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Mathematical Preliminaries

  • Abdo Y. Alfakih
Chapter

Abstract

In this chapter, we briefly review some of the mathematical preliminaries that will be needed throughout the monograph. These include a brief review of the most pertinent concepts and results in the theories of vector spaces, matrices, convexity, and graphs. Proofs of several of these results are included to make this chapter as self-contained as possible.

References

  1. 41.
    D.S. Bernstein, Matrix Mathematics: Theory, Facts and Formulas (Princeton University Press, Princeton, 2009)CrossRefGoogle Scholar
  2. 42.
    N. Biggs, Algebraic Graph Theory (Cambridge University Press, London, 1993)zbMATHGoogle Scholar
  3. 44.
    J.R.S. Blair, B. Peyton, An introduction to chordal graphs and clique trees, in Graph Theory and Sparse Matrix Computation, ed. by J.A. George, J.R. Gilbert, J.W.-H. Liu. IMA Volumes in Mathematics and Its Applications, vol. 56 (Springer, New York, 1993), pp. 1–29Google Scholar
  4. 47.
    J.A. Bondy, U.S.R. Murty, Graph Theory (Springer, New York, 2008)CrossRefGoogle Scholar
  5. 72.
    G.A. Dirac, On rigid circuit graphs. Abh. Math. Sem. Univ. Hamburg 25, 71–76 (1961)MathSciNetCrossRefGoogle Scholar
  6. 78.
    I. Fáry, On straight line representation of planar graphs. Acta Sci. Math. Szeged 11, 229–233 (1948)MathSciNetzbMATHGoogle Scholar
  7. 82.
    D.R. Fulkerson, O.A. Gross, Incidence matrices and interval graphs. Pac. J. Math. 15, 835–855 (1965)MathSciNetCrossRefGoogle Scholar
  8. 89.
    M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs. Annals of Discrete Mathematics, vol. 57 (Elsevier, Amsterdam, 2004)CrossRefGoogle Scholar
  9. 109.
    J.-B. Hiriart-Urruty, C. Lemaréchal, Fundamentals of Convex Analysis (Springer, Berlin, 2001)CrossRefGoogle Scholar
  10. 112.
    R.A. Horn, C.R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1985)CrossRefGoogle Scholar
  11. 113.
    R.A. Horn, C.R. Johnson, Topics in Matrix Analysis (Cambridge University Press, Cambridge, 1991)CrossRefGoogle Scholar
  12. 140.
    G. Marsaglia, G.P.H. Styan, When does rank (a+b)= rank a + rank b? Can. Math. Bull. 15, 451–452 (1972)MathSciNetCrossRefGoogle Scholar
  13. 149.
    J.J. Moreau, Décomposition orthogonale d’un espace hilbertien selon deux cônes mutuellement polaires. C. R. Acad. Sci Paris 255, 238–240 (1962)MathSciNetzbMATHGoogle Scholar
  14. 160.
    R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1970)CrossRefGoogle Scholar
  15. 161.
    D.J. Rose, R.E. Tarjan, G.S. Leuker, Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5, 266–283 (1976)MathSciNetCrossRefGoogle Scholar
  16. 166.
    R. Schneider, Convex Bodies: The Brunn-Minkowski Theory (Cambridge University Press, Cambridge, 1993)CrossRefGoogle Scholar
  17. 180.
    S. Straszewicz, Uber exponierte punkte abgeschlossener punktmengen. Fundam. Math. 24, 139–143 (1935)CrossRefGoogle Scholar
  18. 189.
    R. Thomas, Lecture notes on topology of Euclidean spaces. Georgia Tech., 1993Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Abdo Y. Alfakih
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of WindsorWindsorCanada

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