We define and study the notion of min-wise independent families of permutations. We say that F ⊆ Sn is min-wise independent if for any set X ⊆ [n] and any x ∈ X, when π is chosen at random in F we have Pr(min{π(X)} = π(x)) = 1 |X |. In other words we require that all the elements of any fixed set X have an equal chance to become the minimum element of the image of X under π. Our research was motivated by the fact that such a family (under some relaxations) is essential to the algorithm used in practice by the AltaVista web index software to detect and filter near-duplicate documents. However, in the course of our investigation we have discovered interesting and challenging theoretical questions related to this concept – we present the solutions to some of them and we list the rest as open problems.

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It is shown that the dynamical systems approach to the diffraction properties of model sets can be generalized to regular model sets in arbitrary sigma-compact Abelian groups with arbitrary locally compact Abelian groups as internal spaces. It is then shown that these regular model sets possess pure point diffraction spectra.

Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many features of H cl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the t n have a similar structure to those of the semiclassical En ; in particular, they display random-matrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the t n can be computed accurately from formulae with quantum analogues. The Riemann-Siegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian H cl = XP. Key words. spectral asymptotics, number theory AMS subject classifications. 11M26, 11M06, 35P20, 35Q40, 41A60, 81Q10, 81Q50 PII. S0036144598347497 1.

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We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new. We concentrate primarily on practical computational issues, such issues depending on the domain of the argument, the desired speed of computation, and the incidence of what we call "value recycling".

Arithmetic functions related to number representation systems exhibit various periodicity phenomena. For instance, a well known theorem of Delange expresses the total number of ones in the binary representations of the first n integers in terms of a periodic fractal function. We show that such periodicity phenomena can be analyzed rather systematically using classical tools from analytic number theory, namely the Mellin-Perron formulae. This approach yields naturally the Fourier series involved in the expansions of a variety of digital sums related to number representation systems.

The canonical quantization of any hyperbolic symplectomorphism A of the 2-torus yields a periodic unitary operator on a N-dimensional Hilbert space, N = 1 h . We prove that this quantum system becomes ergodic and mixing at the classical limit (N !1, N prime) which can be interchanged with the time-average limit. The recovery of the stochastic behaviour out of a periodic one is based on the same mechanism under which the uniform distribution of the classical periodic orbits reproduces the Lebesgue measure: the Wigner functions of the eigenstates, supported on the classical periodic orbits, are indeed proved to become uniformly spread in phase space.

Abstract. Mellin transforms and Dirichlet series are useful in quantifying periodicity phenomena present in recursive divide-and-conquer algorithms. This note illustrates the techniques by providing a precise analysis of the standard topdown recursive mergesort algorithm, in the average case, as well as in the worst and best cases. It also derives the variance and shows that the cost of mergesort has a Gaussian limiting distribution. The approach is applicable to a number of divide-and-conquer recurrences. Many algorithms are based on a recursive divide-and-conquer strategy of splitting a problem into two subproblems of equal or almost equal size, separately solving the subproblems, and then knitting their solutions together to find the solution to the original problem. Accordingly, their complexity is expressed by recurrences of the usual divide-and-conquer form where the initial condition,f, , and the ‘‘knitting costs”, e,, depend on the problem being studied. Typical examples are mergesort, heapsort, Karatsuba’s multiprecision multiplication, discrete Fourier transforms, binomial queues, sorting networks, etc. It is relatively easy to determine general orders of growth for solutions to these recurrences as explained in standard texts, see the “master theorem ” of [6, p. 621. However, a precise asymptotic analysis is often appreciably more delicate. At a more detailed level, divide-and-conquer recurrences tend to have solutions that involve periodicities, many of which are of a fractal nature. It is our purpose here to discuss the analysis of such periodicity phenomena while focussing on the analysis of the standard top-down recursive mergesort algorithm. For example, as we shall soon see, the average cost of running mergesort on n keys satisfies u (n) = n lg n + nB (lg n) + 0 (n),

Abstract. We prove an endpoint multilinear estimate for the X s,b spaces associated to the periodic Airy equation. As a consequence we obtain sharp local well-posedness results for periodic generalized KdV equations, as well as some global well-posedness results below the energy norm. 1.