Related papers: On the shadow problem for domains in the Euclidean…
We study the minimal surface equation in the Heisenberg space, Nil_3. A geometric proof of non existence of minimal graphs over non convex, bounded and unbounded domains is achieved (our proof holds in the Euclidean space as well). We solve…
We consider the energy of the torsion problem with Robin boundary conditions in the case where the solution is not a minimizer. Its dependence on the volume of the domain and the surface area of the boundary is discussed. In contrast to the…
We study a variational problem for the perimeter associated with the Grushin plane, called minimal partition problem with trace constraint. This consists in studying how to enclose three prescribed areas in the Grushin plane, using the…
We compare shadows cast by Schwarzschild black holes with those produced by two classes of naked singularities that result from gravitational collapse of spherically symmetric matter. The latter models consist of an interior naked…
It is known since 1954 that every 3-manifold bounds a 4-manifold. Thus, for instance, every 3-manifold has a surgery diagram. There are several proofs of this fact, including constructive proofs, but there has been little attention to the…
For any three nonzero vectors $a,b,c$ in $\mathbb R^2$, we obtain a necessary and sufficient condition for the sum of the three pairwise angles between these vectors to equal $2\pi$. As an easy consequence of this, a proof of Euclid's…
We give the first tight sample complexity bounds for shadow tomography and classical shadows in the regime where the target error is below some sufficiently small inverse polynomial in the dimension of the Hilbert space. Formally we give a…
The system of falling balls is an autonomous Hamiltonian system with a smooth invariant measure and non-zero Lyapunov exponents almost everywhere. Since almost three decades, the question of ergodicity is still open. The subject of this…
A subset of the d-dimensional Euclidean space having nonempty interior is called a spindle convex body if it is the intersection of (finitely or infinitely many) congruent d-dimensional closed balls. The spindle convex body is called a…
We study the shadow cast by a large gravitating sphere, similar to our Sun. For this, we consider the gravitational field produced by a static mass monopole within the first post-Newtonian approximation of the general theory of relativity.…
We introduce the notion of conditional Lipschitz shadowing, which does not aim to shadow every pseudo-orbit, but only those which belong to a certain prescribed set. We establish two types of sufficient conditions under which certain…
We demonstrate that there is a large class of compact metric spaces for which the shadowing property can be characterized as a structural property of the space of dynamical systems. We also demonstrate for this class of spaces, that in…
In any CAT(k) space M, the "shadow" of a tangent vector Z at a point p is the set vectors that form an angle of \pi or more with Z. Taking logarithm maps at points approaching p along a fixed geodesic ray from p with tangent Z collapses the…
We investigate the intersections of balls of radius $r$, called $r$-ball bodies, in Euclidean $d$-space. An $r$-lense (resp., $r$-spindle) is the intersection of two balls of radius $r$ (resp., balls of radius $r$ containing a given pair of…
We prove a sharp bound for the remainder term of the number of lattice points inside a ball, when averaging over a compact set of (not necessarily unimodular) lattices, in dimensions two and three. We also prove that such a bound cannot…
The ball-constrained weighted maximin dispersion problem $(\rm P_{ball})$ is to find a point in an $n$-dimensional Euclidean ball such that the minimum of the weighted Euclidean distance from given $m$ points is maximized. We propose a new…
We present filling as a type of spatial subdivision problem similar to covering and packing. Filling addresses the optimal placement of overlapping objects lying entirely inside an arbitrary shape so as to cover the most interior volume. In…
The paper explores the Rutherford scattering shadow in an entire class of comoving frames -- inertial frames moving along the initial projectile direction -- of which the laboratory frame, where the target is initially at rest, is a…
The Hairy Ball Theorem states that every continuous tangent vector field on an even-dimensional sphere must have a zero. We prove that the associated computational problem of (a) computing an approximate zero is PPAD-complete, and (b)…
We consider the problem of finding the minimal number of points required to intersect all lines in an affine space over the finite field of order 3. We also consider the problem of finding the minimal number of points required to intersect…