 Author
 Title
 Nonobtuse simplices & special matrix classes
 Supervisors
 Cosupervisors
 Award date
 18 March 2016
 Number of pages
 195
 Document type
 PhD thesis
 Faculty
 Faculty of Science (FNWI)
 Institute
 Kortewegde Vries Institute for Mathematics (KdVI)
 Abstract

This thesis focuses on the study of certain special classes of nsimplices 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 ncube, the socalled 0/1simplices.
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 nsimplex.
Recognizing a nonnegative matrix as the inverse of an Mmatrix, without actually performing the inversion, is an open problem. Mmatrices 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 Mmatrices with nonobtuse simplices. This enables to study the inverse Mmatrix 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 0/1simplices, we enumerate all of them modulo the symmetries of the unit ncube, 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 0/1simplex in the unit ncube when n − 3 is a multiple of 4. In particular, we give a full description of acute 0/1simplices that can be represented by an irreducible upper Hessenberg matrix.  Note
 Research conducted at: Universiteit van Amsterdam
 Permalink
 http://hdl.handle.net/11245/1.517833
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