Random Walks with Invariant Loop Probabilities: Stereographic Random Walks

Random walks with invariant loop probabilities comprise a wide family of Markov processes with site-dependent, one-step transition probabilities. The whole family, which includes the simple random walk, emerges from geometric considerations related to the stereographic projection of an underlying geometry into a line. After a general introduction, we focus our attention on the elliptic case: random walks on a circle with built-in reflexing boundaries.     

自然增长下非线性退化椭圆方程的优化Hlder连续指数

利用Moser-Nash迭代和稠密引理,得到了在自然增长下的非线性退化椭圆方程有界弱解具有某一Hlder指数的正则性;在已知数据的进一步正则性下,建立了具有任意γ满足0≤γ<κ的优化Hlder连续性指数,其中κ是A-调和函数的局部Hlder连续指数.     

Minima in elliptic variational problems without convexity assumptions

We prove an abstract existence theorem for the minimum of the functional

A nonlinear oblique derivative boundary value problem for the heat equation Part 2: Singular self-similar solutions

This paper continues the study started in [12]. In the upper half-plane, consider the elliptic equation

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, submitted to the nonlinear oblique derivative boundary condition U_x = UU_z on the axis x = 0. The solution of this problem appears to be the self-similar solution of the heat equation with the same boundary condition. As goes to 0, the function U converges to the non trivial solution U of the corresponding degenerate problem. Moreover there exists z₀ > 0 such that U vanishes for z ≥ z₀, is C^∞ on ]0, z₀[× ₊, is continuous on the boundary x = 0 and is discontinuous on the half-axis {z = 0, x> 0}.     

Numerical modeling of the movement of a rigid particle in viscous fluid

Modeling the movement of a rigid particle in viscous fluid is a problem physicists and smathematicians have tried to solve since the beginning of this century. A general model for an ellipsoidal particle was first published by Jeffery in the twenties. We exploit the fact that Jeffery was concerned with formulae which can be used to compute numerically the velocity field in the neighborhood of the particle during his derivation of equations of motion of the particle. This is our principal contribution to the subject. After a thorough check of Jeffery's formulae, we coded software for modeling the flow around a rigid particle based on these equations. Examples of its applications are given in conclusion. A practical example is concerned with the simulation of sigmoidal inclusion trails in porphyroblast.     

Kato Space for Dirichlet Forms

We study the space of Kato measures relative to a Dirichlet form and we prove that a local solution of a problem relative to a Kato measure is locally continuous. Moreover if the measure of an intrinsic ball is equivalent to a power of the radius we prove also that the density of the form relative to a local solution is locally a Kato measure.     

Elliptic binomial diophantine equations

The complete sets of solutions of the equation are determined for the cases , , , , , , , . In each of these cases the equation is reduced to an elliptic equation, which is solved by using linear forms in elliptic logarithms. In all but one case this is more or less routine, but in the remaining case () we had to devise a new variant of the method.

     

Factorization of the tenth Fermat number

We describe the complete factorization of the tenth Fermat number by the elliptic curve method (ECM). is a product of four prime factors with 8, 10, 40 and 252 decimal digits. The 40-digit factor was found after about 140 Mflop-years of computation. We also discuss the complete factorization of other Fermat numbers by ECM, and summarize the factorizations of .

     

L p Spectral Independence of Elliptic Operators via Commutator Estimates

Let {T _p:q ₁ p q ₂} be a family of consistent C ₀ semigroups on L ^p(), with q ₁,q ₂ [1,) and open. We show that certain commutator conditions on T _p and on the resolvent of its generator A _p ensure the p independence of the spectrum of A _p for p [q ₁,q ₂.Applications include the case of Petrovskij correct systems with Hölder continuous coefficients, Schrödinger operators, and certain elliptic operators in divergence form with real, but not necessarily symmetric, or complex coefficients.     

An index theorem for gauge-invariant families: The case of solvable groups

We define the gauge-equivariant index of a family of elliptic operators invariant with respect to the free action of a family of Lie groups (these families are called ``gauge-invariant families' in what follows). If the fibers of are simply-connected and solvable, we compute the Chern character of the gauge-equivariant index, the result being given by an Atiyah–Singer type formula that incorporates also topological information on the bundle . The algebras of invariant pseudodifferential operators that we study, and , are generalizations of ``parameter dependent' algebras of pseudodifferential operators (with parameter in R q), so our results provide also an index theorem for elliptic, parameter dependent pseudodifferential operators. We apply these results to study Fredholm boundary conditions on a simplex. This revised version was published online in June 2006 with corrections to the Cover Date.