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Register a new userAction of the conformal group on steady state solutions to Maxwells equations and background radiation
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The representation of the conformal group (PSU(2,2)) on the space of solutions to Maxwells equations on the conformal compactification of Minkowski space is shown to break up into four irreducible unitarizable smooth Frechet representations of moderate growth. An explicit inner product is defined on each representation. The frequency spectrum of each of these representations is analyzed. These representations have notable properties; in particular they have positive or negative energy, they are of type $A_{frak q}(lambda)$ and are quaternionic. Physical implications of the results are explained.
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We prove that the steady--state Navier--Stokes problem in a plane Lipschitz domain $Omega$ exterior to a bounded and simply connected set has a $D$-solution provided the boundary datum $a in L^2(partialOmega)$ satisfies ${1over 2pi}|int_{partialOmega}acdot |<1$. If $Omega$ is of class $C^{1,1}$, we can assume $ain W^{-1/4,4}(partialOmega)$. Moreover, we show that for every $D$--solution $(u,p)$ of the Navier--Stokes equations it holds $ abla p = o(r^{-1}), abla_k p = O(r^{epsilon-3/2}), abla_ku = O(r^{epsilon-3/4})$, for all $kin{Bbb N}setminus{1}$ and for all positive $epsilon$, and if the flux of $u$ through a circumference surrounding $complementOmega$ is zero, then there is a constant vector $u_0$ such that $u=u_0+o(1)$.
The contraction of the Poincare group with respect to the space trans- lations subgroup gives rise to a group that bears a certain duality relation to the Galilei group, that is, the contraction limit of the Poincare group with respect to the time translations subgroup. In view of this duality, we call the former the dual Galilei group. A rather remarkable feature of the dual Galilei group is that the time translations constitute a central subgroup. Therewith, in unitary irreducible representations (UIR) of the group, the Hamiltonian appears as a Casimir operator proportional to the identity H = EI, with E (and a spin value s) uniquely characterizing the representation. Hence, a physical system characterized by a UIR of the dual Galilei group displays no non-trivial time evolution. Moreover, the combined U(1) gauge group and the dual Galilei group underlie a non- relativistic limit of Maxwells equations known as the electric limit. The analysis presented here shows that only electrostatics is possible for the electric limit, wholly in harmony with the trivial nature of time evolution governed by the dual Galilei group.
This paper explores a class of non-linear constitutive relations for materials with memory in the framework of covariant macroscopic Maxwell theory. Based on earlier models for the response of hysteretic ferromagnetic materials to prescribed slowly varying magnetic background fields, generalized models are explored that are applicable to accelerating hysteretic magneto-electric substances coupled self-consistently to Maxwell fields. Using a parameterized model consistent with experimental data for a particular material that exhibits purely ferroelectric hysteresis when at rest in a slowly varying electric field, a constitutive model is constructed that permits a numerical analysis of its response to a driven harmonic electromagnetic field in a rectangular cavity. This response is then contrasted with its predicted response when set in uniform rotary motion in the cavity.
An overview is given on recent developments in the affine Weyl group approach to Painleve equations and discrete Painleve equations, based on the joint work with Y. Yamada and K. Kajiwara.
We study smooth SU(2) solutions of the Hitchin equations on R^2, with the determinant of the complex Higgs field being a polynomial of degree n. When n>=3, there are moduli spaces of solutions, in the sense that the natural L^2 metric is well-defined on a subset of the parameter space. We examine rotationally-symmetric solutions for n=1 and n=2, and then focus on the n=3 case, elucidating the moduli and describing the asymptotic geometry as well as the geometry of two totally-geodesic surfaces.
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