Related papers: Archimedes' Revenge
We suggest a method of computing volume for a simple polytope $P$ in three-dimensional hyperbolic space $\mathbb{H}^3$. This method combines the combinatorial reduction of $P$ as a trivalent graph $\Gamma$ (the $1$-skeleton of $P$) by…
Using a recent alternative form of the Kolmogorov-Monin exact relation for fully developed hydrodynamics (HD) turbulence, the incompressible energy cascade rate $\varepsilon$ is computed. Under this current theoretical framework, for…
Let $K$ be a convex body in $\mathbb{R}^{3}$. We denote the volume of $K$ by $Vol(K)$ and the diameter of $K$ by $Diam(K).$ In this paper we prove that there exists a linear bijection $T:\mathbb{R}^{3}\to \mathbb{R}^{3}$ such that…
We introduce a direct method allowing to solve numerically inverse type problems for linear hyperbolic equations. We first consider the reconstruction of the full solution of the wave equation posed in $\Omega\times (0,T)$ - $\Omega$ a…
We study the slices or sections of a convex polytope by affine hyperplanes. We present results on two key problems: First, we provide tight bounds on the maximum number of vertices attainable by a hyperplane slice of $d$-polytope (a sort of…
We use the Legendre polynomials and the Hermite polynomials as two examples to illustrate a simple and systematic technique on deriving asymptotic formulas for orthogonal polynomials via recurrence relations. Another application of this…
In this paper we study geometric coincidence problems in the spirit of the following problems by B. Gr\"unbaum: How many affine diameters of a convex body in $\mathbb R^n$ must have a common point? How many centers (in some sense) of…
We present a method to compute the volume of a solid of revolution as a double integral in a very simple way. Then, we see that the classical methods (disks and shells) are recovered if this double integral is computed by each of the two…
In this note, we investigate the maximal number of intersection points of a line with the contour of hypersurface amoebas in $\mathbb{R}^n$. We define the latter number to be the $\mathbb{R}$-degree of the contour. We also investigate the…
We propose a method to compute the numerical solutions of a polynomial system in complete intersection. This algorithm makes use of Bezout matrices and need only linear algebra computations. All the calculations can be done in floating…
In this paper, we obtain order estimates for the Gelfand widths of intersections of finite-dimensional balls under some conditions on parameters.
We give a volume formula of hyperbolic knot complements using twisted Alexander invariants.
We approximately compute the correspondence degree (as defined by Lazarsfeld and Martin) between two unbalanced complete intersections. This is accomplished by showing that the procedure of taking a subvariety of a product $Y \times Y'$ and…
We construct hyperbolic integer homology 3-spheres where the injectivity radius is arbitrarily large for nearly all points of the manifold. As a consequence, there exists a sequence of closed hyperbolic 3-manifolds which Benjamini-Schramm…
We compute the number of triangulations of a convex $k$-gon each of whose sides is subdivided by $r-1$ points. We find explicit formulas and generating functions, and we determine the asymptotic behaviour of these numbers as $k$ and/or $r$…
These are mostly expository notes based on the course of lectures on arithmetic invariants of hyperbolic manifolds given at the workshop associated with the final "Volume Conference," held at Columbia University, June 2009. Some new results…
We consider the problem of determination of a volume of some bounded space-like hypersurfaces in the case of spherically symmetric spacetimes. In the case when the hypersurfaces is cut or bounded by a light-like hypersurface the problem may…
A method is described to sum multi-dimensional arithmetic functions subject to hyperbolic summation conditions, provided that asymptotic formulae in rectangular boxes are available. In combination with the circle method, the new method is a…
Order estimates for the Kolmogorov widths of a finite-dimensional balls intersection are obtained.
The Hermite interpolation formulas are based on the interpretation of interpolation nodes as roots of suitable polynomials. Therefore, such formulas belong to the class of algebraic interpolations. The article considers a multidimensional…