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О сходимости ветвящихся цепных дробей с целыми элементами.
С. 34–39.
In book
М.: Издательство ИПМ РАН, 2013.
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
Ustinov A., Квант 2010 № 2 С. 32–33
The article discusses applications of continued fractions. ...
Added: October 12, 2025
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
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
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
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
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
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
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
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
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
Ustinov A., Дальневосточный математический журнал 2011 Т. 11 № 1 С. 93–98
The article is devoted to investigation of Gauss — Kuz'min statistics for rational numbers a/b, where b is fixed, 1⩽a⩽b, (a,b)=1. New asymptotic formula for the mean value of Gauss — Kuz'min statistics is proved. It sharpens previous result which is similar to the Porter's theorem. ...
Added: October 9, 2025
Ustinov A., Математический сборник 2013 Т. 204 № 5 С. 143–160
New results related to number theoretic model of spin chains are proved. We solve Arnold's problem on the Gauss-Kuz'min statistics for quadratic irrationals. ...
Added: October 9, 2025
Ustinov A., Дальневосточный математический журнал 2014 Т. 14 № 1 С. 96–99
The article gives a one-parameter family of rational numbers whose expansions in reduced regular continued fractions (Hirzebruch fractions) have equal lengths. ...
Added: October 9, 2025
Smirnov E., М.: МЦНМО, 2022.
В брошюре рассказывается о числовых фризах, определенных Дж. Конвеем и Д. Кокстером в 1970-х гг. Это таблицы целых чисел, построенные по некоторому комбинаторному правилу и обладающие рядом глубоких и неожиданных свойств. В частности, они нумеруются триангуляциями многоугольников, возникают в разложениях чисел в цепные дроби и связаны с соотношениями в модулярной группе. Брошюра написана по материалам ...
Added: August 19, 2022
Smirnov E., Квант 2020 № 5 С. 15–24
В статье рассказывается о числовых фризах Конвея-Кокстера и обсуждаются их основные свойства. ...
Added: September 19, 2020