Issue 2, Volume 26, August 2001

2001 · Open access

1. LINEAR EQUATIONS WITH GENERALIZED RIGHT INVERTIBLE OPERATORS NGUYEN VAN MAU, PHAM THI BACH NGOC PDF

   ▸ Abstract

   Let $X$ be a linear space over a field $\mathcal K$ of scalars and let $R_1(X)$ be the set of all generalized right invertible operators in $L(X)$. Consider the general linear equation with generalized right invertible operator $V$ of the form $$\sum\limits_{m=0}^M\sum\limits_{n=0}^NV^mA_{mn}V^nx=y, \quad y\in X,$$ where $A_{mn}\in L_0(X)$, $A_{MN} = I$, $A_{mn}X_{M+N−n}\subset X_m$, $X_j := \mathrm{dom}V^j$. Similar equations with right invertible operators were studied by Przeworska-Rolewicz, and others (see [1], [2], [3]). In [4], N. V. Mau and N. M. Tuan constructed the generalized right invertible operators. In this paper, we present some new properties of generalized right invertible operators and then apply them to obtain all solutions of the general linear equations for the genezalized right invertible operator $V$ with non-commutative cofficients.

   125-135
2. SOME CONDITIONS FOR NONEMPTINESS OF $\gamma$-SUBDIFFERENTIALS OF $\gamma$-CONVEX FUNCTIONS NGUYEN NGOC HAI PDF

   ▸ Abstract

   $\gamma$-subdifferential is a concept which can be used for global optimization. If $x_*$ is a global minimizer of an arbitrary function f$$ then $0\in \partial_{\gamma}f(x_*)$, where $\partial_{\gamma}f(x_*)$ is the $\gamma$-subdifferential of $f$ at $x_*$. In particular, $\partial_{\gamma}f(x_*)\not = \emptyset$ at a global minimizer $x_*$. In this paper we investigate the nonemptiness and the monotonicity of $\gamma$-subdifferentials of $\gamma$-convex functions. Some sufficient conditions are stated for the nonemptiness of the $\gamma$-subdifferential of a symmetrically $\gamma$-convex function at a point. It is proved that for a symmetrically $\gamma$-convex function, the G\^{a}teaux derivative (when it exists) at a point belongs to the $\gamma$-subdifferential at that point. A relation between the $\gamma$-subdifferential and the Clarke generalized gradient of a symmetrically $\gamma$-convex function is also presented.

   137-145
3. EQUATIONS STOCHASTIQUES D’UN GAZ VISQUEUX ISOTHERME DANS UN DOMAINE MONODIMENSIONNEL INFINI HISAO FUJITA YASHIMA PDF

   ▸ Abstract

   Nous étudions l’équation stochastique pour un gaz visqueux isotherme dans le domaine $] − \infty, +\infty [$ sous une force potentielle telle que la masse totale du gaz en équilibre soit finie. Nous y considérons deux types de perturbation stochastique: celle donnée par rapport aux points matériels représentés par les coordonnées lagrangiennes et celle donnée par rapport aux positions représentées par les coordonnées eulériennes. L' existence et l’unicité de la solution globale sont démontrées pour tous les deux cas, en s’appuyant sur des estimations a priori dans les coordonnées lagrangiennes ainsi que celles dans les coordonnées eulériennes.

   147-168
4. GENERALIZED QUASICONVEXITY VIA PROPERLY CHARACTERISTIC FUNCTIONS ASSOCIATED TO BINARY RELATIONS NICOLAE POPOVICI PDF

   ▸ Abstract

   As shown by Dinh The Luc in his well-known monograph on vector optimization, the cone-quasiconvex vector-valued functions can be characterized in terms of scalar quasiconvexity by means of the smallest strictly monotonic functions. The aim of this paper is to show that similar characterizations can be given for the general class of $(\Gamma,\Omega)$-quasiconvex functions, this time by means of the so-called properly characteristic functions associated to the binary relation $\Omega$.

   169-175
5. ON MARTINGALES IN THE LIMIT AND CONVERGENCE OF THEIR SUBSEQUENCES DINH QUANG LUU, NGUYEN THANH BINH PDF

   ▸ Abstract

   Martingales in the limit and mils were first introduced by Mucci (1976) and Talagrand (1985), respectively. They proved that every $L^1$-bounded mil converges a.s. Recently, Luu (1999) has extended this result to sequential mils. In this note we consider sequences of random variables which are not necessarily integrable. By using a stopping time method we shall give a convergence result for their subsequences.

   177-185
6. ON A GENERALIZED COX-ROSS-RUBINSTEIN OPTION MARKET MODEL NGUYEN VAN HUU, TRAN TRONG NGUYEN PDF

   ▸ Abstract

   This paper considers a generalization of the Cox-Ross-Rubinstein model for an option market. Some limit theorems for the stock price process and their application to approximately determining the rational price and hedging strategies of standard European option are established.

   187-204
7. VECTOR RANDOM STABLE MEASURES AND RANDOM INTEGRALS DANG HUNG THANG PDF

   ▸ Abstract

   In this paper, the definition and basic properties of Banach space-valued symmetric independently scattered stable measures including random Gaussian measures are presented. Random integrals of real-valued deterministic functions with respect to these random measures are also investigated.

   205-218
8. ON SOME HEREDITARY PROPERTIES BETWEEN $I$ AND $\mathrm{IN}(I)$ HENRIK BRESINSKY, LÊ TUÂN HOA PDF

   219-230
9. SOME REMARKS ON FIXED POINTS DO HONG TAN, HA DUC VUONG PDF

   ▸ Abstract

   In this note we establish two independent results on fixed points. The first one is about the continuity of fixed points of limit-compact mappings introduced by Sadovskii in [7]. This result partially generalizes a result of Tan in [10] for collectively condensing mappings. The second result is a new fixed point theorem for the sum of a generalized contraction and a compact mapping, which improves a well-known result of Krasnoselskii [4]. A probabilistic version of this result is also presented here.

   231-239
10. CHU SPACES AND CONDITIONAL PROBABILITY NGUYEN NHUY, NGUYEN VAN QUANG PDF

    ▸ Abstract

    Let $\widetilde{\Omega}=(\Omega, P, \mathcal A)$ and $\widetilde{\Sigma}=(\Sigma, Q, \mathcal B)$ be probability measure spaces, and let $\phi:\Omega\to\Sigma$ and $\psi:\Sigma\to\Omega$ be measurability preserving maps. The maps $\phi$ and $\psi$ induce $\phi^{-1}:\mathcal B\to\mathcal A$ and $\psi^{-1}:\mathcal A\to\mathcal B$. By $(\mathcal A,\mathcal A,f)$ we denote the Chu space associated with the probability measure space $(\Omega,P,\mathcal A)$. Our main results are:

    Theorem 1. Let $P(\widetilde{\Omega})=(\mathcal A, \mathcal A, f)$ and $P(\widetilde{\Sigma})=(\mathcal B,\mathcal B, g)$ be Chu spaces associated with $\widetilde{\Omega}$ and $\widetilde{\Sigma}$, respectively. If $\Phi=(\psi^{-1}, \phi^{-1}):P(\widetilde{\Omega})\to P(\widetilde{\Sigma})$ is a Chu morphism, then both $\phi$ and $\psi$ are measure preserving.

    Theorem 2. The pair $(\phi, \psi)$ is mutually measure preserving if and only if $\Phi=(\psi^{-1}, \phi^{-1}):(\mathcal A,\mathcal A, f)\to (\mathcal B, \mathcal B, g)$ is a Chu morphism.

    241-248