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Applications of saddle-point determinants

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EN
Abstrakty
EN
For a given square matrix $A ∈ M_n({ℝ})$ and the vector $e ∈ (ℝ)^{n}$ of ones denote by (A,e) the matrix
⎡ A e ⎤
⎣ $e^{T}$ 0 ⎦
This is often called the saddle point matrix and it plays a significant role in several branches of mathematics. Here we show some applications of it in: game theory and analysis. An application of specific saddle point matrices that are hollow, symmetric, and nonnegative is likewise shown in geometry as a generalization of Heron's formula to give the volume of a general simplex, as well as a conditions for its existence.
Twórcy
autor
  • Adam Mickiewicz University, Poznań, Poland
  • College of William and Mary, Williamsburg, USA
  • The Jacob of Paradyż University of Applied Sciences, Gorzów Wlkp, Poland
Bibliografia
  • [1] R. Almeida, A mean value theorem for internal functions and an estimation for the differential mean point, Novi Sad J. Math. 38 (2) (2008), 57-64. doi: 10.1.1.399.9060
  • [2] M. Benzi, G.H. Golub and J. Liesen, Numerical solution of saddle point problems, Acta Numerica 14 (2005), 1-137. doi: 10.1017/S0962492904000212
  • [3] I.M. Bomze, On standard quadratic optimization problems, J. Global Optimization, 13 (1998), 369-387. doi: 10.1023/A:1008369322970
  • [4] R.H. Buchholz, Perfect pyramids, Bull. Austral. Math. Soc. 45 (1992), 353-368. doi: 10.1017/S0004972700030252
  • [5] H.S.M. Coxeter and S.L. Greitzer, Geometry Revisited, Washington, DC, Math. Assoc. Amer. 59 (1967), 117-119.
  • [6] H. Diener and I. Loeb, Constructive reverse investigations into differential equations, J. Logic and Analysis 3 (8) (2011), 1-26. doi: 10.4115/jla.2011.3.8
  • [7] W. Dunham, Heron's formula for triangular area, Ch. 5, Journey through Genius: The Great Theorems of Mathematics (New York, Wiley, 1990), 113-132.
  • [8] M. Griffiths, n-dimensional enrichment for further mathematicians, The Mathematical Gazette 89 (516) (2005), 409-416. doi: 10.2307/3621932
  • [9] M. Kline, Mathematical Thought from Ancient to Modern Times (Oxford, England, Oxford University Press, 1990).
  • [10] MathPages, Heron's Formula and Brahmagupta's Generalization, http://www.mathpages.com/home/kmath196.htm.
  • [11] K. Menger, Untersuehungen über allgemeine metrik, Math. Ann. 100 (1928), 75-165. doi: 10.1007/BF01448840
  • [12] T. Ostrowski, Population equilibrium with support in evolutionary matrix games, Linear Alg. Appl. 417 (2006), 211-219. doi: 10.1016/j.laa.2006.03.039
  • [13] T. Ostrowski, On some properties of saddle point matrices with vector blocks, Inter. J. Algebra 1 (2007), 129-138. doi: 10.1.1.518.8476
  • [14] G. Owen, Game Theory, Emerald Group Publishing, 2013.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1239
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