Related papers: Volumes of Solid Objects in Elamite Mathematics
We deduce explicit formulae for the intrinsic volumes of an ellipsoid in $\mathbb R^d$, $d\ge 2$, in terms of elliptic integrals. Namely, for an ellipsoid ${\mathcal E}\subset \mathbb R^d$ with semiaxes $a_1,\ldots, a_d$ we show that…
We consider the moments of the volume of the symmetric convex hull of independent random points in an $n$-dimensional symmetric convex body. We calculate explicitly the second and fourth moments for $n$ points when the given body is $B_q^n$…
We give a linear upper bound on the number of distinct volume-equivalent frameworks of bipyramids, up to rigid motions. As a corollary, we show that global volume rigidity is not a generic property of simplicial complexes.
We compare the volume of a hyperbolic 3-manifold $M$ of finite volume and the complexity of its fundamental group.
For every large enough $n$, we explicitly construct a body of constant width $2$ that has volume less than $0.9^n \text{Vol}(\mathbb{B}^{n}$), where $\mathbb{B}^{n}$ is the unit ball in $\mathbb{R}^{n}$. This answers a question of…
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…
We study spherical tetrahedra with rational dihedral angles and rational volumes. Such tetrahedra occur in the Rational Simplex Conjecture by Cheeger and Simons, and we supply vast families, discovered by computational efforts, of positive…
We present a numerical investigation of the dynamics of one falling oblate ellipsoid particle in a viscous fluid, in three dimensions, using a constrained-force technique \cite{Kai}, \cite{Kaih} and \cite{Esa}. We study the dynamical…
In this paper we investigate the problem of finding the maximum volume polytopes, inscribed in the unit sphere of the $d$-dimensional Euclidean space, with a given number of vertices. We solve this problem for polytopes with $d+2$ vertices…
We present explicit expressions of the thermodynamic volume inside and outside the cosmological horizon of Eguchi-Hanson solitons in general odd dimensions. These quantities are calculable and well-defined regardless of whether or not the…
Let $\mathcal{E}$ be an ellipsoid in $\mathbb{R}^n$. A. Gusakova and D. Zaporozhets conjectured that $\mathcal{E}$ is uniquely (up to rigid motions) determined by its intrinsic volumes. We prove this conjecture for $n = 3$.
Estimating the volume of a convex body is a canonical problem in theoretical computer science. Its study has led to major advances in randomized algorithms, Markov chain theory, and computational geometry. In particular, determining the…
Gromov and Piatetski-Shapiro proved existence of finite volume non-arithmetic hyperbolic manifolds of any given dimension. In dimension four and higher, we show that there are about v^v such manifolds of volume at most v, considered up to…
The main purpose of this paper is to give an explicit formula for the volumes of indefinite ternary lattices with square free discriminant. We also consider several examples of lattices, for which the fundamental domain is well-known and…
For closed and oriented hyperbolic surfaces, a formula of Witten establishes an equality between two volume forms on the space of representations of the surface in a semisimple Lie group. One of the forms is a Reidemeister torsion, the…
The present paper regards the volume function of a doubly truncated hyperbolic tetrahedron. Starting from the previous results of J. Murakami, U. Yano and A. Ushijima, we have developed a unified approach to express the volume in different…
This paper collects some important formulas on hyperbolic volume. To determine concrete values of volume function is a very hard question requiring the knowledge of various methods. Our goal to give a non-elementary integral on the volume…
The aim of this work is to expose some asymptotic series associated to some expressions involving the volume of the n-dimensional unit ball. All proofs and the methods used for improving the classical inequalities announced in the final…
Paralleling our recent computationally-intensive work for the case N=4 (quant-ph/0308037), we undertake the task for N=6 of computing to high numerical accuracy, the formulas of Sommers and Zyczkowski (quant-ph/0304041) for the…
This work rests upon the certainty that only fields of real and complex numbers, quaternions and octonions have algebras of all four arithmetical operations. Also quaternions are good to represent 3-dimensional Euclid space and…