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We present a software suite for the analysis and optimization of ideal convex polyhedra in hyperbolic 3-space $\mathbb{H}^3$. Using Rivin's variational characterization of ideal polyhedra, we develop efficient algorithms for checking…
Despite the relevance of the binomial distribution for probability theory and applied statistical inference, its higher-order moments are poorly understood. The existing formulas are either not general enough, or not structured and…
Let $X$ be an $n$-dimensional manifold and $V_1,\ldots,V_n\subset C^\infty(X,\mathbb R)$ finite-dimensional vector spaces. For systems of equations $\{f_i = a_i\colon\: f_i\in V_i,\:a_i \in\mathbb R,\:i=1,\ldots,n\}$ we discover a…
An identity due to Efron dating from 1965 relates the expected volume of the convex hull of $n$ random points to the expected number of vertices of the convex hull of $n+1$ random points. Forty years later this identity was extended from…
In this note, we derive an asymptotically sharp upper bound on the number of lattice points in terms of the volume of centrally symmetric convex bodies. Our main tool is a generalization of a result of Davenport that bounds the number of…
We consider the Cauchy problem for the 1D generalized Schr\"odinger equation on the whole axis. To solve it, any order finite element in space and the Crank-Nicolson in time method with the discrete transparent boundary conditions (TBCs)…
We consider high dimensional random optimization problems where the dynamical variables are subjected to non-convex excluded volume constraints. We focus on the case in which the cost function is a simple quadratic cost and the excluded…
The simplex was conjectured to be the extremal convex body for the two following "problems of asymmetry":\\ P1) What is the minimal possible value of the quantity $\max_{K'} |K'|/|K|$? Here, $K'$ ranges over all symmetric convex bodies…
We construct a quasi-polynomial time deterministic approximation algorithm for computing the volume of an independent set polytope with restrictions. Randomized polynomial time approximation algorithms for computing the volume of a convex…
The isodiametric inequality states that the Euclidean ball maximizes the volume among all convex bodies of a given diameter. We are motivated by a conjecture of Makai Jr.~on the reverse question: Every convex body has a linear image whose…
Lattice QCD simulations tend to become stuck in a single topological sector at fine lattice spacing or when using chirally symmetric overlap quarks. In such cases physical observables differ from their full QCD counterparts by finite volume…
The present paper gives two concrete formulas for the volume of an arbitrary spherical tetrahedron, which is in a 3-dimensional spherical space of constant curvature +1. One formula is given in terms of dihedral angles, and another one is…
We consider the gravitational scattering of point particles in four dimensions, at Planckian centre of mass energy and low momentum transfer, or the eikonal approximation. The scattering amplitude can be exactly computed by modelling point…
We give a deterministic method of quasi-polynomial complexity to approximate the volume of the intersection of the unit hypercube with two specific sets. The method can actually be applied (without losing the quasi-polynomial complexity) to…
We consider two well-known problems: upper bounding the volume of lower dimensional ellipsoids contained in convex bodies given their John ellipsoid, and lower bounding the volume of ellipsoids containing projections of convex bodies given…
Volume estimates of metric balls in manifolds find diverse applications in information and coding theory. In this paper, some new results for the volume of a metric ball in unitary group are derived via various tools from random matrix…
Central limit theorems for the log-volume of a class of random convex bodies in $\mathbb{R}^n$ are obtained in the high-dimensional regime, that is, as $n\to\infty$. In particular, the case of random simplices pinned at the origin and…
Given five points in a three-dimensional euclidean space, one can consider five tetrahedra, using those points as vertices. We present a pentagon-like formula containing the product of three volumes of those tetrahedra in its l.h.s. and the…
Gravitational properties of a hedge-hog type topological defect in two extra dimensions are considered in General Relativity employing a vector as the order parameter. The developed macroscopic theory of phase transitions with spontaneous…
Consider a uniformly distributed random linear subspace $L$ and a stochastically independent random affine subspace $E$ in $\mathbb{R}^n$, both of fixed dimension. For a natural class of distributions for $E$ we show that the intersection…