\ subject\ to\\\sum_{i,j} ||x_i -x_j ||^2 = n\\x_i \in {\mathbb R}^{*} \end{array}$

An optimization problem is a semidefinite program if every variable is allowed to be an arbitrary vector, and both the objective function and the constraints are linear functions of inner products of pairs of variables. In our case,

$||x_i - x_j||^2 = \langle x_i , x_i \rangle + \langle x_j , x_j \rangle - 2 \langle x_i , x_j \rangle$

is a linear combination of inner products, and so the formulation (5) is a semidefinite program. Another way to define semidefinite programming (indeed, the definition that is usually given first, deriving the one in terms of inner products as an equivalent characterization) is that a semidefinite program is an optimization problem in which the variables are the entries of a positive semidefinite matrix $X$, and the objective function and the constraints are linear in the entries of $X$. A real matrix is positive semidefinite if it is symmetric and all its eigenvalues are non-negative. We can see that an $n\times n$ matrix $X$ is positive semidefinite if and only if there are vectors $x_1,\ldots,x_n$ such that $X(i,j) = \langle x_i,x_j \rangle$.

(The proof is simple: if $X$ is real and symmetric, then all its eigenvalues $\lambda_1,\ldots,\lambda_n$ are real, and there is an orthonormal system of eigenvectors $v_1,\ldots,v_k$ such that

$X=\sum_k \lambda_k v_k \cdot v_k^T$

so that

$X(i,j) = \sum_k \lambda_k v_i (k) v_j(k)$

If the eigenvalues are non-negative, we can let $x_i(k) := \sqrt{\lambda_k} v_i(k)$ and have $X(i,j) = \langle x_i ,x_j\rangle$.

On the other hand, if there are $x_i$ such that $X(i,j) = \langle x_i,x_j\rangle$, let $\lambda$ be any eigenvalue of $X$ and $v$ be a length-1 eigenvector of $\lambda$. Then:

$\lambda = v^T X v = \sum_{i,j} v_n(i)v_n(j) \sum_k x_i(k) x_j(k) = \sum_k \left( \sum_i v_n(i) x_i(k) \right)^2 \geq 0$

which completes the characterization.)

So we have gotten to the point where we can see $d-\lambda_2$ as a semidefinite programming relaxation of sparsest cut. An interesting property of semidefinite programs is that, like linear programs, they admit a dual. From a minimization (primal) semidefinite program, we can write a maximization (dual) semidefinite program such that the cost of any feasible solution for the dual is a lower bound to the cost of any feasible solution for the primal; furthermore, the two programs have the same optimum cost. Returning to our motivating question of certifying the expansion of a given graph, any feasible solution to the dual of (5) gives us a lower bound to the expansion of the graph.

We shall have to suffer a bit longer before we can write the dual of (5). It is best to first move to a formulation in terms of entries of semidefinite matrices. A few calculations show that (5) can be written as

$(6) \ \ \ \begin{array}{l} \min L_G \bullet X\\ subject\ to\\L_n \bullet X =n\\ X \succeq 0 \end{array}$

Where we write $X \succeq 0$ to mean that the matrix $X$ is positive semidefinite (we shall also write $X\succeq Y$ when $X-Y$ is positive semidefinite), we write $A \bullet B$ to mean $\sum_{i,j} A(i,j)B(i,j)$, $L_G$ is the Laplacian of $G$, that is the matrix $dI - A$, and $L_n$ is the Laplacian of the clique on $n$ vertices.

From this, we look up in Wikipedia the rules to form a semidefinite dual, and we get

$(7) \ \ \ \begin{array}{l} \max y\\ subject\ to\\L_G \succeq y\frac 1n L_n \end{array}$

It is possible to make sense of (7) independently of all the above discussion, that is, to see from first principles that the cost of any feasible solution to (7) is a lower bound to the sparsest cut of $G$.

A feasible solution is a value $y$ and a positive semidefinite matrix $Y$ such that $L_G = y\frac 1n L_n + Y$. Equivalently, it is a value $y$ and vectors $y_i$ such that

$L_G(i,j) = y\frac 1n L_n (i,j) + \langle y_i,y_j\rangle$

Take any cut $(S,V-S)$ in $G$. Then

$|S|\cdot |V-S| = 1_S^T L_n 1_S$

and

$edges(S,V-S) = 1_S^T L_G 1_S$

but

$1_S^T L_G 1_S = 1_S^T (y \frac 1n L_n + Y) 1_S \geq 1_S^T y \frac 1n L_n 1_S$

and so the cost of the cut is at least $y$. The justify the last inequality above, we need to prove that if $Y$ is positive semidefinite than

$1_S^T Y 1_S \geq 0$

Indeed

$1_S^T Y 1_S = \sum_{i,j\in S} \langle y_i,y_j \rangle = \langle \sum_{i\in S} y_i,\sum_{i\in S} y_i \rangle = || \sum_{i\in S} y_i ||^2$

We can also see, from first principles, that if $\lambda_1=d,\lambda_2,\ldots,\lambda_n$ are the eigenvalues of $A$ and $v_1,\ldots,v_n$ is a corresponding system of eigenvectors, we can find a feasible solution to (7) of cost $d-\lambda_2$.

We defined $L_G = dI - A$, where we assume that $G$ is $d$-regular. We can write

$dI - A$
$= dI - \sum_i \lambda_i v_i v_i^T$
$= (d-\lambda_2) \cdot\left( I - v_1v_1^T \right) +\lambda_2 I - \left( \lambda_2 v_1v_1^T+ \sum_{i=2}^n \lambda_i v_i v_i^T \right)$

And we are almost there because $\left( I - v_1v_1^T \right) = \frac 1n L_n$, and it remains to prove that the matrix

$Y:= \lambda_2 I - \left( \lambda_2 v_1v_1^T+ \sum_{i=2}^n \lambda_i v_i v_i^T \right)$

is positive semidefinite. This we can see by noting that $v_1,\ldots,v_n$ are eigenvectors of $Y$ with eigenvalues $0,0,\lambda_2-\lambda_3,\ldots,\lambda_2-\lambda_n$. Alternatively, we could construct vectors $y_i$ such that $Y(i,j) = \langle y_i,y_j\rangle$.

Essentially, what is going on is that we are “embedding” a scaled version of the complete graph into our graph. The edge expansion and sparsest cut of the complete graph are known, and the semidefinite constraint in (7) is telling us that the difference between our graph and the scaled complete graph can only add to the sparsest cut, so the sparsest cut of $G$ is at least the known sparsest cut of the scaled complete graph.

Why did we go through all this trouble? Now that we understand the spectral gap inside out, we can derive the Arora-Rao-Vazirani relaxation (ARV) of sparsest cut by just adding the “triangle inequalities” to (5), and we can derive the Leighton-Rao relaxation by removing the semidefinite constraints from ARV. In the dual of Leighton-Rao, we again “embed” a complete graph in our graph, by realizing every edge as a flow. In the dual of ARV, we will find both flows and spectral considerations.

[to be continued]


text: A bullet of mass 4.2 g strikes a ballistic pendulum of mass 3.8 kg. The center of mass of the pendulum rises a vertical distance of 5.9 cm. Assuming that the bullet remains embedded in the pendulum, calculate the bullet's initial speed.