In this note, we study the n×n random Euclidean matrix whose entry (i, j) is equal to f(‖Xi−Xj‖) for some function f and the Xi’s are i.i.d. isotropic vectors inR. In the regime where n and p both grow to infinity and are proportional, we give some sufficient conditions for the empirical distribution of the eigenvalues to converge weakly. We illustrate our result on log-concave random vectors.

Copyright q 2010 Y.-M. Chu and X.-M. Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We prove that G x, y | ∫x y f t dt| is multiplicatively concave on a, b × a, b if f : a, b ⊂ 0,∞ → 0,∞ is continuous and multiplicatively ...

inf u 6 Ttu(x) 6 u(x) 6 Ťtu(x) 6 supu for each t > 0 and each x ∈ H. A function u : H −→ R is called k-semi-concave, k > 0, if the function x −→ u(x)−‖x‖2/k is concave. The function u is called k-semi-convex if −u is k-semiconcave. A bounded function u is t-semi-concave if and only if it belongs to the image of the operator Tt, this follows from Lemma 1 and Lemma 3 below. A function is called s...

A function F defined on all subsets of a finite ground set E is quasiconcave if F (X∪Y ) ≥ min{F (X), F (Y )} for all X, Y ⊂ E. Quasi-concave functions arise in many fields of mathematics and computer science such as social choice, theory of graph, data mining, clustering and other fields. The maximization of quasi-concave function takes, in general, exponential time. However, if a quasi-concav...

Rubin and Narsimhan [1984] proposed a method for formulating fuzzy priorities in goal programming. They used the concept that the membership function based on nested priorities has to be concave to solve the fuzzy goal programming problem by linear programming method. In this paper, we relax the condition of concave membership function by quasi concave membership function. We showed that the fu...

The article presents conditions under which the probability of a linear combination random vectors falling into polyhedral cone is Schur-concave function coefficients combination. It required that contains point 0, its edges are parallel to coordinate axes, and distribution density logarithmically concave sign-invariant function.

Very recently, Ehrenborg and Steingrı́msson [7] studied enumerative properties of the excedance statistic. Let Sn denote the permutation group on the set {1, 2, . . . , n} and π = π1 π2 · · ·πn ∈ Sn . An excedance in π is an index i such that πi > i . Following [7], we encode the excedance set of a permutation as a word in the letters a and b. The excedance word w(π) of π is the ab-word w1w2 · ·...

Log-concave and Log-convex sequences arise often in combinatorics, algebra, probability and statistics. There has been a considerable amount of research devoted to this topic in recent years. Let {xi}i≥0 be a sequence of non-negative real numbers. We say that {xi} is Log-concave ( Log-convex resp.) if and only if xi−1xi+1 ≤ xi (xi−1xi+1 ≥ xi resp.) for all i ≥ 1 (relevant results can see [2] an...