De Morgan classifying toposes

We present a general method for deciding whether a Grothendieck topos satisfies De Morgan's law (resp. the law of excluded middle) or not; applications to the theory of classifying toposes follow. Specifically, we obtain a syntactic characterization of the class of geometric theories whose classifying toposes satisfy De Morgan's law (resp. are Boolean), as well as model-theoretic criteria for theories whose classifying toposes arise as localizations of a given presheaf topos.     

Spin-Hall Effect with Quantum Group Symmetries

We construct a model of spin-Hall effect on a noncommutative four sphere S ⁴ _Θ with isospin degrees of freedom, coming from a noncommutative instanton, and invariance under a quantum group SO _θ. The corresponding representation theory allows to explicitly diagonalize the Hamiltonian and construct the ground state; there are both integer and fractional excitations. Similar models exist on higher dimensional spheres S _Θ ^N and projective spaces . Dedicated to Rafael Sorkin with friendship and respect.     

Ultrasmall Double Junction in Terms of Orthogonal Polynomials

The issue of the form that the energy tensor of the electromagnetic field should be given in matter is reconsidered, and the neat derivation of Abraham’s tensor once provided by W. Gordon is recollected. In order to extend to the high frequency domain the experimental evidence gathered up to now in favour of Abraham’s tensor, a method for detecting the Abraham’s force supposedly exerted by light on a transparent, homogeneous medium is outlined. It avails of the Fresnel-Fizeau effect associated with the motion of matter that should be caused by the above mentioned force.     

Poincaré invariants

We construct an obstruction theory for relative Hilbert schemes in the sense of [K. Behrend, B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1) (1997) 45-88] and compute it explicitly for relative Hilbert schemes of divisors on smooth projective varieties. In the special case of curves on a surface V, our obstruction theory determines a virtual fundamental class , which we use to define Poincaré invariants

     

Induction and Skolemization in saturation theorem proving

We consider a typical integration of induction in saturation-based theorem provers and investigate the effects of Skolem symbols occurring in the induction formulas. In a practically relevant setting we establish a Skolem-free characterization of refutation in saturation-based proof systems with induction. Finally, we use this characterization to obtain unprovability results for a concrete saturation-based induction prover.     

Boundary integrability of nonlinear sigma models

We describe recent work on the classical integrability of the principal chiral model and general sigma models with boundaries in (compact) symmetric spaces.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 2, pp. 322–328, February, 2005.     

Instabilities in nematic elastomers in external electric and magnetic fields

In this paper we study the macroscopic behavior of nematic side-chain liquid single crystal elastomers exposed to an external electric or magnetic field. For this purpose we use the framework of a continuum model. The geometries investigated comprise the bend and the twist geometry known from the classical Frederiks transition in low molecular weight liquid crystals. For the bend geometry we find a laterally homogeneous and a two-dimensional undulatory instability, which may compete at onset. In the case of the twist geometry three instabilities can occur at onset, two of which are two dimensional and clearly show undulations. As a major result we propose how the values of the twist coefficient K(2) and the values of the material parameters D(1) and D(2) connected to relative rotations between the director field and the polymer network can be determined from experimental observations. In addition, we explain why a twist experiment is probably the most suitable set-up in order to measure the material parameter D(1).     

Classical light dispersion theory in a regular lattice

We study the dynamics of an infinite regular lattice of classical charged oscillators. Each individual oscillator is described as a point particle subject to a harmonic restoring potential, to the retarded electromagnetic field generated by all the other particles, and to the radiation reaction expressed according to the Lorentz-Dirac equation. Exact normal mode solutions, describing the propagation of plane electromagnetic waves through the lattice, are obtained for the complete linearized system of infinitely many oscillators. At variance with all the available results, our method is valid for any values of the frequency, or of the ratio between wavelength and lattice parameter. A remarkable feature is that the proper inclusion of radiation reaction in the dynamics of the individual oscillators does not give rise to any extinction coefficient for the global normal modes of the lattice. The dispersion relations resulting from our solution are numerically studied for the case of a simple cubic lattice. New predictions are obtained in this way about the behavior of the crystal at frequencies near the proper oscillation frequency of the dipoles.