The propagation of light through bianisotropic materials is studied in the geometrical optics approximation. For that purpose, we use the quartic general dispersion equation specified by the Tamm-Rubilar tensor, which is cubic in the electromagnetic
response tensor of the medium. A collection of different and remarkable Fresnel (wave) surfaces is gathered, and unified via the projective geometry of Kummer surfaces.
It is known that the Fresnel wave surfaces of transparent biaxial media have 4 singular points, located on two special directions. We show that, in more general media, the number of singularities can exceed 4. In fact, a highly symmetric linear mater
ial is proposed whose Fresnel surface exhibits 16 singular points. Because, for every linear material, the dispersion equation is quartic, we conclude that 16 is the maximum number of isolated singularities. The identity of Fresnel and Kummer surfaces, which holds true for media with a certain symmetry (zero skewon piece), provides an elegant interpretation of the results. We describe a metamaterial realization for our linear medium with 16 singular points. It is found that an appropriate combination of metal bars, split-ring resonators, and magnetized particles can generate the correct permittivity, permeability, and magnetoelectric moduli. Lastly, we discuss the arrangement of the singularities in terms of Kummers (16,6)-configuration of points and planes. An investigation parallel to ours, but in linear elasticity, is suggested for future research.
Variants of fluctuation theorems recently discovered in the statistical mechanics of non-equilibrium processes may be used for the efficient determination of high-dimensional integrals as typically occurring in Bayesian data analysis. In particular f
or multimodal distributions, Monte-Carlo procedures not relying on perfect equilibration are advantageous. We provide a comprehensive statistical error analysis for the determination of the prior-predictive value in a Bayes problem building on a variant of the Jarzynski equation. Special care is devoted to the characterization of the bias intrinsic to the method. We also discuss the determination of averages over multimodal posterior distributions with the help of a variant of the Crooks theorem. All our findings are verified by extensive numerical simulations of two model systems with bimodal likelihoods.
Geometrical optics describes, with good accuracy, the propagation of high-frequency plane waves through an electromagnetic medium. Under such approximation, the behaviour of the electromagnetic fields is characterised by just three quantities: the te
mporal frequency $omega$, the spatial wave (co)vector $k$, and the polarisation (co)vector $a$. Numerous key properties of a given optical medium are determined by the Fresnel surface, which is the visual counterpart of the equation relating $omega$ and $k$. For instance, the propagation of electromagnetic waves in a uniaxial crystal, such as calcite, is represented by two light-cones. Kummer, whilst analysing quadratic line complexes as models for light rays in an optical apparatus, discovered in the framework of projective geometry a quartic surface that is linked to the Fresnel one. Given an arbitrary dispersionless linear (meta)material or vacuum, we aim to establish whether the resulting Fresnel surface is equivalent to, or is more general than, a Kummer surface.
We introduce a family ${cal F}$ of normal Tricomi domains $Om_{a,b}$, $a>0>b$, and we show that its elements are $D$-star-shaped with respect to the vector field $D=-3xpartial_x-2ypartial_y$ if and only if $age1/2$. Provided that the underlying domai
n $Om$ belongs to ${cal F}$ for some $age1/2$, we then establish $L^2$ estimates for the eigenfunctions corresponding to real eigenvalues of the Tricomi operator. In particular, our result highlights a dependency of these estimates on the values of $a$ and $b$ and the parabolic diameter of $Om$.
We derive optimal regularity, in both time and space, for solutions of the Cauchy problem related to a degenerate differential equation in a Banach space X. Our results exhibit a sort of prevalence for space regularity, in the sense that the higher i
s the order of regularity with respect to space, the lower is the corresponding order of regularity with respect to time.
