with quasi-periodic initial data whose Fourier coefficients decay exponentially and prove existence and uniqueness, in the class of functions which have an expansion with exponentially decaying Fourier coefficients, of a solution on a small interval of time, the length of which depends on the given data and the frequency vector involved. For a Diophantine frequency vector and for small quasi-pe...

We construct an explicit family of linear rank-metric codes over any field Fh that enables efficient list decoding up to a fraction ρ of errors in the rank metric with a rate of 1− ρ− ε, for any desired ρ ∈ (0, 1) and ε > 0. Previously, a Monte Carlo construction of such codes was known, but this is in fact the first explicit construction of positive rate rank-metric codes for list decoding bey...

We show how the quasi-Newton least squares method (QN-LS) relates to Krylov subspace methods in general and to GMRes in particular.

In this supplementary material, we give additional details on the two polynomial filters discussed in the main paper. First, we give an example to illustrate how the choice of the degree in the extend McWeeny filter method affects the inflexion point and the rank estimated. Next, we discuss some details on the practical implementation of the Chebyshev polynomial filter method. In section 4, we ...

This is the second part in a series of papers on using spectral sparse grid methods for solving higher-dimensional PDEs. We extend the basic idea in the first part [18] for solving PDEs in bounded higher-dimensional domains to unbounded higher-dimensional domains, and apply the new method to solve the electronic Schrödinger equation. By using modified mapped Chebyshev functions as basis functio...

We introduce the space BLD(Ω) consisting of all fields u : Ω → Rn defined on a domain Ω ⊂ Rn, whose symmetric gradient satisfies ∫ Ω |ε(u)|ln(1 + |ε(u)|) dx < ∞. These fields of bounded logarithmic defomation form a proper subspace of the class BD(Ω) consisting of all functions having bounded deformation. With the help of Reshetnyak’s representation formulas we prove that BLD(Ω) is compactly em...

Let P be a set of n points in Euclidean space and let 0< ε< 1. A wellknown result of Johnson and Lindenstrauss states that there is a projection of P onto a subspace of dimension O(ε−2 logn) such that distances change by at most a factor of 1+ ε. We consider an extension of this result. Our goal is to find an analogous dimension reduction where not only pairs but all subsets of at most k points...

Let P be a set of n points in Euclidean space and let 0 < ε < 1. A well-known result of Johnson and Lindenstrauss states that there is a projection of P onto a subspace of dimension O(ε−2 logn) such that distances change by a factor of 1 + ε at most. We consider an extension of this result. Our goal is to find an analogous dimension reduction where not only pairs, but all subsets of at most k p...

Quaternion non-local means (QNLM) denoising algorithm makes full use of high degree self-similarities inside images to suppress the noise, so similarity metric plays a key role in its performance. In this study, two improvements have been made for QNLM: 1) For low level quaternion quasi-Chebyshev distance is proposed measure image patches and it has used replace Euclidean QNLM algorithm. Since ...

We introduce a new form of composition called weak composition that allows us to obtain polynomial kernelization lower-bounds for several natural parameterized problems. Let d ≥ 2 be some constant and let L1, L2 ⊆ {0, 1}∗ × N be two parameterized problems where the unparameterized version of L1 is NP-hard. Assuming coNP 6⊆ NP/poly, our framework essentially states that composing t L1-instances ...