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A Japanese company sells a magic trick with an interesting mathematical background. Commuting families of “symmetric” permutations play a central role. Our main result states that there are essentially as many such tricks as there are abelian finite groups.

Let ( M 1 , d 1 ) , ( M 2 , d 2 ) be metric spaces. A map f : M 1 → M 2 is said to be locally open at an x 1 ∈ M 1 , if for every e > 0 one finds a δ > 0 such that B ( f ( x 1 ) , δ ) ⊂ f ( B ( x 1 , e ) ) ; here B ( x , r ) stands for the closed ball with center x and radius r . We are particularly interested in the following special case: X , Y , Z are normed spaces, the spaces L ( X , Y ) , L ( Y , Z ) , L ( X , Z ) of linear continuous operators are provided with the operator norm, and the map under consideration is the bilinear map ( S , T ) ↦ S ∘ T (from L ( Y , Z ) × ( L ( X , Y ) to L ( X , Z ) ). For which pairs ( S 0 , T 0 ) ∈ L ( Y , Z ) × ( L ( X , Y ) is it locally open? The main result of the paper gives a complete characterization of pairs ( S , T ) at which this map is locally open in the case of finite-dimensional spaces X , Y , Z .

We obtain two results stating the uniform openness of bilinear operators and multilinear functionals. The first result deals with Banach spaces $L^{p}:=L_{\mathbb{K}}^{p}$ (over $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$ ) and pointwise multiplication from $L^{p}\times L^{q}$ to $L^{r}$ (where $1/p+1/q=1/r$ ). The second result is concerned with the nontrivial $n$ -linear functionals from the product $X_{1}\times\cdots\times X_{n}$ of normed spaces (over $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$ ) to the field $\mathbb{K}$ .

The Spanish physicist Juan Parrondo has provided two stochastic losing games such that for certain stochastic combinations one may obtain a winning game. If a large number of players are involved and if they try to play such that their gain in the next round is maximized one arrives at the problem of investigating a random walk on a certain space of measures. The appropriate abstract setting is as follows. There is given a compact metric space ( M , d ) , and M is written as the union of certain closed subsets A 1 , … , A r . For every ρ = 1 , … , r there is prescribed a strict contraction Γ ρ : A ρ → M . A random walk ( X m ) m ∈ N 0 on M is then defined as follows. The starting position is X 0 = x 0 , where x 0 ∈ M is fixed, and if the walk at the m ’th step is at position X m ∈ M , then one chooses a ρ among the ρ with X m ∈ A ρ (with equal probability, say) and defines X m + 1 as Γ ρ ( X m ) . Associated with the walk is a gain φ ( X m ) in every round, where φ : M → R is a continuous function. The aim of the present investigations is the study of the expectation G m of φ ( X m ) as a function of m . Our main result states that the sequence ( G m ) is “eventually approximately periodic” provided that all A ρ are not only closed but also open in M : for every e there is an l 0 ∈ N such that ( G m ) is l 0 -periodic up to an error of at most e for sufficiently large m . In fact it turns out that the behaviour of our process can be described well with a finite Markov chain. In the general case, however, the process might behave rather chaotically. We give an example where M is the unit interval. M is written as the union of two closed subsets A 1 , A 2 , the contractions Γ 1 , Γ 2 are rather simple, but the expectations of the gains are not even Cesaro convergent.

Let (M, d) be a complete metric space and suppose that there are given finitely many contractions Γρ:M→ M and Lipschitz maps (ρ = 1, ..., r).We consider 'walks' of length m with a given starting point x0 in M. They are defined as follows: one chooses a sequence (ρμ)μ=1,..., m of length m in {1, ..., r}, and this choice induces the 'walk' Associated with x1, ..., xm is the 'reward' We denote by the maximal possible reward.The aim of this paper is to investigate the behaviour of the sequence for large m. It will be shown that the growth is nearly linear: there is a constant γ (which does not depend on x0) such that tends to γ. However, an explicit calculation of γ might be hard. The complexity depends on the fractal dimension of the smallest nonempty compact subset of M which is invariant with respect to all Γρ.In the case of finite M one can say much more. Then—after a suitable rescaling—the sequence is periodic where the length of the period can be described in terms of the length of certain cycles of a graph associated with M.The motivation to study this problem came from a variant of Parrondo's paradox from probability theory: what is the optimal choice of games if a great number of players is involved?

An Astumian game is defined by a finite Markov chain with state space S with precisely two absorbing states, the winning and the losing state. The other states are transient, one of them is the starting position. The game is said to be losing (respectively fair respectively winning) if the probability to be absorbed at the winning state is smaller than 0.5 (respectively = 0.5 respectively larger than 0.5). Astumian's paradox states that there are losing games on the same state space S a stochastic mixture of which is winning. (By "stochastic mixture" we mean that in each step one decides with the help of a fair coin whether to use the transition probabilities of the first or the second game.) Most of our results concern fair games. Mixtures are systematically investigated. Rather surprisingly, the winning probability of the mixture of fair games can be arbitrarily close to zero (or to one). Even more counter-intuitive are examples of definitely losing games (this means that the winning probability is exactly zero) such that the winning probability of the mixture is arbitrarily close to one. We show, however, that such extreme examples are possible only if one tolerates huge running times of the game. As a natural generalization one can also consider arbitrary mixtures: the fair coin is replaced by a biased one, with probability λ respectively 1-λ one plays with the first respectively the second game. It turns out that fair games exist such that — depending on the choice of λ — the λ-mixture can be fair, losing or winning.

We call a sequence (Tn) of measure preserving transformations strongly mixing if \(P(T_n^{ - 1} A \cap B)\) tends to P(A)P(B) for arbitrary measurable A, B. We investigate whether one can pass to a suitable subsequence \(T_{n_k } \) such that \(\frac{1}{K}\sum\limits_{k = 1}^k {f(T_{n_k } )} \to \smallint fdP\) almost surely for all (or "many") integrable f.

A metric space (M;d) is said to have the small ball property (sbp) if for every "0 > 0 it is possible to write M as the union of a sequence (B(xn;rn)) of closed balls such that the rn are smaller than "0 and limrn = 0. We study permanence properties and examples of sbp. The main results of this paper are the following: 1. Bounded convex closed sets in Banach spaces have sbp only if they are compact. 2. Precisely the nite- dimensional Banach spaces have sbp. (More generally: a complete metric group has sbp i it is separable and locally compact.) 3. Let B be a boundary in the bidual of an innite-dimension al Banach space. Then B does not have sbp. In particular the set of extreme points in the unit ball of an innite-dimension al reexiv e Banach space fails to have sbp.

Let x1,..., xn and y1,..., yn be nonnegative numbers. If the lp-norm of (x1,..., xn) coincides with that of (y1,..., yn) for n different values of p > 0, then the xi are a permutation of the xi. This generalizes a well-known algebraic result which concerns the exponents p = 1,..., n.¶Some consequences as well as some generalizations to infinite sequences and to functions are also discussed.

Let X, W be independent real-valued random variables with finite expectation and E(W) = 0. We prove that only in the case W=0 the conditional expectation E(X|X+W) coincides with X+W. The result is a consequence of the following cancellation theorem: Let P, Q, R be Borel probability measures on the real line such that the support of Q resp. R is contained in {x⩽0} resp. {x⩾0}; then P ∗ Q = P ∗ R implies that Q = R(=δ0).

Integer $m\times m$ matrices $A$ with determinant $1$ define diffeomorphisms of the $m$-dimensional torus $T^m=({\Bbb R}/{\Bbb Z})^m$ into itself. Likewise, they define bijective self-maps of the discretized tori $({\Bbb Z}/n{\Bbb Z})^m=({\Bbb z}_n)^m$. We present estimates of the surprisingly low order (or period) $\Per_A(n)$ of the iteration $A^r, r=1,2,3,\ldots,$ on the discretized torus $({\Bbb z}_n)^m$. We obtain $\Per_A(n)\leq 3n$ for dimension $m=2$. In the special case of the Anosov map $A={2\ 1 \atopwithdelims() 1\ 1}$, this result is due to Dyson and Talk [DT92]. For arbitrary dimensions $m>2$ we obtain $$\Per_A(n)\leq {\rm constant}\cdot n^{m-1},$$ provided $n$ is a power of a prime number. For general $n$, number theoretic problems arise.

We point out that the analysis of Piotrowski and Sladkowski of the paradox described by Astumian has a flaw since they consider a different game.

Matrix-closed families of subspaces of a Banach space have been of importance in connection with generalizations of the local reflexivity theorem. The present paper provides sufficient conditions for a family of subspaces to be matrix-closed which can easily be checked. Using this result it is, e.g., simple to decide whether a family consisting of three subspaces is matrix-closed or not. It is also shown that this is in a sense the best possible general assertion, since no similar result can hold for the case of four subspaces.

Everyone uses compact discs nowadays. But why is the musical transfer to a CD purer than that of the traditional vinyl disc? The answer, to adapt a popular slogan, is: There is mathematics inside! More precisely, a branch of discrete mathematics, namely the theory of error correcting codes. This article is an account of the application of such codes to the design of the compact disc audio system, as developed by Philips Electronics and Sony.

Let $C[0,1]$ be the Banach algebra of real valued continuous functions on $[0,1]$, provided with the supremum norm. For $f,g\in C[0,1]$ and balls $B_{f}$, $B_{g}$ with center $f$ and $g$, respectively, it is not necessarily true that $f\cdot g$ is in the interior of $B_{f}\cdot B_{g}$. In the present paper we characterize those pairs $f,g$ where this is the case. The problem is illustrated by using a suitable translation. One studies walks in a landscape with hills and valleys where an accompanying dog can move in a certain prescribed way.