• A
  • A
  • A
  • АБВ
  • АБВ
  • АБВ
  • A
  • A
  • A
  • A
  • A
Обычная версия сайта
  • RU
  • EN
  • HSE University
  • Publications
  • Articles
  • Статистика траекторий частиц в однородной задаче Синая для двумерной решетки
  • RU
  • EN
Расширенный поиск
Высшая школа экономики
Национальный исследовательский университет
Priority areas
  • business informatics
  • economics
  • engineering science
  • humanitarian
  • IT and mathematics
  • law
  • management
  • mathematics
  • sociology
  • state and public administration
by year
  • 2028
  • 2027
  • 2026
  • 2025
  • 2024
  • 2023
  • 2022
  • 2021
  • 2020
  • 2019
  • 2018
  • 2017
  • 2016
  • 2015
  • 2014
  • 2013
  • 2012
  • 2011
  • 2010
  • 2009
  • 2008
  • 2007
  • 2006
  • 2005
  • 2004
  • 2003
  • 2002
  • 2001
  • 2000
  • 1999
  • 1998
  • 1997
  • 1996
  • 1995
  • 1994
  • 1993
  • 1992
  • 1991
  • 1990
  • 1989
  • 1988
  • 1987
  • 1986
  • 1985
  • 1984
  • 1983
  • 1982
  • 1981
  • 1980
  • 1979
  • 1978
  • 1977
  • 1976
  • 1975
  • 1974
  • 1973
  • 1972
  • 1971
  • 1970
  • 1969
  • 1968
  • 1967
  • 1966
  • 1965
  • 1964
  • 1963
  • 1958
  • More
Subject
News
September 18, 2026
When Pictures Hinder Understanding: Illustrations May Impede Learning of Abstract Ideas
Illustrations can help remember specific actions but do not always make abstract ideas easier to learn. Researchers from HSE University and Humboldt University compared how people learn from texts with different levels of abstractness. They found that participants remembered illustrations better and performed better on related tasks after reading a multimedia text about yoga asanas than after reading an abstract text about the Nash equilibrium. The findings could help improve the selection of illustrations for educational and informational materials. The study has been published in Learning and Instruction.
September 17, 2026
'I Wish That People Would Place Greater Trust in Science'
When Tatiana Eremicheva chose Fundamental and Computational Linguistics as her field of study, she thought it would be about learning languages. Instead, she discovered it was about helping people. In this interview for the HSE Young Scientists project, she discusses science as a way of understanding the world, billiards as a team-building activity, and why learning to read is not always as easy as it seems.
September 15, 2026
Immunity to Chaos: How Personal Resources Help Us Cope with the Challenges of a Turbulent World
International conflicts, crises and digital overload—the modern world puts our minds to the test every day. Traditional psychology often focuses on the consequences: anxiety, depression, and psychosomatic disorders. But what if we looked at the problem differently—through the lens of the resources that prevent us from breaking down? Psychological immunity is precisely this set of resources. Alena Zolotareva and her group, Psychological Immunity as a Resource for Positive Functioning, are developing an integrative model of this phenomenon, adapting diagnostic tools and preparing for large-scale empirical research. Why do psychologists need to collaborate with medical professionals, and how could their research transform preventive care in clinics and corporations?

 

Have you spotted a typo?
Highlight it, click Ctrl+Enter and send us a message. Thank you for your help!

Publications
  • Books
  • Articles
  • Chapters of books
  • Working papers
  • Report a publication
  • Research at HSE

?

Статистика траекторий частиц в однородной задаче Синая для двумерной решетки

Функциональный анализ и его приложения. 2008. Т. 42. № 3. С. 10–22.
Быковский В. А., Ustinov A.

 In this paper, we generalize and refine some results by F. P. Boca, R. N. Gologan, and A. Zaharescu on the asymptotic behavior as h→0 of the statistics of the free path length until the first hit of the h-neighborhood (a disk of radius h) of a nonzero integer for a particle issuing from the origin. The established facts imply that the limit distribution function for the free path length and for the sighting parameter (the distance from the trajectory to the integer point in question) does not depend on the particle escape direction (the property of isotropy).

Language: Russian
DOI
Text on another site
Keywords: цепная дробьInteger latticeцелочисленная решеткаKloosterman sumsсуммы Клостермана continued fractions
Similar publications
Hausdorff dimension estimates for Sudler products with positive lower bound
Dmitry Gayfulin, Hauke M., Nonlinearity 2025 Vol. 38 No. 6 Article 065008
Given an irrational number $\alpha$, we study the asymptotic behaviour of the Sudler product denoted by $P_N(\alpha) =\prod_{r=1}^N  2\lvert \sin \pi r \alpha \rvert$. We show that $\liminf_{N \to \infty} P_N(\alpha) >0$ and $\limsup_{N \to \infty} P_N(\alpha)/N < \infty$ whenever the sequence of partial quotients in the continued fraction expansion of $\alpha$ exceeds 3 only finitely ...
Added: March 19, 2026
О равномерных диофантовых экспонентах решеток
German O., Математический сборник 2026 Т. 217 № 2 С. 20–33
The spectrum of values of the weak uniform Diophantine exponents of lattices is studied, and its complete description in the two-dimensional case is given. ...
Added: February 20, 2026
Приложения сумм Клостермана в арифметике и геометрии
Ustinov A., LAP LAMBERT Academic Publishing, 2011.
The book is devoted to applications of Kloosterman sum estimates in various problems of number theory. ...
Added: October 13, 2025
Цепные дроби вокруг нас
Ustinov A., Квант 2010 № 2 С. 32–33
The article discusses applications of continued fractions. ...
Added: October 12, 2025
О статистических свойствах конечных цепных дробей
Ustinov A., Записки научных семинаров ПОМИ РАН 2005 Т. 322 С. 186–211
The article is devoted to the statistical properties of continued fractions for the numbers a/b, for a and b in the sector a,b⩾1, a^2+b^2⩽R^2. ...
Added: October 11, 2025
О статистиках Гаусса–Кузьмина для конечных цепных дробей
Ustinov A., Фундаментальная и прикладная математика 2005 Т. 11 № 6 С. 195–208
The article is devoted to finite continued fractions for numbers a/b when integer points (a,b) are taken from a dilative region. Properties similar to the Gauss–Kuz'min statistics are proved for these continued fractions. ...
Added: October 11, 2025
Короткое доказательство тождества Эйлера для континуантов
Ustinov A., Математические заметки 2006 Т. 79 № 1 С. 155–156
The article presents a short proof of Euler's identity for continuants ...
Added: October 11, 2025
Вычисление дисперсии в одной задаче из теории цепных дробей
Ustinov A., Математический сборник 2007 Т. 198 № 6 С. 139–158
This paper examines a random variable equal to the number of denominators of convergents not exceeding a given bound. Asymptotic formulas with two significant terms are proved for the mathematical expectation of this variable and its variance. ...
Added: October 11, 2025
Асимптотическое поведение первого и второго моментов для числа шагов в алгоритме Евклида
Ustinov A., Известия РАН. Серия математическая 2008 Т. 72 № 5 С. 189–224
We prove asymptotic formulae with two significant terms for the expectation and variance of the random variable s(c/d) when the variables c and d range over the set 1≤c≤d≤R and R→∞, where s(c,d)=s(c/d) is the number of steps in the Euclidean algorithm applied to the numbers c and d. ...
Added: October 11, 2025
О числе решений сравнения xy≡l (mod q) под графиком дважды непрерывно дифференцируемой функции
Ustinov A., Алгебра и анализ 2008 Т. 20 № 5 С. 186–216
A result by V. A.Bykovskiĭ (1981) on the number of solutions of the congruence xy≡l (modq) 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 Euclid algorithm is sharpened and extended to the case of Gauss–Kuzmin statistics. ...
Added: October 11, 2025
Решение задачи Арнольда о слабой асимптотике для чисел Фробениуса с тремя аргументами
Ustinov A., Математический сборник 2009 Т. 200 № 4 С. 131–160
It is shown that on the average the Frobenius numbers f(a,b,c) behave like 8/π√abc . ...
Added: October 11, 2025
Статистика траекторий частиц в неоднородной задаче Синая для двумерной решетки
Ustinov A., Быковский В. А., Известия РАН. Серия математическая 2009 Т. 73 № 4 С. 17–36
In connection with the two-dimensional model known as the ‘periodic Lorentz gas’, we study the asymptotic behaviour of statistical characteristics of a free path interval of a point particle before its first occurrence in an h-neighbourhood (a circle of radius h) of a non-zero integer point as h→0 given that the particle starts from the h-neighbourhood of the origin. We evaluate the limit distribution ...
Added: October 10, 2025
О распределении точек целочисленной решетки
Ustinov A., Дальневосточный математический журнал 2009 Т. 9 № 1-2 С. 176–181
We study distribution of distances from primitive integer points to the origin. ...
Added: October 10, 2025
О статистических свойствах элементов цепных дробей
Ustinov A., Доклады Академии наук 2009 Т. 424 № 4 С. 459–461
The article solves a problem related to the statistical properties of continued fractions that arose during the study of Frobenius numbers with three arguments. ...
Added: October 10, 2025
О распределении чисел Фробениуса с тремя аргументами
Ustinov A., Известия РАН. Серия математическая 2010 Т. 74 № 5 С. 145–170
We prove the existence of the limit density distribution for normalized Frobenius numbers with three arguments. The density is found explicitly. ...
Added: October 9, 2025
  • About
  • About
  • Key Figures & Facts
  • Sustainability at HSE University
  • Faculties & Departments
  • International Partnerships
  • Faculty & Staff
  • HSE Buildings
  • HSE University for Persons with Disabilities
  • Public Enquiries
  • Studies
  • Admissions
  • Programme Catalogue
  • Undergraduate
  • Graduate
  • Exchange Programmes
  • Summer University
  • Summer Schools
  • Semester in Moscow
  • Business Internship
  • Research
  • International Laboratories
  • Research Centres
  • Research Projects
  • Monitoring Studies
  • Conferences & Seminars
  • Academic Jobs
  • Yasin (April) International Academic Conference on Economic and Social Development
  • Media & Resources
  • Publications by staff
  • HSE Journals
  • Publishing House
  • iq.hse.ru: commentary by HSE experts
  • Library
  • Economic & Social Data Archive
  • Video
  • HSE Repository of Socio-Economic Information
  • HSE1993–2026
  • Contacts
  • Copyright
  • Privacy Policy
  • Site Map
Edit