1,354 research outputs found
Geometric aspects of the symmetric inverse M-matrix problem
We investigate the symmetric inverse M-matrix problem from a geometric
perspective. The central question in this geometric context is, which
conditions on the k-dimensional facets of an n-simplex S guarantee that S has
no obtuse dihedral angles. First we study the properties of an n-simplex S
whose k-facets are all nonobtuse, and generalize some classical results by
Fiedler. We prove that if all (n-1)-facets of an n-simplex S are nonobtuse,
each makes at most one obtuse dihedral angle with another facet. This helps to
identify a special type of tetrahedron, which we will call sub-orthocentric,
with the property that if all tetrahedral facets of S are sub-orthocentric,
then S is nonobtuse. Rephrased in the language of linear algebra, this
constitutes a purely geometric proof of the fact that each symmetric
ultrametric matrix is the inverse of a weakly diagonally dominant M-matrix.
Review papers support our belief that the linear algebraic perspective on the
inverse M-matrix problem dominates the literature. The geometric perspective
however connects sign properties of entries of inverses of a symmetric positive
definite matrix to the dihedral angle properties of an underlying simplex, and
enables an explicit visualization of how these angles and signs can be
manipulated. This will serve to formulate purely geometric conditions on the
k-facets of an n-simplex S that may render S nonobtuse also for k>3. For this,
we generalize the class of sub-orthocentric tetrahedra that gives rise to the
class of ultrametric matrices, to sub-orthocentric simplices that define
symmetric positive definite matrices A with special types of k x k principal
submatrices for k>3. Each sub-orthocentric simplices is nonobtuse, and we
conjecture that any simplex with sub-orthocentric facets only, is
sub-orthocentric itself.Comment: 42 pages, 20 figure
Area-angle variables for general relativity
We introduce a modified Regge calculus for general relativity on a
triangulated four dimensional Riemannian manifold where the fundamental
variables are areas and a certain class of angles. These variables satisfy
constraints which are local in the triangulation. We expect the formulation to
have applications to classical discrete gravity and non-perturbative approaches
to quantum gravity.Comment: 7 pages, 1 figure. v2 small changes to match published versio
Maximal rank root subsystems of hyperbolic root systems
A Kac-Moody algebra is called hyperbolic if it corresponds to a generalized
Cartan matrix of hyperbolic type. We study root subsystems of root systems of
hyperbolic algebras. In this paper, we classify maximal rank regular hyperbolic
subalgebras of hyperbolic Kac-Moody algebras.Comment: 16 pages, 19 figures, 1 tabl
Rolling of Coxeter polyhedra along mirrors
The topic of the paper are developments of -dimensional Coxeter polyhedra.
We show that the surface of such polyhedron admits a canonical cutting such
that each piece can be covered by a Coxeter -dimensional domain.Comment: 20pages, 15 figure
Fat 4-polytopes and fatter 3-spheres
We introduce the fatness parameter of a 4-dimensional polytope P, defined as
\phi(P)=(f_1+f_2)/(f_0+f_3). It arises in an important open problem in
4-dimensional combinatorial geometry: Is the fatness of convex 4-polytopes
bounded?
We describe and analyze a hyperbolic geometry construction that produces
4-polytopes with fatness \phi(P)>5.048, as well as the first infinite family of
2-simple, 2-simplicial 4-polytopes. Moreover, using a construction via finite
covering spaces of surfaces, we show that fatness is not bounded for the more
general class of strongly regular CW decompositions of the 3-sphere.Comment: 12 pages, 12 figures. This version has minor changes proposed by the
second refere
- …