308 research outputs found
Statistical Distributions and q-Analogues of k-Fibonacci Numbers
Abstract We study q-analogues of k-Fibonacci numbers that arise from weighted tilings of an n × 1 board with tiles of length at most k. The weights on our tilings arise naturally out of distributions of permutations statistics and set partitions statistics. We use these q-analogues to produce q-analogues of identities involving k-Fibonacci numbers. This is a natural extension of results of the first author and Sagan on set partitions and the first author and Mathisen on permutations. In this paper we give general q-analogues of k-Fibonacci identities for arbitrary weights that depend only on lengths and locations of tiles. We then determine weights for specific permutation or set partition statistics and use these specific weights and the general identities to produce specific identities
Statistics of resonances and of delay times in quasiperiodic Schr"odinger equations
We study the statistical distributions of the resonance widths , and of delay times in one dimensional
quasi-periodic tight-binding systems with one open channel. Both quantities are
found to decay algebraically as , and on
small and large scales respectively. The exponents , and are
related to the fractal dimension of the spectrum of the closed system
as and . Our results are verified for the
Harper model at the metal-insulator transition and for Fibonacci lattices.Comment: 4 pages, 3 figures, submitted to Phys. Rev. Let
Statistics of Resonances and Delay Times in Random Media: Beyond Random Matrix Theory
We review recent developments on quantum scattering from mesoscopic systems.
Various spatial geometries whose closed analogs shows diffusive, localized or
critical behavior are considered. These are features that cannot be described
by the universal Random Matrix Theory results. Instead one has to go beyond
this approximation and incorporate them in a non-perturbative way. Here, we pay
particular emphasis to the traces of these non-universal characteristics, in
the distribution of the Wigner delay times and resonance widths. The former
quantity captures time dependent aspects of quantum scattering while the latter
is associated with the poles of the scattering matrix.Comment: 30 pages, 15 figures (submitted to Journal of Phys. A: Math. and
General, special issue on "Aspects of Quantum Chaotic Scattering"
A Probabilistic Approach to Generalized Zeckendorf Decompositions
Generalized Zeckendorf decompositions are expansions of integers as sums of
elements of solutions to recurrence relations. The simplest cases are base-
expansions, and the standard Zeckendorf decomposition uses the Fibonacci
sequence. The expansions are finite sequences of nonnegative integer
coefficients (satisfying certain technical conditions to guarantee uniqueness
of the decomposition) and which can be viewed as analogs of sequences of
variable-length words made from some fixed alphabet. In this paper we present a
new approach and construction for uniform measures on expansions, identifying
them as the distribution of a Markov chain conditioned not to hit a set. This
gives a unified approach that allows us to easily recover results on the
expansions from analogous results for Markov chains, and in this paper we focus
on laws of large numbers, central limit theorems for sums of digits, and
statements on gaps (zeros) in expansions. We expect the approach to prove
useful in other similar contexts.Comment: Version 1.0, 25 pages. Keywords: Zeckendorf decompositions, positive
linear recurrence relations, distribution of gaps, longest gap, Markov
processe
Why Delannoy numbers?
This article is not a research paper, but a little note on the history of
combinatorics: We present here a tentative short biography of Henri Delannoy,
and a survey of his most notable works. This answers to the question raised in
the title, as these works are related to lattice paths enumeration, to the
so-called Delannoy numbers, and were the first general way to solve Ballot-like
problems. These numbers appear in probabilistic game theory, alignments of DNA
sequences, tiling problems, temporal representation models, analysis of
algorithms and combinatorial structures.Comment: Presented to the conference "Lattice Paths Combinatorics and Discrete
Distributions" (Athens, June 5-7, 2002) and to appear in the Journal of
Statistical Planning and Inference
Recommended from our members
Mini-Workshop: Dynamical versus Diffraction Spectra in the Theory of Quasicrystals
The dynamical (or von Neumann) spectrum of a dynamical system and the diffraction spectrum of the corresponding measure dynamical system are intimately related. While their equivalence in the case of pure point spectra is well understood, this workshop aimed at an appropriate extension to systems with mixed spectra, building on recent developments for systems of finite local complexity and for certain random systems from the theory of point processes. Another focus was the question for connections between Schr¨odinger and dynamical spectra
Artin's primitive root conjecture -a survey -
This is an expanded version of a write-up of a talk given in the fall of 2000
in Oberwolfach. A large part of it is intended to be understandable by
non-number theorists with a mathematical background. The talk covered some of
the history, results and ideas connected with Artin's celebrated primitive root
conjecture dating from 1927. In the update several new results established
after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer
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