This paper is about the rate of convergence of the Markov chain Xn+1 = AXn+Bn (mod p), where A is an integer matrix with nonzero eigenvalues and {Bn}n is a sequence of independent and identically distributed integer vectors, with support not parallel to a proper subspace of Q invariant under A. If |λi| 6 = 1 for all eigenvalues λi of A, then n = O ( (ln p) ) steps are sufficient and n = O(ln p)...

let $g$ be a simple graph‎, ‎and $g^{sigma}$‎ ‎be an oriented graph of $g$ with the orientation ‎$sigma$ and skew-adjacency matrix $s(g^{sigma})$‎. ‎the $k-$th skew spectral‎ ‎moment of $g^{sigma}$‎, ‎denoted by‎ ‎$t_k(g^{sigma})$‎, ‎is defined as $sum_{i=1}^{n}( ‎‎‎lambda_{i})^{k}$‎, ‎where $lambda_{1}‎, ‎lambda_{2},cdots‎, ‎lambda_{n}$ are the eigenvalues of $g^{sigma}$‎. ‎suppose‎ ‎$g^{sigma...

For a nonsingular integer matrix A, we study the growth of the order of A modulo N . We say that a matrix is exceptional if it is diagonalizable, and a power of the matrix has all eigenvalues equal to powers of a single rational integer, or all eigenvalues are powers of a single unit in a real quadratic field. For exceptional matrices, it is easily seen that there are arbitrarily large values o...

The singular solution of the Laplace equation with a straight-crack is represented by a series of eigenpairs, shadows and their associated edge flux intensity functions (EFIFs). We address the computation of the EFIFs associated with the integer eigenvalues by the quasi-dual function method (QDFM).The QDFM is based on the dual eigenpairs and shadows, and we exhibit the presence of logarithmic t...

In this paper we construct trees having only integer eigenvalues with arbitrarily large diameters. In fact, we prove that for every set S of positive integers there exists a tree whose positive eigenvalues are exactly the elements of S. If the set S is different from the set {1} then the constructed tree will have diameter 2|S|.

The definition of the standard differential operator is extended from integer steps to arbitrary stepsize. The classical, nonrelativistic Hamiltonian is quantized, using these new continuous operators. The resulting Schroedinger type equation generates free particle solutions, which are confined in space. The angular momentum eigenvalues are calculated algebraically. It is shown, that the charm...