Schur-convex functions of the 2nd order on $\mathbb {R}^n$
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M. Revyakov
Translated by: M. Revyakov - St. Petersburg Math. J. 31 (2020), 887-902
- DOI: https://doi.org/10.1090/spmj/1627
- Published electronically: September 3, 2020
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Abstract:
In the author’s earlier paper [Revyakov M., J. Multivariate Anal. 116 (2013) 25–34] concerning mathematical statistics, a need arose to employ functions called “Schur-convex functions of the 2nd order with respect to two variables”.
In the present paper, the class of Schur-convex functions of the 2nd order in $n$ variables is introduced. Necessary and sufficient conditions (in the form of analogs of the Sylvester criterion) are established for a function to belong to this class. Examples are given of using Schur-convex functions of the 2nd order for achieving maximal system reliability on the set of all possible allocations of elements into its subsystems.
References
- B. Bernstein and R. A. Toupin, Some properties of the Hessian matrix of a strictly convex function, J. Reine Angew. Math. 210 (1962), 65–72. MR 145024
- Emad El-Neweihi, Frank Proschan, and Jayaram Sethuraman, Optimal assembly of systems using Schur functions and majorization, Naval Res. Logist. 34 (1987), no. 5, 705–712. MR 906431, DOI 10.1002/1520-6750(198710)34:5<705::AID-NAV3220340510>3.0.CO;2-P
- F. R. Gantmacher, The theory of matrices. Vol. 1, AMS Chelsea Publishing, Providence, RI, 1998. Translated from the Russian by K. A. Hirsch; Reprint of the 1959 translation. MR 1657129
- Peter M. Gruber, Convex and discrete geometry, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 336, Springer, Berlin, 2007. MR 2335496
- Albert W. Marshall, Ingram Olkin, and Barry C. Arnold, Inequalities: theory of majorization and its applications, 2nd ed., Springer Series in Statistics, Springer, New York, 2011. MR 2759813, DOI 10.1007/978-0-387-68276-1
- Parametric equation, Encyclopedia of Mathematics. Vol. 4, Soviet Encycl., Moscow, 1984, 221–222. (Russian)
- V. Rajendra Prasad, K. P. K. Nair, and Y. P. Aneja, Optimal assignment of components to parallel-series and series-parallel systems, Oper. Res. 39 (1991), no. 3, 407–414. MR 1111628, DOI 10.1287/opre.39.3.407
- Mikhail Revyakov, Component allocation for a distributed system: reliability maximization, J. Appl. Probab. 30 (1993), no. 2, 471–477. MR 1212680, DOI 10.1017/s0021900200117498
- Mikhail Revyakov, Schur-convexity of 2nd order, certain subclass of multivariate arrangement increasing functions with applications in statistics, J. Multivariate Anal. 116 (2013), 25–34. MR 3049888, DOI 10.1016/j.jmva.2012.11.013
Bibliographic Information
- M. Revyakov
- Affiliation: St. Petersburg Department of Steklov Mathematical Institute, Russian Academy of Sciences , 27 Fontanka, St. Petersburg 191023, Russia
- Email: revyakov.m@gmail.com
- Received by editor(s): June 27, 2018
- Published electronically: September 3, 2020
- © Copyright 2020 American Mathematical Society
- Journal: St. Petersburg Math. J. 31 (2020), 887-902
- MSC (2010): Primary 47A07; Secondary 15B99, 26B25, 90B25
- DOI: https://doi.org/10.1090/spmj/1627
- MathSciNet review: 4022006
Dedicated: Dedicated to the memory of Ingram Olkin