We prove that for a discrete determinantal process the BK inequality occurs for increasing events generated by simple points. We also give some elementary but nonetheless appealing relationships between a discrete determinantal process and the well-known CS decomposition.

We prove that every automorphism of an infinite-dimensional vector space over a field is the product of four involutions, a result that is optimal in the general case. We also characterize the automorphisms that are the product of three involutions. More generally, we study decompositions of automorphisms into three or four factors with prescribed split annihilating polynomials of degree $2$.

It is known that every Toeplitz matrix $T$ enjoys a circulant and skew circulant splitting (denoted $\text{CSCS}$) i.e., $T=C-S$ a circulant matrix and $S$ a skew circulant matrix. Based on the variant of such a splitting (also referred to as $\text{CSCS}$), we first develop classical $\text{CSCS}$ iterative methods and then introduce shifted $\text{CSCS}$ iterative methods for solving hermitian positive definite Toeplitz systems in this paper. The convergence of each method is analyzed. Numerical experiments show that the classical $\text{CSCS}$ iterative methods work slightly better than the Gauss–Seidel $(\text{GS})$ iterative methods if the $\text{CSCS}$ is convergent, and that there is always a constant $\alpha $ such that the shifted $\text{CSCS}$ iteration converges much faster than the Gauss–Seidel iteration, no matter whether the $\text{CSCS}$ itself is convergent or not.

In this paper, double commutativity and the reverse order law for the core inverse are considered. Then new characterizations of the Moore–Penrose inverse of a regular element are given by one-sided invertibilities in a ring. Furthermore, the characterizations and representations of the core and dual core inverses of a regular element are considered.

In this paper a determinant identity is established, from which a simple proof of the multivariate Lagrange–Good inversion formula follows directly. Further discussion on a discrete analogue of the Lagrange–Good inversion formula is also presented.

When a discrete-time homogenous Markov chain is observed at time intervals that correspond to its time unit, then the transition probabilities of the chain can be estimated using known maximum likelihood estimators. In this paper we consider a situation when a Markov chain is observed on time intervals with length equal to twice the time unit of the Markov chain. The issue then arises of characterizing probability matrices whose square root(s) are also probability matrices. This characterization is referred to in the literature as the embedding problem for discrete time Markov chains. The probability matrix which has probability root(s) is called embeddable.

In this paper for two-state Markov chains, necessary and sufficient conditions for embeddability are formulated and the probability square roots of the transition matrix are presented in analytic form. In finding conditions for the existence of probability square roots for (k x k) transition matrices, properties of row-normalized matrices are examined. Besides the existence of probability square roots, the uniqueness of these solutions is discussed: In the case of nonuniqueness, a procedure is introduced to identify a transition matrix that takes into account the specificity of the concrete context. In the case of nonexistence of a probability root, the concept of an approximate probability root is introduced as a solution of an optimization problem related to approximate nonnegative matrix factorization.

We propose a direct solver for the three-dimensional Poisson equation with a variable coefficient, and an algorithm to directly solve the associated sparse linear systems that exploits the sparsity pattern of the coefficient matrix. Introducing some appropriate finite difference operators, we derive a second-order scheme for the solver, and then two suitable high-order compact schemes are also discussed. For a cube containing N nodes, the solver requires arithmetic operations and memory to store the necessary information. Its efficiency is illustrated with examples, and the numerical results are analysed.

A matrix A over a field F is said to be an AJT matrix if there exists a vector x over F such that both x and Ax have no zero component. The Alon–Jaeger–Tarsi (AJT) conjecture states that if F is a finite field, with |F|≥4, and A is an element of GL n (F) , then A is an AJT matrix. In this paper we prove that every nonzero matrix over a field F, with |F|≥3 , is similar to an AJT matrix. Let AJTn (q) denote the set of n×n, invertible, AJT matrices over a field with q elements. It is shown that the following are equivalent for q≥3 : (i) AJTn (q)=GL n (q) ; (ii) every 2n×n matrix of the form (A∣B)t has a nowhere-zero vector in its image, where A,B are n×n, invertible, upper and lower triangular matrices, respectively; and (iii) AJTn (q) forms a semigroup.

In this paper we derive perturbation theorems for the LU and QR factors. Moreover, bounds for κL(A)/κL′(A) and κU(A)/κ′U(A) are given for the LU factorization of a nonsingular matrix. By applying pivoting strategies in the LU factorization, estimates for κL(PAQ)/κL′(PAQ) and κU(PAQ)/κ′U(PAQ) are also obtained.

Let $G$ be a finite group. We classify $G$-equivariant flow equivalence of non-trivial irreducible shifts of finite type in terms of

(i) elementary equivalence of matrices over $ZG$ and

(ii) the conjugacy class in $ZG$ of the group of G-weights of cycles based at a fixed vertex.

In the case $G = Z/2$, we have the classification for twistwise flow equivalence. We include some algebraic results and examples related to the determination of $E(ZG)$ equivalence, which involves $K_1(ZG)$.

Consider the random walk {Sn} whose summands have the distribution P(X=0) = 1-(2/π), and P(X = ± n) = 2/[π(4n2−1)], for n ≥ 1. This random walk arises when a simple random walk in the integer plane is observed only at those instants at which the two coordinates are equal. We derive the fundamental matrix, or Green function, for the process on the integral [0,N] = {0,1,…,N}, and from this, an explicit formula for the mean time xk for the random walk starting from S0 = k to exit the interval. The explicit formula yields the limiting behavior of xk as N → ∞ with k fixed. For the random walk starting from zero, the probability of exiting the interval on the right is obtained. By letting N → ∞ in the fundamental matrix, the Green function on the interval [0,∞) is found, and a simple and explicit formula for the probability distribution of the point of entry into the interval (−∞,0) for the random walk starting from k = 0 results. The distributions for some related random variables are also discovered.

Applications to stress concentration calculations in discrete lattices are briefly reviewed.

A theory of minimal realizations of rational matrix functions $W(\lambda )$ in the “pencil” form $W(\lambda )=C{{(\lambda {{A}_{1}}-{{A}_{2}})}^{-1}}B$ is developed. In particular, properties of the pencil $\text{ }\!\!\lambda\!\!\text{ }{{A}_{1}}\,-\,{{A}_{2}}$ are discussed when $W(\lambda )$ is hermitian on the real line, and when $W(\lambda )$ is hermitian on the unit circle.

The u-invariant u(K) of a field K is the smallest number such that every k-dimensional regular quadratic form over K is universal. Let O(V,f) be the orthogonal group of a finite-dimensional regular metric vector space over a field K of characteristic distinct from 2. Let π ∊ 0(V), B(π) := V(π - 1), dim[B(π) ∩ kernel(π - 1)] > u(K). Given λ1,..., λm ∊ K* where m := dim B(π) - u(K) + 1. Then π = σ1······σk where k := dim B(π) and σi is a symmetry with negative space Kai and f(ai, ai) = λi for i ∊ { 1 , . . . , m}. We prove similar theorems also for symplectic groups where transvections are taken as generators.