周期函数的和、差、积、商的周期性

§1.引言两个周期函数的和、差、积、商是否仍为周期函数?这是一个值得讨论的问题。对于两个具有同一周期t的函数f(x)和g(x),显然它们的和、差、积、商均为以t为周期的函数。这个条件等价于函数f(x)有一周期t_1与g(x)的某一周期t_2是可公度的,即t_1/t_2为有理数。事实上,若f(x)与g(x)有同一周期t,则t/t=1是有理数;反之,若f(x)的周期t_1与g(x)的周期t_2有t_1/t_2=m/n(m和n均为整数),则t=nt_1=mt_2便是它们的公共周期。自然要问:要使两个周期函数的和(或差、积、商)仍为周期函数,是否它们必须有可公度之周期? 关于连续函数,书[1]中指出了(但未证明)下面的结论: 连续周期函数f(x)和g(x)的和仍为周期函数的     

关于函数周期性的一些讨论

设Q表示有理数集 ,R表示实数集 ,C表示复数集 .函数f(t) :R →C称为是T周期的 ,是指存在常数T>0 ,使f(t +T) =f(t) , t∈R .最小的周期T >0 ,称为f(t)的基本周期 .众所周知 ,Dirichlet函数 (它不连续 )和常函数是周期函数 ,但它们没有基本周期 .那么会问什么样的函数会有基本周期呢 ?我们有命题 1　如果f(t)是非常数的连续周期函数 ,则它有基本周期 .证明　用反证法 .若f(t)不存在最小的正周期 ,则存在单调减少的正序列 {kn},kn → 0 ,满足f(t+kn) =f(t) ,t∈R .于是f(t+mkn) =f(t) , t∈R和对任何整数m ,n .设t0 是任何实数 .对任何n…     

利用Dirichlet函数制作的几种反例

Dirichlet函数是如下定义的函数: D(x)=1 x为有理数 0 x为无理数 在此我们利用Dirichlet函数构造的反例驳几个错误的命题.1 驳“在任何周期函数必有最小周期” 反例:f(x)=C·D(x),其中C是任意非零常数. 显然,任何大于0的有理数都是该函数的周     

整函数与其二次迭代周期性之间关系的Gross问题 献给杨乐教授80华诞

1972年, Gross提出了下述有趣的问题:如果f(z)是非常数整函数,而且f(f(z))是周期函数,那么f(z)一定也是周期函数吗?关于这一问题,本文证明了如下结果:如果f(z)是非常数整函数,f(f(z))是以τ为周期的周期函数,且f′(z+nτ)(n=0, 1, 2,...)在复平面C上不局部一致收敛于0,则f(z)也是周期函数.     

师生共话无理数

学生　有理数和无理数有什么区别 ?老师　主要区别有两点 :1.把有理数和无理数都写成小数形式时 ,有理数能写成有限小数或循环小数 ,比如 4 =4 .0 ;45=0 .8;110 =0 .1;13=0 .333…而无理数只能写成无限不循环小数 ,比如 2 =1.4 14 2… ,根据这一点 ,人们把无理数定义为无限不循环小数 .2 .所有的有理数都可以写成两个整数之比 ,而无理数却不能写成两个整数之比 .根据这一点 ,有人建议给无理数摘掉“无理”的帽子 ,把有理数改叫“比数” ,把无理数改叫“非比数” .本来嘛 ,无理数并不是不讲道理 ,只是人们最初对它太不理解罢了 .学生　无限小…     

关于周期函数的再思考

本文一方面基于周期集的性质考虑了周期函数的最小正周期问题,另一方面比较自然地将周期函数推广到概周期函数,并举例说明了两个周期函数的和函数一定为概周期函数     

具有HollingIV类功能性反应时滞扩散捕食模型的多个周期解

考虑一类具有HollingIV类功能性反应时滞扩散捕食模型.该模型的系数为周期函数,这和环境的周期变化相一致.作者应用重合度定理,建立了该模型具有至少两个正周期解的充分条件.     

分数阶微分方程不存在周期解:综述（英文）

周期解的存在性是分数阶动力系统领域中的一个热门话题.分数阶导数不同于经典的整数阶倒数,它们之间的基本区别在于是否存在遗传特性,即分数阶导数是具有弱奇异核的非局部算子,而整数阶导数则为局部算子.简要概述了带分数阶导数(如Grunwald-Letnikov导数、Reimann-Liouville导数、Caputo导数)的分数阶微分方程是否存在周期性解的近期结果.周期函数的经典整数阶导数与该周期函数具有相同周期这一结果已被证明.然而,周期函数的分数阶导数却不是具有相同的周期的周期函数.与此同时,本文也对非常数周期函数分数阶动力系统的周期不存在性以及分数阶动力系统的长时解的周期存在性进行了回顾.     

原函数与导函数周期性和奇偶性联系的探究

笔者在文[1]中对原函数与导函数对称性联系进行了探究,本文就原函数与导函数周期性和奇偶性联系进行探究,得到了几个漂亮的结论.定理1(1)若可导函数f(x)是以T为周期的周期函数,则其导函数f′(x)也是以T为周期的周期函数;     

函数的周期性和对称性

1　周期函数问题设函数 f(x)的定义域为D ,若存在非零常数T ,使得对每个x∈D ,都有 f(x +T) =f(x -T) =f(x)成立 ,则称 f(x)为周期函数 ,T为 f(x)的一个周期 .如果 f(x)的所有正周数中存在最小值T0 ,则称T0 为周期函数 f(x)的最小正周期 .一般说函数的周期通常是指最小正周期 .例 1　判定函数 f(x) =x - [x],x∈R(其中[x]表示不超过x的最大整数 )的周期性并作出其图象 .解　如图 1,我们作出 f(x)的图象 .图 1　例 1图由 f(x)的图像可知 ,当x∈R时 ,f(x) =x -[x]是周期函数 ,且T =1是它的最小正周期 .事实上 ,对x∈R ,有f(x + 1) =x + 1…     

On Periodic Solutions of the Periodic

By treating the periodic Riccati equation ${\rm\dot{z}=a(t)z^2+b(t)z+c(t)}$ as a dynamical system on the sphere S, the number and stability of its periodic solutions are determined. Using properties of Moebius transformations, an exact algebraic relation is obtained between any periodic solution and any complex-valued periodic solution. This leads to a new method for constructing the periodic solutions.     

Stable Periodic Solutions of Periodic Systems of Differential Equations

An infinitely differentiable periodic two-dimensional system of differential equations is considered. It is assumed that there is a hyperbolic periodic solution and there exists a homoclinic solution to the periodic solution. It is shown that, for a certain type of tangency of the stable manifold and unstable manifold, any neighborhood of the nontransversal homoclinic solution contains a countable set of stable periodic solutions such that their characteristic exponents are separated from zero.     

Locally Periodic Versus Globally Periodic Infinite Words

We call a one-way infinite word w over a finite alphabet (ρ,l)-repetitive if all long enough prefixes of w contain as a suffix a ρth power (or more generally a repetition of order ρ) of a word of length at most l. We show that each (2,4)-repetitive word is ultimately periodic, as well as that there exist continuum many, and hence also nonultimately periodic, (2,5)-repetitive words. Further, we characterize nonultimately periodic (2,5)-repetitive words both structurally and algebraically.     

周期系数Ricatti方程之周期解

本文利用Brouwer不动定理，研究了Ricatti方程的周期解的存在性。     

The Existence of Almost Periodic Solution and Periodic Solutions

In this paper, it is obtained that a periodic system has an almost periodic solution if it has a solution x=\phi(t) uniformly stable with respect to \Omega_\phi ,and has a periodic solution if x=\phi(t) is weakly uniformly asymptotically stable with respect to \Omega_\phi. Meanwhile, it is also obtained that a uniformly almost periodic system has an almost periodic solution if it has a solution x=\phi(t) uniformly asymptotically stable with respect to A_\phi^j.     

周期函数和非平稳周期函数的小波变换

探讨了三角函数、周期函数以及一类非平稳周期函数小波变换的一些性质,发现周期函数的小波能谱的峰高和峰宽均正比于信号的周期.提出了一个新的只利用与信号周期有关的一个尺度小波变换系数的重构公式,它可准确地重构三角函数,对一般周期函数的重构结果优于其Fourier级数中的任何一项,对一类均值和振幅变化的非平稳周期函数的重构结果与信号非常吻合.     

Periodic regeneration

We consider stochastic processes Z = (Z_t)_[0,∞), on a general state space, having a certain periodic regeneration property: there is an increasing sequence of random times (S_n)^∞₀ such that the post-S_n process is conditionally independent of S₀,…,S_n given (S_n mod 1) and the conditional distribution does not depend on n. Our basic condition is that the distributions $\text{P}{}_{\mathrm{s}}\text{(A) =}\text{P}\text{(S}{}_{\mathrm{n+1}}\text{? S}{}_{\mathrm{n}}\text{∈ A|S}{}_{\mathrm{n}}\text{= s), s ∈[0, 1)}$, have a common component that is absolutely continuous w.r.t. Lebesgue measure. Then Z has the following time-homogeneous regeneration property: there exists a discrete aperiodic renewal process T = (T_n)^∞₀ such that the post-T_n process is independent of T₀,…,T_n and its distribution does not depend on n; this yields weak ergodicity. Further, the Markov chain (S_nmod 1)^∞₀ has an invariant distribution π_[0,1) and it holds that T_n+1 ? T_n has finite first moment if and only if m = ∫ m(P_s)π_[0,1)(ds) < ∞ where m(P_s) is the first moment of P_s; this yields periodic ergodicity. Also, some distributional properties of T₀ and T_n+1 ? T_n are established leading to improved ergodic regults. Finally, a uniform key periodic renewal theorem is derived.