Mengenwertige Masse und Fortsetzungen

Let (, _i) be a probability space for i=1,2 with and : ^m a correspondence, i.e. () is a non-void subset of ^m for all . We give necessary and sufficient conditions under which it holds, that ₂ extends ₁. iff _A d₂ is equal to _A d₁ for all A, where _A d_i is the set of all integrals _A f d_i of functions f: ^m with f()() _i.-a.e.     

Aperture-Angle Optimization Problems in Three Dimensions

Let [a,b] be a line segment with end points a, b and a point at which a viewer is located, all in R ³. The aperture angle of [a,b] from point , denoted by (), is the interior angle at of the triangle (a,b,). Given a convex polyhedron P not intersecting a given segment [a,b] we consider the problem of computing _max() and _min(), the maximum and minimum values of () as varies over all points in P. We obtain two characterizations of _max(). Along the way we solve several interesting special cases of the above problems and establish linear upper and lower bounds on their complexity under several models of computation.     

The Kronecker Product of Schur Functions Indexed by Two-Row Shapes or Hook Shapes

The Kronecker product of two Schur functions s and s , denoted by s * s , is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the partitions and . The coefficient of s in this product is denoted by , and corresponds to the multiplicity of the irreducible character in .We use Sergeev's Formula for a Schur function of a difference of two alphabets and the comultiplication expansion for s [XY] to find closed formulas for the Kronecker coefficients when is an arbitrary shape and and are hook shapes or two-row shapes.Remmel (J.B. Remmel, J. Algebra 120 (1989), 100–118; Discrete Math. 99 (1992), 265–287) and Remmel and Whitehead (J.B. Remmel and T. Whitehead, Bull. Belg. Math. Soc. Simon Stiven 1 (1994), 649–683) derived some closed formulas for the Kronecker product of Schur functions indexed by two-row shapes or hook shapes using a different approach. We believe that the approach of this paper is more natural. The formulas obtained are simpler and reflect the symmetry of the Kronecker product.     

Measures not charging semipolars and equations of schrödinger type

SupposeX is a Borel right process andm is a -finite excessive measure forX. Given a positive measure not chargingm-semipolars we associate an exact multiplicative functionalM(). No finiteness assumptions are made on . Given two such measures and ,M()=M() if and only if and agree on all finely open measurable sets. The equation (q–L)u+u=f whereL is the generator of (a subprocess of)X may be solved for appropriatef by means of the Feynman-Kac formula based onM(). Both uniqueness and existence are considered.Supported in part by NSF Grant DMS 92-24990.     

Couplings of markov chains by randomized stopping times

Summary We consider a Markov chain on (E, ) generated by a Markov kernel P. We study the question, when we can find for two initial distributions and two randomized stopping times T of (X _n ) _nN and S of ( X _n ) _nN , such that the distribution of X _T equals the one of X _S and T, S are both finite.The answer is given in terms of -, h with h bounded harmonic, or in terms of .For stopping times T, S for two chains ( X _n ) _nN ,( X _n ) _nN we consider measures , on (E, ) defined as follows: (A)=expected number of visits of ( X _n ) toA before T, (A)=expected number of visits of ( X _n ) toA before S.We show that we can construct T, S such that and are mutually singular and ( _v X _T )=( X _S . We relate and to the positive and negative part of certain solutions of the Poisson equation (I-P)(·)=-.     

Formule d'Itô pour des Diffusions Uniformément Elliptiques,et Processus de Dirichlet

Soit X une diffusion uniformément elliptique sur R ^d ,F une fonction dans H _loc ¹(R ^d ) et la loi initiale de la diffusion. On montre que si l'intégrale |F|²(x)U(x)dx est finie, oùU désigne le potentiel de la mesure , alors F(X) est un processus de Dirichlet. Si de plus, F appartient àH ² _loc(R ^d ) et si les intégrales |F|²(x)U(x)dx et |f _k |²(x)U(x)dx sont finies, pour les dérivées faibles f _k de F, alors on peut écrire une formule d'Itô. En particulier, on définit l'intégrale progressive F(X)dX et on prouve l'existence des covariations quadratiques [f _k (X),X ^k ].     

An automatic integration procedure for infinite range integrals involving oscillatory kernels

Let the real functionsK(x) andL(x) be such thatM(x)=K(x)+iL(x)=e^ix g(x), whereg(x) is infinitely differentiable for all largex and is non-oscillatory at infinity. We develop an efficient automatic quadrature procedure for numerically computing the integrals _a K(t)f(t) and _a L(t)f(t)dt, where the functionf(t) is smooth and nonoscillatory at infinity. One such example for which we also provide numerical results is that for whichK(x)=J (x) andL(x)=Y (x), whereJ (x) andY (x) are the Bessel functions of order . The procedure involves the use of an automatic scheme for Fourier integrals and the modified W-transformation which is used for computing oscillatory infinite integrals.     

On the E- and MV-optimality of block designs having k≥v

In this paper we consider the problem of determining and constructing E- and MV-optimal block designs to use in experimental settings where treatments are applied to experimental units occurring in b blocks of size k, k. It is shown that some of the well-known methods for constructing E- and MV-optimal unequally replicated designs having k fail to yield optimal designs in the case where . Some sufficient conditions are derived for the E- and MV-optimality of block designs having and methods for constructing designs satisfying these sufficient conditions are given.     

Compact Toeplitz operators via the Berezin transform on bounded symmetric domains

Let be an irreducible bounded symmetric domain of genusp, h(x, y) its Jordan triple determinant, andA ² () the standard weighted Bergman space of holomorphic functions on square-integrable with respect to the measureh(z, z) ^–p dz. Extending the recent result of Axler and Zheng for =D, =p=2 (the unweighted Bergman space on the unit disc), we show that ifS is a finite sum of finite products of Toeplitz operators onA ² () and is sufficiently large, thenS is compact if and only if the Berezin transform ofS tends to zero asz approaches . An analogous assertion for the Fock space is also obtained.The author's research was supported by GA AV R grant A1019701 and GA R grant 201/96/0411.     

On the strongly spherical Sasakian metric of a spherical tangent bundle

Let Mⁿ denote an n-dimensional Riemannian manifold. Its metric is called -strongly spherical if at every point Q Mⁿ there exists a -dimensional subspace _Q T_QMⁿ such that the curvature operator of the metric of Mⁿ satisfies R(X, Y) Z = k(< Y, Z > X < X, Z > Y), where k = const > 0, Y _Q , X, Z #x2208; T_QMⁿ. The number is called the index of sphericity and k the exponent of sphericity. The following theorems are proved in the paper.THEOREM 1. Let the Sasakian metric of T₁Mⁿ be -strongly spherical with exponent of sphericity k. The following assertions hold: a) = 1 if and only if M² has constant Gaussian curvature K 1 and k = K²/4; b) = 3 if and only if M² has constant curvature K = 1 and k = 1/4; c) = 0, otherwise.THEOREM 2. Let the Sasakian metric of T₁Mⁿ (n Mⁿ) be -strongly spherical with exponent of sphericity k. If k > 1/3 and k 1, then = 0. Let us denote by (Mⁿ, K) a space of constant curvatureK. THEOREM 3. Let the Sasakian metric of T₁(Mⁿ, K) (n 3) be -strongly spherical with exponent of sphericity k. The following assertions hold: a) = 1 if and only if K = 1/4; b) = 0, otherwise. In dimension n = 3 Theorem 2 is true for k {1/4, 1}.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 150–159, 1992.     

Decktransformationen transnormaler Mannigfaltigkeiten

For every transnormal m-manifold V (see [3] or [7]) in ⁿ :VW, mapping pV into its normal plane (p) is a covering map onto a submanifold W of the open Grassmannian H_n,n–m of all (n–m)-dimensional planes in ⁿ. The transnormal frame T:=^–1((p)) admits a transitive operation by a group J of isometries. The group action of the covering transformations of (V,,W) on T commutes with the action of J. The elements of J, which are restrictions of covering transformations to T, are exactly the elements of the centre of J. This property is applied to show the existence of nontrivial covering transformations of (V,,W) for n–m3.

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Diese Arbeit faßt die Kapitel 5, 6 und 7 der von der Fakultät für Allgemeine Ingenieurwissenschaften der TU Berlin genehmigten Dissertation [6] zusammen.     

Monge-Kantorovitch Measure Transportation and Monge-Ampère Equation on Wiener Space

Let (W,,H) be an abstract Wiener space assume two _i ,i=1,2 probabilities on (W,(W)). We give some conditions for the Wasserstein distance between ₁ and ₂ with respect to the Cameron-Martin space to be finite, where the infimum is taken on the set of probability measures on W×W whose first and second marginals are ₁ and ₂. In this case we prove the existence of a unique (cyclically monotone) map T=I _W +, with :WH, such that T maps ₁ to ₂. Moreover, if ₂, then T is stochastically invertible, i.e., there exists S:WW such that ST=I _{W 1} a.s. and TS=I _{W 2} a.s. If, in addition, ₁=, then there exists a 1-convex function in the Gaussian Sobolev space such that =. These results imply that the quasi-invariant transformations of the Wiener space with finite Wasserstein distance from can be written as the composition of a transport map T and a rotation, i.e., a measure preserving map. We give also 1-convex sub-solutions and Ito-type solutions of the Monge-Ampère equation on W. Mathematics Subject Classification (2000)60H07, 60H05,60H25, 60G15, 60G30, 60G35, 46G12, 47H05, 47H1, 35J60, 35B65,35A30, 46N10, 49Q20, 58E12, 26A16, 28C20cf. Theorem 6.1 for the precise hypothesis about ₁ and ₂.In fact this hypothesis is too strong, cf. Theorem 6.1. AcknowledgementThe authors are grateful to Françoise Combelles for all the bibliographical help that she has supplied for the realization of this research. We thank also the anonymous referee for his particular attention and valuable remarks.     

Examples of Saturated Convergence Rates for Tikhonov Regularization

Tikhonov regularization is one of the most popular methods for solving linear operator equations of the first kind Au = f with bounded operator, which are ill-posed in general (Fredholm's integral equation of the first kind is a typical example). For problems with inexact data (both the operator and the right-hand side) the rate of convergence of regularized solutions to the generalised solution u ₊ (i.e.the minimal-norm least-squares solution) can be estimated under the condition that this solution has the source form: u ₊ im(A*A). It is well known that for Tikhonov regularization the highest-possible worst-case convergence rates increase with only for some values of , in general not greater than one. This phenomenon is called the saturation of convergence rate. In this article the analysis of this property of the method with a criterion of a priori regularization parameter choice is presented and illustrated by examples constructed for equations with compact operators.This revised version was published online in October 2005 with corrections to the Cover Date.     

On differentiability of SRB states for partially hyperbolic systems

Consider a one parameter family of diffeomorphisms f such that f ₀ is an Anosov element in a standard abelian Anosov action having sufficiently strong mixing properties. Let be any u-Gibbs state for f . We prove (Theorem 1) that for any C function A the map (A) is differentiable at =0. This implies (Corollary 2.2) that the difference of Birkhoff averages of the perturbed and unperturbed systems is proportional to . We apply this result (Corollary 3.3) to show that a generic perturbation of the time one map of geodesic flow on the unit tangent bundle over a surface of negative curvature has a unique SRB measure with good statistical properties.     

Directed packings with block size 5 and even ν

Let andk be positive integers. A transitively orderedk-tuple (a ₁,a ₂,...,a _k) is defined to be the set {(a _i, a_j) 1i<jk} consisting ofk(k–1)/2 ordered pairs. A directed packing with parameters ,k and index =1, denoted byDP(k, 1; ), is a pair (X, A) whereX is a -set (of points) andA is a collection of transitively orderedk-tuples ofX (called blocks) such that every ordered pair of distinct points ofX occurs in at most one block ofA. The greatest number of blocks required in aDP(k, 1; ) is called packing number and denoted byDD(k, 1; ). It is shown in this paper that for all even integers , where [x] is the floor ofx.     

Primitive Illumination Systems for Families of Convex Bodies in the Plane

Let F= {C₁,C₂,...,C} be a family of ndisjoint convex bodies in the plane. We say that a set Vof exterior light sources illuminates F, if for every boundary point of any member of Fthere is a point in Vsuch that is visible from ,i.e. the open line segment joining and is disjoint from F. An illumination system Vis called primitive if no proper subset of Villuminates F. Let p_max(F) denote the maximum number of points forming a primitive illumination system for F, and letp_max(n) denote the minimum of F) taken over all families Fconsisting of ndisjoint convex bodies in the plane. The aim of this paper is to investigate the quantities p_max(F) and p_max(n).     

The Product of Independent Random Variables with Regularly Varying Tails

A distribution is said to have regularly varying tail with index – (0) if lim _x(kx,)/(x,)=k ^– for each k>0. Let X and Y be independent positive random variables with distributions and , respecitvely. The distribution of product XY is called Mellin–Stieltjes convolution (MS convolution) of and . It is known that D() (the class of distributions on (0,) that have regularly varying tails with index –) is closed under MS convolution. This paper deals with decomposition problem of distributions in D() related to MS convolution. A representation of a regularly varying function F of the following form is investigated: F(x)= _k=0 ^n–1 b _k f(a ^k x), where f is a measurable function and a and b _k (k=1,...,n–1) are real constants. A criterion is given for these constants in order that f be regularly varying. This criterion is applicable to show that there exist two distributions and such that neither nor belongs to D() (>0) and their MS convolution belongs to D().     

Canonical and Boundary Representations on a Hyperboloid of One Sheet

For the hyperboloid of one sheet X=G/H, G=SO₀(1,2), H=SO₀(1,1), canonical representations R _,, C, =0,1, are defined as the restrictions to G of representations of the overgroup =SO₀(2,2) associated with a cone. They act on the torus containing two copies of X as open G-orbits. We study boundary representations generated by R _,. For some , they contain Jordan blocks. The decomposition of R _, into irreducible constituents includes a finite number (depending on ) of irreducible parts of the boundary representations.     

Formulating Measure Differential Inclusions in Infinite Dimensions

Measure differential inclusions were introduced by J. J. Moreau to study sweeping processes, and have since been used to study rigid body dynamics and impulsive control problems. The basic formulation of an MDI is d / d (t) K(t) where is a vector measure, an unsigned measure, and K() is a set-valued map with closed, convex values and is hemicontinuous. Note that need not be absolutely continuous with respect to . Stewart extended Moreau's original concept (which applied only to cone-valued K()) to general convex sets, and gave strong and weak formulations of d / d (t) K(t) where K(t) R ⁿ . Here the strong and weak formulations of Stewart are extended to infinite-dimensional problems where K(t) X where X is a separable reflexive Banach space; they are shown to be equivalent under mild assumptions on K().     

Semilattices of proper inverse semigroups

Given a semilattice Y of inverse semigroups S, there corresponds a semilattice Y of groups G in a natural way. This correspondence is used to study semilattices of proper inverse semigroups. In paticular, it is shown that if S is a semilattice of proper inverse semigroups, then there exists a minimum semilattice congruence such that each -class is proper and there exists a maximum semilattice congruence such that each -class is proper.