A new piecewise spectral homotopy analysisof the Michaelis-Menten enzymatic reactions model

In this paper we report on a novel method for solving systems of nonlinear differential equations which is an extension of the spectral homotopy analysis method (SHAM). The proposed method extends the application of the SHAM to initial value problems that model the Michaelis-Menten enzymatic reaction equation. Results from the proposed method are compared with Runge-Kutta routines as a measure of accuracy and efficiency.     

Comparison of ruin probability estimates in the presence of heavy tails

Four methods of estimation of the ruin probability in the presence of heavy tails are compared in accordance with their effectiveness on the Pareto-distributed claim sizes. The first method, proposed by Embrechis and Veraverbeke, provides an asymptotic expression when the initial capital tends to infinity. The second method, proposed by Goovaerts and De Vylder, provides an algorithm for two-sided estimation based on the solution of the renewal equation through discretization. Its advantage is the perfect applicbility in cases with small initial capital. The third method, proposed initially by Willmoi and improved by Kalashnikov, provides an upper bound of the ruin probability with the help of a test function. The last method, proposed by Kalashnikov, provides two-sided bounds for the ruin probability using truncation of random variables to avoid the Cramér condition. The last two methods enable one to handle cases with intermediate initial capital. Proceedings of the Seminar on Stability Problems for stochastic Models, Hajdúszoboszló, Hungary, 1997, Part III.     

Blow-up analysis for an elliptic equation describing stationary vortex flows with variable intensities in 2D-turbulence

We consider the mean field equation arising in the high-energy scaling limit of point vortices with a general circulation constraint, when the circulation number density is subject to a probability measure. Mathematically, such an equation is a non-local elliptic equation containing an exponential nonlinearity which depends on this probability measure. We analyze the behavior of blow-up sequences of solutions in relation to the circulation numbers. As an application of our analysis we derive an improved Trudinger-Moser inequality for the associated variational functional.     

Assessing the relationship of evolutionary rates and functional variables by mixture estimating equations

In the study of complex organisms, clarifying the association between the evolution of coding genes and the measures of functional variables is of fundamental importance. However, traditional analysis of the evolutionary rate is either built on the assumption of independence between responses or fails to handle a mixture distribution problem. In this paper, we utilize the concept of generalized estimating equations to propose an estimating equation to accommodate continuous and binary probability distributions. The proposed estimate can be shown to have consistency and asymptotic normality. Simulations and data analysis are also presented to illustrate the proposed method.     

Qualitative behaviour of stochastic delay equations with a bounded memory

A dichotomy is proved concerning recurrence properties of the solution of certain stochastic delay equations. If the solution process is recurrent, there exists an invariant measure π on the state space C which is unique (up to a multiplicative constant) and the tail-field is trivial. If π happens to be a probability measure, then for every initial condition, the distribution of the process converges to it as t→∞. We will formulate a sufficient condition for the existence of an invariant probability measure (ipm) in icrnia of Lyapunov junctionals and give two examples, one Heing the stochastic-delay version of the famous logistic equation of population growth. Finally we study approximations of delay equations by Markov chains.     

B—模糊集合代数和广义互信息公式

基于两种概率的区分，推导出了一个广义Shannon熵公式和一个广义互信息公式。后者和模糊性有关，并且柯用于语言和感觉中的信息度量。为了由原子语句为真的条件概率求出合语句为真的条件概率，提出了一个遵循存尔运算的模糊集合代数。所谓的模糊信息被还原为概率信息。新的理论在经典理论－概率论，集合论及Shannon信息论－的基础上容易理解。     

Diffusion Processes Associated with Nonlinear Evolution Equations for Signed Measures

In this paper, we explain how to associate a nonlinear martingale problem with some nonlinear parabolic evolution equations starting at bounded signed measures. Our approach generalizes the classical link made when the initial condition is a probability measure. It consists in giving to each sample-path a signed weight which depends on the initial position. After dealing with the classical McKean-Vlasov equation as an introductory example, we are interested in a viscous scalar conservation law. We prove uniqueness for the corresponding nonlinear martingale problem and then obtain existence thanks to a propagation of chaos result for a system of weakly interacting diffusion processes. Last, we study the behavior of the associated fluctuations and present numerical results which confirm the theoretical rate of convergence.     

Sufficient conditions for the eventual strong Feller property for degenerate stochastic evolutions

The strong Feller property is an important quality of Markov semigroups which helps for example in establishing uniqueness of invariant measure. Unfortunately degenerate stochastic evolutions, such as stochastic delay equations, do not possess this property. However the eventual strong Feller property is sufficient in establishing uniqueness of invariant probability measure. In this paper we provide operator theoretic conditions under which a stochastic evolution equation with additive noise possesses the eventual strong Feller property. The results are used to establish uniqueness of invariant probability measure for stochastic delay equations and stochastic partial differential equations with delay, with an application in neural networks.     

电网综合检修效用概率测度模型研究

本文结合电网检修的工作实际,鉴于电网综合检修适应性的不确定性,借助于概率测度进行电网检修的综合效益分析。为实现检修周期内不再重复停电的"两个减少"目标,提出电网综合检修的概率空间,并定义该概率空间上的概率測度就是综合检修的适应度。     

Viability with Probabilistic Knowledge of Initial Condition, Application to Optimal Control

In this paper we provide an extension of the Viability and Invariance Theorems in the Wasserstein metric space of probability measures with finite quadratic moments in ? ^d for controlled systems of which the dynamic is bounded and Lipschitz. Then we characterize the viability and invariance kernels as the largest viability (resp. invariance) domains. As application of our result we consider an optimal control problem of Mayer type with lower semicontinuous cost function for the same controlled system with uncertainty on the initial state modeled by a probability measure. Following Frankowska, we prove using the epigraphical viability approach that the value function is the unique lower semicontinuous proximal episolution of a suitable Hamilton Jacobi equation.     

Convergence,boundedness, and ergodicity of regime-switching diusion processes with infinite memory

We study a class of diffusion processes, which are determined by solutions X(t) to stochastic functional differential equation with infinite memory and random switching represented by Markov chain Λ(t): Under suitable conditions, we investigate convergence and boundedness of both the solutions X(t) and the functional solutions X_t: We show that two solutions (resp., functional solutions) from different initial data living in the same initial switching regime will be close with high probability as time variable tends to infinity, and that the solutions (resp., functional solutions) are uniformly bounded in the mean square sense. Moreover, we prove existence and uniqueness of the invariant probability measure of two-component Markov-Feller process (X_t,Λ(t)); and establish exponential bounds on the rate of convergence to the invariant probability measure under Wasserstein distance. Finally, we provide a concrete example to illustrate our main results.     

Derivation of the Boltzmann equation and entropy production in functional mechanics

A derivation of the Boltzmann equation from the Liouville equation by the use of the Grad limiting procedure in a finite volume is proposed. We introduce two scales of space-time: macro- and microscale and use the BBGKY hierarchy and the functional formulation of classical mechanics. According to the functional approach to mechanics, a state of a system of particles is formed from the measurements, which are rational numbers. Hence, one can speak about the accuracy of the initial probability density function in the Liouville equation. We assume that the initial data for the microscopic density functions are assigned by the macroscopic one (so one can say about a kind of hierarchy and subordination of the microscale to the macroscale) and derive the Boltzmann equation, which leads to the entropy production.     

Invariant Measures and the Soliton Resolution Conjecture

The soliton resolution conjecture for the focusing nonlinear Schrödinger equation (NLS) is the vaguely worded claim that a global solution of the NLS, for generic initial data, will eventually resolve into a radiation component that disperses like a linear solution, plus a localized component that behaves like a soliton or multisoliton solution. Considered to be one of the fundamental open problems in the area of nonlinear dispersive equations, this conjecture has eluded a proof or even a precise formulation to date. This paper proves a “statistical version” of this conjecture at mass‐subcritical nonlinearity, in the following sense: The uniform probability distribution on the set of all functions with a given mass and energy, if such a thing existed, would be a natural invariant measure for the NLS flow and would reflect the long‐term behavior for “generic initial data” with that mass and energy. Unfortunately, such a probability measure does not exist. We circumvent this problem by constructing a sequence of discrete measures that, in principle, approximate this fictitious probability distribution as the grid size goes to 0. We then show that a continuum limit of this sequence of probability measures does exist in a certain sense, and in agreement with the soliton resolution conjecture, the limit measure concentrates on the unique ground state soliton. Combining this with results from ergodic theory, we present a tentative formulation and proof of the soliton resolution conjecture in the discrete setting. The above results, following in the footsteps of a program of studying the long‐term behavior of nonlinear dispersive equations through their natural invariant measures initiated by Lebowitz, Rose, and Speer and carried forward by Bourgain, McKean, Tzvetkov, Oh, and others, are proved using a combination of techniques from large deviations, PDEs, harmonic analysis, and bare‐hands probability theory. It is valid in any dimension. © 2014 Wiley Periodicals, Inc.     

Analytical solution of stochastic differential equation by multilayer perceptron neural network approximation of Fokker–Planck equation

The Fokker–Planck equation is a useful tool to analyze the transient probability density function of the states of a stochastic differential equation. In this paper, a multilayer perceptron neural network is utilized to approximate the solution of the Fokker–Planck equation. To use unconstrained optimization in neural network training, a special form of the trial solution is considered to satisfy the initial and boundary conditions. The weights of the neural network are calculated by Levenberg–Marquardt training algorithm with Bayesian regularization. Three practical examples demonstrate the efficiency of the proposed method.     

On the approximation of functions satisfying defective renewal equations

Functions satisfying a defective renewal equation arise commonly in applied probability models. Usually these functions do not admit an explicit expression. In this work, we consider their approximation by means of a gamma-type operator given in terms of the Laplace transform of the initial function. We investigate which conditions on the initial parameters of the renewal equation give the optimal order of uniform convergence of the approximation. We apply our results to ruin probabilities in the classical risk model, paying special attention to mixtures of gamma claim amounts.     

On the approximation of functions satisfying defective renewal equations

Functions satisfying a defective renewal equation arise commonly in applied probability models. Usually these functions do not admit an explicit expression. In this work, we consider their approximation by means of a gamma-type operator given in terms of the Laplace transform of the initial function. We investigate which conditions on the initial parameters of the renewal equation give the optimal order of uniform convergence of the approximation. We apply our results to ruin probabilities in the classical risk model, paying special attention to mixtures of gamma claim amounts.     

一种求解物流设施二次分配问题的混合分布估计算法

为了求解物流设施二次分配问题,提出了一种混合分布估计算法(HEDA)。首先,根据QAP的距离和物流量矩阵信息,提出了一种基于假设物流中心启发式规则的种群初始化方法,用于提高初始种群的质量和算法的搜索效率;其次,针对HEDA的概率模型,提出了一种概率矩阵初始构型生成机制和扰动操作,用于提高算法的全局探索能力;最后,在分析QAP的结构性质的基础上,设计了一种基于快速评价的局部搜索策略,用于提高算法的局部开发能力。仿真计算实验和算法比较验证了HEDA的优化性能。     

Wasserstein geometry of porous medium equation

We study the porous medium equation with emphasis on q-Gaussian measures, which are generalizations of Gaussian measures by using power-law distribution. On the space of q-Gaussian measures, the porous medium equation is reduced to an ordinary differential equation for covariance matrix. We introduce a set of inequalities among functionals which gauge the difference between pairs of probability measures and are useful in the analysis of the porous medium equation. We show that any q-Gaussian measure provides a nontrivial pair attaining equality in these inequalities.     

Adaptive multi-element polynomial chaos with discrete measure: Algorithms and application to SPDEs

We develop a multi-element probabilistic collocation method (ME-PCM) for arbitrary discrete probability measures with finite moments and apply it to solve partial differential equations with random parameters. The method is based on numerical construction of orthogonal polynomial bases in terms of a discrete probability measure. To this end, we compare the accuracy and efficiency of five different constructions. We develop an adaptive procedure for decomposition of the parametric space using the local variance criterion. We then couple the ME-PCM with sparse grids to study the Korteweg–de Vries (KdV) equation subject to random excitation, where the random parameters are associated with either a discrete or a continuous probability measure. Numerical experiments demonstrate that the proposed algorithms lead to high accuracy and efficiency for hybrid (discrete–continuous) random inputs.     

On the expected discounted penalty function for the compound Poisson risk model with delayed claims

In this paper, we consider an extension to the compound Poisson risk model for which the occurrence of the claim may be delayed. Two kinds of dependent claims, main claims and by-claims, are defined, where every by-claim is induced by the main claim and may be delayed with a certain probability. Both the expected discounted penalty functions with zero initial surplus and the Laplace transforms of the expected discounted penalty functions are obtained from an integro-differential equations system. We prove that the expected discounted penalty function satisfies a defective renewal equation. An exact representation for the solution of this equation is derived through an associated compound geometric distribution, and an analytic expression for this quantity is given for when the claim amounts from both classes are exponentially distributed. Moreover, the closed form expressions for the ruin probability and the distribution function of the surplus before ruin are obtained. We prove that the ruin probability for this risk model decreases as the probability of the delay of by-claims increases. Finally, numerical results are also provided to illustrate the applicability of our main result and the impact of the delay of by-claims on the expected discounted penalty functions.