scholarly journals On induced permutation matrices and the symmetric group

1936 ◽  
Vol 5 (1) ◽  
pp. 1-13 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Symmetric Group ◽  
Matrix Representation ◽  
Zero Element ◽  
Natural Order ◽  
Third Order ◽  
The Third ◽  
Permutation Matrices ◽  
The Matrix ◽  
Matrix Relation ◽  
Image Position

The n! operations Ai of permutations upon n different ordered symbols correspond to n! matrices Ai of the nth order, which have in each row and in each column only one non-zero element, namely a unit. Such matrices Ai are called permutation matrices, since their effect in premultiplying an arbitrary column vector x = {x1x2….xn} is to impress the permutation Ai upon the elements xi. For example the six matrices of the third orderare permutation matrices. It is convenient to denote them bywhere the bracketed indices refer to the permutations of natural order. Clearly the relation Ai Aj = Ak entails the matrix relation AiAj = Ak; in other words, the n! matrices Ai, give a matrix representation of the symmetric group of order n!.

On Stability of Solutions of Certain Differential Equations of the Third Order

1967 ◽  
Vol 10 (5) ◽  
pp. 681-688 ◽  
Author(s):  
B.S. Lalli
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Stability Of Solutions ◽  
Third Order ◽  
The Third ◽  
Image Position

The purpose of this paper is to obtain a set of sufficient conditions for “global asymptotic stability” of the trivial solution x = 0 of the differential equation1.1using a Lyapunov function which is substantially different from similar functions used in [2], [3] and [4], for similar differential equations. The functions f1, f2 and f3 are real - valued and are smooth enough to ensure the existence of the solutions of (1.1) on [0, ∞). The dot indicates differentiation with respect to t. We are taking a and b to be some positive parameters.

scholarly journals On Appell's Function P (θ, φ)

1932 ◽  
Vol 3 (1) ◽  
pp. 53-55
Author(s):  
Pierre Humbert
Keyword(s):  
Third Order ◽  
The Third ◽  
Direct Generalization ◽  
Circular Functions ◽  
Image Position

§1. Appell's functions, P (θ, φ), Q (θ, φ) and R (θ, φ) are defined by the expansionwhere j3 = 1, affording, both for the third order and the field of two variables, a very direct generalization of the circular functions, as

scholarly journals Asymptotic formulae for linear equations

1972 ◽  
Vol 13 (2) ◽  
pp. 147-152 ◽  
Author(s):  
Don B. Hinton
Keyword(s):  
Asymptotic Behaviour ◽  
Fourth Order ◽  
Linear Equations ◽  
Order Equation ◽  
Third Order ◽  
Fourth Order Equation ◽  
The Third ◽  
Asymptotic Formulae ◽  
Image Position

Numerous formulae have been given which exhibit the asymptotic behaviour as t → ∞solutions ofwhere F(t) is essentially positive and Several of these results have been unified by a theorem of F. V. Atkinson [1]. It is the purpose of this paper to establish results, analogous to the theorem of Atkinson, for the third order equationand for the fourth order equation

A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order

1951 ◽  
Vol 47 (4) ◽  
pp. 699-712 ◽  
Author(s):  
F. W. J. Olver
Keyword(s):  
Differential Equation ◽  
Bessel Functions ◽  
Numerical Technique ◽  
Third Order ◽  
New Approach ◽  
The Third ◽  
Zeros Of Bessel Functions ◽  
Zeros Of Functions ◽  
Linear Differential ◽  
Image Position

In a recent paper (1) I described a method for the numerical evaluation of zeros of the Bessel functions Jn(z) and Yn(z), which was independent of computed values of these functions. The essence of the method was to regard the zeros ρ of the cylinder functionas a function of t and to solve numerically the third-order non-linear differential equation satisfied by ρ(t). It has since been successfully used to compute ten-decimal values of jn, s, yn, s, the sth positive zeros* of Jn(z), Yn(z) respectively, in the ranges n = 10 (1) 20, s = 1(1) 20. During the course of this work it was realized that the least satisfactory feature of the new method was the time taken for the evaluation of the first three or four zeros in comparison with that required for the higher zeros; the direct numerical technique for integrating the differential equation satisfied by ρ(t) becomes unwieldy for the small zeros and a different technique (described in the same paper) must be employed. It was also apparent that no mere refinement of the existing methods would remove this defect and that a new approach was required if it was to be eliminated. The outcome has been the development of the method to which the first part (§§ 2–6) of this paper is devoted.

scholarly journals Tensor Decomposition of KUKA Industrial Robot (KR16-2) in Rotational System

2020 ◽  
Vol 9 (2) ◽  
pp. 688-690
Keyword(s):  
Rotational Motion ◽  
Industrial Robot ◽  
Matrix Decomposition ◽  
Tensor Decomposition ◽  
Tensor Model ◽  
Third Order ◽  
Order Tensor ◽  
The Third ◽  
Decomposition Model ◽  
The Matrix

This paper refers to study of industrial robot (KUKA KR16-2), in which we have considered the matrix decomposition and tensor decomposition model in rotational motion. We have considered robotic matrix & Tensor and defined a modal product between robot rotation matrix and a tensor Further we have proposed the third order tensor for the motion of Industrial robot and tried to find out the useful result. At last we have shown that the tensor model is providing alternate way to find the solution.

scholarly journals Note on a paper of Tsuzuku

1964 ◽  
Vol 6 (4) ◽  
pp. 196-197
Author(s):  
H. K. Farahat
Keyword(s):  
Symmetric Group ◽  
Commutative Ring ◽  
Permutation Group ◽  
Matrix Representation ◽  
Invertible Element ◽  
Unit Element ◽  
Transitive Permutation Group ◽  
Natural Representation ◽  
Correct Proof ◽  
Permutation Matrices

In [2], Tosiro Tsuzzuku gave a proof of the following:THEOREM. Let G be a doubly transitive permutation group of degree n, let K be any commutative ring with unit element and let p be the natural representation of G by n × n permutation matrices with elements 0, 1 in K. Then ρ is decomposable as a matrix representation over K if and only ifn is an invertible element of K.For G the symmetric group this result follows from Theorems (2.1) and (4.12) of [1]. The proof given by Tsuzuku is unsatisfactory, although it is perfectly valid when K is a field. The purpose of this note is to give a correct proof of the general case.

scholarly journals 4. Note on Professor Tait's “Quaternion Path” to Determinants of the Third Order

1869 ◽  
Vol 6 ◽  
pp. 121-125
Author(s):  
Hugh Martin
Keyword(s):  
Third Order ◽  
The Third ◽  
Image Position ◽  
Special Case

I have read with much interest Professor Tait's “Note on Determinants of the Third Order” in the Proceedings of this Session (pp. 59–61), and admire the method of discovering new properties of Determinants. I am not sure, however, that the properties, when discovered, are more difficult of proof by Determinant methods, and I venture to submit the following as simple and elementary:—The first property, namely,is true under greater generality, and the Determinant proof is the same as for the special case.

scholarly journals 1. On the Extraction of the Square Root of a Matrix of the Third Order

1872 ◽  
Vol 7 ◽  
pp. 675-682 ◽  
Author(s):  
Cayley
Keyword(s):  
Second Order ◽  
Square Root ◽  
Third Order ◽  
The Third ◽  
Image Position

Professor Tait has considered the question of finding the square root of a strain, or what is the same thing, that of a matrix of the third order— A mode of doing this is indicated in my “Memoir on the Theory of Matrices” (Phil. Trans., 1858, pp. 17–37), and it is interesting to work out the solution.The notation and method will be understood from the simple case of a matrix of the second order. I write to denote the two equations, x1 = ax + by, y1 = cx + dy.

scholarly journals On Trace Bilinear Forms on Lie-Algebras

1959 ◽  
Vol 4 (2) ◽  
pp. 62-72 ◽  
Author(s):  
Hans Zassenhaus
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Matrix Representation ◽  
Bilinear Forms ◽  
Finite Degree ◽  
Matrix Products ◽  
The Matrix ◽  
Image Position

To what extent is the structure of a Lie-algebra L over a field F determined by the bilinear formon L that is derived from a matrix representationof L with finite degree d(Δ) by forming the trace of the matrix productsSuch a bilinear form is a function with two arguments in L, values in F and the properties:

scholarly journals POSITIVE INTEGER SOLUTION OF THE MATRIX EQUATION Xn = A FOR THIRD-ORDER MATRICES IN THE CASE OF POSITIVE INTEGERS n

2018 ◽  
Vol 54 (2) ◽  
pp. 164-178
Author(s):  
K. L. Yakuto
Keyword(s):  
Positive Integer ◽  
Matrix Equation ◽  
Integer Solution ◽  
Third Order ◽  
Positive Integer Solution ◽  
The Third ◽  
Order Matrix ◽  
The Matrix ◽  
Integer Solutions ◽  
Different Order

The problem of the positive integer solution of the equation Xn = A for different-order matrices is important to solve a large range of problems related to the modeling of economic and social processes. The need to solve similar problems also arises in areas such as management theory, dynamic programming technique for solving some differential equations. In this connection, it is interesting to question the existence of positive and positive integer solutions of the nonlinear equations of the form Xn = A for different-order matrices in the case of the positive integer n. The purpose of this work is to explore the possibility of using analytical methods to obtain positive integer solutions of nonlinear matrix equations of the form Xn = A where A, X are the third-order matrices, n is the positive integer. Elements of the original matrix A are integer and positive numbers. The present study found that when the root of the nth degree of the third-order matrix will have zero diagonal elements and nonzero and positive off-diagonal elements, the root of the nth degree of the third-order matrix will have two zero diagonal elements and nonzero positive off-diagonal elements. It was shown that to solve the problem of finding positive integer solutions of the matrix equation for third-order matrices in the case of the positive integer n, the analytical techniques can be used. The article presents the formulas that allow one to find the roots of positive integer matrices for n = 3,…,5. However, the methodology described in the article can be adopted to find the natural roots of the third-order matrices for large n. 

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