Topological KKM theorems and generalized vector equilibria on G-convex spaces with applications

In the present paper, slightly modifying the topological KKM Theorem of Park and Kim (1996), we obtain a new existence theorem for generalized vector equilibrium problems related to an admissible multifunction. We work here under the general framework of G-convex space which does not have any linear structure. Also, we give applications to greatest element, fixed point and vector saddle point problems. The results presented in this paper extend and unify many results in the literature by relaxing the compactness, the closedness and the convexity conditions.

     

On Subspaces and Quotients of Banach Spaces C(K, X)

 We study the relation of to the subspaces and quotients of Banach spaces of continuous vector-valued functions , where K is an arbitrary dispersed compact set. More precisely, we prove that every infinite dimensional closed subspace of totally incomparable to X contains a copy of complemented in . This is a natural continuation of results of Cembranos-Freniche and Lotz-Peck-Porta. We also improve our result when K is homeomorphic to an interval of ordinals. Next we show that complemented subspaces (resp., quotients) of which contain no copy of are isomorphic to complemented subspaces (resp., quotients) of some finite sum of X. As a consequence, we prove that every infinite dimensional quotient of which is quotient incomparable to X, contains a complemented copy of . Finally we present some more geometric properties of spaces. Received 8 November 2000; in revised form 7 December 2001     

An adaptive finite volume box scheme for solving a class of nonlinear parabolic equations

In this work we are interested in a fully nonconforming (Crouzeix–Raviart) finite element scheme for addressing a nonlinear parabolic equation. In addition to showing the equivalence with a finite volume box scheme we derive a posteriori error indicators in time and space, uncoupled as much as possible.     

Bifunctional-elementary relation algebras

A relation algebra is bifunctional-elementary if it is atomic and for any atom a, the element a;1;a is the join of at most two atoms, and one of these atoms is bifunctional (an element x is bifunctional if ’). We show that bifunctional-elementary relation algebras are representable. Our proof combines the representation theorems for: pair-dense relation algebras given by R. Maddux; relation algebras generated by equivalence elements provided corresponding relativizations are representable by S. Givant; and strong-elementary relation algebras dealt with in our earlier work. It turns out that atomic pair-dense relation algebras are bifunctional elementary, showing that our theorem generalizes the representation theorem of atomic pair-dense relation algebras. The problem is still open whether the related classes of rather elementary, functional-elementary, and strong functional-elementary relation algebras are representable. Received July 15, 2007; accepted in final form March 17, 2008.     

Convergent Algorithm Based on Progressive Regularization for Solving Pseudomonotone Variational Inequalities

In this paper, we extend the Moreau-Yosida regularization of monotone variational inequalities to the case of weakly monotone and pseudomonotone operators. With these properties, the regularized operator satisfies the pseudo-Dunn property with respect to any solution of the variational inequality problem. As a consequence, the regularized version of the auxiliary problem algorithm converges. In this case, when the operator involved in the variational inequality problem is Lipschitz continuous (a property stronger than weak monotonicity) and pseudomonotone, we prove the convergence of the progressive regularization introduced in Refs. 1, 2.     

Factorizations of complete multipartite graphs into generalized cubes

For a positive integer d, the usual d‐dimensional cube Q_d is defined to be the graph (K₂)^d, the Cartesian product of d copies of K₂. We define the generalized cube Q(K_k, d) to be the graph (K_k)^d for positive integers d and k. We investigate the decomposition of the complete multipartite graph K into factors that are vertex‐disjoint unions of generalized cubes Q(K_k, d_i), where k is a power of a prime, n and j are positive integers with j ≤ n, and the d_i may be different in different factors. We also use these results to partially settle a problem of Kotzig on Q_d‐factorizations of K_n. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 144–150, 2000     

Achieving the Reforms Vision: The Effectiveness of a Specialists‐Led Elementary Science Program

This report describes an evaluation project that aimed to assess the potential of two elementary science specialists, as compared to elementary classroom teachers, to realize the reforms vision for science instruction in elementary classrooms. Participant science specialist background, views of elementary science teaching, and planning and assessment practices were compared to those of regular elementary classroom teachers in the specialist district, as well as in a comparable district not employing specialists. Specialists' views and practices were better aligned with those envisioned by current national reform documents in science education. Despite the constraints imposed by the nature of a program evaluation, the present report provides evidence to suggest that students taught by the science specialists (a) were engaged in open‐ended, inquiry‐oriented, science‐based activities of the kind often advocated, but mostly absent, in elementary school, and (b) demonstrated problem solving and higher order and critical thinking skills. This report is the first to provide empirical support for the advocated “effectiveness” of elementary science specialists in achieving the visions espoused by current reform efforts.