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We consider statistical properties of random integer partitions. In order to compute means, variances and higher moments of various partition statistics, one often has to study generating functions of the form P(x)F(x), where P(x) is the generating function for the number of partitions. In this paper, we show how asymptotic expansions can be obtained in a quasi-automatic way from expansions of F(x) around x = 1, which parallels the classical singularity analysis of Flajolet and Odlyzko in many ways. Numerous examples from the literature, as well as some new statistics, are treated via this methodology. In addition, we show how to compute further terms in the asymptotic expansions of previously studied partition statistics.
The asymptotic behaviour of the solutions of Poincaré's functional equation f(λz) = p(f(z)) (λ > 1) for p a real polynomial of degree ≥ 2 is studied in angular regions W of the complex plain. It is known [9, 10] that f(z) ~ exp(zρF(logλz)), if f(z) → ∞ for z ∞ and z ∈ W, where F denotes a periodic function of period 1 and ρ = logλ deg(p). In this paper we refine this result and derive a full asymptotic expansion. The constancy of the periodic function F is characterised in terms of geometric properties of the Julia set of p. For real Julia sets we give inequalities for multipliers of Pommerenke-Levin-Yoccoz type. The distribution of zeros of f is related to the harmonic measure on the Julia set of p.
Signed digit representations with base $q$ and digits $-\frac q2,\dots,\frac q2$ (and uniqueness being enforced by applying a special rule which decides whether $-q/2$ or $q/2$ should be taken) are considered with respect to counting the occurrences of a given (contiguous) subblock of length $r$. The average number of occurrences amongst the numbers $0,\dots,n-1$ turns out to be const$\cdot\log_qn+\delta(\log_qn)+\smallOh(1)$, with a constant and a periodic function of period one depending on the given subblock; they are explicitly described. Furthermore, we use probabilistic techniques to prove a central limit theorem for the number of occurrences of a given subblock.
Imaging critical features by using transmission electron microscopy (TEM) or scanning electron microscopy (SEM) provides a versatile approach for nanostructure characterization. The combination of focused ion beam (FIB) technology for exposing defective sites beneath the surface is shown. Reliability testing and defect analysis by localized characterization of multilayered structures is demonstrated. TEM-imaging of a transistor gate with a locally confined radiation damage demonstrates target preparation by FIB yielding high-resolution TEM samples. The TEM imaging requires a longer sample preparation but provides high image quality (TEM). Investigation of materials previously processed with FIB revealed amorphization damage by the high energetic Ga-ion beam. This damage layer with a thickness in the range of 50 to 100 nm was confirmed in simulation. This disadvantageous damage by amorphization originating from FIB preparation of the cross-section could be removed by soft sputtering with a 250 V Ar+ ion beam. This combined method using FIB for microsample preparation and TEM for imaging and analysis was proven to be a powerful tool the exploitation of nanostructured devices and for defect analysis on a highly localized scale.
The distribution of binomial coefficients in residue classes modulo prime powers and with respect to
the p-adic valuation is studied. For this purpose, general asymptotic results for arithmetic functions
depending on blocks of digits with respect to q-ary expansions are established.
We consider digital expansions with respect to complex integer
We derive precise information
about the length of these expansions and the corresponding
sum-of-digits function. Furthermore we give
an asymptotic formula for the sum-of-digits function in large
circles and prove that this function is
uniformly distributed with respect to the argument. Finally
the summatory function of the sum-of-digits
function along the real axis is analyzed.
We investigate the distribution of the hitting time T defined by the first visit of the Brownian motion on the Sierpiński gasket at geodesic distance r from the origin. For this purpose we perform a precise analysis of the moment generating function of the random variable T. The key result is an explicit description of the analytic behaviour of the Laplace- Stieltjes transform of the distribution function of T. This yields a series expansion for the distribution function and the asymptotics for t →0.
The lifetime of a player is defined to be the time where he gets his
b-th hit, where a hit will
occur with probability p. We consider the maximum statistics of
N independent players.
For b≠1 this is significantly more difficult than the known instance
b=1. The expected value of the maximum lifetime of N players is
given by logQN+(b−1)logQ logQN+ smaller order terms, where Q=1/(1−p).
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