Abstract About two dozens of exactly solvable Markov chains on one-dimensional finite and semi-infinite integer lattices are constructed in terms convolutions orthogonality measures the Krawtchouk, Hahn, Meixner, Charlier, q-Hahn, q-Meixner little q-Jacobi polynomials. By construction, stationary probability distributions, complete sets eigenvalues eigenvectors provided by polynomials measures. A...

Localized states of Harper's equation correspond to strange nonchaotic attractors in the related Harper mapping. In parameter space, these fractal attractors with nonpositive Lyapunov exponents occur in fractally organized tongue-like regions which emanate from the Cantor set of eigenvalues on the critical line epsilon=1. A topological invariant characterizes wave functions corresponding to ene...

Let k be a positive integer and let G be a graph of order n ≥ k. It is proved that the sum of k largest eigenvalues of G is at most 1 2 ( √ k+1)n. This bound is shown to be best possible in the sense that for every k there exist graphs whose sum is 12 ( √ k + 12 )n− o(k−2/5)n. A generalization to arbitrary symmetric matrices is given.

Let A be a t × t invertible matrix with integer entries and with eigenvalues |λi| 6 = 1, i ∈ [1, t]. In this paper we prove explicitly that there exists a vector α, such that discrepancy of the sequence {αA} n=1 is equal to O(N(logN)) for N −→ ∞. This estimate can be improved no more than on the logarithmic factor. Communicated by Robert F. Tichy Dedicated to the memory of Professor Edmund Hlawka

Let Sn be the positive real symmetric matrix of order n with (i, j ) entry equal to ( i + j − 2 j − 1 ) , and let x be a positive real number. Eigenvalues of the Hadamard (or entry wise) power S n are considered. In particular for k a positive integer, it is shown that both the Perron root and the trace of S n are approximately equal to 4 4k−1 ( 2n− 2 n− 1 )k . © 2005 Elsevier Inc. All rights r...

For any even positive integer 2n and any positive integermwe construct a class of regular self-adjoint and non-self-adjoint boundary value problems whose spectrum consists of at most (2n − 1)m + 1 eigenvalues. Our main result reduces to previously known results for the cases n = 1 and n = 2. In the self-adjoint case with separated boundary conditions this upper bound can be improved to n(m + 1)...

The method of symmetric polynomials (MSP) was developed for computation analytical functions of matrices, in particular, integer powers of matrix. MSP does not require for its realization finding eigenvalues of the matrix. A new type of recurrence relations for symmetric polynomials of order n is found. Algorithm for the numerical calculation of high powers of the matrix is proposed.This comput...

In this paper, a fundamentally new method, based on the denition, is introduced for numerical computation of eigenvalues, generalized eigenvalues and quadratic eigenvalues of matrices. Some examples are provided to show the accuracy and reliability of the proposed method. It is shown that the proposed method gives other sequences than that of existing methods but they still are convergent to th...

Let n ? 4 be an even integer and W the wheel graph with vertices. The distance d i j between any two distinct vertices of is length shortest path connecting . D × symmetric matrix diagonal entries equal to zero off-diagonal In this paper, we find a positive semidefinite L ˜ such that rank ( ) = ? 1 , all row sums zero, one w T 2 + An interlacing property eigenvalues also proved.

In this paper we study, both analytically and numerically, questions involving the distribution of eigenvalues of Maass forms on the moonshine groups Γ0(N), where N > 1 is a square-free integer. After we prove that Γ0(N) has one cusp, we compute the constant term of the associated non-holomorphic Eisenstein series. We then derive an “average” Weyl’s law for the distribution of eigenvalues of Ma...