For two given point sets, we present a very simple (almost trivial) algorithm to translate one set so that the Hausdorff distance between the two sets is not larger than a constant factor times the minimum Hausdorff distance which can be achieved in this way. The algorithm just matches the so-called Steiner points of the two sets. The focus of our paper is the general study of reference points (like the Steiner point) and their properties with respect to shape matching. For more general transformations than just translations, our method eliminates several degrees of freedom from the problem and thus yields good matchings with improved time bounds. 1 Introduction This paper is motivated by a problem that is typical in application areas such as computer vision or pattern recognition, namely, given two figures A; B, to determine how much they "resemble each other". Here, a "figure" will be a union of finitely many points and line segments in R 2 or triangles in R 3 . Note t...

this paper is twofold. In the first part (sections 2 - 6) I want to give a survey on recent developments of Monte Carlo complexity. This will include techniques to derive sharp lower bounds as well as the construction of concrete numerical methods which attain these optimal bounds. The field covered here lies at the frontiers of several disciplines, among them theoretical computer science, numerical analysis, probability theory, approximation theory and to a large extent functional analysis. I want to stress the latter aspect and show how new techniques from Banach space and operator theory can be applied to Monte Carlo complexity. In the second part I want to present new results - the solution to a problem concering the Monte Carlo complexity of Fredholm integral equations. This will demonstrate in detail the general approach outlined in part one. We develop a new, fast algorithm - it is a combination of Monte Carlo methods with the Galerkin technique, an approach which seems to be new to this field. The basis functions used for the Galerkin discretization are orthogonal splines of minimal smoothness. They lead to an implementable procedure of minimal computational cost. The paper is organized as follows. In section 2, the main notions of information-based complexity theory are explained. We cover both the deterministic and the stochastic setting in detail, also for the sake of later comparisons. Some relations to s-number theory are presented in section 3. The role of the average case in proofs of lower bounds for Monte Carlo methods is explained in Section 4. In the following three sections, we analyse the complexity of basic numerical problems: Section 5 deals with numerical integration and contains classical results on the complexity of Monte Carlo quadrature, toge...

Abstract. We present an extension of the methods of classical Lie group analysis of differential equations to equations involving generalized functions (in particular: distributions). A suitable framework for such a generalization is provided by Colombeau’s theory of algebras of generalized functions. We show that under some mild conditions on the differential equations, symmetries of classical solutions remain symmetries for generalized solutions. Moreover, we introduce a generalization of the infinitesimal methods of group analysis that allows to compute symmetries of linear and nonlinear differential equations containing generalized function terms. Thereby, the group generators and group actions may be given by generalized functions themselves.

Abstract. We analyze the decomposition rank (a notion of covering dimension for nuclear C ∗-algebras introduced by E. Kirchberg and the author) of subhomogeneous C ∗-algebras. In particular we show that a subhomogeneous C ∗-algebra has decomposition rank n if and only if it is recursive subhomogeneous of topological dimension n and that n is determined by the primitive ideal space. As an application, we use recent results of Q. Lin and N. C. Phillips to show the following: Let A be the crossed product C ∗-algebra coming from a compact smooth manifold and a minimal diffeomorphism. Then the decomposition rank of A is dominated by the covering dimension of the underlying manifold. In [4], E. Kirchberg and the author introduced the decomposition rank; this is a noncommutative generalization of topological covering dimension. If A is a nuclear C∗-algebra, the decomposition rank of A, drA, is defined by imposing a certain condition on systems of completely positive (c.p.) approximations of A; see Section

In this paper we prove two strict insertion theorems for frame homomorphisms. When applied to the frame of all open subsets of a topological space they are equivalent to the insertion statements of the classical theorems of Dowker and Michael regarding, respectively, normal countably paracompact spaces and perfectly normal spaces. In addition, a study of perfect normality for frames is made.

Résumé: Etant donné un espace compact X, nous généralisons la notion d’unitaire multiplicatif introduite par Baaj et Skandalis ([4]) au cadre des C(X)-modules hilbertiens et en étudions les propriétés de continuité ([40]). Nous associons alors à certaines déformations de C ∗-algèbres de Hopf construites par Woronowicz ([51, 53]) des champs continus d’unitaires multiplicatifs et nous montrons que ces déformations correspondent à des déformations topologiques. Abstract: Given a compact space X, we generalize the notion of multiplicative unitary introduced by Baaj and Skandalis ([4]) to the framework of Hilbert C(X)-modules and we study its continuity properties ([40]). We then associate to several deformations of Hopf C ∗-algebras constructed by Woronowicz ([51, 53]) continuous fields of multiplicative unitaries and we prove that those deformations correspond to topological deformations. Classification A.M.S.: 46L05, 46M05, 16W30. Avant propos L’une des constructions de base de l’analyse harmonique est la transformation de Fourier: elle associe à un groupe abélien le groupe abélien de ses caractères et permet d’étudier cette correspondance auto-duale. Cette construction a été généralisée aux groupes localement compacts. Mais dès que l’on sort du cadre commutatif, on est confronté au problème suivant: l’objet dual d’un groupe (l’algèbre de convolution du groupe) n’est plus de même nature. Se pose alors le problème de trouver des objets généralisant les structures des groupes ainsi que celles des objets duaux. C’est pourquoi a été introduite la notion d’algèbre de Hopf de la manière suivante: à un groupe G (compact), on associe l’algèbre A des fonctions sur le groupe. La loi de multiplication G × G − → G induit un morphisme d’algèbres δ: A − → A ⊗ A pour lequel l’associativité s’exprime par le diagramme commutatif

Dopo avere provato un teorema di tipo Lusin per multifunzioni, presentiamo alcuni teoremi sulla propriet`a di Scorza Dragoni. La caratteristica principale di tali teoremi consiste nel fatto che le multifunzioni in esame sono definite e hanno valori in un contesto in cui non e' definita alcuna metrica. Come applicazione, in particolare, si ottiene un teorema di esistenza di selezioni di Carath'eodory. 1 Introduction A Carath'eodory selection of a multifunction F : T \Theta X ! Y is a function f(t; x), defined whenever F (t; x) is non-empty, which is measurable in t, continuous in x and such that f(t; x) 2 F (t; x). As Artstein and Prikry pointed out ([1], x3), Scorza Dragoni type theorems for multifunctions (which give almost lower semicontinuity) enable to derive existence of Carath'eodory selections in situations when lower semicontinuity implies existence of continuous selections. Many Authors have given Scorza Dragoni type theorems for multifunctions (see, e.g., [1], [3], [...

Abstract. We say that a metrizable space M is a Krasinkiewicz space if any map from a metrizable compactum X into M can be approximated by Krasinkiewicz maps (a map g: X → M is Krasinkiewicz provided every continuum in X is either contained in a fiber of g or contains a component of a fiber of g). In this paper we establish the following property of Krasinkiewicz spaces: Let f: X → Y be a perfect map between metrizable spaces and M a Krasinkiewicz complete ANR-space. If Y is a countable union of closed finite-dimensional subsets, then the function space C(X, M) with the source limitation topology contains a dense Gδ-subset of maps g such that all restrictions g|f −1 (y), y ∈ Y, are Krasinkiewicz maps. The same conclusion remains true if M is homeomorphic to a closed convex subset of a Banach space and X is a C-space. 1.

. We study mainly the class (GC) of all real Banach spaces X such that the set E f (a) of the minimizers of the function X 3 x 7! f(kx \Gamma a 1 k; : : : ; kx \Gamma a N k) is nonempty whenever N is a positive integer, a 2 X N , and f is a continuous monotone coercive function on [0; +1[ N . For particular choices of f , the set E f (a) coincides with the set of Chebyshev centers of the set fa i : i = 1; : : : ; Ng or with the set of its medians. The class (GC) is stable under making c 0 -, ` p - and similar sums. Under some geometric conditions on X, the function spaces C b (T; X) or L p (; X) belong to (GC). One of the main tools is a theorem which asserts that, in the definition of the class (GC), one can restrict himself to the functions f of the type f( 1 ; : : : ; N ) = max% i i (% i ? 0). Introduction Let X be a real Banach space, f a real-valued function of N variables defined at least on R N + = [0; +1[ N . Instead of finite sets A = fa 1 ; : : : ; aN g ae X...

We consider the question of simultaneous extension of partial ultrametrics, i.e. continuous ultrametrics defined on nonempty closed subsets of a compact zero-dimenional metrizable space. The main result states that there exists a continuous extension operator that preserves the maximum operation. This extension can also be chosen so that it preserves the Assouad dimension.