Let IH be the n-dimensional hyperbolic space and let P be a simple polytope in IH. P is called an ideal polytope if all vertices of P belong to the boundary of IH. P is called a Coxeter polytope if all dihedral angles of P are submultiples of π. There is no complete classification of hyperbolic Coxeter polytopes. In [6] Vinberg proved that there are no compact hyperbolic Coxeter polytopes in IH...

Suppose that W is an infinite Coxeter group of finite rank n, and suppose that W has a finite parabolic subgroup WJ of rank n 1. Suppose also that the Coxeter diagram of W has no edges with infinite labels. Then any automorphism of W that preserves reflections lies in the subgroup of AutðWÞ generated by the inner automorphisms and the automorphisms induced by symmetries of the Coxeter graph. If...

An example of SDPs (semide nite programs) exhibits a substantial di culty in proving the superlinear convergence of a direct extension of the Mizuno-Todd-Ye type predictorcorrector primal-dual interior-point method for LPs (linear programs) to SDPs, and suggests that we need to force the generated sequence to converge to a solution tangentially to the central path (or trajectory). A Mizuno-Todd...

A novel primal-dual path-following interior-point algorithm for the Cartesian P∗(κ)-linear complementarity problem over symmetric cones is presented. The algorithm is based on a reformulation of the central path for finding the search directions. For a full Nesterov-Todd step feasible interior-point algorithm based on the new search directions, the complexity bound of the algorithm with small-u...

Extended Abstract 1 Preliminaries. Following the spectral graph theory, a graph coloring technique and algebraic methods in graph theory (see [5], [13]), we continue a Coxeter spectral study the category Bigrn of connected non-negative loop-free edge-bipartite (signed) graphs ∆, with n ≥ 1 vertices (bigraphs, in short), and their morsifications introduced by the second named author in [22]-[25]...

In this paper, we show that the center of every Coxeter group is finite and isomorphic to (Z2) n for some n ≥ 0. Moreover, for a Coxeter system (W, S), we prove that Z(W ) = Z(W S\S̃) and Z(W S̃ ) = 1, where Z(W ) is the center of the Coxeter group W and S̃ is the subset of S such that the parabolic subgroup W S̃ is the essential parabolic subgroup of (W, S) (i.e. W S̃ is the minimum parabolic subgr...

Groups defined by presentations of the form 〈s1, . . . , sn | si = 1, (sisj)i,j = 1 (i, j = 1, . . . , n)〉 are called Coxeter groups. The exponents mi,j ∈ N ∪ {∞} form the Coxeter matrix, which characterizes the group up to isomorphism. The Coxeter groups that are most important for applications are the Weyl groups and affine Weyl groups. For example, the symmetric group Sn is isomorphic to the...

In this note we consider a large step modiication of the Mizuno-Todd-Ye O(p nL) predictor-corrector interior-point algorithm for linear programming. We demonstrate that the modiied algorithm maintains its O(p nL)-iteration complexity, while exhibiting superlinear convergence for general problems and quadratic convergence for non-degenerate problems. To our knowledge, this is the rst constructio...