A link between the Perron-Frobenius theorem and Perron’s theorem for difference equations

Let A_n,n∈N, be a sequence of k×k matrices which converge to a matrix A as n→∞. It is shown that if x_n,n∈N, is a sequence of nonnegative nonzero vectors such that

     

Some identities involving theta functions

Let (|q|<1). For k∈N it is shown that there exist k rational numbers A(k,0),…,A(k,k−1) such that

     

Dilation equations and Markov operators

Using the theory of Markov operators, we give a new proof of the known fact saying that for every positive integers N and k>1, and for every nonnegative reals c₀,…,cN satisfying the first sum rule the dilation equation

     

Global stability of some symmetric difference equations

Suppose r∈(0,1],m∈N and 1?k₁<k₂<?<k_2m+1, and let S_2m+1={1,2,…,2m+1}. We show that every positive solution to the difference equation

     

Structural properties of Riordan matrices and extending the matrices

We consider an infinite lower triangular matrix L=[?_n,k]_n,k∈N0 and a sequence Ω=(ω_n)_n∈N0 called the (a,b)-sequence such that every element ?_n+1,k+1 except lying in column 0 can be expressed as

     

Stochastic matrices and a property of the infinite sequences of linear functionals

Our starting point is the proof of the following property of a particular class of matrices. Let T={T_i,j} be a n×m non-negative matrix such that ∑_jT_i,j=1 for each i. Suppose that for every pair of indices (i,j), there exists an index l such that T_i,l≠T_j,l. Then, there exists a real vector k=(k₁,k₂,…,k_m)^T,k_i≠k_j,i≠j;0<k_i?1, such that, if i≠j.Then, we apply that property of matrices to probability theory. Let us consider an infinite sequence of linear functionals , corresponding to an infinite sequence of probability measures {μ_(·)(i)}_i∈N, on the Borel σ-algebra such that, . The property of matrices described above allows us to construct a real bounded one-to-one piecewise continuous and continuous from the left function f such that

     

Super-stationary set, subword problem and the complexity

Let Ω⊂{0,1}^N be a nonempty closed set with N={0,1,2,…}. For N={N₀<N₁<N₂<?}⊂N and ω∈{0,1}^N, define ω[N]∈{0,1}^N by and

     

On the parity of exponents in the standard factorization of n!

Let p₁,p₂,… be the sequence of all primes in ascending order. The following result is proved: for any given positive integer k and any given , there exist infinitely many positive integers n with

     

Precise rates in the law of logarithm for the moment convergence of i.i.d. random variables

Let be a sequence of i.i.d. random variables with EX=0 and EX²=σ²<∞. Set , Mn=maxk?n|Sk|, n?1. Let r>1, then we obtain

     

Maps preserving the spectrum of generalized Jordan product of operators

Let A₁,A₂ be standard operator algebras on complex Banach spaces X₁,X₂, respectively. For k?2, let (i₁,…,i_m) be a sequence with terms chosen from {1,…,k}, and define the generalized Jordan product

     

A short proof of a cross-intersection theorem of Hilton

Families A₁,…,A_k of sets are said to be cross-intersecting if for any A_i∈A_i and A_j∈A_j, i≠j. A nice result of Hilton that generalises the Erd?s-Ko-Rado (EKR) Theorem says that if r≤n/2 and A₁,…,A_k are cross-intersecting sub-families of , then

     

Notes sur la fonction ζ de Riemann, 4

Opération fondamentale de l'arithmétique, familière depuis des millénaires, la division euclidienne n'a pas livré tous ses secrets. Ainsi, notons pour k et a entiers positifs, le reste de la division euclidienne de k par a, et imaginons un instant que, par un choix convenable d'un entier n et de réels c₂,…,c_n, nous sachions rendre arbitrairement petite la quantité

     

The asymptotic number of spanning trees in circulant graphs

Let T(G) be the number of spanning trees in graph G. In this note, we explore the asymptotics of T(G) when G is a circulant graph with given jumps.The circulant graph is the 2k-regular graph with n vertices labeled 0,1,2,…,n−1, where node i has the 2k neighbors i±s₁,i±s₂,…,i±s_k where all the operations are . We give a closed formula for the asymptotic limit as a function of s₁,s₂,…,s_k. We then extend this by permitting some of the jumps to be linear functions of n, i.e., letting s_i, d_i and e_i be arbitrary integers, and examining

     

Generalization of transitive fraternal augmentations for directed graphs and its applications

Let be a directed graph. A transitive fraternal augmentation of is a directed graph with the same vertex set, including all the arcs of and such that for any vertices x,y,z,

1.

if and then or (fraternity);

2.

if and then (transitivity).

In this paper, we explore some generalization of the transitive fraternal augmentations for directed graphs and its applications. In particular, we show that the 2-coloring number col₂(G)≤O(∇₁(G)∇₀(G)²), where ∇_k(G) (k≥0) denotes the greatest reduced average density with depth k of a graph G; we give a constructive proof that ∇_k(G) bounds the distance (k+1)-coloring number col_k+1(G) with a function f(∇_k(G)). On the other hand, ∇_k(G)≤(col_2k+1(G))^2k+1. We also show that an inductive generalization of transitive fraternal augmentations can be used to study nonrepetitive colorings of graphs.     

Perturbation from Dirichlet problem involving oscillating nonlinearities

In this paper we prove that if the potential has a suitable oscillating behavior in any neighborhood of the origin (respectively +∞), then under very mild conditions on the perturbation term g, for every k∈N there exists bk>0 such that

     

Operator algebras associated with unitary commutation relations

We define nonselfadjoint operator algebras with generators Le₁,…,Len,Lf₁,…,Lfm subject to the unitary commutation relations of the form