Sums of products of Apostol–Bernoulli numbers

By expressing the sums of products of the Apostol?CBernoulli polynomials in terms of the special values of multiple Hurwitz?CLerch zeta functions at non-positive integers, we obtain the sums of products identity for the Apostol?CBernoulli numbers which is an analogue of the classical sums of products identity for Bernoulli numbers dating back to Euler.     

Two identities for the Bernoulli–Euler numbers

Two identities for the Bernoulli and for the Euler numbers are derived. These identities involve two special cases of central combinatorial numbers. The approach is based on a set of differential identities for the powers of the secant. Generalizations of the Mittag–Leffler series for the secant are introduced and used to obtain closed-form expressions for the coefficients.     

Optimal sequencing of a set of positive numbers with the variance of the sequence’s partial sums maximized

We consider the problem of sequencing a set of positive numbers. We try to find the optimal sequence to maximize the variance of its partial sums. The optimal sequence is shown to have a nice structure. It is interesting to note that the symmetric problem which aims at minimizing the variance of the same partial sums is proved to be NP-complete in the literature.     

Summation formulae of products of the Apostol–Bernoulli and Apostol–Euler polynomials

In this paper, a further investigation for the Apostol–Bernoulli and Apostol–Euler polynomials is performed, and some summation formulae of products of the Apostol–Bernoulli and Apostol–Euler polynomials are established by applying some summation transform techniques. Some illustrative special cases as well as immediate consequences of the main results are also considered.     

Weierstrass’s construction of the irrational numbers

We present an overview of the development of the irrational numbers due to Karl Weierstrass. This construction was first presented during lectures in the 1860s in Berlin. Weierstrass never published his construction. Several of his students (Kossak, Horwitz, von Dantscher and Pincherle, to name a few) gave accounts in lecture notes from the courses. However, these notes were not published under the direction of Weierstrass.     

Equivalence transformations of the Euler–Bernoulli equation

We give a determination of the equivalence group of the Euler–Bernoulli equation and of one of its generalizations, and thus derive some symmetry properties of this equation.     

Solvability of the clamped Euler–Bernoulli beam equation

In this study, solvability of the initial boundary value problem for general form Euler–Bernoulli beam equation which includes also moving point-loads is investigated. The complete proof of an existence and uniqueness properties of the weak solution of the considered equation with Dirichlet type boundary conditions is derived. The method used here is based on Galerkin approximation which is the main tool for the weak solution theory of linear evolution equations as well as in derivation of a priori estimate for the approximate solutions. All steps of the proposed technique are explained in detail.     

Application of Postnikov’s formula for estimation of short sums of Dirichlet characters over shifted prime numbers

Estimates of short sums of Dirichlet characters over shifted prime numbers are obtained in the case when the modulus of a character is a power of a fixed prime number.     

Dispersive deformations of the Hamiltonian structure of Euler’s equations

Euler’s equations for a two-dimensional fluid can be written in the Hamiltonian form, where the Poisson bracket is the Lie–Poisson bracket associated with the Lie algebra of divergence-free vector fields. For the two-dimensional hydrodynamics of ideal fluids, we propose a derivation of the Poisson brackets using a reduction from the bracket associated with the full algebra of vector fields. Taking the results of some recent studies of the deformations of Lie–Poisson brackets of vector fields into account, we investigate the dispersive deformations of the Poisson brackets of Euler’s equation: we show that they are trivial up to the second order.     

Value distributions of analogues of Kloosterman’s sums

A theorem describing the value distribution of analogues of Kloosterman’s sums is proved. Asymptotic formulas for fractional moments are obtained.     

Arithmetics of the numbers of orbits of the Fermat–Euler dynamical systems

Given positive integers a and n with (a,n)=1, we consider the Fermat–Euler dynamical system defined by the multiplication by a acting on the set of residues modulo n relatively prime to n. Given an integer M>1, the integers n for which the number of orbits of this dynamical system is a multiple of M form an ideal in the multiplicative semigroup of odd integers. We provide new results on the arithmetical properties of these ideals by using the topological properties of some directed graphs (the monads).      

Minkowski’s inequality and sums of squares

Positive polynomials arising from Muirhead’s inequality, from classical power mean and elementary symmetric mean inequalities and from Minkowski’s inequality can be rewritten as sums of squares.     

Rebuttal of Kowalenko’s Paper As Concerns the Irrationality of Euler’s Constant γ

We rebut Kowalenko??s claims in this journal that he proved the irrationality of Euler??s constant ??, and that his rational series for ?? is new.     

Estimates of Weyl’s exponential sums for very small values of the variable

We consider Weyls exponential sums for very small values of the variable. In this case we can give an asymptotic transformation formula. Weyls exponential sums will be usually estimated by means of Weyls or Vinogradovs method. Here we use van der Corputs method and obtain sufficiently good results in the present case.Received: 14 January 2002     

A generalization of Euler’s constant

The purpose of this paper is to evaluate the limit γ(a) of the sequence , where a ∈ (0, + ∞ ).      

Triple products of Coleman’s families

We discuss modular forms as objects of computer algebra and as elements of certain p-adic Banach modules. We discuss a problem-solving approach in number theory, which is based on the use of generating functions and their connection with modular forms. In particular, the critical values of various L-functions of modular forms produce nontrivial but computable solutions of arithmetical problems. Namely, for a prime number we consider three classical cusp eigenforms

Summation formulas for the products of the Frobenius–Euler polynomials

We present here a further investigation for the classical Frobenius–Euler polynomials. By making use of the generating function methods and summation transform techniques, we establish some summation formulas for the products of an arbitrary number of the classical Frobenius–Euler polynomials. The results presented here are generalizations of the corresponding known formulas for the classical Bernoulli polynomials and the classical Euler polynomials.     

Generalizations of Arnold’s version of Euler’s theorem for matrices

A recent result, conjectured by Arnold and proved by Zarelua, states that for a prime number p, a positive integer k, and a square matrix A with integral entries one has ${\textrm tr}(A^{p^k}) \equiv {\textrm tr}(A^{p^{k-1}}) ({\textrm mod}{p^k})${\textrm tr}(A^{p^k}) \equiv {\textrm tr}(A^{p^{k-1}}) ({\textrm mod}{p^k}). We give a short proof of a more general result, which states that if the characteristic polynomials of two integral matrices A,  B are congruent modulo p ^k then the characteristic polynomials of A ^p and B ^p are congruent modulo p ^k+1, and then we show that Arnold’s conjecture follows from it easily. Using this result, we prove the following generalization of Euler’s theorem for any 2 × 2 integral matrix A: the characteristic polynomials of A ^Φ(n) and A ^Φ(n)-ϕ(n) are congruent modulo n. Here ϕ is the Euler function, ?_i=1^l p_i^a_i\prod_{i=1}^{l} p_i^{\alpha_i} is a prime factorization of n and $\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2$\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2.