Stochastic analysis and simulation of spin star systems

Methods and apparatus for detecting periodicity and/or for determining (Fig. 10) fundamental period of a signal such as speech. The methods include embedding a portion of a sampled digitized signal into an m-dimensional state space to obtain a sequence of m-dimensional vectors, selecting closest pairs of vectors in state space from a plurality of possible pairs of m-dimensional vectors in said sequence of m-dimensional vectors, accumulating total numbers of selected closest pairs of vectors having the same time separation values to produce a histogram of accumulated numbers, and locating at least a highest peak in a portion of said histogram to obtain a value indicating the fundamental period of the signal. Various embodiments are directed to speech and audio signal processing and other speech related applications. However, the methods have a general nature and can be applied to other types of periodic or quasi-periodic signals as well.

A new approach to theory and simulation of the non-Markovian dynamics of open quantum systems is presented. It is based on identification of a parameter which is uniformly bounded on wide time intervals: the occupation of the virtual cloud of quanta. By ‘virtual’ we denote those bath excitations which were emitted by the open system, but eventually will be reabsorbed before any measurement of the bath state. A useful property of the virtual cloud is that the number of its quanta is expected to saturate on long times, since physically this cloud is a (retarded) polarization of the bath around the system. Therefore, the joint state of open system and virtual cloud (we call it dressed state) can be accurately represented in a truncated basis of Fock states, on a wide time scale. At the same time, there can be an arbitrarily large number of the observable quanta (which survive up to measurement), especially if the open system is under driving. However, it turns out that the statistics of the bath-measurement outcomes is classical (in a suitable measurement basis): one can employ a Monte Carlo sampling of these outcomes. Therefore, it is possible to efficiently simulate the dynamics of the observable quantum field. In this work we consider the bath measurement with respect to the coherent states, which yields the Husimi function as the positive (quasi)probability distribution of the outcomes. The joint evolution of the dressed state and the corresponding outcome is called the dressed quantum trajectory. The Monte Carlo sampling of these trajectories yields a stochastic simulation method with promising convergence properties on wide time scales.

Many background error correlation (BEC) models in data assimilation are formulated in terms of a positive-definite smoothing operator B that is employed to simulate the action of correlation matrix on a vector in state space. In this chapter, a general procedure for constructing a BEC model as a rational function of the diffusion operator D is presented and analytic expressions for the respective correlation functions in the homogeneous case are obtained. It is shown that this class of BEC models can describe multi-scale stochastic fields whose characteristic scales can be expressed in terms of the polynomial coefficients of the model. In particular, the connection between the inverse binomial model and the well-known Gaussian model \(\mathbf{\mathsf{B}}_{g} =\exp \mathbf{\mathsf{D}}\) is established and the relationships between the respective decorrelation scales are derived.By its definition, the BEC operator has to have a unit diagonal and requires appropriate renormalization by rescaling. The exact computation of the rescaling factors (diagonal elements of B) is a computationally expensive procedure, therefore an efficient numerical approximation is needed. Under the assumption of local homogeneity of D, a heuristic method for computing the diagonal elements of B is proposed. It is shown that the method is sufficiently accurate for realistic applications, and requires 102 times less computational resources than other methods of diagonal estimation that do not take into account prior information on the structure of B.

Bisimilarity is one of the most important relations for comparing the behaviour of formal systems in concurrency theory. Decision algorithms for bisimilarity in finite state systems are usually classified into two kinds: global algorithms are generally efficient but require to generate the whole state spaces in advance, and local algorithms combine the verification of a system's behaviour with the generation of the system's state space, which is often more effective to determine that one system fails to be related to another. Although local algorithms are well established in the classical concurrency theory, the study of local algorithms in probabilistic concurrency theory is not mature. In this paper we propose a polynomial time local algorithm for checking probabilistic bisimilarity. With mild modification, the algorithm can be easily adapted to decide probabilistic similarity with the same time complexity.

State space models are greatly favored by scientific researchers on account of particular superiority. Aiming at the minimized error criterion and using the matrix differential theory, the paper represented a global optimal state space model-identifying algorithm for stochastic state space model. This is a new identifying algorithm that integrates system parameters identification, structural identification and state estimation. In this method, hypothesis state space model is disturbed by the measuring noise and process noise. First, state space vector x is identified according to error criterion J(Es TEs). Then, system parameters matrix A, B, C and D is identified by criterion function J(Ei TEi). Results of mathematical simulation proved that this identifying method is characterized as simple calculation and higher identifying precision.

State-Partition-Based Control of Discrete Event Systems for Enforcement of Regular Language Specifications

Chaos is the most complex dynamic behavior a system can show. Yet, it is produced by two simple, though nonlinear, deterministic mechanisms, hence the more proper name Deterministic Chaos. The first mechanism, stretching, acts locally and linearly, expanding small sets in the system's state space in (at least) one direction and contracting in the others. The second mechanism, folding, acts globally and is the nonlinear operation of bending the elongated sets to form the forward image of the original sets. These two simple operations are invertible and form a chaotic set for the system's forward and backward map, containing an uncountable infinity of astonishingly complex system's trajectories. The simplest explanation of chaos I ever heard is James A. Yorke “pizza machine” (plenary lecture at the SIAM DS, Snowbird, 2003): kneading the pizza dough is nothing but stretching and folding. The mathematical formalization is Stephen Smale's horseshoe map. Using this nice tool, I revisit the hallmarks of chaos, one of which (in the title) still misses a constructive proof.

In linearly stable shear flows, turbulence spontaneously decays with a characteristic lifetime that varies with Reynolds number. The lifetime sharply increases with Reynolds number so that a possible divergence marking the transition to sustained turbulence at a critical point has been discussed. We present a mechanism by which the lifetimes increase: in the system's state space, turbulent motion is supported by a chaotic saddle. Inside this saddle a locally attracting periodic orbit is created and undergoes a traditional bifurcation sequence generating chaos. The formed new "turbulent bubble" is initially an attractor supporting persistent chaotic dynamics. Soon after its creation, it collides with its own boundary, by which it becomes leaky and dynamically connected with the surrounding structures. The complexity of the chaotic saddle that supports transient turbulence hence increases by incorporating the remnant of a new bubble. As a a result, the time it takes for a trajectory to leave the saddle and decay to the laminar state is increased. We demonstrate this phenomenon in plane Couette flow and show that characteristic lifetimes vary nonsmoothly and nonmonotonically with Reynolds number.

Inventory Management with Stochastic Lead Times

This article analyzes a continuous time back-ordered inventory system with stochastic demand and stochastic delivery lags for placed orders. This problem in general has an infinite dimensional state space and is hence intractable. We first obtain the set of minimal conditions for reducing such a system's state space to one dimension and show how this reduction is done. Next, by modeling demand as a diffusion process, we reformulate the inventory control problem as an impulse control problem. We simplify the impulse control problem to a Quasi-Variation Inequality (QVI). Based on the QVI formulation, we obtain the optimality of the (s, S) policy and the limiting distribution of the inventory level. We also obtain the long run average cost of such an inventory system. Finally, we provide a method to solve the QVI formulation. Using a set of computational experiments, we show that significant losses are incurred in approximating a stochastic lead-time system with a fixed lead-time system, thereby highlighting the need for such stochastic lead-time models. We also provide insights into the dependence of this value loss on various problem parameters.

The second law of thermodynamics imposes a fundamental asymmetry in the flow of events. The so-called thermodynamic arrow of time introduces an ordering that divides the system's state space into past, future, and incomparable regions. In this work, we analyze the structure of the resulting thermal cones, i.e., sets of states that a given state can thermodynamically evolve to (the future thermal cone) or evolve from (the past thermal cone). Specifically, for a d-dimensional classical state of a system interacting with a heat bath, we find explicit construction of the past thermal cone and the incomparable region. Moreover, we provide a detailed analysis of their behavior based on thermodynamic monotones given by the volumes of thermal cones. Results obtained apply also to other majorization-based resource theories (such as that of entanglement and coherence), since the partial ordering describing allowed state transformations is then the opposite of the thermodynamic order in the infinite temperature limit. Finally, we also generalize the construction of thermal cones to account for probabilistic transformations.

Scaling verification to large circuits requires some form of abstraction relative to the asserted property. We describe a safety analysis of finite-state systems that generalizes from counterexamples to the inductiveness of the safety specification to inductive invariants. It thus abstracts the system's state space relative to the property. The analysis either strengthens a safety specification to be inductive or discovers a counterexample to its correctness. The analysis is easily made parallel. We provide experimental data showing how the analysis time decreases with the number of processes on several hard problems.

This paper deals with the optimal control of a one-machine two-product manufacturing system with setup changes, operating in a continuous time dynamic environment. The system is deterministic. When production is switched from one product to the other, a known constant setup time and a setup cost are incurred. Each product has specified constant processing time and constant demand rate, as well as an infinite supply of raw material. The problem is formulated as a feedback control problem. The objective is to minimize the total backlog, inventory and setup costs incurred over a finite horizon. The optimal solution provides the optimal production rate and setup switching epochs as a function of the state of the system (backlog and inventory levels). For the steady state, the optimal cyclic schedule is determined. To solve the transient case, the system's state space is partitioned into mutually exclusive regions such that with each region, the optimal control policy is determined analytically.

A one-dimensional object is used in considering control systems in state space or operator space. From the complexity viewpoint (number of sensors and number of elements in data-processing system), operator space is preferable, and certain tasks are more readily handled than in state space. Examples are given.

After providing an intrinsic definition of the adjoint system of a linear system given in state space form, we characterize the adjoint system in terms of the system’s external behavior. Conversely we show how this external characterization allows one to recover the state space definition by constructing an isomorphism from the abstractly defined minimal state space of the adjoint system to the dual of the minimal state space of the system itself. It is shown how a coordinate expression of this isomorphism can be obtained from a modified form of the Bezoutian matrix.