50,776 research outputs found
Central limit theorem for Fourier transforms of stationary processes
We consider asymptotic behavior of Fourier transforms of stationary ergodic
sequences with finite second moments. We establish a central limit theorem
(CLT) for almost all frequencies and also an annealed CLT. The theorems hold
for all regular sequences. Our results shed new light on the foundation of
spectral analysis and on the asymptotic distribution of periodogram, and it
provides a nice blend of harmonic analysis, theory of stationary processes and
theory of martingales.Comment: Published in at http://dx.doi.org/10.1214/10-AOP530 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Converse Lyapunov theorems for discrete-time linear switching systems with regular switching sequences
We present a stability analysis framework for the general class of
discrete-time linear switching systems for which the switching sequences belong
to a regular language. They admit arbitrary switching systems as special cases.
Using recent results of X. Dai on the asymptotic growth rate of such systems,
we introduce the concept of multinorm as an algebraic tool for stability
analysis.
We conjugate this tool with two families of multiple quadratic Lyapunov
functions, parameterized by an integer T >= 1, and obtain converse Lyapunov
Theorems for each.
Lyapunov functions of the first family associate one quadratic form per state
of the automaton defining the switching sequences. They are made to decrease
after every T successive time steps. The second family is made of the
path-dependent Lyapunov functions of Lee and Dullerud. They are parameterized
by an amount of memory (T-1) >= 0.
Our converse Lyapunov theorems are finite. More precisely, we give sufficient
conditions on the asymptotic growth rate of a stable system under which one can
compute an integer parameter T >= 1 for which both types of Lyapunov functions
exist.
As a corollary of our results, we formulate an arbitrary accurate
approximation scheme for estimating the asymptotic growth rate of switching
systems with constrained switching sequences
Esthetic Numbers and Lifting Restrictions on the Analysis of Summatory Functions of Regular Sequences
When asymptotically analysing the summatory function of a -regular
sequence in the sense of Allouche and Shallit, the eigenvalues of the sum of
matrices of the linear representation of the sequence determine the "shape" (in
particular the growth) of the asymptotic formula. Existing general results for
determining the precise behavior (including the Fourier coefficients of the
appearing fluctuations) have previously been restricted by a technical
condition on these eigenvalues.
The aim of this work is to lift these restrictions by providing a insightful
proof based on generating functions for the main pseudo Tauberian theorem for
all cases simultaneously. (This theorem is the key ingredient for overcoming
convergence problems in Mellin--Perron summation in the asymptotic analysis.)
One example is discussed in more detail: A precise asymptotic formula for the
amount of esthetic numbers in the first~ natural numbers is presented. Prior
to this only the asymptotic amount of these numbers with a given digit-length
was known.Comment: to appear in "2019 Proceedings of the Sixteenth Meeting on Analytic
Algorithmics and Combinatorics (ANALCO)
Automatic enumeration of regular objects
We describe a framework for systematic enumeration of families combinatorial
structures which possess a certain regularity. More precisely, we describe how
to obtain the differential equations satisfied by their generating series.
These differential equations are then used to determine the initial counting
sequence and for asymptotic analysis. The key tool is the scalar product for
symmetric functions and that this operation preserves D-finiteness.Comment: Corrected for readability; To appear in the Journal of Integer
Sequence
On the non-holonomic character of logarithms, powers, and the n-th prime function
We establish that the sequences formed by logarithms and by "fractional"
powers of integers, as well as the sequence of prime numbers, are
non-holonomic, thereby answering three open problems of Gerhold [Electronic
Journal of Combinatorics 11 (2004), R87]. Our proofs depend on basic complex
analysis, namely a conjunction of the Structure Theorem for singularities of
solutions to linear differential equations and of an Abelian theorem. A brief
discussion is offered regarding the scope of singularity-based methods and
several naturally occurring sequences are proved to be non-holonomic.Comment: 13 page
Asymptotics of classical spin networks
A spin network is a cubic ribbon graph labeled by representations of
. Spin networks are important in various areas of Mathematics
(3-dimensional Quantum Topology), Physics (Angular Momentum, Classical and
Quantum Gravity) and Chemistry (Atomic Spectroscopy). The evaluation of a spin
network is an integer number. The main results of our paper are: (a) an
existence theorem for the asymptotics of evaluations of arbitrary spin networks
(using the theory of -functions), (b) a rationality property of the
generating series of all evaluations with a fixed underlying graph (using the
combinatorics of the chromatic evaluation of a spin network), (c) rigorous
effective computations of our results for some -symbols using the
Wilf-Zeilberger theory, and (d) a complete analysis of the regular Cube
spin network (including a non-rigorous guess of its Stokes constants), in the
appendix.Comment: 24 pages, 32 figure
Period-luminosity relations in evolved red giants explained by solar-like oscillations
Solar-like oscillations in red giants have been investigated with CoRoT and
Kepler, while pulsations in more evolved M giants have been studied with
ground-based microlensing surveys. After 3.1 years of observation with Kepler,
it is now possible to make a link between these different observations of
semi-regular variables. We aim to identify period-luminosity sequences in
evolved red giants identified as semi-regular variables. Then, we investigate
the consequences of the comparison of ground-based and space-borne
observations. We have first measured global oscillation parameters of evolved
red giants observed with Kepler with the envelope autocorrelation function
method. We then used an extended form of the universal red giant oscillation
pattern, extrapolated to very low frequency, to fully identify their
oscillations. From the link between red giant oscillations observed by Kepler
and period-luminosity sequences, we have identified these relations in evolved
red giants as radial and non-radial solar-like oscillations. We were able to
expand scaling relations at very low frequency. This helped us to identify the
different sequences of period-luminosity relations, and allowed us to propose a
calibration of the K magnitude with the observed frequency large separation.
Interpreting period-luminosity relations in red giants in terms of solar-like
oscillations allows us to investigate, with a firm physical basis, the time
series obtained from ground-based microlensing surveys. This can be done with
an analytical expression that describes the low-frequency oscillation spectra.
The different behavior of oscillations at low frequency, with frequency
separations scaling only approximately with the square root of the mean stellar
density, can be used to address precisely the physics of the semi-regular
variables.Comment: Accepted in A&
An exactly solvable self-convolutive recurrence
We consider a self-convolutive recurrence whose solution is the sequence of
coefficients in the asymptotic expansion of the logarithmic derivative of the
confluent hypergeometic function . By application of the Hilbert
transform we convert this expression into an explicit, non-recursive solution
in which the th coefficient is expressed as the th moment of a
measure, and also as the trace of the th iterate of a linear operator.
Applications of these sequences, and hence of the explicit solution provided,
are found in quantum field theory as the number of Feynman diagrams of a
certain type and order, in Brownian motion theory, and in combinatorics
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