Asymptotic Frequency Research Articles

Hadamard–Perron theorems and effective hyperbolicity

We prove several new versions of the Hadamard–Perron theorem, which relates infinitesimal dynamics to local dynamics for a sequence of local diffeomorphisms, and in particular establishes the existence of local stable and unstable manifolds. Our results imply the classical Hadamard–Perron theorem in both its uniform and non-uniform versions, but also apply much more generally. We introduce a notion of ‘effective hyperbolicity’ and show that if the rate of effective hyperbolicity is asymptotically positive, then the local manifolds are well behaved with positive asymptotic frequency. By applying effective hyperbolicity to finite-orbit segments, we prove a closing lemma whose conditions can be verified with a finite amount of information.

Asymptotic frequencies of various damped nonlocal beams and plates

A striking difference between the conventional local and nonlocal dynamical systems is that the later possess finite asymptotic frequencies. The asymptotic frequencies of four kinds of nonlocal viscoelastic damped structures are derived, including an Euler–Bernoulli beam with rotary inertia, a Timoshenko beam, a Kirchhoff plate with rotary inertia and a Mindlin plate. For these undamped and damped nonlocal beam and plate models, the analytical expressions for the asymptotic frequencies, also called the maximum or escape frequencies, are obtained. For the damped nonlocal beams or plates, the asymptotic critical damping factors are also obtained. These quantities are independent of the boundary conditions and hence simply supported boundary conditions are used. Taking a carbon nanotube as a numerical example and using the Euler–Bernoulli beam model, the natural frequencies of the carbon nanotubes with typical boundary conditions are computed and the asymptotic characteristics of natural frequencies are shown.

Dynamics of multiple viscoelastic carbon nanotube based nanocomposites with axial magnetic field

Nanocomposites and magnetic field effects on nanostructures have received great attention in recent years. A large amount of research work was focused on developing the proper theoretical framework for describing many physical effects appearing in structures on nanoscale level. Great step in this direction was successful application of nonlocal continuum field theory of Eringen. In the present paper, the free transverse vibration analysis is carried out for the system composed of multiple single walled carbon nanotubes (MSWCNT) embedded in a polymer matrix and under the influence of an axial magnetic field. Equivalent nonlocal model of MSWCNT is adopted as viscoelastically coupled multi-nanobeam system (MNBS) under the influence of longitudinal magnetic field. Governing equations of motion are derived using the Newton second low and nonlocal Rayleigh beam theory, which take into account small-scale effects, the effect of nanobeam angular acceleration, internal damping and Maxwell relation. Explicit expressions for complex natural frequency are derived based on the method of separation of variables and trigonometric method for the “Clamped-Chain” system. In addition, an analytical method is proposed in order to obtain asymptotic damped natural frequency and the critical damping ratio, which are independent of boundary conditions and a number of nanobeams in MNBS. The validity of obtained results is confirmed by comparing the results obtained for complex frequencies via trigonometric method with the results obtained by using numerical methods. The influence of the longitudinal magnetic field on the free vibration response of viscoelastically coupled MNBS is discussed in detail. In addition, numerical results are presented to point out the effects of the nonlocal parameter, internal damping, and parameters of viscoelastic medium on complex natural frequencies of the system. The results demonstrate the efficiency of the suggested methodology to find the closed form solutions for the free vibration response of multiple nanostructure systems under the influence of magnetic field.

On the approximation by three consecutive continued fraction convergents

Denote by pn/qn,n=1,2,3,…, the sequence of continued fraction convergents of the real irrational number x. Define the sequence of approximation coefficients by θn:=qn|qnx−pn|,n=1,2,3,…. A laborious way of determining the mean value of the sequence |θn+1−θn−1|,n=2,3,…, is simplified. The method involved also serves for showing that for almost all x the pattern θn−1<θn<θn+1 occurs with the same asymptotic frequency as the pattern θn+1<θn<θn−1, namely 0.12109⋯. All the four other patterns have the same asymptotic frequency 0.18945⋯. The constants are explicitly given.

Tsunami Recurrence Function: Structure, Methods of Creation, and Application for Tsunami Hazard Estimates

This paper considers a theoretical basement for a Poissonian probability model for tsunami run-up heights, with emphasis on the tsunami recurrence function. It is shown that the tsunami recurrence function of a general type contains at least two scale parameters: asymptotic frequency of big tsunamis f related to the considered region and characteristic tsunami height H* for the considered location in the region. A method for the correct statistical evaluation of the parameters f and H*, and their variations, using observational data from tsunami catalogues, is created. The paper considers some theoretical and applied problems related to the tsunami recurrence function, an example of a two-parameter tsunami hazard map, and also the problem of probabilistic tsunami hazard estimation.

Quantization of Horizon Entropy and the Thermodynamics of Spacetime

This is a review of my work published in the papers of Skakala (JHEP 1201:144, 2012; JHEP 1206:094, 2012) and Chirenti et al. (Phys. Rev. D 86:124008, 2012; Phys. Rev. D 87:044034, 2013). It offers a more detailed discussion of the results than the accounts in those papers, and it links my results to some conclusions recently reached by other authors. It also offers some new arguments supporting the conclusions in the cited articles. The fundamental idea of this work is that the semiclassical quantization of the black hole entropy, as suggested by Bekenstein (Phys. Rev. D 7:2333–2346, 1973), holds (at least) generically for the spacetime horizons. We support this conclusion by two separate arguments: (1) we generalize Bekenstein’s lower bound on the horizon area transition to a much wider class of horizons than only the black-hole horizon, and (2) we obtain the same entropy spectra via the asymptotic quasi-normal frequencies of some particular spherically symmetric multi-horizon spacetimes (in the way proposed by Maggiore (Phys. Rev. Lett. 100:141301, 2008)). The main result of this paper supports the conclusions derived by Kothawalla et al. (Phys. Rev. D 78:104018, 2008) and Kwon and Nam (Class. Quant. Grav. 28:035007, 2011), on the basis of different arguments.

Regenerative tree growth: structural results and convergence

We introduce regenerative tree growth processes as consistent families of random trees with n labelled leaves, n>=1, with a regenerative property at branch points. This framework includes growth processes for exchangeably labelled Markov branching trees, as well as non-exchangeable models such as the alpha-theta model, the alpha-gamma model and all restricted exchangeable models previously studied. Our main structural result is a representation of the growth rule by a sigma-finite dislocation measure kappa on the set of partitions of the natural numbers extending Bertoin's notion of exchangeable dislocation measures from the setting of homogeneous fragmentations. We use this representation to establish necessary and sufficient conditions on the growth rule under which we can apply results by Haas and Miermont for unlabelled and not necessarily consistent trees to establish self-similar random trees and residual mass processes as scaling limits. While previous studies exploited some form of exchangeability, our scaling limit results here only require a regularity condition on the convergence of asymptotic frequencies under kappa, in addition to a regular variation condition.

On fractal properties of non-normal numbers with respect to Rényi f-expansions generated by piecewise linear functions

The paper is devoted to the study of fractal properties of subsets of the set of non-normal numbers with respect to Rényi f-expansions generated by continuous increasing piecewise linear functions defined on [0,+∞). All such expansions are expansions for real numbers generated by infinite linear IFS f={f0,f1,…,fn,…} with the following list of ratios Q∞=(q0,q1,…,qn,…).We prove the superfractality of the set of Q∞-essentially non-normal numbers, i.e. real numbers having no asymptotic frequencies of any digits from the alphabet A={0,1,…,n,…}, for any infinite stochastic vector Q∞, independently of the finiteness resp. infiniteness of its entropy and independently of the faithfulness resp. non-faithfulness of the family of cylinders generated by these expansions.

Asymptotic quasi-normal frequencies, horizon area spectra and multi-horizon spacetimes

There exists a long term debate about a possible link of the asymptotic quasinormal (QNM) modes to the black hole thermodynamics. The first conjecture providing a link between the classical asymptotic black hole oscillation frequencies and black hole thermodynamics was formulated more than a decade ago by Hod. The conjecture was later modified by Maggiore. We analyse the behavior of the asymptotic frequencies of the spherically symmetric multi-horizon spacetimes (in particular Reissner-Nordström, Schwarzschild-deSitter, Reissner-Nordström-deSitter spacetime) and provide some suggestions for how to interpret the results following the spirit of the modified Hod's conjecture. The interpretation suggested is in some sense analogical to the Schwarzschild black hole case, but has some new specific features. This paper1 refers to work done over a longer period of time contained in papers [1, 2, 3, 4] and also to some extent in [5].

Non-normal numbers with respect to Markov partitions

We call a real number normal if for any block of digits the asymptoticfrequency of this block in the $N$-adic expansion equals the expectedone. In the present paper we consider non-normal numbers and, inparticular, essentially and extremely non-normal numbers. We call areal number essentially non-normal if for each single digit thereexists no asymptotic frequency of its occurrence. Furthermore we calla real number extremely non-normal if all possible probability vectorsare accumulation points of the sequence of frequency vectors. Our aimnow is to extend and generalize these results to Markov partitions.

Pulsations of rapidly rotating stars with compositional discontinuities

AbstractRecent observations of rapidly rotating stars have revealed the presence of regular patterns in their pulsation spectra. This has raised the question as to their physical origin, and, in particular, whether they can be explained by an asymptotic frequency formula for low-degree acoustic modes, as recently discovered through numerical calculations and theoretical considerations. In this context, a key question is whether compositional/density gradients can adversely affect such patterns to the point of hindering their identification. To answer this question, we calculate frequency spectra using two-dimensional ESTER stellar models. These models use a multi-domain spectral approach, allowing us to easily insert a compositional discontinuity while retaining a high numerical accuracy. We analyse the effects of such discontinuities on both the frequencies and eigenfunctions of pulsation modes in the asymptotic regime. We find that although there is more scatter around the asymptotic frequency formula, the semi-large frequency separation can still be clearly identified in a spectrum of low-degree acoustic modes.

Tests of the asymptotic large frequency separation of acoustic oscillations in solar-type and red-giant stars

Asteroseismology, i.e. the study of the internal structures of stars via their global oscillations, is a valuable tool to obtain stellar parameters such as mass, radius, surface gravity and mean density. These parameters can be obtained using certain scaling relations which are based on an asymptotic approximation. Usually the observed oscillation parameters are assumed to follow these scaling relations. Recently, it has been questioned whether this is a valid approach, i.e., whether the order of the observed oscillation modes are high enough to be approximated with an asymptotic theory. In this work we use stellar models to investigate whether the differences between observable oscillation parameters and their asymptotic estimates are indeed significant. We compute the asymptotic values directly from the stellar models and derive the observable values from adiabatic pulsation calculations of the same models. We find that the extent to which the atmosphere is included in the models is a key parameter. Considering a larger extension of the atmosphere beyond the photosphere reduces the difference between the asymptotic and observable values of the large frequency separation. Therefore, we conclude that the currently suggested discrepancies in the scaling relations might have been overestimated. Hence, based on the results presented here we believe that the suggestions of Mosser et al. (2013) should not be followed without careful consideration.

Transients in Porous Media: Exact and Modelled Time-Domain Green's Functions

Time domain responses of porous media have been studied by some authors, but generally the possible descriptions have been given in the frequency domain. The aim of this paper, limited to materials with rigid skeleton considered as equivalent fluids, is to compare the descriptions by Johnson-Allard ($JA$%) as well as by Pride-Lafarge ($PL$) with i) some analytical, approximate formulas, based upon asymptotic high frequency expansion ; ii) the exact formula by Zwikker and Kosten for the case of cylindrical pores. The paper starts with a short summary of the statement of the different general full frequency models ($JA$ and $PL).$ The Green function in the time domain is shown to exhibit interesting properties of materials. In particular the maximum response depends on one dimensionless parameter only, which is denoted $\xi $ and is the ratio of the travelled distance to the product of the \textquotedblleft frozen\textquotedblright\ sound speed and a characteristic viscous relaxation time. The distance $\xi $ is related to a time domain Stokes number. The numerical computation of the Green function is done by FFT, with some precautions, because of the importance of the higher frequencies on the response shape. The $PL$ description is shown to be the best full frequency general model, but some discrepancies with the exact model appear at short times or short distances. When the distance $\xi $ increases from zero, the asymptotic expansion shows that the maximum of the Green function decreases first as $1/\xi ^{2}$, then exponentially.

New approaches to source–sink metapopulations decoupling demography and dispersal

Source–sink systems are metapopulations of patches with different, and possibly temporally varying, habitat qualities, which are commonly used in ecology to study the fate of spatially extended populations.We propose new techniques that disentangle the respective contributions of demography and dispersal to the dynamics and fate of a single species in a source–sink system. Our approach is valid for a general class of stochastic, individual-based, stepping-stone models, with density-independent demography and dispersal, provided that the metapopulation is finite or else enjoys some transitivity property.We provide (1) a simple criterion of persistence, by studying the motion of a single random disperser until it returns to its initial position; (2) a joint characterization of the long-term growth rate and of the asymptotic occupancy frequencies of the ancestral lineage of a random survivor, by using large deviations theory. Both techniques yield formulas decoupling demography and dispersal, and can be adapted to the case of periodic or random environments, where habitat qualities are autocorrelated in space and possibly in time.In this last case, we display examples of coupled time-averaged sinks allowing survival, as was previously known in the absence of demographic stochasticity for fully mixing (Jansen and Yoshimura, 1998) or partially mixing (Evans et al., 2012; Schreiber, 2010) metapopulations.

Vibration of nonlocal Kelvin–Voigt viscoelastic damped Timoshenko beams

This paper investigates the dynamic behavior of nonlocal viscoelastic damped nanobeams. The Kelvin–Voigt viscoelastic model, velocity-dependent external damping and Timoshenko beam theory are employed to establish the governing equations and boundary conditions for the bending vibration of nanotubes. Using transfer function methods (TFM), the natural frequencies and frequency response functions (FRF) are computed for beams with different boundary conditions. Unlike local structures, taking into account rotary inertia and shear deformation, the nonlocal beam has maximum frequencies, called the escape frequencies or asymptotic frequencies, which are obtained for undamped and damped nonlocal Timoshenko beams. Damped nonlocal beams are also shown to possess an asymptotic critical damping factor. Taking a carbon nanotube as a numerical example, the effects of the nonlocal parameter, viscoelastic material constants, the external damping ratio, and the beam length-to-diameter ratio on the natural frequencies and the FRF are investigated. The results demonstrate the efficiency of the proposed modeling and analysis methods for the free vibration and frequency response analysis of nonlocal viscoelastic damped Timoshenko beams.

Convergence rates of Markov chains on spaces of partitions

We study the convergence rate to stationarity for a class of exchangeable partition-valued Markov chains called cut-and-paste chains. The law governing the transitions of a cut-and-paste chain are determined by products of i.i.d. stochastic matrices, which describe the chain induced on the simplex by taking asymptotic frequencies. Using this representation, we establish upper bounds for the mixing times of ergodic cut-and-paste chains, and under certain conditions on the distribution of the governing random matrices we show that the cutoff phenomenon holds.

Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto--Daido model with inertia

We present the global existence and long-time behavior of measure-valued solutions to the kinetic Kuramoto--Daido model with inertia. For the global existence of measure-valued solutions, we employ a Neunzert's mean-field approach for the Vlasov equation to construct approximate solutions. The approximate solutions are empirical measures generated by the solution to the Kuramoto--Daido model with inertia, and we also provide an a priori local-in-time stability estimate for measure-valued solutions in terms of a bounded Lipschitz distance. For the asymptotic frequency synchronization, we adopt two frameworks depending on the relative strength of inertia and show that the diameter of the projected frequency support of the measure-valued solutions exponentially converge to zero.

Asymptotic frequencies of beams left fixed and right supported

The purpose of this paper is to construct the asymptotic for natural frequencies of the Euler-Bernoulli beam problem with non–uniform section using the method of Wentzel-Kramers-Brillouin (WKB) and find the secular equation for the eigenvalues a non–uniform beam embedded at one end and supported at the other end.

Quasi-normal modes, area spectra and multi-horizon spacetimes

We suggest an interpretation for the highly damped QNM frequencies of the spherically symmetric multi-horizon spacetimes (Reissner-Nordstrom, Schwarzschild-deSitter, Reissner-Nordstrom-deSitter) following Maggiore's proposal about the link between the asymptotic QNM frequencies and the black hole thermodynamics. We show that the behavior of the asymptotic frequencies is easy to understand if one assumes that all of the horizons have the same equispaced area spectra. The QNM analysis is then consistent with the choice of the area spectra to be the one originally proposed for the black hole's horizon by Bekenstein: A=8\pi n (in Planck units). The interpretation of the highly damped QNM frequencies in the multi-horizon case is based on the similar grounds as in the single horizon (Schwarzschild) case, but it has some new features that are discussed in the paper.

A NONCONVENTIONAL STRONG LAW OF LARGE NUMBERS AND FRACTAL DIMENSIONS OF SOME MULTIPLE RECURRENCE SETS

We provide conditions which yield a strong law of large numbers for expressions of the form [Formula: see text] where X(n), n ≥ 0's is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, F is a continuous function with polinomial growth and certain regularity properties and qi, i &gt; m are positive functions taking on integer values on integers with some growth conditions. Applying these results we study certain multifractal formalism type questions concerning Hausdorff dimensions of some sets of numbers with prescribed asymptotic frequencies of combinations of digits at places q1(n), …, qℓ(n).