Related papers: Generalizations of the shadow problem

M\"uhlig et. al. propose and fabricate a "cloak" comprised of nano-particles on the surface of a sub-wavelength silica sphere. However, the coating only reduces the scattered fields. This is achieved by increased absorption, such that total…

At a first glance, the problem of illuminating the boundary of a convex body by external light sources and the problem of covering a convex body by its smaller positive homothetic copies appear to be quite different. They are in fact two…

Full generalization of Kasner metric for the case of $n+1$ dimensions and $m\le n+1$ essential variables is obtained. Any solution is defined by the corresponding constant matrix of Kasner parameters. This parameters form in euclidian space…

Global stability of the spherically symmetric nonisentropic compressible Euler equations with positive density around global-in-time background affine solutions is shown in the presence of free vacuum boundaries. Vacuum is achieved despite…

Let two distinct $N$-simplexes be given in an Euclidean or pseudo-Euclidean $N+1$ dimensional space as each is defined by the coordinates of its $N+1$ vertexes. We consider the two families of $N$-spheres passing through the vertexes of the…

We study smoothness of generalized solutions of nonlocal elliptic problems in plane bounded domains with piecewise smooth boundary. The case where the support of nonlocal terms can intersect the boundary is considered. We announce…

We address various notions of shadowing and expansivity for continuous maps restricted to a proper subset of their domain. We prove new equivalences of shadowing and expansive properties, we demonstrate under what conditions certain…

Polytopes are the basic finite data structures for convex sets: they appear as feasible regions in linear optimization, as geometric summaries in algorithms, and as random objects in stochastic geometry. A natural geometric question is…

Estimating the number of vertices of a two dimensional projection, called a shadow, of a polytope is a fundamental tool for understanding the performance of the shadow simplex method for linear programming among other applications. We prove…

We consider general relativity with cosmological constant minimally coupled to electromagnetic field and assume that four-dimensional space-time manifold is the warped product of two surfaces with Lorentzian and Euclidean signature metrics.…

We study the $\overline{\partial}$-Neumann problem using the Sobolev space inner product. We show that the problem can be solved on any smoothly bounded, pseudoconvex domain. We further formulate estimates and the basic results of a Sobolev…

A large class of Type II fluid solutions to Einstein field equations in N-dimensional spherical spacetimes is found, wich includes most of the known solutions. A family of the generalized collapsing Vaidya solutions with homothetic…

Affine transformations in Euclidean space generates a correspondence between integrable systems on cotangent bundles to the sphere, ellipsoid and hyperboloid embedded in $R^n$. Using this correspondence and the suitable coupling constant…

An approach is developed to find approximate solutions to the classical Newtonian problem of N bodies. Sets of N gravitating bodies having spherically symmetric mass distributions, small angular velocities (< 1 rad/s) and bounded position…

Inspired by the Kruskal-Katona theorem a minimization problem is studied, where the role of the shadow is replaced by the image of the action of the monoid of increasing functions. One of our main results shows that compressed sets are a…

In this paper, we investigate the existence and nonexistence of entire solutions to a general class of Cauchy problems in the positive half line. Our results provide a unified approach to proving sharp local and entire solvability of…

Motivated by applications in robotics and computer vision, we study problems related to spatial reasoning of a 3D environment using sublevel sets of polynomials. These include: tightly containing a cloud of points (e.g., representing an…

In this paper we prove the existence of a solution to the Dirichlet problem for harmonic maps into a geodesic ball on which the squared distance function from the origin is strictly convex. This improves a celebrated theorem obtained by S.…

We use non-perturbative U-duality symmetries of type II strings to construct new vacuum solutions. In some ways this generalizes the F-theory vacuum constructions. We find the possibilities of new vacuum constructions are very limited.…

In this paper, we derive general conditions for a shadow to occur, without a photon sphere in a spacetime, caused by central nulllike or timelike naked singularities. Using these conditions, we propose classes of spacetimes which have…