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
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
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 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
§Ⅰ. 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 con...§Ⅰ. 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
The relation between continued fractions and Berlekamp’s algorithm was studied bysome reseachers. The latter is an iterative procedure proposed for decoding BCH codes. However,there remains an unanswered question whe...The relation between continued fractions and Berlekamp’s algorithm was studied bysome reseachers. The latter is an iterative procedure proposed for decoding BCH codes. However,there remains an unanswered question whether each of the iterative steps in the algorithm canbe interpreted in terms of continued fractions. In this paper, we first introduce the so-calledrefined convergents to the continued fraction expansion of a binary sequence s, and then give athorough answer to the question in the context of Massey’s linear feedback shift register synthesisalgorithm which is equivalent to that of Berlekamp, and at last we prove that there exists a one-to-one correspondence between the n-th refined convergents and the length n segments.展开更多
A novel method of renormalization called Pacman renormalization allows us to study (unicritical) Siegel functions through Pacman-type functions. It has been used to investigate the Siegel parameters with combinatorial...A novel method of renormalization called Pacman renormalization allows us to study (unicritical) Siegel functions through Pacman-type functions. It has been used to investigate the Siegel parameters with combinatorially periodic rotation number in the main cardioid of the Mandelbrot set. It is already known that it can be defined a Pacman renormalization operator such that for Siegel pacmen, with combinatorially periodic rotation numbers, the operator is compact, analytic and has a unique fixed point, at which it is hyperbolic with one-dimensional unstable manifold. In this paper we observe that this Pacman renormalization operator is compact and analytic at any Siegel Pacman or Siegel map with combinatorially bounded rotation number. This allows us to define a renormalization operator on the hybrid classes of the standard Siegel pacmen to which we built its horseshoe where the operator is topologically semiconjugated to the left shift on the space of bi-infinite sequences of natural numbers bounded by some constant.展开更多
The four-dimensional character of Einstein’s spacetime is generally accepted in mainstream physics as beyond reasonable doubt correct. However the real problem is when we require scale invariance and that this spacet...The four-dimensional character of Einstein’s spacetime is generally accepted in mainstream physics as beyond reasonable doubt correct. However the real problem is when we require scale invariance and that this spacetime be four-dimensional on all scales. It is true that on our classical scale, the 4D decouples into 3D plus one time dimension and that on very large scale only the curvature of spacetime becomes noticeable. However the critical problem is that such spacetime must remain 4D no matter how small the scale we are probing is. This is something of crucial importance for quantum physics. The present work addresses this basic, natural and logical requirement and shows how many contradictory results and shortcomings of relativity and quantum gravity could be eliminated when we “complete” Einstein’s spacetime in such a geometrical gauge invariant way. Concurrently the work serves also as a review of the vast Literature on E-Infinity theory used here.展开更多
As in our previous work [14], a function is said to be of theta-type when its asymptotic behavior near any root of unity is similar to what happened for Jacobi theta functions. It is shown that only four Euler infinit...As in our previous work [14], a function is said to be of theta-type when its asymptotic behavior near any root of unity is similar to what happened for Jacobi theta functions. It is shown that only four Euler infinite products have this property. That this is the case is obtained by investigating the analyticity obstacle of a Laplace-type integral of the exponential generating function of Bernoulli numbers.展开更多
Fractional-order differentiator is a principal component of the fractional-order controller.Discretization of fractional-order differentiator is essential to implement the fractionalorder controller digitally.Discreti...Fractional-order differentiator is a principal component of the fractional-order controller.Discretization of fractional-order differentiator is essential to implement the fractionalorder controller digitally.Discretization methods generally include indirect approach and direct approach to find the discrete-time approximation of fractional-order differentiator in the Z-domain as evident from the existing literature.In this paper,a direct approach is proposed for discretization of fractional-order differentiator in delta-domain instead of the conventional Z-domain as the delta operator unifies both analog system and digital system together at a high sampling frequency.The discretization of fractional-order differentiator is accomplished in two stages.In the first stage,the generating function is framed by reformulating delta operator using trapezoidal rule or Tustin approximation and in the next stage,the fractional-order differentiator has been approximated by expanding the generating function using continued fraction expansion method.The proposed method has been compared with two well-known direct discretization methods taken from the existing literature.Two examples are presented in this context to show the efficacy of the proposed discretization method using simulation results obtained from MATLAB.展开更多
Computing upper bounds of the positive real roots of some polynomials is a key step of those real root isolation algorithms based on continued fraction expansion and Vincent's theorem.The authors give a new algori...Computing upper bounds of the positive real roots of some polynomials is a key step of those real root isolation algorithms based on continued fraction expansion and Vincent's theorem.The authors give a new algorithm for computing an upper bound of positive roots in this paper.The complexity of the algorithm is O(n log(uH-l))additions and multiplications where u is the optimal upper bound satisfying Theorem 3.1 of this paper and n is the degree of the polynomial.The method together w辻h some tricks have been implemented as a software package logcf using C language.Experiments on many benchmarks show that logcf is competitive with Root Intervals of Mathematica and the function realroot of Maple averagely and it is much faster than existing open source real root solvers in many test cases.展开更多
We consider an M/M/2 queueing system with two-heterogeneous servers and multiple vacations. Customers arrive according to a Poisson process. However, customers become impatient when the system is on vacation. We obtai...We consider an M/M/2 queueing system with two-heterogeneous servers and multiple vacations. Customers arrive according to a Poisson process. However, customers become impatient when the system is on vacation. We obtain explicit expressions for the time dependent probabilities,mean and variance of the system size at time t by employing probability generating functions, continued fractions and properties of the modified Bessel functions. Finally, two special cases are provided.展开更多
Let p≥2 be a prime number and Zp be the ring of p-adic intergers.Let G be a semigroup generated by infinitely many contractive maps on p Zp.It is shown that if G satisfies the open tiling conditions,then there exists...Let p≥2 be a prime number and Zp be the ring of p-adic intergers.Let G be a semigroup generated by infinitely many contractive maps on p Zp.It is shown that if G satisfies the open tiling conditions,then there exists a shift transformation on the limit set of G and the shift transformation is ergodic with respect to the Haar measure on p Zp.As an application,we can generalize p-adic Khinchin’s Theorem and p-adic Lochs’Theorem to any infinitely generated semigroup by use of the ergodicity of the shift transformation.展开更多
This paper studies the time-dependent analysis of an M/M/1 queueing model with single,multiple working vacation,balking and vacation interruptions.Whenever the system becomes empty,the server commences working vacatio...This paper studies the time-dependent analysis of an M/M/1 queueing model with single,multiple working vacation,balking and vacation interruptions.Whenever the system becomes empty,the server commences working vacation.During the working vacation period,if the queue length reaches a positive threshold value‘k’,the working vacation of the server is interrupted and it immediately starts the service in an exhaustive manner.During working vacations,the customers become discouraged due to the slow service and possess balking behavior.The transient system size probabilities of the proposed model are derived explicitly using the method of generating function and continued fraction.The performance indices such as average and variance of system size are also obtained.Further,numerical simulations are presented to analyze the impact of system parameters.展开更多
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 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§Ⅰ. 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.
文摘The relation between continued fractions and Berlekamp’s algorithm was studied bysome reseachers. The latter is an iterative procedure proposed for decoding BCH codes. However,there remains an unanswered question whether each of the iterative steps in the algorithm canbe interpreted in terms of continued fractions. In this paper, we first introduce the so-calledrefined convergents to the continued fraction expansion of a binary sequence s, and then give athorough answer to the question in the context of Massey’s linear feedback shift register synthesisalgorithm which is equivalent to that of Berlekamp, and at last we prove that there exists a one-to-one correspondence between the n-th refined convergents and the length n segments.
文摘A novel method of renormalization called Pacman renormalization allows us to study (unicritical) Siegel functions through Pacman-type functions. It has been used to investigate the Siegel parameters with combinatorially periodic rotation number in the main cardioid of the Mandelbrot set. It is already known that it can be defined a Pacman renormalization operator such that for Siegel pacmen, with combinatorially periodic rotation numbers, the operator is compact, analytic and has a unique fixed point, at which it is hyperbolic with one-dimensional unstable manifold. In this paper we observe that this Pacman renormalization operator is compact and analytic at any Siegel Pacman or Siegel map with combinatorially bounded rotation number. This allows us to define a renormalization operator on the hybrid classes of the standard Siegel pacmen to which we built its horseshoe where the operator is topologically semiconjugated to the left shift on the space of bi-infinite sequences of natural numbers bounded by some constant.
文摘The four-dimensional character of Einstein’s spacetime is generally accepted in mainstream physics as beyond reasonable doubt correct. However the real problem is when we require scale invariance and that this spacetime be four-dimensional on all scales. It is true that on our classical scale, the 4D decouples into 3D plus one time dimension and that on very large scale only the curvature of spacetime becomes noticeable. However the critical problem is that such spacetime must remain 4D no matter how small the scale we are probing is. This is something of crucial importance for quantum physics. The present work addresses this basic, natural and logical requirement and shows how many contradictory results and shortcomings of relativity and quantum gravity could be eliminated when we “complete” Einstein’s spacetime in such a geometrical gauge invariant way. Concurrently the work serves also as a review of the vast Literature on E-Infinity theory used here.
基金The author was supported by Labex CEMPI(Centre Europeen pour les Mathematiques,la Physique et leurs Interaction).
文摘As in our previous work [14], a function is said to be of theta-type when its asymptotic behavior near any root of unity is similar to what happened for Jacobi theta functions. It is shown that only four Euler infinite products have this property. That this is the case is obtained by investigating the analyticity obstacle of a Laplace-type integral of the exponential generating function of Bernoulli numbers.
文摘Fractional-order differentiator is a principal component of the fractional-order controller.Discretization of fractional-order differentiator is essential to implement the fractionalorder controller digitally.Discretization methods generally include indirect approach and direct approach to find the discrete-time approximation of fractional-order differentiator in the Z-domain as evident from the existing literature.In this paper,a direct approach is proposed for discretization of fractional-order differentiator in delta-domain instead of the conventional Z-domain as the delta operator unifies both analog system and digital system together at a high sampling frequency.The discretization of fractional-order differentiator is accomplished in two stages.In the first stage,the generating function is framed by reformulating delta operator using trapezoidal rule or Tustin approximation and in the next stage,the fractional-order differentiator has been approximated by expanding the generating function using continued fraction expansion method.The proposed method has been compared with two well-known direct discretization methods taken from the existing literature.Two examples are presented in this context to show the efficacy of the proposed discretization method using simulation results obtained from MATLAB.
基金supported by the National Science Foundation of China under Grant Nos.61802318,61732001and 61532019
文摘Computing upper bounds of the positive real roots of some polynomials is a key step of those real root isolation algorithms based on continued fraction expansion and Vincent's theorem.The authors give a new algorithm for computing an upper bound of positive roots in this paper.The complexity of the algorithm is O(n log(uH-l))additions and multiplications where u is the optimal upper bound satisfying Theorem 3.1 of this paper and n is the degree of the polynomial.The method together w辻h some tricks have been implemented as a software package logcf using C language.Experiments on many benchmarks show that logcf is competitive with Root Intervals of Mathematica and the function realroot of Maple averagely and it is much faster than existing open source real root solvers in many test cases.
基金Supported by the National Natural Science Foundation of China(11671204)
文摘We consider an M/M/2 queueing system with two-heterogeneous servers and multiple vacations. Customers arrive according to a Poisson process. However, customers become impatient when the system is on vacation. We obtain explicit expressions for the time dependent probabilities,mean and variance of the system size at time t by employing probability generating functions, continued fractions and properties of the modified Bessel functions. Finally, two special cases are provided.
基金Supported by Natural Science Foundation of China(Grant Nos.11301510,11671092)。
文摘Let p≥2 be a prime number and Zp be the ring of p-adic intergers.Let G be a semigroup generated by infinitely many contractive maps on p Zp.It is shown that if G satisfies the open tiling conditions,then there exists a shift transformation on the limit set of G and the shift transformation is ergodic with respect to the Haar measure on p Zp.As an application,we can generalize p-adic Khinchin’s Theorem and p-adic Lochs’Theorem to any infinitely generated semigroup by use of the ergodicity of the shift transformation.
文摘This paper studies the time-dependent analysis of an M/M/1 queueing model with single,multiple working vacation,balking and vacation interruptions.Whenever the system becomes empty,the server commences working vacation.During the working vacation period,if the queue length reaches a positive threshold value‘k’,the working vacation of the server is interrupted and it immediately starts the service in an exhaustive manner.During working vacations,the customers become discouraged due to the slow service and possess balking behavior.The transient system size probabilities of the proposed model are derived explicitly using the method of generating function and continued fraction.The performance indices such as average and variance of system size are also obtained.Further,numerical simulations are presented to analyze the impact of system parameters.