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Imagine a set S of people, the inhabitants of your home town, say. For every s I S there is a subset 𝒩 s of S: the people whom s knows, his or her neighbours, friends or colleagues. It happens that some people are infected by a dangerous disease D, the probability that a particular person s has D will naturally depend on the number of t ∈𝒩 s with D. What can be said about the distribution of infected people? Will D eventually disappear or will everybody be infected sooner or later?

In this chapter we are going to generalize the preceding considerations to the case of an arbitrary finite group (G, o), the approach will be similar: relate the abstract group with something more concrete and solve problems for G by transforming them into problems concerning numbers. However, it is not to be expected that the notions introduced up to now will suffice, the reason is simple: Whenever a group homomorphism, say ∅, on G has a commutative range, then $$ \phi \left( {goh} \right) = \phi \left( g \right)\phi \left( h \right) = \phi \left( h \right)\phi \left( g \right) = \phi \left( {hog} \right) $$ for all g, h. Therefore there is no hope that such distinguish between different elements of a non-commutative group. In particular, homomorphisms from G to T will not suffice.

The results of the preceding chapter will enable us to complete our picture of finite Markov chains. We will proceed as follows: Investigation of the behaviour of the p ii (k) for “large” k in the case of recurrent states i. Proof of the existence of an equilibrium distribution for irreducible aperiodic chains Discussion of irreducible chains with an arbitrary period. Calculation of the first passage time matrix.

The technique we are going to describe now was introduced in the eighties by Aldous and Diaconis (see [24], [3], [4]). As in the previous chapter stopping times play an important role, the reader can find the necessary prerequisites in chapter 12.

In order to understand the simple idea which underlies Markov chains we remind the reader of the well-known random walks from elementary probability. We have in mind a walk on a finite interval {1,2,..., N} of integers. The walk starts at 2, say, and every step is to the left or to the right with equal probability 0.5; if the walk is in position 1 or N, some extra rule has to be applied (e.g., one may prescribe that the next position is the starting position).

The setting is as at the beginning of the previous chapter: we are given a finite set \( \tilde{S} \) and a function1 \( H:\tilde{S} \to \mathbb{R} \), and this function gives rise to a probability measure \( {\mathbb{P}_{H}} \) on \( \tilde{S} \) by $$ {\mathbb{P}_{H}}(\{ x\} ): = \frac{{{e^{{ - H(x)}}}}}{z}\quad with\;Z:\sum\limits_{z} {{e^{{ - H(z)}}}} $$ .

What is essential? It is one thing to fix a set of axioms as the starting point of a hopefully interesting new theory. However, it is usually much more difficult to find the relevant notions which enable one to study — sometimes even to completely classify — the new objects. Examples are very rare where this has been achieved by a single mathematician. Generally it takes years or decades where many suggestions are under consideration, where it turns out to be necessary to modify the axioms and where many researchers are involved.

Schwingungen sind allgegenwartig: Musik gelangt uber Luftschwingungen an unser Ohr, Licht- und Radiowellen sind — genauso wie Rontgenstrahlen — hochfrequente elektromagnetische Schwingungen, Schaukeln schwingen, Wolkenkratzer schwingen im Wind usw. All das ist wohlbekannt. Es hat sich jedoch noch nicht allzuweit herumgesprochen, dass das Thema seit mehr als 2000 Jahren wichtige Impulse fur die Entwicklung der Mathematik gibt. Daruber soll in diesem Beitrag berichtet werden.

Let K ⊂ C2 be a compact convex set such that f(K) is a disk for every complex linear functional f. Such sets occurred in D. Yost's investigation of certain approximation problems in Banach spaces. Here we present some general results concerning sets K with the above property. Also we sketch an example showing that the K under consideration do not necessarily have a center of symmetry.

Professor Zbigniew Lipecki from the University of Wroclaw has told me that the definition of “small ball property” introduced in the paper “Metric spaces with the small ball property” by E. Behrends and V. Kadets (Studia Math. 148, No. 3, 275-287, 2001), has been investigated under different names earlier in the literature and that some of the results in this paper and in my paper quoted in the title are rediscoveries of known facts. In particular this applies to Proposition 2.3. I am grateful to Professor Lipecki for providing the following references to us: – Ch. Bandt, Mathematika 28 (1981), 206-210. – R. Duda and R. Telgarsky, Czechoslovak Math. J. 18(93) (1968), 66-82. – E. Szpilrajn-Marczewski, Fund. Math. 15 (1930), 126-27. – E. Szpilrajn-Marczewski, Fund. Math. 22 (1934), 303-311. – B. Tsaban, Real Analysis Exchange 27 (2001/2002), 757-764.

An operatorT on a complex Banach spaceX is callednormal if there exists an operatorS such that (1/2)(T+S) and (1/2i)(T−S) are hermitian, andTS=ST. We show thatT: X→X is normal iffT′: X′ » X′ is normal. Using a generalization of the principle of local reflexivity this result enables us to prove that multipliers on complexL1-predual spaces are always normal.

Let x 1 , l , x n and y 1 , l , y n be vectors in the finite dimensional euclidean space R d . We investigate the problem of how one can find a U such that ∑ i 〈 U x i , y i 〉 is maximal when U runs through the orthogonal group or the special orthogonal group, i.e., we are looking for a U such that the U x 1 , l ,U x n and the y 1 , l , y n are “as parallel as possible”. For d =3 this problem arises, for instance, from the data analysis of crystallographic diffraction experiments on orientationally disordered systems: the x 's stand for the atom positions of fragments M 1 of the crystal structure, the y 's are taken from the set M 2 of maxima of the electron-density distribution reconstructed from diffraction data, and one must know the above transformations U in order to determine the M 1 – M 2 -configurations of minimum distance since they are responsible for the shape of the density distribution. It turns out that one can associate a d × d -matrix G with this problem in such a way that the relevant U are precisely those for which the trace of GU is maximal. Using this transformation we are able to provide an explicit solution by means of the singular value decomposition of G . Several further topics in connection with this problem are also discussed. In particular we investigate in which cases the optimal position is unique, and we study generalizations to nondiscrete situations and to the infinite dimensional setting.

Bisher haben wir uns bei unseren Untersuchungen im wesentlichen auf jeweils eine Funktion konzentriert. In diesem Kapitel sollen die wichtigsten Raume mesbarer Funktionen, die Lp-Raume fur 1 ≤ p ≤ ∞, behandelt werden.

Preliminaries topology.- Lp-projections.- The Cunningham p-algebra.- The integral module representation.- The classical Lp-spaces.- Integral modules and duality.- Spectral theory for Lp-operators.- The Lp-structure of the bochner spaces and related results.

Ausgangspunkt unserer mastheoretischen Untersuchungen war bisher ein MaBraum (S,A,µ), wobei S ublicherweise irgendeine Menge ohne Strukturen war; zwar spielten hin und wieder — etwa im Zusammenhang mit Borelmengen — topologische Begriffe schon eine gewisse Rolle, doch wurde der Zusammenhang zwischen Mas und Topologie nicht naher untersucht.

Ziel dieses Anhangs ist der Beweis der Aussage, das es eine Borelmenge in [0,1]2 gibt, deren erst Projektion keine Borelmenge ist; diese Projektion ist aufgrund von Anhang I eine Souslinmenge, so das wir insbesondere eine echte Souslinmenge gefunden hatten. 1)

In diesem Kapitel sollen die grundlegenden Fragen der Mas- und Integrationstheorie gestellt und beantwortet werden. Um der Gefahr zu begegnen, die Theorie in einem Brei technischer Details ersticken zu lassen, werden wir uns auf das Notwen-digste beschranken, ausfuhrlichere und alternative Ausfuhrungen zu diesen Grundlagen finden sich reichlich in der Literatur.

Inhaltsubersicht: Grundlagen der Mass- und Integrationstheorie.- Die fundamentalen Satze der Masstheorie.- Masse auf dem , Riemann contra Lebesgue.- Raume messbarer Funktionen.- Masse auf topologischen Raumen.- Anhang I.- Anhang II.- Zeittabelle.- Lebensdaten einiger fur die Masstheorie relevanter Mathematiker.- Literatur.- Bezeichnungen.- Register.

In diesem Kapitel soil zunachst uberblicksartig zusammengestellt werden, was zur Mastheorie des ℝP bisher erarbeitet wurde. Nach einigen Erganzungen wird es dann um die Beziehungen zwischen Riemann- und Lebesgueintegral gehen.