Advertisement

Applications of Mathematics

, Volume 64, Issue 1, pp 1–31 | Cite as

On the combinatorial structure of 0/1-matrices representing nonobtuse simplices

  • Jan BrandtsEmail author
  • Abdullah Cihangir
Article

Abstract

A 0/1-simplex is the convex hull of n+1 affinely independent vertices of the unit n-cube In. It is nonobtuse if none of its dihedral angles is obtuse, and acute if additionally none of them is right. Acute 0/1-simplices in In can be represented by 0/1-matrices P of size n× n whose Gramians G = PP have an inverse that is strictly diagonally dominant, with negative off-diagonal entries.

In this paper, we will prove that the positive part D of the transposed inverse P−⊤ of P is doubly stochastic and has the same support as P. In fact, P has a fully indecomposable doubly stochastic pattern. The negative part C of P−⊤ is strictly row-substochastic and its support is complementary to that of D, showing that P−⊤ = D−C has no zero entries and has positive row sums. As a consequence, for each facet F of an acute 0/1-facet S there exists at most one other acute 0/1-simplex Ŝ in In having F as a facet. We call Ŝ the acute neighbor of S at F.

If P represents a 0/1-simplex that is merely nonobtuse, the inverse of G = PΤP is only weakly diagonally dominant and has nonpositive off-diagonal entries. These matrices play an important role in finite element approximation of elliptic and parabolic problems, since they guarantee discrete maximum and comparison principles. Consequently, P−⊤ can have entries equal to zero. We show that its positive part D is still doubly stochastic, but its support may be strictly contained in the support of P. This allows P to have no doubly stochastic pattern and to be partly decomposable. In theory, this might cause a nonobtuse 0/1-simplex S to have several nonobtuse neighbors Ŝ at each of its facets.

In this paper, we study nonobtuse 0/1-simplices S having a partly decomposable matrix representation P. We prove that if S has such a matrix representation, it also has a block diagonal matrix representation with at least two diagonal blocks. Moreover, all matrix representations of S will then be partly decomposable. This proves that the combinatorial property of having a fully indecomposable matrix representation with doubly stochastic pattern is a geometrical property of a subclass of nonobtuse 0/1-simplices, invariant under all n-cube symmetries. We will show that a nonobtuse simplex with partly decomposable matrix representation can be split in mutually orthogonal simplicial facets whose dimensions add up to n, and in which each facet has a fully indecomposable matrix representation. Using this insight, we are able to extend the one neighbor theorem for acute simplices to a larger class of nonobtuse simplices.

Keywords

acute simplex nonobtuse simplex orthogonal simplex 0/1-matrix doubly stochastic matrix fully indecomposable matrix partly decomposable matrix 

MSC 2010

05B20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R. B. Bapat, T. E. S. Raghavan: Nonnegative Matrices and Applications. Encyclopedia of Mathematics and Applications 64, Cambridge University Press, Cambridge, 1997.CrossRefzbMATHGoogle Scholar
  2. [2]
    A. Berman, R. J. Plemmons: Nonnegative Matrices in the Mathematical Sciences. Classics in Applied Mathematics 9, SIAM, Philadelphia, 1994.CrossRefzbMATHGoogle Scholar
  3. [3]
    D. Braess: Finite Elements. Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press, Cambridge, 2001.zbMATHGoogle Scholar
  4. [4]
    J. Brandts, A. Cihangir: Counting triangles that share their vertices with the unit n-cube. Proc. Conf. Applications of Mathematics 2013 (J. Brandts et al., eds. ). Institute of Mathematics AS CR, Praha, 2013, pp. 1–12.zbMATHGoogle Scholar
  5. [5]
    J. Brandts, A. Cihangir: Geometric aspects of the symmetric inverse M-matrix problem. Linear Algebra Appl. 506 (2016), 33–81.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    J. Brandts, A. Cihangir: Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group. Spec. Matrices 5 (2017), 158–201.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    J. Brandts, S. Dijkhuis, V. de Haan, M. Křížek: There are only two nonobtuse triangulations of the unit n-cube. Comput. Geom. 46 (2013), 286–297.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    J. Brandts, S. Korotov, M. Křížek: Dissection of the path-simplex in Rn into n pathsubsimplices. Linear Algebra Appl. 421 (2007), 382–393.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    J. Brandts, S. Korotov, M. Křížek: The discrete maximum principle for linear simplicial finite element approximations of a reaction-diffusion problem. Linear Algebra Appl. 429 (2008), 2344–2357.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    J. Brandts, S. Korotov, M. Křížek, J. Šolc: On nonobtuse simplicial partitions. SIAM Rev. 51 (2009), 317–335.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    S. C. Brenner, L. R. Scott: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics 15, Spinger, New York, 1994.CrossRefGoogle Scholar
  12. [12]
    R. A. Brualdi: Combinatorial Matrix Classes. Encyclopedia of Mathematics and Its Applications 108, Cambridge University Press, Cambridge, 2006.CrossRefGoogle Scholar
  13. [13]
    R. A. Brualdi, H. J. Ryser: Combinatorial Matrix Theory. Encyclopedia of Mathematics and Its Applications 39, Cambridge University Press, Cambridge, 1991.Google Scholar
  14. [14]
    M. Fiedler: Über qualitative Winkeleigenschaften der Simplexe. Czech. Math. J. 7 (1957), 463–478. (In German. )zbMATHGoogle Scholar
  15. [15]
    N. A. Grigor’ev: Regular simplices inscribed in a cube and Hadamard matrices. Proc. Steklov Inst. Math. 152 (1982), 97–98.zbMATHGoogle Scholar
  16. [16]
    J. Hadamard: Résolution d’une question relative aux déterminants. Darboux Bull. (2) 17 (1893), 240–246. (In French. )zbMATHGoogle Scholar
  17. [17]
    C. R. Johnson: Inverse M-matrices. Linear Algebra Appl. 47 (1982), 195–216.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    C. R. Johnson, R. L. Smith: Inverse M-matrices II. Linear Algebra Appl. 435 (2011), 953–983.CrossRefGoogle Scholar
  19. [19]
    G. Kalai, G. M. Ziegler, (Eds. ): Polytopes—Combinatorics and Computation. DMV Seminar 29, Birkhäuser, Basel, 2000.zbMATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2019

Authors and Affiliations

  1. 1.Korteweg-de Vries Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands

Personalised recommendations