Let n and n be respectively the uniform empirical and quantile processes , and deene R n = n + n , which usually is referred to as the Bahadur{Kiefer process. The well-known Bahadur{Kiefer theorem connrms the following remarkable equivalence: kR n k= p k n k n ?1=4 (log n) 1=2 almost surely, as n goes to innnity, where kfk = sup 0t1 jf(t)j is the L 1-norm. We prove that kR n k 2 = p k n k 1 n ?...

We propose a convex optimization formulation with the Ky Fan 2-k-norm and `1-norm to find k largest approximately rank-one submatrix blocks of a given nonnegative matrix that has low-rank block diagonal structure with noise. We analyze low-rank and sparsity structures of the optimal solutions using properties of these two matrix norms. We show that, under certain hypotheses, with high probabili...

This paper introduces an inequality on vectors in $mathbb{R}^n$ which compares vectors in $mathbb{R}^n$ based on the $p$-norm of their projections on $mathbb{R}^k$ ($kleq n$). For $p>0$, we say $x$ is $d$-projectionally less than or equal to $y$ with respect to $p$-norm if $sum_{i=1}^kvert x_ivert^p$ is less than or equal to $ sum_{i=1}^kvert y_ivert^p$, for every $dleq kleq n$. For...

p . It follows that inequality (1.2) holds for any a ∈ lp when U1/p ≥ ||C||p,p and fails to hold for some a ∈ lp when U1/p < ||C||p,p. Hardy’s inequality thus asserts that the Cesáro matrix operator C, given by cn,k = 1/n, k ≤ n and 0 otherwise, is bounded on l p and has norm ≤ p/(p−1). (The norm is in fact p/(p− 1).) We say a matrix A = (an,k) is a lower triangular matrix if an,k = 0 for n < k...

p . It follows that inequality (1.2) holds for any a ∈ lp when U1/p ≥ ||C||p,p and fails to hold for some a ∈ lp when U1/p < ||C||p,p. Hardy’s inequality thus asserts that the Cesáro matrix operator C, given by cn,k = 1/n, k ≤ n and 0 otherwise, is bounded on l p and has norm ≤ p/(p−1). (The norm is in fact p/(p− 1).) We say a matrix A = (an,k) is a lower triangular matrix if an,k = 0 for n < k...

We study the spectra of p×p random matrices K with off-diagonal (i, j) entry equal to n−1/2k(XT i Xj/n ), where Xi’s are the rows of a p× n matrix with i.i.d. entries and k is a scalar function. It is known that under mild conditions, as n and p increase proportionally, the empirical spectral measure of K converges to a deterministic limit μ. We prove that if k is a polynomial and the distribut...

Let ‖ · ‖ be a unitary similarity invariant norm on the set Mn of n× n complex matrices. A description is obtained for surjective maps φ on Mn satisfying ‖AB− BA‖ = ‖φ(A)φ(B)−φ(B)φ(A)‖ for all A,B ∈ Mn . The general theorem covers the special cases when the norm is one of the Schatten p -norms, the Ky-Fan k -norms, or the k -numerical radii. Mathematics subject classification (2000): Primary 15...

Assuming the Generalized Riemann Hypothesis (GRH), we show that the norm-Euclidean Galois cubic fields are exactly those with discriminant ∆ = 7, 9, 13, 19, 31, 37, 43, 61, 67, 103, 109, 127, 157 . A large part of the proof is in establishing the following more general result: Let K be a Galois number field of odd prime degree ` and conductor f . Assume the GRH for ζK(s). If 38(`− 1)(log f) log...

In the recent years, the trace norm of graphs has been extensively studied under the name of graph energy. The trace norm is just one of the Ky Fan k-norms, given by the sum of the k largest singular values, which are studied more generally in the present paper. Several relations to chromatic number, spectral radius, spread, and to other fundamental parameters are outlined. Some results are ext...