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
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
We consider the block matrices and 3-dimensional graph manifolds associated with a special type of tree graphs. We demonstrate that the linking matrices of these graph manifolds coincide with the reduced matrices obta...We consider the block matrices and 3-dimensional graph manifolds associated with a special type of tree graphs. We demonstrate that the linking matrices of these graph manifolds coincide with the reduced matrices obtained from the Laplacian block matrices by means of Gauss partial diagonalization procedure described explicitly by W. Neumann. The linking matrix is an important topological invariant of a graph manifold which is possible to interpret as a matrix of coupling constants of gauge interaction in Kaluza-Klein approach, where 3-dimensional graph manifold plays the role of internal space in topological 7-dimensional BF theory. The Gauss-Neumann method gives us a simple algorithm to calculate the linking matrices of graph manifolds and thus the coupling constants matrices.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
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
In recent years, a variety of chaos-based cryptosystems have been proposed. Some of these systems are used in designing a pseudo random bit generator (PRBG) for stream cipher applications. Most of the chaotic systems ...In recent years, a variety of chaos-based cryptosystems have been proposed. Some of these systems are used in designing a pseudo random bit generator (PRBG) for stream cipher applications. Most of the chaotic systems used in cryptography have good chaotic properties like ergodicity, sensitivity to initial values and sensitivity to control parameters. However, some of them are not very suitable for use in cryptography because of their non-uniform density function, and their relatively small key space. To be used in cryptography, a PRBG may need to meet stronger requirements than for other applications. In particular, various statistical tests can be applied to the outputs of such generators to conclude whether the generator produces a truly random sequence or not. In this paper, we propose a PRBG based on the use of the standard chaotic map with large key space and the Engle Continued Fractions (ECF) map. The outputs of the standard map are used as the inputs of ECF-map. The chaotic nature of the standard map and the good statistical properties of the ECF map motivate us to design a new PRBG for stream cipher applications. The numerical simulation analysis indicates that our PRBG produces bit sequences possessing excellent statistical and cryptographic properties.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
A kind of triple branched continued fractions is defined by making use of Samel- son inverse and Thiele-type partial inverted di?erences [1]. In this paper, a levels-recursive algorithm is constructed and a numerical ...A kind of triple branched continued fractions is defined by making use of Samel- son inverse and Thiele-type partial inverted di?erences [1]. In this paper, a levels-recursive algorithm is constructed and a numerical example is given.展开更多
This paper constructs a new kind of block based bivariate blending rational interpolation via symmetric branched continued fractions. The construction process may be outlined as follows. The first step is to divide th...This paper constructs a new kind of block based bivariate blending rational interpolation via symmetric branched continued fractions. The construction process may be outlined as follows. The first step is to divide the original set of support points into some subsets (blocks). Then construct each block by using symmetric branched continued fraction. Finally assemble these blocks by Newton’s method to shape the whole interpolation scheme. Our new method offers many flexible bivariate blending rational interpolation schemes which include the classical bivariate Newton’s polynomial interpolation and symmetric branched continued fraction interpolation as its special cases. The block based bivariate blending rational interpolation is in fact a kind of tradeoff between the purely linear interpolation and the purely nonlinear interpolation. Finally, numerical examples are given to show the effectiveness of the proposed method.展开更多
【目的】叶绿素含量可以用来评价棉花的长势情况,快速、准确和大面积监测棉花叶绿素含量,有助于实现精准农业。【方法】分别用0~2阶(步长为0.2)的分数阶微分处理和1~10尺度下的小波变换对田间测定的陆地棉和海岛棉等2种棉花的高光谱反...【目的】叶绿素含量可以用来评价棉花的长势情况,快速、准确和大面积监测棉花叶绿素含量,有助于实现精准农业。【方法】分别用0~2阶(步长为0.2)的分数阶微分处理和1~10尺度下的小波变换对田间测定的陆地棉和海岛棉等2种棉花的高光谱反射率进行处理,提高棉花叶绿素含量反演精度。通过分析不同处理方式的光谱与叶绿素含量之间的相关性,筛选得出敏感波段;并运用支持向量机回归和随机森林回归模型分别构建棉花叶绿素含量高光谱估算模型。【结果】(1)在全波段范围内,2种棉花325~1075 nm光谱反射率曲线整体变化趋势基本相同,其反射率均随着叶绿素含量的增加而增大。(2)经连续小波变换和分数阶微分变换后,2种棉花高光谱数据和叶绿素含量的相关性有所增强。使用随机森林回归和小波能量系数7对陆地棉叶绿素含量的反演效果最好,建模集决定系数(coefficient of determination,R^(2))为0.931,均方根误差(root mean square error,RMSE)为0.782,剩余预测偏差(residual prediction deviation,RPD)为2.162;使用随机森林回归和小波能量系数6对海岛棉叶绿素含量的反演效果最佳,建模集R^(2)为0.932,RMSE为1.198,RPD为2.687。【结论】本研究可为棉花叶绿素含量遥感估算提供技术参考。展开更多
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.
文摘We consider the block matrices and 3-dimensional graph manifolds associated with a special type of tree graphs. We demonstrate that the linking matrices of these graph manifolds coincide with the reduced matrices obtained from the Laplacian block matrices by means of Gauss partial diagonalization procedure described explicitly by W. Neumann. The linking matrix is an important topological invariant of a graph manifold which is possible to interpret as a matrix of coupling constants of gauge interaction in Kaluza-Klein approach, where 3-dimensional graph manifold plays the role of internal space in topological 7-dimensional BF theory. The Gauss-Neumann method gives us a simple algorithm to calculate the linking matrices of graph manifolds and thus the coupling constants matrices.
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)<∞}.
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In recent years, a variety of chaos-based cryptosystems have been proposed. Some of these systems are used in designing a pseudo random bit generator (PRBG) for stream cipher applications. Most of the chaotic systems used in cryptography have good chaotic properties like ergodicity, sensitivity to initial values and sensitivity to control parameters. However, some of them are not very suitable for use in cryptography because of their non-uniform density function, and their relatively small key space. To be used in cryptography, a PRBG may need to meet stronger requirements than for other applications. In particular, various statistical tests can be applied to the outputs of such generators to conclude whether the generator produces a truly random sequence or not. In this paper, we propose a PRBG based on the use of the standard chaotic map with large key space and the Engle Continued Fractions (ECF) map. The outputs of the standard map are used as the inputs of ECF-map. The chaotic nature of the standard map and the good statistical properties of the ECF map motivate us to design a new PRBG for stream cipher applications. The numerical simulation analysis indicates that our PRBG produces bit sequences possessing excellent statistical and cryptographic properties.
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.
基金Supported by the National Natural Science Foundation of China under Grant No. 10171026.
文摘A kind of triple branched continued fractions is defined by making use of Samel- son inverse and Thiele-type partial inverted di?erences [1]. In this paper, a levels-recursive algorithm is constructed and a numerical example is given.
基金Project supported by the National Natural Science Foundation of China (No. 10171026, No. 60473114) the AnhuiProvincial Natural Science Foundation, China (No. 03046102)the Research Funds for Young InnovationGroup, Education Department of Anhui Province (No. 2005TD03).
文摘This paper constructs a new kind of block based bivariate blending rational interpolation via symmetric branched continued fractions. The construction process may be outlined as follows. The first step is to divide the original set of support points into some subsets (blocks). Then construct each block by using symmetric branched continued fraction. Finally assemble these blocks by Newton’s method to shape the whole interpolation scheme. Our new method offers many flexible bivariate blending rational interpolation schemes which include the classical bivariate Newton’s polynomial interpolation and symmetric branched continued fraction interpolation as its special cases. The block based bivariate blending rational interpolation is in fact a kind of tradeoff between the purely linear interpolation and the purely nonlinear interpolation. Finally, numerical examples are given to show the effectiveness of the proposed method.
文摘【目的】叶绿素含量可以用来评价棉花的长势情况,快速、准确和大面积监测棉花叶绿素含量,有助于实现精准农业。【方法】分别用0~2阶(步长为0.2)的分数阶微分处理和1~10尺度下的小波变换对田间测定的陆地棉和海岛棉等2种棉花的高光谱反射率进行处理,提高棉花叶绿素含量反演精度。通过分析不同处理方式的光谱与叶绿素含量之间的相关性,筛选得出敏感波段;并运用支持向量机回归和随机森林回归模型分别构建棉花叶绿素含量高光谱估算模型。【结果】(1)在全波段范围内,2种棉花325~1075 nm光谱反射率曲线整体变化趋势基本相同,其反射率均随着叶绿素含量的增加而增大。(2)经连续小波变换和分数阶微分变换后,2种棉花高光谱数据和叶绿素含量的相关性有所增强。使用随机森林回归和小波能量系数7对陆地棉叶绿素含量的反演效果最好,建模集决定系数(coefficient of determination,R^(2))为0.931,均方根误差(root mean square error,RMSE)为0.782,剩余预测偏差(residual prediction deviation,RPD)为2.162;使用随机森林回归和小波能量系数6对海岛棉叶绿素含量的反演效果最佳,建模集R^(2)为0.932,RMSE为1.198,RPD为2.687。【结论】本研究可为棉花叶绿素含量遥感估算提供技术参考。