In this paper we show some very interesting properties of weak Chebyshev subspaces and use them to simplify Pinkus's characterization of Asubspaces of C[a, b]. As a consequence we obtain that if the metric projection PG from C[a, b] onto a finite-dimensional subspace G has a continuous selection and elements of G have no common zeros on (a, b), then G is an /4-subspace.

Let T be a compact Hausdorff topological space and let M denote an n–dimensional subspace of the space C(T ), the space of real–valued continuous functions on T and let the space be equipped with the uniform norm. Zukhovitskii [7] attributes the Basic Theorem to E.Ya.Remez and gives a proof by duality. He also gives a proof due to Shnirel’man, which uses Helly’s Theorem, now the paper obtains a...

It is widely believed that Krylov subspace iterative methods are better than Chebyshev semi-iterative methods. When the solution of a linear system with a symmetric and positive definite coefficient matrix is required then the Conjugate Gradient method will compute the optimal approximate solution from the appropriate Krylov subspace, that is, it will implicitly compute the optimal polynomial. ...

The real ε-pseudospectrum of a real matrix A consists of the eigenvalues of all real matrices that are ε-close in spectral norm to A. The real pseudospectral abscissa, which is the largest real part of these eigenvalues for a prescribed value ε, measures the structured robust stability of A w.r.t. real perturbations. In this report, we introduce a criss-cross type algorithm to compute the real ...

In the present paper, we discuss the novel concept of super-compressed tensor-structured data formats in high dimensional applications. We describe the multi-folding or quantics based tensor approximation method of O(d logN)-complexity (logarithmic scaling in the volume size), applied to the discrete functions over the product index set {1, ..., N}⊗d, or briefly N -d tensors of size N, and to t...

Let P be a set of n points in R. In the projective clustering problem, given k, q and norm ρ ∈ [1,∞], we have to compute a set F of k q-dimensional flats such that ( ∑ p∈P d(p,F)) is minimized; here d(p,F) represents the (Euclidean) distance of p to the closest flat in F . We let f k (P, ρ) denote the minimal value and interpret f k (P,∞) to be maxr∈P d(r,F). When ρ = 1, 2 and ∞ and q = 0, the ...

Let V be an n-dimensional affine space over the field with pd elements, p 6= 2. Then for every ε > 0 there is an n(ε) such that if n = dim(V ) n(ε) then any subset of V with more than εjV j elements must contain 3 collinear points (i.e., 3 points lying in a one-dimensional affine subspace).

Hydrodynamic interactions play an important role in the dynamics of macromolecules. The most common way to take into account hydrodynamic effects in molecular simulations is in the context of a Brownian dynamics simulation. However, the calculation of correlated Brownian noise vectors in these simulations is computationally very demanding and alternative methods are desirable. This paper studie...

Hydrodynamic interactions play an important role in the dynamics of macromolecules. The most common way to take into account hydrodynamic effects in molecular simulations is in the context of a brownian dynamics simulation. However, the calculation of correlated brownian noise vectors in these simulations is computationally very demanding and alternative methods are desirable. This paper studie...

We apply the polynomial method—specifically, Chebyshev polynomials—to obtain a number of new results on geometric approximation algorithms in low constant dimensions. For example, we give an algorithm for constructing ε-kernels (coresets for approximate width and approximate convex hull) in close to optimal time O(n + (1/ε)(d−1)/2), up to a small near-(1/ε)3/2 factor, for any d-dimensional n-po...