Alexey Ustinov - Academia.edu (original) (raw)

Papers by Alexey Ustinov

Izvestiya: Mathematics, 2010

Sbornik: Mathematics, 2013

$Research paper thumbnail of Geometry and combinatoric of Minkowski–Voronoi 3-dimensional continued fractions$

Math Notes Engl Tr, 1998

In the paper, an estimate of the number of summands in the asymptotic formula for the number of s... more In the paper, an estimate of the number of summands in the asymptotic formula for the number of solutions to Waring's equation is obtained. This is achieved by means of a recurrent process leading to a greater reduction than that in Vinogradov's mean value theorem.

Given a triangle ABC with sides a, b, c. We want to construct a triangle A ′ B ′ C ′ such that se... more Given a triangle ABC with sides a, b, c. We want to construct a triangle A ′ B ′ C ′ such that segments AA ′ , BB ′ and CC ′ are its angle bisectors. This problem was discussed in [3, 2] (see problem 138) and remained open. Article [4] gives conic solution of this problem. Using formulae from [4] it is easy to prove that in the general case construction of a triangle from the feet of its angle bisectors with compass and ruler is impossible. Following this article we denote by (x : y : z) barycentric coordinates of incenter of triangle A ′ B ′ C ′ with respect to triangle ABC. Then vertices of triangle A ′ B ′ C ′ will have coordinates (−x, y, z), (x, −y, z), (x, y, −z). As it was proved in [4], numbers x, y, z satisfy following equations:

Известия Российской академии наук. Серия математическая, 2008

Асимптотическое поведение первого и второго моментов для числа шагов в алгоритме Евклида Доказаны... more Асимптотическое поведение первого и второго моментов для числа шагов в алгоритме Евклида Доказаны асимптотические формулы с двумя значащими членами для математического ожидания и дисперсии случайной величины s(c/d), когда переменные c и d меняются в пределах 1 c d R и R → ∞, где s(c, d) = s(c/d)-число шагов в алгоритме Евклида, примененном к числам c и d. Библиография: 20 наименований. Работа выполнена при финансовой поддержке INTAS (грант № 03-51-5070), РФФИ (грант № 07-01-00306), проекта ДВО РАН 06-III-А-01-017 и Фонда содействия отечественной науке. c

In this paper we investigate the combinatorial structure of Minkowski-Voronoi continued fractions... more In this paper we investigate the combinatorial structure of Minkowski-Voronoi continued fractions. Our main goal is to prove the asymptotic stability of Minkowski-Voronoi complexes in special two-parametric families of rank-1 lattices. In addition we construct explicitly the complexes for the case of White's rank-1 lattices and provide with a hypothetic description in a more complicated settings.

Известия Российской академии наук. Серия математическая, 2010

О распределении чисел Фробениуса с тремя аргументами Доказано существование предельной плотности ... more О распределении чисел Фробениуса с тремя аргументами Доказано существование предельной плотности распределения нормированных чисел Фробениуса от трех аргументов. Плотность найдена явно. Библиография: 22 наименования.

ABSTRACT The first part of this paper is concerned with the proof of a discrete analog of the Poi... more ABSTRACT The first part of this paper is concerned with the proof of a discrete analog of the Poisson summation formula. In the second part, we describe an elementary proof of a functional equation for the function , based on the summation formula derived in the paper.

In this paper, we prove a discrete analog of Euler's summation formula. The difference from the c... more In this paper, we prove a discrete analog of Euler's summation formula. The difference from the classical Euler formula is in that the derivatives are replaced by finite differences and the integrals by finite sums. Instead of Bernoulli numbers and Bernoulli polynomials, special numbers P n and special polynomials P n (x) introduced by Korobov in 1996 appear in the formula.

Математические заметки, 2015

$Research paper thumbnail of On the number of solutions of the congruence <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mi>y</mi><mo>≡</mo><mi>l</mi><mspace></mspace><mspace width="0.4444em"/><mo stretchy="false">(</mo><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi></mrow><mspace width="0.3333em"/><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">xy\equiv l\pmod {q}</annotation></semantics></math>xy≡l(modq) under the graph of a twice continuously differentiable function$

St. Petersburg Mathematical Journal, 2009

A result by V. A. Bykovskiȋ (1981) on the number of solutions of the congruence xy ≡ l (mod q) un... more A result by V. A. Bykovskiȋ (1981) on the number of solutions of the congruence xy ≡ l (mod q) under the graph of a twice continuously differentiable function is refined. As an application, Porter's result (1975) on the mean number of steps in the Euclidean algorithm is sharpened and extended to the case of Gauss-Kuzmin statistics.

Sbornik: Mathematics, 2009

It is shown that on the average the Frobenius numbers f (a, b, c) behave like 8 π √ abc. Bibliogr... more It is shown that on the average the Frobenius numbers f (a, b, c) behave like 8 π √ abc. Bibliography: 28 titles.

$Research paper thumbnail of Calculation of the variance in a problem in the theory of continued fractions$

Sbornik: Mathematics, 2007

We study the random variable N (α, R) = #{j 1 : Qj(α) R}, where α ∈ [0; 1) and Pj(α)/Qj(α) is the... more We study the random variable N (α, R) = #{j 1 : Qj(α) R}, where α ∈ [0; 1) and Pj(α)/Qj(α) is the jth convergent of the continued fraction expansion of the number α = [0; t1, t2,. .. ]. For the mean value N (R) = Z 1 0 N (α, R) dα and variance D(R) = Z 1 0`N (α, R) − N (R)´2 dα of the random variable N (α, R), we prove the asymptotic formulae with two significant terms N (R) = N1 log R+N0 +O(R −1+ε), D(R) = D1 log R+D0 +O(R −1/3+ε). Bibliography: 13 titles.

Математические заметки, 2009

Математические заметки, 2010

Длина разложения рационального числа в цепную дробь с нечетными неполными частными выражается чер... more Длина разложения рационального числа в цепную дробь с нечетными неполными частными выражается через статистики Гаусса-Кузьмина для классической цепной дроби. Благодаря этому для средней длины алгоритма Евклида с нечетными неполными частными удается доказать асимптотические формулы аналогичные тем, которые ранее были известны для классического алгоритма Евклида. Библиография: 12 названий.

Mathematical Notes, 2010

The length of the continued-fraction expansion of a rational number with odd partial quotients is... more The length of the continued-fraction expansion of a rational number with odd partial quotients is expressed via the Gauss-Kuz’min statistics for the classical continued fraction. This has made it possible to prove asymptotic formulas, similar to those already known for the classical Euclidean algorithm, for the mean length of the Euclidean algorithm with odd partial quotients.