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Quantile-Locating Functions And The Distance Between The Mean And Quantiles, D. Gilat, Theodore P. Hill

Research Scholars in Residence

Given a random variable X with finite mean, for each 0 < p < 1, a new sharp bound is found on the distance between a p-quantile of X and its mean in terms of the central absolute first moment of X. The new bounds strengthen the fact that the mean of X is within one standard deviation of any of its medians, as well as a recent quantile-generalization of this fact by O'Cinneide.

Isosymmetric Linear Transformations On Complex Hilbert Space, Mark Stankus

Mathematics

We explore the elementary operator theory of the equation
(0.1) ∑ c_m,nT^*nT^m=0
for c_m,n E C, c_m,n nonzero for only finitely many m, n and Ta bounded linear transformation on a complex Hilbert space in Chapters 1 and 2. We explore the equation

(0.2) T^*2T - T^*T² + T - T^* = 0
in greater depth in Chapters 4 and 5.

Chapter 1 explores the algebraic and C^*-algebraic aspects of the equation (0.1) and both the spectral picture of and growth conditions on …

Orbifolds Of Maximal Diameter, Joseph Borzellino

Mathematics

In this paper the Maximal Diameter Theorem of Riemannian geometry is proven for Riemannian orbifolds. In particular, it is shown that a complete Riemannian orbifold with Ricci curvature bounded below by (n−1) and diameter = π, must have constant sectional curvature 1, and must be a quotient of the sphere (Sn, can) of constant sectional curvature 1 by a subgroup of the orthogonal group O(n+1) acting discontinuously and isometrically on Sn. It is also shown that the singular locus of the orbifold forms a geometric barrier to the length minimization property of geodesics. We also extend the Bishop relative volume …

Partitioning Inequalities In Probability And Statistics, Theodore P. Hill

Research Scholars in Residence

This article surveys fair-division or cake-cutting inequalities in probability statistics, including bisection inequalities, basic fairness inequalities, convexity tools, superfairness inequalities, and partitioning inequalities hypotheses testing and optimal stopping theory. The emphasis is measure theoretic, as opposed to game theoretic or economic, and a number of open problems in the area are mentioned.