Primitive Matrices Research Articles

Twisted Maaß–Koecher series and spinor zeta functions

AbstractIt is shown that a Siegel-Hecke eigenform of integral weight k and genus 2 is uniquely determined by its Fourier coefficients indexed by nT where T runs over all half-integral positive definite primitive matrices of size 2 and n over all squarefree positive integers. The proof uses analytic arguments involving Koecher-Maaß series and spinor zeta functions.

Graphs with Odd Girth at Least Seven and High Minimum Degree

The extreme matrices of the kth upper multiexponents of primitive matrices are characterized completely and the existence of gaps in the kth upper multiexponent set is proved.

Algebraic Computations with Continued Fractions

General algorithms, viewed as transducers, are introduced for computing rational expressions with continued fraction expansions. Moreover, expansions of some algebraic numbers, like2or those related to primitive matrices are considered.

The exponent of the primitive Cayley digraphs on finite Abelian groups

Let G be finite group and let S be a subset of G. We prove a necessary and sufficient condition for the Cayley digraph X ( G, S) to be primitive when S contains the central elements of G. As an immediate consequence we obtain that a Cayley digraph X ( G, S) on an Abelian group is primitive if and only if S −1 S is a generating set for G. Moreover, it is shown that if a Cayley digraph X ( G, S) on an Abelian group is primitive, then its exponent either is n − l, [ n 2 ], [ n 2 ] − 1 or is not exceeding [ n 3 ] + 1 . Finally, we also characterize those Cayley digraphs on Abelian groups with exponent n − 1, [ n 2 ], [ n 2 ] − 1 . In particular, we generalize a number of well-known results for the primitive circulant matrices.

K-System Generator of Pseudorandom Numbers on Galois Field

We analyze the structure of the periodic trajectories of the K-system generator of pseudorandom numbers on a rational sublattice which coincides with the Galois field GF[p]. The period of the trajectories increases as a function of the lattice size p and the dimension of the K-matrixd. We emphasize the connection of this approach with the one which is based on primitive matrices over Galois fields.

Primitivity of generalized circulant Boolean matrices

Let G u be the semigroup of n × n generalized circulant Boolean matrices and G( J n ) the set of all primitive matrices in G n . Necessary and sufficient conditions for an r-circulant Boolean matrix to be a primitive matrix are given. We also show that G( J n ) is a subsemigroup of G n .

A conjecture about the strict fully indecomposable exponents of primitive matrices

The conjecture about the strict fully indecomposable exponents of nxn primitive Boolean matrices are proved as follows:

The characterization of symmetric primitive matrices with exponent 2n−2r(n)∗

In 1986, Shao [3] proved that the exponent set of symmetric primitive matrices is {1,2,…,2n−2}\S, where S consists of all odd numbers among {n,n∣1,…,2n−2}, This paper gives the complete description of symmetric primitive matrices with exponent 2n−2r(n).

Proof of a conjecture about the exponent of primitive matrices

An n × n nonnegative matrix A is called primitive if for some positive integer k, every entry in the matrix A k is positive ( A k ⪢ 0). The exponent of primitivity of A is defined to be γ( A) = min{ k ∈ Z + : A k ⪢ 0}, where Z + denotes the set of positive integers. The upper bound on γ( A) due to Wielandt is γ( A) ≤ ( n − 1) 2 + 1, and a better bound for γ( A) due to Hartwig and Neumann is γ( A) ≤ m( m − 1), where m is the degree of the minimal polynomial of A. Also, Hartwig and Neumann conjecture that γ( A) ≤ ( m − 1) 2 + 1, which had been suggested in 1984. In this paper, we prove this conjecture.

On a conjecture about the generalized exponent of primitive matrices

AbstractIn this paper the conjecture on the kth upper multiexponent of primitive matrices proposed by R.A. Brualdi and Liu are completely proved.

The index set problem for Boolean (or nonnegative) matrices

The index set problem for a class of Boolean (or nonnegative) matrices is a generalization of the exponent set problem for n × n primitive matrices. We survey the recent advances on the index set problem for various classes of Boolean matrices. Some related research problems are suggested.

The drift of a one-dimensional self-avoiding random walk

We prove that a self-avoiding random walk on the integers with bounded increments grows linearly. We characterize its drift in terms of the Frobenius eigenvalue of a certain one parameter family of primitive matrices. As an important tool, we express the local times as a two-block functional of a certain Markov chain, which is of independent interest.

Algebraic Shift Equivalence and Primitive Matrices

Algebraic Shift Equivalence and Primitive Matrices

Algebraic shift equivalence and primitive matrices

Motivated by symbolic dynamics, we study the problem, given a unital subring S S of the reals, when is a matrix A A algebraically shift equivalent over S S to a primitive matrix? We conjecture that simple necessary conditions on the nonzero spectrum of A A are sufficient, and establish the conjecture in many cases. If S S is the integers, we give some lower bounds on sizes of realizing primitive matrices. For Dedekind domains, we prove that algebraic shift equivalence implies algebraic strong shift equivalence.

Generalized exponents of primitive simple graphs

The exponent of a primitive digraph has been generalized in [2]. In this paper we obtain new parameters on generalized exponent of primitive simple graphs (symmetric primitive (0,1) matrices with zero trace) completely.

Exponents of classes of non-negative matrices

AbstractA square non-negative matrix is called primitive if all the elements of some power of this matrix are positive. The exponent of a primitive matrix is the minimum power satisfying this condition. The exponent of a class of primitive matrices is the minimum power such that all the matrices of the class to this power have positive elements only. In this paper, bounds for the exponents of primitive matrices and classes of matrices are obtained.

On fully indecomposable exponent for primitive boolean matrices with symmetric ones

If A is a primitive matrix, then there is a smallest power of A (its fully indecomposable exponent) which is fully indecomposable, and a smallest power of A (its strict fully indecomposable exponent) starting from which all powers are fully indecomposable. We obtain bounds on these two exponents for primitive Boolean matrices with symmetric one's.

Strong shift equivalence of boolean and positive rational matrices

Firstly, we show that two primitive Boolean matrices are strong shift equivalent if and only if the Boolean traces of each of their powers are equal. Secondly, we prove that if two matrices over Q + are strong shift equivalent through positive matrices over R +, then they are strong shift equivalent through matrices over Q +.

On the growth of infinite products of slowly varying primitive matrices

Let { A( t)} be a sequence of bounded nonnegative matrices [each with Perron root λ( t)] satisfying a condition of uniform primitivity and having nonzero entries bounded away from 0. In a generalization of a result on powers of primitive matrices it is shown that if the matrices A( t) vary slowly enough (i.e., ‖ A( t+1)− A( t)‖⩽ ε for all t and some small ε), then, as soon as the entries p i,j ( t) of the backward product A( t) A( t−1), ⋯, A(1) are positive, the ratios p i, j ( t)/ p i, j ( t−1) are equal to λ( t)+ h i, j ( t), where for all t 0 each h i,j ( t 0) is small in a precisely quantified manner: the larger t 0 is and the slower the probability-normed Perron vectors V( t) of A( t) have changed in the recent past preceding t 0, the smaller h i,j ( t 0) will be. (Other similar results are derived.) The resulting approximation p i, j ( t)/ p i, j ( t−1)≈ λ( t) is illustrated with two numerical examples.

Introduction of a switch reduced circuit for the analysis of variable topology networks

A systematic and unified procedure is developed to minimize the computation time necessary for the generation of transformation tensors needed to formulate automatically the equations of variable topology networks. This procedure is used whenever a change in circuit configuration has occured due to the presence of switching devices. A novel feature of this work is the introduction of a new concept, that of the Switch Reduced Circuit (SRC). This is an auxiliary “circuit”, containing only all switch branches of the graph of the network under consideration. Then, the transformation tensors are more easily produced as a function of the resultant circuit configuration, by making use of the new concept of SRC, plus elementary concepts from classical circuit theory, and primitive incidence matrices. Finally, the necessary on-switch current and off-switch voltage relationships are also presented.