Multiscale stochastic homogenization of convection-diffusion equations

Multiscale stochastic homogenization is studied for convection-diffusion problems. More specifically, we consider the asymptotic behaviour of a sequence of realizations of the form ∂u _ɛ ^ω / ∂t+1 / ɛ ₃ C(T ₃(x/ɛ ₃)ω ₃) · ∇u _ɛ ^ω − div(α(T ₂(x/ɛ ₂)ω ₂, t) ∇u _ɛ ^ω ) = f. It is shown, under certain structure assumptions on the random vector field C(ω ₃) and the random map α(ω ₁, ω ₂, t), that the sequence {u _ɛ ^ω } of solutions converges in the sense of G-convergence of parabolic operators to the solution u of the homogenized problem ∂u/∂t − div (B(t)∇u= f).     

Approximation of the effective conductivity of ergodic media by periodization

 This paper is concerned with the approximation of the effective conductivity σ(A, μ) associated to an elliptic operator ∇ _xA (x,η)∇ _x where for xℝ ^d , d≥1, A(x,η) is a bounded elliptic random symmetric d×d matrix and η takes value in an ergodic probability space (X, μ). Writing A ^N (x, η) the periodization of A(x, η) on the torus T ^d _N of dimension d and side N we prove that for μ-almost all η

Homogenization of some parabolic operators with several time scales

The main focus in this paper is on homogenization of the parabolic problem ∂ _t uɛ − ∇ · (a(x/ɛ,t/ɛ,t/ɛ ^r )∇u ^ɛ ) = f. Under certain assumptions on a, there exists a G-limit b, which we characterize by means of multiscale techniques for r > 0, r ≠ 1. Also, an interpretation of asymptotic expansions in the context of two-scale convergence is made.     

Uniqueness of Nelsons diffusions

Let ℒ≔Δ/2+(∇φ/φ) ·∇ be a generalized Schr?dinger operator or generator of Nelsons diffusion, defined on C ^∞ ₀(D) where φ is a continuous and strictly positive function on an open domain D⊂ℝ ^d such that ∇φ∈L _loc ²(D). Some results are given about the two questions below: (i) Whether does ℒ generate a unique semigroup in L ¹(D, φ² dx)? (ii) Whether the semigroup determined by ℒ is strong Feller? Received: 21 October 1997 / Revised version: 3 September 1998     

A Harnack inequality for the equation ∇(a∇u)+b∇u=0, when |b|∈Kn+1

By proving a lower and an upper bound of the fundamental solutions of certain parabolic equations, we establish a Harnack inequality for some parabolic and elliptic equations which include: ∇(a∇u)+b∇u=0, |b|∈K _n+1.     

An information geometry algorithm for distribution control

In this paper, we consider the problem of distribution control from the viewpoint of information geometry. Different from most existing models used in stochastic control, it is assumed that the control input directly affects the distribution of the system output in probability sense. Here, we set up a new manifold (S), meanwhile the B-spline manifold (B) and the system output manifold (M) can be referred to as its submanifolds. We give an information geometrical algorithm which can be called as geodesic-projection algorithm using the properties of manifold. In the geodesic step, we can obtain the geodesic equation from the initial point V₀ = (ω₁₀, ω₂₀, ··· , ω_(n−1)0) to the specified point V_g = (ω_1g, ω_2g, ··· , ω_(n−1)g) in B. This gives us an optimal trajectory for the points changing along in B. In the projection step, we project the sample points selected from the geodesic onto M. The coordinates of the projections in M give the trajectory of the control input u.     

Optimal control of a coefficient in modification Navier-Stokes equations

This paper deals with the optimal control of a coefficient in the modification of Navier-Stokes equations. Namely, the motion of the viscous incompressible fluid for a small gradient of velocity is described by Navier-Stokes equations where the coefficient of the kinematic viscosity ν is the positive constant (ν ₀). For a greater gradient of velocity the coefficient of kinematic viscosity is a positive function of the gradient of velocity, that is ν (|∇u|). In our case ν (|∇u|) = ν ₀ + ν ₁ a (|∇u|) where ν ₀, ν ₁ ∈ ℝ₊. The function a is positive and monotone and it is taken as a control variable. The existence of a solution of the optimal control problem is proved. Further, the approximation of the control problem by the finite-dimensional control problem is performed. The proof of the existence of a solution of that aproximate problem has been brought into light. Finally, the connection between the solution of the control problem and the solution of the approximate control problem is established.     

Interactive epistemology II: Probability

Formal Interactive Epistemology deals with the logic of knowledge and belief when there is more than one agent or “player.” One is interested not only in each person's knowledge and beliefs about substantive matters, but also in his knowledge and beliefs about the others' knowledge and beliefs. This paper examines two parallel approaches to the subject. The first is the semantic, in which knowledge and beliefs are represented by a space Ω of states of the world, and for each player i, partitions ℐ_i of Ω and probability distributions π_i(·; ω) on Ω for each state ω of the world. The atom of ℐ_i containing a given state ω represents i's knowledge at that state – the set of those other states that i cannot distinguish from ω; the probability distributions π_i(·; ω) represents i's beliefs at the state ω. The second is the syntactic approach, in which beliefs are embodied in sentences constructed according to certain syntactic rules. This paper examines the relation between the two approaches, and shows that they are in a sense equivalent.  In game theory and economics, the semantic approach has heretofore been most prevalent. A question that often arises in this connection is whether, in what sense, and why the space Ω, the partitions ℐ_i, and the probability distributions π_i(·; ω) can be taken as given and commonly known by the players. An answer to this question is provided by the syntactic approach.     

Estimates of the weak distance between finite-dimensional Banach spaces

We prove a variant of a theorem of N. Alon and V. D. Milman. Using it we construct for everyn-dimensional Banach spacesX andY a measure space Ω and two operator-valued functionsT: Ω→L(X, Y),S: Ω→L(Y, X) so that ∫_Ω S(ω)oT(ω)dω is the identity operator inX and ∫_Ω||S(ω)||·||T(ω)||dω=O(n ^α ) for some absolute constantα<1. We prove also that any subset of the unitn-cube which is convex, symmetric with respect to the origin and has a sufficiently large volume possesses a section of big dimension isomorphic to ak-cube. Research supported in part by a grant of the Israel Academy of Sciences.     

Scattering theory for perturbed stratified media

We study the operatorH = -c ² x,y)Μx,y)∇ · Μ ^-1 (x,y)∇, wherec andΜ are perturbations of functionsc ₀(y) andΜ ₀(y) which depend only on the one-dimensional variabley. In particular, we study the spatial asymptotics of lim_ε↺0(H - (λ +iε)²)^-1 applied to functions which have compact support or are otherwise well-behaved at infinity and relate the scattering matrix to the asymptotics of the generalized eigenfunctions. We then prove a trace formula for the operatorH in terms of the scattering phase, and, in a very special situation, use the trace formula to find spectral asymptotics forH. Partially supported by an NSF Postdoctoral Fellowship and the University of Missouri Research Board.     

The ultrametric corona problem

Let K be a complete ultrametric algebraically closed field and let A be the K-Banach algebra of bounded analytic functions in the disk D: |x| < 1. Let Mult(A, ∥ · ∥) be the set of continuous multiplicative semi-norms of A, let Mult _m (A, ∥ · ∥) be the subset of the ϕ ∈ Mult(A, ∥ · ∥) whose kernel is a maximal ideal and let Mult _a (A, ∥ · ∥) be the subset of the ϕ ∈ Mult _m (A, ∥ · ∥) whose kernel is of the form (x − a)A, a ∈ D ( if ϕ ∈ Mult _m (A, ∥ · ∥) \ Mult _a (A, ∥ · ∥), the kernel of ϕ is then of infinite codimension). We examine whether Mult _a (A, ∥ · ∥) is dense inside Mult _m (A, ∥ · ∥) with respect to the topology of simple convergence. This a first step to the conjecture of density of Mult _a (A, ∥ · ∥) in the whole set Mult(A, ∥ · ∥): this is the corresponding problem to the well-known complex corona problem. We notice that if ϕ ∈ Mult _m (A, ∥ · ∥) is defined by an ultrafilter on D, then ϕ lies in the closure of Mult _a (A, ∥ · ∥). Particularly, we show that this is case when a maximal ideal is the kernel of a unique ϕ ∈ Multm(A, ∥ · ∥). Particularly, when K is strongly valued all maximal ideals enjoy this property. And we can prove this is also true when K is spherically complete, thanks to the ultrametric holomorphic functional calculus. More generally, we show that if ψ ∈ Mult(A, ∥ · ∥) does not define the Gauss norm on polynomials (∥ · ∥), then it is defined by a circular filter, like on rational functions and analytic elements. As a consequence, if ψ ∈ Multm(A, ∥ · ∥) \ Multa(A, ∥ · ∥) or if φ does not lie in the closure of Mult _a (A, ∥ · ∥), then its restriction to polynomials is the Gauss norm. The first situation does happen. The second is unlikely. The text was submitted by the authors in English.     

On linear forms whose coefficients are in arithmetic progression

Let Ω be the set of positive integers that are omitted values of the form where thea _i are fixed positive integers with g.c.d. 1 and thex _i are variable nonnegative integers. Set ω=|Ω| and κ=max Ω+1. Using an expression of Roberts [4] for κ when thea _i form an arithmetic progression, we determine ω in this case.     

Spectral structure of Anderson type Hamiltonians

We study self adjoint operators of the form?H _ω = H ₀ + ∑λ_ω(n) <δ _n ,·>δ _n ,?where the δ _n ’s are a family of orthonormal vectors and the λ_ω(n)’s are independently distributed random variables with absolutely continuous probability distributions. We prove a general structural theorem saying that for each pair (n,m), if the cyclic subspaces corresponding to the vectors δ _n and δ _m are not completely orthogonal, then the restrictions of H _ω to these subspaces are unitarily equivalent (with probability one). This has some consequences for the spectral theory of such operators. In particular, we show that “well behaved” absolutely continuous spectrum of Anderson type Hamiltonians must be pure, and use this to prove the purity of absolutely continuous spectrum in some concrete cases. Oblatum 27-V-1999 & 6-I-2000?Published online: 8 May 2000     

Strict topology over the space of measures with continuous translations on a locally compact foundation semigroup

Let $ \mathfrak{S} $ \mathfrak{S} be a locally compact semigroup, ω be a weight function on $ \mathfrak{S} $ \mathfrak{S} , and M _a ($ \mathfrak{S} $ \mathfrak{S} , ω) be the weighted semigroup algebra of $ \mathfrak{S} $ \mathfrak{S} . Let L ₀^∞ ($ \mathfrak{S} $ \mathfrak{S} ; M _a ($ \mathfrak{S} $ \mathfrak{S} , ω)) be the C*-algebra of all M _a ($ \mathfrak{S} $ \mathfrak{S} , ω)-measurable functions g on $ \mathfrak{S} $ \mathfrak{S} such that g/ω vanishes at infinity. We introduce and study a strict topology β ¹($ \mathfrak{S} $ \mathfrak{S} , ω) on M _a ($ \mathfrak{S} $ \mathfrak{S} , ω) and show that the Banach space L ₀^∞ ($ \mathfrak{S} $ \mathfrak{S} ; M _a ($ \mathfrak{S} $ \mathfrak{S} , ω)) can be identified with the dual of M _a ($ \mathfrak{S} $ \mathfrak{S} , ω) endowed with β ¹($ \mathfrak{S} $ \mathfrak{S} , ω). We finally investigate some properties of the locally convex topology β ¹($ \mathfrak{S} $ \mathfrak{S} , ω) on M _a ($ \mathfrak{S} $ \mathfrak{S} , ω).     

On the multiplicative representation of integers

Leta ₁<a ₂<··· be an infinite sequence of integers. Denote byg(n) the number of solutions ofn=a _i···a _j. Ifg(n)>0 for a sequencen of positive upper density then lim supg(n)=∞. Dedicated to my friend A. D. Wallace on the occasion of his 60th birthday.     

The Riemann–Hilbert Problem in Weighted Classes of Cauchy Type Integrals with Density from L P( · )(Γ)

We consider the Riemann–Hilbert problem in the following setting: find a function whose boundary values ϕ⁺(t) satisfy the condition a.e. on Γ. Here D is a simply connected domain bounded by a simple closed curve Γ, and K ^{p( · )}(D;ω) is the set of functions ϕ(z) representable in the form , where ω(z) is a weight function and (K _Γφ )(z) is a Cauchy type integral whose density φ is integrable with a variable exponent p(t). It is assumed that Γ is a piecewise-Lyapunov curve without zero angles, ω(z) is an arbitrary power function and p(t) satisfies the Log-H?lder condition. The solvability conditions are established and solutions are constructed. These solutions largely depend on the coefficients a, b, c, the weight ω, on the values of p(t) at the angular points of Γ and on the values of angles at these points. Submitted: May 13, 2007. Revised: August 8, 2007 and August 28, 2007. Accepted: November 8, 2007.     

On periodic solutions of even-order ordinary differential equations

Optimal in a certain sense sufficient conditions are given for the existence and uniqueness of ω-periodic solutions of the nonautonomous ordinary differential equation u ^(2m) =f(t,u,...,u ^(m-1) ), where the function f:ℝ×ℝ ^m →ℝ is periodic with respect to the first argument with period ω. Received: December 21, 1999; in final form: August 12, 2000?Published online: October 2, 2001     

Ein Beitrag Zum Ricci-Kalkül

Riassunto Sianos, t dei campi tensoriali antisi metrici sopra unan-varietà riamanniana orientata. Siano, rispettivamente,a eb i gradi dis et. Allora rot(s·t)=±(a+1)(grads)·(dual _{n−(b−a)−1} dual _b−a t) ±s·(dual _{n−(b−a)−1} div dual _b−a t), dove dual _i sono delle modificazioni dell’operatore ben noto dual. Cons⋎t=(duals)·t, il prodottos⋎t possiede delle proprità, sotto certi aspetti duali a quelle dei prodotto esterno,s⋏t. Discutendo il prodottos⋏t, si vede: l'operatore div ed il prodotto ⋎ corrispondono all’operatore rot e al prodotto ⋏.

---

Résumé Soients, t des champs tensoriels antisy métriques sur unen-variété riemannienne orientée. Soient, respectivement,a etb les degrés des ett. Alors rot(s·t)=±(a+1)(grads)·(dual _{n−(b−a)−1} dual _b−a t) ±s·(dual _{n−(b−a)−1} div dual _b−a t), où dual _i sont des modifications de l'opérateur connu dual. Avecs⋎t=(duali)·t, le produits⋎t possède des propriétés à certains égards duales à ceux du produit extérieur,s⋏t. En discutant le produits⋎t, l'on voit de plus: l'opérateur div et le produit ⋎ correspondent à l'opérateur rot et au produit ⋏.

     

Existence,uniqueness and regularity of stationary solutions to inhomogeneous Navier-Stokes equations in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

For a bounded domain Ω ⊂ ℝ ⁿ , n ⩾ 3, we use the notion of very weak solutions to obtain a new and large uniqueness class for solutions of the inhomogeneous Navier-Stokes system − Δu + u · ∇u + ∇p = f, div u = k, u _|aΩ = g with u ∈ L ^q , q ⩾ n, and very general data classes for f, k, g such that u may have no differentiability property. For smooth data we get a large class of unique and regular solutions extending well known classical solution classes, and generalizing regularity results. Moreover, our results are closely related to those of a series of papers by Frehse & Růžička, see e.g. Existence of regular solutions to the stationary Navier-Stokes equations, Math. Ann. 302 (1995), 669–717, where the existence of a weak solution which is locally regular is proved.      

Classification locale simultanée de deux formes symplectiques compatibles

Consider a symplectic form ω and a closed 2-form ω₁ on a real or complex manifold. Suppose that the Nijenhuis torsion of the tensor fieldJ defined by ω₁(X,Y) = ω(JX,Y) vanishes. In this paper we give the complete local classification of the couple {ω, ω₁} on a dense open set, defined by some minor conditions of regularity. Around each point of this open set we can find coordinates on wich ω is written with constant coefficients and ω₁ with affine ones. Projet de recherche DGICYT PB91-0412