Openness of the Galois image for τ-modules of dimension 1

Let C be a smooth projective absolutely irreducible curve over a finite field , F its function field and A the subring of F of functions which are regular outside a fixed point ∞ of C. For every place ? of A, we denote the completion of A at ? by .In [Pi2], Pink proved the Mumford-Tate conjecture for Drinfeld modules. Let φ be a Drinfeld module of rank r defined over a finitely generated field K containing F. For every place ? of A, we denote by Γ_? the image of the representation

     

Moments of combinatorial and Catalan numbers

In this paper we obtain the moments {Φm}_m?0 defined by

     

Asymptotics of zeros of polynomials arising from rational integrals

We prove that the zeros of the polynomials Pm(a) of degree m, defined by Boros and Moll via

     

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

     

Positive solutions of m-point boundary value problems

The second-order m-point boundary value problem

     

Irregular scaling functions with orthogonal translations

For all integers m,k>1 with m≡1modk we construct, among others, a function with dense graph in the set [0,k]×R such that

     

Simultaneous Diophantine approximation on manifolds and Hausdorff dimension

Let M be an m-dimensional, C^k manifold in , for any , and for any τ>0 let

     

Global asymptotic stability in a class of generalized Putnam equations

It was conjectured that for every integer m?3 the unique equilibrium c=1 of the generalized Putnam equation

     

Endpoint estimates for some maximal operators associated to the circular conical surface

For and m>0 we consider the maximal operator given by

     

Three positive solutions for higher order m-point boundary value problems

In this paper, we consider the existence of at least three positive solutions for the 2nth order m-point boundary value problem

     

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.

     

Multiple positive solutions of singular second-order m-point boundary value problems

The existence of multiple positive solutions about the singular second-order m-point boundary value problem

     

Mean value of some exponential sums and applications to Kloosterman sums

Let q, m, n, k be integers with q?3 and k?1, define the exponential sum

     

Parabolic approach to nonlinear elliptic eigenvalue problems

We consider the asymptotic profiles of the nonlinear parabolic flows ut=Δum to show the geometric properties of the following elliptic nonlinear eigenvalue problems:

     

On the existence of m-point boundary value problem at resonance for higher order differential equation

By means of Mawhin's continuation theorem, we study m-point boundary value problem at resonance in the following form:

     

A combinatorial identity with application to Catalan numbers

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

     

A note on the edge cover number and independence number in hypergraphs

Consider a hypergraph of rank r>2 with m edges, independence number α and edge cover number ρ. We prove the inequality

     

On a conjecture of Alon

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Let f(n,m) be the cardinality of largest subset of {1,2,…,n} which does not contain a subset whose elements sum to m. In this note, we show that