Continuous-time Versions Research Articles

Quantum simulation of discrete linear dynamical systems and simple iterative methods in linear algebra

Quantum simulation is capable of simulating certain dynamical systems in continuous time—Schrödinger’s equations being the most direct and well known—more efficiently than classical simulation. Any linear dynamical system can in fact be transformed into a system of Schrödinger’s equations via a method called Schrödingerisation (Jin et al. 2022. Quantum simulation of partial differential equations via Schrödingerisation. ( https://arxiv.org/abs/2212.13969 ) and Jin et al. 2023. Phys. Rev. A 108 , 032603. ( doi:10.1103/PhysRevA.108.032603 )). We show how Schrödingerisation allows quantum simulation to be directly used for the simulation of continuous-time versions of general (explicit) iterative schemes or discrete linear dynamical systems. In particular, we use this new method to solve linear systems of equations and for estimating the maximum eigenvector and eigenvalue of a matrix, respectively. This method is applicable using either discrete-variable quantum systems or on hybrid continuous-variable and discrete-variable quantum systems. This framework provides an interesting alternative to solve linear algebra problems using quantum simulation.

Perturbative solution of fermionic sign problem in quantum Monte Carlo computations

We have developed a strong-coupling perturbation scheme for a general doped Hubbard model around a particle-hole-symmetric reference system, which is free from the fermionic sign problem. Our approach is based on the lattice determinantal Quantum Monte Carlo (QMC) method in both continuous and discrete time versions for large periodic clusters in a fermionic bath. By considering the first-order perturbation in the shift of the chemical potential and the second-neighbor hopping, we are able to obtain an accurate electronic spectral function for a range of parameters that correspond to optimally doped cuprate systems at temperatures of up to T = 0.1t, which are challenging to access using straightforward lattice QMC calculations. We also discuss the formation of the pseudogap and the nodal-antinodal dichotomy for a doped Hubbard system with the interaction parameter U equal to the bandwidth and the optimal value of the next-nearest-neighbor hopping parameter t′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${t}^{{\\prime} }$$\\end{document} for high-temperature superconducting cuprates.

Coupled dynamics of endemic disease transmission and gradual awareness diffusion in multiplex networks

Understanding the interplay between human behavioral phenomena and infectious disease dynamics has been one of the central challenges of mathematical epidemiology. However, socio-cognitive processes critical for the initiation of desired behavioral responses during an outbreak have often been neglected or oversimplified in earlier models. Combining the microscopic Markov chain approach with the law of total probability, we herein institute a mathematical model describing the dynamic interplay between stage-based progression of awareness diffusion and endemic disease transmission in multiplex networks. We analytically derived the epidemic thresholds for both discrete-time and continuous-time versions of our model, and we numerically demonstrated the accuracy of our analytic arguments in capturing the time course and the steady state of the coupled disease-awareness dynamics. We found that our model is exact for arbitrary unclustered multiplex networks, outperforming a widely adopted probability-tree-based method, both in the prediction of the time-evolution of a contagion and in the final epidemic size. Our findings show that informing the unaware individuals about the circulating disease will not be sufficient for the prevention of an outbreak unless the distributed information triggers strong awareness of infection risks with adequate protective measures, and that the immunity of highly-aware individuals can elevate the epidemic threshold, but only if the rate of transition from weak to strong awareness is sufficiently high. Our study thus reveals that awareness diffusion and other behavioral parameters can nontrivially interact when producing their effects on epidemiological dynamics of an infectious disease, suggesting that future public health measures should not ignore this complex behavioral interplay and its influence on contagion transmission in multilayered networked systems.

Non-Markovian SIR epidemic spreading model of COVID-19

We introduce non-Markovian SIR epidemic spreading model inspired by the characteristics of the COVID-19, by considering discrete- and continuous-time versions. The distributions of infection intensity and recovery period may take an arbitrary form. By taking corresponding choice of these functions, it is shown that the model reduces to the classical Markovian case. The epidemic threshold is analytically determined for arbitrary functions of infectivity and recovery and verified numerically. The relevance of the model is shown by modeling the first wave of the epidemic in Italy, Spain and the UK, in the spring, 2020.

Locating platforms and scheduling a fleet of drones for emergency delivery of perishable items

Motivated by issues dealing with delivery of emergency medical products during humanitarian disasters, this paper addresses the general problem of delivering perishable items to remote demands accessible only by helicopters or drones. Each drone operates out of platforms that may be moved when not in use and each drone has a limited delivery range to service a demand point. Associated with each demand point is a disutility function, or a cost function, with respect to time that reflects preferred delivery clock time for the demanded item, as well as the item’s perishability characteristic that models nonincreasing quality with time. The paper first addresses the problem of locating the platforms as well concurrently determining which platform serves which demand points and in what order – to minimize total disutility for product delivery. The second scenario addresses the two-period problem where the platforms can be relocated, using useable road network, after the first period. It can be easily proven that continuous time versions of these problems are NP-Hard. However, a practical “time-slot” version of the problem, where time is discretized into slots, can be solved by standard optimization software. Extensive computational experiments, using different drone delivery ranges as well as different drone fleet sizes, provide valuable insights on the performance of such drone delivery systems.

Decomposition dynamical systems for solving variational inequalities

In this paper, we propose two decomposition dynamical systems for solving strongly pseudomonotone variational inequalities given by a sum of two mappings. We prove that trajectories of these dynamical systems converge to the unique solution of the variational inequality. Also, we extend these algorithms for equilibrium problems and study their convergence rate. This approach is novel and the new algorithms can be considered as continuous-time versions of existing decomposition algorithms for variational inequalities and equilibrium problems.

On the Continuous-time and Discrete-Time Versions of an Alternative Epidemic Model of the SIR Class

The well-known SIR epidemic model is revisited. Continuous-time and discrete-time versions of an alternative model of this class are presented, discussed and validated with actual data. The proposed model follows from the calculation of the mean number of new infected cases due to the eventual meeting of susceptible and infected individuals, based on a simple probabilistic argument. Determination of the invariant set in the state space and convergence conditions towards equilibrium are established. For numerical analysis, data of daily number of new diagnosed cases provided by the Brazilian Ministry of Health and World Health Organization of COVID-19 outbreak that currently occurs respectively in Brazil and in the UK are used. Illustrations and model prediction analysis are provided and discussed from full data of both aforementioned countries which include more than 400 epidemic days. Three different and complementary strategies for parameter identification including the impact of causality on the optimal solution of the nonlinear mean square problem are discussed.

Parameter Identification With Finite-Convergence Time Alertness Preservation

In this letter note we present two new parameter identifiers whose estimates converge in finite time under weak interval excitation assumptions. The main novelty is that, in contrast with other finite-convergence time (FCT) estimators, our schemes preserve the FCT property when the parameters change. The previous versions of our FCT estimators can track the parameter variations only asymptotically. Continuous-time and discrete-time versions of the new estimators are presented.

Protein-DNA target search relies on quantum walk

Protein-DNA interactions play a fundamental role in all life systems. A critical issue of such interactions is given by the strategy of protein search for specific targets on DNA. The mechanisms by which the protein are able to find relatively small cognate sequences, typically 15–20 base pairs (bps) for repressors, and 4–6 bps for restriction enzymes among the millions of bp of non-specific chromosomal DNA have hardly engaged researchers for decades. Recent experimental studies have generated new insights on the basic processes of protein-DNA interactions evidencing the underlying complex dynamic phenomena involved, which combine three-dimensional and one-dimensional motion along the DNA chain. It has been demonstrated that protein molecules have an extraordinary ability to find the target very quickly on the DNA chain, in some cases, with two orders of magnitude faster than the diffusion limit. This unique property of protein-DNA search mechanism is known as facilitated diffusion. Several theoretical mechanisms have been suggested to describe the origin of facilitated diffusion. However, none of such models currently has the ability to fully describe the protein search strategy.In this paper, we suggest that the ability of proteins to identify consensus sequences on DNA is based on the entanglement of π-π electrons between DNA nucleotides and protein amino acids. The π-π entanglement is based on Quantum Walk (QW), through Coin-position entanglement (CPE). First, the protein identifies a dimer belonging to the consensus sequence, and localize a π on such dimer, hence, the other π electron scans the DNA chain until the sequence is identified. Focusing on the example of recognition of consensus sequences of EcoRV or EcoRI, we will describe the quantum features of QW on protein-DNA complexes during the search strategy, such as walker quadratic spreading on a coherent superposition of different vertices and environment-supported long-time survival probability of the walker. We will employ both discrete- or continuous-time versions of QW. Biased and unbiased classical Random Walk (CRW) have been used for a long time to describe the Protein-DNA search strategy. QW, the quantum version of CRW, has been widely studied for its applications in quantum information applications. In our biological application, the walker (the protein) resides at a vertex in a graph (the DNA structural topology). Differently to CRW, where the walker moves randomly, the quantum walker can hop along the edges in the graph to reach other vertices entering coherently a superposition across different vertices spreading quadratically faster than CRW analogous evidencing the typical speed up features of the QW. When applied to a protein-DNA target search problem, QW gives the possibility to achieve the experimental diffusional motion of proteins over diffusion classical limits experienced along DNA chains exploiting quantum features such as CPE and long-time survival probability supported by the environment. In turn, we come to the conclusion that, under quantum picture, the protein search strategy does not distinguish between one-dimensional (1D) and three-dimensional (3D) cases.

Inference for Nonlinear State Space Models: A Comparison of Different Methods applied to Markov-Switching Multifractal Models

Nonlinear, non-Gaussian state space models have found wide applications in many areas. These models usually do not allow for an analytical representation of their likelihood function and thus, sequential Monte Carlo or particle filter methods are mostly applied to estimate their parameters. Finding the best-fitting parameters of a model is a non-trivial task since stochastic approximations lead to non-smooth likelihood functions. Recently proposed iterative filtering algorithms developed for this purpose are compared with simpler on-line filters and more traditional methods of inference. A highly nonlinear class of Markov-switching models, the so called Markov-switching multifractal model (MSM) is used as illustrative example in the comparison of different optimisation routines. Besides the well-established univariate discrete-time MSM, univariate and multivariate continuous-time versions of MSM are considered. Monte Carlo simulation experiments indicate that across a variety of MSM specifications, the classical Nelder-Mead or simplex algorithm appears still as more efficient and robust compared to a number of online and iterated filters. A very close competitor is one of the recently proposed iterated filters while other alternatives are mostly dominated by these two algorithms. An empirical application of both discrete and continuous-time MSM to seven financial time series shows that both models dominate GARCH and FIGARCH models in terms of in-sample goodness-of-fit. Out-of-sample forecast comparisons show in the majority of cases a clear dominance of the continuous-time MSM under a mean absolute error criterion, and less conclusive results under a mean squared error criterion.

Large deviations of a long-time average in the Ehrenfest urn model

Since its inception in 1907, the Ehrenfest urn model (EUM) has served as a test bed of key concepts of statistical mechanics. Here we employ this model to study large deviations of a time-additive quantity. We consider two continuous-time versions of the EUM with K urns and N balls: with and without interactions between the balls in the same urn. We evaluate the probability distribution that the average number of balls in one urn over time T, , takes any specified value aN, where . For long observation time, , a Donsker–Varadhan large deviation principle holds: , where … denote additional parameters of the model. We calculate the rate function exactly by two different methods due to Donsker and Varadhan and compare the exact results with those obtained with a variant of WKB approximation (after Wentzel, Kramers and Brillouin). In the absence of interactions the WKB prediction for is exact for any N. In the presence of interactions the WKB method gives asymptotically exact results for . The WKB method also uncovers the (very simple) time history of the system which dominates the contribution of different time histories to .

On sensitivity of mixing times and cutoff

A sequence of chains exhibits (total-variation) cutoff (resp., pre-cutoff) if for all $0<\epsilon< 1/2$, the ratio $t_{\mathrm{mix}}^{(n)}(\epsilon)/t_{\mathrm{mix}}^{(n)}(1-\epsilon)$ tends to 1 as $n \to \infty $ (resp., the $\limsup$ of this ratio is bounded uniformly in $\epsilon$), where $t_{\mathrm{mix}}^{(n)}(\epsilon)$ is the $\epsilon$-total-variation mixing-time of the $n$th chain in the sequence. We construct a sequence of bounded degree graphs $G_n$, such that the lazy simple random walks (LSRW) on $G_n$ satisfy the "product condition" $\mathrm{gap}(G_n) t_{\mathrm{mix}}^{(n)}(\epsilon) \to \infty $ as $n \to \infty$, where $\mathrm{gap}(G_n)$ is the spectral gap of the LSRW on $G_n$ (a known necessary condition for pre-cutoff that is often sufficient for cutoff), yet this sequence does not exhibit pre-cutoff. Recently, Chen and Saloff-Coste showed that total-variation cutoff is equivalent for the sequences of continuous-time and lazy versions of some given sequence of chains. Surprisingly, we show that this is false when considering separation cutoff. We also construct a sequence of bounded degree graphs $G_n=(V_{n},E_{n})$ that does not exhibit cutoff, for which a certain bounded perturbation of the edge weights leads to cutoff and increases the order of the mixing-time by an optimal factor of $\Theta (\log |V_n|)$. Similarly, we also show that "lumping" states together may increase the order of the mixing-time by an optimal factor of $\Theta (\log |V_n|)$. This gives a negative answer to a question asked by Aldous and Fill.

Zero-Error Capacity of $ P $ -ary Shift Channels and FIFO Queues

The objects of study of this paper are communication channels in which the dominant type of noise are symbol shifts, the main motivating examples being timing and bit-shift channels. Two channel models are introduced and their zero-error capacities and zero-error-detection capacities determined by explicit constructions of optimal codes. Model A can be informally described as follows: 1) The information is stored in an $ n $-cell register, where each cell is either empty or contains a particle of one of $ P $ possible types, and 2) due to the imperfections of the device each of the particles may be shifted several cells away from its original position over time. Model B is an abstraction of a single-server queue: 1) The transmitter sends packets from a $ P $-ary alphabet through a queuing system with an infinite buffer and a First-In-First-Out (FIFO) service procedure, and 2) each packet is being processed by the server for a random number of time slots. More general models including additional types of noise that the particles/packets can experience are also studied, as are the continuous-time versions of these problems.

Theoretical Foundations and Practical Applications of Within-Cycle Correction Methods.

Modeling guidelines recommend applying a half-cycle correction (HCC) to outcomes from discrete-time state-transition models (DTSTMs). However, there is still no consensus on why and how to perform the correction. The objective was to provide theoretical foundations for HCC and to compare (both mathematically and numerically) the performance of different correction methods in reducing errors in outcomes from DTSTMs. We defined 7 methods from the field of numerical integration: Riemann sum of rectangles (left, midpoint, right), trapezoids, life-table, and Simpson's 1/3rd and 3/8th rules. We applied these methods to a standard 3-state disease progression Markov chain to evaluate the cost-effectiveness of a hypothetical intervention. We solved the discrete- and continuous-time (our gold standard) versions of the model analytically and derived expressions for various outcomes including discounted quality-adjusted life-years, discounted costs, and incremental cost-effectiveness ratios. The standard HCC method gave the same results as the trapezoidal rule and life-table method. We found situations where applying the standard HCC can do more harm than good. Compared with the gold standard, all correction methods resulted in approximation errors. Contrary to conventional wisdom, the errors need not cancel each other out or become insignificant when incremental outcomes are calculated. We found that a wrong decision can be made with a less accurate method. The performance of each correction method vastly improved when a shorter cycle length was selected; Simpson's 1/3rd rule was the fastest method to converge to the gold standard. Cumulative outcomes in DTSTMs are prone to errors that can be reduced with more accurate methods like Simpson's rules. We clarified several misconceptions and provided recommendations and algorithms for practical implementation of these methods.

Higher Moments and Prediction‐Based Estimation for the COGARCH(1,1) Model

AbstractCOGARCH models are continuous time versions of the well‐known GARCH models of financial returns. The first aim of this paper is to show how the method of prediction‐based estimating functions can be applied to draw statistical inference from observations of a COGARCH(1,1) model if the higher‐order structure of the process is clarified. A second aim of the paper is to provide recursive expressions for the joint moments of any fixed order of the process. Asymptotic results are given, and a simulation study shows that the method of prediction‐based estimating function outperforms the other available estimation methods.

Analysis of distributed power control under constant and time-varying delays

This work studies the distributed power control algorithm proposed in 1993 by Foschini-Miljanic, standardised for universal mobile telecommunication systems. Continuous and discrete time versions of this algorithm are analysed. First, the stability of the distributed power allocation schemes was studied, where sufficient conditions to guarantee stability and convergence to a desired quality of service were provided. In this study, the channel gains are assumed to be slowly time-varying or piece-wise constant. For closed-loop control, a proportional controller is then employed under integral action in order to achieve good tracking despite time-varying and unknown channel gains. Next, the effects of constant and time-varying time delays in the closed-loop structure are studied. Explicit stability regions for the control gains in the Foschini-Miljanic scheme are derived for both the continuous and discrete-time versions of the algorithm, under constant and time-varying delays. For time-varying scenario, the resulting stability regions do not impose limitations on the rate change of the time-varying profiles. A comprehensive evaluation using simulations is performed to validate the analytical derivations described in the paper.

Gauge subsystems, separability and robustness in autonomous quantum memories

Quantum error correction provides a fertile context for exploring the interplay of feedback control, microscopic physics and non-commutative probability. In this paper we deepen our understanding of this nexus through high-level analysis of a class of quantum memory models that we have previously proposed, which implement continuous-time versions of well-known stabilizer codes in autonomous nanophotonic circuits that require no external clocking or control. We demonstrate that the presence of the gauge subsystem in the nine-qubit Bacon–Shor code allows for a loss-tolerant layout of the corresponding nanophotonic circuit that substantially ameliorates the effects of optical propagation losses, argue that code separability allows for simplified restoration feedback protocols, and propose a modified fidelity metric for quantifying the performance of realistic quantum memories. Our treatment of these topics exploits the homogeneous modeling framework of autonomous nanophotonic circuits, but the key ideas translate to the traditional setting of discrete time, measurement-based quantum error correction.

On small-controllability of a class of Hammerstein-nonlinear systems – an implicit function approach

In this paper, an implicit function approach is applied to controllability of a class of discrete-time Hammerstein-nonlinear systems. Based on this approach, the systems are shown to own a strong property, namely, small-controllability. As an immediate application, the obtained small-controllability results are generalised to the continuous-time Hammerstein-nonlinear systems and some continuous-time versions of small-controllability are presented.

Pontryagin Maximum Principle for Finite Dimensional Nonlinear Optimal Control Problems on Time Scales

In this paper we derive a strong version of the Pontryagin maximum principle for general nonlinear optimal control problems on time scales in finite dimension. The final time can be fixed or not, and in the case of general boundary conditions we derive the corresponding transversality conditions. Our proof is based on Ekeland's variational principle. Our statement and comments clearly show the distinction between right-dense points and right-scattered points. At right-dense points a maximization condition of the Hamiltonian is derived, similarly to the continuous-time case. At right-scattered points a weaker condition is derived, in terms of so-called stable $\Omega$-dense directions. We do not make any specific restrictive assumption on the dynamics or on the set $\Omega$ of control constraints. Our statement encompasses the classical continuous-time and discrete-time versions of the Pontryagin maximum principle, and holds on any general time scale, that is, any closed subset of $\mathbb{R}$.

Parameter and uncertainty estimation for process‐oriented population and distribution models: data, statistics and the niche

AbstractThe spatial distribution of a species is determined by dynamic processes such as reproduction, mortality and dispersal. Conventional static species distribution models (SDMs) do not incorporate these processes explicitly. This limits their applicability, particularly for non‐equilibrium situations such as invasions or climate change. In this paper we show how dynamic SDMs can be formulated and fitted to data within a Bayesian framework. Our focus is on discrete state‐space Markov process models which provide a flexible framework to account for stochasticity in key demographic processes, including dispersal, growth and competition. We show how to construct likelihood functions for such models (both discrete and continuous time versions) and how these can be combined with suitable observation models to conduct Bayesian parameter inference using computational techniques such as Markov chain Monte Carlo. We illustrate the current state‐of‐the‐art with three contrasting examples using both simulated and empirical data. The use of simulated data allows the robustness of the methods to be tested with respect to deficiencies in both data and model. These examples show how mechanistic understanding of the processes that determine distribution and abundance can be combined with different sources of information at a range of spatial and temporal scales. Application of such techniques will enable more reliable inference and projections, e.g. under future climate change scenarios than is possible with purely correlative approaches. Conversely, confronting such process‐oriented niche models with abundance and distribution data will test current understanding and may ultimately feedback to improve underlying ecological theory.