The Skorohod oblique reflection problem in domains with corners and application to stochastic differential equations

Summary The Skorohod oblique reflection problem for (D, , w) (D a general domain in ^d , (x),xD, a convex cone of directions of reflection,w a function inD(⁺, ^d )) is considered. It is first proved, under a condition on (D, ), corresponding to (x) not being simultaneously too large and too much skewed with respect to D, that given a sequence {w ⁿ} of functions converging in the Skorohod topology tow, any sequence {(x ⁿ, ⁿ)} of solutions to the Skorohod problem for (D, , w ⁿ) is relatively compact and any of its limit points is a solution to the Skorohod problem for (D, , w). Next it is shown that if (D, ) satisfies the uniform exterior sphere condition and another requirement, then solutions to the Skorohod problem for (D, , w) exist for everywD(⁺, ^d ) with small enough jump size. The requirement is met in the case when D is piecewiseC _b ¹ , is generated by continuous vector fields on the faces ofD and (x) makes and angle (in a suitable sense) of less than /2 with the cone of inward normals atD, for everyxD. Existence of obliquely reflecting Brownian motion and of weak solutions to stochastic differential equations with oblique reflection boundary conditions is derived.     

Fonctions séparément harmoniques,un théorème de type Terada

Given D a domain in , G an open set in and E a subset of D verifying the harmonic analogue of Local Polynomial Condition of Leja at some point in D. We prove that if f(x, y) is a complex function defined on D × G such that– f(x, ) is harmonic on G for every fixed x E,– f(, y) is harmonic on D for every fixed y G,then f is harmonic in (x, y) on D × G.     

Large-Scale Regularities of Lattice Embeddings of Posets

Let N denote the set of natural numbers and let P =(N ^k , ) be a countably infinite poset on the k-dimensional lattice N ^k . Given x N ^k , we write max(x) (min(x)) for the maximum (minimum) coordinate of x. Let be the directed-incomparability graph of P which is defined to be the graph with vertex set equal to N ^k and edge set equal to the set of all (x, y) such that max(x) max(y) and x and y not comparable in P. For any subset D N ^k , we let P _D and _D denote the restrictions of P and to D. Points x N ^k with min(x) = 0 will be called boundary points. We define a geometrically natural notion of when a point is interior to P or relative to the lattice N ^k , and an analogous notion of monotone interior with respect to or _D . We wish to identify situations where most of these interior points are exposed to the boundary of the lattice or, in the case of monotone interior points, not concealed very much from the boundary. All of these ideas restrict to finite sublattices F ^k and/or infinite sublattices E ^k of N ^k . Our main result shows that for any poset P and any arbitarily large integer M > 0, there is an F E with F = M where, relative to the sublattices F ^k E ^k , the ideal situation of total exposure of interior points and very little concealment of monotone interior points must occur. Precisely, we prove that for any P =(N ^k , ) and any integer M > 0, there is an infinite E N and a finite D F ^k with F E and F = M such that (1) every interior vertex of P _{E k} or _{E k} is exposed and (2) there is a fixed set C E, C k ^k , such that every monotone-interior point of _D belonging to F ^k has its monotone concealment in the set C. In addition, we show that if P ¹ =(N ^k , ₁),..., P ^r =(N ^k , _r ) is any sequence of posets, then we can find E,D, and F so that the properties (1) and (2) described above hold simultaneously for each P ⁱ . We note that the main point of (2) is that the bound k ^k depends only on the dimension of the lattice and not on the poset P. Statement (1) is derived from classical Ramsey theory while (2) is derived from a recent powerful extension of Ramsey theory due to H. Friedman and shown by Friedman to be independent of ZFC, the usual axioms of set theory. The fact that our result is proved as a corollary to a combinatorial theorem that is known to be independent of the usual axioms of mathematics does not, of course, mean that it cannot be proved using ZFC (we just couldn"t find such a proof). This puts our geometrically natural combinatorial result in a somewhat unusual position with regard to the axioms of mathematics.     

A characterization of brownian motion in a lipschitz domain by its killing distributions

Let (Y _t, Q^x) be a strong Markov process in a bounded Lipschitz domainD with continuous paths up to its lifetime , and let (X _t, P^x) be a Brownian motion inD. IfY _– exists in D andQ ^x(Y_– C)=P^x(X_– C) for all Borel subsetsC of D and allx, thenY is a time change ofX.     

Minimization of the Conformal Radius under Circular Cutting of a Domain

Let D be a simply connected domain on the complex plane such that 0 D. For r > 0 , let D _r be the connected component of D {z : |z| < r} containing the origin. For fixed r, we solve the problem on minimization of the conformal radius R(D _r, 0) among all domains D with given conformal radius R(D, 0). This also leads to the solution of the problem on maximization of the logarithmic capacity of the local -extension E (a) of E among all continua E with given logarithmic capacity. Here, E (a) = E {z : |z – a| }, a E, > 0. Bibliography: 12 titles.     

Balanced sets andQ-polynomial association schemes

The notion of a balanced set of vectors is defined, and the classification of such sets suggested. A slightly stronger condition is considered for association schemes, with the following result. LetX be ad-class symmetric association scheme with Bose-Mesner algebraM and Krein parametersq _ij ^h , and let E = E_t (0 t d) be any primitive idempotent ofM. For eachx X letx _E denote the diagonal matrix withy, y entryE _xy(y X). Define the representation diagramD _E on the nodes 0, 1, ...,d by drawing an undirected arc between any nodesh, j for whichq _tj ^h > 0. CallD _E an augmented tree if it has a single loop, whose removal yields a tree. Writexy = i ifx, y X are ith associates (0 i d).     

Propriété de stabilité de la fonction extrémale relative

Nous donnons une caractérisation des domaines DX pour lesquels la fonction extrémale relative ^*(,E,D) a la propriété de stabilité pour tout ED, i.e. lim _k^*(,E,D _k )=^*(,E,D), ED. Ensuite, nous étudions la relation entre cette propriété et les enveloppes pluripolaires. Nous concluons par quelques remarques sur la propriété de stabilité lim _k^*(,E _k ,D)=^*(,E,D).     

Ouverts d"harmonicité pour les fonctions séparément harmoniques

Let D ^N , G ^M be two open sets, E D and F G two compact sets which satisfy the condition (H) (that is a harmonic condition similar to Leja"s condition). We find an open set ^N+M such that each separately harmonic function f : X : = (D× F) (E × G) (i.e.: for all x in E, f(x,.) is harmonic on G; for all y in F, f(., y) is harmonic on D) extends to a harmonic function on .     

Building ofGL(m, D) and centralizers

LetG=GL(m, D) whereD is a central division algebra over a commutative nonarchimedean local fieldF. LetE/F be a field extension contained inM(m, D). We denote byI (resp.I _E) the nonextended affine building ofG (resp. of the centralizer ofE ^x inG). In this paper we prove that there exists a uniqueG _E-equivariant affine mapj _EI_EI. It is injective and its image coincides with the set ofE ^x-fixed points inI. Moreover, we prove thatj _E is compatible with the Moy-Prasad filtrations.This author's contribution was written while he was a post-doctoral student at King's College London and supported by an european TMR grant     

Analogues of hadwiger's theorem in space ℝ n —Sufficient conditions for a convex domain to enclose another

We estimate the kinematic measure of one convex domain moving to another under the groupG of rigid motions in ⁿ . We first estimate the kinematic formula for the total scalar curvature D ₀gD ₁ Rdv of then–2 dimensional intersection submanifold D ₀gD ₁. Then we use Chern and Yen's kinematic fundamental formula and our integral inequality to obtain a sufficient condition for one convex domain to contain another in ⁿ (4). Forn=4, we directly obtain another sufficient condition in ⁴.     

Generalized sentinels defined via least squares

We address the problem of monitoring a linear functional (c, x)_Eof an unknown vectorx of a Hilbert spaceE, the available data being the observationz, in a Hilbert spaceF, of a vectorAx depending linearly onx through some known operatorA(E; F). WhenE=E ₁×E ₂,c=(c ₁ 0), andA is injective and defined through the solution of a partial differential equation, Lions ([6]–[8]) introduced sentinelssF such that (s, Ax)_Fis sensitive to x₁ E ₁ but insensitive to x₂ E₂. In this paper we prove the existence, in the general case, of (i) a generalized sentinel (s, ) ×E, where F withF dense in 80, such that for anya priori guess x₀ ofx, we have s, Ax + (, x₀)_E=(c, x)_E, where x is the least-squares estimate ofx closest tox ₀, and (ii) a family of regularized sentinels (s _n , _n ) F×E which converge to (s, ). Generalized sentinels unify the least-squares approach (by construction !) and the sentinel approach (whenA is injective), and provide a general framework for the construction of sentinels with special sensitivity in the sense of Lions [8]).     

Application of Topological Degree to the Study of Bifurcation in von Kármán Equations

The aim of this paper is to illustrate the use of topological degree for the study of bifurcation in von Kármán equations with two real positive parameters and for a thin elastic disk lying on the elastic base under the action of a compressing force, which may be written in the form of an operator equation F(x, , ) = 0 in some real Banach spaces X and Y. The bifurcation problem that we study is a mathematical model for a certain physical phenomenon and it is very important in the mechanics of elastic constructions. We reduce the bifurcation problem in the solution set of equation F(x, , ) = 0 at a point (0, ₀, ₀) X × IR ₊ ² to the bifurcation problem in the solution set of a certain equation in IR ⁿ at a point (0, ₀, ₀) IR ⁿ × IR ₊ ², where n = dim Ker F _x (0, ₀, ₀) and F _x (0, ₀, ₀): X Y is a Fréchet derivative of F with respect to x at (0, ₀, ₀). To solve the bifurcation problem obtained as a result of reduction, we apply homotopy and degree theory.     

On some functional equations of Goląb-Schinzel type

Summary LetE be a real Hausdorff topological vector space. We consider the following binary law * on ·E:(, ) * (, ) = (, ^k + ) for(, ), (, ) × E where is a nonnegative real number,k andl are integers.In order to find all subgroupoids of ( ·E, *) which depend faithfully on a set of parameters, we have to solve the following functional equation:f(f(y) ^k x +f(x) ^l y) =f(x)f(y) (x, y E). (1)In this paper, all solutionsf: of (1) which are in the Baire class I and have the Darboux property are obtained. We obtain also all continuous solutionsf: E of (1). The subgroupoids of (^* ·E, *) which dapend faithfully and continuously on a set of parameters are then determined in different cases. We also deduce from this that the only subsemigroup ofL _n ¹ of the form {(F(x ₂,x ₃, ,x _n ),x ₂,x ₃, ,x _n ); (x ₂, ,x _n ) ^{n – 1} }, where the mappingF: ^{n – 1 *} has some regularity property, is {1} × ^{n – 1} .We may noitice that the Gob-Schinzel functional equation is a particular case of equation (1)(k = 0, l = 1, = 1). So we can say that (1) is of Gob—Schinzel type. More generally, whenE is a real algebra, we shall say that a functional equation is of Gob—Schinzel type if it is of the form:f(f(y) ^k x +f(x) ^l y) =F(x,y,f(x),f(y),f(xy)) wherek andl are integers andF is a given function in five variables. In this category of functional equations, we study here the equation:f(f(y) ^k x +f(x) ^l y) =f(xy) (x, y f: ). (4)This paper extends the results obtained by N. Brillouët and J. Dhombres in [3] and completes some results obtained by P. Urban in his Ph.D. thesis [11] (this work has not yet been published).Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth     

Convex programs with an additional reverse convex constraint

A method is presented for solving a class of global optimization problems of the form (P): minimizef(x), subject toxD,g(x)0, whereD is a closed convex subset ofR ⁿ andf,g are convex finite functionsR ⁿ . Under suitable stability hypotheses, it is shown that a feasible point is optimal if and only if 0=max{g(x):xD,f(x)f( )}. On the basis of this optimality criterion, the problem is reduced to a sequence of subproblemsQ _k ,k=1, 2, ..., each of which consists in maximizing the convex functiong(x) over some polyhedronS _k . The method is similar to the outer approximation method for maximizing a convex function over a compact convex set.     

Infinitesimal obstructions to weakly mixing

Let ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M ⁿ , whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d ^t and, for each x M, we define a smooth real function _x (t) : (1 + _i (t)), where the _i(t) are the eigenvalues of AA ^T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d^2t , restricted to v at the point _x ^-t M ⁿ.Among other things, we prove the Theorem (Theorem II, below). Assume v is also volume preserving and that _x ^' (t) 0 for all x M and real t; then, if _x ^t : M M is weakly missng for some t, it is necessary that vx 0 at all x M.     

On the Uniqueness ofL 2-solutions in half-space of certain differential equations

Square integrable solutions to the equation{– ²/y² + P(D_x)+b(y)–}u(x, y) = f(x, y) are considered in the half-spacey>0, x ⁿ , whereP(D _x) is a constant coefficient operator. Under suitable conditions on lim_y0u(x, y), b(y), f(x, y) and , it is shown that suppu = suppf. This generalizes a result due to Walter Littman.Research partially supported by USNSF Grant 79-02538-A02.     

A Bernstein Property of Affine Maximal Hypersurfaces

Let x: M A ^{n + 1} be a locally strongly convex hypersurface, given as a graph of a locally strongly convexfunction x _{n + 1} =f(x ₁, ..., x _n )defined in a domain A ⁿ . We introduce a Riemannian metricG ^# = (² f/x _i x _j )dx _i dx _j on M. In this paper, we investigate the affine maximalhypersurfaces which are complete with respect to the metricG ^# and prove a Bernstein property for the affine maximalhypersurfaces.     

Geometric methods in the study of irregularities of distribution

Let be a signed measure on E ^d with E ^d =0 and ¦¦E^d<. DefineD _s() as sup ¦H¦ whereH is an open halfspace. Using integral and metric geometric techniques results are proved which imply theorems such as the following.Theorem A. Let be supported by a finite pointsetp _i. ThenD _s()>c _d(₁/ ₂)^1/2{ _{i(p i})²}^1/2 where ₁ is the minimum distance between two distinctp _i, and ₂ is the maximum distance. The numberc _d is an absolute dimensional constant. (The number .05 can be chosen forc ₂ in Theorem A.)Theorem B. LetD be a disk of unit area in the planeE ², andp ₁,p ₂,...,p _n be a set of points lying inD. If m if the usual area measure restricted toD, while nP _i=1/n defines an atomic measure _n, then independently of _n,nD _s(m – _n) .0335n ^1/4. Theorem B gives an improved solution to the Roth disk segment problem as described by Beck and Chen. Recent work by Beck shows thatnD _s(m – _n)cn ^1/4(logn)^–7/2.     

Symmetrische Permutationsmengen

A permutation set (M, I) consisting of a setM and a set of permutations ofM, is calledsymmetric, if for any two permutations, the existence of anx M with (x) (x) and ^–1 (x) = ^–1 (x) implies ^–1 = ^–1 , andsharply 3-transitive, if for any two triples (x ₁,x ₂,x ₃), (y ₁,y ₂,y ₃) M ³ with|{x ₁,x ₂,x ₃ }| = |{y ₁,y ₂,y ₃ }| = 3 there is exactly one permutation with(x ₁) =y ₁,(x ₂) =y ₂,(x ₃) =y ₃. The following theorem will be proved.THEOREM.Let (M, ) be a sharply 3-transitive symmetric permutation set with |M|3, such that contains the identity. Then is a group and there is a commutative field K such that and the projective linear group PGL(2, K) are isomorphic.     

An Inequality Involving the Local Eigenvalues of a Distance-Regular Graph

Let denote a distance-regular graph with diameter D 3, valency k, and intersection numbers a _i, b _i, c _i. Let X denote the vertex set of and fix x X. Let denote the vertex-subgraph of induced on the set of vertices in X adjacent X. Observe has k vertices and is regular with valency a ₁. Let _{1 2} ··· _k denote the eigenvalues of and observe ₁ = a ₁. Let denote the set of distinct scalars among ₂, ₃, ..., _k . For let mult denote the number of times appears among ₂, ₃,..., _k . Let denote an indeterminate, and let p ₀, p₁, ...,p _D denote the polynomials in [] satisfying p ₀ = 1 andp _i = c _i+1 p _i+1 + (a _i – c _i+1 + c _i)p _i + b _i p _i–1 (0 i D – 1),where p _–1 = 0. We show where we abbreviate = –1 – b ₁(1+)^–1. Concerning the case of equality we obtain the following result. Let T = T(x) denote the subalgebra of Mat _X ( ) generated by A, E*₀, E*₁, ..., E* _D , where A denotes the adjacency matrix of and E* _i denotes the projection onto the ith subconstituent of with respect to X. T is called the subconstituent algebra or the Terwilliger algebra. An irreducible T-module W is said to be thin whenever dimE* _i W 1 for 0 i D. By the endpoint of W we mean min{i|E* _i W 0}. We show the following are equivalent: (i) Equality holds in the above inequality for 1 i D – 1; (ii) Equality holds in the above inequality for i = D – 1; (iii) Every irreducible T-module with endpoint 1 is thin.