323 research outputs found
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 . 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 -invariant function on ,
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> 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 > 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
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