The asymptotic stability of the oscillations of a two-mass resonance sifter

The asymptotic stability of the periodic oscillations in a model of a two-mass resonance sifter with a unilateral limiter without a gap is proved, on the assumption that the linear generating system allows of oscillations with frequencies of ω and 2ω and the frequency of the external motor is identical with ω. This formulation corresponds to the widely used mode of operation of the sifter – resonance. The presence of a limiter leads to nondifferentiability along certain planes of the right-hand sides of the corresponding differential equations. The averaging principle, the applicability of which in the case considered has previously been justified, is employed. It is proved that the resonance mode of operation obtained is subharmonic.     

An invariant‐region‐preserving limiter for DG schemes to isentropic Euler equations

In this article, we introduce an invariant‐region‐preserving (IRP) limiter for the p‐system and the corresponding viscous p‐system, both of which share the same invariant region. Rigorous analysis is presented to show that for smooth solutions the order of approximation accuracy is not destroyed by the IRP limiter, provided the cell average stays away from the boundary of invariant region. Moreover, this limiter is explicit, and easy for computer implementation. A generic algorithm incorporating the IRP limiter is presented for high order finite volume type schemes as long as the evolved cell average of the underlying scheme stays strictly within the invariant region. For high order discontinuous Galerkin (DG) schemes to the p‐system, sufficient conditions are obtained for cell averages to stay in the invariant region. For the viscous p‐system, we design both second and third order IRP DG schemes. Numerical experiments are provided to test the proven properties of the IRP limiter and the performance of IRP DG schemes.     

Logistic模型阈值控制悖论及复杂性分析

研究了Logistic模型稳态均值在阈值控制下产生反直觉变化现象与复杂响应的机理.结果表明,阈值控制能引起映射区间改变和映射分布变化,二者之间的竞争导致了受控Logistic模型稳态均值的反直觉变化现象与复杂响应.在阈值下限接近0或阈值上限接近1时,映射分布变化是影响稳态均值复杂响应的主导因素.阈值下限大于其临界值或阈值上限小于其临界值时,稳态均值变化主要由映射区间改变决定,此时,受控Logistic模型稳态均值会出现反直觉变化现象.理论分析结果通过数值仿真得到进一步证实.     

On third-order limiter functions for finite volume methods

In this article, we propose a finite volume limiter function for a reconstruction on the three-point stencil. Compared to classical limiter functions in the MUSCL framework, which yield 2^nd-order accuracy, the new limiter is 3^rd-order accurate for smooth solution. In an earlier work, such a 3^rd-order limiter function was proposed and showed successful results [2]. However, it came with unspecified parameters. We close this gap by giving information on these parameters.     

High-order accurate monotone difference schemes for solving gasdynamic problems by Godunov’s method with antidiffusion

An approach to the construction of high-order accurate monotone difference schemes for solving gasdynamic problems by Godunov’s method with antidiffusion is proposed. Godunov’s theorem on monotone schemes is used to construct a new antidiffusion flux limiter in high-order accurate difference schemes as applied to linear advection equations with constant coefficients. The efficiency of the approach is demonstrated by solving linear advection equations with constant coefficients and one-dimensional gasdynamic equations.     

A new technique for obtaining diophantine representations via elimination of bounded universal quantifiers

M. Davis proved in the early 1950s that every recursively enumerable set has an arithmetic representation with a unique bounded universal quantifier, known today as the Davis normal form. Davis, H. Putnam, and J. Robinson showed in 1961 how the Davis normal form can be transformed into a purely existential exponential Diophantine representation which uses not only addition and multiplication, but also exponentiation. The present author eliminated the exponentiation in 1970 and thus obtained the unsolvability of Hilbert's tenth problem. The paper presents a new method for transforming the Davis normal form into the exponential Diophantine representation. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 220, 1995, pp. 83–92. Original article Translated by Yu. V. Matiyasevich. The research described in this work was made possible in part by Grant No. 94-01-01030 from the Russsian Foundation for Fundamental Research and Grant No. R43000 from the International Science Foundation.     

Even- vs. Odd-dimensional Charney–Davis Conjecture

More than once we have heard that the Charney–Davis Conjecture makes sense only for odd-dimensional spheres. This is to point out that in fact it is also a statement about even-dimensional spheres.     

From factorizations of noncommutative polynomials to combinatorial topology

This is an extended version of a talk given by the author at the conference “Algebra and Topology in Interaction” on the occasion of the 70th Anniversary of D.B. Fuchs at UC Davis in September 2009. It is a brief survey of an area originated around 1995 by I. Gelfand and the speaker.     

The Charney–Davis Conjecture for Certain Subdivisions of Spheres

Notions of sesquiconstructible complexes and odd iterated stellar subdivisions are introduced, and some of their basic properties are verified. The Charney–Davis conjecture is then proven for odd iterated stellar subdivisions of sesquiconstructible balls and spheres.     

Stability analysis and a priori error estimate of explicit Runge-Kutta discontinuous Galerkin methods for correlated random walk with density-dependent turning rates

In this paper,we analyze the explicit Runge-Kutta discontinuous Galerkin(RKDG)methods for the semilinear hyperbolic system of a correlated random walk model describing movement of animals and cells in biology.The RKDG methods use a third order explicit total-variation-diminishing Runge-Kutta(TVDRK3)time discretization and upwinding numerical fluxes.By using the energy method,under a standard CourantFriedrichs-Lewy(CFL)condition,we obtain L2stability for general solutions and a priori error estimates when the solutions are smooth enough.The theoretical results are proved for piecewise polynomials with any degree k 1.Finally,since the solutions to this system are non-negative,we discuss a positivity-preserving limiter to preserve positivity without compromising accuracy.Numerical results are provided to demonstrate these RKDG methods.