Linear Versus Nonlinear Rules for Mixture Normal Priors

The problem under consideration is the -minimax estimation, under L₂loss, of a multivariate normal mean when the covariance matrix is known. The family of priors is induced by mixing zero mean multivariate normals with covariance matrix I by nonnegative random variables , whose distributions belong to a suitable family G. For a fixed family G, the linear (affine) -minimax rule is compared with the usual -minimax rule in terms of corresponding -minimax risks. It is shown that the linear rule is "good", i.e., the ratio of the risks is close to 1, irrespective of the dimension of the model. We also generalize the above model to the case of nonidentity covariance matrices and show that independence of the dimensionality is lost in this case. Several examples illustrate the behavior of the linear -minimax rule.     

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.     

On some locally finite groups which are sharply triply transitive

We classify the groups satisfying the following conditions: i) is locally finite; ii) is a sharply triply transitive permutation group; iii) all elements of have fixpoints.Published in Ukraninskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 1060–1065, July–August, 1991.     

On graphs and Lie rings

From a finite oriented graph , finite-dimensional graded nilpotent Lie rings () and () are naturally constructed; these rings are related to subtrees and connected subgraphs of , respectively. Diverse versions of these constructions are also suggested. Moreover, an embedding of Lie rings of the form () in the adjoint Lie rings of finite-dimensional associative rings (also determined by the graph ) is indicated.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 449–459.Original Russian Text Copyright © 2005 by Yu. S. Sernenov.This revised version was published online in April 2005 with a corrected issue number.     

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.     

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.     

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.     

Cohomology of Compact Locally Symmetric Spaces

We obtain a necessary condition for a cohomology class on a compact locally symmetric space S()=X (a quotient of a symmetric space X of the non-compact type by a cocompact arithmetic subgroup of isometries of X) to restrict non-trivially to a compact locally symmetric subspace S _H()=Y of X. The restriction is in a 'virtual' sense, i.e. it is the restriction of possibly a translate of the cohomology class under a Hecke correspondence. As a consequence we deduce that when X and Y are the unit balls in ⁿ and ^m , then low degree cohomology classes on the variety S() restrict non-trivially to the subvariety S _H (); this proves a conjecture of M. Harris and J-S. Li. We also deduce the non-vanishing of cup-products of cohomology classes for the variety S().     

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 .     

Lax Projective Embeddings of Polar Spaces

Let be a non-degenerate polar space of rank n 3 where all of its lines have at least three points. We prove that, if admits a lax embedding e : in a projective space defined over a skewfield K, then is a classical and defined over a sub-skewfield K₀ of K. Accordingly, admits a full embedding e₀ in a K₀-projective space ₀. We also prove that, under suitable hypotheses on e and e₀, there exists an embedding such that and preserves dimensions.Received: March, 2004     

Reconstruction of a wiener field from its realizations on a curve

Let w(x, y), x 0 and y 0 be a Wiener field on the plane; be a curve given parametrically x=x() and y=y(), [0, 1], where x() is a positive, continuous, nondecreasing function; y() is a positive, continuous, nonincreasing function. A best estimate in the mean-square sense is constructed for w(u, v)(u, v) , based on the values w(x, y), (x, y) and its error is found.Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 87–93, 1988.     

Splitting with Continuous Control in Algebraic K-Theory

In this work, the continuously controlled assembly map in algebraic K-theory, as developed by Carlsson and Pedersen, is proved to be a split injection for groups that satisfy certain geometric conditions. The group is allowed to have torsion, generalizing a result of Carlsson and Pedersen. Combining this with a result of John Moody, K₀(k) is proved to be isomorphic to the colimit of K₀(kH) over the finite subgroups H of , when is a virtually polycyclic group and k is a field of characteristic zero.     

The Laplacian with Wentzell-Robin Boundary Conditions on Spaces of Continuous Functions

We investigate the Laplacian on a smooth bounded open set Rn with Wentzell-Robin boundary condition $\beta u+\frac{\partial u}{\partial \nu} + \Delta u=0$ on the boundary . Under the assumption $\memb$ C() with $\geq$ 0 , we prove that generates a differentiable positive contraction semigroup on $C(\bar{\Omega})$ and study some monotonicity properties and the asymptotic behaviour.     

Statistical Cluster Points of Sequences in Finite Dimensional Spaces

In this paper we study the set of statistical cluster points of sequences in m-dimensional spaces. We show that some properties of the set of statistical cluster points of the real number sequences remain in force for the sequences in m-dimensional spaces too. We also define a notion of -statistical convergence. A sequence xis -statistically convergent to a set Cif Cis a minimal closed set such that for every > 0 the set has density zero. It is shown that every statistically bounded sequence is -statistically convergent. Moreover if a sequence is -statistically convergent then the limit set is a set of statistical cluster points.     

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.     

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}.     

Convexity theories III. Classification of certain real convexity theories

In this paper we classify all real convexity theories that contain the standard convexity theory _c. For this purpose we consider three subcases: finitary; infinitary and (_sc\_c)Ø; infinitary and _sc=_c. In each of these subcases one encounters a phenomenon resembling bifurcation.This research was supported by the Deutsche Forschungsgemeinschaft.     

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