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Filliman duality expresses (the characteristic measure of) a convex polytope P containing the origin as an alternating sum of simplices that share supporting hyperplanes with P. The terms in the alternating sum are given by a triangulation…
Electromagnetic response of PT-dipole is studied both analytically and numerically. In analytical approach, dipole is represented by two point scatterers. Within the first Born approximation, the asymmetry of the scattering field with…
We introduce the vertex index, vein(K), of a given centrally symmetric convex body K, which, in a sense, measures how well K can be inscribed into a convex polytope with small number of vertices. This index is closely connected to the…
This paper is concerned with the Floating Body Problem of S. Ulam: the existence of objects other than the sphere, which can float in a liquid in any orientation. Despite recent results of F. Wegner pointing towards an affirmative answer, a…
We study a version of the Busemann-Petty problem for $\log$-concave measures with an additional assumption on the dilates of convex, symmetric bodies. One of our main tools is an analog of the classical large deviation principle applied to…
We study two-body correlations in systems of identical bosons. We use a Faddeev type of decomposition of the wave function where all pairs of particles are treated equally. We focus on a new multi-particle Efimov effect at large scattering…
Astrophysical polarization measurements provide some of the strongest constraints on the photon sector of the non-minimal Standard-Model Extension. This paper reviews some recent results obtained by combining optical linear and circular…
The interaction of compact objects with an infinitely extended mirror plane due to quantum fluctuations of a scalar or electromagnetic field that scatters off the objects is studied. The mirror plane is assumed to obey either Dirichlet or…
We show that the limit, by rescaling, of the `new supersymmetric index' attached to the Fourier-Laplace transform of a polarized variation of Hodge structure on a punctured affine line is equal to the spectral polynomial attached to the…
The illumination conjecture is a classical open problem in convex and discrete geometry, asserting that every compact convex body~$K$ in $\mathbb R^n$ can be illuminated by a set of no more than $2^n$ points. If $K$ has smooth boundary, it…
In this article we consider the affinely-rigid body moving in the three-dimensional physical space and subject to the Kirchhoff-Love constraints, i.e., while it deforms homogeneously in the two-dimensional central plane of the body it…
We prove that, up to isometric congruence, there are exactly 2n+1 homogeneous polar foliations of the complex hyperbolic space. We also give an explicit description of each of these foliations.
An affine invariant point on the class of convex bodies in R^n, endowed with the Hausdorff metric, is a continuous map p which is invariant under one-to-one affine transformations A on R^n, that is, p(A(K))=A(p(K)). We define here the new…
We present numerical simulations, obtained with the Versatile Advection Code, of the oscillations of an inverse polarity prominence. The internal prominence equilibrium, the surrounding corona and the inert photosphere are well represented.…
We introduce the flag-approximability of a convex body to measure how easy it is to approximate by polytopes. We show that the flag-approximability is exactly half the volume entropy of the Hilbert geometry on the body, and that both…
Interpolating between the classic notions of intersection and polar centroid bodies, (real) $L_p$-intersection bodies, for $-1<p<1$, play an important role in the dual $L_p$-Brunn--Minkowski theory. Inspired by the recent construction of…
The relative equilibria of planar Newtonian $N$-body problem become coorbital around a central mass in the limit when all but one of the masses becomes zero. We prove a variety of results about the coorbital relative equilibria, with an…
A general theory of polariton parametric photoluminescence in spatially inhomogeneous systems is developed. The quantum Langevin equations are solved in a generalized Bogoliubov de Gennes approximation. We apply the formalism to the…
We present a probabilistic model of illuminating a convex body by independently distributed light sources. In addition to recovering C.A. Rogers' upper bounds for the illumination number, we improve previous estimates of J. Januszewski and…
We consider the process of light scattering by optical solitons in a planar waveguide with homogeneous and inhomogeneous refractive index core. We observe resonant reflection (Fano resonances) as well as resonant transmission of light by…