Issue 1, Volume 29, April 2004

2004 · Open access

1. LYAPUNOV REGULARITY OF LINEAR DIFFERENTIAL ALGEBRAIC EQUATIONS OF INDEX 1 NGUYEN DINH CONG, HOANG NAM PDF

   ▸ Abstract

   In this paper, we introduce a concept of Lyapunov regularity of linear differential algebraic equations (DAEs) based on the notion of Lyapunov exponents of DAEs. It was proved that under certain conditions a DAE of index 1 is Lyapunov regular if and only if the corresponding ordinary differential equation is Lyapunov regular.

   1-21
2. CONNECTIONS BETWEEN IMPLICIT DIFFERENCE EQUATIONS AND DIFFERENTIAL-ALGEBRAIC EQUATIONS PHAM KY ANH, NGUYEN HUU DU, LE CONG LOI PDF

   ▸ Abstract

   Recently, a notion of index-1 linear implicit difference equations (LIDEs) has been introduced and the solvability of initial value problems (IVPs) as well as multipoint boundary-value problems (MPBVPs) for index-1 LIDEs has been studied. In this note, we show that the explicit Euler method (EEM) applied to linear transferable differential-algebraic equations (DAEs) leads to index-1 LIDEs. Besides, we discuss the convergence of solutions of IVPs (MPBVPs) for index-1 LIDEs to the solutions of the corresponding problems for transferable DAEs.

   23-39
3. PROBABILITY CAPACITIES IN $\mathbb R^d$ AND THE CHOQUET INTEGRAL FOR CAPACITIES NGUYEN NHUY, LE XUAN SON PDF

   ▸ Abstract

   A notion of capacity in $\mathbb R^d$ is introduced and a concept of Choquet integral of measurable functions $f : \mathbb R^d \to \mathbb R$ is defined and investigated.

   41-56
4. $nX$-COMPLEMENTARY GENERATIONS OF THE SPORADIC GROUP $Co_1$ M. R. DARAFSHEH, A. R. ASHRAFI, G. A. MOGHANI PDF

   ▸ Abstract

   A finite group $G$ with conjugacy class nX is said to be $nX$-complementary generated if, for any arbitrary $x\in G-\{1\}$, there is an element $y\in nX$ such that $G = \langle x, y\rangle$. In this paper we prove that the Conway's group $Co_1$ is $nX$-complementary generated for all $n\in \Pi_e(Co_1)$. Here $\Pi_e(Co_1)$ denotes the set of all element orders of the group $Co_1$.

   57-75
5. A CONVERSE OF THE JENSEN INEQUALITY FOR CONVEX MAPPINGS OF SEVERAL VARIABLES AND APPLICATIONS S. S. DRAGOMIR PDF

   ▸ Abstract

   In this paper we point out a converse result of the celebrated Jensen inequality for differentiable convex mappings of several variables and apply it to counterpart well-known analytic inequalities. Applications to Shannon's and Rényi's entropy mappings are also given.

   77-88
6. ON SOME BADLY-SOLVED PROBLEMS WITH INVEXITY HOANG XUAN PHU PDF

   ▸ Abstract

   Using invexity, D. V. Luu and N. X. Ha investigated in [4], [5], and [8] the problem
   $$\begin{array}{l l} \text{minimize }& f(x)=\inf\limits_{\alpha\in A}f_{\alpha}(x),\\ \text{subject to }& g(x)=\inf\limits_{\beta\in B}g_{\beta}(x)\leq 0,\quad x\in C.\end{array}$$ We show that their results concerned with this problem are wrong or too weak or redundant, while a lot of too strong and unnecessary assumptions are used. For correction, a necessary condition and a sufficient condition for local minima are given in this paper. Some comments and sufficient conditions for local minima of the other problems considered in [5] and [8] are also presented.

   89-106
7. ON A NECESSARY OPTIMALITY CONDITION WITH INVEXITY HOANG XUAN PHU PDF

   ▸ Abstract

   N. X. Ha and D. V. Luu presented in Bull. Austral. Math. Soc. 68 (2003) some su±cient conditions for invexity and a necessary optimality condition, in which invexity is used for ensuring the positivity of the Lagrange multiplier $\lambda$ corresponding to the objective function. We show that the necessary optimality condition is wrong and invexity is misplaced there. Although $\lambda > 0$ follows immediately from the so-called Jourani constraint qualification and no additional explanation is needed, they spent almost three pages to prove the sufficiency of this condition for an invexity property, and then almost one page to derive $\lambda > 0$ from the invexity property and an additional condition of Slater type, while the latter ones are strong enough to yield the Jourani constraint qualification. Moreover, the other two main sufficient conditions for invexity in the mentioned paper yield the Jourani constraint qualification.

   141-148