Ergodic Structure Research Articles

Numerical studies of plasma edge in W7-X with 3D FINDIF code

Abstract Modelling of the plasma transport for inherently three-dimensional (3D) problems as in stellarators requires dedicated complex codes. FINDIF is a 3D multifluid plasma edge transport code that has been previously successfully used for the analysis of energy transport in the TEXTOR-DED tokamak [1], where 3D perturbations led to an ergodic structure of field lines in the plasma edge. The ongoing efforts to apply it meaningfully to Wendelstein 7-X (W7-X) plasma problems resulted in advancements in the main model and accompanying tools for mesh generation and post-processing. In order to verify the applicability of the code and to compare with the reported simulation (EMC3-EIRENE) and experimental (OP1.1) results, a series of simulations for varying plasma density, temperature and anomalous transport coefficients as well as for fixed input power were performed. The connection length pattern of FINDIF traced magnetic field lines on the limiter was reproduced and its impact on heat loads was confirmed. An increase in the peak heat load on the limiter with a rise in plasma density, temperature and anomalous plasma transport coefficients was observed. The decay lengths of density, electron temperature and heat flux did not change with density, and were decreasing with temperature and increasing with anomalous plasma transport coefficient, which was compared to the simple scrape-off layer (SOL) model.

A quasi-ergodic approach to non-integer base expansions

P. Erdős et al. proved in 1990 that every nontrivial number has a continuum of expansions on two-letter alphabets in every base smaller than the Golden ratio, and that this property fails for the Golden ratio base. It was shown in a recent paper of Baiocchi et al. that if we replace the powers of the Golden ratio by the closely related Fibonacci sequence, then the resulting integer base expansions still have the continuum expansion property. The proof heavily relied on the special properties of the Golden ratio. The difficulty came from the fact that the new expansions do not have any more the ergodic structure of non-integer base expansions. In this paper we introduce a new “quasi-ergodic” approach that allows us to handle many more general cases. We apply this approach to Baker's generalized Golden ratios.

Regularity and inverse theorems for uniformity norms on compact abelian groups and nilmanifolds

Abstract We prove a general form of the regularity theorem for uniformity norms, and deduce an inverse theorem for these norms which holds for a class of compact nilspaces including all compact abelian groups, and also nilmanifolds; in particular we thus obtain the first non-abelian versions of such theorems. We derive these results from a general structure theorem for cubic couplings, thereby unifying these results with the Host–Kra Ergodic Structure Theorem. A unification of this kind had been propounded as a conceptual prospect by Host and Kra. Our work also provides new results on nilspaces. In particular, we obtain a new stability result for nilspace morphisms. We also strengthen a result of Gutman, Manners and Varjú, by proving that a k-step compact nilspace of finite rank is a toral nilspace (in particular, a connected nilmanifold) if and only if its k-dimensional cube set is connected. We also prove that if a morphism from a cyclic group of prime order into a compact finite-rank nilspace is sufficiently balanced (i.e. equidistributed in a certain quantitative and multidimensional sense), then the nilspace is toral. As an application of this, we obtain a new proof of a refinement of the Green–Tao–Ziegler inverse theorem.

A Novel Broad Learning System Based Leakage Detection and Universal Localization Method for Pipeline Networks

The security of pipeline systems draws attention increasingly; therefore, a novel method based on neural network and graph theory is proposed for the detection and localization of pipeline networks in this paper. First, the detection algorithm based on the broad learning system (BLS) is used to distinguish abnormities under large-scale pipeline network environments. During the process, the varied BLS models result in indeterminate performance and fast ergodic structure search is executed via adaptive mutation particle swarm algorithm (APSO) to generate an appropriate structure, succinct parameters, speedability, and accuracy. And manual features are implanted into the BLS feature layer to targetedly improve performance for complex pipeline network signals. Second, based on the detection results, a universal Dijkstra-based applicable localization method is proposed for diverse topological pipeline structures, including mesh-form networks, which have fewer sensors than anchors. The synchronous approximation is adopted to shun local minimum, and the shrinkage of search domain economizes time. Revised BLS was contrasted with several networks trained by real pipeline data and the system was integrated into SCADA and applied on an operational large-scale pipeline network successfully.

Cut Points and Diffusions in Random Environment

In this article we investigate the asymptotic behavior of a new class of multidimensional diffusions in random environment. We introduce cut times in the spirit of the work done by Bolthausen et al. (Ann. Inst. Henri Poincare 39(5):527–555, 2003) in the discrete setting providing a decoupling effect in the process. This allows us to take advantage of an ergodic structure to derive a strong law of large numbers with possibly vanishing limiting velocity and a central limit theorem under the quenched measure.

Strictly ordered minimal subsets of a class of convex monotone skew-product semiflows

We study the topological and ergodic structure of a class of convex and monotone skew-product semiflows. We assume the existence of two strongly ordered minimal subsets K 1, K 2 and we obtain an ergodic representation of their upper Lyapunov exponents. In the case of null upper Lyapunov exponents, we obtain a lamination into minimal subsets of an intermediate region where the restriction of the semiflow is affine. In the hyperbolic case, we deduce the long-time behaviour of every trajectory ordered with K 2. Some examples of skew-product semiflows generated by non-autonomous differential equations and satisfying the assumptions of monotonicity and convexity are also presented.

The structure of the bounded trajectories set of a scalar convex differential equation

The present paper describes the topological and ergodic structure of the set of bounded trajectories of the flow defined by a scalar convex differential equation. We characterize the minimal subsets, the ergodic measures concentrated on them, and study the longtime behaviour of the bounded trajectories in terms of the Lyapunov exponents of the linearized equations. In particular, we obtain conditions that guarantee the existence of almost-periodic, almost-automorphic and recurrent solutions.

Value iteration and approximately optimal stationary policies in finite-state average Markov decision chains

This work concerns finte-state Markov decision chains endowed with the long-run average reward criterion. Assuming that the optimality equation has a solution, it is shown that a nearly optimal stationary policy, as well as an approximation to the optimal average reward within a specified error, can be obtained in a finite number of steps of the value iteration method. These results extend others already available in the literature, which were established under more stringent restrictions on the ergodic structure of the decision process.

Effect of limiter recycling on measured poloidal impurity emission profiles in Tore Supra

Poloidal impurity emission profiles measured with the Tore Supra grazing incidence duochromator exhibit a complex spatial structure during ergodic divertor operation with an outboard poloidal guard limiter. As previous measurements with inboard-wall limited plasmas have shown that these profiles give important information about the ergodic field structure, so the contribution of local neon recycling from the limiter-induced plume has been modeled. This permits a discrimination of edge and core transport effects. The BBQ 3D scrape-off layer code calculates the asymmetric contribution to the emission amd MIST 1D simulation gives the symmetric part. A systematic increase is observed in the decay rate of neon emission after injection as the ergodic divertor strength is increased. The calculations permit identification of the limiter plume contribution to the profile structure, and, with this identification, the effect of the divertor to enhance impurity efflux can be seen from the decay data.

A pragmatic approach to adaptive antennas

This paper presents a novel approach for efficient computation of adaptive weights in phased-array antennas. The fundamental philosophical differences between adaptive antennas and adaptive signal-processing methodology are also delineated. This approach, unlike the conventional statistical techniques, eliminates the requirement for an interference covariance matrix, and represents a rethinking of the entire conventional approach to adaptive processing. This approach provides greater flexibility in solving a wider class of problems, at the expense of a slightly reduced number of degrees of freedom. It is important to note that the application of a deterministic approach to address stochastic problems with an ergodic structure can be seen in the works of Wiener (1949) and Kolmogorov (1939). This paper presents examples to illustrate the effectiveness and uniqueness of this new pragmatic approach.

Continuity of Optimal Values and Solutions for Control of Markov Chains with Constraints

We consider in this paper constrained Markov decision processes. This type of control model has many applications in telecommunications and other fields [E. Altman and A. Shwartz, IEEE Trans. Automat. Control, 34 (1989), pp. 1089--1102, E. A. Feinberg and M. I. Reiman, Probab. Engrg. Inform. Sci., 8 (1994), pp. 463--489, A. Hordijk and F. Spieksma, Adv. in Appl. Probab., 21 (1989), pp. 409--431, A. Lazar, IEEE Trans. Automat. Control, 28 (1983), pp. 1001--1007, P. Nain and K. W. Ross, IEEE Trans. Automat. Control, 31 (1986), pp. 883--888, K. W. Ross and B. Chen, IEEE Trans. Automat. Control, 33 (1988), pp. 261--267]. We address the issue of the convergence of the value and optimal policies of the problem with discounted costs, to the ones for the problem with expected average cost. We consider the general multichain ergodic structure. We present two stability results in this paper. We establish the continuity of optimal values and solutions of as well as some type of robustness of some suboptimal solutions in the discount factor. Our proof relies on same general theory on continuity of values and solutions in convex optimization that relies on well-known notions of $\Gamma$-convergence.

Time averages for continuous functions on distal flows

We study the time averages of continuous functions along the trajectories of the distal projective flow induced by an ergodic family of Schrödinger equations. General conditions guaranteeing that the set of nonconvergence points is a residual subset are found. Applications to the study of the ergodic structure of the projective flow are given.

Ergodic structure and invariant densities of non-Markovian interval maps with indifferent fixed points

We consider piecewise twice differentiable maps T on [0,1] with indifferent fixed points giving rise to infinite invariant measures. Without assuming the existence of a Markov partition and only requiring that the first image of the fundamental partition is finite, we prove that the interval decomposes into a finite number of ergodic cycles with exact powers plus a dissipative part. T is shown to be exact on components containing indifferent fixed points. We also determine the order of the singularities of the invariant densities.

New cases of the cutting stock problem having MIRUP

In this paper we consider a singularly perturbed Markov decision process with finitely many states and actions and the limiting expected average reward criterion. We make no assumptions about the underlying ergodic structure. We present algorithms for the computation of a uniformly optimal deterministic control, that is, a control which is optimal for all values of the perturbation parameter that are sufficiently small. Our algorithms are based on Jeroslow's Asymptotic Linear Programming.

Considerations on diagnostics during operation with the dynamic ergodic divertor on TEXTOR-94

In this contribution, at first the required diagnostics are described according to the physics goals such as the study of particle removal, performance optimization, near field divertor option, island and ergodic structure and flows and possible confinement improvement due to the field rotation. In a second part, details of the different diagnostics are discussed.

Characteristic parameters of the magnetic field due to the dynamic ergodic divertor

Resonant perturbations of sufficient amplitude impose an ergodic structure on the field line trajectories. In the standard theory, these features are related to the amplitude of the resonant Fourier components of the Br-field. The characteristic parameters of an ergodized field are the width of magnetic islands, the Chirikov parameter (amount of overlapping of islands) and the coherence length (Kolmogorov length). The perturbation current amplitude allows for Chirikov parameters of up to 1.6 for the pure 12/4 excitation mode. The Chirikov value is very sensitive to a possible mixing of 12/4 and 6/2 modes and can considerably be enhanced.

Ergodic theorems for some classical problems in combinatorial optimization

We show that the stochastic versions of some classical problems in combinatorial optimization may be imbedded in multiparameter subadditive processes having an intrinsic ergodic structure. A multiparameter generalization of Kingman's subadditive ergodic theorem is used to capture strong laws for these optimization problems, including the traveling salesman and minimal spanning tree processes. In this way we make progress on some open problems and provide alternate proofs of some well known asymptotic results.

An ergodic classification of bidimensional linear systems

The ergodic structure of the projective flow induced by a family of bidimensional linear systems is studied. It is shown that the existence of a continuous invariant measure guarantees the existence of another measure, called linear by the authors, which provides substantial information upon the properties of the complex bundle. Some examples are given to illustrate the applicability of these results.

Scheduling and stability aspects of a general class of parallel processing systems

In this paper we study the following general class of concurrent processing systems. There are several different classes of processors (servers) and many identical processors within each class. There is also a continuous random flow of jobs, arriving for processing at the system. Each job needs to engage concurrently several processors from various classes in order to be processed. After acquiring the needed processors the job begins to be executed. Processing is done non-preemptively, lasts for a random amount of time, and then all the processors are released simultaneously. Each job is specified by its arrival time, its processing time, and the list of processors that it needs to access simultaneously. The random flow (sequence) of jobs has a stationary and ergodic structure. There are several possible policies for scheduling the jobs on the processors for execution; it is up to the system designer to choose the scheduling policy to achieve certain objectives. We focus on the effect that the choice of scheduling policy has on the asymptotic behavior of the system at large times and especially on its stability, under general stationary and ergocic input flows.

Scheduling and stability aspects of a general class of parallel processing systems

In this paper we study the following general class of concurrent processing systems. There are several different classes of processors (servers) and many identical processors within each class. There is also a continuous random flow of jobs, arriving for processing at the system. Each job needs to engage concurrently several processors from various classes in order to be processed. After acquiring the needed processors the job begins to be executed. Processing is done non-preemptively, lasts for a random amount of time, and then all the processors are released simultaneously. Each job is specified by its arrival time, its processing time, and the list of processors that it needs to access simultaneously. The random flow (sequence) of jobs has a stationary and ergodic structure. There are several possible policies for scheduling the jobs on the processors for execution; it is up to the system designer to choose the scheduling policy to achieve certain objectives.We focus on the effect that the choice of scheduling policy has on the asymptotic behavior of the system at large times and especially on its stability, under general stationary and ergocic input flows.