Proof. For the m = d case, first rotate the subspace W to become span(e1, . . . , ed) (via multiplication by an orthogonal matrix), and then project to the first d coordinates. This clearly preserves norms in W exactly. Now, assume there is an ε-subspace embedding Π ∈ Rm×n for m < d. Then, the map Π : W → Rm has a nontrivial kernel, in particular there is some w ∈ W,w 6= 0 such that Πw = 0. On ...

We give new constructions of two classes of algebraic code families which are efficiently list decodable with small output list size from a fraction 1 − R − ε of adversarial errors where R is the rate of the code, for any desired positive constant ε. The alphabet size depends only ε and is nearly-optimal. The first class of codes are obtained by folding algebraic-geometric codes using automorph...

It is shown that the four vector extrapolation methods, minimal polynomial extrapolation, reduced rank extrapolation, modified minimal polynomial extrapolation, and topological epsilon algorithm, when applied to linearly generated vector sequences, are Krylov subspace methods, and are equivalent to some well known conjugate gradient type methods. A unified recursive method that includes the con...

It follows from [1] and [7] that any closed n-codimensional subspace (n ≥ 1 integer) of a real Banach space X is the kernel of a projection X → X, of norm less than f(n) + ε (ε > 0 arbitrary), where f(n) = 2 + (n − 1) √ n + 2 n + 1 . We have f(n) < √ n for n > 1, and f(n) = √ n − 1 √ n + O (

Solving the electronic structure problem for nanoscale systems remains a computationally challenging problem. The numerous degrees of freedom, both electronic and nuclear, make the problem impossible to solve without some effective approximations. Here we illustrate some advances in algorithm developments to solve the Kohn-Sham eigenvalue problem, i.e. we solve the electronic structure problem ...

Let X be a normed linear space, x ∈ X an element of norm one, and ε > 0 and δ(x,ε) the local modulus of convexity of X . We denote by ρ(x,ε) the greatest ρ≥ 0 such that for each closed linear subspace M of X the quotient mapping Q : X → X/M maps the open ε-neighbourhood of x in U onto a set containing the open ρ-neighbourhood of Q(x) in Q(U). It is known that ρ(x,ε) ≥ (2/3)δ(x,ε). We prove that...

In the first part of the paper we study the structure of Banach spaces with a conditional spreading basis. The geometry of such spaces exhibit a striking resemblance to the geometry of James’ space. Further, we show that the averaging projections onto subspaces spanned by constant coefficient blocks with no gaps between supports are bounded. As a consequence, every Banach space with a spreading...

We study Chebyshev filter diagonalization as a tool for the computation of many interior eigenvalues of very large sparse symmetric matrices. In this technique the subspace projection onto the target space of wanted eigenvectors is approximated with high-order filter polynomials obtained from a regularized Chebyshev expansion of a window function. After a short discussion of the conceptual foun...

We study weakly perturbed integrodifferential equations with degenerate kernel in Banach spaces and establish conditions for the bifurcation of solutions at point ε = 0. A convergent iterative procedure is proposed determination form series $$ {\sum}_{i=1}^{+\infty }{\varepsilon}^i{z}_i(t) powers ε..

Numerical methods for strongly oscillatory and singular functions are given in this paper. We present an alternative numerical solution for of oscillatory integrals of the general form Where and are smooth functions in the interval [a, b]. Also ε ω and . In order to achieve this goal and avoid singularity, the mentioned integral is solved by the modified moments using interpolating at the Cheby...