矩形域上分形插值研究

该文给出了矩形域上分形插值数学模型, 分形插值曲面的计算公式, 证明了分形插值曲面迭代函数系唯一性定理, 导出了分形插值曲面的维数定理,并应用实际数据进行了分形插值曲面的实例研究. 为工程中长期寻求的粗糙表面模拟提供了理论基础和实用方法.     

一类Ekeland变分原理的推广

本文给出一个向量值形式的Ekeland变分原理,其目标函数是从完备的度量空间映到锥序拓扑线性空间.此结果是对钟承奎建立的Ekeland变分原理的推广.     

带误差的合成隐迭代新算法

该文参照Banach压缩映象原理合理地引进了一涉及有限族渐进非扩张映象的具误差的合成隐迭代式. 在适当条件下 证得了该迭代序列给出的序列弱收敛与强收敛到有限族渐进非扩张映象的一公共不动点, 并由此得出该合成隐迭代式导出的一非隐迭代算法的弱收敛与强收敛的新定理. 值得一提的是, 这是在未增加任何附加条件的情况下将2006年一文献的主要结果由隐迭代算法改进为非隐的显式迭代算法.     

Extremal principle based on the linear subdifferential

An extremal principle in the vein of Mordukhovich is proved based on the linear generalized gradient.     

一类具有扩散的捕食-食饵模型正解的存在性和惟一性

考虑了一类带Beddington-DeAngelis反应项的捕食-食饵模型正解的存在性和惟一性. 首先利用分歧理论给出正解存在的充分条件, 并刻画了整体分歧的结构. 然后利用特征值变分原理将所给的条件简化, 从而得到正解存在的更为简单更好计算的充分条件和必要条件. 最后利用隐函数定理和广义极大值原理在一维情况下给出了正解的存在惟一性.     

Perron's method for second order hyperbolic PDE. Part I

In parts I, II, and III combined of this paper, we define a notion of viscosity solution for these equations and existence is proved by a Perron-like method. Here, in part I, we prove useful identities, and a maximum-like principle for smooth sub(super) solutions of the standard wave equation. We define a new potential theoretic (P) notion of solution, subsolution and supersolution, and a related potential type (P) Cauchy problem for semilinear second order hyperbolic equations.     

Averaging principle for stochastic quasi-geostrophic flow equation with a fast oscillation

This work is focused on the quasi-geostrophic flow equation with a fast oscillation governed by a stochastic reaction–diffusion equation. It derives the well-posedness of the slow–fast system, in which the fast component is ergodic and the slow component is tight. Applying the averaging principle, it is further proved that there exists a limit process, with respect to the singular perturbing parameter ε, where the fast component is averaged out. Moreover, the slow component of the slow–fast system converges to the solution of the averaged equation in some strong sense as ε tends to zero.     

Oscillation criteria for forced second-order nonlinear differential equations with damping

In this paper, we are concerned with oscillations in a class of forced second-order differential equations with nonlinear damping terms. By using a classical variational principle and an averaging technique, several new interval oscillation criteria for the equations are established, which improve and extend some known results. An example is also given to illustrate the results.     

Large Deviation Principle for Optimal Filtering

   Abstract. We derive a large deviation principle for the optimal filter where the signal and the observation processes take values in conuclear spaces. The approach follows from the framework established by the author in an earlier paper. The key is the verification of the exponential tightness for the optimal filtering process and the exponential continuity of the coefficients in the Zakai equation.     

Existence of Optimal Controls for Semilinear Parabolic Equations without Cesari-Type Conditions

   Abstract. Optimal control problems governed by semilinear parabolic partial differential equations are considered. No Cesari-type conditions are assumed. By proving the existence theorem and the Pontryagin maximum principle of optimal ``state-control" pairs for the corresponding relaxed problems, an existence theorem of optimal pairs for the original problem is established.     

Empirical distributions in marked point processes

We study the asymptotic behaviour of the empirical distribution function derived from a stationary marked point process when a convex sampling window is expanding without bounds in all directions. We consider a random field model which assumes that the marks and the points are independent and admits dependencies between the marks. The main result is the weak convergence of the empirical process under strong mixing conditions on both independent components of the model. Applying an approximation principle weak convergence can be also shown for appropriately weighted empirical process defined from a stationary d-dimensional germ-grain process with dependent grains.     

The Maximum Principle for Optimal Control Problems with State Constraints by R.V. Gamkrelidze: Revisited

A maximum principle in the form given by R.V. Gamkrelidze is obtained, although without a priori regularity assumptions to be satisfied by the optimal trajectory. After its formulation and proof, we propose various regularity concepts that guarantee, in one sense or another, the nondegeneracy of the maximum principle. Finally, we show how the already known first-order necessary conditions can be deduced from the proposed theorem.     

非线性发展方程的Jacobi椭圆函数解

借助齐次平衡原则,提出了一种新的构造非线性发展方程的Jacobi椭圆函数精确解的方法. 并利用之得到了KdV方程,Boussinesq方程,KGS方程组的新形式 Jacobi椭圆函数解.     

A Coupled System of Integrodifferential Equations Arising in Liquidity Risk Model

We study the mathematical aspects of the portfolio/consumption choice problem in a market model with liquidity risk introduced in (Pham and Tankov, Math. Finance, 2006, to appear). In this model, the investor can trade and observe stock prices only at exogenous Poisson arrival times. He may also consume continuously from his cash holdings, and his goal is to maximize his expected utility from consumption. This is a mixed discrete/continuous time stochastic control problem, nonstandard in the literature. We show how the dynamic programming principle leads to a coupled system of Integro-Differential Equations (IDE), and we prove an analytic characterization of this control problem by adapting the concept of viscosity solutions. This coupled system of IDE may be numerically solved by a decoupling algorithm, and this is the topic of a companion paper (Pham and Tankov, Math. Finance, 2006, to appear).     

On Bellman’s principle with inequality constraints

We consider an example by Haviv (1996) [10] of a constrained Markov decision process that, in some sense, violates Bellman’s principle. We resolve this issue by showing how to preserve a form of Bellman’s principle that accounts for a change of constraint at states that are reachable from the initial state.     

A hydrodynamical model for holes in silicon semiconductors: The case of non-parabolic warped bands

This paper can be considered as the natural prosecution of Mascali and Romano (2009) [5]. Here, we describe the motion of holes in silicon by also taking into account the non-parabolicity of the heavy and light bands. The model is still based on the moment method and the closure of the system of equations is obtained by using the maximum entropy principle. Comparisons are made with the results in [5], in the case of bulk silicon, in order to establish the importance of non-parabolicity.     

Maximum Principle for Stochastic Control in Continuous Time with Hard End Constraints

A maximum principle is proved for certain problems of continuous–time stochastic control with hard end constraints (end constraints satisfied a.s.). In the problems, the dynamics (the state differential equation) changes at certain stochastic points in time.     

A new zero-finder for Tikhonov regularization

Tikhonov regularization with the regularization parameter determined by the discrepancy principle requires the computation of a zero of a rational function. We describe a cubically convergent zero-finder for this purpose. AMS subject classification (2000)  65F22, 65H05, 65R32     

An iterative scheme for solving nonlinear equations with monotone operators

An iterative scheme for solving ill-posed nonlinear operator equations with monotone operators is introduced and studied in this paper. A discrete version of the Dynamical Systems Method (DSM) algorithm for stable solution of ill-posed operator equations with monotone operators is proposed and its convergence is proved. A discrepancy principle is proposed and justified. A priori and a posteriori stopping rules for the iterative scheme are formulated and justified. AMS subject classification (2000)  47J05, 47J06, 47J35, 65R30     

A basic problem in the correlations between statistics and thermodynamics

By using a functional of probability distributions, several different statistical physics including extensive and nonextensive statistics are unified in a general formalism. It’s shown that the equivalence between the Maxent approach and the relation dU = TdS can be seen from the virtual work principle of mechanics. From dU = TdS, any entropic form which can exists for thermodynamical equilibrium system can be maximized in order to derive the probability distribution functions.