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A Lower Bound for Linear Forms in Values of a continued fraction

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Acta Mathematica Sinica,English Series 2011年第10期27卷 2033-2038页

作者： Xiu Yuan YU Zhong Hua SHEN Department of Mathematics Hangzhou Normal University Hangzhou 310036 P. R. China Department of Computer Science Taizhou Vocational and Technical College Taizhou 318000 P. R. China

Let y = y（x） be a function defined by a continued fraction. A lower bound for │A│ =│β1y1 ＋β2y2 ＋α│ is given, where y1 = y（x1）, y2 = y（x2）, x1 and x2 are positive integers, α,β and β2 are algebraic ir... 详细信息

关键词： continued fraction linear form lower bound

Gravity Field Imaging by continued fraction Downward Continuation: A Case Study of the Nechako Basin(Canada)

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Acta Geologica Sinica(English Edition) 2021年第S01期95卷 102-105页

作者： ZHANG Chong ZHOU Wenna LV Qingtian YAN Jiayong State Key Laboratory of Lunar and Planetary Sciences Macao University of Science and TechnologyMacao 999078China China Deep Exploration Center-SinoProbe Center Chinese Academy of Geological SciencesBeijing 100037China School of Earth Sciences Key Laboratory of Mineral Resources in Western ChinaLanzhou UniversityLanzhou 730000China

Interpretation of gravity data plays an important role in the study of geologic structure and resource exploration in the deep part of the earth,like the lower crust,the upper mantle(Lüet al.,2013,2019).The gravity a... 详细信息

关键词： depth estimation downward continuation gravity data continued fraction

The distribution of the large partial quotients in continued fraction expansions

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Science China Mathematics 2023年第5期66卷 935-956页

作者： Bo Tan Chen Tian Baowei Wang School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074China

The existence of large partial quotients destroys many limit theorems in the metric theory of continued *** achieve some variant forms of limit theorems,a common approach mostly used in practice is to discard the largest partial quotient,while this approach works in obtaining limit theorems only when there cannot exist two terms of large partial quotients in a metric *** by this,we are led to consider the metric theory of points with at least two large partial *** precisely,denoting by[a1(x),a2(x),...]the continued fraction expansion of x∈[0,1)and lettingψ:N→R+be a positive function tending to in nity as n→∞,we present a complete characterization on the metric properties of the set,i.e.,E(ψ)={x∈[0,1):∃16 k̸=ℓ6 n,ak(x)>ψ(n),aℓ(x)>ψ(n)for in nitely many n∈N}in the sense of the Lebesgue measure(the Borel-Bernstein type result)and the Hausdor dimension(the Jarnik type result).The main result implies that any nite deletion from a1(x)+……+an(x)cannot result in a law of large numbers.

关键词： continued fraction Hausdor dimension Borel-Bernstein theorem

Quantitative Poincar recurrence in continued fraction dynamical system

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Science China Mathematics 2012年第1期55卷 131-140页

作者： PENG Li TAN Bo WANG BaoWei School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074China

Let T ： X → X be a transformation. For any x C [0, 1） and r 〉 O, the recurrence time Tr（x） of x under T in its r-neighborhood is defined as Tr（X） = inf{k ≥ 1： d（Tk（x）,x）〈 r}.For 0 ≤ α ≤ β ∞ co, le... 详细信息

Let T ： X → X be a transformation. For any x C [0, 1） and r 〉 O, the recurrence time Tr（x） of x under T in its r-neighborhood is defined as Tr（X） = inf{k ≥ 1： d（Tk（x）,x）〈 r}.For 0 ≤ α ≤ β ∞ co, let E（α,β） be the set of points with prescribed recurrence time as follows E（α,β）={x∈X：lim inf r→0 logTr（x）/-logr=α,lim sup r→0 logTr（x）/-logr=β}.In this note, we consider the Gauss transformation T on [0, 1）, and determine the size of E（α,β）by showing that dimH E（α,β） = 1 no matter what a and/~ are. This can be compared with Feng and Wu＇s result [Nonlinearity, 14 （2001）, 81-85] on the symbolic space.

关键词： recurrence continued fraction Hausdorff dimension

On continued fractions and Their Applications

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Journal of Applied Mathematics and Physics 2022年第1期10卷 142-159页

作者： Zakiya M. Ibran Efaf A. Aljatlawi Ali M. Awin Department of Mathematics Faculty of Science University of Tripoli Tripoli Libya

continued fractions constitute a very important subject in mathematics. Their importance lies in the fact that they have very interesting and beautiful applications in many fields in pure and applied sciences. This review article will reveal some of these applications and will reflect the beauty behind their uses in calculating roots of real numbers, getting solutions of algebraic Equations of the second degree, and their uses in solving special ordinary differential Equations such as Legendre, Hermite, and Laguerre Equations;moreover and most important, their use in physics in solving Schrodinger Equation for a certain potential. A comparison will also be given between the results obtained via continued fractions and those obtained through the use of well-known numerical methods. Advances in the subject will be discussed at the end of this review article.

关键词： continued fraction Equation Numerical Method Roots Series Finite

OBTAINING EXACT INTERPOLATION MULTIVARIATE POLYNOMIAL BY APPROXIMATION

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Journal of Systems Science & Complexity 2011年第4期24卷 803-815页

作者： Yong FENG Xiaolin QIN Jingzhong ZHANG Xun YUAN Laboratory of Computer Reasoning and Trustworthy Computation University of Electronic b'cience and 'l~ech-nology of China Chengdu 611731 China. Laboratory for Automated Reasoning and Programming Chengdu Institute of Computer Applications ChineseAcademy of Sciences Chengdu 610041 China Graduate University of Chinese Academy of Sciences Beijing100049 China. Laborutory of Computer Reasoning and 7Yustworthy Computation University of Electronic Science and Tech-nology of China Chengdu 611731 China. Laboratory for Automated Reasoning and Programming Chengdu Institute of Computer Applications ChineseAcademy of Sciences Chengdu 610041 China.

In some fields such as Mathematics Mechanization, automated reasoning and Trustworthy Computing, etc., exact results are needed. Symbolic computations are used to obtain the exact results. Symbolic computations are of high complexity. In order to improve the situation, exact interpolating methods are often proposed for the exact results and approximate interpolating methods for the ap- proximate ones. In this paper, the authors study how to obtain exact interpolation polynomial with rational coefficients by approximate interpolating methods.

关键词： continued fraction multivariate interpolation numerical approximate computation symbolic-numerical computation Vandermonde determinant.

Algebraic Independence of Certain Values of Exponential Function

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Acta Mathematica Sinica,English Series 2006年第2期22卷 571-576页

作者： Yao Chen ZHU Academy of Mathematics and Systems Science Chinese Academy of Sciences

The algebraic independence of e^θ1,…,e^θs is proved, where θ1,… ,θs are certain gap series or power series of algebraic numbers, or certain transcendental continued fractions with algebraic elements.

关键词： Algebraic independence Exponential function Gap series Power series continued fraction Algebraic number

Algebraic Independence of Certain Numbers

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Acta Mathematica Sinica,English Series 1999年第4期15卷 507-514页

作者： Yonggao Chen Department of Mathematics,Nanjing Normal University,Nanjing 210097,P.R.ChinaYaochen Zhu Institute of Applied Mathematics,Academia Sinica,Beijing 100080,P.R.China Department of Mathematics Nanjing Normal University Nanjing P. R. China Institute of Applied Mathematics Academia Sinica Beijing P. R. China

In this note the algebraic independence of values of general Mahler series with different parameters at algebraic points is established.

关键词： Algebraic independence General Mahler series continued fraction

On an attack on RSA with small CRT-exponents

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Science China(Information Sciences) 2010年第8期53卷 1511-1518页

作者： HAN LiDong 1,2 , WANG XiaoYun 3 & XU GuangWu 3,4 1 Key Laboratory of Cryptologic Technology and Information Security, Ministry of Education, Shandong University, Jinan 250100, China 2 School of Mathematics, Shandong University, Jinan 250100, China 3 Institute for Advanced Study, Tsinghua University, Beijing 100084, China 4 Department of Electrical Engineering and Computer Science, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA 1. Key Laboratory of Cryptologic Technology and Information Security Ministry of Education Shandong University Jinan 250100 China2. School of Mathematics Shandong University Jinan 250100 China3. Institute for Advanced Study Tsinghua University Beijing 100084 China4. Department of Electrical Engineering and Computer Science University of Wisconsin-Milwaukee Milwaukee WI 53201 USA

This paper concerns the RSA system with private CRT-exponents. Since Chinese remainder representation provides efficiency in computation, such system is of some practical significance. In this paper, an existing attack to small private CRT-exponents is analyzed. It is indicated that this attack makes nice use of lattice in RSA analysis, but some argument does not hold in general. Several counterexamples are constructed. Refinements and more precise statements of the attack are given.

关键词： RSA CRT cryptanalysis continued fraction lattice reduction