PROPERTIES OF SOME SETS OF SEQUENCES DEFINED BY A MODULUS FUNCTION

In this article, the author introduces the generalized difference paranormed sequence spaces c （△v^m, f, p, q, s）, c0 （△v^m, f, p, q, s）, and l∞ （△v^m, f, p, q, s） defined over a seminormed sequence space （X, q）. The author also studies their properties like completeness, solidity, symmetricitv, etc.     

数学娱乐圈

黄鹤楼对联之数学版　　六千年数学 ,旷世传来 ,想初等代数 ,解析几何 ,集合理论 ,积分方程 ,何其博大精深 ,极目古今悠 ,莫惊疑数海茫茫 ,形山隐隐 ;一万里长江 ,几人淘尽 ,仰割圆刘徽 ,算率冲之 ,流数牛顿 ,过桥欧拉 ,真是超凡入圣 ,放怀天地阔 ,须礼赞勋名赫赫 ,伟业煌煌 .陈　武提供 (湖南省祁东文武学校　 4 2 16 0 0 )念奴娇·数学怀古大江东去 ,浪淘尽 ,千古风流人物 .文明天空 ,吾景仰 ,历代数学大师 .刘徽冲之 ,牛顿欧拉 ,还有笛沙格 .英贤辈出 ,世界多少豪杰 .遥观数学之树 ,枝繁叶茂 ,硕果辉五色 .代数精深 ,几何奇 ,分析功效卓…     

多位数乘法巧解法

多位数相乘,按常规,无论是留头乘,破头乘,中心乘,空盘乘,均离不掉逐位乘,操作程序多,如三位数相乘须九次操作,四位数相乘就得十六次计算,相当繁琐,费工,费劲,费心,一旦某一计算程序之间发生误差会使全题错误,前功尽弃,而用巧解法,不仅提高计算速度,而且提高计算的准确率,因为巧解法     

Pascal定理的复数证明

Pascal定理　设ABCDEF为圆内接六边形,相对的边AB和DE,BC和EF,CD和FA(或延长线)的交点分别为P,Q,R,则此三点共线(见图1,图2).Pascal定理是平面几何中的著名定理,有多种证法,可见[1].本文,我们在平面上建立复坐标,计算出P,Q,R三点的坐标,用(zQ-zP)/(zR-zP)是实数这一事实来说明P,Q,R三点共线.这种做法的难点是得到(zQ-zP)的因子分解式(见下文(1)式),优点是思路简单,除了计算外,无须添助任何辅助线.证明　在复平面上,设圆的中心为原点,半径为1.A,B,C,D,E,F,P,Q,R九点的复数分别为t1,t2,t3,t4,t5,t6,zP,zQ,zR.先给出直线AB…     

例说数学思维的困境及其摆脱

<正>思维受挫就是在解题时,按一般的思路介入后,或者浅尝辄止,或者眼花缭乱,或者误入歧途,一错再错,或者前途渺茫,无法推进,就算使尽了浑身解数,左冲右突,终觉"山重水复疑无路".笔者以为,要修复受挫的思维,只要仔细读题审题,寻找蛛丝马迹,类比联想相关知识点,分析可能的思想方法,探索恰当的思维路径,就会"峰回路转,柳暗花明".下面举例说明,供同学们参考.     

双非线性抛物方程初始迹与Harnack不等式

在最优的初始值条件下考虑如下拟线性抛物方程的柯西问题u_t-diva(x,t,u,Du)=b(x,t,u,Du),(x,t)属于S_T=R~N×(0,T).令a(x,t,u,Du)={a_i(x,t,u,Du)},假设a_i(x,t,u,Du)与b(x,t,u,Du)皆为Caratheodory函数,并且假设它们满足Du的单调性,关于u,|Du|等一定的增长阶条件下,得到了解的比较定理,证明了解的存在性,并得到了相关的Harnack不等式.     

例说数学思维的困境及其摆脱

思维受挫就是在解题时,按一般的思路介入后,或者浅尝辄止,或者眼花缭乱,或者误入歧途,一错再错,或者前途渺茫,无法推进,就算使尽了浑身解数,左冲右突,终觉“山重水复疑无路”.笔者以为,要修复受挫的思维,只要仔细读题审题,寻找蛛丝马迹,类比联想相关知识点,分析可能的思想方法,探索恰当的思维路径,就会“峰回路转,柳暗花明”.下面举例说明,供同学们参考.     

Hardy spaces H~p over non-homogeneous metric measure spaces and their applications

Let(X,d,)be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions.Let ρ∈(1,∞),0p≤1≤q≤∞,p≠q,γ∈[1,∞)and ∈∈(0,∞).In this paper,the authors introduce the atomic Hardy space Hp,q,γ atb,ρ(μ)and the molecular Hardy space Hp,q,γ,mb,ρ∈(μ)via the discrete coefficient K(ρ),p B,S,and prove that the Calder′on-Zygmund operator is bounded from Hp,q,γ,δmb,ρ(μ)(or Hp,q,γatb,ρ(μ))into Lp(μ),and from Hp,q,γ+1atb,ρ(ρ+1)(μ)into H p,q,γ,12(δ-νp+ν)mb,ρ(μ).The boundedness of the generalized fractional integral Tβ(β∈(0,1))from Hp1,q,γ,θmb,ρ(μ)(or Hp1,q,γatb,ρ(μ))into Lp2(μ)with 1/p2=1/p1-β is also established.The authors also introduce theρ-weakly doubling condition,withρ∈(1,∞),of the measure and construct a non-doubling measure satisfying this condition.If isρ-weakly doubling,the authors further introduce the Campanato space Eα,qρ,η,γ(μ)and show that Eα,qρ,η,γ(μ)is independent of the choices ofρ,η,γand q;the authors then introduce the atomic Hardy space Hp,q,γatb,ρ(μ)and the molecular Hardy space Hp,q,γ,mb,ρ(μ),which coincide with each other;the authors finally prove that Hp,q,γatb,ρ(μ)is the predual of E1/p-1,1ρ,ρ,1(μ).Moreover,if is doubling,the authors show that Eα,qρ,η,γ(μ)and the Lipschitz space Lipα,q(μ)(q∈[1,∞)),or Hp,q,γatb,ρ(μ)and the atomic Hardy space Hp,q at(μ)(q∈(1,∞])of Coifman and Weiss coincide.Finally,if(X,d,)is an RD-space(reverse doubling space)with(X)=∞,the authors prove that Hp,q,γatb,ρ(μ),Hp,q,γ,mb,ρ(μ)and Hp,q at(μ)coincide for any q∈(1,2].In particular,when(X,d,):=(RD,||,dx)with dx being the D-dimensional Lebesgue measure,the authors show that spaces Hp,q,γatb,ρ(μ),Hp,q,γ,mb,ρ(μ),Hp,q,γatb,ρ(μ)and Hp,q,γ,mb,ρ(μ)all coincide with Hp(RD)for any q∈(1,∞).     

方程的等效

大家知道,形式為 F_1(x,y,…,z)=F_2(x,y,…,z) (1)的等式,其中F_1(x,y,…,z)和F_2(x,y,…,z)是變量x,y,…,z的函數,叫做未知量x,y,…,z的方程。在初等數學中,只考慮特殊形式的方程,其中F_1和F_2是藉助於某種解析表達式,即藉助於公式而給定的,以後我們只限於這種情况。未知量的數值組,能使方程的左邊和右邊相等者,就叫做方程的解。     

THE SUFFICIENT EFFICIENCY CONDITIONS IN SEMIINFINITE MULTIOBJECTIVE FRACTIONAL PROGRAMMING UNDER HIGHER ORDER EXPONENTIAL TYPE HYBRID TYPE INVEXITIES

First,a class of higher order exponential type hybrid(α,β,γ,η,ρ,h(·,·),k(·,·),ω(·,·,·),ω(·,·,·),θ)-invexities is introduced,second,some parametrically sufficient efficiency conditions based on the higher order exponential type hybrid invexities are established,and finally some parametrically sufficient efficiency results under the higher order exponential type hybrid(α,β,γ,η,ρ,h(·,·),k(·,·),ω(·,·,·),ω(·,·,·),θ)-invexities are investigated to the context of solving semiinfinite multiobjective fractional programming problems.The notions of the higher order exponential type hybrid(α,β,γ,η,ρ,h(·,·),k(·,·),ω(·,·,·),ω(·,·,·),θ)-invexities encompass most of the generalized invexities in the literature.To the best of our knowledge,the results on semiinfinite multiobjective fractional programming problems established in this communication are new and application-oriented toward multitime multiobjectve problems as well as multiobjective control problems.     

Large deviation of some infinite-dimensional markov processes ∗

In this paper it is shown that a large deviation principle in the sense of Donsker and Varadhan is satisfied by the Ornstein-Uhlenbeck process on C[0,1].     

An ornstein-uhlenbeck model for random oscillations

We consider the effect of a random "noise" on an n-dimensional simple harmonic oscillator with time-dependent damping. The noise in the system is modelled by incorporating a Brownian motion term in the equation for the velocity process of the simple harmonic oscillator, giving a stochastic differential equation similar to that of an Ornstein-Uhlenbeck proces. Necessary and sufficient conditions for the convergence of the solution of this SDE to an orbit of simple harmonic motion (satisfying the usual ODE) are then obtained     

两参数Ornstein－Uhlenbeck过程的连续模

对［２］中引入的两参数Ｏｒｎｓｔｅｉｎ－Ｕｈｌｅｎｂｅｃｋ过程Ｘ（ｕ，ｖ），如果我们定义增量△ｕ，ｓ（ｖ）＝｜Ｘ（ｕ＋ｘ，ｖ）－Ｘ（ｕ，ｖ）｜，则我们可以找到正则化因子βｖ，使得若，则若，则这里     

矩阵值Ornstein-Uhlenbeck过程的大偏差上界

本文研究了矩阵值Ornstein-Uhlenbeck过程的大偏差问题.通过构造指数鞅,得到了矩阵值Ornstein-Uhlenbeck过程的经验谱过程的大偏差上界,推广了厄米特布朗运动相应的结果.     

ON THE FRACTAL NATURE OF INCREMENTS OF THE INFINITE SERIES OF OU PROCESSES RELATED TO THE CHUNG LIL

The author proves that the set of points where the Chung type LIL fails for the path of the infinite series of independent Ornstein-Uhlenbeck processes is a random fraetal, and evaluates its Hausdorff dimension.     

QUASI-STATIONARY DISTRIBUTIONS FOR THE RADIAL ORNSTEIN-UHLENBECK PROCESSES

The purpose of this article is to obtain the quasi-stationary distributions of the δ（δ 〈 2）-dimensional radial Ornstein-Uhlenbeck process with parameter -λ by using the methods of Martinez and San Martin （2001）. It is described that the law of this process conditioned on first hitting 0 is just the probability measure induced by a （4 - δ）- dimensional radial Ornstein-Uhlenbeck process with parameter -λ. Moreover, it is shown that the law of the conditioned process associated with the left eigenfunction of the process conditioned on first hitting 0 is induced by a one-parameter diffusion.     

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For the Ornstein-Uhlenbeck process, we prove that if the classical Kalman's condition doesn't hold, then it can be divided into a deterministic process plus a stochastic process after a linear and invertible transformation.     

Between local and global logarithmic averages

We obtain an approximation for the logarithmic averages of I{k^1/2a(k) S(k) k^1/2b(k)}, where a(k) → 0, b(k) → 0 (k → ∞) and S(k) is partial sum of independent, identically distributed random variables.     

MODERATE DEVIATIONS FOR PARAMETER ESTIMATORS IN FRACTIONAL ORNSTEIN-UHLENBECK PROCESS

We study moderate deviations for estimators of the drift parameter of the fractional Ornstein-Uhlenbeck process. Two moderate deviation principles are obtained.     

ON LOCAL CONTINUITY MODULI FOR GAUSSIAN PROCESSES

ＬＵＣＨＵＡＮＲＯＮＧ（陆传荣）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨａｎｇｚｈｏｕＵｎｉｖｅｒｓｉｔｙ，Ｈａｎｇｚｈｏｕ３１００２８，Ｃｈｉｎａ）（ＴｈｉｓｗｏｒｋｉｓｓｕｐｐｏｒｔｅｄｂｙｔｈｅＮａｔｉｏｎａｌＮａｔｕｒａｌＳｃｉｅ...