For the Sylvester continued fraction expansions of real numbers,FAN et al.(2007)proved that,for almost all real numbers,the nth partial quotient grows exponentially with respect to the product of the first n-1 partial...For the Sylvester continued fraction expansions of real numbers,FAN et al.(2007)proved that,for almost all real numbers,the nth partial quotient grows exponentially with respect to the product of the first n-1 partial quotients.In this paper,we establish the Hausdorff dimension of the exceptional set where the growth rate is a general function.展开更多
For each real number x∈(0,1),let[a_(1)(x),a_(2)(x),…,a_n(x),…]denote its continued fraction expansion.We study the convergence exponent defined byτ(x)=inf{s≥0:∞∑n=1(a_(n)(x)a_(n+1)(x))^(-s)<∞},which reflect...For each real number x∈(0,1),let[a_(1)(x),a_(2)(x),…,a_n(x),…]denote its continued fraction expansion.We study the convergence exponent defined byτ(x)=inf{s≥0:∞∑n=1(a_(n)(x)a_(n+1)(x))^(-s)<∞},which reflects the growth rate of the product of two consecutive partial quotients.As a main result,the Hausdorff dimensions of the level sets ofτ(x)are determined.展开更多
Branched continued fractions are one of the multidimensional generalization of the continued fractions. Branched continued fractions with not equivalent variables are an analog of the regular C-fractions for multiple ...Branched continued fractions are one of the multidimensional generalization of the continued fractions. Branched continued fractions with not equivalent variables are an analog of the regular C-fractions for multiple power series. We consider 1-periodic branched continued fraction of the special form which is an analog fraction with not equivalent variables if the values of that variables are fixed. We establish an analog of the parabola theorem for that fraction and estimate truncation error bounds for that fractions at some restrictions. We also propose to use weight coefficients for obtaining different parabolic regions for the same fraction without any additional restriction for first element.展开更多
A famous theorem of Szemer＇edi asserts that any subset of integers with posi- tive upper density contains arbitrarily arithmetic progressions. Let Fq be a finite field with q elements and Fq（（X^-1）） be the power ...A famous theorem of Szemer＇edi asserts that any subset of integers with posi- tive upper density contains arbitrarily arithmetic progressions. Let Fq be a finite field with q elements and Fq（（X^-1）） be the power field of formal series with coefficients lying in Fq. In this paper, we concern with the analogous Szemeredi problem for continued fractions of Laurent series： we will show that the set of points x ∈ Fq（（X-1）） of whose sequence of degrees of partial quotients is strictly increasing and contain arbitrarily long arithmetic progressions is of Hausdorff dimension 1/2.展开更多
Let x∈(0,1)be a real number with continued fraction expansion[a_(1)(x),a_(2)(x),a_(3)(x),⋯].This paper is concerned with the multifractal spectrum of the convergence exponent of{a_(n)(x)}_(n≥1) defined by τ(x):=in...Let x∈(0,1)be a real number with continued fraction expansion[a_(1)(x),a_(2)(x),a_(3)(x),⋯].This paper is concerned with the multifractal spectrum of the convergence exponent of{a_(n)(x)}_(n≥1) defined by τ(x):=inf{s≥0:∑n≥1an^(-s)(x)<∞}.展开更多
In the paper, firstly, based on new non-tensor-product-typed partially inverse divided differences algorithms in a recursive form, scattered data interpolating schemes are constructed via bivariate continued fractions...In the paper, firstly, based on new non-tensor-product-typed partially inverse divided differences algorithms in a recursive form, scattered data interpolating schemes are constructed via bivariate continued fractions with odd and even nodes, respectively. And equivalent identities are also obtained between interpolated functions and bivariate continued fractions. Secondly, by means of three-term recurrence relations for continued fractions, the characterization theorem is presented to study on the degrees of the numerators and denominators of the interpolating continued fractions. Thirdly, some numerical examples show it feasible for the novel recursive schemes. Meanwhile, compared with the degrees of the numera- tors and denominators of bivariate Thiele-typed interpolating continued fractions, those of the new bivariate interpolating continued fractions are much low, respectively, due to the reduc- tion of redundant interpolating nodes. Finally, the operation count for the rational function interpolation is smaller than that for radial basis function interpolation.展开更多
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 gravi...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 anomaly reflects the lateral resolution of the underground mass distribution.展开更多
For a univariate function given by its Taylor series expansion, a continued fraction expansion can be obtained with the Viscovatov＇s algorithm, as the limiting value of a Thiele interpolating continued fraction or by...For a univariate function given by its Taylor series expansion, a continued fraction expansion can be obtained with the Viscovatov＇s algorithm, as the limiting value of a Thiele interpolating continued fraction or by means of the determinantal formulas for inverse and reciprocal differences with coincident data points. In this paper, both Viscovatov-like algorithms and Taylor-like expansions are incorporated to yield bivariate blending continued expansions which are computed as the limiting value of bivariate blending rational interpolants, which are constructed based on symmetric blending differences. Numerical examples are given to show the effectiveness of our methods.展开更多
A recursive rational algorithm for matrix exponentials was obtained by making use of the generalized inverse of a matrix in this paper. On the basis of the n th convergence of Thiele type continued fraction expa...A recursive rational algorithm for matrix exponentials was obtained by making use of the generalized inverse of a matrix in this paper. On the basis of the n th convergence of Thiele type continued fraction expansion, a new type of the generalized inverse matrix valued Padé approximant (GMPA) for matrix exponentials was defined and its remainder formula was proved. The results of this paper were illustrated by some examples.展开更多
A new method for the construction of bivariate matrix valued rational interpolants on a rectangulargrid is introduced. The rational interpolants are expressed in the continued fraction form with scalardenominator. Til...A new method for the construction of bivariate matrix valued rational interpolants on a rectangulargrid is introduced. The rational interpolants are expressed in the continued fraction form with scalardenominator. Tile matrix quotients are based oil the generalized inverse for a matrix, Which is found to beeffective in continued fraction interpolation. In this paper, tWo dual expansions for bivariate matrix valuedThiele-type interpolating continued fractions are presented, then, tWo dual rational interpolants are definedout of them.展开更多
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 re...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.展开更多
The definition of vector valned continned fraction interpolating splines is at first introduced by means of generalized inverse of a vector. In the computation of the interpolating splines,which are of representation ...The definition of vector valned continned fraction interpolating splines is at first introduced by means of generalized inverse of a vector. In the computation of the interpolating splines,which are of representation of the convergences for Thiele-type continned fraction.the three relation is avioded and a new effective recursive algorithm is constrncted. A sufficient condition for existence is given. Some interpolation results incluing uniqueness are given. In the end. a exact interpolation remainder formula is obtained.展开更多
The minimal continued fraction of m (where 0<m∈Z is not a square) is connected with the corresponding simple continued fraction, from which it can be written out.In this paper, it is shown that the minimal conti...The minimal continued fraction of m (where 0<m∈Z is not a square) is connected with the corresponding simple continued fraction, from which it can be written out.In this paper, it is shown that the minimal continued fraction is periodic, its period is shorter than twice of the period of the corresponding simple continued fraction, its absolute\|period is not greater than the period of the corresponding simple continued fraction.Several properties of the minimal continued fraction are also obtained.展开更多
By means of the determinantal formulae for inverse and reciprocal differences with coincident data points, the limiting case of Thiele's interpolating continued fraction expansion is studied in this paper and give...By means of the determinantal formulae for inverse and reciprocal differences with coincident data points, the limiting case of Thiele's interpolating continued fraction expansion is studied in this paper and given numerical example shows that the limiting Thiele's continued fraction expansion can be determined once for all instead of carrying out computations for each step to obtain each convergent as done in [3].展开更多
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...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.展开更多
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...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 irrational numbers.展开更多
The existence of large partial quotients destroys many limit theorems in the metric theory of continued fractions.To achieve some variant forms of limit theorems,a common approach mostly used in practice is to discard...The existence of large partial quotients destroys many limit theorems in the metric theory of continued fractions.To 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 sense.Motivated by this,we are led to consider the metric theory of points with at least two large partial quotients.More 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.展开更多
1.Introduction In order to discuss the irrationality, the transcendence and the algebraic independence for p-adic numbers, the first author introduced in two previous papers [1, 2] a simple form for p-adic continued f...1.Introduction In order to discuss the irrationality, the transcendence and the algebraic independence for p-adic numbers, the first author introduced in two previous papers [1, 2] a simple form for p-adic continued fraction which is called p-adic simple continued fraction by making use of the algebraic theory of continued fraction in the real field mentioned by Schmidt, and gave a sufficient condition for certain p-adic integers which and whose sum, defference, product and quotient are all p-adic transcendental numbers.展开更多
In the light of multi-continued fraction theories, we make a classification and counting for multi-strict continued fractions, which are corresponding to multi-sequences of multiplicity m and length n. Based on the ab...In the light of multi-continued fraction theories, we make a classification and counting for multi-strict continued fractions, which are corresponding to multi-sequences of multiplicity m and length n. Based on the above counting, we develop an iterative formula for computing fast the linear complexity distribution of multi-sequences. As an application, we obtain the linear complexity distributions and expectations of multi-sequences of any given length n and multiplicity m less than 12 by a personal computer. But only results of m=3 and 4 are given in this paper.展开更多
基金Supported by Projects from Chongqing Municipal Science and Technology Commission(CSTB2022NSCQ-MSX0445)。
文摘For the Sylvester continued fraction expansions of real numbers,FAN et al.(2007)proved that,for almost all real numbers,the nth partial quotient grows exponentially with respect to the product of the first n-1 partial quotients.In this paper,we establish the Hausdorff dimension of the exceptional set where the growth rate is a general function.
基金supported by the Scientific Research Fund of Hunan Provincial Education Department(21B0070)the Natural Science Foundation of Jiangsu Province(BK20231452)+1 种基金the Fundamental Research Funds for the Central Universities(30922010809)the National Natural Science Foundation of China(11801591,11971195,12071171,12171107,12201207,12371072)。
文摘For each real number x∈(0,1),let[a_(1)(x),a_(2)(x),…,a_n(x),…]denote its continued fraction expansion.We study the convergence exponent defined byτ(x)=inf{s≥0:∞∑n=1(a_(n)(x)a_(n+1)(x))^(-s)<∞},which reflects the growth rate of the product of two consecutive partial quotients.As a main result,the Hausdorff dimensions of the level sets ofτ(x)are determined.
文摘Branched continued fractions are one of the multidimensional generalization of the continued fractions. Branched continued fractions with not equivalent variables are an analog of the regular C-fractions for multiple power series. We consider 1-periodic branched continued fraction of the special form which is an analog fraction with not equivalent variables if the values of that variables are fixed. We establish an analog of the parabola theorem for that fraction and estimate truncation error bounds for that fractions at some restrictions. We also propose to use weight coefficients for obtaining different parabolic regions for the same fraction without any additional restriction for first element.
文摘A famous theorem of Szemer＇edi asserts that any subset of integers with posi- tive upper density contains arbitrarily arithmetic progressions. Let Fq be a finite field with q elements and Fq（（X^-1）） be the power field of formal series with coefficients lying in Fq. In this paper, we concern with the analogous Szemeredi problem for continued fractions of Laurent series： we will show that the set of points x ∈ Fq（（X-1）） of whose sequence of degrees of partial quotients is strictly increasing and contain arbitrarily long arithmetic progressions is of Hausdorff dimension 1/2.
基金This research was supported by National Natural Science Foundation of China(11771153,11801591,11971195,12171107)Guangdong Natural Science Foundation(2018B0303110005)+1 种基金Guangdong Basic and Applied Basic Research Foundation(2021A1515010056)Kunkun Song would like to thank China Scholarship Council(CSC)for financial support(201806270091).
文摘Let x∈(0,1)be a real number with continued fraction expansion[a_(1)(x),a_(2)(x),a_(3)(x),⋯].This paper is concerned with the multifractal spectrum of the convergence exponent of{a_(n)(x)}_(n≥1) defined by τ(x):=inf{s≥0:∑n≥1an^(-s)(x)<∞}.
基金Supported by the Special Funds Tianyuan for the National Natural Science Foundation of China(Grant No.11426086)the Fundamental Research Funds for the Central Universities(Grant No.2016B08714)the Natural Science Foundation of Jiangsu Province for the Youth(Grant No.BK20160853)
文摘In the paper, firstly, based on new non-tensor-product-typed partially inverse divided differences algorithms in a recursive form, scattered data interpolating schemes are constructed via bivariate continued fractions with odd and even nodes, respectively. And equivalent identities are also obtained between interpolated functions and bivariate continued fractions. Secondly, by means of three-term recurrence relations for continued fractions, the characterization theorem is presented to study on the degrees of the numerators and denominators of the interpolating continued fractions. Thirdly, some numerical examples show it feasible for the novel recursive schemes. Meanwhile, compared with the degrees of the numera- tors and denominators of bivariate Thiele-typed interpolating continued fractions, those of the new bivariate interpolating continued fractions are much low, respectively, due to the reduc- tion of redundant interpolating nodes. Finally, the operation count for the rational function interpolation is smaller than that for radial basis function interpolation.
基金the National Natural Science Foundation(Grant nos.41904122,42004068)China Geological Survey’s project(Grant nos.DD20190012,DD20190435,and DD 20190129)+2 种基金the Special Project for Basic Scientific Research Service(Grant No.JKY202007)the Macao Young Scholars Program(Grant No.AM2020001)the Science and Technology Development Fund,Macao SAR
文摘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 anomaly reflects the lateral resolution of the underground mass distribution.
基金The NNSF(10171026 and 60473114)of Chinathe Research Funds(2005TD03) for Young Innovation Group,Education Department of Anhui Province.
文摘For a univariate function given by its Taylor series expansion, a continued fraction expansion can be obtained with the Viscovatov＇s algorithm, as the limiting value of a Thiele interpolating continued fraction or by means of the determinantal formulas for inverse and reciprocal differences with coincident data points. In this paper, both Viscovatov-like algorithms and Taylor-like expansions are incorporated to yield bivariate blending continued expansions which are computed as the limiting value of bivariate blending rational interpolants, which are constructed based on symmetric blending differences. Numerical examples are given to show the effectiveness of our methods.
文摘A recursive rational algorithm for matrix exponentials was obtained by making use of the generalized inverse of a matrix in this paper. On the basis of the n th convergence of Thiele type continued fraction expansion, a new type of the generalized inverse matrix valued Padé approximant (GMPA) for matrix exponentials was defined and its remainder formula was proved. The results of this paper were illustrated by some examples.
文摘A new method for the construction of bivariate matrix valued rational interpolants on a rectangulargrid is introduced. The rational interpolants are expressed in the continued fraction form with scalardenominator. Tile matrix quotients are based oil the generalized inverse for a matrix, Which is found to beeffective in continued fraction interpolation. In this paper, tWo dual expansions for bivariate matrix valuedThiele-type interpolating continued fractions are presented, then, tWo dual rational interpolants are definedout of them.
文摘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.
文摘The definition of vector valned continned fraction interpolating splines is at first introduced by means of generalized inverse of a vector. In the computation of the interpolating splines,which are of representation of the convergences for Thiele-type continned fraction.the three relation is avioded and a new effective recursive algorithm is constrncted. A sufficient condition for existence is given. Some interpolation results incluing uniqueness are given. In the end. a exact interpolation remainder formula is obtained.
基金Supported by the Science Foundation of Tsinghua Uni-versity
文摘The minimal continued fraction of m (where 0<m∈Z is not a square) is connected with the corresponding simple continued fraction, from which it can be written out.In this paper, it is shown that the minimal continued fraction is periodic, its period is shorter than twice of the period of the corresponding simple continued fraction, its absolute\|period is not greater than the period of the corresponding simple continued fraction.Several properties of the minimal continued fraction are also obtained.
基金Supported by the Foundation for Excellent Young Teachers of the Ministry of Education of China and inpart by the Foundation f
文摘By means of the determinantal formulae for inverse and reciprocal differences with coincident data points, the limiting case of Thiele's interpolating continued fraction expansion is studied in this paper and given numerical example shows that the limiting Thiele's continued fraction expansion can be determined once for all instead of carrying out computations for each step to obtain each convergent as done in [3].
基金supported by National Natural Science Foundation of China (Grant Nos.10631040,10901066)
文摘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.
基金Supported by National Natural Science Foundation of China （Grant No. 10671051）, Natural Science Foundation of Zhejiang Province （Grant No. 103060） and Foundation of Zhejiang Educational Committee （Grant No. 20061069）
文摘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 irrational numbers.
基金supported by National Natural Science Foundation of China(Grant Nos.12171172 and 11831007)。
文摘The existence of large partial quotients destroys many limit theorems in the metric theory of continued fractions.To 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 sense.Motivated by this,we are led to consider the metric theory of points with at least two large partial quotients.More 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.
基金Project Supported by the Science Fund of the Chinese Academy of Science
文摘1.Introduction In order to discuss the irrationality, the transcendence and the algebraic independence for p-adic numbers, the first author introduced in two previous papers [1, 2] a simple form for p-adic continued fraction which is called p-adic simple continued fraction by making use of the algebraic theory of continued fraction in the real field mentioned by Schmidt, and gave a sufficient condition for certain p-adic integers which and whose sum, defference, product and quotient are all p-adic transcendental numbers.
基金the National Natural Science Foundation of China (Grants Nos. 60173016 and 90604011)
文摘In the light of multi-continued fraction theories, we make a classification and counting for multi-strict continued fractions, which are corresponding to multi-sequences of multiplicity m and length n. Based on the above counting, we develop an iterative formula for computing fast the linear complexity distribution of multi-sequences. As an application, we obtain the linear complexity distributions and expectations of multi-sequences of any given length n and multiplicity m less than 12 by a personal computer. But only results of m=3 and 4 are given in this paper.
