Critical points of essential norms of singular integral operators in weighted spaces

We show that for any simple piecewise Ljapunov contour there exists a power weight such that the essential norm |S | in the spaceL ₂(, ) does not depend on the angles of the contour and it is given by formula (2). All such weights are described. For the union =₁₂ of two simple piecewise Lyapunov curves we prove that the essential norm |S | inL ₂() is minimal if both ₁ and ₂ are smooth in some neighborhoods of the common points. It is the case when the norm |S | in the spaceL ₂() as well as inL ₂(, ) does not depend on the values of the angles and it can be calculated by formula (5).     

The Spectral Theory of Amenable Actions and Invariants of Discrete Groups

Let G denote a semisimple group, a discrete subgroup, B=G/P the Poisson boundary. Regarding invariants of discrete subgroups we prove, in particular, the following:(1) For any -quasi-invariant measure on B, and any probablity measure on , the norm of the operator () on L ²(B,) is equal to (), where is the unitary representation in L ²(X,), and is the regular representation of .(2) In particular this estimate holds when is Lebesgue measure on B, a Patterson–Sullivan measure, or a -stationary measure, and implies explicit lower bounds for the displacement and Margulis number of (w.r.t. a finite generating set), the dimension of the conformal density, the -entropy of the measure, and Lyapunov exponents of .(3) In particular, when G=PSL₂() and is free, the new lower bound of the displacement is somewhat smaller than the Culler–Shalen bound (which requires an additional assumption) and is greater than the standard ball-packing bound.We also prove that ()=_G() for any amenable action of G and L ¹(G), and conversely, give a spectral criterion for amenability of an action of G under certain natural dynamical conditions. In addition, we establish a uniform lower bound for the -entropy of any measure quasi-invariant under the action of a group with property T, and use this fact to construct an interesting class of actions of such groups, related to 'virtual' maximal parabolic subgroups. Most of the results hold in fact in greater generality, and apply for instance when G is any semi-simple algebraic group, or when is any word-hyperbolic group, acting on their Poisson boundary, for example.     

On the Neumann-Poincaré operator

Let be a rectifiable Jordan curve in the finite complex plane which is regular in the sense of Ahlfors and David. Denote by L _C ² () the space of all complex-valued functions on which are square integrable w.r. to the arc-length on . Let L ²() stand for the space of all real-valued functions in L _C ² () and put Since the Cauchy singular operator is bounded on L _C ² (), the Neumann-Poincaré operator C ₁ sending each h L ²() into , is bounded on L ²(). We show that the inclusion characterizes the circle in the class of all AD-regular Jordan curves .     

Convexity theories 0. Foundations

For an arbitrary prenormed semiring, the closed unit ball functor from the category _R pnSmod ¹ ofR-prenormedR-semimodules with contractions to the category of sets has a left adjoint. For such a semiringR the notion of finitary convexity theory overR is introduced and the category C of -modules is defined. It is shown that the canonical functor _R pnSmod ¹ C has a left adjoint. In caseR is a banach semiring one has infinitary convexity theories, in addition to the finitary ones, and again the canonical functor _R bnSmod ¹ C has a left adjoint.Many more happy returns, Nico. Sixty is forever.     

A remark on Nehari's problem

An example of a strickly contractive Hankel matrix is given such that the central contractive extensionf _c of satisfies f _c =1. This way we answer a problem raised by Ciprian Foias.Partially supported by a Georgia State University Research Grant.     

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.     

Orbit counting for some discrete groups acting on simply connected manifolds with negative curvature

Summary In this article we develop a method of deriving asymptotic formulae for the orbital counting function for the action of certain discrete groups of isometries of simply connected negatively curved manifolds. We consider the particular case of normal subgroups ₀ of a co-compact group ₀ for which the quotient ₀/ ^k . Even in the special case of manifolds ofconstant negative curvature, this leads to new results. In particular, we have asymptotic estimates for some groups which arenot geometrically finite.Oblatum III-1993 & 18-VIII-1993The first author was supported by The Royal Society through a University Research Fellowship. The second author was supported by the UK SERC under grant number GR/G51930     

An Arithmetic Reduction of Finite Rank 3 Geometries with Linear Spaces as Plane Residues and with Dual Linear Spaces as Point Residues

Let be a rank three incidence geometry of points, lines and planes whose planes are linear spaces and whose point residues are dual linear spaces (notice that we do not require anything on the line residues). We assume that the residual linear spaces of belong to a natural class of finite linear spaces, namely those linear spaces whose full automorphism group acts flag-transitively and whose orders are polynomial functions of some prime number. This class consists of six families of linear spaces. In the amalgamation of two such linear spaces imposes an equality on their orders leading, in particular, to a series of diophantine equations, the solutions of which provide a reduction theorem on the possible amalgams of linear spaces that can occur in .We prove that one of the following holds (up to a permutation of the words point and plane):A) the planes of and the dual of the point residues belong to the same family and have the same orders,B) the diagram of is in one of six families,C) the diagram of belongs to a list of seven sporadic cases.Finally, we consider the particular case where the line residues of are generalized digons.     

A finite element method for the nonlinear Tricomi problem

Summary We consider a finite element procedure for numerical solution of the nonlinear problem:L[u]=yu _xx +u _yy +r(x,y)u=f(x, y, u) in a simply connected regionG in thex-y plane. The boundary ofG consists of ₀, ₁, and ₂ and we impose the boundary condition . ₀ is assumed to be a piecewises smooth curve lying in the half-planey>0 with endpointsA(–1, 0) andB(0, 0). ₁ and ₂ are characteristics of the operatorL issued fromA andB which intersect at the pointC(–1/2,y _c). An error analysis of the method is also given.     

Embedded minimal annuli solving an exterior problem

Given a compact, strictly convex body in ³ and a closed Jordan curve ³ satisfying several additional assumptions, the existence of a parametric, annulus type minimal surface is proved, which parametrizes along one boundary component, has a free boundary onX along the other boundary component, and which stays in ³. As a consequence of this and a reasoning developed by W. H. Meeks and S. -T. Yau we find an embedded minimal surface with these properties. Another application is the existence of an embedded minimal surface with a flat end, free boundary onX and controlled topology.This article was processed by the author using the LATEX style filepljourlm from Springer-Verlag.     

On the Normality of Cayley Digraphs of Valency 2 on Non-Abelian Group of Odd Order

In this paper, we prove that the Cayley digraph = Cay(G, S) of valency 2 on non-abelian group G of odd order is normal if the automorphism group of A(), a graph constructed from by using the method presented in the paper, is primitive on the vertices set V(A(). We also prove that the Cayley digraphs of valency 2 on non-abelian group of order pq² are normal, where p and q are distinct odd primes.AMS Subject Classification (2000) 05C25 20B25Supported by the National Natural Science Foundation of China (Grant no. 19971086) and the Doctoral Program Foundation of the National Education Department of China.     

Estimation of a sum along an algebraic curve

Let be an algebraic curve determined over a finite field k = [q]; e,x are subsidiary additive and multiplicative characters of the field k;, are functions in determined over k and satisfying some natural conditions. If P passes through the points of curve , rational over k, then where constant C depends only on the powers of ,,.Translated from Matematicheskie Zametki, Vol. 5, No. 3, pp. 373–380, March, 1969.     

Continuous Quotients for Lattice Actions on Compact Spaces

Let < SL _n ( ) be a subgroup of finite index, where n 5. Suppose acts continuously on a manifold M, where ₁(M) = ⁿ , preserving a measure that is positive on open sets. Further assume that the induced action on H ¹(M) is non-trivial. We show there exists a finite index subgroup < and a equivariant continuous map : M ⁿ that induces an isomorphism on fundamental group. We prove more general results providing continuous quotients in cases where ₁(M) surjects onto a finitely generated torsion free nilpotent group. We also give some new examples of manifolds with actions.     

Extended generalized octagons and the groupHe

Let be an extended generalized octagon such that the points of a triple {u, v, w} not on a block are pairwise adjacent if and only if the distance betweenv andw in the local generalized octagon _u equals 3 and there is a thick line through any point of _u . Then is one of the two examples related to the groups 2·L ₃(4).2² andHe. It is also shown that does not admit further extensions.     

A condition for nonnegativity of an operator related to the Dirichlet operator

Let L₀ be a positive definite closed linear operator with domain of definition D(L₀) dense in the Hilbert space H; let(, ₁, ₂) be the positive boundary value space of the operator L₀ such that the restriction of L ₀ ^* to ker ₂ is the Friedrichs extension of the operator L₀. We establish a test for nonnegativity of an operator T of the form Ty=L ₀ ^* y+^*(₁–C)y, y D(T)= ker(₂+), where :H and C: are respectively a compact operator and a bounded nonnegative operator.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 30–33.     

Two Exterior Algebras for Orthogonal and Symplectic Quantum Groups

Let be one of the N ²-dimensional bicovariant first-order differential calculi on the quantum groups O _q (N) or Sp _q (N), where q is not a root of unity. We show that the second antisymmetrizer exterior algebra _s is the quotient of the universal exterior algebra _u by the principal ideal generated by . Here denotes the unique up to scalars bi-invariant 1-form. Moreover, is central in _u and _u is an inner differential calculus.     

Integral Congruence Two Hyperbolic 5-Manifolds

In this paper, we classify all the orientable hyperbolic 5-manifolds that arise as a hyperbolic space form H ⁵/ where is a torsion-free subgroup of minimal index of the congruence two subgroup ⁵ ₂ of the group ⁵ of positive units of the Lorentzian quadratic form x 2/1 +... +x 5/2 -x 6/2. We also show that ⁵ ₂ is a reflection group with respect to a 5-dimensional right-angled convex polytope in H ⁵. As an application, we construct a hyperbolic 5-manifold of smallest known volume 7 (3)/4.     

Exact boundary controllability onL 2(Ω) ×H −1(Ω) of the wave equation with dirichlet boundary control acting on a portion of the boundary∂Ω, and related problems

Consider the wave equation defined on a smooth bounded domain R ⁿ with boundary = _{0 1}. The control action is exercised in the Dirichlet boundary conditions only on ₁ and is of classL ₂(0,T: L ₂(₁)); instead, homogeneous boundary conditions of Dirichlet (or Neumann) type are imposed on the complementary part ₀. The main result of the paper is a theorem which, under general conditions on the triplet {, ₀, ₁} with ₀ , guarantees exact controllability on the spaceL ₂() ×H ^–1() of maximal regularity forT greater than a computable timeT ₀>0, which depends on the triplet. This theorem generalizes prior results by Lasiecka and the author [L-T.3] (obtained via uniform stabilization) and by Lions [L.5], [L.6] (obtained by a direct approach, different from the one followed here). The key technical issue is a lower bound on theL ₂(₁)-norm of the normal derivative of the solution to the corresponding homogeneous problem, which extends to a larger class of triplets {, ₀, ₁} prior results by Lasiecka and the author [L-T.3] and by Ho [H.1].This research was partially supported by the National Science Foundation under Grant NSF-DMS-8301668 and by the Air Force Office of Scientific Research under Grant AFOSR-84-0365. This paper was presented at the IFIP WG7.2 Conference on Boundary Control and Boundary Variations held at the Départment de Mathématiques, Universite de Nice, 10–13 June 1986; at the International Conference on Control of Distributed Parameter Systems held at Vorau (Austria), 6–12 July 1986; and at the Second Workshop on Control of Systems Governed by Partial Differential Equations held at Val David, Quebec, 5–9 October 1986.     

Canonical contact forms on spherical CR manifolds

We construct the CR invariant canonical contact form can(J) on scalar positive spherical CR manifold (M,J), which is the CR analogue of canonical metric on locally conformally flat manifold constructed by Habermann and Jost. We also construct another canonical contact form on the Kleinian manifold ()/, where is a convex cocompact subgroup of Aut_CRS²ⁿ⁺¹=PU(n+1,1) and () is the discontinuity domain of . This contact form can be used to prove that ()/ is scalar positive (respectively, scalar negative, or scalar vanishing) if and only if the critical exponent ()<n (respectively, ()>n, or ()=n). This generalizes Nayatanis result for convex cocompact subgroups of SO(n+1,1). We also discuss the connected sum of spherical CR manifolds.