A set of n triangles sharing a common edge is called a book with n pages and is denoted by Bn. It is known that the Ramsey number r (Bn) satisfies r (Bn) = (4 + o (1))n. We show that every red-blue edge coloring of Kb(4−ε)nc with no monochromatic Bn exhibits quasi-random properties when ε tends to 0. This implies that there is a constant c > 0 such that for every red-blue edge coloring of Kr(Bn...

In this paper, we study the computational complexity and approximation complexity of the connected set-cover problem. We derive necessary and sufficient conditions for the connected set-cover problem to have a polynomial-time algorithm. We also present a sufficient condition for the existence of a (1 + ln δ)-approximation. In addition, one such (1 + ln δ)-approximation algorithm for this proble...

We consider the smoothness of solutions of a system of refinement equations written in the form φ = ∑ α∈Z a(α)φ(2 · − α), where the vector of functions φ = (φ1, . . . , φr) is in (Lp(R)) and a is a finitely supported sequence of r× r matrices called the refinement mask. We use the generalized Lipschitz space Lip∗(ν, Lp(R)), ν > 0, to measure smoothness of a given function. Our method is to rela...

Motivated by a question of Vincent Lafforgue, we study the Banach spaces X satisfying the following property: there is a function ε → ∆ X (ε) tending to zero with ε > 0 such that every operator T : L 2 → L 2 with T ≤ ε that is simultaneously contractive (i.e. of norm ≤ 1) on L 1 and on L ∞ must be of norm ≤ ∆ X (ε) on L 2 (X). We show that ∆ X (ε) ∈ O(ε α) for some α > 0 iff X is isomorphic to ...

We present two characterizations of Lagrange interpolation sets for weak Chebyshev spaces. The rst of them is valid for an arbitrary weak Chebyshev space U and is based on an analysis of the structure of zero sets of functions in U extending Stockenberg's theorem. The second one holds for all weak Chebyshev spaces that possess a locally linearly independent basis. x1. Introduction Let U denote ...

We present a study and application of quasi-stationarity of electroencephalogram for intraoperative neurophysiological monitoring (IONM) and an application of Chebyshev time windowing for preconditioning SSEP trials to retain the morphological characteristics of somatosensory evoked potentials (SSEP). This preconditioning was followed by the application of a principal component analysis (PCA)-b...

In this paper we present a complete perturbation analysis for the Hamiltonian Schurform of a Hamiltonian matrix under similarity transformations with unitary symplectic matrices.Both linear asymptotic and non-linear perturbation bounds are presented. The same analysis isalso carried out for two less condensed block-Schur forms. It suggests that the block forms areless sensitive ...

An infinite dimensional notion of asymptotic structure is considered. This notion is developed in terms of trees and branches on Banach spaces. Every countably infinite countably branching tree T of a certain type on a space X is presumed to have a branch with some property. It is shown that then X can be embedded into a space with an FDD (Ei) so that all normalized sequences in X which are alm...

A new method is presented for estimating the column space (signal subspace) of a low rank data matrix distorted by additive noise. It is based on a tangible expression for the set of all matrices of minimal rank that are ε-close to the data matrix in matrix 2-norm. The usual truncated SVD approximant is contained in this set. Features of the algorithm are (1) it has the same computational struc...

An infinite Markov system {f0, f1, . . . } of C2 functions on [a, b] has dense span in C[a, b] if and only if there is an unbounded Bernstein inequality on every subinterval of [a, b]. That is if and only if, for each [α, β] ⊂ [a, b] and γ > 0, we can find g ∈ span{f0, f1, . . . } with ‖g′‖[α,β] > γ‖g‖[a,b]. This is proved under the assumption (f1/f0)′ does not vanish on (a, b). Extension to hi...