Related papers: On the shadow problem for domains in the Euclidean…
This paper is concerned with the problem of finding $n$ distinct squares such that, on excluding any one of them, the sum of the remaining $n-1$ squares is a square. While parametric solutions are known when $n=3$ and $n=4$, when $n > 4$,…
Cosmic expansion is expected to influence on the size of black hole shadow observed by comoving observer. Except the simplest case of Schwarzschild black hole in de Sitter universe, analytical approach for calculation of shadow size in…
For all $N \geq 9$, we find smooth entire epigraphs in $\R^N$, namely smooth domains of the form $\Omega : = \{x\in \R^N\ / \ x_N > F (x_1,\ldots, x_{N-1})\}$, which are not half-spaces and in which a problem of the form $\Delta u + f(u) =…
Given a finite grid in $\mathbb{R}^2$, how many lines are needed to cover all but one point at least $k$ times? Problems of this nature have been studied for decades, with a general lower bound having been established by Ball and Serra. We…
This paper proposes a mathematical justification of the phenomenon of extreme congestion at a very limited number of nodes in very large networks. It is argued that this phenomenon occurs as a combination of the negative curvature property…
Consider a bin containing $n$ balls colored with two colors. In a $k$-query, $k$ balls are selected by a questioner and the oracle's reply is related (depending on the computation model being considered) to the distribution of colors of the…
Isoperimetric inequalities have been studied since antiquity, and in recent decades they have been studied extensively on discrete objects, such as the hypercube. An important special case of this problem involves bounding the size of the…
In the space of holomorphic functions in a convex domain it is studied the interpolation problem by means of sums of the series of exponentials converging uniformly on all compact sets of the domain. The discrete set of the interpolation…
Compositing an object into an image involves multiple non-trivial sub-tasks such as object placement and scaling, color/lighting harmonization, viewpoint/geometry adjustment, and shadow/reflection generation. Recent generative image…
In this paper, we study the Dirichlet problem for the minimal surface equation in $\rm Sol_3$ with possible infinite boundary data, where $\rm Sol_3$ is the non-abelian solvable $3$-dimensional Lie group equipped with its usual…
A black hole embedded within a bright, optically thin emitting region imprints a nearly circular "shadow" on its image, corresponding to the observer's line-of-sight into the black hole. The shadow boundary depends on the black hole's mass…
For each i = 1, ..., n constructions are given for convex bodies K and L in n-dimensional Euclidean space such that each rank i orthogonal projection of K can be translated inside the corresponding projection of L, even though K has…
A subset of the sphere is said short if it is contained in an open hemisphere. A short closed set which is geodesically convex is called a cap. The following theorem holds: 1. The minimal number of short closed sets covering the $n$-sphere…
The problem of finding provably maximal sets of mutually unbiased bases in $\mathbb{C}^d$, for composite dimensions $d$ which are not prime powers, remains completely open. In the first interesting case, $d=6$, Zauner predicted that there…
We will consider the resolution of the 3D non linear wave equation under the assumption of spherical symmetry on the Euclidian space. For this purpose, we will build a non trivial measure on distributions such that there exists a set of…
In this article, we provide a review of the current state of the research of the black hole shadow, focusing on analytical (as opposed to numerical and observational) studies. We start with particular attention to the definition of the…
We study the variety of common tangents for up to four quadric surfaces in projective three-space, with particular regard to configurations of four quadrics admitting a continuum of common tangents. We formulate geometrical conditions in…
We consider a deposition model in which balls rain down at random towards a 2-dimensional surface, roll downwards over existing adsorbed balls, are adsorbed if they reach the surface, and discarded if not. We prove a spatial law of large…
It is a well-known fact -- which can be shown by elementary calculus -- that the volume of the unit ball in $\mathbb{R}^n$ decays to zero and simultaneously gets concentrated on the thin shell near the boundary sphere as $n \nearrow…
The Euclidean concentration inequality states that, among sets with fixed volume, balls have $r$-neighborhoods of minimal volume for every $r>0$. On an arbitrary set, the deviation of this volume growth from that of a ball is shown to…