Criteria For Ergodicity Research Articles

Applying Statistical Criteria to Choose Optimal Metaparameters in Gene Fragment Recognition

We analyze the problem of choosing the optimal order of hidden Markov model for recognizing functional gene fragments. We propose four statistical criteria to determine the optimal order, which are based on likelihood ratio test, ergodicity, Markov property, and Akaike's information criterion. Additionally, we confirm the efficiency of Bayesian mixtures of Markov models for solving the problem in question and determine the optimal mixture size using statistical criteria.

Criteria of measure-preservation for 1-Lipschitz functions on Fq[[T]] in terms of the van der Put basis and its applications

We give a characterization of measure-preservation of 1-Lipschitz functions on Fq[[T]] in terms of the van der Put expansion and use this result to give sufficient conditions for measure-preserving 1-Lipschitz functions on Fq[[T]] in terms of the three well known bases, Carlitz polynomials, digit derivatives and digit shifts. We show that these conditions are also necessary for F2[[T]]. Moreover, we give an alternate, unified proof of ergodicity criteria of F2[[T]] in terms of the aforementioned three bases.

Many-body localization and quantum ergodicity in disordered long-range Ising models

Ergodicity in quantum many-body systems is---despite its fundamental importance---still an open problem. Many-body localization provides a general framework for quantum ergodicity and may therefore offer important insights. However, the characterization of many-body localization through simple observables is a difficult task. In this article, we introduce a measure for distances in Hilbert space for spin-$\frac{1}{2}$ systems that can be interpreted as a generalization of the Anderson localization length to many-body Hilbert space. We show that this many-body localization length is equivalent to a simple local observable in real space, which can be measured in experiments of superconducting qubits, polar molecules, Rydberg atoms, and trapped ions. By using the many-body localization length and a necessary criterion for ergodicity that it provides, we study many-body localization and quantum ergodicity in power-law-interacting Ising models subject to disorder in a transverse field. Based on the nonequilibrium dynamical renormalization group, numerically exact diagonalization, and an analysis of the statistics of resonances, we find a many-body localized phase at infinite temperature for small power-law exponents. Within the applicability of these methods, we find no indications of a delocalization transition.

Modeling of vibration tests of winding elements of power electric equipment

The possibilities of the use of lateral and longitudinal vibrations excited in metal samples while passing the impulse currents through them to model the operation upon the conditions of high current loads and high temperatures are examined. The possibility to perform tests for the dynamic gigacyclic loads is specified in conditions close to actual ones using test methods without complex and expensive benches. The correspondence of the tested processes to stationarity and ergodicity criteria is shown, which permits a significant reduction in test volume.

Joint ergodicity along generalized linear functions

A criterion of joint ergodicity of several sequences of transformations of a probability measure space $X$ of the form $T_{i}^{\unicode[STIX]{x1D711}_{i}(n)}$ is given for the case where $T_{i}$ are commuting measure-preserving transformations of $X$ and $\unicode[STIX]{x1D711}_{i}$ are integer-valued generalized linear functions, that is, the functions formed from conventional linear functions by an iterated use of addition, multiplication by constants, and the greatest integer function. We also establish a similar criterion for joint ergodicity of families of transformations depending on a continuous parameter, as well as a condition of joint ergodicity of sequences $T_{i}^{\unicode[STIX]{x1D711}_{i}(n)}$ along primes.

Characterization of the ergodicity of 1-Lipschitz functions on [formula omitted] using the q-Mahler basis

The ergodicity of 1-Lipschitz functions on Z2 represented by the Mahler basis was characterized by V.S. Anashin (1994) in [1]. His results are mainly based on the so-called folklore criterion for ergodicity, depending on the algebraic normal form of Boolean functions associated with coordinate functions. In this paper, we employ the q-Mahler basis to provide q-analogues of Anashin's results whose proof does not rely on this criterion.

Studies of nonergodic criterion based on the fractional heat bath model

The generalized Langevin equation with a power law memory kernel is derived via the gas/solid-surface model with fractional heat bath. Using Lapalce transformation, the dynamic evolution and long-time asymptotic behaviors of the gas particles occurring either in free or harmonic potentials are then investigated. In particular, the validity of three kinds of ergodic criteria is analyzed in detail, including the Khinchin criterion, Lee criterion, and the intrinsic and external behaviors. It is found that the Khinchin criterion holds for all ranges of diffusion and transport processes described by a generalized Langevin equation. Lee criterion is just applied to distinguish diffusion processes. Meanwhile, the intrinsic criterion and external behaviors can not only divide the nonergodicity into two classes but also reveal the underlying physical origins.

Ergodic currents dual to a real tree

Let $T$ be an $\mathbb{R}$-tree with dense orbits in the boundary of outer space. When the free group $\mathbb{F}_{N}$ acts freely on $T$, we prove that the number of projective classes of ergodic currents dual to $T$ is bounded above by $3N-5$. We combine Rips induction and splitting induction to define unfolding induction for such an $\mathbb{R}$-tree $T$. Given a current ${\it\mu}$ dual to $T$, the unfolding induction produces a sequence of approximations converging towards ${\it\mu}$. We also give a unique ergodicity criterion.

Analysis of an MMAP/PH1, PH2/N/∞ queueing system operating in a random environment

Abstract A multi-server queueing system with two types of customers and an infinite buffer operating in a random environment as a model of a contact center is investigated. The arrival flow of customers is described by a marked Markovian arrival process. Type 1 customers have a non-preemptive priority over type 2 customers and can leave the buffer due to a lack of service. The service times of different type customers have a phase-type distribution with different parameters. To facilitate the investigation of the system we use a generalized phase-type service time distribution. The criterion of ergodicity for a multi-dimensional Markov chain describing the behavior of the system and the algorithm for computation of its steady-state distribution are outlined. Some key performance measures are calculated. The Laplace-Stieltjes transforms of the sojourn and waiting time distributions of priority and non-priority customers are derived. A numerical example illustrating the importance of taking into account the correlation in the arrival process is presented

Individual Pooling for Group-Based Modeling Under the Assumption of Ergodicity

Psychology principally utilizes nomothetic, interindividual approaches to model phenomena of interest. However, it is the case that these approaches do not always capture the processes for each individual in the sample. If the research is focused on individual processes, confining analysis to the idiographic level may be more appropriate. One way to overcome the nomothetic inability to capture idiographic processes is to identify those participants who meet the criteria of ergodicity and restrict analysis to the resulting sample. Under these conditions it is quantitatively justifiable to create a group model without concern that it may fail to represent each member's idiographic process. In this study we explore the utility of such a method by (a) applying an ergodic pooling test to a sample of dyads (N = 128) who provided daily (T = 50) self-reports of affect, (b) applying an ergodic pooling test to samples (N = 4) of simulated ergodic time series data (T = 50, 250, and 1,000), (c) modeling dyads and simulated subgroups identified as ergodic, and (d) comparing the results from a model specified at the group level with those from models specified at the individual level.

Criteria of ergodicity for p-adic dynamical systems in terms of coordinate functions

This paper is devoted to the problem of ergodicity of p-adic dynamical systems. We solved the problem of characterization of ergodicity and measure preserving for (discrete) p-adic dynamical systems for arbitrary prime p for iterations based on 1-Lipschitz functions. This problem was open since long time and only the case p=2 was investigated in details. We formulated the criteria of ergodicity and measure preserving in terms of coordinate functions corresponding to digits in the canonical expansion of p-adic numbers. (The coordinate representation can be useful, e.g., for applications to cryptography.) Moreover, by using this representation we can consider non-smooth p-adic transformations. The basic technical tools are van der Put series and usage of algebraic structure (permutations) induced by coordinate functions with partially frozen variables. We illustrate the basic theorems by presenting concrete classes of ergodic functions. As is well known, p-adic spaces have the fractal (although very special) structure. Hence, our study covers a large class of dynamical systems on fractals. Dynamical systems under investigation combine simplicity of the algebraic dynamical structure with very high complexity of behavior.

On the ergodicity of flat surfaces of finite area

We prove some ergodic theorems for flat surfaces of finite area. The first result concerns such surfaces whose Teichmuller orbits are recurrent to a compact set of \({SL(2,\mathbb{R})/SL(S,\alpha)}\) , where SL(S,α) is the Veech group of the surface. In this setting, this means that the translation flow on a flat surface can be renormalized through its Veech group. This result applies in particular to flat surfaces of infinite genus and finite area. Our second result is an criterion for ergodicity based on the control of deforming metric of a flat surface. Applied to translation flows on compact surfaces, it improves and generalizes a theorem of Cheung and Eskin et al. (Partially hyperbolic dynamics, laminations, and Teichmuller flow, Fields Inst. Commun., Vol. 51. Amer. Math. Soc., Providence, pp. 213–221, 2007).

Stochastic Stability of Markovianly Switched Systems

This technical note examines the stochastic stability of noisy dynamics in discrete and continuous time. The notion of moment stability in the wide sense (MSWS) is presented as a generalization of ϵ-moment stability. MSWS is intentionally not based on stochastic convergence properties, since in most practically appearing systems convergence to any equilibrium is not present. A sufficient criterion for both MSWS and ergodicity is presented for a class of systems comprising a finite set of noisy dynamical systems among which switching is governed by a Markov chain. Stability/instability properties for each separate subsystem are assumed to be quantified by a Lyapunov function candidate together with an associated growth rate equation. For the set of Lyapunov functions, a compatibility criterion is assumed to be fulfilled, bounding the ratio between pairs of Lyapunov functions.

Ergodicity criteria for non-expanding transformations of 2-adic spheres

In the paper, we obtain necessary and sufficient conditions for ergodicity (with respect to the normalized Haar measure) of discrete dynamical systems $\langle f;\mathbf S_{2^-r}(a)\rangle$ on 2-adic spheres $\mathbf S_{2^-r}(a)$ of radius $2^{-r}$, $r\ge 1$, centered at some point $a$ from the ultrametric space of 2-adic integers $\mathbb Z_2$. The map $f\colon\mathbb Z_2\to\mathbb Z_2$ is assumed to be non-expanding and measure-preserving; that is, $f$ satisfies a Lipschitz condition with a constant 1 with respect to the 2-adic metric, and $f$ preserves a natural probability measure on $\mathbb Z_2$, the Haar measure $\mu_2$ on $\mathbb Z_2$ which is normalized so that $\mu_2(\mathbb Z_2)=1$.

Strong ergodicity of the regime-switching diffusion processes

We provide criteria for the strong ergodicity of regime-switching diffusion processes. Our conditions are imposed on the coefficients of the processes. Particularly, we show that for regime-switching diffusions on the half line, if the corresponding diffusion on each fixed environment is strongly ergodic, then the regime-switching diffusion is strongly ergodic as well, which does not depend on the changing rate of the environment. Moreover, the converse is not always true, which is shown by an example. For transience, recurrence and positive recurrence, there is no such good consistency [R. Pinsky and M. Scheutzow, Some remarks and examples concerning the transience and recurrence of random diffusions, Ann. Inst. Henri. Poincaré 28 (1992) 519–536].

Long-Term Optimal Investment Strategies in the Presence of Adjustment Costs

We consider the problem of determining in a dynamical way the optimal capacity level of an investment project that operates within a random economic environment. In particular, we consider an investment project that yields payoff at a rate that depends on its installed capacity level and on a random economic indicator such as the price of the project's output commodity. We model this economic indicator by means of a general one-dimensional ergodic diffusion. At any time, the project's capacity level can be increased or decreased at given proportional costs. The aim is to maximize an ergodic performance criterion that reflects the long-term average payoff resulting from the project's management. We solve this genuinely two-dimensional stochastic control problem by constructing an explicit solution to an appropriate Hamilton--Jacobi--Bellman equation and by fully characterizing an optimal investment strategy.

Stationary product measures for conservative particle systems and ergodicity criteria

We study conservative particle systems on W^S, where S is countable and W = {0, ..., N} or the natural numbers. The rate of a particle moving from site x to site y is given by p(x,y) b(eta_x, eta_y), where eta_z is the number of particles at site z. Under assumptions on b and the assumption that p is finite range, which allow for the exclusion, zero range and misanthrope processes, we show exactly what the stationary product measures are. Furthermore we show that a stationary measure mu is ergodic if and only if the tail sigma algebra of the partial sums is trivial under mu. This is a consequence of a more general result on interacting particle systems that shows that a stationary measure is ergodic if and only if the sigma algebra of sets invariant under the transformations of the process is trivial. We apply this result combined with a coupling argument on the stationary product measures to determine which product measures are ergodic. For the case that W is finite this gives a complete characterisation. For the case that W is the set of natural numbers we show that for nearly all functions b a stationary product measure is ergodic if and only if it is supported by configurations with an infinite amount of particles. We show that this picture is not complete, we give an example of a system where b is such that there is a stationary product measure which is not ergodic, even though it concentrates on configurations with an infinite number of particles.

A local ergodic theorem for non-uniformly hyperbolic symplectic maps with singularities

AbstractIn this paper, we prove a criterion for the local ergodicity of non-uniformly hyperbolic symplectic maps with singularities. Our result is an extension of a theorem of Liverani and Wojtkowski.

New criteria for ergodicity and nonuniform hyperbolicity

In this work we obtain a new criterion to establish ergodicity and nonuniform hyperbolicity of smooth measures of diffeomorphisms of closed connected Riemannian manifolds. This method allows us to give a more accurate description of certain ergodic components. The use of this criterion in combination with topological devices such as blenders lets us obtain global ergodicity and abundance of nonzero Lyapunov exponents in some contexts. In the partial hyperbolicity context, we obtain that stably ergodic diffeomorphisms are C1-dense among volume-preserving partially hyperbolic diffeomorphisms with 2-dimensional center bundle. This is motivated by a well-known conjecture of Pugh and Shub.

Cellular Downlink Performance with Covariance-CSIT-Based MIMO Precoding

The nature of the trade-off between reduced overhead of channel state information (CSI) and resultant performance losses influences the design of frequency-division duplexed practical cellular systems. One candidate for CSI feedback reduction is the use of covariance-matrix-based CSI at the transmitter (CSIT) in conjunction with linear precoding techniques. This paper analyzes the performance of such systems in the downlink for both single-user (SU-) and multiuser (MU-) multiple-input multiple output (MIMO) in comparison to those using optimal perfect-instantaneous-CSIT-based precoding. In addition, the effectiveness of techniques enforcing frequency domain diversity versus those based on the maximal ergodic channel capacity criterion is evaluated. A novel precoding scheme using covariance matrix information that supports spatial multiplexing in both SU- and MU-MIMO is proposed. Simulation results show that the spectral efficiency loss from covariance-CSIT-based techniques from those utilizing perfect, instantaneous CSIT is shown to be about 1 dB in a highly correlated urban channel environment for both SU- and MU-MIMO, whereas for microcell environments it is between 3 and 4 dB.