29,831 research outputs found
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
Computation of Composition Functions and Invariant Vector Fields in Terms of Structure Constants of Associated Lie Algebras
Methods of construction of the composition function, left- and
right-invariant vector fields and differential 1-forms of a Lie group from the
structure constants of the associated Lie algebra are proposed. It is shown
that in the second canonical coordinates these problems are reduced to the
matrix inversions and matrix exponentiations, and the composition function can
be represented in quadratures. Moreover, it is proven that the transition
function from the first canonical coordinates to the second canonical
coordinates can be found by quadratures
Distribution of Behaviour into Parallel Communicating Subsystems
The process of decomposing a complex system into simpler subsystems has been
of interest to computer scientists over many decades, for instance, for the
field of distributed computing. In this paper, motivated by the desire to
distribute the process of active automata learning onto multiple subsystems, we
study the equivalence between a system and the total behaviour of its
decomposition which comprises subsystems with communication between them. We
show synchronously- and asynchronously-communicating decompositions that
maintain branching bisimilarity, and we prove that there is no decomposition
operator that maintains divergence-preserving branching bisimilarity over all
LTSs.Comment: In Proceedings EXPRESS/SOS 2019, arXiv:1908.0821
Wave function statistics and multifractality at the spin quantum Hall transition
The statistical properties of wave functions at the critical point of the
spin quantum Hall transition are studied. The main emphasis is put onto
determination of the spectrum of multifractal exponents governing
the scaling of moments with the system
size and the spatial decay of wave function correlations. Two- and
three-point correlation functions are calculated analytically by means of
mapping onto the classical percolation, yielding the values and
. The multifractality spectrum obtained from numerical
simulations is given with a good accuracy by the parabolic approximation
but shows detectable deviations. We also study
statistics of the two-point conductance , in particular, the spectrum of
exponents characterizing the scaling of the moments . Relations
between the spectra of critical exponents of wave functions (),
conductances (), and Green functions at the localization transition with a
critical density of states are discussed.Comment: 16 pages, submitted to J. Phys. A, Special Issue on Random Matrix
Theor
Fidelity and Concurrence of conjugated states
We prove some new properties of fidelity (transition probability) and
concurrence, the latter defined by straightforward extension of Wootters
notation. Choose a conjugation and consider the dependence of fidelity or of
concurrence on conjugated pairs of density operators. These functions turn out
to be concave or convex roofs. Optimal decompositions are constructed. Some
applications to two- and tripartite systems illustrate the general theorem.Comment: 10 pages, RevTex, Correction: Enlarged, reorganized version. More
explanation
- …