Approximations in Sobolev spaces by prolate spheroidal wave functions

Recently, there is a growing interest in the spectral approximation by the Prolate Spheroidal Wave Functions (PSWFs) ${\psi }_{n,c},c>0$. This is due to the promising new contributions of these functions in various classical as well as emerging applications from Signal Processing, Geophysics, Numerical Analysis, etc. The PSWFs form a basis with remarkable properties not only for the space of band-limited functions with bandwidth c, but also for the Sobolev space $H^s([?1,1])$. The quality of the spectral approximation and the choice of the parameter c when approximating a function in $H^s([?1,1])$ by its truncated PSWFs series expansion, are the main issues. By considering a function $f\in H^s([?1,1])$ as the restriction to $[?1,1]$ of an almost time-limited and band-limited function, we try to give satisfactory answers to these two issues. Also, we illustrate the different results of this work by some numerical examples.     

Regularized Principal Component Analysis for Spatial Data

In many atmospheric and earth sciences, it is of interest to identify dominant spatial patterns of variation based on data observed at p locations and n time points with the possibility that p > n. While principal component analysis (PCA) is commonly applied to find the dominant patterns, the eigenimages produced from PCA may exhibit patterns that are too noisy to be physically meaningful when p is large relative to n. To obtain more precise estimates of eigenimages, we propose a regularization approach incorporating smoothness and sparseness of eigenimages, while accounting for their orthogonality. Our method allows data taken at irregularly spaced or sparse locations. In addition, the resulting optimization problem can be solved using the alternating direction method of multipliers, which is easy to implement, and applicable to a large spatial dataset. Furthermore, the estimated eigenfunctions provide a natural basis for representing the underlying spatial process in a spatial random-effects model, from which spatial covariance function estimation and spatial prediction can be efficiently performed using a regularized fixed-rank kriging method. Finally, the effectiveness of the proposed method is demonstrated by several numerical examples.     

Gabor systems and almost periodic functions

Inspired by results of Kim and Ron, given a Gabor frame in $L^2(\mathbb{R})$, we determine a non-countable generalized frame for the non-separable space ${\mathit{AP}}_2(\mathbb{R})$ of the Besicovic almost periodic functions. Gabor type frames for suitable separable subspaces of ${\mathit{AP}}_2(\mathbb{R})$ are constructed. We show furthermore that Bessel-type estimates hold for the AP norm with respect to a countable Gabor system using suitable almost periodic norms of sequences.     

Permutations via linear translators

We show that many infinite classes of permutations over finite fields can be constructed via translators with a large choice of parameters. We first characterize some functions having linear translators, based on which several families of permutations are then derived. Extending the results of [9], we give in several cases the compositional inverse of these permutations. The connection with complete permutations is also utilized to provide further infinite classes of permutations. Moreover, we propose new tools to study permutations of the form $x\mapsto x+{(x^{p^m}-x+\delta )}^s$ and a few infinite classes of permutations of this form are proposed.     

Indefinite integrals of incomplete elliptic integrals from Jacobi elliptic functions

Integration formulas are derived for the three canonical Legendre elliptic integrals. These formulas are obtained from the differential equations satified by these elliptic integrals when the independent variable u is the argument of Jacobian elliptic function theory. This allows a limitless number of indefinite integrals with respect to the amplitude to be derived for these three elliptic integrals. Sample results are given, including the integrals derived from powers of the 12 Glaisher elliptic functions. New recurrence relations and integrals are also given for the 12 Glaisher elliptic functions.     

Arithmetic properties of overpartition triples

Let p_3(n) be the number of overpartition triples of n. By elementary series manipulations,we establish some congruences for p_3(n) modulo small powers of 2, such as p_3(16 n + 14) ≡ 0(mod 32), p_3(8 n + 7) ≡ 0(mod 64).We also find many arithmetic properties for p_3(n) modulo 7, 9 and 11, involving the following infinite families of Ramanujan-type congruences: for any integers α≥ 1 and n ≥ 0, we have p_3 (3~(2α+1)(3n + 2))≡ 0(mod 9 · 2~4), p_3(4~(α-1)(56 n + 49)) ≡ 0(mod 7),p_3 (7~(2α+1)(7 n + 3))≡ p_3 (7~(2α+1)(7 n + 5))≡ p_3 (7~(2α+1)(7 n + 6))≡ 0(mod 7),and for r ∈ {1, 2, 3, 4, 5, 6},p_3(11 · 7~(4α-1)(7 n + r)≡ 0(mod 11).