Reinhard Winkler

Finitely additive measures on groups and rings

On topological groups a natural finitely additive measure can be defined via compactifications. It is closely related to Hartmans concept of uniform distribution on non-compact groups (cf. [3]). Applications to several situations are possible. Some results of M. Paštéka and other authors on uniform distribution with respect to translation invariant finitely additive probability measures on Dedekind domains are transferred to more general situations. Furthermore it is shown that the range of a polynomial of degree ≥2 on a ring of algebraic integers has measure 0.

Topological automorphisms of modified Sierpiński gaskets realize arbitrary finite permutation groups

Abstract The n-dimensional Sierpinski gasket X, spanned by n +1 vertices, has ( n +1)! symmetries acting as the symmetric group on the vertices. The object of this note is the remarkable observation that for n ≥2 every topological automorphism of X is one of these symmetries. A modification of the arguments yields that, given any finite permutation group G ≤ S n +1 acting on an ( n +1)-element set, there is a finite subset T ⫅ X such that G is the group of topological automorphisms of X \ T considered as a group acting faithfully on the vertices.

Metric, Fractal Dimensional and Baire Results on the Distribution of Subsequences

Let X be a locally compact metric space. One important object connected with the distribution behavior of an arbitrary sequence x on X is the set M(x) of limit measures of x. It is defined as the set of accumulation points of the sequence of the discrete measures induced by x. Using binary representation of reals one gets a natural bijective correspondence between infinite subsets of the set ℕ of positive integers and numbers in the unit interval I = 〈0, 1]. Hence to each sequence x = (xn)n∈ℕ ∈ Xℕ and every a I there corresponds a subsequence denoted by ax. We investigate the set M(ax) for given x with emphasis on the behavior for “typical” a in the sense of Baire category, Lebesgue measure and Hausdorff dimension.

Distribution Preserving Sequences of Maps and Almost Constant Sequences on Finite Sets.

LetX andY be finite sets and, for eachnεN,fn:X→Y. If λ and μ are probability measures onX andY resp., we ask the following question: How has the sequence (fn)nεN to look like such that every sequence (xn)nεN with distribution λ induces a sequence (fn(xn))nεN with distribution μ? A satisfactory description of these so called (λ,μ)-uniform distribution preserving sequences of maps is the object of this paper. Almost constant sequences and related notions are the key for an adequate understanding of the problem.

Mathematik als zentraler Teil des Projektes Aufklärung auf breiter Front

Methodisch fust die Mathematik auf strenger Logik und bildet somit ein vorbildliches Spielfeld fur folgerichtiges Denken. Daruber herrscht zweifellos breiter Konsens. Dass damit aber nicht nur sterile Glasperlenspiele innerhalb in sich geschlossener Systeme moglich sind, sondern wesentliche Beitrage zu einer im besten Sinne aufgeklarten Gesellschaft geleistet werden konn(t)en, verdient durchaus fundierter Begrundungen und exemplarischer Illustrationen. Solche zu geben, ist das Anliegen des vorliegenden Artikels – sowohl in Form allgemeiner Uberlegungen als auch anhand konkreter Beispiele.

Mathematische Prozesse im Widerstreit

Bei der Untersuchung von Prozessen, in denen Mathematik im Spiel ist, ergibt sich die Notwendigkeit einer sorgfaltigen Unterscheidung verschiedener Arten und Betrachtungsweisen. Naher untersucht sollen hier werden: historische, gesellschaftliche, technologische, psychologische und innermathematische Prozesse, die jeweils selbst wieder in verschiedenen Auspragungen auftreten.

Der Organismus der Mathematik – mikro-, makro- und mesoskopisch betrachtet

Meist enden ahnliche Gesprache uber Mathematik etwa an diesem Punkt, ohne dass der Nichtmathematiker von der Sinnhaftigkeit mathematischer Forschung, ja mathematischer Tatigkeit generell uberzeugt werden konnte. Ich glaube nicht, dass dem Laien Blindheit fur die Grosartigkeit unserer Wissenschaft vorzuwerfen ist, wenn hier keine befriedigendere Kommunikation zustande kommt. Ich sehe als Ursache eher ein stark verkurztes Bild von der Mathematik, welches auch Fachleute oft zeichnen, weil ihnen eine angemessenere Darstellung ihres Faches zu viel Muhe macht – und das obwohl Mathematik nur betreiben kann, wer geistige Muhen sonst keineswegs scheut. Ich will versuchen, den Ursachen dieses eigentumlichen Phanomens auf den Grund zu gehen.

Distribution preserving transformations of sequences on compact metric spaces

Let X be a compact metric space. For x E X let μx denote the point measure in x, i.e. μx(B) = 1 if x E B and μx(B) = 0 else for each Borel set B ⊆ X. For each sequence x = (Xn)n Ge N consider the Borel measures μx,N = 1N∑Nn = 1 μxn. Let the weak topology on the space of all Borel probability measures be induced by the metric d. Call two sequences x and x′ equivalent (x ~ x′) if limN → ∝ d(μx,N, μx′,N) = 0. Let furthermore Y be another compact metric space. Consider sequences f = (fn)n E N of maps fn: X → Y and the induced sequences f(x) = (fn(xn))n E N. The object of this paper is to characterize all f such that x ~ x′ always implies f(x) ~ f(x′). The topic is closely related to more elementary questions on Cesaro means.

Monotonic functions and an inequality of Myerson on point distributions

Abstract In the paper ‘Discrepancy and distance between sets’ G. Myerson studied several notions of distances and discrepancies of point distributions on the unit interval. Among several results he proves an inequality between p-discrepancy (p ≥ 1) and the ‘distance’ between n-element subsets of the unit interval. This paper contains a proof of his conjecture that this inequality holds in a stronger version. Furthermore it can be transferred to arbitrary probability distributions on the unit interval which are Borel measures. The results can be embedded into a topological context. The last section contains a corollary which is a further expansion of the inequalitys domain of validity.

Algebraic endomorphisms of LCA-groups that preserve closed subgroups

The object of this paper is the classification of those algebraic (i.e. not necessarily continuous) endomorphisms of a locally compact abelian group leaving invariant all closed subgroups. In a canonical way they turn out to form again a locally compact abelian group which can be determined up to isomorphism. If the group is totally disconnected or not periodic all endomorphisms with this property are continuous and form a topological ring.