Scattering of electromagnetic waves on a planar surface in a model with the Chern-Simons potential

The Chern-Simons potential occurs in quantum electrodynamics in constructing a model for the interaction of a material surface with the photon field if the locality, gauge invariance, and renormalizability requirements are imposed. In the framework of such a model, we consider the problem of an electromagnetic wave scattering on a plane.     

A “planar” representation for generalized transition kernels

In [9], Mauldin, Preiss and von Weizsäcker have given a theorem representing transition kernels (atomless and between standard Borel spaces) by a planar model. Here, motivated by measure-theoretic as well as probabilistic considerations, we generalize by allowing the parametrizing spaceX to be arbitrary, with an arbitrary σ-field of “Borel” subsets, and allowing the corresponding measures to have atoms. (We also, for convenience rather than generality, allow arbitrary finite measures rather than probability ones.) The transition kernel is replaced by a substantially equivalent one fromX toX ×I that is “sectioned”, hence completely orthogonal. This is shown to be isomorphic to a model in which the image space consists of 3 specifically defined subsets ofX × ?: an ordinate set (in which vertical sections have Lebesgue measure), an “atomic” set contained inX × (??), and a “singular” set with null sections. The method incidentally produces and exploits a “reverse” transition kernel fromX toX ×I. Some further extensions are briefly discussed; in particular, allowing “uniformly σ-finite” measures (in the “standard” case) leads to a generalization that includes the planar representation theorem of Rokhlin [10] and the author [5]; cf. also [7, 2].     

Nonlinear waves in ferromagnets

Tbilisi State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 80, No. 3, pp. 461–469, September, 1989.     

On the dynamics of deformable ferromagnets

The paper begins with a fairly detailed presentation of a general mathematical model for the dynamics of ferromagnetic bodies undergoing arbitrarily large deformations. Next, a mathematical study is presented of the evolution problem for the magnetization field in a «soft» ferromagnetic body which is «mechanically at rest». No matter how special and simple this problem within the frame-work of the full theory, the governing equation is interesting: it is identical to the dynamic version of the harmonic-map equation usually referred to in the mathematical literature as the Gilbert form of the Landau-Lifshitz equation. Motivated by recent nonuniqueness results for the dynamic harmonicmap equation, we give a new proof of global existence of weak solutions to the Gilbert-Landau-Lifshitz equation.     

An analysis of the two-vortex case in the Chern-Simons Higgs model

Extending work of Caffarelli-Yang and Tarantello, we present a variational existence proof for two-vortex solutions of the periodic Chern-Simons Higgs model and analyze the asymptotic behavior of these solutions as the parameter coupling the gauge field with the scalar field tends to 0. Received September 24, 1997 / Accepted October 2, 1997     

Superfield Quantization in the Lagrangian Formalism

We generalize the superfield BRST quantization method for general gauge theories to the case of gauge fixing by the corresponding generating equation. We find a superfield form of the BRST symmetry of the vacuum functional and prove the gauge independence of the S-matrix. We show that the vacuum functional of the BV quantization method corresponds to a particular solution of the gauge-fixing generating equation. We discuss a modified version of the Ward identities related to the proposed generalized procedure of gauge fixing.     

Domain branching in uniaxial ferromagnets: asymptotic behavior of the energy

In this article we analyze the ground state energy of a ferromagnetic bulk sample with strong uniaxial anisotropy in a regime featuring domain branching. We derive the convergence of the micromagnetic energy towards a sharp interface energy in the spirit of Γ-limits. Then we use this convergence to rigorously justify the notion of a minimal energy per cross-section area. Compared to known results, the scaling bounds for the minimal energy are improved to an asymptotic equality.     

Quantization and the method of orbits

A survey is given of the method of orbits which makes it possible to construct irreducible unitary representations of an arbitrary Lie group proceeding from mechanical considerations. After a brief introduction to symplectic geometry, a construction of a representation associated with an orbit of a group in the dual space of its Lie algebra is given. Various generalizations of this construction are discussed.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 22, pp. 37–58, 1984.     

The Quantization of the Cantor Distribution

For a real-valued random variable whose distribution is the classical Cantor probability, the n - quantization error and the n - optimal quantization rules are calculated for every natural number n. Moreover, the connection between the rate of convergence of the logarithms of the quantization errors for n going to infinity and the Hausdorff dimension of the Cantor set is indicated.     

Asymptotic limit for condensate solutions in the Abelian Chern-Simons Higgs model

In this paper we show that the maximal solutions in the Abelian Chern-Simons Higgs model on a 't Hooft type periodic domain converges to and is a harmonic map. We also study asymptotic behaviors of the energy density.

     

A probabilistic Poisson representation for positive solutions of Δu = u2 in a planar domain

We use the stochastic process called the Brownian snake to investigate solutions of the partial differential equation Δu = u² in a domain D of class C² of the plane. We prove that nonnegative solutions are in one-to-one correspondence with pairs (K, v) where K is a closed subset of ∂D and v is a Radon measure on ∂D\K. Both Kand v are determined from the boundary behavior of the solution u. On the other hand, u can be expressed in terms of the pair (K, v) by an explicit probabilistic representation formula involving the Brownian snake. © 1997 John Wiley & Sons, Inc.     

A note on the Chern-Simons invariant of hyperbolic 3-manifolds

In this note we study how the Chern-Simons invariant behaves when two hyperbolic 3-manifolds are glued together along incompressible
thrice-punctured spheres. Such an operation produces many hyperbolic 3-manifolds with different numbers of cusps sharing the same volume and the same Chern-Simons invariant. The results in this note, combined with those of Meyerhoff and Ruberman, give an algorithm for determining the unknown constant in Neumann's simplicial formula for the Chern-Simons invariant of hyperbolic 3-manifolds.

     

Rate‐independent behavior of ferromagnets

We review properties of a model allowing for rate‐independent evolution ofbulk ferromagnets. The model is based on Brown's micromagnetic theory specialized to a large‐body limit by DeSimone [2] which is enriched by a rate‐independent dissipative mechanism. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A system of elliptic equations arising in Chern-Simons field theory

We prove the existence of topological vortices in a relativistic self-dual Abelian Chern-Simons theory with two Higgs particles and two gauge fields through a study of a coupled system of two nonlinear elliptic equations over R². We present two approaches to prove existence of solutions on bounded domains: via minimization of an indefinite functional and via a fixed point argument. We then show that we may pass to the full R² limit from the bounded-domain solutions to obtain a topological solution in R².