Infinite State Research Articles

Air-Core Non-Zero Dispersion-Shifted Fiber With High-Index Ring for OAM Mode

It is well-known now that orbital angular momentum (OAM) modes can be a potential technology to take full advantages of the space resources in the fiber-based optical communication system because of its infinite states. However, the corresponding fiber design for OAM modes to meet various requirements in the fiber-based communication system is still not enough. For the dispersion property, there are various attributes given in the standard of ITU-T series, which has not yet been proposed for the OAM modes. In this paper, we propose and design a non-zero dispersion-shifted air-core ring fiber for OAM mode, which can offer more options on dispersion values. A negative dispersion of -1.246 ps/(nm·km) or a positive dispersion of 1.541 ps/(nm ·km) at 1550 nm with <; 1.5 ps/(nm ·km) dispersion variation across the C band or <; 4 ps/(nm ·km) dispersion variation across the C and L bands for OAM <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1,1</sub> mode is achieved in the numerical simulation.

Enhancing an Intelligent Digital Twin with a Self-organized Reconfiguration Management based on Adaptive Process Models

Shorter product life cycles and increasing individualization of production leads to an increased reconfiguration demand in the domain of industrial automation systems, which will be dominated by cyber-physical production systems in the future. In constantly changing systems, however, not all configuration alternatives of the almost infinite state space are fully understood. Thus, certain configurations can lead to process instability, a reduction in quality or machine failures. Therefore, this paper presents an approach that enhances an intelligent Digital Twin with a self-organized reconfiguration management based on adaptive process models in order to find optimized configurations more comprehensively.

Symbolic Refinement of Extended State Machines with Applications to the Automatic Derivation of Sub-Components and Controllers

Nowadays, extended state machines are prominent requirements specification techniques due to their capabilities of modeling complex systems in a compact way. These machines extend the standard state machines with variables and have transitions guarded by enabling predicates and may include variable update statements. Given a system modeled as an extended state machine, with possibly infinite state space and some non-controllable (parameterized) interactions, a pruning procedure is proposed to symbolically derive a maximal sub-machine of the original system that satisfies certain conditions; namely, some safeness and absence of undesirable deadlocks which could be produced during pruning. In addition, the user may specify, as predicates associated with states, some general goal assertions that should be preserved in the obtained sub-machine. Further, one may also specify some specific requirements such as the elimination of certain undesirable deadlocks at states, or fail states that should never be reached. Application examples are given considering deadlock avoidance and loops including infinite loops over non-controllable interactions showing that the procedure may not terminate. In addition, the procedure is applied for finding a controller of a system to be controlled. The approach generalizes existing work in respect to the considered extended machine model and the possibility of user defined control objectives written as assertions at states.

Time crystallinity and finite-size effects in clean Floquet systems

A cornerstone assumption that most literature on discrete time crystals has relied on is that homogeneous Floquet systems generally heat to a featureless infinite temperature state, an expectation that motivated researchers in the field to mostly focus on many-body localized systems. Some works have however shown that the standard diagnostics for time crystallinity apply equally well to clean settings without disorder. This fact raises the question whether an homogeneous discrete time crystal is possible in which the originally expected heating is evaded. Studying both a localized and an homogeneous model with short-range interactions, we clarify this issue showing explicitly the key differences between the two cases. On the one hand, our careful scaling analysis confirms that, in the thermodynamic limit and in contrast to localized discrete time crystals, homogeneous systems indeed heat. On the other hand, we show that, thanks to a mechanism reminiscent of quantum scars, finite-size homogeneous systems can still exhibit very crisp signatures of time crystallinity. A subharmonic response can in fact persist over timescales that are much larger than those set by the integrability-breaking terms, with thermalization possibly occurring only at very large system sizes (e.g., of hundreds of spins). Beyond clarifying the emergence of heating in disorder-free systems, our work casts a spotlight on finite-size homogeneous systems as prime candidates for the experimental implementation of nontrivial out-of-equilibrium physics.

Round robin differential quadrature phase shift quantum key distribution by using odd coherent states

In this paper, we propose a round robin differential quadrature phase shift quantum key distribution by using one decoy state and odd coherent states (RRDQPS-QKD-OCS). By comparing existing RRDPS-QKD and RRDQPS-QKD, our proposed protocol, RRDQPS-QKD-OCS, has an outstanding performance, where not only the key generation rate is much larger but also the maximum transmission distance is much longer. Additionally, a one decoy state method is sufficient for RRDQPS-QKD-OCS to be close to the asymptotic limit performance with RRDQPS-QKD-OCS by using infinite decoy states.

Emergence of nonlinear crossover under epidemic dynamics in heterogeneous networks.

Potential diffusion processes of real-world systems are relevant to the underlying network structure and dynamical mechanisms. The vast majority of the existing work on spreading dynamics, in response to a large-scale network, is built on the condition of the infinite initial state, i.e., the extremely small seed size. The impact of an increasing seed size on the persistent diffusion has been less investigated. Based on classical epidemic models, this paper offers a framework for studying such impact through observing a crossover phenomenon in a two-diffusion-process dynamical system. The two diffusion processes are triggered by nodes with a high and low centrality, respectively. Specifically, given a finite initial state in networks with scale-free degree distributions, we demonstrate analytically and numerically that the diffusion process triggered by low centrality nodes pervades faster than that triggered by high centrality nodes from a certain point. The presence of the crossover phenomenon reveals that the dynamical process under the finite initial state is far more than the vertical scaling of the spreading curve under an infinite initial state. Further discussion emphasizes the persistent infection of individuals in epidemic dynamics as the essential reason rooted in the crossover, while the finite initial state is the catalyst directly leading to the emergence of this phenomenon. Our results provide valuable implications for studying the diversity of hidden dynamics on heterogeneous networks.

Bond-Alternation-Induced Topological Quantum Gaussian Transition and Topological Quantum Crossover in Ising Chains with the Dzyaloshinskii-Moriya Interaction

Non-local orders, entanglement entropy, and quantum fidelity are investigated in an infinite-size bond-alternating Ising chain with the Dzyaloshinskii-Moriya interaction by employing the infinite matrix product state representation with the infinite time evolving block decimation method. Directly computing two distinct types of finite string correlations for very large lattice distances, in contrast to an extrapolated extreme value for finite size chains, reveals two topologically ordered phases. As the bond alternation varies, a topological quantum phase transition with continuously variable critical exponents along the phase boundary occurs between the two Haldane phases for a Dzyaloshinskii-Moriya interaction stronger than the Ising interaction while a topological quantum crossover between them happens through an intermediate antiferromagentic phase demonstrated with the quantum fidelity for a Dzyaloshinskii-Moriya interaction weaker than the Ising interaction. The critical exponents of the order parameters and the central charges from the entanglement entropy quantify the universality classes of the phase transition points. Anisotropic Heisenberg types of spin chains with bond alternations are finally discussed to share the same criticality.

Recording optical vortices in azo polymer films by applying holographic method

In this paper the optical vortices with topological charge &lt;i&gt;q&lt;/i&gt; = –1, 1, 2, 4 are recorded in azo polymer films by using holographic technology. The forked holographic gratings formed by the Gaussian beam and optical vortex beam are recorded in the sample films, the original forked holographic grating and the recording rate are analyzed. The vortex beam is reconstructed by illuminating the sample film with a reference beam, and the recording quality is analyzed. Also the erasability and durability of the sample are tested. The experimental results show that the recording rates of vortex beams with different topological charges are relatively uniform, which means that the optical vortices with different topological charges can be recorded at the same speed. The forked holographic grating of the high-order optical vortex splits in the recording process due to the disturbances, such as anisotropic nonlinear light, atmospheric turbulence, and background light field. However, the split vortex beam still maintains a stable ring structure. The reconstructed optical vortex and the original optical vortex are highly consistent in morphology, and the interference fringes of the reconstructed optical vortices are highly consistent with the original vortex holographic gratings, indicating that the topological charge information in the optical vortices can be effectively recorded and read out. The recorded information can be erased by heating the sample to about 97 ℃, and new information can be re-recorded after cooling. There appears no fatigue in the sample after the information has been erased 100 times and good durability is still retained. Optical vortices theoretically have infinite states of topological charges, based on which great success is achieved in optical communication and information encoding. Therefore, storing and reading information of topological charges in optical vortices may have potential applications in optical information storage.

The direct method of Lyapunov for nonlinear dynamical systems with fractional damping

In this paper, we introduce a generalization of Lyapunov’s direct method for dynamical systems with fractional damping. Hereto, we embed such systems within the fundamental theory of functional differential equations with infinite delay and use the associated stability concept and known theorems regarding Lyapunov functionals including a generalized invariance principle. The formulation of Lyapunov functionals in the case of fractional damping is derived from a mechanical interpretation of the fractional derivative in infinite state representation. The method is applied on a single degree-of-freedom oscillator first, and the developed Lyapunov functionals are subsequently generalized for the finite-dimensional case. This opens the way to a stability analysis of nonlinear (controlled) systems with fractional damping. An important result of the paper is the solution of a tracking control problem with fractional and nonlinear damping. For this problem, the classical concepts of convergence and incremental stability are generalized to systems with fractional-order derivatives of state variables. The application of the related method is illustrated on a fractionally damped two degree-of-freedom oscillator with regularized Coulomb friction and non-collocated control.

Model predictive control of a non-isothermal axial dispersion tubular reactor with recycle

This paper addresses the model predictive output controller design for a non-isothermal axial dispersion tubular reactor accounting for energy and mass transport in a recycle stream. The underlying transport-reaction process is characterized by different mass and energy Peclet numbers. Open-loop analysis reveals unstable operating conditions based on the reactor’s parameters. The discrete model of the system is obtained by considering a Crank-Nicolson type of discretization method without any model reduction and/or spatial approximation for the system of coupled parabolic PDEs. The proposed predictive controller accounts for asymptotic close-loop system stabilization and/or naturally present input and state constraints with the rejection of possible disturbances arising from reactor operations. To account for the output controller design, a discrete observer is developed to reconstruct the infinite dimensional states in the predictive control realization. Finally, the controller’s performance is assessed via simulation studies, implying proper state stabilization and constraints satisfaction with input disturbance rejection.

Robust State Estimation for Positive Real Infinite-Dimensional Systems With Actuator and Sensor Faults

In this article, we propose a novel framework on the state estimation for a class of positive real infinite-dimensional systems, which suffer from unknown actuator and sensor faults and uncertain exogenous disturbance. Four cases of fault occurrence are taken into account: single multiplicative actuator fault, single multiplicative sensor fault, simultaneous actuator and sensor faults, and single actuator fault with a dead-zone nonlinearity. We develop new adaptive laws for the estimation of fault parameters and embed the proposed laws into the infinite dimensional state observer design. Moreover, by considering the actuator fault with a dead-zone nonlinearity, an infinite dimensional observer associated with coupled parameter update laws is developed to help to complete the fault accommodation. This is achieved by reconfiguring the actuator input based on the estimated information of parameters in the actuator fault map with a dead-zone nonlinearity.

An adaptive solution to the chemical master equation using quantized tensor trains with sliding windows

To cope with an extremely large or even infinite state space when solving the chemical master equation in biological problems, a potent strategy is to restrict to a finite state projection (FSP) and represent the transition matrix and probability vector in quantized tensor train (QTT) format, leading to savings in storage while retaining accuracy. In an earlier adaptive FSP–QTT algorithm, the multidimensional state space was downsized and kept in the form of a hyper rectangle that was updated when needed by selectively doubling some of its side dimensions. However, this could result in a much larger state space than necessary, with the effect of hampering both the execution time and stepping scheme. In this work, we improve the algorithm by enabling sliding windows that can dynamically slide, shrink or expand, with updates driven by a number of stochastic simulation algorithm trajectories. The ensuing state space is a considerably reduced hyper rectangle containing only the most probable states at each time step. Three numerical experiments of varying difficulty are performed to compare our approach with the original adaptive FSP–QTT algorithm.

A natural extension of Markov processes and applications to singular SDEs

On établit une méthode générale pour élargir l’espace d’états d’un processus de Markov, de telle façon que l’ensemble des points ajoutés est polaire. Cette extension est une amélioration de l’extension triviale, dans quel cas le processus est bloqué dans les points ajoutés, et elle produit une technique nouvelle de construction des solutions étendues pour des ED(P) stochastiques, à partir de tous les points de départ, telle qu’elles soient des solutions au moins après tout moment de temps strictement positif. Concrètement, on adopte cette stratégie pour étudier des ED stochastiques avec des coefficients singuliers, sur un espace d’états de dimension infinie (par example des EDP stochastiques de type evolution), pour lesquelles on rencontre des situations où tous les points de l’espace ne sont pas des conditions initiales autorisées. La même chose peut se passer dans la construction des solutions pour des problèmes de martingales ou pour des processus de Markov à partir de formes de Dirichlet (généralisées), pour lesquelles notre nouvelle technique s’applique aussi.

Bifurcation analysis of a general activator-inhibitor model with nonlocal dispersal

In this paper, we are mainly concerned with the effect of nonlocal diffusion and dispersal spread on bifurcations of a general activator-inhibitor system in which the activator has a nonlocal dispersal. We find that spatially inhomogeneous patterns always exist if the dispersal rate of the activator is sufficiently small, while a larger dispersal spread and an increase of the activator diffusion inhibit the formation of patterns. Compared with the spatial averaging nonlocal dispersal model, our model admits a larger parameter region supporting pattern formations, which is also true if compared with the local reaction-diffusion one when the dispersal spread is small. We also study the existence of nonconstant positive steady states through bifurcation theory and find that there could exist finite or infinite steady state bifurcation points of the inhibitor diffusion constant. As an example of our results, we study a water-biomass model with nonlocal dispersal of plants and show that the water and plant distributions could be inphase and antiphase.

Decentralized Optimal Stabilization of Active Loads in Islanded Microgrids

This article proposes a decentralized, online optimal feedback control strategy to optimally stabilize active loads in islanded DC microgrids (DCMGs). Each active load is modeled as a control affine dynamical system with an interconnected coupling term in the energy and admittance domain. Then the decentralized, constrained input of each active load is obtained online in the feedback form to minimize the infinite horizon quadratic state cost and non quadratic control effort while supplying the active load demand. The approximate dynamic programming (ADP) approach is employed to solve the feedback optimal control problem online by successive approximation of the value function via two linear in the parameter (LIP) neural networks (NNs). A concurrent reinforcement learning (RL) based method is introduced to update the unknown weights in the NN. The convergence of the unknown weights to a neighborhood of the optimal weights and the uniformly ultimately bounded stability of the system are analyzed by a Lyapunov direct method. Series of simulation results are presented to demonstrate the effectiveness and the applicability of the proposed concept.

Melting of the critical behavior of a Tomonaga-Luttinger liquid under dephasing

Strongly correlated quantum systems often display universal behavior as, in certain regimes, their properties are found to be independent of the microscopic details of the underlying system. An example of such a situation is the Tomonaga-Luttinger liquid description of one-dimensional strongly correlated bosonic or fermionic systems. Here, we investigate how such a quantum liquid responds under dissipative dephasing dynamics and, in particular, we identify how the universal Tomonaga-Luttinger liquid properties melt away. Our study, based on adiabatic elimination, shows that dephasing first translates into the damping of the oscillations present in the density-density correlations, a behavior accompanied by a change in the Tomonaga-Luttinger liquid exponent. This first regime is followed by a second one characterized by the diffusive propagation of featureless correlations as expected for an infinite temperature state. We support these analytical predictions by numerically exact simulations carried out using a number-conserving implementation of the matrix product states algorithm adapted to open systems.

Triggering parametric-down conversion-based quantum key distribution via radiation field

We discuss the influence of radiation field on the secure key size and the maximum safety distance during QKD by using a set of photons produced via a spontaneous parametric-down conversion (SPDC) photon source. Four implementations that use multiple-photons and decoy states are discussed, these include nondecoy state, infinite active decoy state, passive decoy state with threshold detector and passive decoy state with perfect photon-number resolving detector. Results show that the radiation field significantly improves both the secure key size and the maximum secure communication distance. Therefore, the radiation field is found to be a good candidate to reduce unwanted interactions of photons with the quantum channel and hence, to increase the secure key rate and the maximum safety distance between legitimate users.

Characterizing the Non-equilibrium Dynamics of Field-Driven Correlated Quantum Systems

Recent experimental advances in ultrafast phenomena have triggered renewed interest in the dynamics of correlated quantum systems away from equilibrium. We review nonequilibrium dynamical mean-field theory studies of both the transient and the steady states of a DC field-driven correlated quantum system. In particular, we focus on the nonequilibrium behavior and how it relates to the fluctuation-dissipation theorem. The fluctuation-dissipation theorem emerges as an indicator for how the system thermalizes and for how it reaches a steady state. When the system thermalizes in an infinite temperature steady state it can pass through a succession of quasi-thermal states that approximately obey the fluctuation-dissipation theorem. We also discuss the Wigner distribution and what its evolution tells us about the nonequilibrium many-body problem.

On the representation theory of the infinite Temperley–Lieb algebra

We begin the study of the representation theory of the infinite Temperley–Lieb algebra. We fully classify its finite-dimensional representations, then introduce infinite link state representations and classify when they are irreducible or indecomposable. We also define a construction of projective indecomposable representations for TL[Formula: see text] that generalizes to give extensions of TL[Formula: see text] representations. Finally, we define a generalization of the spin chain representation and conjecture a generalization of Schur–Weyl duality.

Construction of the concept of pairability in comparing infinite sets in the individuals’ mind: an APOS approach

Abstract This study focuses on the concept of pairability as a fundamental metaphorical concept in the Cantorian set theory regarding comparison of infinite sets. In this theory, pairability appears in a hierarchical manner of generating and developing as a one-to-one correspondence by arrows in finite, and then in infinite states, a bijection map through an explicit formula and the concept of cardinality. Adapting these hierarchical components of pairability and the APOS theory, about description of constructing and understanding a mathematical concept in a hierarchical manner, this study examines the construction of the concept of pairability in the individuals’ mind. In this way, their imaginations of pairability and practical performances in different situations will also be surveyed. In so doing, a total of 20 mathematics teachers and university lecturers holding at least an M.Sc. degree in mathematics were chosen. To collect the data, interviews were conducted in which the participants not only answered questions about the concept of the various types of pairability but also compared infinite countable sets in different situations. Our findings revealed that a bijection map via an explicit formula was the individuals’ dominant conception of pairability and most of the incorrect answers were related to unsuccessful attempts to recall a formula or method as the only possible way, and the encapsulated concept of cardinality was used less frequently in practice. Therefore, there was not a total schema of actions, processes and objects of the concept of pairability in the individuals’ mind.