For 0 < q ≤ 2, 1 ≤ k < n, let X = (X1, ...,Xn) and Y = (Y1, ..., Yn) be symmetric q-stable random vectors so that the joint distributions of X1, ...,Xk and Xk+1, ...,Xn are equal to the joint distributions of Y1, ..., Yk and Yk+1, ..., Yn, respectively, but Yi and Yj are independent for every 1 ≤ i ≤ k, k + 1 ≤ j ≤ n. We prove that E(f(X)) ≥ E(f(Y )) where f is any continuous, positive, homogen...

We investigate a Maclaurin inequality for vectors and its connection to an Aleksandrov-type parallelepipeds.

The Gaussian product inequality is an important conjecture concerning the moments of random vectors. While all attempts to prove in full generality have been unsuccessful date, numerous partial results derived recent decades and we provide here further on problem. Most importantly, establish a strong version for multivariate gamma distributions case nonnegative correlations, thereby extending r...

This note gives a simple analysis of a randomized approximation scheme for matrix multiplication proposed by [Sar06] based on a random rotation followed by uniform column sampling. The result follows from a matrix version of Bernstein’s inequality and a tail inequality for quadratic forms in subgaussian random vectors.

Using a sharp version of the reverse Young inequality, and a Rényi entropy comparison result due to Fradelizi, Madiman, and Wang, the authors are able to derive a Rényi entropy power inequality for log-concave random vectors when Rényi parameters belong to (0, 1). Furthermore, the estimates are shown to be somewhat sharp.

We report 96 inequalities with common structure, all elementary to state but many not elementary to prove. If n is a positive integer, a = (a1,..., an) and b = (b1,..., bn) are arbitrary vectors in R(+)n=[0,infinity)n, and rho(mij) is the spectral radius of an n x n matrix with elements m(ij), then, for example: [equation: see text]. The second inequality is obtained from the first inequality b...

in conjunction with Euler’s formula, f1 = f0 + f2 − 2; see also [6, Sect. 10.3]. Moreover, the cone of flag vectors of 3-polytopes is also given by these two inequalities, together with the usual Dehn– Sommerville relations, such as f02 = 2f1 and f012 = 4f1. Here the extreme cases, polytopes whose f or flag vectors lie at the boundary of the cone, are given by the simplicial polytopes (for whic...

The paper presents ab abstract definition of linear inequality concepts leading to linearly invariant measures and characterizes the class of linear concepts completely. Two general methods of deriving ethical measures are proposed. They imply an Atkinson-Kolm-Sen index and a new dual index reflecting the inequality of living standard. Then all separable social welfare orderings which generate ...

Some genearlizations of Precupanu's inequality for orthornormal families of vectors in real or complex inner product spaces and applications related to Buzano's, Richard's and Kurepa's results are given.

We define a new concept, the “1/2-power of plane vectors,” and use it to provide another proof of Erdős’ inequality for triangles.