Distribution of lattice points on surfaces of second order

Let f₁(x₁,..., xl₁) and f₂(y₁,...,yl₂) be positive definite primitive quadratic forms in l₁ and l₂ variables, respectively. We obtain new results in the well-known problem on the number of lattice points on the cone f₁(x₁,...,xl₁)=f₂(y₁,...,yl₂), in the domain f₁(x₁,...,xl₁)≦N for N»∞. Our technical tool is the Rankin-Selberg convolution. In several special cases the results can be sharpened by other methods. In addition, new facts concerning the uniform distribution of lattice points on ellipsoids in l variables, l odd, l≧5 are obtained.     

Orbital classification of geodesic flows on two-dimensional ellipsoids. The Jacobi problem is orbitally equivalent to the integrable Euler case in rigid body dynamics

Moscow State University, Department of Mathematics and Mechanics. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 29, No. 3, pp. 1–15, July–September, 1995.     

Distribution of the roots of a quadratic congruence

The polynomial, , is considered, where , r² + . Let (f/d) be the Kronecker symbol and It is proved that Some number theoretic consequences of this result are given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 183, pp. 155–164, 1990.     

Asymptotic distribution of integral points on the three-dimensional sphere

Let Q(X)=x ₁ ² +x ₂ ² +x ₃ ² , X=(x₁, x₂, x₃); r(n) let r(n) be the number of integral solutions of the equation

Siegel's formula for genus 2

Let S be a semi-integral, symmetric, positive-definite m x m matrix; mn1. By the Siegel fundamental formula we mean the identity between the Siegel theta series of genus n, associated with the genus of the matrix S, and the correspond ing Eisenstein-Siegel series (C. L. Siegel, Lectures on the Analytical Theory of Quadratic Forms, 3rd rev. edition, Peppmüller, Göttingen, 1963). The validity of the mentioned formula for m/2n+1 is an open problem in the general case. In this paper we prove Siegel's formula for n=2, m=6.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 76, pp. 210–215, 1978.     

Multivalued Homotopy on an Ordered Set,Fixed and Coincidence Points of Mappings,and Applications in Game Theory

Mathematical Notes - The article develops results of the authors’ previous papers on the topic. The notion of the homotopy of a multivalued mapping of an ordered set is introduced. We study...     

Study of the microstructure and distribution of elements in diffusion welded joints of tantalum with copper and VT14 titanic alloy

The microstructure of transition areas of welded joints of tantalum with M1 copper and VT14 alloy is investigated. The concentration curve distributions of chemical elements are obtained and the width of transition areas is computed. The coefficients of interdiffusion and the isothermic holding time for obtaining the required width of transition area are computed.