143 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
Factorization of CP-rank-3 completely positive matrices
A symmetric positive semi-definite matrix A is called completely positive if
there exists a matrix B with nonnegative entries such that A=BB^T. If B is such
a matrix with a minimal number p of columns, then p is called the cp-rank of A.
In this paper we develop a finite and exact algorithm to factorize any matrix A
of cp-rank 3. Failure of this algorithm implies that A does not have cp-rank 3.
Our motivation stems from the question if there exist three nonnegative
polynomials of degree at most four that vanish at the boundary of an interval
and are orthonormal with respect to a certain inner product.Comment: 13 pages, 10 figure
There are only two nonobtuse binary triangulations of the unit -cube
Triangulations of the cube into a minimal number of simplices without
additional vertices have been studied by several authors over the past decades.
For this so-called simplexity of the unit cube is now
known to be , respectively. In this paper, we study
triangulations of with simplices that only have nonobtuse dihedral
angles. A trivial example is the standard triangulation into simplices. In
this paper we show that, surprisingly, for each there is essentially
only one other nonobtuse triangulation of , and give its explicit
construction. The number of nonobtuse simplices in this triangulation is equal
to the smallest integer larger than .Comment: 17 pages, 7 figure
Generalization of the Zlámal condition for simplicial finite elements in
summary:The famous Zlámal's minimum angle condition has been widely used for construction of a regular family of triangulations (containing nondegenerating triangles) as well as in convergence proofs for the finite element method in . In this paper we present and discuss its generalization to simplicial partitions in any space dimension
Consolation : Trost Im Leid
https://digitalcommons.library.umaine.edu/mmb-ps/2849/thumbnail.jp
Generalization of the Zlámal condition for simplicial finite elements in ℝ d
The famous Zlámal's minimum angle condition has been widely used for construction of a regular family of triangulations (containing nondegenerating triangles) as well as in convergence proofs for the finite element method in 2d. In this paper we present and discuss its generalization to simplicial partitions in any space dimension
- …