Two natural metrics and their covariant derivatives on a manifold of embeddings

OnE(M, ⁿ ), the Fréchet-manifold of all smooth embeddings of a smooth, compact, closed, orientable manifoldM (of dimensionn-1) into ⁿ two natural metricsG and are considered. The metric plays a central rôle in elasticity theory. Using a generalised notion of the Fréchet derivative their respective spraysS and and the correspoonding Levi-Civita connections are computed. BothS and are smooth in a well defined sense. In contrast toS the spray turns out to be trivial.     

Corona Extendibility and Asymptotic Multiplicativity

We study outer multiplier algebras, C(E)=M(E)/E, also known as corona algebras, and *-homomorphisms A C(E) . We prove in several instances that for all such maps there must exist an extension to a largerC ^* -algebra . The Kasparov Technical Theorem gives one class of examples where . Our theorems apply to subhomogeneous C ^* -algebras, such as , the algebra used in Cuntz's picture of K-theory. Where such an extension theorem exists, there must exist an asymptotic morphism whose restriction to A is equivalent to the identity. We also use extension results to prove closure properties for the collection of C ^*-algebras that have stable relations.     

Linear extension majority cycles on partial orders

Let P define a partial order on a set X of cardinalityn. A linear extensionL ofP is a linear order withP G L, and is the set of all linear extensions ofP. denotes that subset of withxLy forx, y X. A linear extension majority (LEM) relationM onX is defined byxMy if . Similarly,M is defined byxMy if . An LEM cycle exists if there arex, y, z X withxMyMzMx, and an LEM quasi-cycle exists ifxMyMzMx and the equality part of the definition ofM holds for exactly one pair in the triple. The study shows that no semiorders have LEM cycles or LEM quasi-cycles, and that every interval order has a maximal element under theM relation. LEM cycles and LEM quasi-cycles are also considered for partial orders with specific structures. Simulation is used to determine the relative likelihood with which LEM cycles and LEM quasi-cycles are observed when connected partial orders are generated at random by a specific procedure.Dr. Gehrlein's research was supported through a fellowship from the Center for Advanced Study of the University of Delaware.     

An infinite order operator on the lattice of varieties of completely regular semigroups

Let be a variety of completely regular semigroups. Define C ^* to be the class of all completely regular semigroupsS whose least full and self-conjugate subsemigroupC ^*(S) belongs to . ThenC ^* is an operator on the lattice of varieties of completely regular semigroups. In this note we show that the order ofC ^* is infinite. This fact yields that the Mal'cev project is not associative on . We describe (C ^*)¹, andi 0, in terms of -invariant normal subgroups of the free group over a countably infinite set. The lattice theoretic properties ofC ^* are also studied.Presented by W. Taylor.     

On the structure of initial segments of models of arithmetic

For any countable nonstandard modelM of a sufficiently strong fragment of arithmeticT, and any nonstandard numbersa, c M, Mca, there is a modelK ofT which agrees withM up toa and such that inK there is a proof of contradiction inT with Gödel number .     

The Eta Invariant and the Real Connective K-Theory of the Classifying Space for Quaternion Groups

We express the real connective K-theory groups o_4k–1(B Q ) ofthe quaternion group Q of order = 2 ^j 8 in terms of therepresentation theory of Q by showing o_4k–1(B Q ) = Sp(S ^4k+3/Q )where is any fixed point free representation of Q in U(2k + 2).     

Differentiability of Inverse Operators and Limit Theorems for Inverse Functions

Let H be a map from a set SR ^d to R ^d . For tR ^d let _H (t) denote the distance from t to the set H(S). Consider sequences {s _n} _n1 in S such that . Any limit point of any such sequence (finite or infinite) is considered as a possible value of the inverse H ^–1(t). Any map defined in such a way will be called an SC-inverse (a selected closest inverse) to H. In the paper we study differentiability of the nonlinear operator at H=G, where G is a one-to-one map from S onto a set TR ^d with good analytic properties (specifically, a diffeomorphism). We establish compact differentiability of this operator tangentially to continuous functions and introduce a family of norms such that it is Fréchet differentiable with respect to them. We also obtain optimal bounds for the remainder of the differentiation, extending to the multivariate case recent results of Dudley. These differentiability results are applied to random maps , which could be statistical estimators of an unknown map G. For a function J on R ^d , let (J) _T be its restriction to T. It is shown that for a diffeomorphism G and for an increasing sequence of positive numbers {a _n } _n1 weak convergence of the sequence {a _n (G _n – G)} _n1 (locally in S) is equivalent to weak convergence of the sequence (locally in T) along with the convergence of the sequence to 0 in probability (locally uniformly in S). The equivalence holds for all SC-inverses and all double SC-inverses and it extends to the multivariate case a theorem of Vervaat. Moreover, each of these equivalent statements implies a kind of Taylor expansion of the SC-inverse at G (locally uniformly in T) where inv(A) denotes the inverse of a nonsingular linear transformation A in R ^d . Such limit theorems for functional inverses can be used to study asymptotic behavior of statistical estimators defined implicitly (as solutions of equations involving the empirical distribution P _n ). We show how to apply this approach to get asymptotic normality of M-estimators in the multivariate case under minimal assumptions. We consider an extension of the quantile function to the multivariate case related to M-parameters of a distribution P in R ^d (an M-quantile function)and use limit theorems for functional inverses to study limit behavior of the empirical M-quantile process. We also show how to use these theorems to study asymptotics of regression quantiles.     

A Characterisation of the Generalized Quadrangle Q (5, q) Using Cohomology

If a GQ S of order (s, s) is contained in a GQ S of order (s, s ²) as a subquadrangle, then for each point X of S\S the set of points of S collinear with X form an ovoid of S. Thas and Payne proved that if S= (4,q),q even, and is an elliptic quadric for each XS\S,thenS (5,q). In this paper we provide a single proof for the q odd and q even cases by establishing a link between the geometry involved and the first cohomology group of a related simplicial complex.     

A polynomial-time algorithm for a class of linear complementarity problems

Given ann × n matrixM and ann-dimensional vectorq, the problem of findingn-dimensional vectorsx andy satisfyingy = Mx + q, x 0,y 0,x _i y _i = 0 (i = 1, 2,,n) is known as a linear complementarity problem. Under the assumption thatM is positive semidefinite, this paper presents an algorithm that solves the problem in O(n ³ L) arithmetic operations by tracing the path of centers,{(x, y) S: x _i y _i = (i = 1, 2,,n) for some > 0} of the feasible regionS = {(x, y) 0:y = Mx + q}, whereL denotes the size of the input data of the problem.     

Hyperplane Sections of Kantor's Unitary Ovoids

The projective plane is embedded as a variety of projective points in , where M is a nine dimensional -module for the groupG=GL(3,q ²). The hyperplane sections of thisvariety and their stabilizers in the group G aredetermined. When q 2 (mod 3) one such hyperplanesection is a member of the family of Kantor's unitary ovoids.We furtherdetermine all sections whereD has codimension two in M and demonstratethat these are never empty. Consequences are drawn for Kantor'sovoids.     

Zur Interpolation und Integration differenzierbarer periodischer Funktionen

Summary Letx ₀<x ₁<...<x _n–1<x ₀+2 be nodes having multiplicitiesv ₀,...,v _n–1, 1v _k r (0k<n). We approximate the evaluation functional ,x fixed, and the integral respectively by linear functionals of the form and determine optimal weights for the Favard classesW _r C ₂. In the even case of optimal interpolation these weights are unique except forr=1,x(x _k +x _k–1)/2 mod 2. Moreover we get periodic polynomial splinesw _{k, j} (0k<n, 0j<v _k ) of orderr such that are the optimal weights. Certain optimal quadrature formulas are shown to be of interpolatory type with respect to these splines. For the odd case of optimal interpolation we merely have obtained a partial solution.

Bojanov hat in [4, 5] ähnliche Resultate wie wir erzielt. Um Wiederholungen zu vermeiden, werden Resultate, deren Beweise man bereits in [4, 5] findet, nur zitiert     

Robust tests in group sequential analysis: One- and two-sided hypotheses in the linear model

Consider the linear modelY=X+E in the usual matrix notation where the errors are independent and identically distributed. We develop robust tests for a large class of one- and two-sided hypotheses about when the data are obtained and tests are carried out according to a group sequential design. To illustrate the nature of the main results, let and be anM- and the least squares estimator of respectively which are asymptotically normal about with covariance matrices ²(X ^t X)^–1 and ²(X ^t X)^–1 respectively. Let the Wald-type statistics based on and be denoted byRW andW respectively. It is shown thatRW andW have the same asymptotic null distributions; here the limit is taken with the number of groups fixed but the numbers of observations in the groups increase proportionately. Our main result is that the asymptotic Pitman efficiency ofRW relative toW is (²/²). Thus, the asymptotic efficiency-robustness properties of relative to translate to asymptotic power-robustness ofRW relative toW. Clearly, this is an attractive result since we already have a large literature which shows that is efficiency-robust compared to . The results of a simulation study show that with realistic sample sizes,RW is likely to have almost as much power asW for normal errors, and substantially more power if the errors have long tails. The simulation results also illustrate the advantages of group sequential designs compared to a fixed sample design, in terms of sample size requirements to achieve a specified power.     

Arithmetic distance on compact symmetric spaces

Let M be a compact Riemannian symmetric space. Then M=G/K, where G is the identity component of the isometry group of M and K is the isotropy subgroup of G at a point. In 1965 Nagano studied and classified the geometric transformation groups of compact symmetric spaces. Roughly speaking they are larger groups L that act on M, (i) G/L; (ii) L is a Lie transformation group acting effectively on M; (iii) L preserves the symmetric structure of M; and (iv) L is simple.Using Helgason spheres, S(), the minimal totally geodesic spheres in a compact irreducible symmetric space, we define an arithmetic distance for compact irreducible symmetric spaces and prove: THEOREM. Let M=G ^p(K ⁿ ), K=, H, or R, or M=AI(n), of rank greater that 1 and dimension greater that 3, let L be the geometric transformation group of M. Let L={: MM: is a diffeomorphism and preserves arithmetic distance}. Then L=L     

Differentiable structures with zero entropy on simply connected 4-manifolds

We show that a closed 4-dimensional simply connected topological manifoldM admits a differentiable structure with aC Riemannian metric whose geodesic flow has zero topological entropy if and only ifM is homeomorphic toS ⁴, ²,S ²×S ², or ²#².     

Separation of hypersurfaces

We give a generalization of results obtained in [15]. LetK ⁿ denote the set of embedded hypersurfaces in ⁿ⁺¹; for all xSⁿ and MK ⁿ we denote by C _x ^M the apparent contour ofM in the directionx. Then we give a sufficient condition on WSⁿ such that the map _W ^{K n}:K ⁿ P(T Sⁿ) , defined by _W ^{K n} (M)={C _w ^M ¦ wW}, is injective.     

Some dimension results for super-Brownian motion

Summary The Dawson-Watanabe super-Brownian motion has been intensively studied in the last few years. In particular, there has been much work concerning the Hausdorff dimension of certain remarkable sets related to super-Brownian motion. We contribute to this study in the following way. Let (Y _t)_t0 be a super-Brownian motion on ^d (d2) andH be a Borel subset of ^d . We determine the Hausdorff Dimension of {t0; SuppY _tHØ}, improving and generalizing a result of Krone. We also obtain a new proof of a result of Tribe which gives, whend4, the Hausdorff dimension of SuppY _t as a function of the dimension ofB.     

Large deviations for the invariant measure of a reaction-diffusion equation with non-Gaussian perturbations

Summary In this paper we establish a large deviations principle for the invariant measure of the non-Gaussian stochastic partial differential equation (SPDE) _t v =v +f(x,v )+(x,v ) . Here is a strongly-elliptic second-order operator with constant coefficients, h:=DH _xx-h, and the space variablex takes values on the unit circleS ¹. The functionsf and are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic PDE, and in particular we require that 0<mM wherem andM are some finite positive constants. The perturbationW is a Brownian sheet. It is well-known that under some simple assumptions, the solutionv ² is aC ^k (S ¹)-valued Markov process for each 0<1/2, whereC (S ¹) is the Banach space of real-valued continuous functions onS ¹ which are Hölder-continuous of exponent . We prove, under some further natural assumptions onf and which imply that the zero element ofC (S ¹) is a globally exponentially stable critical point of the unperturbed equation _t ⁰ = ⁰ +f(x,⁰), that has a unique stationary distributionv ^K, on (C (S ¹), (C ^K (S ¹))) when the perturbation parameter is small enough. Some further calculations show that as tends to zero,v ^K, tends tov ^K,0, the point mass centered on the zero element ofC (S ¹). The main goal of this paper is to show that in factv ^K, is governed by a large deviations principle (LDP). Our starting point in establishing the LDP forv ^K, is the LDP for the process , which has been shown in an earlier paper. Our methods of deriving the LDP forv ^K, based on the LDP for are slightly non-standard compared to the corresponding proofs for finite-dimensional stochastic differential equations, since the state spaceC (S ¹) is inherently infinite-dimensional.This work was performed while the author was with the Department of Mathematics, University of Maryland, College Park, MD 20742, USA     

Divisibility of Certain Partition Functions by Powers of Primes

Let be the prime factorization of a positive integer k and let b _k (n) denote the number of partitions of a non-negative integer n into parts none of which are multiples of k. If M is a positive integer, let S _k (N; M) be the number of positive integers N for which b _k(n ) 0(mod M). If we prove that, for every positive integer j In other words for every positive integer j, b _k(n) is a multiple of for almost every non-negative integer n. In the special case when k=p is prime, then in representation-theoretic terms this means that the number ofp -modular irreducible representations of almost every symmetric groupS _n is a multiple of p ^j. We also examine the behavior of b _k(n) (mod ) where the non-negative integers n belong to an arithmetic progression. Although almost every non-negative integer n (mod t) satisfies b _k(n) 0 (mod ), we show that there are infinitely many non-negative integers n r (mod t) for which b _k(n) 0 (mod ) provided that there is at least one such n. Moreover the smallest such n (if there are any) is less than 2 .     

Über Abstand 1 erhaltende Abbildungen in Minkowski-Ebenen

By using a computer the following theorem is proved: Consider K=GF(q), q {32,64,81,128}, :K² K² bijective such that P,Q K². Then is a semi-isometry. The assumption bijective can be dropped if q {32,128}.     

On the connectivity of unit distance graphs

For a number fieldK , consider the graphG(K^d), whose vertices are elements ofK ^d, with an edge between any two points at (Euclidean) distance 1. We show thatG(K²) is not connected whileG(K^d) is connected ford 5. We also give necessary and sufficient conditions for the connectedness ofG(K³) andG(K⁴).