A combinatorial identity with application to Catalan numbers

By a very simple argument, we prove that if l,m,n∈{0,1,2,…} then

     

On various restricted sumsets

For finite subsets A₁,…,A_n of a field, their sumset is given by . In this paper, we study various restricted sumsets of A₁,…,A_n with restrictions of the following forms:

     

On sums of binomial coefficients and their applications

In this paper we study recurrences concerning the combinatorial sum and the alternate sum , where m>0, n?0 and r are integers. For example, we show that if n?m-1 then

     

On the number of representations of certain integers as sums of 11 or 13 squares

Let r_k(n) denote the number of representations of an integer n as a sum of k squares. We prove that for odd primes p,

     

On the residue classes of integer sequences satisfying a linear three term recurrence formula

In this paper we investigate linear three-term recurrence formulae with sequences of integers (T(n))_n?0 and (U(n))_n?0, which are ultimately periodic modulo m, e.g.

     

Large matchings from eigenvalues

We find lower bounds on the difference between the spectral radius λ₁ and the average degree of an irregular graph G of order n and size e. In particular, we show that, if n ? 4, then

     

Congruences for sums of binomial coefficients

Let q>1 and m>0 be relatively prime integers. We find an explicit period νm(q) such that for any integers n>0 and r we have

     

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

     

Inequalities for the median of the gamma distribution

We provide several new inequalities involving λ_n, the median of the gamma distribution of order n+1 with parameter 1. Among others, we present sharp upper and lower bounds for the arithmetic mean of λ₁,λ₂,…,λ_n. For all integers n?1 we have

     

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

     

Asymptotic properties of generalized Laguerre orthogonal polynomials

In the present paper we deal with the polynomials L_n^(α,M,N) (x) orthogonal with respect to the Sobolev inner product

     

Relaxation and regularity in the calculus of variations

In this work we prove that, if L(t,u,ξ) is a continuous function in t and u, Borel measurable in ξ, with bounded non-convex pieces in ξ, then any absolutely continuous solution to the variational problem

     

On random perturbation of some intersective sets

Given a subset S of Z and a sequence I = (I_n)_n=1^∞ of intervals of increasing length contained in Z, let

     

4-Books of three pages

Let H be a 4-uniform hypergraph on an n-element vertex set V containing no 4-book of 3 pages, i.e., a hypergraph of 4 quadruples with vertices {1,2,…,7} and edges {1234,1235,1236,4567}. Then for n>n₀

     

The boundary of higher rank numerical ranges

The rank-k-numerical range of an n×n matrix A is defined as

     

Le problème de Yamabe avec singularités

Let (Mn,g) be a compact riemannian manifold of dimension n?3. Under some assumptions, we prove that there exists a positive function φ solution of the Yamabe equation

     

Young’s inequality and trace

Let φ be a positive linear functional on M_n(C) and f,g mutually conjugate in the sense of Young. In this note we show a necessary and sufficient condition for the inequality