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REFINEMENTS OF TRANSFORMATION INEQUALITIES FOR ZERO-BALANCED HYPERGEOMETRIC FUNCTIONS 被引量:9
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作者 王淼坤 褚玉明 《Acta Mathematica Scientia》 SCIE CSCD 2017年第3期607-622,共16页
In the article, we present some refinements of three classes of transformation inequalities for zero-balanced hypergeometric functions by use of the updated monotonicity criterion for the quotient of power series.
关键词 Gauss hypergeometric function complete elliptic integrals quadratic transfor-mation ramanujan cubic transformation
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Singular values, quasiconformal maps and the Schottky upper bound 被引量:9
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作者 裘松良 《Science China Mathematics》 SCIE 1998年第12期1241-1247,共7页
Some properties and asymptotically sharp bounds are obtained for singdar values of Ramanujan’s generalized modular equation. from which infinite-product representations of the Hersch-Pfluger ?dimtortion function ? K ... Some properties and asymptotically sharp bounds are obtained for singdar values of Ramanujan’s generalized modular equation. from which infinite-product representations of the Hersch-Pfluger ?dimtortion function ? K (r) and the Agard η-distortion function η K (t) follow. By these results, the explicit quasiconformal Schwan lemma is improved, several properties are obtained for the Schottky upper bound, and a conjecture on the linear distortion function λ (K) is proved to be true. 展开更多
关键词 ramanujan’s modular equation singular value quasicodformal mappings Schottky’s theorem special functions
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Grstzsch环与Ramanujan的模方程 被引量:8
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作者 裘松良 《数学学报(中文版)》 SCIE CSCD 北大核心 2000年第2期283-290,共8页
本文建立了拟共形映照理论中平面Grtzsch环的模μ(r)与由(8)式定义的出现于数论中Ramanujan广义模方程的函数μa(γ)之间的关系,并将μ(γ)的若干已知结果推广到μa(γ);为研究模方程理论提供了一种... 本文建立了拟共形映照理论中平面Grtzsch环的模μ(r)与由(8)式定义的出现于数论中Ramanujan广义模方程的函数μa(γ)之间的关系,并将μ(γ)的若干已知结果推广到μa(γ);为研究模方程理论提供了一种方法. 展开更多
关键词 ramanujan Groetzsch环 拟共形映照 模方程
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A Number Theoretic Analysis of the Enthalpy, Enthalpy Energy Density, Thermodynamic Volume, and the Equation of State of a Modified White Hole, and the Implications to the Quantum Vacuum Spacetime, Matter Creation and the Planck Frequency
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作者 Michele Nardelli Amos S. Kubeka Alizera Amani 《Journal of Modern Physics》 2024年第1期1-50,共50页
In this paper, we analyze the enthalpy, enthalpy energy density, thermodynamic volume, and the equation of state of a modified white hole. We obtain new possible mathematical connections with some sectors of Number Th... In this paper, we analyze the enthalpy, enthalpy energy density, thermodynamic volume, and the equation of state of a modified white hole. We obtain new possible mathematical connections with some sectors of Number Theory, Ramanujan Recurring Numbers, DN Constant and String Theory, that enable us to extract the quantum geometrical properties of these thermodynamic equations and the implication to the quantum vacuum spacetime geometry of our early universe as they act as the constraints to the nature of quantum gravity of the universe. 展开更多
关键词 Number Theory ramanujan Recurring Numbers DN Constant String Theory Loop Quantum Gravity Matter Creation Enthalpy Energy Density Thermodynamic Volume ENTHALPY
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On Some Mathematical Connections between the Cyclic Universe, Inflationary Universe, p-Adic Inflation, p-Adic Cosmology and Various Sectors of Number Theory
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作者 Michele Nardelli 《Journal of Modern Physics》 2024年第11期1869-1958,共90页
This paper is a review, a thesis, of some interesting results that have been obtained in various research concerning the “brane collisions in string and M-theory” (Cyclic Universe), p-adic inflation and p-adic cosmo... This paper is a review, a thesis, of some interesting results that have been obtained in various research concerning the “brane collisions in string and M-theory” (Cyclic Universe), p-adic inflation and p-adic cosmology. In Section 2, we have described some equations concerning cosmic evolution in a Cyclic Universe. In Section 3, we have described some equations concerning the cosmological perturbations in a Big Crunch/Big Bang space-time, the M-theory model of a Big Crunch/Big Bang transition and some equations concerning the solution of a braneworld Big Crunch/Big Bang Cosmology. In Section 4, we have described some equations concerning the generating ekpyrotic curvature perturbations before the Big Bang, some equations concerning the effective five-dimensional theory of the strongly coupled heterotic string as a gauged version of N=1five-dimensional supergravity with four-dimensional boundaries, and some equations concerning the colliding branes and the origin of the Hot Big Bang. In Section 5, we have described some equations regarding the “null energy condition” violation concerning the inflationary models and some equations concerning the evolution to a smooth universe in an ekpyrotic contracting phase with w>1. In Section 6, we have described some equations concerning the approximate inflationary solutions rolling away from the unstable maximum of p-adic string theory. In Section 7, we have described various equations concerning the p-adic minisuperspace model, zeta strings, zeta nonlocal scalar fields and p-adic and adelic quantum cosmology. In Section 8, we have shown various and interesting mathematical connections between some equations concerning the p-adic inflation, the p-adic quantum cosmology, the zeta strings and the brane collisions in string and M-theory. Furthermore, in each section, we have shown the mathematical connections with various sectors of Number Theory, principally the Ramanujan’s modular equations, the Aurea Ratio and the Fibonacci’s numbers. 展开更多
关键词 String Theory M-THEORY Cyclic Cosmology p-Adic and Adelic Analysis Number Theory ramanujan’s Modular Equations
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A Kind of Diophantine Equations in Finite Simple Groups 被引量:3
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作者 曹珍富 《Northeastern Mathematical Journal》 CSCD 2000年第4期391-397,共7页
In this paper, we prove that if p, q are distinct primes, (p,q)≡(1,7) (mod 12) and Legendres symbol pq=1 , then the equation 1+p a=2 bq c+2 dp eq f has only solutions of the form (a,b,c,d,e,f)=... In this paper, we prove that if p, q are distinct primes, (p,q)≡(1,7) (mod 12) and Legendres symbol pq=1 , then the equation 1+p a=2 bq c+2 dp eq f has only solutions of the form (a,b,c,d,e,f)=(t,0,0,0,t,0), where t is a non negative integer. We also give all solutions of a kind of generalized Ramanujan Nagell equations by using the theories of imaginary quadratic field and Pells equation. 展开更多
关键词 exponential Diophantine equation generalized ramanujan Nagell equation finite simple group difference set
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有限长Ramanujan-Fourier快速变换及频率估计 被引量:2
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作者 郭旭静 王祖林 《北京航空航天大学学报》 EI CAS CSCD 北大核心 2011年第10期1317-1320,1325,共5页
近来出现的Ramnujan-Fourier变换(RFT,Ramanujan Fourier Transformation)是以"Ramanujan和"为基向量的算术变换,该变换可提供分数频率分辨力.首先分析了有限长Ramanujan频谱特点,给出了基向量分布情况,推导了该变换的快速算... 近来出现的Ramnujan-Fourier变换(RFT,Ramanujan Fourier Transformation)是以"Ramanujan和"为基向量的算术变换,该变换可提供分数频率分辨力.首先分析了有限长Ramanujan频谱特点,给出了基向量分布情况,推导了该变换的快速算法,比较了有限长RFT与快速傅里叶变换的乘法计算量;其次,给出了利用RFT的递归峰值检测频率估计算法,并分析了RFT的频率分辨率和适用特点,在非高斯噪声条件下,仿真比较了RFT与傅里叶变换对信号进行频率估计的性能,得到在信噪比为-20 dB的非高斯噪声情况下,频率估计的归一化均方误差可以达到10-3. 展开更多
关键词 ramanujan FOURIER变换 ramanujan 傅里叶变换 频率估计
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APPROXIMATE CONVEXITY AND CONCAVITY OFGENERALIZED GROTZSCH RING FUNCTION 被引量:2
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作者 王根娣 裘松良 +1 位作者 张孝惠 褚玉明 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2006年第2期203-206,共4页
In this paper, the so-called approximate convexity and concavity properties of generalized Groetzsch ring function μa (r) by studying the monotonieity,convexity or concavity of certain composites of μa(r) are ob... In this paper, the so-called approximate convexity and concavity properties of generalized Groetzsch ring function μa (r) by studying the monotonieity,convexity or concavity of certain composites of μa(r) are obtained. 展开更多
关键词 generalized Grotzsch ring function ramanujan's modular equation CONVEXITY concawty.
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对一个Ramanujan型连分数的研究
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作者 何宗健 《内江科技》 2024年第6期104-106,共3页
Ramanujan在他丢失的笔记本中记录了很多美丽的连分数c(q),在这篇论文中研究其中一个类似的连分数。主要通过thate函数以及模关系进行研究,得出关于c(q)的表达式以及c(-q),c(-q^(n)),c(q^(n))的模关系,最后得到c(q)的某些具体数值。
关键词 连分数 ramanujan 笔记本 表达式
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Some Identities Involving Certain Hardy Sums and Ramanujan Sum 被引量:1
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作者 HongYanLIU WenPengZHANG 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2005年第1期109-116,共8页
The main purpose of this paper is, using the mean-value theorem of Dirichletl-functions, to study the distribution properties of the hybrid mean value involving certain Hardysums and Ramanujan sum, and give four inter... The main purpose of this paper is, using the mean-value theorem of Dirichletl-functions, to study the distribution properties of the hybrid mean value involving certain Hardysums and Ramanujan sum, and give four interesting identities. 展开更多
关键词 Certain Hardy sums ramanujan sum Hybrid mean value IDENTITIES
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基于Jacobi三重乘积恒等式的Ramanujan函数χ(-q)的剖分研究
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作者 李紫微 《绵阳师范学院学报》 2023年第5期18-24,共7页
首先,介绍了剖分的定义以及相关方法,然后引入了Ramanujan theta函数定义和Jacobi三重乘积恒等式,基于Ramanujan theta函数的剖分现状进行了说明,利用Jacobi三重乘积恒等式给出了一个新的特殊函数χ(-q)及其倒数的2-剖分的形式,根据χ(... 首先,介绍了剖分的定义以及相关方法,然后引入了Ramanujan theta函数定义和Jacobi三重乘积恒等式,基于Ramanujan theta函数的剖分现状进行了说明,利用Jacobi三重乘积恒等式给出了一个新的特殊函数χ(-q)及其倒数的2-剖分的形式,根据χ(-q)及其倒数的2-剖分得到关于χ(-q)剖分的两个推论. 展开更多
关键词 Jacobi三重乘积恒等式 2-剖分 ramanujan theta函数 χ(-q)
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A Short Proof of Ramanujan's 1ψ1 Formula 被引量:1
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作者 CHEN Xiao-fing LIU Qi-qun MA Xin-rong 《Journal of Mathematical Research and Exposition》 CSCD 北大核心 2008年第1期46-48,共3页
In this paper,we give a short proof of the celebrated Ramanujan's 1ψ1 formula.
关键词 q-binomial theorem ramanujan's 1ψ1 formula bilateral q-series.
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一个Ramanujan Tau函数的新表达式 被引量:1
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作者 程开敏 《纯粹数学与应用数学》 2017年第2期129-133,共5页
通过对Ramanujan Tau函数的研究,并借助Ewell的一个关于q-级数的恒等式结果,发现了关于Ramanujan Tau函数的生成函数的恒等式,该恒等式中含有两类重要的算术函数,即表正整数为若干三角数的和的表法数及表正整数为若干平方数的和的表法数... 通过对Ramanujan Tau函数的研究,并借助Ewell的一个关于q-级数的恒等式结果,发现了关于Ramanujan Tau函数的生成函数的恒等式,该恒等式中含有两类重要的算术函数,即表正整数为若干三角数的和的表法数及表正整数为若干平方数的和的表法数.从而得到了一个关于Ramanujan Tau函数新的显式表达式.最后作为结果的应用,该文还给出了一个关于Ramanujan Tau函数新的简洁同余恒等式. 展开更多
关键词 ramanujan Tau函数 恒等式 同余式
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Distribution of a Certain Partition Function Modulo Powers of Primes
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作者 Hei-Chi CHAN 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2011年第4期625-634,共10页
In this paper, we study a certain partition function a(n) defined by ∑n≥0 a(n)qn := ∏n=1(1- qn)-1(1 -2n)-1. We prove that given a positive integer j 〉 1 and a prime m _〉 5, there are infinitely many cong... In this paper, we study a certain partition function a(n) defined by ∑n≥0 a(n)qn := ∏n=1(1- qn)-1(1 -2n)-1. We prove that given a positive integer j 〉 1 and a prime m _〉 5, there are infinitely many congruences of the type a(An + B) ≡ 0 (rood m3). This work is inspired by Ono's ground breaking result in the study of the distribution of the partition function p(n). 展开更多
关键词 Distribution of partition function Ono's theorem ramanujan's congruences
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有关Ramanujan Tau函数的注记
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作者 程开敏 《数学杂志》 2018年第2期337-344,共8页
本文研究了若干特殊的theta函数和q-级数.利用它们定义的特殊形式,建立了几类特殊的q-级数与Ramanujan Tau函数的生成函数的关系,从而得到了几个Ramanujan Tau函数新的显式表达式.最后作为定理的应用,还得到了一个有关Ramanujan Tau函... 本文研究了若干特殊的theta函数和q-级数.利用它们定义的特殊形式,建立了几类特殊的q-级数与Ramanujan Tau函数的生成函数的关系,从而得到了几个Ramanujan Tau函数新的显式表达式.最后作为定理的应用,还得到了一个有关Ramanujan Tau函数的同余恒等式. 展开更多
关键词 THETA函数 Q-级数 ramanujan Tau函数
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Apropos 1+2+3+4+5+...=<img src="http://latex.codecogs.com/gif.latex?-\frac{1}{12}"title="-\frac{1}{12}"/>: Mapping Infinity in Light of the Number Circle (or Cycle), in L. Euler’s Footsteps and with the Aid of Two Dimensional Infinite Series, and Replacing Negative Infinity and Positive Infinity with Just Infinity
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作者 Leo Depuydt 《Advances in Pure Mathematics》 2017年第1期75-133,共59页
The number circle—that is, the notion that the largest possible positive numbers are followed by infinity and then by the smallest possible negative numbers—is not new. L. Euler defended it in the eighteenth century... The number circle—that is, the notion that the largest possible positive numbers are followed by infinity and then by the smallest possible negative numbers—is not new. L. Euler defended it in the eighteenth century and, before him, J. Wallis considered something vaguely similar. However, in the nineteenth century, the number circle was for the most part abandoned—even if something similar is on occasion accepted in geometry, in the sense that space is circular. The design of the present paper is to present positive proof of the veracity of the number circle and therefore, at the same time, to falsify the number line. Verifying the number circle implies falsifying negative infinity and positive infinity—infinity instead being neither negative nor positive, just like 0. Part of said proof involves showing that infinity can be defined both as 1+1+1+1+1+1+... and as -1-1-1-1-1-... and that the following Equation applies: 1+1+1+1+1+1+...=-1-1-1-1-1-... The principal mathematical technique that will be used to provide said proof is introduced here for the first time. It is called the two dimensional infinite series. It is an infinite series of infinite series. Some additional observations regarding the geography of infinity will be made. A more detailed description of the geography of infinity will be reserved for other papers. The Equation is discussed in this paper only to the extent that the attention that has been paid to it has necessitated the construction of a theory of infinity that, upon closer inspection, makes the Equation more self-evident and intuitively apparent;a fuller discussion will take place in a later paper. 展开更多
关键词 Euler L. INFINITE SERIES INFINITE SERIES of INFINITE INFINITY Geography of NEGATIVE INFINITY Invalidity of NUMBER Circle Veracity of NUMBER Cycle NUMBER Line Positive INFINITY Two dimensional INFINITE SERIES ramanujan Rational Human Intelligence Wallis J.
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Prime-Composition Approach to Ramanujan-Fourier Transform Computation
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作者 周丽娜 王祖林 赵磊 《Transactions of Tianjin University》 EI CAS 2014年第3期197-202,共6页
Ramanujan sums (RS) and their Fourier transforms have attracted more and more attention in signal processing in recent years. Due to their non-periodic and non-uniform spectrum, RS are widely used in low-frequency n... Ramanujan sums (RS) and their Fourier transforms have attracted more and more attention in signal processing in recent years. Due to their non-periodic and non-uniform spectrum, RS are widely used in low-frequency noise processing, Doppler spectrum estimation and time-frequency analysis. However, the traditional method for calculating RS values is rather complex since it requires two numbers' factorization in two arithmetic functions. For a length-n vector, its Ramanujan-Fourier transform usually involves a series of RS values which will occupy O(n2) memory units. Thus, in this paper an approach based on prime-composition is proposed to reduce the complexity of RS calculation to O(n). Meanwhile, the complexity of Ramanujan-Fourier transform can be further reduced from O(n2) to O(n In(In(n))) . 展开更多
关键词 ramanujan sums ramanujan-Fourier transform prime-composition
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The Products of Three Theta Functions and the General Cubic Theta Functions
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作者 Xiao Mei YANG 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2010年第6期1115-1124,共10页
In this paper we establish two theta function identities with four parameters by the theory of theta functions. Using these identities we introduce common generalizations of Hirschhorn-Garvan-Borwein cubic theta funct... In this paper we establish two theta function identities with four parameters by the theory of theta functions. Using these identities we introduce common generalizations of Hirschhorn-Garvan-Borwein cubic theta functions, and also re-derive the quintuple product identity, one of Ramanujan's identities, Winquist's identity and many other interesting identities. 展开更多
关键词 Jacobi theta function quintuple product identity Winquist's identity ramanujan identity
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Ramanujan模3和模5的模恒等式的推广
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作者 闫庆伦 《南京邮电大学学报(自然科学版)》 北大核心 2014年第4期124-126,共3页
在模形式理论中,模3、模5的模等式起着核心的作用。文中主要是利用多节级数方法,借助Jacobi三重积恒等式和筛法原理,得到了Ramanujan模3和模5的模方程的推广形式。
关键词 Jacobi三重积恒等式 ramanujan THETA函数 模函数方程
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一类超几何函数比的几个不等式
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作者 王根娣 《杭州师范学院学报(自然科学版)》 CAS 2005年第2期87-90,共4页
获得了由(4)式定义的出现于数论Ramanujan广义模方程中的这类超几何函数比所满足的一些不等式,改进了广义Grotzsch环函数原有的一些界,所得结果可应用于Ramanujan广义模方程的研究.
关键词 超几何函数 不等式 ramanujan Groetzsch环 模方程 广义 数论
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