We show that nonlinear tight-binding lattices of different geometries and dimensionalities, display an universal selftrapping behavior. First, we consider the single nonlinear impurity problem in various tight-binding lattices, and use the Green's function formalism for an exact calculation of the minimum nonlinearity strength to form a stationary bound state. For all lattices, we find that thi...

A short, concise proof is given for that k≥5 there exists a k-uniform hypergraph H without exponent, i.e., when the Turán function not polynomial in n. More precisely, we have ex(n,H)=o(nk−1) but it exceeds nk−1−c any positive c n>n0(k,c). We conjecture this true k∈{3,4} as well.

In 1978, Apéry [2] proved the irrationality of ζ(3) by constructing two explicit sequences of integers (un)n and (vn)n such that 0 6= unζ(3)− vn → 0 and un → +∞, both at geometric rates. He also deduced from this an upper bound for the irrationality exponent μ(ζ(3)) of ζ(3). In general, the irrationality exponent μ(ξ) of an irrational number ξ is defined as the infimum of all real numbers μ suc...

Article history: Received 29 June 2010 Received in final revised form 9 September 2010 Available online 30 October 2010

This paper deals with the number of subgroups of a given exponent in a finite abelian group. Explicit formulas are obtained in the case of rank two and rank three abelian groups. An asymptotic formula is also presented.

We consider a perturbed system of exponents { en } n∈Z, when the sequence {λn }n∈Z has a definite asymptotics. We study basis properties (completeness, minimality, basicity) of this system in Lebesgue space of functions Lp(·) (−π, π) with variable summability exponent p(·), subject to parameters contained in the asymptotics {λn}n∈Z.

Primitive digraphs on n vertices, k arcs and girth s are considered. By a(n, k, s) we mean the minimum exponent taken over all such digraphs. We estimate the number a(n, k, s) using the Frobenius number for special values of k and s.

In this note we use variational arguments {namely Ekeland's Principle and the Mountain Pass Theorem{ to study the equation ?u + a(x)u = u q + u 2 ?1 in R N : The main concern is overcoming compactness diiculties due both to the unboundedness of the domain R N , and the presence of the critical exponent 2 = 2N=(N ? 2).

Let Pn,d be the set of n x n non-symmetric primitive matrices with exactly d nonzero diagonal entries. For each positive integer 2 ~ k ~ n -1, we determine the kth upper generalized exponent set for Pn,d and characterize the extremal matrices by using a graph theoretical method.

We show that if the private exponent d used in the RSA public-key cryptosystem is less than N then the system is insecure. This is the first improvement over an old result of Wiener showing that when d < N the RSA system is insecure. We hope our approach can be used to eventually improve the bound to d < N.