No Arabic abstract
In this paper we find a realization for the DB-boundary conditions, which imposes vanishing normal derivatives of the normal components of the D and B fields. The implementation of the DB boundary, requiring vanishing normal components of D and B, is known. It is shown that the realization of the DB boundary can be based on a layer of suitable metamaterial, called the wave-guiding quarter-wave transformer, which transforms the DB boundary to the DB boundary. In an appendix, the mixed-impedance boundary, which is a generalization of both DB and DB boundaries, is shown to transform to another mixed-impedance boundary through the same transformer.
We consider the free boundary condition Gibbs measure of the Potts model on a random tree. We provide an explicit temperature interval below the ferromagnetic transition temperature for which this measure is extremal, improving older bounds of Mossel and Peres. In information theoretic language extremality of the Gibbs measure corresponds to non-reconstructability for symmetric q-ary channels. The bounds are optimal for the Ising model and appear to be close to what we conjecture to be the true values up to a factor of 0.0150 in the case q = 3 and 0.0365 for q = 4. Our proof uses an iteration of random boundary entropies from the outside of the tree to the inside, along with a symmetrization argument.
The Bargmann-Wigner (BW) framework describes particles of spin-j in terms of Dirac spinors of rank 2j, obtained as the local direct product of n Dirac spinor copies, with n=2j. Such spinors are reducible, and contain also (j,0)+(0,j)-pure spin representation spaces. The 2(2j+1) degrees of freedom of the latter are identified by a projector given by the n-fold direct product of the covariant parity projector within the Dirac spinor space. Considering totally symmetric tensor spinors one is left with the expected number of 2(2j+1) independent degrees of freedom. The BW projector is of the order $partial ^{2j}$ in the derivatives, and so are the related spin-j wave equations and associated Lagrangians. High order differential equations can not be consistently gauged, and allow several unphysical aspects, such as non-locality, acausality, ghosts and etc to enter the theory. In order to avoid these difficulties we here suggest a strategy of replacing the high order of the BW wave equations by the universal second order. To do so we replaced the BW projector by one of zeroth order in the derivatives. We built it up from one of the Casimir invariants of the Lorentz group when exclusively acting on spaces of internal spin degrees of freedom. This projector allows one to identify anyone of the irreducible sectors of the primordial rank-2j spinor, in particular (j,0)+(0,j), and without any reference to the external space-time and the four-momentum. The dynamics is then introduced by requiring the (j,0)+ (0,j) sector to satisfy the Klein-Gordon equation. The scheme allows for a consistent minimal gauging.
Implementing the modal method in the electromagnetic grating diffraction problem delivered by the curvilinear coordinate transformation yields a general analytical solution to the 1D grating diffraction problem in a form of a T-matrix. Simultaneously it is shown that the validity of the Rayleigh expansion is defined by the validity of the modal expansion in a transformed medium delivered by the coordinate transformation.
We introduce and study the mechanical system which describes the dynamics and statics of rigid bodies of constant density floating in a calm incompressible fluid. Since much of the standard equilibrium theory, starting with Archimedes, allows bodies with vertices and edges, we assume the bodies to be convex and take care not to assume more regularity than that implied by convexity. One main result is the (Liapunoff) stability of equilibria satisfying a condition equivalent to the standard metacentric criterion.
The geodesic has a fundamental role in physics and in mathematics: roughly speaking, it represents the curve that minimizes the arc length between two points on a manifold. We analyze a basic but misinterpreted difference between the Lagrangian that gives the arc length of a curve and the one that describes the motion of a free particle in curved space. Although they provide the same formal equations of motion, they are not equivalent. We explore this difference from a geometrical point of view, where we observe that the non-equivalence is nothing more than a matter of symmetry. As applications, some distinct models are studied. In particular, we explore the standard free relativistic particle, a couple of spinning particle models and also the forceless mechanics formulated by Hertz.