Exact calculation of Fourier series in nonconforming spectral-element methods

In this note is presented a method, given nodal values on multidimensional nonconforming spectral elements, for calculating global Fourier-series coefficients. This method is ``exact'' in that given the approximation inherent in the spectral-element method (SEM), no further approximation is introduced that exceeds computer round-off error. The method is very useful when the SEM has yielded an adaptive-mesh representation of a spatial function whose global Fourier spectrum must be examined, e.g., in dynamically adaptive fluid-dynamics simulations.Comment: 7 pages, 4 figures, submitted to J. Comp. Phys. 2005 June

From a Kac-like particle system to the Landau equation for hard potentials and Maxwell molecules

We prove a quantitative result of convergence of a conservative stochastic particle system to the solution of the homogeneous Landau equation for hard potentials. There are two main difficulties: (i) the known stability results for this class of Landau equations concern regular solutions and seem difficult to extend to study the rate of convergence of some empirical measures; (ii) the conservativeness of the particle system is an obstacle for (approximate) independence. To overcome (i), we prove a new stability result for the Landau equation for hard potentials concerning very general measure solutions. Due to (ii), we have to couple, our particle system with some non independent nonlinear processes, of which the law solves, in some sense, the Landau equation. We then prove that these nonlinear processes are not so far from being independent. Using finally some ideas of Rousset [25], we show that in the case of Maxwell molecules, the convergence of the particle system is uniform in time

Smoothness of the law of some one-dimensional jumping S.D.E.s with non-constant rate of jump

We consider a one-dimensional jumping Markov process $\{X^x_t\}_{t \geq 0}$, solving a Poisson-driven stochastic differential equation. We prove that the law of $X^x_t$ admits a smooth density for $t>0$, under some regularity and non-degeneracy assumptions on the coefficients of the S.D.E. To our knowledge, our result is the first one including the important case of a non-constant rate of jump. The main difficulty is that in such a case, the map $x \mapsto X^x_t$ is not smooth. This seems to make impossible the use of Malliavin calculus techniques. To overcome this problem, we introduce a new method, in which the propagation of the smoothness of the density is obtained by analytic arguments

The great cultural divide

In recent years, Roger Williams University has experienced a great deal of debate regarding some of the most controversial political and cultural issues that are confronting contemporary America society. Many of the speakers representing the various viewpoints on these issues have been criticized as espousing either a left - or right - wing agenda, or as being too inflammatory to propel genuine civil discourse. RWU and the Commission on Civil Discourse have attempted to remedy this situation by bringing a wide variety of diverse speakers to campus, including the president of the Campaign for Working Families, Gary Bauer

A new regularization possibility for the Boltzmann equation with soft potentials

We consider a simplified Boltzmann equation: spatially homogeneous, two-dimensional, radially symmetric, with Grad's angular cutoff, and linearized around its initial condition. We prove that for a sufficiently singular velocity cross section, the solution may become instantaneously a function, even if the initial condition is a singular measure. To our knowledge, this is the first regularization result in the case with cutoff: all the previous results were relying on the non-integrability of the angular cross section. Furthermore, our result is quite surprising: the regularization occurs for initial conditions that are not too singular, but also not too regular. The objective of the present work is to explain that the singularity of the velocity cross section, which is often considered as a (technical) obstacle to regularization, seems on the contrary to help the regularization