On the power values of power sums

Using several currently available techniques, including Baker's method, Frey curves and modular forms, we prove that for odd values of k with 1?k<170, the equation

k¹+k²+?+xk=y2n     

Trigonometric sums over one-dimensional quasilattices of arbitrary codimension

A new class of one-dimensional quasilattices parametrized by the translations of the torus is introduced. For this class, parameter-dependent trigonometric sums over points of quasilattice are considered. Nontrivial estimates of the trigonometric sums under consideration are obtained. For a number of trigonometric sums of special form, asymptotic formulas are derived. It is proved that the distribution of points of quasilattices is uniform modulo h for almost all h. Earlier similar results were obtained in the particular case of quasilattices parametrized by the rotations of the circle.     

On multidimensional expansions of distributions of sums by the saddle point method

Many problems of applied statistics, in particular, hypotheses testing, contain no information about true distributions of the statistics under consideration. Therefore, limit distributions of those statistics are used, which may give inadequate results in actual practice. Problems on the refinement of the distributions thus arise. In the present paper, we extend the results of H. Daniels, R. Lugannani, and S. Rice to the multidimensional case. We prefer to use the Lugannani-Rice method because it yields approximations which are as good as those of other forms and do not depend on variation of statistics. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 195–202, Perm, 1993.     

Some identities on the Bernoulli, Euler and Genocchi polynomials via power sums and alternate power sums

In this paper, by the generating function method, we establish various identities concerning the (higher order) Bernoulli polynomials, the (higher order) Euler polynomials, the Genocchi polynomials and the degenerate higher order Bernoulli polynomials. Particularly, some of these identities are also related to the power sums and alternate power sums. It can be found that, many well known results, especially the multiplication theorems, and some symmetric identities demonstrated recently, are special cases of our results.     

Deforming the point spectra of one-dimensional Dirac operators

We provide a method of inserting and removing any finite number of prescribed eigenvalues into spectral gaps of a given one-dimensional Dirac operator. This is done in such a way that the original and deformed operators are unitarily equivalent when restricted to the complement of the subspace spanned by the newly inserted eigenvalue. Moreover, the unitary transformation operator which links the original operator to its deformed version is explicitly determined.

     

Seventh power moments of Kloosterman sums

Evaluations of the n-th power moments S _n of Kloosterman sums are known only for n ⩽ 6. We present here substantial evidence for an evaluation of S ₇ in terms of Hecke eigenvalues for a weight 3 newform on Γ_O(525) with quartic nebentypus of conductor 105. We also prove some congruences modulo 3, 5 and 7 for the closely related quantity T ₇, where T _n is a sum of traces of n-th symmetric powers of the Kloosterman sheaf.     

On inequalities for products of power sums

In the present note we investigate several inequalities concerning the product of power sums . The only restriction which is used is ₁ + ... + _k = 0. The conditions obtained are necessary and sufficient everywhere.     

Enumeration of power sums modulo a prime

We consider, for odd primes p, the function N(p, m, α) which equals the number of subsets S?{1,…,p ? 1} with the property that Σ_∞∈Sx^m ≡ α (mod p). We obtain a closed form expression for N(p, m, α). We give simple explicit formulas for N(p, 2, α) (which in some cases involve class numbers and fundamental units), and show that for a fixed m, the difference between N(p, m, α) and its average value p^?12^p?1 is of the order of $\text{exp}\text{(p}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ or less. Finally, we obtain the curious result that if p ? 1 does not divide m, then N(p, m, 0) > N(p, m, α) for all α ? 0 (mod p).     

Test of symmetry of a one-dimensional distribution against positive biasedness

Summary Definitions of different strengths are given to the notion of ‘a positively biased random variable’. This notion is related to that of ‘a stochastically larger component of a two-dimensional random vector’, which was introduced previously by the authors. Properties of common rank tests of symmetry about zero against our specification of alternatives are studied in detail. The positive biasedness is extended to ‘positively more biased’. Test of symmetry of a two-dimensional random vector is also referred to.     

Group analysis of the one-dimensional Boltzmann equation: I. symmetry groups

We consider the one-dimensional Boltzmann equation f _t + cf _x + Ff _c = 0, where the functions f and F are assumed to depend on three variables t, x, and c. We obtain relations defining the symmetry algebra in the general case and also under the additional conditions of conservation of the relations dx = c dt and dc = F dt, which arise from physical considerations. We show that the widest symmetry algebra is obtained in the case of conservation of both relations. This algebra is infinite-dimensional, and its structure is independent of the form of the function F.     

A generalization of power moments of Kloosterman sums

We find an expression for a sum which can be viewed as a generalization of power moments of Kloosterman sums studied by Kloosterman and Salié. Received: 24 March 2006     

On the fourth power mean of the generalized quadratic Gauss sums

The main purpose of this paper is to use elementary methods and properties of the classical Gauss sums to study the computational problem of one kind of fourth power mean of the generalized quadratic Gauss sums mod q (a positive odd number), and give an exact computational formula for it.     

Root and critical point behaviors of certain sums of polynomials

It is known that no two roots of the polynomial equation

$$\begin{aligned} \begin{aligned} \prod _{j=1}^n (x-r_j) + \prod _{j=1}^n (x+r_j) =0, \end{aligned} \end{aligned}$$

(1)

where \(0 < r_1 \le r_2 \le \cdots \le r_n\), can be equal and the gaps between the roots of (1) in the upper half-plane strictly increase as one proceeds upward, and for \(0< h< r_k\), the roots of

$$\begin{aligned} (x-r_k-h)\prod _{\begin{array}{c} j=1\\ j\ne k \end{array}}^n(x-r_j) + (x+r_k+h)\prod _{\begin{array}{c} j=1\\ j\ne k \end{array} }^n (x+r_j) = 0 \end{aligned}$$

(2)

and the roots of (1) in the upper half-plane lie alternatively on the imaginary axis. In this paper, we study how the roots and the critical points of (1) and (2) are located.