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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.

Ein Zuschauer und der Zauberer spielen ein Spiel mit 2n Karten, das offensichtlich fair ist. Uberraschender Weise sind die Gewinnwahrscheinlichkeiten aber fur nicht zu grose n recht weit von 0,5 entfernt, und sie hangen von n modulo 4 ab. Die Chancen sind fur den Zauberer am besten, wenn n modulo 4 gleich 1 ist. Fur die – teilweise recht technischen – Berechnungen der Gewinnwahrscheinlichkeiten wird nur elementare Stochastik benotigt.

The “classical” Australian under-down shuffle starts with a deck of n cards. Then one proceeds as follows: one card under the deck, one on the table, one under the deck, one on the table, etc. One continues until only one card remains. There is an explicit formula to calculate the number of the card in the original deck that survives, and this is the basis of several mathematical magic tricks.

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.

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.

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.

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.