Two-sided combinatorial volume bounds for non-obtuse hyperbolic polyhedra

We give a method for computing upper and lower bounds for the volume of a non-obtuse hyperbolic polyhedron in terms of the combinatorics of the 1-skeleton. We introduce an algorithm that detects the geometric decomposition of good 3-orbifolds with planar singular locus and underlying manifold S ³. The volume bounds follow from techniques related to the proof of Thurston’s Orbifold Theorem, Schläfli’s formula, and previous results of the author giving volume bounds for right-angled hyperbolic polyhedra.     

Notes on right-angled Coxeter polyhedra in hyperbolic spaces

We show that (finite volume) right-angled Coxeter polyhedra may exist in hyperbolic spaces only in dimension at most 12. This is a slight improvement on a theorem of Potyagailo and Vinberg stated in Potyagailo and Vinberg (Comment Math Helv 80:1–12, 2005).     

On the volume and surface area of hyperbolic polyhedra

A truncated tetrahedron is a building block of hyperbolic 3-manifolds with totally geodesic boundary. We study the relation between the volume of a truncated tetrahedron and the area of its faces which form the boundary of manifolds.     

On hyperbolic right-angled polygons

In this paper we find conditions in order to construct hyperbolic right-angledN-gons with the lengths ofN-3 sides given.Explicit formulae for the length of a side in terms of the lengths ofN-3 non-adjacent sides are obtained.Partially supported by CICYT.     

Two-sided hyperbolic SVD

In this paper, we propose the two-sided hyperbolic SVD (2HSVD) for square matrices, i.e., A=UΣV^[∗], where U and V^[∗] are J-unitary (J=diag(±1)) and Σ is a real diagonal matrix of “double-hyperbolic” singular values. We show that, with some natural conditions, such decomposition exists without the use of hyperexchange matrices. In other words, U and V^[∗] are really J-unitary with regard to J and not some matrix which is permutationally similar to matrix J. We provide full characterization of 2HSVD and completely relate it to the semidefinite J-polar decomposition.     

Two-sided bounds for the logarithmic capacity of multiple intervals

Potential theory on the complement of a subset of the real axis attracts much attention in both function theory and applied sciences. This paper discusses one aspect of the theory — the logarithmic capacity of closed subsets of the real line. We give simple but precise upper and lower bounds for the logarithmic capacity of multiple intervals and a lower bound valid also for closed sets comprising an infinite number of intervals. Using some known methods to compute the exact values of capacity, we demonstrate graphically how our estimates compare with them. The main machinery behind our results are the separating transformation and dissymmetrization developed by V. N. Dubinin and a version of the latter due to K. Haliste, as well as some classical symmetrization and projection results for the logarithmic capacity. The results of the paper improve some previous achievements by A. Yu. Solynin and K. Shiefermayr.     

Two-sided a posteriori error bounds for electro-magnetostatic problems

This paper is concerned with the derivation of computable and guaranteed upper and lower bounds of the difference between exact and approximate solutions of a boundary value problem for static Maxwell equations. Our analysis is based upon purely functional argumentation and does not invoke specific properties of the approximation method. For this reason, the estimates derived in the paper at hand are applicable to any approximate solution that belongs to the corresponding energy space. Such estimates (also called error majorants of the functional type) have been derived earlier for elliptic problems. Bibliography: 24 titles.     

Two-sided a posteriori error bounds for incompressible quasi-Newtonian flows

Endre Süli ^{We develop a posteriori upper and lower error bounds for mixedfinite-element approximations of a general family of steady,viscous, incompressible quasi-Newtonian flows in a bounded Lipschitzdomain ; thefamily includes degenerate models such as the power law model,as well as non-degenerate ones such as the Carreau model. Theunified theoretical framework developed herein yields residual-baseda posteriori bounds which measure the error in the approximationof the velocity in the W1, r() norm and that of the pressurein the Lr'() norm, 1/r + 1/r' = 1, r (1, ).}     

Two-sided bounds for linked unknown nonlinear boundary conditions of reaction-diffusion

Two enzymes bound at opposite ends of a finite interval affect each other via activation and/or inhibition by their respective products. The local concentrations of the diffusing products, in the vicinity of the other enzyme, determines the rate of production by that enzyme of its product. A mathematical model (cf. Thames and Elster [J. Theor. Biol. 59 (1976), 415–427]) consists of linear diffusion equations coupled through unknown and nonlinear boundary conditions. When the (nonlinear) functions describing the boundary conditions have certain monotone properties it is shown that the boundary values can be found iteratively by means of convergent two sided bounds. Some results for reaction chains involving more that two enzymes are presented.     

Dirichlet polyhedra for parabolic cyclic groups in complex hyperbolic space

We consider cyclic groupsG generated by an ellipto-parabolic isometry of complex hyperbolic space. We show that the Dirichlet fundamental polyhedron forG centred atz ₀ has two faces ifz ₀ is on the axis of the generator, otherwise it has infinitely many faces.     

A new formula for the volume of lattice polyhedra

LetL _n be the lattice consisting of all pointsx inR ^N such thatnx belongs to the fundamental latticeL ₁ of points with integer coordinates. When the vertices of a polyhedronP inR ^N are restricted to lie inL ₁ there is a formula which relates the volume ofP to the numbers of points ofL ₁,...,L _N in the interior and on the boundary ofP. The aim of this note is to show that the volume ofP can be determined only by means of the numbers of points ofL ₁,...,L _N lying in the interior ofP and cannot be expressed by the numbers of points ofL ₁,...,L _N lying on the boundary ofP. The latter numbers in turn can be used to compute to comopute the Euler characteristic of the boundary ofP.     

Upper bounds for the diameter and height of graphs of convex polyhedra

Let Δ(d, n) be the maximum diameter of the graph of ad-dimensional polyhedronP withn-facets. It was conjectured by Hirsch in 1957 that Δ(d, n) depends linearly onn andd. However, all known upper bounds for Δ(d, n) were exponential ind. We prove a quasi-polynomial bound Δ(d, n)≤n ^{2 logd+3}. LetP be ad-dimensional polyhedron withn facets, let ϕ be a linear objective function which is bounded onP and letv be a vertex ofP. We prove that in the graph ofP there exists a monotone path leading fromv to a vertex with maximal ϕ-value whose length is at most . This research was supported in part by a BSF grant, by a GIF grant, and by ONR-N00014-91-C0026.     

Dirichlet polyhedra for dihedral groups acting on complex hyperbolic space

We consider groups Γ generated by inversions in a pair of asymptotic complex hyperplanes in complex hyperbolic spaceH _? ⁿ . We show that there exists a Γ-invariant real hypersurfaceF ?H _? ⁿ such that the Dirichlet fundamental polyhedron for Γ centered at z₀ has two sides (resp. infinitely many sides) if and only ifz ₀ ∈F (resp.z ₀ ?F). The Dirichlet regions are determined explicitly in terms of coordinates on Γ-invariant horospheres and the geometry ofH _? ⁿ is developed in terms of these horospherical coordinates.     

Two-sided bounds of eigenvalues of second-and fourth-order elliptic operators

This article presents an idea in the finite element methods (FEMs) for obtaining two-sided bounds of exact eigenvalues. This approach is based on the combination of nonconforming methods giving lower bounds of the eigenvalues and a postprocessing technique using conforming finite elements. Our results hold for the second and fourth-order problems defined on two-dimensional domains. First, we list analytic and experimental results concerning triangular and rectangular nonconforming elements which give at least asymptotically lower bounds of the exact eigenvalues. We present some new numerical experiments for the plate bending problem on a rectangular domain. The main result is that if we know an estimate from below by nonconforming FEM, then by using a postprocessing procedure we can obtain two-sided bounds of the first (essential) eigenvalue. For the other eigenvalues λ_l, l = 2, 3, …, we prove and give conditions when this method is applicable. Finally, the numerical results presented and discussed in the paper illustrate the efficiency of our method.     

Two-sided bounds on the rate of convergence to a stable law

Summary We derive necessary and sufficient conditions for several characterizations of the rate of convergence of a sum of independent variables to a stable law. The technique used is to obtain upper and lower bounds on the rate in terms of functions depending in a very simple way on the common underlying distribution. This permits a general approach to the problem of rates of convergence.     

Convex polyhedra in hyperbolic space and interpolation of functions

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