Majorization

Abstract: 1 Introduction to Locally Convex Topological Vector Spaces and Dual Pairs- 1 Locally Convex Vector Spaces- 2 Hahn-Banach Theorems- 3 Dual Pairs- Notes and Remarks- 2 Radon Measures and Integral Representations- 1 Introduction to Radon Measures on Hausdorff Spaces- 2 The Riesz Representation Theorem- 3 Weak Convergence of Finite Radon Measures- 4 Vague Convergence of Radon Measures on Locally Compact Spaces- 5 Introduction to the Theory of Integral Representations- Notes and Remarks- 3 General Results on Positive and Negative Definite Matrices and Kernels- 1 Definitions and Some Simple Properties of Positive and Negative Definite Kernels- 2 Relations Between Positive and Negative Definite Kernels- 3 Hubert Space Representation of Positive and Negative Definite Kernels- Notes and Remarks- 4 Main Results on Positive and Negative Definite Functions on Semigroups- 1 Definitions and Simple Properties 86 2 Exponentially Bounded Positive Definite Functions on Abelian Semigroups- 3 Negative Definite Functions on Abelian Semigroups- 4 Examples of Positive and Negative Definite Functions- 5 T-Positive Functions- 6 Completely Monotone and Alternating Functions- Notes and Remarks- 5 Schoenberg-Type Results for Positive and Negative Definite Functions- 1 Schoenberg Triples- 2 Norm Dependent Positive Definite Functions on Banach Spaces- 3 Functions Operating on Positive Definite Matrices- 4 Schoenberg's Theorem for the Complex Hilbert Sphere- 5 The Real Infinite Dimensional Hyperbolic Space- Notes and Remarks- 6 Positive Definite Functions and Moment Functions- 1 Moment Functions- 2 The One-Dimensional Moment Problem- 3 The Multi-Dimensional Moment Problem- 4 The Two-Sided Moment Problem- 5 Perfect Semigroups- Notes and Remarks- 7 Hoeffding's Inequality and Multivariate Majorization- 1 The Discrete Case- 2 Extension to Nondiscrete Semigroups- 3 Completely Negative Definite Functions and Schur-Monotonicity- Notes and Remarks- 8 Positive and Negative Definite Functions on Abelian Semigroups Without Zero- 1 Quasibounded Positive and Negative Definite Functions- 2 Completely Monotone and Completely Alternating Functions- Notes and Remarks- References- List of Symbols

Abstract: Fundamentals of operators and matrices.- Mappings and algebras.- Functional calculus and derivation.- Matrix monotone functions and convexity.- Matrix means and inequalities.- Majorization and singular values.- Some applications.

Abstract: We investigate balls-into-bins processes allocating m balls into n bins based on the multiple-choice paradigm. In the classical single-choice variant each ball is placed into a bin selected uniformly at random. In a multiple-choice process each ball can be placed into one out of $d \ge 2$ randomly selected bins. It is known that in many scenarios having more than one choice for each ball can improve the load balance significantly. Formal analyses of this phenomenon prior to this work considered mostly the lightly loaded case, that is, when $m \approx n$. In this paper we present the first tight analysis in the heavily loaded case, that is, when $m \gg n$ rather than $m \approx n$.The best previously known results for the multiple-choice processes in the heavily loaded case were obtained using majorization by the single-choice process. This yields an upper bound of the maximum load of bins of $m/n + {\mbox{$\cal O$}}(\sqrt{m \ln n \,/\, n})$ with high probability. We show, however, that the multiple-choice...