Optimal Runge–Kutta smoothers for the p-multigrid discontinuous Galerkin solution of the 1D Euler equations

This work presents a family of original Runge–Kutta methods specifically designed to be effective relaxation schemes in the numerical solution of the steady state solution of purely advective problems with a high-order accurate discontinuous Galerkin space discretization and a p-multigrid solution algorithm. The design criterion for the construction of the Runge–Kutta methods here developed is different form the one traditionally used to derive optimal Runge–Kutta smoothers for the h-multigrid algorithm, which are designed in order to provide a uniform damping of the error modes in the high-frequency range only. The method here proposed is instead designed in order to provide a variable amount of damping of the error modes over the entire frequency spectrum. The performance of the proposed schemes is assessed in the solution of the steady state quasi one-dimensional Euler equations for two test cases of increasing difficulty. Some preliminary results showing the performance for multidimensional applications are also presented.     

Tracer diffusion in lattice gases

It has been proved that a tracer particle in a reversible lattice gas converges to Brownian motion. However, only in a few particular cases has a strictly positive self-diffusion coefficientD been established. Here we supply the missing piece and show thatD>0 in general. The exceptions are one-dimensional lattice gases with nearest neighbor jumps only, for whichD=0. The proof establishes a variational formula forD which could be used to obtain realistic bounds.     

Modularity of a certain Calabi-Yau threefold and combinatorial congruences

We prove a certain Calabi-Yau threefold is modular in the sense that the number of points on its reduction modulo p is expressed in terms of the pth coefficient of a weight 4 newform in S₄(Γ₀(6)). We also give a mod p² combinatorial expression for these coefficients. 2000 Mathematics Subject Classification: 11G25, 11G40     

Line integrals and Green's formula on time scales

In this paper we study curves parametrized by a time scale parameter, introduce line delta and nabla integrals along time scale curves, and obtain an analog of Green's formula in the time scale setting.     

Contributions of non-extremum critical points to the semi-classical trace formula

We study the semi-classical trace formula at a critical energy level for a h-pseudo-differential operator on whose principal symbol has a totally degenerate critical point for that energy. We compute the contribution to the trace formula of isolated non-extremum critical points under a condition of “real principal type”. The new contribution to the trace formula is valid for all time in a compact subset of but the result is modest since we have restrictions on the dimension.     

Stochastic calculus for Markov processes associated with non-symmetric Dirichlet forms

Nakao’s stochastic integrals for continuous additive functionals of zero energy are extended from the symmetric Dirichlet forms setting to the non-symmetric Dirichlet forms setting.It? ’s formula in terms of the extended stochastic integrals is obtained.     

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??A class of backward doubly stochastic differential equations driven by white noises and Poisson random measures are studied in this paper. The definitions of solutions and Yamada-Watanabe type theorem to this equation are established.     

On Connes’ trace formula for the Hankel transformation of order-1/2

A local Hankel transformation of order 1/2 is defined for every finite place of the field of rational numbers.Its inversion formula and the Plancherel type theorem are obtained.A Connes type trace formula is given for each local Hankel transformation of order 1/2.An S-local Connes type trace formula is derived for the S-local Hankel transformation of order 1/2.These formulas are generalizations of Connes’ corresponding trace formulas in 1999.     

K3 surfaces,Picard numbers and Siegel disks

If a K3 surface admits an automorphism with a Siegel disk, then its Picard number is an even integer between 0 and 18. Conversely, using the method of hypergeometric groups, we are able to construct K3 surface automorphisms with Siegel disks that realize all possible Picard numbers. The constructions involve extensive computer searches for appropriate Salem numbers and computations of algebraic numbers arising from holomorphic Lefschetz-type fixed point formulas and related Grothendieck residues.