Duality and optimality conditions for generalized equilibrium problems involving DC functions

We consider a generalized equilibrium problem involving DC functions which is called (GEP). For this problem we establish two new dual formulations based on Toland-Fenchel-Lagrange duality for DC programming problems. The first one allows us to obtain a unified dual analysis for many interesting problems. So, this dual coincides with the dual problem proposed by Martinez-Legaz and Sosa (J Glob Optim 25:311–319, 2006) for equilibrium problems in the sense of Blum and Oettli. Furthermore it is equivalent to Mosco’s dual problem (Mosco in J Math Anal Appl 40:202–206, 1972) when applied to a variational inequality problem. The second dual problem generalizes to our problem another dual scheme that has been recently introduced by Jacinto and Scheimberg (Optimization 57:795–805, 2008) for convex equilibrium problems. Through these schemes, as by products, we obtain new optimality conditions for (GEP) and also, gap functions for (GEP), which cover the ones in Antangerel et al. (J Oper Res 24:353–371, 2007, Pac J Optim 2:667–678, 2006) for variational inequalities and standard convex equilibrium problems. These results, in turn, when applied to DC and convex optimization problems with convex constraints (considered as special cases of (GEP)) lead to Toland-Fenchel-Lagrange duality for DC problems in Dinh et al. (Optimization 1–20, 2008, J Convex Anal 15:235–262, 2008), Fenchel-Lagrange and Lagrange dualities for convex problems as in Antangerel et al. (Pac J Optim 2:667–678, 2006), Bot and Wanka (Nonlinear Anal to appear), Jeyakumar et al. (Applied Mathematics research report AMR04/8, 2004). Besides, as consequences of the main results, we obtain some new optimality conditions for DC and convex problems.     

Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line

Consider an evolution family U=(U(t,s))_t?s?0 on a half-line R₊ and an integral equation . We characterize the exponential dichotomy of the evolution family through solvability of this integral equation in admissible function spaces which contain wide classes of function spaces like function spaces of Lp type, the Lorentz spaces Lp,q and many other function spaces occurring in interpolation theory. We then apply our results to study the robustness of the exponential dichotomy of evolution families on a half-line under small perturbations.     

To the theory of operator monotone and operator convex functions

We prove that a real function is operator monotone (operator convex) if the corresponding monotonicity (convexity) inequalities are valid for some normal state on the algebra of all bounded operators in an infinite-dimensional Hilbert space. We describe the class of convex operator functions with respect to a given von Neumann algebra in dependence of types of direct summands in this algebra. We prove that if a function from ℝ⁺ into ℝ⁺ is monotone with respect to a von Neumann algebra, then it is also operator monotone in the sense of the natural order on the set of positive self-adjoint operators affiliated with this algebra.     

Linear Approximation and Asymptotic Expansion of Solutions in Many Small Parameters for a Nonlinear Kirchhoff Wave Equation with Mixed Nonhomogeneous Conditions

In this paper, we consider the following nonlinear Kirchhoff wave equation

$\left\{\begin{array}{l}u_{tt}-\frac{\partial }{\partial x}(\mu (u,\Vert u_{x}\Vert ^{2})u_{x})=f(x,t,u,u_{x},u_{t}),\quad 0

(1)

where \(\widetilde{u}_{0}\), \(\widetilde{u}_{1}\), μ, f, g are given functions and \(\Vert u_{x}\Vert ^{2}=\int_{0}^{1}u_{x}^{2}(x,t)dx.\) To the problem (1), we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved by applying the Faedo–Galerkin method and the weak compact method. In particular, motivated by the asymptotic expansion of a weak solution in only one, two or three small parameters in the researches before now, an asymptotic expansion of a weak solution in many small parameters appeared on both sides of (1)₁ is studied.

     

A generalized distance and enhanced Ekeland’s variational principle for vector functions

We propose a definition of lower closed transitive relations and prove the existence of minimal elements for such a relation. This result is shown to contain probably a large part of existing versions of Ekeland’s variational principle (EVP). We introduce the notion of a weak τ-function p as a generalized distance and use it together with the above result on minimal elements to establish enhanced EVP for various settings, under relaxed lower semicontinuity assumptions. These principles conclude the existence not only of p-strict minimizers of p-perturbations of the considered vector function, but also p-sharp and p-strong minimizers. Our results are proved to be stronger than the classical EVP and many generalizations in the literature, even in the usual one-dimensional case, by numerous corollaries and examples. We include equivalent formulations of our enhanced EVP as well.     

The Existence of SA 1, SA d,SA 2 and S∼C Smectic Phases in the New Series 4-Alkoxyphenyl-4′- [4″-Cyanobenzyloxv] Benzoates

A homologous series of 4 - alkoxyphenyl-4′-[4″-cyano-benzyloxy] benzoates have been synthesized. Within this series three smectic A phases : S_{A 1}, S_{A d} and S_{A 2} were identified by mean of small angle X-Ray scattering. Furthermore three derivatives exhibit the novel smectic phase made of ribbons S～_C.     

Mesomorphic Properties of Di- (4 - Alkoxyphenyl) and Di- (4 - Alkanoyloxyphenyl) Tetrathiafulvalenes

The polymorphism of the di- (4 - alkoxyphenyl) and di- (4 - alkanoyloxyphenyl) tetrathiafulvalenes (TTF) is investigated. In the dialkoxy series, the short chain derivative exhibits a smectic G phase and a nematic phase. The latter phase disappears in the long chain derivative. The dialkanoyloxy derivatives exhibit a smectic G phase (for short chains) and a smectic C phase (for long chains). A plot of the mesormorphic-isotropic transition temperatures against the number of carbon atoms of the substituent shows the usual odd-even effect. Some tetra (alkoxyphenyl) tetrathiafulvalenes are described, unfortunately they have no mesomorphic properties.     

Raman Scattering from Liquid Crystals

Abstract

The use of Raman scattering technique as a tool for studying local order in complex liquid crystal systems is illustrated by three examples, namely the locally anisotropic liquid L-phase, lipid membranes, and PDLC's.     

Effects of constrained chain conformations on polymer-solute interactions in semicrystalline polymers

The chemical potential of a solute in a solid polymer includes contributions from solute-polymer chain conformations, Flory-Huggins type interaction, and elastic energy of swelling. Presence of impermeable and rigid crystallites in such systems is expected to affect all these contributions. Theoretical calculations have been performed to check the direct effects of constrained chain conformations in the amorphous domains in semicrystalline polymers. Experimental results are used to determine Flory-Huggins coefficient and elastic modulus. From all these, the primary effects are shown to be on the entropic part of the Flory-Huggins coefficient and an increase in the elastic modulus by one or two order of magnitude. Finally, these results are used to calculate the rates of solvent-induced crystallization to show that these rates can drop to negligible values as the amount of crystals formed rises. Thus, the actual degree of crystallization can lie well below the Flory-Yoon limit.