Existence of Weak Solutions of Two-dimensional Euler Equations with Initial Vorticity in Lorentz Space L （2,1） （ R^2）

Some estimates on 2-D Euler equations are given when initial vorticity ω0 belongs to a Lorentz space L(2, 1). Then based on these estimates, it is proved that there exist global weak solutions of two dimensional Euler equations when ωo ∈ L(2, 1).     

一类生化反应模型的极限环

本文研究了一类多分子生化反应模型Dx＝δ-ax-x^py^p,Dy=x^py^p-by当a≠0,p=3,q=2的情形,得出了不存在闭轨的参数区域;至少分支出两个极限环的参数区域以及极限的唯一性的参数区域的一些结果。     

Quasi-maximum likelihood estimators in generalized linear models with autoregressive processes

The paper studies a generalized linear model(GLM)y_t = h(x_t~T β) + ε_t,t = l,2,...,n,where ε_1 = η_1,ε_1 =ρε_t +η_t,t = 2,3,...;n,h is a continuous differentiable function,η_t's are independent and identically distributed random errors with zero mean and finite variance σ~2.Firstly,the quasi-maximum likelihood(QML) estimators of β,p and σ~2 are given.Secondly,under mild conditions,the asymptotic properties(including the existence,weak consistency and asymptotic distribution) of the QML estimators are investigated.Lastly,the validity of method is illuminated by a simulation example.     

Estimator of a change point in single index models

This paper considers the problem of change point in single index models. In order to obtain asymptotically valid confidence intervals for the estimation of the change point, the convergence rate and asymptotic distribution of the change point estimate is studied. Some simulation results are presented which show that the numerical performance of our estimator is satisfactory.     

Convergence of the Finite Difference Scheme for a Fast Diffusion Equation in Porous Media

We are concerned with an implicit scheme for the finite difference solution to a nonlinear parabolic equation with a multivalued coefficient that describes the fast diffusion in a porous medium. The boundary conditions contain the multivalued function as well. We prove the stability and the convergence of the scheme, emphasizing the precise nature of convergence in this specific case, and compute the error level of the approximating solution. The method is aimed to simplify the numerical computations for the solutions to equations of this type, without performing an approximation of the multivalued function. The theory is illustrated by numerical results.     

A Convergence Analysis of Newton-Like Method for Singular Equations Using Recurrent Functions

We provide new semilocal convergence results for Newton-like method involving outer or generalized inverses in a Banach space setting. Using our new idea of recurrent functions and the same or weaker conditions than before [5-19 A. Ben-Israel and N.E. Greville ( 1974 ). Generalized Inverses: Theory and Applications, Pure and Applied Mathematics . Wiley-Interscience , New York . X. Chen and T. Yamamoto ( 1989 ). Convergence domains of certain iterative methods for solving nonlinear equations . Numer. Funct. Anal. Optimiz. 10 : 37 – 48 . J.E. Dennis , Jr. ( 1968 ). On Newton-like methods . Numer. Math. 11 : 324 – 330 . P. Deuflhard and C. Heindl ( 1979 ). Convergence theorems for Newton's method and extensions to related methods . SIAM J. Numer. Anal. 16 : 1 – 10 . J.M. Gutiérrez ( 1997 ). A new semilocal convergence theorem for Newton's method . J. Comp. Appl. Math. 79 : 131 – 145 . J.M. Gutiérrez , M.A. Hernández , and M.A. Salanova ( 1995 ). Accessibility of solutions by Newton's method . Internat. J. Comput. Math. 57 : 239 – 247 . W.M. Häubler ( 1986 ). A Kantorovich-type convergence analysis for the Gauss–Newton methods . Numer. Math. 48 : 119 – 125 . L.V. Kantorovich and G.P. Akilov ( 1964 ). Functional Analysis . Pergamon Press , Oxford . M.Z. Nashed and X. Chen ( 1993 ). Convergence of Newton-like methods for singular operator equations using outer inverses . Numer. Math. 66 : 235 – 257 . F.A. Potra and V. Ptàk ( 1980 ). Sharp error bounds for Newton's process . Numer. Math. 34 : 67 – 72 . W.C. Rheinboldt ( 1968 ). A unified convergence theory for a class of iterative processes . SIAM J. Numer. Anal. 5 : 42 – 63 . W.C. Rheinboldt ( 1977 ). An adaptive continuation process for solving systems of nonlinear equations . Polish Academy of Sciences, Banach Ctr. Publ. 3 : 129 – 142 . T. Yamamoto ( 1987 ). A method for finding sharp error bounds for Newton's method under the Kantorovich assumptions . Numer. Math. 49 : 203 – 230 . T. Yamamoto ( 1987 ). A convergence theorem for Newton-like methods in Banach spaces . Numer. Math. 51 : 545 – 557 . T. Yamamoto ( 1989 ). Uniqueness of the solution in a Kantorovich-type theorem of Haubler for the Gauss–Newton method . Japan J. Appl. Math. 6 : 77 – 81 . ], we provide more precise information on the location of the solution and finer bounds on the distances involved. Moreover, since our Newton–Kantorovich-type hypothesis is weaker than before, we can now cover cases not previously possible.

Applications and numerical examples involving a nonlinear integral equation of Chandrasekhar-type and a differential equation with Green's function are also provided in this study.     

Analysis of a Parabolic Cross-Diffusion Semiconductor Model with Electron-Hole Scattering

The global-in-time existence of non-negative solutions to a parabolic strongly coupled system with mixed Dirichlet–Neumann boundary conditions is shown. The system describes the time evolution of the electron and hole densities in a semiconductor when electron-hole scattering is taken into account. The parabolic equations are coupled to the Poisson equation for the electrostatic potential. Written in the quasi-Fermi potential variables, the diffusion matrix of the parabolic system contains strong cross-diffusion terms and is only positive semi-definite such that the problem is formally of degenerate type. The existence proof is based on the study of a fully discretized version of the system, using a backward Euler scheme and a Galerkin method, on estimates for the free energy, and careful weak compactness arguments.     

On the solution of stochastic ordinary differential equations via small delays

dx(t)=g(x{t))dW(t) is proved using an approximating sequence of stochastic delay equationsGeneralizations of the approximation scheme are indicated for the Stratonovich case and when the Brownian motion W is replaced by a continuous semi-martingale.     

Existence and uniqueness of an optimum in the infinite-horizon portfolio-cum-saving model with semimartingale investments

The model considered here is essentially that formulated in the author's previous paper Conditions for Optimality in the Infinite-Horizon Portfolio-cum-Saving Problem with Semimartingale Investments, Stochastics and Stochastics Reports 29 (1990), 133-171. In this model, the vector process representing returns to investments is a general semimartingale. Processes defining portfolio plans arc here required only to be predictable and non-negative. Existence of an optimal portfolio-cum-saving plan is proved under slight conditions of integrability imposed on the welfare functional; the proofs rely on properties of weak precompactness of portfolio and utility sequences in suitable L _p spaces together with dominated and monotone convergence arguments. Conditions are also obtained for the uniqueness of the portfolio plan generating a given returns process (i.e. for the uniqueness of the integrands generating a given sum of semimartingale integrals) and for the uniqueness of an optimal plan; here use is made of random measures associated with the jumps of a semimartingale     

Using Random Quasi-Monte-Carlo Within Particle Filters,With Application to Financial Time Series

This article presents a new particle filter algorithm which uses random quasi-Monte-Carlo to propagate particles. The filter can be used generally, but here it is shown that for one-dimensional state-space models, if the number of particles is N, then the rate of convergence of this algorithm is N^?1. This compares favorably with the N^?1/2 convergence rate of standard particle filters. The computational complexity of the new filter is quadratic in the number of particles, as opposed to the linear computational complexity of standard methods. I demonstrate the new filter on two important financial time series models, an ARCH model and a stochastic volatility model. Simulation studies show that for fixed CPU time, the new filter can be orders of magnitude more accurate than existing particle filters. The new filter is particularly efficient at estimating smooth functions of the states, where empirical rates of convergence are N^?3/2; and for performing smoothing, where both the new and existing filters have the same computational complexity.