Limits of One-dimensional Interacting Particle Systems with Two-scale Interaction*

This paper characterizes the limits of a large system of interacting particles distributed on the real line. The interaction occurring among neighbors involves two kinds of independent actions with different rates. This system is a generalization of the voter process, of which each particle is of type A or a. Under suitable scaling, the local proportion functions of A particles converge to continuous functions which solve a class of stochastic partial differential equations driven by Fisher-Wrig...     

毗邻沙漠地区微分博弈模型与宏观环境治理策略研究

工业化发展导致了脆弱的土地进一步沙漠化,沙漠地区政府应综合考虑经济发展与土地沙漠化导致的生态质量降低。通过构建两个沙漠化地区微分博弈模型,研究了两个地区的沙漠治理问题。研究结果表明,提高工业生产排放技术或沙漠治理技术均能够提高均衡经济产出水平,在工业生产收益系数较高的情况下也能够提高沙漠治理投入水平,并且两种技术提高存在交互效应;只有对生产排放技术水平和沙漠治理技术水平较高的地区,延长地方政府领导任期有利于沙漠治理;实现沙漠完全治理情况下控制土地再沙化的治理投入低于控制沙漠稳定在一定规模的治理投入。     

Triple Positive Solutions of Boundary Value Problems for High-order Fractional Differential Equation at Resonance with Singularities

In this paper, we investigate the existence of triple positive solutions of boundary value problems for high-order fractional differential equation at resonance with singularities by using the fixed point index theory and the Leggett-Williams theorem. The spectral theory and some new height functions are also employed to establish the existence of triple positive solutions. The nonlinearity involved is arbitrary fractional derivative, and permits singularity.     

Generalized solutions of a stochastic partial differential equation

We discuss the Cauchy problem of a certain stochastic parabolic partial differential equation arising in the nonlinear filtering theory, where the initial data and the nonhomogeneous noise term of the equation are given by Schwartz distributions. The generalized (distributional) solution is represented by a partial (conditional) generalized expectation ofT(t)° _0,t ^–1 , whereT(t) is a stochastic process with values in distributions and _s,t is a stochastic flow generated by a certain stochastic differential equation. The representation is used for getting estimates of the solution with respect to Sobolev norms.Further, by applying the partial Malliavin calculus of Kusuoka-Stroock, we show that any generalized solution is aC -function under a condition similar to Hörmander's hypoellipticity condition.     

Boundary value estimates for harmonic forms

We prove a bound for the -norm of harmonic forms in terms of certain -norms of their normal and tangential components. In turn, this is used to show the -norm equivalence of the normal and tangential components of harmonic forms on manifolds.

     

Basic differential forms for actions of Lie groups

A section of a Riemannian -manifold is a closed submanifold which meets each orbit orthogonally. It is shown that the algebra of -invariant differential forms on which are horizontal in the sense that they kill every vector which is tangent to some orbit, is isomorphic to the algebra of those differential forms on which are invariant with respect to the generalized Weyl group of , under some condition.

     

WKB-type approximations for second-order differential equations in -algebras

Rigorous asymptotic approximations of the WKB (or Liouville-Green) type are obtained for a basis of solutions to in the framework of -algebras. Both cases and are included, thus generalizing the classical theory for scalar equations developed by F.W.J. Olver to matrix as well as to infinite-dimensional equations.

     

Curvature invariants, differential operators and local homogeneity

We first prove that a Riemannian manifold with globally constant additive Weyl invariants is locally homogeneous. Then we use this result to show that a manifold whose Laplacian commutes with all invariant differential operators is a locally homogeneous space.

     

Derivative of the exponential mapping for infinite dimensional lie groups

It is proved that for infinite dimensional Lie groups in the sense of the differential calculus of Frölicher and Kriegl the derivative of the exponential mappings is given by the formula d(exp)(X)Y=d_exp(X)(e) ₀ ¹ Ad_exp(–tX) Y dt, where stands for the left translation ande is the neutral element.This work was supported by the Alexander von Humboldt-Stiftung.     

Liquids and liquid mixtures: Deduction of the BGY equation considering the angular distribution function

The Born→Green→Yvon equation for molecular fluid has been deduced considering the orientational distribution functions. The isotropic and anisotropic parts of the distribution function have been separated. The expressions deduced can be used in the case of mixtures and for the non-central type of intermolecular potential energy.