Pseudoprocesses on a circle and related Poisson kernels

Pseudoprocesses, constructed by means of the solutions of higher-order heat-type equations, have been developed by several authors and many related functionals have been analysed by applying the Feynman–Kac functional or by means of the Spitzer identity. We here examine pseudoprocesses wrapped up on circles and derive their explicit signed density measures. By composing the circular pseudoprocesses with positively skewed stable processes, we arrive at genuine circular processes whose distribution is obtained in the form of Poisson kernels. The distribution of circular even-order pseudoprocesses is similar to the Von Mises (or Fisher) circular normal law and to the wrapped up law of Brownian motion. Time-fractional and space-fractional equations related to processes and pseudoprocesses on the unit radius circumference are introduced and analysed.     

Riesz Potentials and Liouville Operators on Fractals

An analogue to the theory of Riesz potentials and Liouville operators in R ⁿ for arbitrary fractal d-sets is developed. Corresponding function spaces agree with traces of Euclidean Besov spaces on fractals. By means of associated quadratic forms we construct strongly continuous semigroups with Liouville operators as infinitesimal generator. The case of Dirichlet forms is discussed separately. As an example of related pseudodifferential equations the fractional heat-type equation is solved.     

Numerical solutions to time-fractional stochastic partial differential equations

Numerical Algorithms - This study is concerned with numerical approximations of time-fractional stochastic heat-type equations driven by multiplicative noise, which can be used to model the...     

Infinite Dimensional Oscillatory Integrals with Polynomial Phase and Applications to Higher-Order Heat-Type Equations

The definition of infinite dimensional Fresnel integrals is generalized to the case of polynomial phase functions of any degree and applied to the construction of a functional integral representation for solutions of a general class of higher-order heat-type equations.     

Differential Harnack estimates for backward heat equations with potentials under the Ricci flow

In this paper, we derive a general evolution formula for possible Harnack quantities. As a consequence, we prove several differential Harnack inequalities for positive solutions of backward heat-type equations with potentials (including the conjugate heat equation) under the Ricci flow. We shall also derive Perelman's Harnack inequality for the fundamental solution of the conjugate heat equation under the Ricci flow.     

Die Kalkulation von PKV-Tarifen unter Einbeziehung des Übertragungswertes

German law requires that private health insurers credit a transferral value (“Übertragungswert”) to the costumer in case of surrender. This benefit must be taken into account when premiums and reserves are calculated. It leads to a non-linear system of equations whose solvability is investigated in this article. The authors show that despite the non-linearity a unique solution always exists which can be computed by a suitable algorithm.     

Heat kernels for non-divergence operators of Hörmander type

We prove the existence of a fundamental solution for a class of Hörmander heat-type operators. For this fundamental solution and its derivatives we obtain sharp Gaussian bounds that allow to prove an invariant Harnack inequality. To cite this article: M. Bramanti et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

一种经典时空理论(Ⅰ)——基础

尽管广义相对论形式优美,成果辉煌,但在以下几个方面却未尽完善:(1)它不能容纳不对称的总能量-动量张量,这种不对称性已经在电磁理论中被证明是存在的.(2)场方程可以导出线动量平衡定律,却不能导出角动量平衡定律的精确方程.(3)如果没有附加(非物理)的假设,缩并的第二Bianchi恒等式的四度任意性使场方程无法获得唯一解.为了解决这些问题,我们在本文提出,把纤维丛P[M,SU(2)]定律作为四维时空的基本几何结构.于此,结构群SU(2)是特殊二维复酉群的实表示.SU(2)同时使定义在整个M上的度规型dS2=g_αβdxαdxβ和基本二型φ=(1/2₁)a_αβdxα∧dxβ不变.以SU(2)连络定义的爱因斯坦方程利用了时空流形以及把非齐次麦克斯韦方程作为辅助条件.于此,电磁张量与曲率张量的缩并形式是等价的.我们得到的结果是关于16个未知场变量(g_αβ,a_αβ)的16个独立的基本方程.另外,角动量平衡定律恰好是推广的爱因斯坦方程的斜对称部分.这里,自旋角动量张量直接被证明与扭转张量成比例.     

Large deviations and mean-field theory for asymmetric random recurrent neural networks

 In this article, we study the asymptotic dynamics of a noisy discrete time neural network, with random asymmetric couplings and thresholds. More precisely, we focus our interest on the limit behaviour of the network when its size grows to infinity with bounded time. In the case of gaussian connection weights, we use the same techniques as Ben Arous and Guionnet (see [3]) to prove that the image law of the distribution of the neurons' activation states by the empirical measure satisfies a temperature free large deviation principle. Moreover, we prove that if the connection weights satisfy a general condition of domination by gaussian tails, then the distribution of the activation potential of each neuron converges weakly towards an explicit gaussian law, the characteristics of which are contained in the mean-field equations stated by Cessac-Doyon-Quoy-Samuelides (see [4–6]). Furthermore, under this hypothesis, we obtain a law of large numbers and a propagation of chaos result. Finally, we show that many classical distributions on the couplings fulfill our general condition. Thus, this paper provides rigorous mean-field results for a large class of neural networks which is currently investigated in neural network literature. Received: 10 January 2000 / Revised version: 15 June 2001 / Published online: 13 May 2002     

Mathematical analysis of a viscoelastic problem with temperature‐dependent coefficients—Part I: Existence and uniqueness

The aim of this article is to study the quasistatic evolution of a thermoviscoelastic problem whose behaviour law is of the Maxwell–Norton type with coefficients depending on temperature. In this law, the deformation rate tensor is a superposition of viscoelastic and thermal contributions. The existence and uniqueness of the solution is proved. Copyright © 2007 John Wiley & Sons, Ltd.     

Forward and backward stochastic differential equations with normal constraints in law

In this paper we investigate the well-posedness of backward or forward stochastic differential equations whose law is constrained to live in an a priori given (smooth enough) set and which is reflected along the corresponding “normal” vector. We also study the associated interacting particle system reflected in mean field and asymptotically described by such equations. The case of particles submitted to a common noise as well as the asymptotic system is studied in the forward case. Eventually, we connect the forward and backward stochastic differential equations with normal constraints in law with partial differential equations stated on the Wasserstein space and involving a Neumann condition in the forward case and an obstacle in the backward one.     

Effective Dynamics of a Magnetic Vortex in a Local Potential

We study a simplified mean field model of superconductor dynamics in the presence of impurities or for variable superconductor depth. This model is given by the gradient-flow version of the Ginzburg-Landau equations (Gorkov-Eliashberg equations) with an addition of a potential term. We find a dynamical law of motion of the vortex center, involving the potential, such that for datum close to a (static) magnetic vortex the solution is close, for all times, to a magnetic vortex whose center obeys this law.     

Geometry of conservation laws for a class of parabolic PDE's, II: Normal forms for equations with conservation laws

We consider conservation laws for second-order parabolic partial differential equations for one function of three independent variables. An explicit normal form is given for such equations having a nontrivial conservation law. It is shown that any such equation whose space of conservation laws has dimension at least four is locally contact equivalent to a quasi-linear equation. Examples are given of nonlinear equations that have an infinite-dimensional space of conservation laws parameterized (in the sense of Cartan-K?hler) by two arbitrary functions of one variable. Furthermore, it is shown that any equation whose space of conservation laws is larger than this is locally contact equivalent to a linear equation.     

Optimal Decay Rates for the Highest-Order Derivatives of Solutions for the Compressible MHD Equations with Coulomb Force

For the Cauchy problem of the 3D compressible MHD equations with Coulomb force, the large time behavior of this model is further investigated in this article. Compared to the previous related works in Tan-Tong-Wang [\emph{J. Math. Anal. Appl.} 427 (2015) 600--617], the main novelty of this paper is that we prove the optimal decay rates for the highest-order spatial derivatives of the solutions to the compressible MHD equations with Coulomb force, which are the same as those of the heat equation.     

Explicit-implicit schemes for convection-diffusion-reaction problems

Boundary value problems for time-dependent convection-diffusion-reaction equations are basic models of problems in continuum mechanics. To study these problems, various numerical methods are used. With a finite difference, finite element, or finite volume approximation in space, we arrive at a Cauchy problem for systems of ordinary differential equations whose operator is asymmetric and indefinite. Explicit-implicit approximations in time are conventionally used to construct splitting schemes in terms of physical processes with separation of convection, diffusion, and reaction processes. In this paper, unconditionally stable schemes for unsteady convection-diffusion-reaction equations are constructed with explicit-implicit approximations used in splitting the operator reaction. The schemes are illustrated by a model 2D problem in a rectangle.     

Remarques sur l'équation de Schrödinger non linéaire avec potentiel harmoniqueRemarks on the nonlinear Schrödinger equation with harmonic potential

Bose–Einstein condensation is usually modeled by nonlinear Schrödinger equations with harmonic potential. We study the Cauchy problem for these equations, in particular the wave collapse phenomenon. For this, we establish an evolution law, which is the analogue of the pseudo-conformal conservation law for the nonlinear Schrödinger equation. We state wave collapse criteria, allowing a range of positive values for the energy. To cite this article: R. Carles, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 763–766.     

常系数线性分数阶微分方程组的解

本文研究了常系数线性分数阶微分方程组的求解问题.利用逆Laplace变换,Jordan标准矩阵和最小多项式,得到矩阵变量Mittag-Leffler函数的三种不同的计算方法,包含了常系数线性一阶微分方程组的解.     

SPATIAL PATTERN INDUCED BY ASYMMETRIC COMPETITION: A MODELING APPROACH

ABSTRACT. The paper addresses the question: how does asymmetric competition for light affect the spatial pattern of trees? It is based on an individual-based spatially explicit model of forest dynamics, whose growth equations are derived from gap models. The model is calibrated on a stand of natural rainforest in French Guiana, where the tree pattern exhibits regularity at short distances (< 10 m) and clustering at medium distances (∼ 30 m). The model reproduces the regularity but not the clustering. As mortality and recruitment have been modeled so as to favor a random pattern, we conclude that regularity emerges from the asymmetric competition in the growth submodel. Also the scale at which regularity appears is linked to the range of interactions between trees.     

On Sojourn Distributions of Processes Related to Some Higher-Order Heat-Type Equations

It is well known that the sojourn time of Brownian motion B(t), t>0 on the positive half-line, during the interval [0, t] and under the condition B(t)=0, is uniformly distributed, while it has the form of the corrected arc-sine law when the condition B(t)>0 is assumed. We find the analogues of these laws for processes X(t), t>0 governed by signed measures whose densities are the fundamental solutions of third and fourth-order heat-type equations. Surprisingly, both laws hold for the fourth-order process. The uniform law is still valid for the third-order process but a different law emerges when the condition X(t)>0 is considered.