A system of linear equations related to the transportation problem with application to probability theory

Let P=[p_ij] be a m×n matrix and let C be the coefficient matrix of Σ_j=1ⁿ p_ijx_ij=u_i, 1≤i≤m, Σ_i=1^mp_ijx_ij=vj, 1≤j≤n. The relation between the reducibility of P and the rank of C is investigated. An application to martingale extension is given.     

Minimum norm problems over transportation polytopes

We consider the problem of updating input-output matrices, i.e., for given (m,n) matrices A ? 0, W ? 0 and vectors u ? $\text{R}$^m, v?$\text{R}$ⁿ, find an (m,n) matrix X ? 0 with prescribed row sums Σⁿ_j=1X_ij = u_i (i = 1,…,m) and prescribed column sums Σ^m_i=1X_ij = v_j (j = 1,…,n) which fits the relations X_ij = A_ij + λ_iW_ij + W_ij + W_ijμ_j for all i,j and some λ?$\text{R}$^m, μ?$\text{R}$ⁿ. Here we consider the question of existence of a solution to this problem, i.e., we shall characterize those matrices A, W and vectors u,v which lead to a solvable problem. Furthermore we outline some computational results using an algorithm of [2].     

Least squares solutions to AX = B for bisymmetric matrices under a central principal submatrix constraint and the optimal approximation

A matrix A∈R^n×n is called a bisymmetric matrix if its elements a_i,j satisfy the properties a_i,j=a_j,i and a_i,j=a_n-j+1,n-i+1 for 1?i,j?n. This paper considers least squares solutions to the matrix equation AX=B for A under a central principal submatrix constraint and the optimal approximation. A central principal submatrix is a submatrix obtained by deleting the same number of rows and columns in edges of a given matrix. We first discuss the specified structure of bisymmetric matrices and their central principal submatrices. Then we give some necessary and sufficient conditions for the solvability of the least squares problem, and derive the general representation of the solutions. Moreover, we also obtain the expression of the solution to the corresponding optimal approximation problem.     

Utilizing Shelve Slots: Sufficiency Conditions for Some Easy Instances of Hard Problems

In this paper we present a minimal set of conditions sufficient to assure the existence of a solution to a system of nonnegative linear diophantine equations. More specifically, suppose we are given a finite item set U = {u₁, u₂, . . . , u_k} together with a "size" v_i ≡ v(u_i) ∈ Z⁺, such that v_i ≠ v_j for i ≠ j, a "frequency" a_i ≡ a(u_i) ∈ Z⁺, and a positive integer (shelf length) L ∈ Z⁺ with the following conditions: (i) L = ∏ⁿ_j=1p_j(p_j ∈ Z⁺ ∀j, p_j ≠ p_l for j ≠ l) and v_i = ∏ _j∈Aip_j, A_i ⊆ {l, 2, . . . , n} for i = 1, . . . , n; (ii) (A_i\{⋂^k_j=1A_j}) ∩ (A_l\{⋂^k_j=1A_j}) = ⊘∀i ≠ l. Note that v_i|L (divides L) for each i. If for a given m ∈ Z⁺, ∑ⁿ_i=1a_iv_i = mL (i.e., the total size of all the items equals the total length of the shelf space), we prove that conditions (i) and (ii) are sufficient conditions for the existence of a set of integers {b₁₁, b₁₂, . . . , b_1m, b₂₁, . . . , b_n1, . . . , b_nm}⊆ N such that ∑^m_j=1b_ij = a_i, i = 1, . . . , k, and ∑^k_i=1b_ijv_i = L, j =1, . . . , m (i.e., m shelves of length L can be fully utilized). We indicate a number of special cases of well known NP-complete problems which are subsequently decided in polynomial time.     

Extensions of band matrices with band inverses

Let R_ij be a given set of μ_i× μ_j matrices for i, j=1,…, n and |i?j| ?m, where 0?m?n?1. Necessary and sufficient conditions are established for the existence and uniqueness of an invertible block matrix =[F_ij], i,j=1,…, n, such that F_ij=R_ij for |i?j|?m, F admits either a left or right block triangular factorization, and (F^?1)_ij=0 for |i?j|>m. The well-known conditions for an invertible block matrix to admit a block triangular factorization emerge for the particular choice m=n?1. The special case in which the given R_ij are positive definite (in an appropriate sense) is explored in detail, and an inequality which corresponds to Burg's maximal entropy inequality in the theory of covariance extension is deduced. The block Toeplitz case is also studied.     

Sign-nonsingular matrices and even cycles in directed graphs

A sign-nonsingular matrix or L-matrix A is a real m× n matrix such that the columns of any real m×n matrix with the same sign pattern as A are linearly independent. The problem of recognizing square L-matrices is equivalent to that of finding an even cycle in a directed graph. In this paper we use graph theoretic methods to investigate L-matrices. In particular, we determine the maximum number of nonzero elements in square L-matrices, and we characterize completely the semicomplete L-matrices [i.e. the square L-matrices (a_ij) such that at least one of a_ij and a_ij is nonzero for any i,j] and those square L-matrices which are combinatorially symmetric, i.e., the main diagonal has only nonzero entries and a_ij=0 iff a_ji=0. We also show that for any n×n L-matrix there is an i such that the total number of nonzero entries in the ith row and ith column is less than n unless the matrix has a completely specified structure. Finally, we discuss the algorithmic aspects.     

The Jordan canonical form for a class of zero-one matrices

Let f:N→N be a function. Let A_n=(a_ij) be the n×n matrix defined by a_ij=1 if i=f(j) for some i and j and a_ij=0 otherwise. We describe the Jordan canonical form of the matrix A_n in terms of the directed graph for which A_n is the adjacency matrix. We discuss several examples including a connection with the Collatz 3n+1 conjecture.     

Zwei unzulässige verstärkungen der vermutung von Wilkinson

In the task of factoring a real n × n matrix A according to Gauss elimination with complete pivoting for size, Wilkinson conjectured in 1963 that the absolute values of the intermediate coefficients should not exceed the n-Max norm of the matrix, i.e. n Max_ij{|a_ij|}. Such a conjecture would be wrong for the 1-norm Max_j{Σ_i |a_ij|}, and for the Schur-Frobenius norm (Σ_ija²_ij)^{$\text{1}\text{2}$}, as is shown by providing counterexamples. However, these stronger versions of Wilkinson's conjecture do hold in the case of matrices of low order.     

A note on a conjecture on permanents

Let Kn denote the set of all n × n nonnegative matrices whose entries have sum n, and let ϕ be a real function on K_n defined by ϕ (X) = Πⁿ_i=1Σⁿ_j=1x_ij + Πⁿ_j=1Σⁿ_i=1x_ij − per X for X = [x_ij] ϵ K_n. A matrix A ϵ K_n is called a ϕ -maximizing matrix on K_n if ϕ (A) ⩾ ϕ (X) for all X ϵ K_n. It is conjectured that J_n = [1/n]_{n × n} is the unique ϕ-maximizing matrix on K_n. In this note, the following are proved: (i) If A is a positive ϕ-maximizing matrix, then A = J_n. (ii) If A is a row stochastic ϕ-maximizing matrix, then A = J_n. (iii) Every row sum and every column sum of a ϕ-maximizing matrix lies between 1 − √2·n!/nⁿ and 1 + (n − 1)√2·n!/nⁿ. (iv) For any p.s.d. symmetric A ϵ K_n, ϕ (A) ⩽ 2 − n!/nⁿ with equality iff A = J_n. (v) ϕ attains a strict local maximum on K_n at J_n.     

Distance matrices for points on a line,on a circle,and at the vertices of ann-dimensional cube

Forn pointsA _i ,i=1, 2, ...,n, in Euclidean space ℝ ^m , the distance matrix is defined as a matrix of the form D=(D _i ,j) _i ,j=1,...,n, where theD _i ,j are the distances between the pointsA _i andA _j . Two configurations of pointsA _i ,i=1, 2,...,n, are considered. These are the configurations of points all lying on a circle or on a line and of points at the vertices of anm-dimensional cube. In the first case, the inverse matrix is obtained in explicit form. In the second case, it is shown that the complete set of eigenvectors is composed of the columns of the Hadamard matrix of appropriate order. Using the fact that distance matrices in Euclidean space are nondegenerate, several inequalities are derived for solving the system of linear equations whose matrix is a given distance matrix. Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 127–138, July, 1995.     

On discount residue classes

It is shown that, whenever m₁, m₂,…, m_n are natural numbers such that the pairwise greatest common divisors, d_ij=(m_i, m_j), i≠j are distinct and different from 1, then there exist integers a₁, a₂,…,a_n such that the solution sets of the congruences x≡_i (modm_i), i= 1,2,…,n are disjoint.     

On the g-circulant solutions to the matrix equation Am = λJ

Let g and n be positive integers and let a₁ = (g, n) and a_m = (g^m, n). If h ≡ 0 (mod $\text{na}{}_1\text{a}{}_{\mathrm{m}}$), then the g-circulant whose Hall polynomial is equal to Σ_i=0^h?1xⁱ satisfies the matrix equation A^m = λJ, where n is the size of matrix J.     

r-Numbers completion problems of partial upper canonical form I

A pair (A, B), where A is an n × n matrix and B is an n × m matrix, is said to have the nonnegative integers sequence {r_j}_j=1^p as the r-numbers sequence if r₁ = rank(B) and r_j = rank[B AB … A^j−1 B] − rank[B AB … A^j−2B], 2 ≤ j ≤ p. Given a partial upper triangular matrix A of size n × n in upper canonical form and an n × m matrix B, we develop an algorithm that obtains a completion A_c of A, such that the pair (A^c, B) has an r-numbers sequence prescribed under some restrictions.     

On a Conjecture of E. Dittert

Let K_n denote the set of all n X n nonnegative matrices whose entries have sum n, and let φ be a real valued function defined on K_n by φ(X) = π_iⁿ⁼¹ n, + π_j=1^c_j⁻ⁿ per X for X E K_n with row sum vector (r₁,…, r_n) and column sum vector (c_l,hellip;, c_n). For the same X, let φ_ij(X)= π_k≠ir_k + π_1≠jc₁ - per X(i| j). A ϵK_n is called a φ-maximizing matrix if φ(A) > φ(X) for all X ϵ K_n. Dittert's conjecture asserts that J_n = [1/n]_n×n is the unique (φ-maximizing matrix on K_n. In this paper, the following are proved: (i) If A = [a_ij] is a φ-maximizing matrix on K_n then φ_ij(A) = φ (A) if a_ij > 0, and φ_ij (A) ⩽ φ (A)if aij = 0. (ii) The conjecture is true for n = 3.     

Presentations of alternating semigroups

One presentation of the alternating groupA _n hasn?2 generatorss ₁,…,s_n?2 and relationss ₁ ³ =s _i ² =(s_1?1s_i)³=(s_js_k)²=1, wherei>1 and |j?k|>1. Against this backdrop, a presentation of the alternating semigroupA _n ^c )A _n is introduced: It hasn?1 generatorss ₁,…,S _n?2,e, theA _n-relations (above), and relationse ²=e, (es ₁)⁴, (es _j)²=(es _j)⁴,es _i=s _i s ₁ ^-1 es ₁, wherej>1 andi≥1.     

Least‐squares solutions of matrix inverse problem for bi‐symmetric matrices with a submatrix constraint

An n × n real matrix A = (a_ij)_{n × n} is called bi‐symmetric matrix if A is both symmetric and per‐symmetric, that is, a_ij = a_ji and a_ij = a_n+1?1,n+1?i (i, j = 1, 2,..., n). This paper is mainly concerned with finding the least‐squares bi‐symmetric solutions of matrix inverse problem AX = B with a submatrix constraint, where X and B are given matrices of suitable sizes. Moreover, in the corresponding solution set, the analytical expression of the optimal approximation solution to a given matrix A^* is derived. A direct method for finding the optimal approximation solution is described in detail, and three numerical examples are provided to show the validity of our algorithm. Copyright © 2007 John Wiley & Sons, Ltd.     

Comportement periodique des fonctions a seuil binaires et applications

Soit (α_ij) une matrice symétrique n×n, à éléments réels, b un vecteur réel à n composantes et ? l'application de {0, 1)ⁿ dans lui-même, où la i-ème composante est une fonction à seuil avec séparateurΣⁿ_j=1α_ijy_j<b_i(y_j=0,1). Dans ce papier nous démontrons que la composition successive de △ par elle-même, n'a, en régime stationnaire, que des points fixes où des cycles de longueur deux. Ceci englobe le comportement périodique d'une certaine classe d'automates cellulaires et des modèles en dynamique des groupes pour lesquels existaient seulement des résultats particuliers (1,4,5,6).Let (α_ij) be a symmetric real n×n matrix and b a real n-vector. Let △ be a function from {0, 1}ⁿ to itself, whose ith component is the threshold function with separator Σⁿ_j=1α_ijy_j<b_i(y_j=0,1). It is shown that the repeated application of △, leads either to a fixed point or to a cycle of length two. This includes the periodic behaviour of a class of cellular automata and some models in groups dynamics (1,4,5,6).     

A class of Fibonacci-type sequences

Let {L_n:n ? 1} be a sequence of the form

where {a_j} and {b_j} are positive integers, and e=max_i,j{a_i, b_j}. A necessary and sufficient condition on the integers {a_j} and {b_j} is given so that, for all choices of positive initial values L₁, L₂,…,L_e,L_n=Σ^p_j=1L_n?aj for all large enough n.     

The inverse M-matrix problem

One aspect of the inverse M-matrix problem can be posed as follows. Given a positive n × n matrix A=(a_ij) which has been scaled to have unit diagonal elements and off-diagonal elements which satisfy 0 < y ? a_ij ? x < 1, what additional element conditions will guarantee that the inverse of A exists and is an M-matrix? That is, if A^?1=B=(b_ij), then b_ii> 0 and b_ij ? 0 for i≠_j. If n=2 or x=y no further conditions are needed, but if n ? 3 and y < x, then the following is a tight sufficient condition. Define an interpolation parameter s via x²=sy+(1?s)y²; then B is an M-matrix if s^?1 ? n?2. Moreover, if all off-diagonal elements of A have the value y except for a_ij=a_jj=x when i=n?1, n and 1 ? _j ? n?2, then the condition on both necessary and sufficient for B to be an M-matrix.     

Generalized latin square

LetX be a n-set and letA = [a_ij] be an xn matrix for whichaij ?X, for 1 ≤i, j ≤n. A is called a generalized Latin square onX, if the following conditions is satisfied: $ \cup _{i = 1}^n a_{ij} = X = \cup _{j = 1}^n a_{ij} $ . In this paper, we prove that every generalized Latin square has an orthogonal mate and introduce a H_v -structure on a set of generalized Latin squares. Finally, we prove that every generalized Latin square of ordern, has a transversal set.