Spacer length dependence of TGBC* and SmA phases in cholesteryl and dihydrocholesteryl-containing trimer liquid crystals

Two series of trimer liquid crystals were investigated that contain a biphenylyl central group and two cholesteryl or dihydrocholesteryl terminal mesogenic groups. Only compounds with even spacers were investigated. The dihydrocholesteryl-containing trimers show a triply intercalated smectic A (SmA) phase when the spacer lengths are greater than 8, whereas the cholesteryl-containing trimers exhibit this triply intercalated SmA phase when the spacer lengths are more than 6. With shorter spacers, a twist grain boundary C* (TGBC*) phase was found. This is revealed by the formation of a typical dotted square grid pattern upon cooling from the chiral nematic (N*) phase in the planar texture. The dots are spaced by a distance of about 1.5-1.8 µm. Upon cooling from the N* phase in the focal conic texture a striped pattern is observed with the same spacing. X-ray diffraction revealed a repeat distance for the TGBC* phase that corresponds with a monolayer ordering. The results show that the weaker interaction between the dihydrocholesteryl groups compared with cholesteryl groups or longer spacers destabilize the monolayer TGBC* phase.     

Tight bounds and 2-approximation algorithms for integer programs with two variables per inequality

The problem of integer programming in bounded variables, over constraints with no more than two variables in each constraint is NP-complete, even when all variables are binary. This paper deals with integer linear minimization problems inn variables subject tom linear constraints with at most two variables per inequality, and with all variables bounded between 0 andU. For such systems, a 2-approximation algorithm is presented that runs in time O(mnU ² log(Un ² m)), so it is polynomial in the input size if the upper boundU is polynomially bounded. The algorithm works by finding first a super-optimal feasible solution that consists of integer multiples of 1/2. That solution gives a tight bound on the value of the minimum. It furthermore has an identifiable subset of integer components that retain their value in an integer optimal solution of the problem. These properties are a generalization of the properties of the vertex cover problem. The algorithm described is, in particular, a 2-approximation algorithm for the problem of minimizing the total weight of true variables, among all truth assignments to the 2-satisfiability problem.This paper is dedicated to Phil Wolfe on the occasion of his 65th birthday.Research supported in part by ONR contracts N00014-88-K-0377 and N00014-91-J-1241.Research supported in part by ONR contract N00014-91-C-0026.Part of this work was done while the author was visiting the International Computer Science Institute in Berkeley, CA and DIMACS, Rutgers University, New Brunswick, NJ.     

Realization of the transformation operator in the Jansen—Byers brown perturbation theory for exchange interactions

We obtain the general form and exhibit simple expressions for the transformation operator in the Jansen—Byers Brown exchange perturbation theory, without the use of an orthonormalization process.     

Ω-Invariance in control systems with bounded controls

The concepts of weakly Ω-invariant sets and strictly weakly Ω-invariant sets in control systems with bounded controls are defined and analyzed. Computable conditions for weak Ω-invariance are derived, and the question of existence of rest points and stationary points in weakly Ω-invariant sets is considered. For linear dynamics, properties of weakly Ω-invariant sets are studied, and questions of constrained reachability are investigated.     

Ethers of indoxylie acid

O-alkylations of indoxylie acid (2b) to produce 3-indolyl-alkyl ethers and a synthesis of 5-melhoxy-2-earboxyindol-3-oxyaeetic acid (25) are described.     

The use of jacobi's lemma in unimodularity theory

The paper applies Jacobi's fundamental result on minors of the adjoint matrix to obtain properties on determinants of unimodular matrices.     

Multiple Čerenkov second-harmonic waves in a two-dimensional nonlinear photonic structure

We report simultaneous generation of multiple conical second-harmonic waves in a two-dimensional nonlinear photonic structure when illuminated by two overlapping noncollinear fundamental beams. We show that this phenomenon is caused by the nonlinear ?erenkov radiation emitted due to the interaction of photons from each constituent fundamental beam as well as the virtual one propagating along the bisector of the two beams. In addition, by studying the asymmetric geometry of the interaction, we uniquely verify the effects of reciprocal vectors on the ?erenkov-type second-harmonic generation in nonlinear photonic structures.