On consecutive happy numbers

Let e?1 and b?2 be integers. For a positive integer with 0?aj<b, define

     

Positive clustered layered solutions for the Gierer-Meinhardt system

We consider the stationary Gierer-Meinhardt system in a ball of RN:

     

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

     

The local variational principle of topological pressure for sub-additive potentials

Let (X,T) be a topological dynamical system and be a sub-additive potential on C(X,R). Let U be an open cover of X. Then for any T-invariant measure μ, let . The topological pressure for open covers U is defined for sub-additive potentials. Then we have a variational principle:

     

Idéaux fermés d'algèbres de Beurling analytiques sur le bidisque

We study the closed ideal in the Beurling algebras of holomorphic function f in the bidisc such that

     

New solutions for nonlinear Schrödinger equations with critical nonlinearity

We consider the following nonlinear Schrödinger equations in Rn

     

Minimal pseudocompact group topologies on free abelian groups

A Hausdorff topological group G is minimal if every continuous isomorphism f:G→H between G and a Hausdorff topological group H is open. Significantly strengthening a 1981 result of Stoyanov, we prove the following theorem: For every infinite minimal abelian group G there exists a sequence of cardinals such that

     

Congruences involving Bernoulli and Euler numbers

Let [x] be the integral part of x. Let p>5 be a prime. In the paper we mainly determine , , and in terms of Euler and Bernoulli numbers. For example, we have

     

On Delannoy numbers and Schröder numbers

The nth Delannoy number and the nth Schröder number given by

     

On the Cauchy problem for a reaction-diffusion equation with a singular nonlinearity

We consider the following Cauchy problem with a singular nonlinearity

(P)

     

A class of nonlinear evolution equations subjected to nonlocal initial conditions

We prove the existence of C⁰-solutions for a class of nonlinear evolution equations subjected to nonlocal initial conditions, of the form:

     

Critical points between varieties generated by subspace lattices of vector spaces

We denote by the semilattice of all compact congruences of an algebra A. Given a variety V of algebras, we denote by the class of all semilattices isomorphic to for some A∈V. Given varieties V and W of algebras, the critical point of V under W is defined as . Given a finitely generated variety V of modular lattices, we obtain an integer ?, depending on V, such that for any n≥? and any field F.In a second part, using tools introduced in Gillibert (2009) [5], we prove that:

     

Menger curvature and rectifiability in metric spaces

We show that for any metric space X the condition

     

Inverse spectral problems for Bessel operators

We study the inverse spectral problem for a class of Bessel operators given in L₂(0,1) by the differential expression