We prove that if X is a subspace of Lp (2 < p < ∞), then either X embeds isomorphically into `p ⊕ `2 or X contains a subspace Y, which is isomorphic to `p(`2). We also give an intrinsic characterization of when X embeds into `p⊕`2 in terms of weakly null trees in X or, equivalently, in terms of the “infinite asymptotic game” played in X. This solves problems concerning small subspaces of Lp ori...

When solving the Symmetric Positive Definite (SPD) linear system Ax = b with the conjugate gradient method, the smallest eigenvalues in the matrix A often slow down the convergence. Consequently if the smallest eigenvalues in A could be somehow “removed”, the convergence may be improved. This observation is of importance even when a preconditioner is used, and some extra techniques might be inv...

where σ is the Lipschitz constant of the extension of f and P is the natural probability on {0, 1}. Here we extend this inequality to more general product probability spaces; in particular, we prove the same inequality for {0, 1} with the product measure ((1− η)δ0 + ηδ1) . We believe this should be useful in proofs involving random selections. As an illustration of possible applications we give...

We present two computationally inexpensive techniques for estimating the numerical rank of a matrix, combining powerful tools from computational linear algebra. These techniques exploit three key ingredients. The first is to approximate the projector on the non-null invariant subspace of the matrix by using a polynomial filter. Two types of filters are discussed, one based on Hermite interpolat...

We investigate the effects of annealing microstructure on low-cycle fatigue (LCF) life (Nf), cyclic stress behaviour and a Fe–15Mn–10Cr–8Ni–4Si alloy that exhibits deformation-induced transformation austenite (γ-phase) into ε-martensite (ε-phase). The rolled at 800°C was annealed 600°C, 700°C, 900°C to vary grain size, fraction recrystallised grains texture intensity. Fully reversed axial strai...

Let ξ(u), u ∈ R be an ergodic stationary Markov chain, taking a finite number of values, and consider the diffusion process generated by the SDE dX t = b(X ε t )dt+ ε ξ ( X t /ε ) dBt with a small positive scaling parameter ε, where B = (Bt)t∈R+ is a Brownian motion, independent of ξ, and κ ≥ 0 is a fixed constant. Such model describes evolution of a particle, perturbed by a small white noise d...

We study the eigenvalue trajectories of a time dependent matrix Gt=H+itvv∗ for t≥0, where H is an N×N Hermitian random and v unit vector. In particular, we establish that with high probability, outlier can be distinguished at all times t>1+N−1∕3+ε, any ε>0. The this natural process combines elements non-Hermitian analysis, illustrates some aspects intrinsic instability (even weakly) matrices.

We show that if Y is a separable subspace of a Banach space X such that both X and the quotient X/Y have Cp-smooth Lipschitz bump functions, and U is a bounded open subset of X, then, for every uniformly continuous function f : Y ∩U → R and every ε > 0, there exists a Cp-smooth Lipschitz function F : X → R such that |F (y)− f(y)| ≤ ε for every y ∈ Y ∩U . If we are given a separable subspace Y o...

We obtain two results concerning the Feichtinger conjecture for systems of normalized reproducing kernels in the model subspace KΘ = H 2 ⊖ ΘH of the Hardy space H, where Θ is an inner function. First, we verify the Feichtinger conjecture for the kernels k̃λn = kλn/‖kλn‖ under the assumption that sup n |Θ(λn)| < 1. Secondly, we prove the Feichtinger conjecture in the case where Θ is a one-compone...