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

     

Lower bounds of Copson type for Hausdorff matrices

Let 1 ? p ? ∞, 0 < q ? p, and A = (a_n,k)_n,k?0 ? 0. Denote by L_p,q(A) the supremum of those L satisfying the following inequality:

     

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

     

Sum rules for Jacobi matrices and divergent Lieb-Thirring sums

Let E_j be the eigenvalues outside [-2,2] of a Jacobi matrix with a_n-1∈?² and b_n→0, and μ^′ the density of the a.c. part of the spectral measure for the vector δ₁. We show that if b_n∉?⁴, b_n+1-b_n∈?², then

     

Quasi-convex sequences in the circle and the 3-adic integers

In this paper, we present families of quasi-convex sequences converging to zero in the circle group T, and the group J₃ of 3-adic integers. These sequences are determined by increasing sequences of integers. For an increasing sequence , put gn=an+1−an. We prove that

(a)

the set {0}∪{±³−(an+1)|n∈N} is quasi-convex in T if and only if a₀>0 and gn>1 for every n∈N;

(b)

the set {0}∪{±an³|n∈N} is quasi-convex in the group J₃ of 3-adic integers if and only if gn>1 for every n∈N.

Moreover, we solve an open problem from [D. Dikranjan, L. de Leo, Countably infinite quasi-convex sets in some locally compact abelian groups, Topology Appl. 157 (8) (2010) 1347-1356] providing a complete characterization of the sequences such that {0}∪{±²−(an+1)|n∈N} is quasi-convex in T. Using this result, we also obtain a characterization of the sequences such that the set {0}∪{±²−(an+1)|n∈N} is quasi-convex in R.     

Locally compact abelian groups admitting non-trivial quasi-convex null sequences

In this paper, we show that, for every locally compact abelian group G, the following statements are equivalent:

(i)

G contains no sequence such that {0}∪{±x_n∣n∈N} is infinite and quasi-convex in G, and x_n?0;

(ii)

one of the subgroups {g∈G∣2g=0} or {g∈G∣3g=0} is open in G;

(iii)

G contains an open compact subgroup of the form or for some cardinal κ.

     

Erd?s-Ko-Rado for three sets

Fix integers k?3 and n?3k/2. Let F be a family of k-sets of an n-element set so that whenever A,B,C∈F satisfy |A∪B∪C|?2k, we have A∩B∩C≠∅. We prove that with equality only when ?_F∈FF≠∅. This settles a conjecture of Frankl and Füredi [2], who proved the result for n?k²+3k.     

Note on integer powers in sumsets

Let k,m,n?2 be integers. Let A be a subset of {0,1,…,n} with 0∈A and the greatest common divisor of all elements of A is 1. Suppose that

     

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

     

Extremal and Barabanov semi-norms of a semigroup generated by a bounded family of matrices

Let S={Si}_i∈I be an arbitrary family of complex n-by-n matrices, where 1?n<∞. Let denote the joint spectral radius of S, defined as

     

Some relations between rank of a graph and its complement

Let G be a graph of order n and rank(G) denotes the rank of its adjacency matrix. Clearly, . In this paper we characterize all graphs G such that or n + 2. Also for every integer n ? 5 and any k, 0 ? k ? n, we construct a graph G of order n, such that .     

A note on the spectral singularities of non-selfadjoint matrix-valued difference operators

Let denote the non-selfadjoint operator generated in ?₂(N,E) by the matrix difference expression (?y)_n=A_n−1y_n−1+B_ny_n+A_ny_n+1, n∈N, and the boundary condition y₀=0. In this paper we investigate the Jost solution, the continuous spectrum, the eigenvalues and the spectral singularities of L.     

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

     

Subordination results for classes of analytic functions related to conic domains defined by a fractional operator

Let a fractional operator (n∈N₀={0,1,2,…}, 0?α<1, λ?0) be defined by

     

On the Turán number for the hexagon

A long-standing conjecture of Erd?s and Simonovits is that ex(n,C_2k), the maximum number of edges in an n-vertex graph without a 2k-gon is asymptotically as n tends to infinity. This was known almost 40 years ago in the case of quadrilaterals. In this paper, we construct a counterexample to the conjecture in the case of hexagons. For infinitely many n, we prove that

     

Convergence of subdivision schemes and smoothness of limit functions

Starting with vector λ=(λ(k))_k∈Z∈?p(Z), the subdivision scheme generates a sequence of vectors by the subdivision operator