Columnar liquid crystals based on amino-1,3,4-thiadiazole derivatives

Novel liquid crystal materials base on amino-thiadiazoles, specifically 5-(3,4-di-, and/or 3,4,5-tri-n-alkoxy)phenyl-2-amino-1,3,4-thiadiazoles (3a–f, n?=?10, 12, 14) and 5-(3,4-di- and/or 3,4,5-tri-n-alkoxy)phenyl-2-(4-amino)phenyl-1,3,4-thiadiazoles (6a–f, n?=?10, 12, 14) were synthesised. The mesomorphic properties of these compounds were studied by differential scanning calorimetry and polarising optical microscopy. All the compounds in series 3a–f and series 6a–f display an enantiotropic columnar phase. The mesomorphic properties were found to be dependent on the number of the side alkoxy chains, and on the length of the rigid core, and on the position of the thiadiazole ring. The best results were obtained with compounds of series 3a–f.     

Novel chiral liquid crystals based on amides and azo compounds derived from 2-amino-1,3,4-thiadiazoles: synthesis and mesomorphic properties

Novel chiral amides (Ia-Ie, II) and azo compounds (III, IV) with a 1,3,4-thiadiazole unit in the rigid core were synthesized and their liquid crystalline properties investigated by polarizing optical microscopy and differential scanning calorimetry. The amides Ia-Ie contain a chiral alkoxy chain derived from (R)-2-octanol and an achiral chain varying from 6 to 10 carbon atoms at the end of the rigid core. In amide II one of the terminal group is a chiral alkoxy chain derived from (S)-isoleucine and the other terminal substituent is an achiral n-decyloxy chain. Azo compounds III and IV contain an achiral n-decyloxy chain and a chiral alkoxy chain derived from (R)-2-octanol and (S)-isoleucine, respectively, at the end of the rigid core. The first homologue in the series of amides (Ia) exhibits enantiotropic smectic X (SmX)-chiral nematic (N*) dimorphism and the homologues Ib-Ie display enantiotropic SmX-chiral smectic C (SmC*)-N* mesomorphism. Amide II displays an enantiotropic smectic A phase. The azo compounds III and IV do not show smectic order and only an enantiotropic N* phase was observed. Thus, the mesomorphic behaviour depends on the nature of the central linkage and on the nature of the chiral alkoxy chain.     

Uniqueness of Positive Solutions of Δu+f(u)=0 in ℝN, N≦3

We study the uniqueness of radial ground states for the semilinear elliptic partial differential equation in ℝ ^N . We assume that the function f has two zeros, the origin and u ₀>0. Above u ₀ the function f is positive, is locally Lipschitz continuous and satisfies convexity and growth conditions of a superlinear nature. Below u ₀, f is assumed to be non-positive, non-identically zero and merely continuous. Our results are obtained through a careful analysis of the solutions of an associated initial‐value problem, and the use of a monotone separation theorem. It is known that, for a large class of functions f, the ground states of (*) are radially symmetric. In these cases our result implies that (*) possesses at most one ground state. (Accepted July 3, 1996)     

An averaging method applied to a duffing equation

We approximate a Duffing equation by an averaged system. We solve the system explicitely and draw bifurcation diagrams in dependence of the forcing term. We discuss the goodness of the averaging method, relative to the behavior of the solutions in dependence of involved parameters, by comparing with results obtained in [6,10] for the original Duffing equation.     

How to Approximate the Heat Equation with Neumann Boundary Conditions by Nonlocal Diffusion Problems

We present a model for nonlocal diffusion with Neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary. We study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero. We prove that the solutions of this family of problems converge to a solution of the heat equation with Neumann boundary conditions.     

A Nonlocal Diffusion Equation whose Solutions Develop a Free Boundary

Let J:\mathbbR ? \mathbbRJ:\mathbb{R} \to \mathbb{R} be a nonnegative, smooth compactly supported function such that ò_\mathbbR J(r)dr = 1. \int_\mathbb{R} {J(r)dr = 1.} We consider the nonlocal diffusion problem

Synthesis of new benzobisthiazole materials and a study of their mesomorphic and luminescence properties

Novel tetra- and hexacatenar materials based on benzobisthiazole derivatives have been synthesised (4a–c, 5a–c and 6a–c) and their mesomorphic properties have been studied by differential scanning calorimetry, polarised optical microscopy and X-ray diffraction. All compounds in series 4a–c display an enantiotropic hexagonal columnar phase, whereas their analogous compounds in series 5a–c did not show mesomorphic behaviour. Compound 6a displays an unidentified monotropic columnar phase, whereas compounds 6b and 6c show an enantiotropic rectangular columnar mesophase. The mesomorphic properties were found to be dependent on the length, position and number of alkoxy chains attached at the end of the rigid core. These compounds also show photoluminescence properties in the visible region and have good quantum yields. A thermogravimetric analysis was also performed.     

Subdirect representation theory for classes without equality

This paper wants to be a contribution to the development of an algebraic theory of structures without equality. The basic framework for a suitable approach to this theory is provided and the problem of subdirect representability is addressed. The approach to the theory is largely borrowed from recent developments in abstract algebraic logic. Received March 4, 1996; accepted in final form July 10, 1998.