Formalism of two potentials for the numerical solution of Maxwell’s equations

A new formulation of Maxwell’s equations based on the introduction of two vector and two scalar potentials is proposed. As a result, the electromagnetic field equations are written as a hyperbolic system that contains, in contrast to the original Maxwell system, only evolution equations and does not involve equations in the form of differential constraints. This makes the new equations especially convenient for the numerical simulation of electromagnetic processes. Specifically, they can be solved by applying powerful modern shock-capturing methods based on the approximation of spatial derivatives by upwind differences. The cases of an electromagnetic field in a vacuum and an inhomogeneous material are considered. Examples are given in which electromagnetic wave propagation is simulated by solving the formulated system of equations with the help of modern high-order accurate schemes.     

Weyl formula for the negative dissipative eigenvalues of Maxwell’s equations

Let \(V(t) = e^{tG_b},\, t \ge 0,\) be the semigroup generated by Maxwell’s equations in an exterior domain \(\Omega \subset {\mathbb R}^3\) with dissipative boundary condition \(E_{tan}- \gamma (x) (\nu \wedge B_{tan}) = 0, \gamma (x) > 0, \forall x \in \Gamma = \partial \Omega .\) We study the case when \(\Omega = \{x \in {\mathbb R}^3:\, |x| > 1\}\) and \(\gamma \ne 1\) is a constant. We establish a Weyl formula for the counting function of the negative real eigenvalues of \(G_b.\)     

Maxwell’s equations,the Euler index,and Morse theory

We show that the singularities of the Fresnel surface for Maxwell’s equation on an anisotrpic material can be accounted from purely topological considerations. The importance of these singularities is that they explain the phenomenon of conical refraction predicted by Hamilton. We show how to desingularise the Fresnel surface, which will allow us to use Morse theory to find lower bounds for the number of critical wave velocities inside the material under consideration. Finally, we propose a program to generalise the results obtained to the general case of hyperbolic differential operators on differentiable bundles.     

About inclusion of Maxwell’s equations in systems relativistic of the invariant equations

Maxwell’s equations, relativistic invariant equations, foundations of difference schemes.     

Theoretical analysis of boundary control extremal problems for Maxwell’s equations

Under study are the extremal problems of multiplicative boundary control for timeharmonic Maxwell’s equations considered with the impedance boundary condition for the electric field. The solvability of the original extremal problem is proved. Some sufficient conditions are derived on the original data which guarantee the stability of solutions to concrete extremal problems with respect to certain perturbations of both the quality functional and one of the known functions that has the meaning of the density of the electric current.     

A sweeping preconditioner for Yee’s finite difference approximation of time-harmonic Maxwell’s equations

This paper is concerned with the fast iterative solution of linear systems arising from finite difference discretizations in electromagnetics. The sweeping preconditioner with moving perfectly matched layers previously developed for the Helmholtz equation is adapted for the popular Yee grid scheme for wave propagation in inhomogeneous, anisotropic media. Preliminary numerical results are presented for typical examples.     

On a class of semilinear evolution equations for vector potentials associated with Maxwell’s equations in Carnot groups

In this paper we prove existence and regularity results for a class of semilinear evolution equations that are satisfied by vector potentials associated with Maxwell’s equations in Carnot groups (connected, simply connected, stratified nilpotent Lie groups). The natural setting for these equations is provided by the so-called Rumin’s complex of intrinsic differential forms.     

An analogue of Beurling’s theorem for the Jacobi transform

In this paper, we prove Beurling＇s theorem for the Jacobi transform, from which we derive some other versions of uncertainty principles.     

Harnack’s inequality for a class of non-divergent equations in the Heisenberg group

We prove an invariant Harnack’s inequality for operators in non-divergence form structured on Heisenberg vector fields when the coe?cient matrix is uniformly positive definite, continuous, and symplectic. The method consists in constructing appropriate barriers to obtain pointwise-to-measure estimates for supersolutions in small balls, and then invoking the axiomatic approach developed by Di Fazio, Gutiérrez, and Lanconelli to obtain Harnack’s inequality.     

Spectral theory and Wiener-Itô decomposition for the image of a Jacobi field

Assume that K ⁺: H _? → T _? is a bounded operator, where H _? and T _? are Hilbert spaces and ρ is a measure on the space H _?. Denote by ρ_K the image of the measure ρ under K ⁺. We study the measure ρ_K under the assumption that ρ is the spectral measure of a Jacobi field and obtain a family of operators whose spectral measure is equal to ρ_K. We also obtain an analog of the Wiener-Itô decomposition for ρ_K. Finally, we illustrate the results obtained by explicit calculations carried out for the case, where ρ_K is a Lévy noise measure.     

Galilei’s relativity principle for a system of reaction-convection-diffusion equations

The representations of the Galilean algebra and its extensions relative to which the system of nonlinear reaction-convection-diffusion equations can be invariant are investigated. The kinds of nonlinearities at which this system is invariant relative to those algebras are determined to within continuous equivalence transformations.     

The strength of Martin-Löf type theory with a superuniverse. Part I

Universes of types were introduced into constructive type theory by Martin-L?f [12]. The idea of forming universes in type theory is to introduce a universe as a set closed under a certain specified ensemble of set constructors, say . The universe then “reflects”. This is the first part of a paper which addresses the exact logical strength of a particular such universe construction, the so-called superuniverse due to Palmgren (cf. [16, 18, 19]). It is proved that Martin-L?f type theory with a superuniverse, termed MLS, is a system whose proof-theoretic ordinal resides strictly above the Feferman-Schütte ordinal but well below the Bachmann-Howard ordinal. Not many theories of strength between and the Bachmann-Howard ordinal have arisen. MLS provides a natural example for such a theory. Received: 14 October 1997     

Seeking a proof of Xie’s inequality: analogues for series and Fourier series

Seeking to free the existence and regularity theory for the Navier–Stokes equations from assumptions about the regularity of a fluid’s boundary, we continue efforts of Wenzheng Xie and myself to prove a certain domain independent inequality for solutions of the steady Stokes equations. For the Laplacian, Xie proved an analogue of the desired inequality by using the maximum principle in obtaining an intermediary result. His conjecture that an analogue of this intermediary result is also valid for the Stokes equations remains unproven. My efforts to circumvent the need for it have led, so far, only to further interesting conjectures. Here, we seek to better understand both Xie’s arguments and mine by applying them to simpler problems concerning series and Fourier series. First, a bound is proven for a series of real numbers that can be interpreted as a bound for the sup-norm of a Fourier cosine series, in terms of the \(L^{2}\) -norms of its fractional-order derivatives of orders 1/3 and 2/3. This is generalized to a bound for a weighted sum of a sequence of real numbers. We conjecture that the hypotheses concerning the weights are satisfied by the sequence of numbers \(\{ \sin ny\}\) , for any nonzero \(y\in (-\pi ,\pi )\) . If so, we obtain an inequality for the sup-norm of a Fourier sine series, similar to that for a cosine series. Remarkably, the hypotheses for the weights are analogous to those we have been seeking to verify in trying to prove the original inequality for the Stokes equations. We conclude with a remark showing that Xie’s central argument provides a possibly new, very straightforward, proof of Hölder’s inequality for series.