On the boundedness of Cauchy singular operator from the spaceL p toL q,p>q>-1

It is proved that for a Cauchy type singular operator, given by equality (1), to be bounded from the Lebesgue spaceL _p () tol _q (), as = _n=1 ^Ȟ _n , _n ={z:|z|=r _n }, it is necessary and sufficient that either condition (4) or (5) be fulfilled.     

Stability results for an inverse problem in potential theory

Sunto Nel presente lavoro consideriamo un problema inverso, connesso a questioni di elettrocardiologia, relativo a un potenziale di doppio strato obliquo logaritmico, u, portato da una curva piana . Il problema consiste nel determinare conoscendo u sulla frontiera di un dominio T contenente . Il problema non è ben posto secondo Hadamard. Introduciamo delle limitazioni a priori su e dimostriamo alcuni risultati di stabilità.     

On the removable singularities of CR functions given on a generic manifold

Summary In this paper we consider the problem of holomorphic continuation and removal of singularities of the CR functions given on K, where is a generic manifold with nondegenerate Levi form and K is a meromorphically p-convex compactum. We get some conditions on , relative to p-convexity and q-concavity, under which every integrable CR function given on K extends holomorphically in some domain \K, where is a wedge domain with edge . Our Results are local.Authors had a support of Russian Fund of Fundamental Investigations (grant 93-011-258).     

The lower algebraicK-theory of virtually infinite cyclic groups

This paper gives a proof of a conjecture of W.-C. Hsiang for the negativeK-theory of integral grouprings , when the group is a subgroup of a uniform lattice in a Lie group. The authors' earlier paper reduced this result to the very special cases where either is finite or is virtually infinite cyclic. The finite case was done much earlier by Carter extending results of Bass and Murthy. The major work of the present paper consists of proving the conjecture when is virtually infinite cyclic.Both authors were supported in part by the National Science Foundation.     

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.     

Free Algebras of Automorphic Forms on the Upper Half-Plane

We find all pairs (,a) consisting of a cocompact Fuchsian group of genus zero and an automorphy factor a of for which the graded algebra of a--automorphic forms is free.     

GAMMA－MINIMAX ESTIMATORS FOR THE MEAN OF A MULTIVARIATE NORMAL DISTRIBUTION WITH PARTIALLY UNKNOWN COVARIANCE MATRIX

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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).     

An algebra of integral operators with fixed singularities in kernels

We continue the study of algebras generated by the Cauchy singular integral operator and integral operators with fixed singularities on the unit interval, started in R.Duduchava, E.Shargorodsky, 1990. Such algebras emerge when one considers singular integral operators with complex conjugation on curves with cusps. As one of possible applications of the obtained results we find an explicit formula for the local norms of the Cauchy singular integral operator on the Lebesgue spaceL ₂ (, ), where is a curve with cusps of arbitrary order and is a power weight. For curves with angles and cusps of order 1 the formula was already known (see R.Avedanio, N.Krupnik, 1988 and R.Duduchava, N.Krupnik, 1995). Dedicated to Professor Israel Gohberg on the occasion of his 70-th birthday Supported by EPSRC grant GR/K01001     

Subsets with Restricted Movement

Let G be a finite permutation group on a set with no fixed points in and let m and k be integers with 0 < m < k. For a finite subset of the movement of is defined as move() = max_gG| ^g \ |. Suppose further that G is not a 2-group and that p is the least odd prime dividing |G| and move() m for all k-element subsets of . Then either || k + m or k (7m – 5) / 2, || (9m – 3)/2. Moreover when || > k + m, then move() m for every subset of .     

Die Automorphismengruppen hyperelliptischer Kurven

Let be a hyperelliptic curve over an algebraically closed field of characteristic O and G a group of automorphisms of that contains the canonical involution of . We describe the group-theoretical structure of G by means of the arithmetical structure of the morphism ^G.     

Singular Integral Operators with Carleman Shift and Discontinuous Coefficients in the Spaces H^{\omega}_0(\Gamma,\rho)

For the singular integral operators with Carleman shift, preserving or changing orientation, and piecewise continuous coefficients we prove the theorem on Fredholmness and obtain the formula for index in the generalized Hölder spaces defined by an arbitrary continuity modulus from the Bari-Stechkin class and some general weights on a closed or open finite Lyapunov curve .     

On groups of polynomial subgroup growth

Summary Let be a finitely generated group anda _n ()=the number of its subgroups of indexn. We prove that, assuming is residually nilpotent (e.g., linear), thena _n () grows polynomially if and only if is solvable of finite rank. This answers a question of Segal. The proof uses a new characterization ofp-adic analytic groups, the theory of algebraic groups and the Prime Number Theorem. The method can be applied also to groups of polynomial word growth.Oblatum 1-VII-1989 & 7-VI-1990     

On Locally Projective Graphs of Girth 5

Let be a graph and G be a 2-arc transitive automorphism group of . For a vertex x let G(x)^(x) denote the permutation group induced by the stabilizer G(x) of x in G on the set (x) of vertices adjacent to x in . Then is said to be a locally projective graph of type (n,q) if G(x)^(x) contains PSL_n(q) as a normal subgroup in its natural doubly transitive action. Suppose that is a locally projective graph of type (n,q), for some n 3, whose girth (that is, the length of a shortest cycle) is 5 and suppose that G(x) acts faithfully on (x). (The case of unfaithful action was completely settled earlier.) We show that under these conditions either n=4, q=2, has 506 vertices and , and contains the Wells graph on 32 vertices as a subgraph. In the latter case if, for a given n, at least one graph satisfying the conditions exists then there is a universal graph W(n) of which all other graphs for this n are quotients. The graph W(3) satisfies the conditions and has 2²⁰ vertices.     

Toeplitz and singular integral operators on Carleson curves with logarithmic whirl points

We consider Toeplitz operators with piecewise continuous symbols and singular integral operators with piecewise continuous coefficients onL ^p (,w) where 1<p<,w is a Muckenhoupt weight and belongs to a large class of Carleson curves. This class includes curves with corners and cusps as well as curves that look locally like two logarithmic spirals scrolling up at the same point. Our main result says that the essential spectrum of a Toeplitz operator is obtained from the essential range of its symbol by joining the endpoints of each jump by a certain spiralic horn, which may degenerate to a usual horn, a logarithmic spiral, a circular arc or a line segment if the curve and the weightw behave sufficiently well at the point where the symbol has a jump. This result implies a symbol calculus for the closed algebra of singular integral operators with piecewise continuous coefficients onL ^p (,w).Research supported by the Alfried Krupp Förderpreis für junge Hochschullehrer of the Krupp Foundation.     

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.     

Banach algebras of singular integral operators with piecewise continuous coefficients. General contour and weight

The Banach algebra generated by one-dimensional linear singular integral operators with matrix valued piecewise continuous coefficients in the spaceL _p (,) with an arbitrary weight is studied. The contour consists of a finite number of closed curves and open arcs with satisfy the Carleson condition. The contour may have a finite number of points of selfintersection. The symbol calculus in this algebra is the main result of the paper.     

On Embeddings of the Flag Geometries of Projective Planes in Finite Projective Spaces

The flag geometry =( ) of a finite projective plane of order s is the generalized hexagon of order (s, 1) obtained from by putting equal to the set of all flags of , by putting equal to the set of all points and lines of and where I is the natural incidence relation (inverse containment), i.e., is the dual of the double of in the sense of Van Maldeghem Mal:98. Then we say that is fully and weakly embedded in the finite projective space PG(d, q) if is a subgeometry of the natural point-line geometry associated with PG(d, q), if s = q, if the set of points of generates PG(d, q), and if the set of points of not opposite any given point of does not generate PG(d, q). Preparing the classification of all such embeddings, we construct in this paper the classical examples, prove some generalities and show that the dimension d of the projective space belongs to {6,7,8}.     

On the stochastic independence properties of hard-core distributions

A probability measurep on the set of matchings in a graph (or, more generally 2-bounded hypergraph) ishard-core if for some : [0,), the probabilityp(M) ofM is proportional to . We show that such distributions enjoy substantial approximate stochastic independence properties. This is based on showing that, withM chosen according to the hard-core distributionp, MP () the matching polytope of , and >0, if the vector ofmarginals, (Pr(AM):A an edge of ), is in (1–) MP (), then the weights (A) are bounded by someA(). This eventually implies, for example, that under the same assumption, with fixed, as the distance betweenA, B tends to infinity.Thought to be of independent interest, our results have already been applied in the resolutions of several questions involving asymptotic behaviour of graphs and hypergraphs (see [14, 16], [11]–[13]).Supported in part by NSFThis work forms part of the author's doctoral dissertation [16]; see also [17]. The author gratefully acknowledges NSERC for partial support in the form of a 1967 Science and Engineering Scholarship.