Critical points and symmetries of a free energy function for biaxial nematic liquid crystals

We describe a general model for the free energy function for a homogeneous medium of mutually interacting molecules, based on the formalism for a biaxial nematic liquid crystal set out by Katriel {\em et al.} (1986) in an influential paper in {\em Liquid Crystals} {\bf 1} and subsequently called the KKLS formalism. The free energy is expressed as the sum of an entropy term and an interaction (Hamiltonian) term. Using the language of group representation theory we identify the order parameters as averaged components of a linear transformation, and characterise the full symmetry group of the entropy term in the liquid crystal context as a wreath product $SO(3)\wr Z_2$. The symmetry-breaking role of the Hamiltonian, pointed out by Katriel {\em et al.}, is here made explicit in terms of centre manifold reduction at bifurcation from isotropy. We use tools and methods of equivariant singularity theory to reduce the bifurcation study to that of a $D_3\,$-invariant function on ${\bf R}^2$, ubiquitous in liquid crystal theory, and to describe the 'universal' bifurcation geometry in terms of the superposition of a familiar swallowtail controlling uniaxial equilibria and another less familiar surface controlling biaxial equilibria. In principle this provides a template for {\em all} nematic liquid crystal phase transitions close to isotropy, although further work is needed to identify the absolute minima that are the critical points representing stable phases.Comment: 74 pages, 17 figures : submitted to Nonlinearit

Perturbed hedgehogs: continuous deformation of point defects in biaxial nematic liquid crystals

A spherically symmetric isolated point defect in a 3D uniaxial nematic liquid crystal sample is often called a {\it radial hedgehog}. We use topological methods to describe local configurations of uniaxial and biaxial states into which a hedgehog naturally deforms under small perturbations: these include the biaxial torus and split core configurations studied in the literature using analytical and numerical methods. The topological results here take no account of the governing physical laws but provide a library of options from which the physics must make a choice.<br/

HR issues evolution along the market lifecycle and the value chain: Case of the hi-tech industry

In this article, we shall provide arguments that proper matching of HR management with evolving market conditions should also contribute substantially to the survival and long run success of the firm

A presentation for the mapping class group of the closed non-orientable surface of genus 4

Finite presentations for the mapping class group M(F) are known for arbitrary orientable compact surface F. If F is non-orientable, then such presentations are known only when F has genus at most 3 and few boundary components. In this paper we obtain finite presentation for the mapping class group of the closed non-orientable surface of genus 4 from its action on the so called ordered complex of curves.Comment: 28 pages, 7 figure

A linear representation of the mapping class group and the theory of winding numbers

This paper describes a linear representation F of the mapping class group , of an orientable surface S with one boundary component. The representation F extends the symplectic representation, and is defined for surfaces of arbitrary genus g&gt; 1. The main tools used to define F are crossed homomorphisms which are defined using nonvanishing vector fields X on S, and the theory of winding numbers of curves on surfaces described by Chillingworth in [1,2]. These crossed homomorphisms were essentially described by Morita in [6]. A geometric interpretation of F is then given. If T1S denotes the unit tangent bundle of S1 then F records the action of on H1(T1S;Z). The kernel of F is then characterized using knowledge of the crossed homomorphisms ex. If matrix entries are taken modulo 2g-2, the representation F factors through the mapping class group of a closed orientable surface of genus g &gt; 1. Thus F induces representations of Dn of for any n[-45 degree rule]2g-2. The Dn were discovered by Sipe in [7, 8], and it is noted that her characterization of the image of Dn carries over to the integer valued case. The structure found in characterizing ker F is then used to study ker Dn. In particular, it is shown that a uotient of ker Dn is a semidirect product for each even n dividing 2g-2.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30247/1/0000642.pd

Topology of Vibro-Impact Systems in the Neighborhood of Grazing

The grazing bifurcation is considered for the Newtonian model of vibro-impact systems. A brief review on the conditions, sufficient for existence of a grazing family of periodic solutions, is given. The properties of these periodic solutions are discussed. A plenty of results on the topological structure of attractors of vibro-impact systems is known. However, since the considered system is strongly nonlinear, these attractors may be invisible or, at least, very sensitive to changes of parameters of the system. On the other hand, they are observed in experiments and numerical simulations. We offer (Theorem 2) an approach which allows to explain this contradiction and give a new robust mathematical model of the non-hyperbolic dynamics in the neighborhood of grazing.Comment: Submitted to Physica