Partial regularity of weak solutions of the liquid crystal equilibrium system

For a parameter, we consider the modified relaxed energy of the liquid crystal system. Each minimizer of the modified relaxed energy is a weak solution to the liquid crystal equilibrium system. We prove the partial regularity of minimizers of the modified relaxed energy. We also prove the existence of infinitely many weak solutions for the special boundary value x.

Existence of solutions to some equilibrium problems

The concept of a monotone family of functions, which need not be countable, and the solution of an equilibrium problem associated with the family are introduced. A fixed-point theorem is applied to prove the existence of solutions to the problem.

Regularity and relaxed problems of minimizing biharmonic maps into spheres

For n >= 5 and k >= 4, we show that any minimizing biharmonic map from Omega subset of R-n to S-k is smooth off a closed set whose Hausdorff dimension is at most n - 5. When n = 5 and k = 4, for a parameter lambda is an element of [0, 1] we introduce lambda-relaxed energy H-lambda of the Hessian energy for maps in W-2,W-2 (Omega; S-4) so that each minimizer u(lambda) of H-lambda is also a biharmonic map. We also establish the existence and partial regularity of a minimizer of H-lambda for lambda is an element of [0, 1).

A new system of variational inclusions with (H,eta)-monotone operators in Hilbert spaces

In this paper, we introduce and study a new system of variational inclusions involving (H, eta)-monotone operators in Hilbert space. Using the resolvent operator associated with (H, eta)monotone operators, we prove the existence and uniqueness of solutions for this new system of variational inclusions. We also construct a new algorithm for approximating the solution of this system and discuss the convergence of the sequence of iterates generated by the algorithm. (c) 2005 Elsevier Ltd. All rights reserved.

Stability for parametric implicit vector equilibrium problems

In this paper, we consider a class of parametric implicit vector equilibrium problems in Hausdorff topological vector spaces where a mapping f and a set K are perturbed by parameters is an element of and lambda respectively. We establish sufficient conditions for the upper semicontinuity and lower semicontinuity of the solution set mapping S : Lambda(1) x A(2) -> 2(X) for such parametric implicit vector equilibrium problems. (c) 2005 Elsevier Ltd. All rights reserved.

Investigating ontogenetic space with developmental cell lineages

Development plays a significant role in biological evolution, and is likely to prove an effective route to overcoming the limitations of direct genotype-phenotype mappings in artificial evolution. Nonetheless, the relationship between development and evolution is complex and still poorly understood. One question of current interest concerns the possible role that developmental processes may play in orienting evolution. A first step towards exploring this issue from a theoretical perspective is understanding the structure of ontogenetic space: the space of possible genotype-phenotype mappings. Using a quantitative model of development that enables ontogenetic space to be characterised in terms of complexity, we show that ontogenetic landscapes have a characteristic structure that varies with genotypic properties.

Incremental derivation of abstraction relations for data refinements

Data refinements are refinement steps in which a program’s local data structures are changed. Data refinement proof obligations require the software designer to find an abstraction relation that relates the states of the original and new program. In this paper we describe an algorithm that helps a designer find an abstraction relation for a proposed refinement. Given sufficient time and space, the algorithm can find a minimal abstraction relation, and thus show that the refinement holds. As it executes, the algorithm displays mappings that cannot be in any abstraction relation. When the algorithm is not given sufficient resources to terminate, these mappings can help the designer find a suitable abstraction relation. The same algorithm can be used to test an abstraction relation supplied by the designer.