On some Diophantine fourier series

We continue our study on arithmetical Fourier series by considering two Fourier series which are related to Diophantine analysis. The first one was studied by Hardy and Littlewood in connection with the classification of numbers and the second one was studied by Hartman and Wintner by Lebesgue integration theory.     

On Representations of Non-Symmetric Dirichlet Forms

In this paper, we present some structure results on non-symmetric Dirichlet forms. These include, in particular, an analogue of LeJan’s transformation rule for their diffusion parts and a Lévy-Khintchine type formula for regular non-symmetric Dirichlet forms on R ^d .     

Regularizing effects for the classical solutions to the Landau equation in the whole space

The Landau equation describes the binary collisional effects (through long range coulombian interaction) in a plasma. In this paper, we prove that the known classical solutions to the Landau equation near Maxwellian in the whole space have a regularizing effect in all (time, space and velocity) variables, that is, become immediately smooth with respect to all variables.     

Hollow Gaussian beam generated by beam shaping with phase-only liquid crystal spatial light modulator

Based on the refractive beam shaping system, the transformation of a quasi-Gaussian beam into a dark hollow Gaussian beam by a phase-only liquid crystal spatial light modulator (LC-SLM) is proposed. According to the energy conservation and constant optical path principle, the phase distribution of the aspheric lens and the phase-only LC-SLM can modulate the wave-front properly to generate the hollow beam. The numerical simulation results indicate that, the dark hollow intensity distribution of the output shaped beam can be maintained well for a certain propagation distance during which the dark region will not decrease whereas the ideal hollow Gaussian beam will do. By designing the phase modulation profile, which loaded into the LC-SLM carefully, the experimental results indicate that the dark hollow intensity distribution of the output shaped beam can be maintained well even at a distance much more than 550 mm from the LC-SLM, which agree with the numerical simulation results.     

Composite Legendre-Laguerre pseudospectral approximation in unbounded domains

A composite Legendre–Laguerre pseudospectral approximationin unbounded domains is developed. Some approximation resultsare obtained. As an application, a composite pseudospectralscheme is proposed for the Burgers equation on the half-line.The stability and convergence of the scheme are proved. By choosingappropriate base functions, the resulting system of this methodhas a sparse structure and can be solved in parallel. Numericalresults are given to show the efficiency of this new method.     

Inequalities for a Simplex and an Interior Point

We establish some inequalities for the inradius, circumradius and distances between an interior point and facets of an n-simplex, and prove a recursion inequality for the radii of the circumscribed spheres of an n-simplex and its facets.     

Some statistics on Stirling permutations and Stirling derangements

A permutation of the multiset $\{1,1,2,2,\dots ,n,n\}$ is called a Stirling permutation of order $n$ if every entry between the two occurrences of $i$ is greater than $i$ for each $i\in \{1,2,\dots ,n\}$. In this paper, we introduce the definitions of block, even indexed entry, odd indexed entry, Stirling derangement, marked permutation and bicolored increasing binary tree. We first study the joint distribution of ascent plateaux, even indexed entries and left-to-right minima over the set of Stirling permutations of order $n$. We then present an involution on Stirling derangements.     

Global Large Solutions to the Navier-Stokes-Nernst-Planck-Poisson Equations

In this article, we obtain a new wellposedness result to the \(d\)-dimensional Navier-Stokes-Nernst-Planck-Poisson equations. Our result implies that, if the initial charge densities of a negatively and positively charged species are close enough, we can get global solutions only needing smallness condition imposed on initial velocity. The structure coming from equations and the weighted Chemin-Lerner type space are crucial in our arguments.