Generalized (f,U s,u t , λ s t )-structures

iauliai Pedagogical Institute, P. Viinskio str. 25, 5419 iauliai, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 33, No. 2, pp. 135–141, April–June, 1993.     

Exact Distribution of the Product of N Student’s t RVs

Exact expressions are derived for the probability density function, cumulative distribution function and moment properties of the product of N independent Student??s t random variables. A program is provided for computing the associated percentage points. Estimation issues by the methods of moments and maximum likelihood are discussed.     

(s,t,d)-bi-Koszul代数

本文关注具有non-pure分解的1次生成正分次代数,主要讨论作为bi-Koszul代数的推广的一类新的代数(s,t,d)-bi-Koszul代数.用两个具有pure分解的周期代数可以获得(s,t,d)-bi-Koszul代数.本文讨论了(s,t,d)-bi-Koszul代数的Koszul对偶的生成性,在此基础上,提出了强(s,t,d)-bi-Koszul代数的概念并且进一步讨论了它们的同调性质.     

The Existence of a Positive Solution of D α[x(t) − x(0)] = x(t)f(t,x t )

This paper is devoted to the problem of existence of positive solution for a delay differential equation with fractional order. Some sufficient conditions for its existence of positive solution are given. This work was supported in part by the Key Project of the National Nature Science Foundation of China (No. 60534020), the National Nature Science Foundation of China (No. 60474037), and Program for New Century Excellent Talents in University (No. NCET-04-415).     

Transformations z(t) = L(t)y(ϕ(t)) of ordinary differential equations

The paper describes the general form of an ordinary differential equation of an order n + 1 (n ≥ 1) which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form $f\left( {s,w_{00} \upsilon _0 ,...,\sum\limits_{j = 0}^n {w_{nj\upsilon _j } } } \right) = \sum\limits_{j = 0}^n {w_{n + 1j\upsilon j} + w_{n + 1n + 1} f\left( {x,\upsilon ,\upsilon _1 ,...,\upsilon _n } \right),}$ where $w_{n + 10} = h\left( {s,x,x_1 ,u,u_1 ,...,u_n } \right),w_{n + 11} = g\left( {s,x,x_1 ,...,x_n ,u,u_1 ,...,u_n } \right){\text{ and }}w_{ij} = a_{ij} \left( {x_i ,...,x_{i - j + 1} ,u,u_1 ,...,u_{i - j} } \right)$ for the given functions a _ij is solved on $\mathbb{R},u \ne {\text{0}}$ .     

On the solution set of an equation of the type f(t, Φ(u(t)) = 0

We investigate the solution set Γ of an equation of the type f(t, Φ(u(t)) = 0, where Φ is a linear homeomorphism from a topological vector space X onto L ¹(T) and f: T×R → R is a Carathéodory function. More precisely, we characterize the property of Γ of intersecting each closed hyperplane of X.     

方程(t)+p(t)(t)+q(t)x(t)=0的稳定性

本文应用交换技巧以及直接方法研究了二阶线性系统(1)的平凡解的稳定性,得到若干充分条件、必要条件以及充分必要条件,推广了文献[1—8]中的有关结果。     

THE STABILITY OF RDDE (?)(t) p_1(t)x(t) q_1(t)x(t) p_2(t)x(t-r(t)) q_2(t)x(t-r(t))=0

In this paper, the author considered the stability of zero solution of linear RDDEx(t) p_1(t)x(t) q_1(t)x(t) p_2(t)x(t-r(t)) q_2(t)x(t-r(t))=0,(1)x(t) p_1(t)x(t) q_1(t)x(t) q_2(t)x(t-r(t))=0 (2)using Liapunov-Razumikhin functional and transformations and obtained some sufficient condi-tions for the stability of Eqs.(1) and (2). These results are suitable both for bounded p_i(t), q_i(t)and r(t).i =1, 2.     

Variétés linéaires par morceaux et variétés combinatoires

Résumé On conna?t le r?le important joué par la notion de variété combinatoire dans dans les développements récents de la topologie différentielle [6], [8]. Une structure combinatoire sur une variété topologiqueU s'identifie à une classe de triangulations isomorphes deU (i.e. deux quelconques d'entre elles sont obtenues à partir de complexes simpliciaux qui admettent des subdivisions isomorphes). Il convient de considerer en outre une relation d'équivalence plus forte définissant des structures de “variétés linéaires par morceaux”, une structure combinatoire correspondant à une classe de structures linéaires par morceaux “homéomorphes”. Nous montrons que cette notion est équivalente, sur une variété paracompacte, à celle de structure d'espace localement isomorphe à Rⁿ munie du pseudogroupe des homéomorphismes locaux linéaires par morceaux. On montre aussi que les variétés linéaires par morceaux forment une catégorie qui admet un foncteur covariant associant à chaque objetU de la catégorie un espace fibré C(U) de “c?nes tangents”. Au Professeur Enrico Bompiani, pour son Jubilé scientifique, en témoignage de mon admiration. Ce travail développe des exposés faits au Séminaire mathématique de l'Ecole de Physique et Mathématique de l'Université Centrale du Vénézuéla (Caracas) en mai 1961. (Voir [4b]).     

On transformations z(t) = y(ϕ(t)) of ordinary differential equations

The paper describes the general form of an ordinary differential equation of the order n + 1 (n 1) which allows a nontrivial global transformation consisting of the change of the independent variable. A result given by J. Aczél is generalized. A functional equation of the form where are given functions, is solved on .     

Properties of (θ,s)-continuous functions

Joseph and Kwack [Proc. Amer. Math. Soc. 80 (1980) 341–348] introduced the notion of (θ,s)-continuous functions in order to investigate S-closed spaces due to Thompson [Proc. Amer. Math. Soc. 60 (1976) 335–338]. In this paper, further properties of (θ,s)-continuous functions are obtained and relationships between (θ,s)-continuity, contra-continuity and regular set-connectedness defined by Dontchev et al. [Internat. J. Math. Math. Sci. 19 (1996) 303–310 and elsewhere] are investigated.     

Sur l'équation fonctionnelle ∫T(ψ(t+s)−ψ(s))3ds=sint

Clearly the equation given in the title has no solution in $C^{\alpha }$, $\alpha >\frac{1}{3}$. We give an explicit solution in $C^{1/3}$. The motivation for this question, and other questions of the same type, comes from the forthcoming article by H. Brezis. To cite this article: J.-P. Kahane, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

二阶线性方程(t)+p(t)(t)+q(t)x(t)=0一致稳定的一个充分条件

研究了二阶非定常线性微分方程的稳定性,利用分析技巧、Bellman不等式及稳定性有关理论得到了该方程一致稳定的一个充分性条件,获得了新的结论.     

The region of attraction of the zero solution of y′(t)=−(y(t−1))2m+1

Estimates for the size of the region of attraction of the zero solution of y(t)=–(y(t–1))^2m+1 are given.     

Variétés instables d'ensembles hyperboliques

Here the unstable manifolds of hyperbolic systems admitting a one-rectangle Markov partition are characterized up to homeomorphism and up to topological conjugacy (of the underlying dynamics) in a very easy-to-compute way. The geometrical behaviour of the unstable manifolds is described with the help of an algebraic setting.