Nonobtuse simplices & special matrix classes

This thesis focuses on the study of certain special classes of n-simplices that occur in the context of numerical analysis, linear algebra, abstract algebra, geometry, and combinatorics. The type of simplex that is of central interest is the nonobtuse simplex, a simplex without any obtuse dihedral angles. Nonobtuse simplices without right dihedral angles are called acute. Special attention will be paid to acute and nonobtuse simplices whose vertices are vertices of the unit n-cube, the so-called 0/1-simplices. Several qualitative properties of finite element approximations of PDEs do not allow simplices in the triangulation of the physical domain to have obtuse or even right dihedral angles. This motivates to investigate whether such triangulations of a given computational domain into nonobtuse or acute simplices actually exist. As a possible tool to tackle related restricted triangulation problems, we prove sharp upper and lower bounds on the sum of all the dihedral angles of a nonobtuse n-simplex. Recognizing a nonnegative matrix as the inverse of an M-matrix, without actually performing the inversion, is an open problem. M-matrices figure in iterative methods for linear systems, are used in numerical linear algebra to yield eigenvalue bounds, and appear in finite Markov chains. In this thesis, we associate the set of inverses of symmetric diagonally dominant M-matrices with nonobtuse simplices. This enables to study the inverse M-matrix problem from a geometric viewpoint, and we prove several results, in particular for simplices whose facets are all nonobtuse. To facilitate the study of acute and nonobtuse 0/1-simplices, we enumerate all of them modulo the symmetries of the unit n-cube, for some larger values of n. Our interest in this type of simplices comes from Hadamard’s maximal determinant conjecture which is equivalent to showing that there exists a regular 0/1-simplex in the unit n-cube when n − 3 is a multiple of 4. In particular, we give a full description of acute 0/1-simplices that can be represented by an irreducible upper Hessenberg matrix