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# Vector and matrix norms

### bookdepositoryImage bookdepositoryIn mathematics,a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices(of given dimensions).{displaystyle |A|} ).Thus,the matrix norm is a functionMatrix norm - Wikipedia

Was this helpful?People also askWhat is 2 norm of matrix?What is 2 norm of matrix?In MatLab,the 1-norm,2-norm and -norm are invoked by the statements norm(A,1) ,norm(A,2) ,and norm(A,inf) ,respectively.The 2-norm is the defaultin MatLab.The statement norm(A) is interpreted as norm(A,2) by MatLab.Since the 2-norm used in the majority of applications,we will adopt it as our default.Chapter 4 Matrix Norms - UAH - EngineeringVector and matrix norms in SAS - The DO LoopApr 07,2014 Vector and matrix norms#0183;Matrix norms might be less familiar to some readers.A matrix norm tells you how much a matrix can stretch a vector.For a matrix A,the matrix norm of A is the maximum value of the ratio |Av |/ |v ||,where v is any nonzero vector.The matrix norm of A is denoted |A |and depends on (or subordinate to) a vector norm.Vector and matrix norms in SAS - The DO LoopApr 07,2014 Vector and matrix norms#0183;Matrix norms might be less familiar to some readers.A matrix norm tells you how much a matrix can stretch a vector.For a matrix A,the matrix norm of A is the maximum value of the ratio |Av |/ |v ||,where v is any nonzero vector.The matrix norm of A is denoted |A |and depends on (or subordinate to) a vector norm.

### Vector and Matrix Norms

ECS231 Handout Vector and Matrix Norms Norms are an indispensable tool to provide vectors and matrices with measures of size,length and distance.I.Vector norms 1.A vector norm on Cnis a mapping that maps each x2Cnto a real number kxk,satisfying (a)Vector and Matrix Norms - Universitetet i osloon Cn be vector norms,and let kkbe a matrix norm on Cm;n.I We say that the matrix norm kkis subordinate to the vector norm kk if kAxk kAkkxk for all A 2Cm;n and all x 2Cn.I The Frobenius norm is subordinate to the Euclidian vector norm.I The p matrix norm is subordinate to the p vector normVector and Matrix Norms - Iowa State UniversityVector and Matrix Norms 5.1 Vector Norms A vector norm is a measure for the size of a vector.De nition 5.1.A norm on a real or complex vector space V is a mapping V !R with properties (a) kvk 0 8v (b) kvk= 0 ,v= 0 (c) k vk= j jkvk (d) kv+ wk kvk+ kwk (triangle inequality) De nition 5.2.The vector p-norm

### Vector Norms - USM

Any vector norm induces a matrix norm.It can be shown that given a vector norm,de ned appropriately for m-vectors and n-vectors,the function kk Rm n!R de ned by kAk= sup x6=0 kAxk kxk = max kxk=1 kAxk is a matrix norm.It is called the natural,or induced,matrix norm.Furthermore,if the vector norm is a Vector Norm and Matrix Norm (L0,L1,L2) Develop PaperIntuitive understanding In the real field,the size of a number and the distance between two numbers are measured by absolute values.In analytic geometry,the size of a vector and the difference between two vectors are measured by the concepts of length and distance.In order to make numerical analysis of matrix operations,we []Vector Norm - an overview ScienceDirect TopicsWe used vector norms to measure the length of a vector,and we will develop matrix norms to measure the size of a matrix.The size of a matrix is used in determining whether the solution,x,of a linear system Ax = b can be trusted,and determining the convergence rate of a vector sequence,among other things.We define a matrix norm in the same way we defined a vector norm.

### VECTOR AND MATRIX NORMS - University of Arizona

Jan 17,2004 Vector and matrix norms#0183;vector and matrix norms The question of how big? is a vector is not very complicated,and theres no reason not to do it early on in a course like this.In fact,when you took Vector Calculus (Math 223 or some equivalent),one of the first things you did was learn the Pythagorean formula for the length of a vector.VECTOR AND MATRIX NORMS - University of ArizonaJan 17,2004 Vector and matrix norms#0183;vector and matrix norms The question of how big? is a vector is not very complicated,and theres no reason not to do it early on in a course like this.In fact,when you took Vector Calculus (Math 223 or some equivalent),one of the first things you did was learn the Pythagorean formula for the length of a vector.VECTOR AND MATRIX NORMS - University of ArizonaJan 17,2004 Vector and matrix norms#0183;vector and matrix norms The question of how big? is a vector is not very complicated,and theres no reason not to do it early on in a course like this.In fact,when you took Vector Calculus (Math 223 or some equivalent),one of the first things you did was learn the Pythagorean formula for the length of a vector.

### PPT Vector and matrix norms PowerPoint presentation

Vector and matrix norms.A norm is the measure of the magnitude of a matrix/vector (matrix size) Denoted ; The same symbol is used to denote both vector and matrix norms,the specific type should be inferred from the context ; Properties ; 7 Norms.There are infinitely many norms that can be constructed.Most common norms (Euclidian norm Notes on Vector and Matrix Norms3 Matrix Norms It is not hard to see that vector norms are all measures of how \big the vectors are.Similarly,we want to have measures for how \big matrices are.We will start with one that are somewhat arti cial and then move on to the important class of induced matrix norms.3.1 Frobenius norm De nition 12.The Frobenius norm kk F Cm n!R Notes on Vector and Matrix Norms - PeopleNov 14,1999 Vector and matrix norms#0183;Notes on Vector and Matrix Norms These notes survey most important properties of norms for vectors and for linear maps from one vector space to another,and of maps norms induce between a vector space and its dual space.Dual Spaces and Transposes of Vectors Along with any space of real vectors x comes its dual space of linear functionals w T

### Notes on Vector and Matrix Norms - People

Nov 14,1999 Vector and matrix norms#0183;Notes on Vector and Matrix Norms These notes survey most important properties of norms for vectors and for linear maps from one vector space to another,and of maps norms induce between a vector space and its dual space.Dual Spaces and Transposes of Vectors Along with any space of real vectors x comes its dual space of linear functionals w TMatrix and vector norms A norm is a value that provides Matrix and vector norms A norm is a value that provides some measure of the size or length of a vector or matrix The norm of a vector or matrix is expressed as or Often a subscript is used to specify the type of norm. The Euclidean norm is common Compute the Euclidean norm forMatrix NormsEquivalent norms Adapting some general results on vector norms to matrix norms give Theorem 2.x 1.All matrix norms are equivalent.Thus,if kk and kk are two matrix norms on Cm,n then there are positive constants Vector and matrix norms#181; and M such that Vector and matrix norms#181;kAk kAk MkAk holds for all A Cm,n.2.

### MATH2071 LAB 5 Norms,Errors and Whatnot

But that vector-bound matrix norm is not always the only choice.In particular,the matrix norm is difficult (time-consuming) to compute,but there is a simple alternative.Note that The ,and matrix norms can be shown to be vector-bound to the corresponding vector norms and hence are guaranteed to be compatible with them;MATH2071 LAB 5 Norms,Errors and WhatnotBut that vector-bound matrix norm is not always the only choice.In particular,the matrix norm is difficult (time-consuming) to compute,but there is a simple alternative.Note that The ,and matrix norms can be shown to be vector-bound to the corresponding vector norms and hence are guaranteed to be compatible with them;Lectures - Week 4 Matrix norms,Conditioning,Vector Matrix norms,Conditioning,Vector Spaces,Linear Independence,Spanning sets and Basis,Null space and Range of a Matrix Matrix Norms Now we turn to associating a number to each matrix.We could choose our norms anal-ogous to the way we did for vector norms; e.g.,

### Lectures - Week 4 Matrix norms,Conditioning,Vector

Matrix norms,Conditioning,Vector Spaces,Linear Independence,Spanning sets and Basis,Null space and Range of a Matrix Matrix Norms Now we turn to associating a number to each matrix.We could choose our norms anal-ogous to the way we did for vector norms; e.g.,Gentle Introduction to Vector Norms in Machine LearningAug 09,2019 Vector and matrix norms#0183;Calculating the length or magnitude of vectors is often required either directly as a regularization method in machine learning,or as part of broader vector or matrix operations.In this tutorial,you will discover the different ways to calculate vector lengths or magnitudes,called the vector norm.After completing this tutorial,you will know The L1 norm that is calculated as theFile Size 248KBPage Count 42Vectors,Matrices and Norms - CS 357A general matrix norm is a real valued function that satisfies the following properties Positivity Definiteness if and only if ; Homogeneity for all scalars ; Triangle inequality Induced (or operator) matrix norms are associated with a specific vector norm and are defined as An induced matrix norm is a particular type of a general matrix

### Chapter 6 Vector Norms and Matrix Norms

matrix norms is that they should behave well with re-spect to matrix multiplication.Denition 6.3.A matrix norm kkon the space of square nn matrices in Mn(K),with K = R or K = C,is a norm on the vector space Mn(K),with the additional property called submultiplicativity that kABk kAkkBk,for all A,B 2 Mn(K).A norm on matrices Chapter 4 Matrix Norms - UAH - EngineeringThe Matrix 1-Norm Recall that the vector 1-norm is given by r X i n 1 1 = = xi.(4-7) Subordinate to the vector 1-norm is the matrix 1-norm A a j ij i 1 = F HG I max KJ.(4-8) That is,the matrix 1-norm is the maximum of the column sums .To see this,let m Vector and matrix norms#215;n matrixChapter 4 Matrix Norms - UAH - EngineeringThe Matrix 1-Norm Recall that the vector 1-norm is given by r X i n 1 1 = = xi.(4-7) Subordinate to the vector 1-norm is the matrix 1-norm A a j ij i 1 = F HG I max KJ.(4-8) That is,the matrix 1-norm is the maximum of the column sums .To see this,let m Vector and matrix norms#215;n matrix

### C.5 Vector and Matrix Norms

The observation that the $\ell_{2}$ norm of the vector of singular values of a matrix is identical to its Frobenius norm motivates the use of other $\ell_{p}$ norms on the vector $\mathbf{s}$.In particular the $\ell_{1}$ norm of $\mathbf{s}$ defines the nuclear norm of $\mathbf{X}$ denoted by $\left\Vert \mathbf{X}\right\Vert _{*}$C.5 Vector and Matrix NormsThe observation that the $\ell_{2}$ norm of the vector of singular values of a matrix is identical to its Frobenius norm motivates the use of other $\ell_{p}$ norms on the vector $\mathbf{s}$.In particular the $\ell_{1}$ norm of $\mathbf{s}$ defines the nuclear norm of $\mathbf{X}$ denoted by $\left\Vert \mathbf{X}\right\Vert _{*}$7.1 Norms of Vectors and Matrices4 The distance between matrices and with respect to a matrix norm is Theorem 7.9.If is a vector norm,the induced (or natural) matrix norm is given by Exampleduced the ,norm.the ,induced norm.Alternative definition For any vector ,the vector has Since

### 7.1 Norms of Vectors and Matrices

4 The distance between matrices and with respect to a matrix norm is Theorem 7.9.If is a vector norm,the induced (or natural) matrix norm is given by Exampleduced the ,norm.the ,induced norm.Alternative definition For any vector ,the vector has Since7.1 Norms of Vectors and Matrices4 The distance between matrices and with respect to a matrix norm is Theorem 7.9.If is a vector norm,the induced (or natural) matrix norm is given by Exampleduced the ,norm.the ,induced norm.Alternative definition For any vector ,the vector has Since4.7 Vector and Matrix Norms Department of Electrical and The 1- and -Matrix Norms.It can be shown that the 1-norm of a matrix M,if we use the same definition from above,may be given by the maximum column sum of the absolute values of the entries of the matrix,or using Matlab Vector and matrix normsgt; Vector and matrix normsgt; max( sum( abs( M ) ) )

### 4.7 Vector and Matrix Norms Department of Electrical and

The 1- and -Matrix Norms.It can be shown that the 1-norm of a matrix M,if we use the same definition from above,may be given by the maximum column sum of the absolute values of the entries of the matrix,or using Matlab Vector and matrix normsgt; Vector and matrix normsgt; max( sum( abs( M ) ) )1 Norms and Vector Spaces - Stanford University1 Norms and Vector Spaces 2008.10.07.01 The induced 2-norm.Suppose A Rm Vector and matrix norms#215;n is a matrix,which denes a linear map from Rn to Rm in the usual way.Then the induced 2-norm of A is kAk = 1(A) where 1 is the largest singular value of the matrix A.This is also called the spectral norm results for this questionWhat is the norm of matrix in MATLAB?What is the norm of matrix in MATLAB?MATLAB Function Reference.The norm of a matrix is a scalar that gives some measure of the magnitude of the elements of the matrix.The norm function calculates several different types of matrix norms n = norm(A)returns the largest singular value of A,max(svd(A)).norm (MATLAB Functions)

### results for this questionWhat is the norm of a matrix?What is the norm of a matrix?The norm of a matrix is a scalar that gives some measure of the magnitude of the elements of the matrix.norm (MATLAB Functions) results for this questionWhat is a subordinate matrix norm?What is a subordinate matrix norm?The matrix norm A , {\displaystyle \|A\|_{\alpha ,\beta }} is sometimes called a subordinate norm.Subordinate norms are consistent with the norms that induce them,giving.Matrix norm - Wikipedia,the free encyclopedia results for this questionFeedbackChapter 4 Vector Norms and Matrix Norms

Vector Norms and Matrix Norms 4.1 Normed Vector Spaces In order to dene how close two vectors or two matrices are,and in order to dene the convergence of sequences of vectors or matrices,we can use the notion of a norm.Recall that R + = {x R x 0}.

### bookdepositoryImage bookdepositoryIn mathematics,a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices(of given dimensions).{displaystyle |A|} ).Thus,the matrix norm is a functionMatrix norm - Wikipedia

Was this helpful?People also askWhat is 2 norm of matrix?What is 2 norm of matrix?In MatLab,the 1-norm,2-norm and -norm are invoked by the statements norm(A,1) ,norm(A,2) ,and norm(A,inf) ,respectively.The 2-norm is the defaultin MatLab.The statement norm(A) is interpreted as norm(A,2) by MatLab.Since the 2-norm used in the majority of applications,we will adopt it as our default.Chapter 4 Matrix Norms - UAH - EngineeringVector and matrix norms in SAS - The DO LoopApr 07,2014 Vector and matrix norms#0183;Matrix norms might be less familiar to some readers.A matrix norm tells you how much a matrix can stretch a vector.For a matrix A,the matrix norm of A is the maximum value of the ratio |Av |/ |v ||,where v is any nonzero vector.The matrix norm of A is denoted |A |and depends on (or subordinate to) a vector norm.Vector and matrix norms in SAS - The DO LoopApr 07,2014 Vector and matrix norms#0183;Matrix norms might be less familiar to some readers.A matrix norm tells you how much a matrix can stretch a vector.For a matrix A,the matrix norm of A is the maximum value of the ratio |Av |/ |v ||,where v is any nonzero vector.The matrix norm of A is denoted |A |and depends on (or subordinate to) a vector norm.

### Vector and Matrix Norms

ECS231 Handout Vector and Matrix Norms Norms are an indispensable tool to provide vectors and matrices with measures of size,length and distance.I.Vector norms 1.A vector norm on Cnis a mapping that maps each x2Cnto a real number kxk,satisfying (a)Vector and Matrix Norms - Universitetet i osloon Cn be vector norms,and let kkbe a matrix norm on Cm;n.I We say that the matrix norm kkis subordinate to the vector norm kk if kAxk kAkkxk for all A 2Cm;n and all x 2Cn.I The Frobenius norm is subordinate to the Euclidian vector norm.I The p matrix norm is subordinate to the p vector normVector and Matrix Norms - Iowa State UniversityVector and Matrix Norms 5.1 Vector Norms A vector norm is a measure for the size of a vector.De nition 5.1.A norm on a real or complex vector space V is a mapping V !R with properties (a) kvk 0 8v (b) kvk= 0 ,v= 0 (c) k vk= j jkvk (d) kv+ wk kvk+ kwk (triangle inequality) De nition 5.2.The vector p-norm

### Vector Norms - USM

Any vector norm induces a matrix norm.It can be shown that given a vector norm,de ned appropriately for m-vectors and n-vectors,the function kk Rm n!R de ned by kAk= sup x6=0 kAxk kxk = max kxk=1 kAxk is a matrix norm.It is called the natural,or induced,matrix norm.Furthermore,if the vector norm is a Vector Norm and Matrix Norm (L0,L1,L2) Develop PaperIntuitive understanding In the real field,the size of a number and the distance between two numbers are measured by absolute values.In analytic geometry,the size of a vector and the difference between two vectors are measured by the concepts of length and distance.In order to make numerical analysis of matrix operations,we []Vector Norm - an overview ScienceDirect TopicsWe used vector norms to measure the length of a vector,and we will develop matrix norms to measure the size of a matrix.The size of a matrix is used in determining whether the solution,x,of a linear system Ax = b can be trusted,and determining the convergence rate of a vector sequence,among other things.We define a matrix norm in the same way we defined a vector norm.

### VECTOR AND MATRIX NORMS - University of Arizona

Jan 17,2004 Vector and matrix norms#0183;vector and matrix norms The question of how big? is a vector is not very complicated,and theres no reason not to do it early on in a course like this.In fact,when you took Vector Calculus (Math 223 or some equivalent),one of the first things you did was learn the Pythagorean formula for the length of a vector.VECTOR AND MATRIX NORMS - University of ArizonaJan 17,2004 Vector and matrix norms#0183;vector and matrix norms The question of how big? is a vector is not very complicated,and theres no reason not to do it early on in a course like this.In fact,when you took Vector Calculus (Math 223 or some equivalent),one of the first things you did was learn the Pythagorean formula for the length of a vector.VECTOR AND MATRIX NORMS - University of ArizonaJan 17,2004 Vector and matrix norms#0183;vector and matrix norms The question of how big? is a vector is not very complicated,and theres no reason not to do it early on in a course like this.In fact,when you took Vector Calculus (Math 223 or some equivalent),one of the first things you did was learn the Pythagorean formula for the length of a vector.

### PPT Vector and matrix norms PowerPoint presentation

Vector and matrix norms.A norm is the measure of the magnitude of a matrix/vector (matrix size) Denoted ; The same symbol is used to denote both vector and matrix norms,the specific type should be inferred from the context ; Properties ; 7 Norms.There are infinitely many norms that can be constructed.Most common norms (Euclidian norm Notes on Vector and Matrix Norms3 Matrix Norms It is not hard to see that vector norms are all measures of how \big the vectors are.Similarly,we want to have measures for how \big matrices are.We will start with one that are somewhat arti cial and then move on to the important class of induced matrix norms.3.1 Frobenius norm De nition 12.The Frobenius norm kk F Cm n!R Notes on Vector and Matrix Norms - PeopleNov 14,1999 Vector and matrix norms#0183;Notes on Vector and Matrix Norms These notes survey most important properties of norms for vectors and for linear maps from one vector space to another,and of maps norms induce between a vector space and its dual space.Dual Spaces and Transposes of Vectors Along with any space of real vectors x comes its dual space of linear functionals w T

### Notes on Vector and Matrix Norms - People

Nov 14,1999 Vector and matrix norms#0183;Notes on Vector and Matrix Norms These notes survey most important properties of norms for vectors and for linear maps from one vector space to another,and of maps norms induce between a vector space and its dual space.Dual Spaces and Transposes of Vectors Along with any space of real vectors x comes its dual space of linear functionals w TMatrix and vector norms A norm is a value that provides Matrix and vector norms A norm is a value that provides some measure of the size or length of a vector or matrix The norm of a vector or matrix is expressed as or Often a subscript is used to specify the type of norm. The Euclidean norm is common Compute the Euclidean norm forMatrix NormsEquivalent norms Adapting some general results on vector norms to matrix norms give Theorem 2.x 1.All matrix norms are equivalent.Thus,if kk and kk are two matrix norms on Cm,n then there are positive constants Vector and matrix norms#181; and M such that Vector and matrix norms#181;kAk kAk MkAk holds for all A Cm,n.2.

### MATH2071 LAB 5 Norms,Errors and Whatnot

But that vector-bound matrix norm is not always the only choice.In particular,the matrix norm is difficult (time-consuming) to compute,but there is a simple alternative.Note that The ,and matrix norms can be shown to be vector-bound to the corresponding vector norms and hence are guaranteed to be compatible with them;MATH2071 LAB 5 Norms,Errors and WhatnotBut that vector-bound matrix norm is not always the only choice.In particular,the matrix norm is difficult (time-consuming) to compute,but there is a simple alternative.Note that The ,and matrix norms can be shown to be vector-bound to the corresponding vector norms and hence are guaranteed to be compatible with them;Lectures - Week 4 Matrix norms,Conditioning,Vector Matrix norms,Conditioning,Vector Spaces,Linear Independence,Spanning sets and Basis,Null space and Range of a Matrix Matrix Norms Now we turn to associating a number to each matrix.We could choose our norms anal-ogous to the way we did for vector norms; e.g.,

### Lectures - Week 4 Matrix norms,Conditioning,Vector

Matrix norms,Conditioning,Vector Spaces,Linear Independence,Spanning sets and Basis,Null space and Range of a Matrix Matrix Norms Now we turn to associating a number to each matrix.We could choose our norms anal-ogous to the way we did for vector norms; e.g.,Gentle Introduction to Vector Norms in Machine LearningAug 09,2019 Vector and matrix norms#0183;Calculating the length or magnitude of vectors is often required either directly as a regularization method in machine learning,or as part of broader vector or matrix operations.In this tutorial,you will discover the different ways to calculate vector lengths or magnitudes,called the vector norm.After completing this tutorial,you will know The L1 norm that is calculated as theFile Size 248KBPage Count 42Vectors,Matrices and Norms - CS 357A general matrix norm is a real valued function that satisfies the following properties Positivity Definiteness if and only if ; Homogeneity for all scalars ; Triangle inequality Induced (or operator) matrix norms are associated with a specific vector norm and are defined as An induced matrix norm is a particular type of a general matrix

### Chapter 6 Vector Norms and Matrix Norms

matrix norms is that they should behave well with re-spect to matrix multiplication.Denition 6.3.A matrix norm kkon the space of square nn matrices in Mn(K),with K = R or K = C,is a norm on the vector space Mn(K),with the additional property called submultiplicativity that kABk kAkkBk,for all A,B 2 Mn(K).A norm on matrices Chapter 4 Matrix Norms - UAH - EngineeringThe Matrix 1-Norm Recall that the vector 1-norm is given by r X i n 1 1 = = xi.(4-7) Subordinate to the vector 1-norm is the matrix 1-norm A a j ij i 1 = F HG I max KJ.(4-8) That is,the matrix 1-norm is the maximum of the column sums .To see this,let m Vector and matrix norms#215;n matrixChapter 4 Matrix Norms - UAH - EngineeringThe Matrix 1-Norm Recall that the vector 1-norm is given by r X i n 1 1 = = xi.(4-7) Subordinate to the vector 1-norm is the matrix 1-norm A a j ij i 1 = F HG I max KJ.(4-8) That is,the matrix 1-norm is the maximum of the column sums .To see this,let m Vector and matrix norms#215;n matrix

### C.5 Vector and Matrix Norms

The observation that the $\ell_{2}$ norm of the vector of singular values of a matrix is identical to its Frobenius norm motivates the use of other $\ell_{p}$ norms on the vector $\mathbf{s}$.In particular the $\ell_{1}$ norm of $\mathbf{s}$ defines the nuclear norm of $\mathbf{X}$ denoted by $\left\Vert \mathbf{X}\right\Vert _{*}$C.5 Vector and Matrix NormsThe observation that the $\ell_{2}$ norm of the vector of singular values of a matrix is identical to its Frobenius norm motivates the use of other $\ell_{p}$ norms on the vector $\mathbf{s}$.In particular the $\ell_{1}$ norm of $\mathbf{s}$ defines the nuclear norm of $\mathbf{X}$ denoted by $\left\Vert \mathbf{X}\right\Vert _{*}$7.1 Norms of Vectors and Matrices4 The distance between matrices and with respect to a matrix norm is Theorem 7.9.If is a vector norm,the induced (or natural) matrix norm is given by Exampleduced the ,norm.the ,induced norm.Alternative definition For any vector ,the vector has Since

### 7.1 Norms of Vectors and Matrices

4 The distance between matrices and with respect to a matrix norm is Theorem 7.9.If is a vector norm,the induced (or natural) matrix norm is given by Exampleduced the ,norm.the ,induced norm.Alternative definition For any vector ,the vector has Since7.1 Norms of Vectors and Matrices4 The distance between matrices and with respect to a matrix norm is Theorem 7.9.If is a vector norm,the induced (or natural) matrix norm is given by Exampleduced the ,norm.the ,induced norm.Alternative definition For any vector ,the vector has Since4.7 Vector and Matrix Norms Department of Electrical and The 1- and -Matrix Norms.It can be shown that the 1-norm of a matrix M,if we use the same definition from above,may be given by the maximum column sum of the absolute values of the entries of the matrix,or using Matlab Vector and matrix normsgt; Vector and matrix normsgt; max( sum( abs( M ) ) )

### 4.7 Vector and Matrix Norms Department of Electrical and

The 1- and -Matrix Norms.It can be shown that the 1-norm of a matrix M,if we use the same definition from above,may be given by the maximum column sum of the absolute values of the entries of the matrix,or using Matlab Vector and matrix normsgt; Vector and matrix normsgt; max( sum( abs( M ) ) )1 Norms and Vector Spaces - Stanford University1 Norms and Vector Spaces 2008.10.07.01 The induced 2-norm.Suppose A Rm Vector and matrix norms#215;n is a matrix,which denes a linear map from Rn to Rm in the usual way.Then the induced 2-norm of A is kAk = 1(A) where 1 is the largest singular value of the matrix A.This is also called the spectral norm results for this questionWhat is the norm of matrix in MATLAB?What is the norm of matrix in MATLAB?MATLAB Function Reference.The norm of a matrix is a scalar that gives some measure of the magnitude of the elements of the matrix.The norm function calculates several different types of matrix norms n = norm(A)returns the largest singular value of A,max(svd(A)).norm (MATLAB Functions)

### results for this questionWhat is the norm of a matrix?What is the norm of a matrix?The norm of a matrix is a scalar that gives some measure of the magnitude of the elements of the matrix.norm (MATLAB Functions) results for this questionWhat is a subordinate matrix norm?What is a subordinate matrix norm?The matrix norm A , {\displaystyle \|A\|_{\alpha ,\beta }} is sometimes called a subordinate norm.Subordinate norms are consistent with the norms that induce them,giving.Matrix norm - Wikipedia,the free encyclopedia results for this questionFeedbackChapter 4 Vector Norms and Matrix Norms

Vector Norms and Matrix Norms 4.1 Normed Vector Spaces In order to dene how close two vectors or two matrices are,and in order to dene the convergence of sequences of vectors or matrices,we can use the notion of a norm.Recall that R + = {x R x 0}.

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# Tensor

## Basics

Tensor is a parent object type for all mathematical expressions used in Redberry. Thus, all expressions in Redberry are tensors. From the programming point of view that means, that Tensor is a parent class for all types of mathematical expressions used in Redberry (sums, products, etc.). Tensor defines common properties and functional shared in all its subtypes.

## Common properties

There are several unifying properties of all mathematical expressions like size, indices, container properties etc. All these properties can be obtained from any expression using methods defined in Tensor. Let us consider each property in detail.

### Indices

Presence of indices it a key feature of all tensorial expressions. Indices can be obtained for any expression using .indices property (or equivalently getIndices() method):

println 'g_ab * T^cd'.t.indices

   > ^cd_ab

println 'T_ba^dc'.t.indices

   > _ba^dc

(in case of pure scalars .indices will return Indices with zero size).

Indices is a feature which is inherent in any type of expression (sum, product, function, simple tensor etc.). Meanwhile, indices of different types of expressions can have slightly different nature. For example, indices of SimpleTensor will appear exactly in that order as they were specified by the user, while indices of e.g. Product will be always sorted in some way; indices of Sum are sorted free indices of its summands etc. Thus, Tensor defines only abstract method .indices while each particular subtype of Tensor is responsible for implementation. For more details see Tensors and Indices.

### Container

Any tensor has a content (summands in case of Sum, multipliers in case of Product etc.). The following table summarises, how one can access container properties of expressions:

 T.size() returns a size of tensor, i.e. number of summands in case of sum, number of arguments in case of function etc. T[i] returns element at i-th position T[i..j] returns a list of elements from i-th (inclusive) to j-th (exclusive) T[i, j, k] returns a list of elements at specified positions T.set(i, expr) returns a new tensor with expr at i-th position
(one can also find more special methods like remove in Sum and Product)

One can also construct Tensor from a list in the following way

def list = ['a', 'b', 'c']
println list as Sum // same as list.sum()

> a + c + b

def t = 'a + b + c + d + e + f + g'.t
println t[1..4] as Product // same as t[1..4].multiply()

> b*d*e

Note, that the ordering of expressions (summands in sum, or multipliers in product) can be different from that one in which the expression was inputted by the user. So, if one inputted e.g.

def t = 'a + b + c'.t

and then take t[0], the result will not necessary be a but also can be b or c. This ordering will be always same during a single run of Redberry, by can be changed from run to run. In order to make this ordering same in all runs one can put the following line in the beginning of the code:

CC.resetTensorNames(1)

### Iterable

This simply means that one can iterate through expression content using for loop:

def expr = 'a*(b+e)*Sin[c]*d*(e+f)'.t
def sums = []
for (def t in expr) //iterating through expr
if (t instanceof Sum) //taking only sums
sums << t
println sums

   > [b+e, e+f]

println sums as Sum

   > b + 2*e + f

In such a way one can iterate only through direct "children" of expression. In order to iterate through all expression parts, one should use Tree traversal.

## Creating tensors programmatically

In most cases one will input expression by hand, but often it becomes necessary to construct tensors programmatically from a given content. There are a variety ways to do that in Redberry.

First of all, Redberry allows to use simple algebraic operators on tensors:

def x = 'x'.t, y = 'y'.t
println x/y + sin(x*y) + y**3

   > x/y + Sin[x] + y**3

(however such syntax may be inconvenient for indexed expressions in some cases, since it does not reflect presence of indices)

Sometimes we need to construct expression from a given content. Here is a some useful methods that allow to do that:

 sum(exprs) returns a sum of specified expressions multiply(exprs) returns a product of specified expressions simpleTensor(head, indices) returns a simple tensor with specified name and simple indices field(head, indices, args) returns a tensor field with specified name, simple indices and arguments fieldDerivative(field, dIndices, arg, order) returns a derivative of field with respect to specified argument, argument indices and order

Consider examples:

def list = ['a', 'b', 'c'].t
println sum(list)

   > a+b+c

println multiply(list)

   > a*b*c

println simpleTensor('f', '_abc'.si)

   > f_abc

println field('f', '_abc'.si, list)

   > f_abc[a,b,c]

println fieldDerivative('f_a[x_a, y_b]'.t, '_ij'.si, 1, 2)

   > f~(0,2)_{aij}[x_a, y_b]

One should be careful when using multiply() to multiply tensors with dummy indices. By default, multiply() will not rename conflicting dummy indices:

println multiply('A^a_a'.t, 'B^a_a'.t)

   > InconsistentIndicesException: Inconsistent index ^a.

If one need to rename such dummies, one can use multiplyAndRenameConflictingDummies():
println multiplyAndRenameConflictingDummies('A^a_a'.t, 'B^a_a'.t)

   > A^{a}_{a}*B^{b}_{b}

## Advanced features: builders and factories

In some cases (especially when writing custom transformations) it becomes necessary to rebuild expression without knowing it particular type in advance. For this purpose, each expression provide a “builder” (via .builder property), which allows to construct tensor of same type.

For example, suppose we need to implement a “toy” transformation which simply removes all simple tensors from product or sum. This can be done in the following way:

def tr = { expr ->
def builder = expr.builder
for (def t in expr)
if (t.class != SimpleTensor)
builder << t
builder.build()
} as Transformation

def product = 'a*(b+e)*Sin[c]*d*(e+f)'.t
println tr >> product

   > Sin[c]*(e+b)*(f+e)

def sum = 'a+b*e+Sin[c]+d+e*f'.t
println tr >> sum

   > Sin[c]+b*e+e*f

When the length of result is known in advance, one can use .factory. Consider a “toy” transformation which appends tensor a to expression:

def tr = { Tensor expr ->
def data = expr as List
data << 'a'.t
expr.factory.create(data)
} as Transformation

def product = 'f[x]*Sin[a]'.t
println tr >> product

   > a*Sin[a]*f[x]

println tr >> (product as List).sum()

   > a+Sin[a]+f[x]

The general convention on builders and factories is that if t is any expression, then the result of rebuild is same:

assert t == (t.builder << (t as List)).build()
assert t == t.factory.create(t as List)

format: plain


text:
# Homework Help: Parametric Equation of a sphere

1. Jan 15, 2010

### anubis01

1. The problem statement, all variables and given/known data
Find parametric equations for the part of sphere x2+y2+z2=9 that lies between the planes y=1 and y=2.

2. Relevant equations

3. The attempt at a solution

Okay knowing that the p=3 I wrote the parametric equations for a sphere as
x=3sin$$\phi$$cos$$\theta$$ y=3sin$$\phi$$sin$$\theta$$
z=3cos$$\phi$$ now the phi bound is 0<$$\phi$$<$$\pi$$
but I'm not sure what to write for the $$\theta$$ bound.

2. Jan 15, 2010

### Dick

It would be easier if they said between z=2 and z=1, right? Then you would just have to restrict phi. There's no reason why you can't interchange say, y and z in your parametrization.

3. Jan 15, 2010

### anubis01

Oh okay so I can say z=3cos$$\phi$$
and from there plug in the values to find the $$\phi$$ bound to be cos(1/3)-1<$$\phi$$<cos(2/3)-1 and the $$\theta$$ bound should just be 0< $$\theta$$<2$$\pi$$

4. Jan 15, 2010

### Dick

Right. But they did say y=1 and y=2. You'll also have to tweak your parametrization so y=3*cos(theta) instead of z=3*cos(theta).

5. Jan 15, 2010

### anubis01

oh so then phi bound should be 0<$$\phi$$<$$\pi$$ and since y=3cos$$\theta$$ the bounds for $$\theta$$ are
cos(1/3)-1<$$\theta$$<cos(2/3)-1

6. Jan 15, 2010

### Dick

Sorry, sorry!!! I meant to say your parametrization has z=3*cos(phi) and you want to change it so that y=3*cos(phi). Leave theta as it is.

7. Jan 15, 2010

### anubis01

alright i think i got it now. So bound for theta 0<$$\theta$$<2$$\pi$$
y=3cos$$\phi$$ the bounds for phi are then
cos(1/3)-1<$$\phi$$<cos(2/3)-1

and the parametric equation for x=3sin$$\phi$$cos$$\theta$$
and z=3sin$$\phi$$sin$$\theta$$

8. Jan 15, 2010

### Dick

That looks good to me.

9. Jan 15, 2010

### anubis01

alright thank you for all the help, its much appreciated.


text:
# VirtualBox Tips

There is a lot of installation guides and configuration suggestion dedicated to the VirtualBox. There are also many problems solved that can be easily found in your favorite search engine. So I will not make you bored and I not discuss a well-written installation and configuration which you will find in the official documentation. I will summarize instead what should be done after installing VirtualBox so as to avoid basic errors. I will also mention a few solutions that every VirtualBox user is looking for, sooner or later.

I wrote down the following steps to make it easier for myself to work with VirtualBox and not to wonder every time, why something does not work, just because I forgot about a few obvious settings.

## Three simple steps

Do this after every Virtual Box installation on Linux host to make sure everything works correctly.

1. Add user to vboxusers group on host machine and do the same on guest machine

2. Install Extension Pack.

On guest machine install required packages

then Insert Guest Additions CD and run as root ./VBoxLinuxAdditions.run

With above 3 suggestion you should not have any problems with shared folder access, usb devices, copy & paste between host & guest, graphics adjustment.

## Reduce .vdi size

After some time, each VDI disk created with dynamically allocated space has the maximum size even though the system does not really occupy the entire virtual surface. It is worth then to compact its size. This method works only for VDI files.

### Linux

#### On guest

Cleanup whole empty space by nullify free space:

It can take some time.

#### On host

Use vboxmanage tool to compact disk

this command on VirtualBox version >=6.1.4 looks now:

### Windows

#### On guest

Run disk defragmentation.

Once done download SDelete tool and run it from command line with parameter:

replace c: with the drive letter of the VDI

#### On host

this command on VirtualBox version >=6.1.4 looks now:

## Disk conversion

If you would like to compact your disk but its in VMDK format you can easily covert it to VDI.

Just in case make a copy of drive before conversion.

## Resize disk

In case where your virtual disk is out of space you can resize it. This method also works for VDI files only.

Resize value should be set to the new size of whole disk, not the amount of new space you want to add.

Next, on guest host resize partition using your favorite partition manager, for example Gparted.

## Slow Windows guest

Since VirtualBox 6 if Hyper-V is enabled, then icon for green turtle will be shown in the notification area of VirtualBox, as shown below, and Virtual Machine will be slow.

In my case I had to turn off one of the option in Microsoft Windows Defender called Core isolation.

Also turn off Hyper-V using command in PowerShell (don’t forget to run PowerShell as Admin).

then reboot:


text:
# What are practical uses of over-damping? [closed]

We have been given this task of preparing some small research on critical damping and comparing its behaviour and uses with over-damping. I am done with everything else but have been unable to find practical uses of over damping. It'd be great if someone could explain where it's desired.

## closed as off-topic by JMac, Ruslan, ZeroTheHero, John Rennie, Jon CusterAug 22 '17 at 12:34

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question appears to be about engineering, which is the application of scientific knowledge to construct a solution to solve a specific problem. As such, it is off topic for this site, which deals with the science, whether theoretical or experimental, of how the natural world works. For more information, see this meta post." – JMac, Ruslan, ZeroTheHero, John Rennie, Jon Custer
If this question can be reworded to fit the rules in the help center, please edit the question.

• – Ruslan Aug 21 '17 at 17:31
• Hello, I have been through this page but I kind of found it to be vague and more of a convo between two people rather than something that I can understand as a newbie. – DmRo912 Aug 21 '17 at 17:35
• This question seems to be more of an engineering question, and potentially doing homework for you. My only hint: What happens in objects where you don't know it's exact characteristics (like mass/stiffness may vary to an extent, such as in a car) and overshoot must be avoided? – JMac Aug 21 '17 at 17:36
• The practical uses are situations where "overshooting" would be a disaster. To repeat the examples in the link, if a plane is landing and controlled by an autopilot, an overshoot which causes the plane to attempt to fly below the level of the runway probably won't end well! The same is true for controlling a machine tool (which is subject to random vibration caused by cutting the material) - if you cut too deep, you end up with scrap metal. Controlling an autonomous vehicle would be another example - you don't want to overshoot when stopping in a confined space for example. – alephzero Aug 22 '17 at 5:10
• @alephzero how about the automatic control of a foundry crucible pouring molten iron! Imagine what overshoot might lead to. – docscience Aug 22 '17 at 22:00

• Every textbook I've seen defines the critical damping factor as $1.0$ (exactly), not $\sim 0.7$. Is $0.7$ an approximation to $1/\sqrt{2}$, in some alternative formulation of the math? Or is it just an empirical number used in some application area (control engineering or whatever)? – alephzero Aug 22 '17 at 5:12


text:
# RS Markov Chain Generator

This project was developed in 2010, mainly for fun. It is now deprecated. This page only exists for archival purposes.

## Project summary

This application tries to generate somewhat correct text by using an algorithm called Markov chain. Simply put, Markov chain is an algorithm that makes its next state dependent on the previous one. Even more simpler, it analyzes correct text written by humans, makes a link between the used words, then tries to generate new text, knowing which word can follow which.

The software was written in C# and makes heavy use of the LINQ extensions to browse the database.

[Note from 2016 me: While this project is nothing serious, just me screwing around during an afternoon, I did get more serious with natural language processing during my university years. I co-created and published a sentiment analysis engine for the Hungarian language, see paper 10.1515/ausi-2015-0018.]

## Output

Well, obviousily, a simple algorithm won't be able to somehow turn into Skynet overnight and master the human language. So, rather than rating the text by correctness, rate it by lulz or epic fails. It's much more entertaining that way. However, don't try to rate the text by cuils. I tried it, the result is $943 \cdot 10^{1024}$ cuils.

## Dictionaries

For one-click fun, there are 3 included dictionaries: English, Hungarian and Romanian. For maximum lols, they're all scientifical, so they can generate gems like "The antigravity force ate the tortoise".

If you want to use these dictionaries for other purposes, you're free to use them. They're not encrypted, just compressed with System.IO.Compression.DeflateStream. After decompression deserialize them with System.Runtime.Serialization.Formatters.Binary.BinaryFormatter. (Actually, you can directly provide the DeflateStream to the BinaryFormatter, and save memory and time!)

## Markov Chain implementation

This is how I did it. Also you will need these if you want to deserialize the dictionaries.

[Serializable]
public class MarkovChain : Dictionary<string, Word>
{
public MarkovChain() : base() { }
public MarkovChain(SerializationInfo info, StreamingContext context) : base(info, context) { }

public new Word this[string key]
{
get
{
if (base.ContainsKey(key))
{
return base[key];
}
else
{
return base[key] = new Word(key);
}
}
set
{
base[key] = value;
}
}
}

[Serializable]
public class Word
{
public string Value;
public bool Starts;     // was it seen to start a sentence?
public bool Ends;       // ...or to end one?
public List<Word> Next;

public Word(string value, List<Word> next)
{
Value  = value;
Starts = false;
Ends   = false;
Next   = next ?? new List<Word>();
}

public Word(string value, bool starts = false, bool ends = false, List<Word> next = null)
{
Value  = value;
Starts = starts;
Ends   = ends;
Next   = next ?? new List<Word>();
}

public Word GetNext()
{
if (Next.Count == 0)
{
throw new Exception("Fun's over. Out of words.");
}

return Next[Utils.Random.Next(Next.Count)];
}
}

You will need to use C# 4.0 to compile this code, because of the optional parameters on the second constructor of the Word class. Once you have a dictionary, it's pretty easy to play with it. Pick a sentence starter word from the MarkovChain, then just call the GetNext() on the last Word to get the words. To actually look like a sentence, you will need to implement some grammar functions, and use Starts and Ends properties to determine whether a word can start or end a sentence.

If you make new dictionaries or you succeed in perfecting the text generator make sure to drop me an email! I'd like to see how better can get an algorithm at speaking.

### Ideas for perfecting the algorithm

• Create a larger and more accurate dictionary.
• ...which can be achieved by perfecting the text parser at RoliSoft.MarkovChain.Tokenizer.Tokenize()
• While spellchecking is not required, grammatical analysis is. You can prehaps use Microsoft Office's API for this?
• Create a deeper list, so it won't change the subject with its every word.
• Analyze and fix verb tenses. This issue is very annoying, especially in languages like Hungarian.


text:
# Elastic-Plastic Cross-Section Design

### Technical Article

New

001576

12 June 2019

The following article describes designing a two-span beam subjected to bending by means of the RF-/STEEL EC3 add-on module according to EN 1993-1-1. The global stability failure will be excluded due to sufficient stabilizing measures.

#### Design of Cross-Section Class

The area of the inner support of the two-span beam is governing for designing the cross-section class and performing the cross-section design.

Design for the web (ψ = -1):
[1] Table 5.2, cross-section parts supported on two sides
$\begin{array}{l}\mathrm c\;=\;590\;-\;2\;⋅\;(25\;+\;27)\;=\;486\;\mathrm{mm}\\\mathrm{existing}\;\frac{\mathrm c}{{\mathrm t}_{\mathrm w}}\;=\;\frac{486}{13}\;=\;37.38\\\mathrm{limit}\;\frac{\mathrm c}{{\mathrm t}_{\mathrm w}}\;=\;72\;⋅\;\mathrm\varepsilon\;=\;72\;⋅\;1\;=\;72\;>\;37.38\end{array}$
The web thus fulfills the requirements of cross-section class 1.

Design for the bottom chord (ψ = 1):
[1] Table 5.2, cross-section parts supported on one side
$\begin{array}{l}\mathrm c\;=\;\frac{300\;-\;(13\;+\;2\;\cdot\;27)}2\;=\;116.5\;\mathrm{mm}\\\mathrm{existing}\;\frac{\mathrm c}{{\mathrm t}_{\mathrm f}}\;=\;\frac{116.5}{25}\;=\;4.66\;<\;9\;⋅\;\mathrm\varepsilon\;=\;9\;⋅\;1\;=\;9\end{array}$
The chords thus fulfill the requirements of cross-section class 1. The cross-section has to be assigned to cross-section class 1.

#### Elastic-Plastic Cross-Section Design

$\begin{array}{l}{\mathrm W}_{\mathrm{pl},\mathrm y}\;=\;2\;\cdot\;\left(30\;\cdot\;2.5\;\cdot\;28.25\;+\;27\;\cdot\;1.3\;\cdot\;16\right)\;=\;5,360\;\mathrm{cm}^3\\{\mathrm M}_{\mathrm{pl},\mathrm y,\mathrm{Rd}}\;=\;{\mathrm W}_{\mathrm{pl},\mathrm y\;}\;\cdot\;\frac{{\mathrm f}_{\mathrm y}}{{\mathrm\gamma}_{\mathrm M0}}\;=\;5,360\;\cdot\;\frac{23.5}1\;=\;125,960\;\mathrm{kNcm}\;=\;1,259.6\;\mathrm{kNm}\\{\mathrm A}_{\mathrm v,\mathrm z}\;=\;\mathrm A\;–\;2\;\cdot\;\mathrm b\;\cdot\;{\mathrm t}_{\mathrm f\;}+\;({\mathrm t}_{\mathrm w}\;+\;2\;\cdot\;\mathrm r)\;\;\cdot\;{\mathrm t}_{\mathrm f}\;=\;226\;–\;2\;\cdot\;30\;\cdot\;2.5\;+\;(1.3\;+\;2\;\cdot\;2.7)\;\cdot\;2.5\;=\;92.75\;\mathrm{cm}^2\\{\mathrm V}_{\mathrm{pl},\mathrm{Rd}}\;=\;{\mathrm A}_{\mathrm v,\mathrm z}\;\cdot\;\frac{{\mathrm f}_{\mathrm y}}{\sqrt3\;\cdot\;{\mathrm\gamma}_{\mathrm M0}}\;=\;92.75\;\cdot\;\frac{23.5}{\sqrt3\;\cdot\;1}\;=\;1,258.41\;\mathrm{kN}\end{array}$

The cross-section design is performed for the cross-section of class 1. Above the inner support, the beam is subjected to bending and shear force, at the location of the maximum moment in the span only to bending. The influence of the M-V interaction is checked before determining the ultimate limit state of the structure. If VEd is not more than 0.5 ⋅ Vpl,Rd, it is not required to reduce the moment resistance according to [1] Section 6.2.8 (2).

$\frac{{\mathrm V}_{\mathrm z,\mathrm{Ed}}}{{\mathrm V}_{\mathrm{pl},\mathrm z,\mathrm{Rd}}}\;=\;\frac{853.55}{1,258.41}\;=\;0.678\;>\;0.5$

It is necessary to reduce the moment resistance.

For I-cross-sections with the same flanges and uniaxial bending about the principal axis, it is allowed to determine the reduction of the design value of the plastic moment resistance due to the shear loading as follows:

$\begin{array}{l}\mathrm\rho\;=\;\left(\frac{2\;\cdot\;{\mathrm V}_{\mathrm{Ed}}}{{\mathrm V}_{\mathrm{pl},\mathrm{Rd}}}\;-\;1\right)^2\;=\;\left(\frac{2\;\cdot\;853.55}{1,258.41}\;-\;1\right)^2\;=\;0.127\\{\mathrm A}_{\mathrm w}\;=\;{\mathrm h}_{\mathrm w}\;\cdot\;{\mathrm t}_{\mathrm w}\;=\;\left(59\;-\;2\;\cdot\;2.5\right)\;\cdot\;1.3\;=\;70.2\;\mathrm{cm}\\{\mathrm M}_{\mathrm{pl},\mathrm V,\mathrm{Rd}}\;=\;\left({\mathrm W}_{\mathrm{pl},\mathrm y}\;-\;\frac{\mathrm\rho\;\cdot\;\mathrm A_{\mathrm w}^2}{4\;\cdot\;{\mathrm t}_{\mathrm w}}\right)\;\cdot\;\frac{{\mathrm f}_{\mathrm y}}{{\mathrm\gamma}_{\mathrm M0}}\;=\;\left(5,360\;-\;\frac{0.127\;\cdot\;70.2^2}{4\;\cdot\;1.3}\right)\;\cdot\;\frac{23.5}{1\;\cdot\;100}\;=\;1,231.32\;\mathrm{kNm}\\{\mathrm M}_{\mathrm{Ed}}\;<\;{\mathrm M}_{\mathrm{pl},\mathrm V,\mathrm{Rd}}\;\rightarrow\;1,068.36\;<\;1,231.32\;\mathrm{kNm}\end{array}$

#### Reference

 [1] Eurocode 3: Design of steel structures - Part 1‑1: General rules and rules for buildings; EN 1993‑1‑1:2010‑12 [2] Manual RF-/STEEL EC3. (2018). Tiefenbach: Dlubal Software. [3] Albert, A.: Schneider - Bautabellen für Ingenieure mit Berechnungshinweisen und Beispielen, 23. Auflage. Köln: Bundesanzeiger, 2018


text:
# Newbie: What is the difference between hypothesis class and models?

I am new to machine learning and I am confused with the terminology. Thus far, I used to view a hypothesis class as different instance of hypothesis function... Example: If we are talking about linear classification then different lines characterized by different weights would together form the hypothesis class.

Is my understanding correct or can a hypothesis class represent anything which could approximate the target function? For instance, can a linear or quadratic function that approximates the target function together form a single hypothesis class or both are from different hypothesis classes?

2. Hypothesis classes also don't have to consist of only simple functions. If you manage to search over all piecewise-$\tanh^2$ functions, then those functions are what your hypothesis class includes.


text:
Measurement of the tt¯ production cross-section in the lepton+jets channel at s=13 TeV with the ATLAS experiment

@inproceedings{Collaboration2020MeasurementOT,
title={Measurement of the tt¯ production cross-section in the lepton+jets channel at s=13 TeV with the ATLAS experiment},
author={Atlas Collaboration},
year={2020}
}
Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be… Expand
3 Citations

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References

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The PDF4LHC Working Group Interim Recommendations
This note provides an interim summary of the current recommendations of the PDF4LHC working group for the use of parton distribution functions (PDFs) and of PDF uncertainties at the LHC, for crossExpand
ATLAS data quality operations and performance for 2015–2018 data-taking
The ATLAS detector at the Large Hadron Collider reads out particle collision data from over 100 million electronic channels at a rate of approximately 100 kHz, with a recording rate for physicsExpand
Measurement of the tt¯ production cross-section and lepton differential distributions in eμ dilepton events from pp collisions at √s = 13TeV with the ATLAS detector
Author(s): Aad, G; Abbott, B; Abbott, DC; Abud, AA; Abeling, K; Abhayasinghe, DK; Abidi, SH; AbouZeid, OS; Abraham, NL; Abramowicz, H; Abreu, H; Abulaiti, Y; Acharya, BS; Achkar, B; Adachi, S; Adam,Expand
ATLAS b-jet identification performance and efficiency measurement with tt¯ events in pp collisions at √s = 13 TeV
The algorithms used by the ATLAS Collaboration during Run 2 of the Large Hadron Collider to identify jets containing b-hadrons are presented. The performance of the algorithms is evaluated in the sExpand
Electron and photon performance measurements with the ATLAS detector using the 2015-2017 LHC proton-proton collision data
This paper describes the reconstruction of electrons and photons with the ATLAS detector, employed for measurements and searches exploiting the complete LHC Run 2 dataset. An improved energyExpand
Event Generation with Sherpa 2.2
Sherpa is a general-purpose Monte Carlo event generator for the simulation of particle collisions in high-energy collider experiments. We summarize essential features and improvements of the SherpaExpand
Event generation with Sherpa
• SciPost Phys
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Measurement of the production cross section, the top quark mass, and the strong coupling constant using dilepton events in pp collisions at
A measurement of the top quark–antiquark pair production cross section in proton–proton collisions at a centre-of-mass energy of 13 is presented. The data correspond to an integrated luminosity of ,Expand
Measurement of the top quark mass in the $$t\bar{t}\rightarrow$$tt¯→ lepton+jets channel from $$\sqrt{s}=8$$s=8 TeV ATLAS data and combination with previous results
The top quark mass is measured using a template method in the $$t\bar{t} \rightarrow \mathrm {lepton+jets}$$tt¯→lepton+jets channel (lepton is e or $$\mu$$μ) using ATLAS data recorded in 2012 at theExpand
Measurements of top-quark pair differential and double-differential cross-sections in the ℓ +jets channel with pp collisions at √{s }=13 TeV using the ATLAS detector
Single- and double-differential cross-section measurements are presented for the production of top-quark pairs, in the lepton + jets channel at particle and parton level. Two topologies, resolved aExpand


text:
# Lesson 6

Distinguishing between Two Types of Situations

## 6.1: Which One Doesn’t Belong: Seeing Structure (5 minutes)

### Warm-up

This warm-up prompts students to compare four equations. It encourages students to explain their reasoning, hold mathematical conversations, and gives you the opportunity to hear how they use terminology and talk about characteristics of the equations in comparison to one another. To allow all students to access the activity, each equation has one obvious reason it does not belong. Encourage students to find reasons based on the structure of the equation (MP7). During the discussion, listen for important ideas and terminology.

### Launch

Arrange students in groups of 2–4. Display the equations for all to see. Ask students to indicate when they have noticed one that does not belong and can explain why. Give students 1 minute of quiet think time and then time to share their thinking with their group. In their groups, tell each student to share their reasoning about why a particular question does not belong, and together, find at least one reason each question doesn’t belong.

### Student Facing

Which equation doesn’t belong?

$$4(x + 3) = 9$$

$$4 \boldcdot x + 12 = 9$$

$$4 + 3x = 9$$

$$9 = 12 + 4x$$

### Activity Synthesis

Ask each group to share one reason why a particular equation does not belong. Record and display the responses for all to see. After each response, ask the class whether they agree or disagree. Since there is no single correct answer to the question asking which one does not belong, attend to students’ explanations and ensure the reasons given are correct. During the discussion, ask students to explain the meaning of any terminology they use, such as coefficient or solution. Also, press students on unsubstantiated claims.

## 6.2: Card Sort: Categories of Equations (15 minutes)

### Activity

The goal of this activity is for students to notice the structure of equations. Any way of sorting is fine, but the discussion should land on explaining how equations involving an expression like $$p(x+q)$$ are different from ones that have an expression like $$px+q$$. Monitor for different ways groups choose to categorize the equations, but especially for categories that distinguish between these two types of expressions. As students work, encourage them to refine their descriptions of equations using more precise language and mathematical terms (MP6).

### Launch

Arrange students in groups of 2. Tell them that in this activity, they will sort some cards into categories of their choosing. When they sort the equations, they should work with their partner to come up with categories, and then take turns sorting each equation into one of their categories, explaining why they are doing so. If necessary, demonstrate this protocol before students start working.

Distribute one set of cards to each group of students. Give students 5 minutes to work with their partner, followed by a whole-class discussion.

### Student Facing

Your teacher will give you a set of cards that show equations. Sort the cards into 2 categories of your choosing. Be prepared to explain the meaning of your categories. Then, sort the cards into 2 categories in a different way. Be prepared to explain the meaning of your new categories.

### Activity Synthesis

Select groups of students to share their categories and how they sorted their equations. You can choose as many different types of categories as time allows, but ensure that one set of categories distinguishes between equations that involved expressions of the form $$px+q$$ vs $$p(x+q)$$. Attend to the language that students use to describe their categories and equations, giving them opportunities to describe their equations more precisely. Highlight the use of terms like coefficient, sum, product, variable, and solution

Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion when students share the categories they created to sort their equations. After groups describe the set of categories that distinguish between equations that involved expressions of the form $$px + q$$ vs. $$p(x + q)$$, call on students to restate and/or revoice their peers' descriptions using mathematical terms (e.g., coefficient, sum, product, variable, and solution). This will provide more students with an opportunity to produce language that describes the form and structure of the two equations.
Design Principle(s): Support sense-making; Maximize meta-awareness

## 6.3: Even More Situations, Diagrams, and Equations (15 minutes)

### Activity

This activity is an opportunity to put together the learning of the past several lessons: correspondences between tape diagrams, equations, and stories, and using representations to reason about a solution. The focus of this activity is still contrasting the two main types of equations that students encounter in this unit.

### Launch

Keep students in the same groups. 5 minutes of quiet work time followed by sharing with a partner and a whole-class discussion.

Representation: Internalize Comprehension. Provide appropriate reading accommodations and supports to ensure students access to written directions, word problems, and other text-based content.
Supports accessibility for: Language; Conceptual processing
Reading, Representing: MLR6 Three Reads. Use this routine with the first story to support students’ understanding of the situation and how to represent it with a tape diagram or equation. Use the first read to orient students to the situation (Lin and volunteers are hanging flyers at school). Use the second read to identify the important quantities (number of volunteers, number of flyers each). After the third read, ask students to brainstorm how the situation can be represented in a tape diagram or equation. This will help students make connections between equations and diagrams and understand the language of a situation that will distinguish which form of and equation would be an appropriate representation.
Design Principle(s): Support sense-making; Maximize meta-awareness

### Student Facing

Story 1: Lin had 90 flyers to hang up around the school. She gave 12 flyers to each of three volunteers. Then she took the remaining flyers and divided them up equally between the three volunteers.

Story 2: Lin had 90 flyers to hang up around the school. After giving the same number of flyers to each of three volunteers, she had 12 left to hang up by herself.

1. Which diagram goes with which story? Be prepared to explain your reasoning.
2. In each diagram, what part of the story does the variable represent?
3. Write an equation corresponding to each story. If you get stuck, use the diagram.
4. Find the value of the variable in the story.

### Student Facing

#### Are you ready for more?

A tutor is starting a business. In the first year, they start with 5 clients and charge \$10 per week for an hour of tutoring with each client. For each year following, they double the number of clients and the number of hours each week. Each new client will be charged 150% of the charges of the clients from the previous year.

1. Organize the weekly earnings for each year in a table.
2. Assuming a full-time week is 40 hours per week, how many years will it take to reach full time and how many new clients will be taken on that year?
3. After reaching full time, what is the tutor’s annual salary if they take 2 weeks of vacation?
4. Is there another business model you’d recommend for the tutor? Explain your reasoning.

### Activity Synthesis

For each story, select 1 or more groups to present the matching diagram, their equation, and their solution method. Possible questions to ask:

• “How were the diagrams alike? How were they different?” (They have the same numbers and a letter. One has 3 equal groups and an extra bit, the other just has 3 equal groups, but each group is a sum.)
• “How were the stories alike? How were they different?” (They were both about distributing 90 flyers. In one story, Lin makes a series of moves to each volunteer. In the other story, she gives each volunteer the same amount, but then there are some left over.)
• “What parts of the story made you think that one diagram represented it?”
• “Explain how you reasoned about the story, diagram, or equation to find the value of the variable.”

## Lesson Synthesis

### Lesson Synthesis

Display the two equations from the last activity for all to see:

$$\displaystyle 3x+12=90$$

$$\displaystyle 3(y+12)=90$$

Tell students, “These equations have lots of things in common. They each have a 3, a letter, a 12, a 90, an equal sign, multiplication, and addition. Explain how these equations are different.” Ask students to think about this question quietly for a moment and share with a partner, then ask a few students to share with the whole class.

Highlight any responses that speak in general terms about the structure of the equations. For example, one equation is the sum of a product and a number and the other is the product of a number and a sum. Alternatively, if we evaluated one expression for a value of the variable, we would multiply it by 3 first and then add 12. For the other, we would add 12 first and then multiply by 3. One has three equal groups and an extra bit, and the other just has 3 equal groups, but the groups are each the result of adding 12 to an unknown.

## Student Lesson Summary

### Student Facing

In this unit, we encounter two main types of situations that can be represented with an equation. Here is an example of each type:

1. After adding 8 students to each of 6 same-sized teams, there were 72 students altogether.

2. After adding an 8-pound box of tennis rackets to a crate with 6 identical boxes of ping pong paddles, the crate weighed 72 pounds.

The first situation has all equal parts, since additions are made to each team. An equation that represents this situation is $$6(x+8)=72$$, where $$x$$ represents the original number of students on each team. Eight students were added to each group, there are 6 groups, and there are a total of 72 students.

In the second situation, there are 6 equal parts added to one other part. An equation that represents this situation is $$6x+8=72$$, where $$x$$ represents the weight of a box of ping pong paddles, there are 6 boxes of ping pong paddles, there is an additional box that weighs 8 pounds, and the crate weighs 72 pounds altogether.

In the first situation, there were 6 equal groups, and 8 students added to each group. $$6(x+8)=72$$.

In the second situation, there were 6 equal groups, but 8 more pounds in addition to that. $$6x+8=72$$.


text:
# Chapter 6 Cluster analysis

Cluster analysis is the collection of techniques designed to find subgroups or clusters in a dataset of variables $$X_1,\ldots,X_k$$. Depending on the similarities between the observations, these are partitioned in homogeneous groups as separated as possible between them. Clustering methods can be classified into two main categories:

• Partition methods. Given a fixed number of cluster $$k$$, these methods aim to assign each observation of $$X_1,\ldots,X_k$$ to a unique cluster, in such a way that the within-cluster variation is as small as possible (the clusters are as homogeneous as possible) while the between cluster variation is as large as possible (the clusters are as separated as possible).
• Hierarchical methods. These methods construct a hierarchy for the observations in terms of their similitudes. This results in a tree-based representation of the data in terms of a dendogram, which depicts how the observations are clustered at different levels – from the smallest groups of one element to the largest representing the whole dataset.

We will see the basics of the most well-known partition method, namely $$k$$-means clustering, and of the agglomerative hierarchical clustering.


text:
## Halting problem

Judging by a source code of a function, can you say, will it terminate at some point, or will it spin into an infinite loop? Alan Turing famously proved that this is impossible ("Halting Problem").

But let's see...

Let's say, you have a very basic 8-bit CPU, like 8080 or 6502. Say, there are only 5 8-bit registers, or 2 16-bit registers + one 8-bit register. Not uncommon for cheap CPUs of that era.

And there is no RAM attached. So all memory you can use is 40 bits. This is not uncommon for geeks of the time, when RAM chips were expensive and bought after CPU.

You can simulate such a basic CPU on any decent desktop computer. To track all states, you can use a dictionary. Key = values-from-all-registers-including-PC, value = boolean. The key has a size of 40 bits. The size of the whole dictionary is $2^{40}$ bits or $2^{32}$ bytes (just 4GB of RAM).

Now you can simulate any code running on this toy CPU using your desktop computer and track all states. When the code stops, you stop. When you hit a state that already encountered (checked using the dictionary), you stop and report an infinite loop.

Hence, to tell if the program will stop or not, you just need a computer with bigger RAM. To simulate a program running on your desktop computer with 4GB of RAM, you would need a computer with at least $2^{2^{64}}$ bits of RAM.

$log_{10}(2^{2^{64}}) \approx 5.55 \cdot 10^{18}$, but still theoretically possible.

These toy CPUs and computers are in fact LBAs (Linear bounded automaton) -- a Turing machine with limited tape (or memory).

While Turing's Halting Problem is about Turing machine with infinite tape (or RAM).


text:
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## An exact algorithm for stable instances of the $k$-means problem with penalties in fixed-dimensional Euclidean space

 1 Department of Operations Research and Information Engineering, Beijing University of Technology, Beijing 100124, China 2 Beijing Institute for Scientific and Engineering Computing, Beijing University of Technology, Beijing 100124, China 3 Faculty of Management, University of New Brunswick, Fredericton, NB Canada E3B 5A3, Canada 4 School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China

* Corresponding author: Dachuan Xu

Received  October 2020 Revised  May 2021 Early access July 2021

We study stable instances of the $k$-means problem with penalties in fixed-dimensional Euclidean space. An instance of the problem is called $\alpha$-stable if this instance exists a sole optimal solution and the solution keeps unchanged when distances and penalty costs are scaled by a factor of no more than $\alpha$. Stable instances of clustering problem have been used to explain why certain heuristic algorithms with poor theoretical guarantees perform quite well in practical. For any fixed $\epsilon > 0$, we show that when using a common multi-swap local-search algorithm, a $(1+\epsilon)$-stable instance of the $k$-means problem with penalties in fixed-dimensional Euclidean space can be solved accurately in polynomial time.

Citation: Fan Yuan, Dachuan Xu, Donglei Du, Min Li. An exact algorithm for stable instances of the $k$-means problem with penalties in fixed-dimensional Euclidean space. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021122
##### References:
 [1] S. Ahmadian, A. Norouzi-Fard, O. Svensson and J. Ward, Better guarantees for $k$-means and Euclidean $k$-median by primal-dual algorithms, 58th Annual IEEE Symposium on Foundations of Computer Science-FOCS, (2017), 61-72. doi: 10.1137/18M1171321.  Google Scholar [2] D. Aloise, A. Deshpande, P. Hansen and P. Popat, NP-hardness of Euclidean sum-of-squares clustering, Machine Learning, 75 (2009), 245-248.  doi: 10.1007/s10994-009-5103-0.  Google Scholar [3] H. Angelidakis, K. Makarychev and Y. Makarychev, Algorithms for stable and perturbation-resilient problems, STOC'17-Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, ACM, New York, (2017), 438-451. doi: 10.1145/3055399.3055487.  Google Scholar [4] V. Arya, N. Garg, R. Khandekar, A. Meyerson, K. Munagala and V. Pandit, Local search heuristics for $k$-median and facility location problems, SIAM J. Comput., 33 (2004), 544-562.  doi: 10.1137/S0097539702416402.  Google Scholar [5] P. Awasthi, A. Blum and O. Sheffet, Stability yields a PTAS for $k$-median and $k$-means clustering, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science-FOCS, (2010), 309-318. doi: 10.1109/FOCS. 2010.36.  Google Scholar [6] P. Awasthi, A. Blum and O. Sheffet, Center-based clustering under perturbation stability, Inform. Process. Lett., 112 (2012), 49-54.  doi: 10.1016/j.ipl.2011.10.006.  Google Scholar [7] M. F. Balcan and Y. Liang, Clustering under perturbation resilience, SIAM J. Comput., 45 (2016), 102-155.  doi: 10.1137/140981575.  Google Scholar [8] Y. Bilu and N. Linial, Are stable instances easy?, Combin. Probab. Comput., 21 (2012), 643-660.  doi: 10.1017/S0963548312000193.  Google Scholar [9] M. Charikar and S. Guha, Improved combinatorial algorithms for the facility location and $k$-median problems, 40th Annual Symposium on Foundations of Computer Science (New York, 1999), (1999), 378-388. doi: 10.1109/SFFCS. 1999.814609.  Google Scholar [10] V. Cohen-Addad, P. N. Klein and C. Mathieu, Local search yields approximation schemes for $k$-means and $k$-median in Euclidean and minor-free metrics, SIAM J. Comput., 48 (2019), 644-667.  doi: 10.1137/17M112717X.  Google Scholar [11] P. Drineas, A. Frieze, R. Kannan, S. Vempala and V. Vinay, Clustering large graphs via the singular value decomposition, Machine Learning, 56 (2004), 9-33.  doi: 10.1023/B:MACH.0000033113.59016.96.  Google Scholar [12] D. Du, X. Wang and D. Xu, An approximation algorithm for the $k$-level capacitated facility location problem, J. Comb. Optim., 20 (2010), 361-368.  doi: 10.1007/s10878-009-9213-1.  Google Scholar [13] Q. Feng, Z. Zhang, F. Shi and J. Wang, An improved approximation algorithm for the $k$-means problem with penalties, Proceedings of FAW, (2019), 170-181. doi: 10.1007/978-3-030-18126-0_15.  Google Scholar [14] Z. Friggstad, K. Khodamoradi and M. R. Salavatipour, Exact algorithms and lower bounds for stable instances of Euclidean $k$-means, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, (2019), 2958-2972. doi: 10.1137/1.9781611975482.183.  Google Scholar [15] Z. Friggstad, M. Rezapour and M. R. Salavatipour, Local search yields a PTAS for $k$-means in doubling metrics, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), (2016), 365-374. doi: 10.1109/focs. 2016.47.  Google Scholar [16] S. Ji, D. Xu, L. Guo, M. Li and D. Zhang, The seeding algorithm for spherical $k$-means clustering with penalties, Journal of Combinatorial Optimization, (2020, Accepted). doi: 10.1007/s10878-020-00569-1.  Google Scholar [17] T. Kanungo, D. M. Mount, N. S. Netanyahu, C. D. Piatko, R. Silverman and A. Y. Wu, A local search approximation algorithm for $k$-means clustering, Comput. Geom., 28 (2004), 89-112.  doi: 10.1016/j.comgeo.2004.03.003.  Google Scholar [18] M. Li, The bi-criteria seeding algorithms for two variants of $k$-means problem, J. Comb. Optim., (2020, Accepted). doi: 10.1007/s10878-020-00537-9.  Google Scholar [19] M. Li, D. Xu, J. Yue, D. Zhang and P. Zhang, The seeding algorithm for $k$-means problem with penalties, J. Comb. Optim., 39 (2020), 15-32.  doi: 10.1007/s10878-019-00450-w.  Google Scholar [20] A.-Y. Liang and D. Lin, Crossover iterated local search for SDCARP, J. Oper. Res. Soc. China, 2 (2014), 351-367.  doi: 10.1007/s40305-014-0056-9.  Google Scholar [21] M. Mahajan, P. Nimbhorkar and K. Varadarajan, The planar $k$-means problem is NP-hard, Proceedings of WALCOM, 5431 (2009), 274-285.  doi: 10.1007/978-3-642-00202-1_24.  Google Scholar [22] J. Matoušek, On approximate geometric $k$-clustering, Discrete Comput. Geom., 24 (2000), 61-84.  doi: 10.1007/s004540010019.  Google Scholar [23] G. C. Tseng, Penalized and weighted $k$-means for clustering with scattered objects and prior information in high-throughput biological data, Bioinformatics, (2007), 2247-2255. doi: 10.1093/bioinformatics/btm320.  Google Scholar [24] H. Yang, F. Li, D. Yu, Y. Zou and J. Yu, Reliable data storage in heterogeneous wireless sensor networks by jointly optimizing routing and storage node deployment, Tsinghua Science and Technology, 26 (2021), 230-238.  doi: 10.26599/TST.2019.9010061.  Google Scholar [25] D. Ye, L. Mei and Y. Zhang, Strategy-proof mechanism for obnoxious facility location on a line, Proceedings of COCOON, 9198 (2015), 45-56.  doi: 10.1007/978-3-319-21398-9_4.  Google Scholar [26] D. Zhang, C. Hao, C. Wu, D. Xu and Z. Zhang, Local search approximation algorithms for the $k$-means problem with penalties, J. Comb. Optim., 37 (2019), 439-453.  doi: 10.1007/s10878-018-0278-6.  Google Scholar [27] Y. Zhang, F. Y. L. Chin and H. Zhu, A 1-local asymptotic 13/9-competitive algorithm for multicoloring hexagonal graphs, Algorithmica, 54 (2009), 557-567.  doi: 10.1007/s00453-008-9203-1.  Google Scholar

show all references

##### References:
 [1] S. Ahmadian, A. Norouzi-Fard, O. Svensson and J. Ward, Better guarantees for $k$-means and Euclidean $k$-median by primal-dual algorithms, 58th Annual IEEE Symposium on Foundations of Computer Science-FOCS, (2017), 61-72. doi: 10.1137/18M1171321.  Google Scholar [2] D. Aloise, A. Deshpande, P. Hansen and P. Popat, NP-hardness of Euclidean sum-of-squares clustering, Machine Learning, 75 (2009), 245-248.  doi: 10.1007/s10994-009-5103-0.  Google Scholar [3] H. Angelidakis, K. Makarychev and Y. Makarychev, Algorithms for stable and perturbation-resilient problems, STOC'17-Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, ACM, New York, (2017), 438-451. doi: 10.1145/3055399.3055487.  Google Scholar [4] V. Arya, N. Garg, R. Khandekar, A. Meyerson, K. Munagala and V. Pandit, Local search heuristics for $k$-median and facility location problems, SIAM J. Comput., 33 (2004), 544-562.  doi: 10.1137/S0097539702416402.  Google Scholar [5] P. Awasthi, A. Blum and O. Sheffet, Stability yields a PTAS for $k$-median and $k$-means clustering, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science-FOCS, (2010), 309-318. doi: 10.1109/FOCS. 2010.36.  Google Scholar [6] P. Awasthi, A. Blum and O. Sheffet, Center-based clustering under perturbation stability, Inform. Process. Lett., 112 (2012), 49-54.  doi: 10.1016/j.ipl.2011.10.006.  Google Scholar [7] M. F. Balcan and Y. Liang, Clustering under perturbation resilience, SIAM J. Comput., 45 (2016), 102-155.  doi: 10.1137/140981575.  Google Scholar [8] Y. Bilu and N. Linial, Are stable instances easy?, Combin. Probab. Comput., 21 (2012), 643-660.  doi: 10.1017/S0963548312000193.  Google Scholar [9] M. Charikar and S. Guha, Improved combinatorial algorithms for the facility location and $k$-median problems, 40th Annual Symposium on Foundations of Computer Science (New York, 1999), (1999), 378-388. doi: 10.1109/SFFCS. 1999.814609.  Google Scholar [10] V. Cohen-Addad, P. N. Klein and C. Mathieu, Local search yields approximation schemes for $k$-means and $k$-median in Euclidean and minor-free metrics, SIAM J. Comput., 48 (2019), 644-667.  doi: 10.1137/17M112717X.  Google Scholar [11] P. Drineas, A. Frieze, R. Kannan, S. Vempala and V. Vinay, Clustering large graphs via the singular value decomposition, Machine Learning, 56 (2004), 9-33.  doi: 10.1023/B:MACH.0000033113.59016.96.  Google Scholar [12] D. Du, X. Wang and D. Xu, An approximation algorithm for the $k$-level capacitated facility location problem, J. Comb. Optim., 20 (2010), 361-368.  doi: 10.1007/s10878-009-9213-1.  Google Scholar [13] Q. Feng, Z. Zhang, F. Shi and J. Wang, An improved approximation algorithm for the $k$-means problem with penalties, Proceedings of FAW, (2019), 170-181. doi: 10.1007/978-3-030-18126-0_15.  Google Scholar [14] Z. Friggstad, K. Khodamoradi and M. R. Salavatipour, Exact algorithms and lower bounds for stable instances of Euclidean $k$-means, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, (2019), 2958-2972. doi: 10.1137/1.9781611975482.183.  Google Scholar [15] Z. Friggstad, M. Rezapour and M. R. Salavatipour, Local search yields a PTAS for $k$-means in doubling metrics, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), (2016), 365-374. doi: 10.1109/focs. 2016.47.  Google Scholar [16] S. Ji, D. Xu, L. Guo, M. Li and D. Zhang, The seeding algorithm for spherical $k$-means clustering with penalties, Journal of Combinatorial Optimization, (2020, Accepted). doi: 10.1007/s10878-020-00569-1.  Google Scholar [17] T. Kanungo, D. M. Mount, N. S. Netanyahu, C. D. Piatko, R. Silverman and A. Y. Wu, A local search approximation algorithm for $k$-means clustering, Comput. Geom., 28 (2004), 89-112.  doi: 10.1016/j.comgeo.2004.03.003.  Google Scholar [18] M. Li, The bi-criteria seeding algorithms for two variants of $k$-means problem, J. Comb. Optim., (2020, Accepted). doi: 10.1007/s10878-020-00537-9.  Google Scholar [19] M. Li, D. Xu, J. Yue, D. Zhang and P. Zhang, The seeding algorithm for $k$-means problem with penalties, J. Comb. Optim., 39 (2020), 15-32.  doi: 10.1007/s10878-019-00450-w.  Google Scholar [20] A.-Y. Liang and D. Lin, Crossover iterated local search for SDCARP, J. Oper. Res. Soc. China, 2 (2014), 351-367.  doi: 10.1007/s40305-014-0056-9.  Google Scholar [21] M. Mahajan, P. Nimbhorkar and K. Varadarajan, The planar $k$-means problem is NP-hard, Proceedings of WALCOM, 5431 (2009), 274-285.  doi: 10.1007/978-3-642-00202-1_24.  Google Scholar [22] J. Matoušek, On approximate geometric $k$-clustering, Discrete Comput. Geom., 24 (2000), 61-84.  doi: 10.1007/s004540010019.  Google Scholar [23] G. C. Tseng, Penalized and weighted $k$-means for clustering with scattered objects and prior information in high-throughput biological data, Bioinformatics, (2007), 2247-2255. doi: 10.1093/bioinformatics/btm320.  Google Scholar [24] H. Yang, F. Li, D. Yu, Y. Zou and J. Yu, Reliable data storage in heterogeneous wireless sensor networks by jointly optimizing routing and storage node deployment, Tsinghua Science and Technology, 26 (2021), 230-238.  doi: 10.26599/TST.2019.9010061.  Google Scholar [25] D. Ye, L. Mei and Y. Zhang, Strategy-proof mechanism for obnoxious facility location on a line, Proceedings of COCOON, 9198 (2015), 45-56.  doi: 10.1007/978-3-319-21398-9_4.  Google Scholar [26] D. Zhang, C. Hao, C. Wu, D. Xu and Z. Zhang, Local search approximation algorithms for the $k$-means problem with penalties, J. Comb. Optim., 37 (2019), 439-453.  doi: 10.1007/s10878-018-0278-6.  Google Scholar [27] Y. Zhang, F. Y. L. Chin and H. Zhu, A 1-local asymptotic 13/9-competitive algorithm for multicoloring hexagonal graphs, Algorithmica, 54 (2009), 557-567.  doi: 10.1007/s00453-008-9203-1.  Google Scholar
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# Rotation of helium balloons [closed]

A number of balloons are attached to a circular disk with string. Some balloons are filled with air and some balloons are filled with helium. The disk is hung freely from ceiling of a room and is disk is rotated about its center. Assuming that the disk remains horizontal while rotating, describe what happens to the balloons?

(a) All the balloons move away from the center of the disk.

(b) All the balloons move towards the center of the disk.

(c) Air balloons move outwards away from the center and helium balloon move towards the center.

(d) Air balloons move inwards towards the center and helium balloon move out- wards the away from the center. I am in a doubt that is the centrifugal force will dominating and all balloon goes outside or some special effect(like pressure) will affect to helium balloon to go inside(like the behaviour of helium balloon in a car at a turn)

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Since the balloons are rotating, there must be a centripetal force that is keeping them going around the circle.

This resultant force is coming from a component of the tension in the string.

The centripetal force must point inwards towards the centre of rotation, so therefore the balloon must swing outwards so that part of the tension points inwards.

I can't see any difference in the behaviour between the helium and air balloons - in the helium balloons, the vertical component of the tension counters upthrust, whereas in the air balloons, the vertical component counters weight.

I made the following diagram to aid my explanation.

• $$T$$ is the overall tension in the string.
• $$F_c$$ is the centripetal force (resultant).
• $$F_b$$ is what I am calling the "balancing force" - in the case of the air balloon, this is the opposite of the weight vector; for the helium balloon, it opposes the resultant upthrust (total thrust minus weight).

• What about wind resistance, which will be greater on the outer part of the balloons, pushing them inwards to where it is negligible in the center of rotation. – Adrian Howard Sep 16 at 21:15
• @AdrianHoward Interesting though - I do not know enough about wind resistance to comment. – Joe Iddon Sep 17 at 19:03
• I am no expert either it just seems that balloons are so light but bulky that wind force could overcome centrifugal force? – Adrian Howard Sep 17 at 21:54
• @AdrianHoward The wind would provide an orthogonal force to the centrifugal force, so I think they would still hang out as I describe, but would "lag behind" the rotation of the disk. – Joe Iddon Sep 19 at 18:47


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Linear partial differential operators (or, in the language of quantum mechanics, quantum observables) on, say, ${\bf R}^n$, are (in principle, at least) generated by the position operators $x_j$ and the momentum operators $\frac{1}{i} \frac{\partial}{\partial x_j}$, which are then related to each other by the basic commutation relations $$\frac{1}{i} [x_j, \frac{1}{i} \frac{\partial}{\partial x_k}] = \delta_{jk}$$ where $[,]$ here is the commutator $[A,B] = AB-BA$.
Meanwhile, classical observables on the phase space $T^* {\bf R}^n$ are (again in principle) generated by the position functions $q_j$ and momentum functions $p_j$, which are related to each other by the basic commutation relations $$\{ q_j, p_k \} = \delta_{jk}$$ where ${,}$ \{,\}$is now the Poisson bracket (I may have the sign conventions reversed here). One of the great insights of quantum mechanics (or, on the mathematical side, semi-classical analysis) is the correspondence principle that asserts, roughly speaking, that the behaviour of quantum observables converges in the high-frequency limit (or, after rescaling, the semi-classical limit) to the analogous behaviour of classical observables. The correspondence is easier to see on the observable side than on the physical space side, for instance by connecting the von Neumann algebra of bounded quantum observables with smooth symbol with the Poisson algebra of smooth classical observables. The former is connected to linear PDE and the latter to symplectic (or Hamiltonian) geometry. Another way to see the connection is to investigate what happens when one applies a linear partial differential (or pseudodifferential) operator to a high-frequency function (or "quantum state"), when viewing that function through its Wigner transform, which can be viewed as approximately describing the quantum state by a classical one. A standard calculation shows (under Weyl quantisation) that the top order contribution of the operator on this transform is given by its symbol, and the next order term is basically given by the Hamiltonian vector field associated to that symbol. (This is discussed for instance in Folland's "Harmonic analysis on phase space".) This suggests that the dynamics of linear PDE at high frequencies are going to be driven by the associated Hamiltonian dynamics of the symbol of that PDE. 1 Linear partial differential operators (or, in the language of quantum mechanics, quantum observables) on, say,${\bf R}^n$, are (in principle, at least) generated by the position operators$x_j$and the momentum operators$\frac{1}{i} \frac{\partial}{\partial x_j}$, which are then related to each other by the basic commutation relations $$\frac{1}{i} [x_j, \frac{1}{i} \frac{\partial}{\partial x_k}] = \delta_{jk}$$ where$[,]$here is the commutator$[A,B] = AB-BA$. Meanwhile, classical observables on the phase space$T^* {\bf R}^n$are (again in principle) generated by the position functions$q_j$and momentum functions$p_j$, which are related to each other by the basic commutation relations $$\{ q_j, p_k \} = \delta_{jk}$$ where${,}\$ is now the Poisson bracket (I may have the sign conventions reversed here).