Discrete Time Research Articles

Development of discrete opposition-based NSGA-III model for optimizing trade-off between discrete time, cost, and resource in construction projects

Development of discrete opposition-based NSGA-III model for optimizing trade-off between discrete time, cost, and resource in construction projects

Biased versus unbiased numerical methods for stochastic simulations

Approximate numerical methods are one of the most used strategies to extract information from many-interacting-agents systems. In particular, numerical approximations are of extended use to deal with epidemic, ecological and biological models, since unbiased methods like the Gillespie algorithm can become unpractical due to high CPU time usage required. However, the use of approximations has been debated and there is no clear consensus about whether unbiased methods or biased approach is the best option. In this work, we derive scaling relations for the errors in approximations based on binomial extractions. This finding allows us to build rules to compute the optimal values of both the discretization time and number of realizations needed to compute averages with the biased method with a target precision and minimum CPU-time usage. Furthermore, we also present another rule to discern whether the unbiased method or biased approach is more efficient. Ultimately, we will show that the choice of the method should depend on the desired precision for the estimation of averages.

On V-Geometric Ergodicity Markov Chains of the Two-Inertia Systems

This study employs the diffusion process to construct Markov chains for analyzing the common two-inertia systems used in industry. Two-inertia systems are prevalent in commonly used equipment, where the load is influenced by the coupling of external force and the drive shaft, leading to variations in the associated output states. Traditionally, the control of such systems is often guided by empirical rules. This paper examines the equilibrium distribution and convergence rate of the two-inertia system and develops a predictive model for its long-term operation. We explore the qualitative behavior of the load end at discrete time intervals. Our findings are applicable not only in control engineering, but also provide insights for small-scale models incorporating dual-system variables.

Degree evolution in a general growing network

We consider the preferential attachment model introduced by Deijfen and Lindholm (2009) in which, at every discrete time step: (i) either we add a vertex and connect it to an older vertex; or (ii) we add an edge between two random vertices; or (iii) we delete one edge. We show that, when the deletion probability equals 1/3, the expected degree of any given vertex grows logarithmically, thus correcting a statement made in Lindholm and Vallier (2011). Moreover we show that, when the deletion probability is strictly less than 1/3, then the function which scales the expected degree of a given vertex, identified in Lindholm and Vallier (2011), also guarantees almost sure convergence for the degree process of a given vertex.

Variational data assimilation with finite-element discretization for second-order parabolic interface equation

Abstract In this paper, we propose and analyze a finite-element method of variational data assimilation for a second-order parabolic interface equation on a two-dimensional bounded domain. The Tikhonov regularization plays a key role in translating the data assimilation problem into an optimization problem. Then the existence, uniqueness and stability are analyzed for the solution of the optimization problem. We utilize the finite-element method for spatial discretization and backward Euler method for the temporal discretization. Then based on the Lagrange multiplier idea, we derive the optimality systems for both the continuous and the discrete data assimilation problems for the second-order parabolic interface equation. The convergence and the optimal error estimate are proved with the recovery of Galerkin orthogonality. Moreover, three iterative methods, which decouple the optimality system and significantly save computational cost, are developed to solve the discrete time evolution optimality system. Finally, numerical results are provided to validate the proposed method.

Methods for implementing integrated step-selection functions with incomplete data

Integrated step-selection analyses (iSSAs) are versatile and powerful frameworks for studying habitat and movement preferences of tracked animals. iSSAs utilize integrated step-selection functions (iSSFs) to model movements in discrete time, and thus, require animal location data that are regularly spaced in time. However, many real-world datasets are incomplete due to tracking devices failing to locate an individual at one or more scheduled times, leading to slight irregularities in the duration between consecutive animal locations. To address this issue, researchers typically only consider bursts of regular data (i.e., sequences of locations that are equally spaced in time), thereby reducing the number of observations used to model movement and habitat selection. We reassess this practice and explore four alternative approaches that account for temporal irregularity resulting from missing data. Using a simulation study, we compare these alternatives to a baseline approach where temporal irregularity is ignored and demonstrate the potential improvements in model performance that can be gained by leveraging these additional data. We also showcase these benefits using a case study on a spotted hyena (Crocuta crocuta).

A Net Present Value Analysis of Opportunity-Based Age Replacement Models in Discrete Time

Two important opportunistic age replacement models, under replacement first and last disciplines, are generalized in discrete time. The net present value (NPV) is applied to formulate the expected total costs. The priority of multiple replacement options is considered to classify the cost model with discounting into six cases. Since the NPV method accurately calculates the expected replacement costs over an infinite horizon in an unstable economic environment, we discuss some optimal opportunistic age replacement policies which minimize the expected total discounted costs over an infinite time horizon. Furthermore, we formulate a unified model under each discipline, merging six discrete time replacement models with probabilistic priority. Finally, a case study on optimal replacement first and last policies for pole air switches in a Japanese power company is presented.

An Empirical Study On Imaging Sensitivity Of SKA1-LOW Based on Point Source Sensitivity

The Square Kilometre Array (SKA) radio telescope will achieve unparalleled sensitivity and angular resolution, substantially progressing our research into the formation and development of the early universe. Sensitivity is one of the most critical telescope metrics, often used to determine the observing mode and observing time of a radio source. It is also used to determine imaging parameters to balance imaging sensitivity and spatial resolution during data processing. In this paper, to meet the needs of continuum imaging pipeline data processing for the SKA1-LOW telescope, we derive equations for point-source-sensitivity (PSS) calculations under multichannel, multipolarization, discrete frequency sampling, discrete time sampling, and different weighting schemes. After validating the correctness of the equation, we improved and refined the sensitivity calculation application in the Radio Astronomy Simulation, Calibration, and Imaging Library, and further calculated the variation of the PSS versus angular resolution for the full-scale SKA1-LOW with different weighting schemes, and provided SKA1-LOW broadband image performance as functions of angular scale for four frequencies. These results can provide not only a theoretical basis for the construction and commissioning of the SKA1-LOW telescope but also guidance for imaging processing in future scientific data processing.

Temporal approximation of stochastic evolution equations with irregular nonlinearities

In this paper, we prove convergence for contractive time discretisation schemes for semi-linear stochastic evolution equations with irregular Lipschitz nonlinearities, initial values, and additive or multiplicative Gaussian noise on 2-smooth Banach spaces X. The leading operator A is assumed to generate a strongly continuous semigroup S on X, and the focus is on non-parabolic problems. The main result concerns convergence of the uniform strong errorEk∞:=(Esupj∈{0,…,Nk}‖U(tj)-Uj‖Xp)1/p→0(k→0),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \ extrm{E}_{k}^{\\infty } {:}{=}\\Big (\\mathbb {E}\\sup _{j\\in \\{0, \\ldots , N_k\\}} \\Vert U(t_j) - U^j\\Vert _X^p\\Big )^{1/p} \\rightarrow 0\\quad (k \\rightarrow 0), \\end{aligned}$$\\end{document}where p∈[2,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\in [2,\\infty )$$\\end{document}, U is the mild solution, Uj\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U^j$$\\end{document} is obtained from a time discretisation scheme, k is the step size, and Nk=T/k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_k = T/k$$\\end{document} for final time T>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T>0$$\\end{document}. This generalises previous results to a larger class of admissible nonlinearities and noise, as well as rough initial data from the Hilbert space case to more general spaces. We present a proof based on a regularisation argument. Within this scope, we extend previous quantified convergence results for more regular nonlinearity and noise from Hilbert to 2-smooth Banach spaces. The uniform strong error cannot be estimated in terms of the simpler pointwise strong errorEk:=(supj∈{0,…,Nk}E‖U(tj)-Uj‖Xp)1/p,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \ extrm{E}_k {:}{=}\\bigg (\\sup _{j\\in \\{0,\\ldots ,N_k\\}}\\mathbb {E}\\Vert U(t_j) - U^{j}\\Vert _X^p\\bigg )^{1/p}, \\end{aligned}$$\\end{document}which most of the existing literature is concerned with. Our results are illustrated for a variant of the Schrödinger equation, for which previous convergence results were not applicable.

Well‐posedness and error analysis of wave equations with Markovian switching

Compared to traditional partial differential equation modeling methods, Markov switching models can accurately capture the abrupt changes or jumps that complex systems often experience in the real world. In this paper, we propose a novel wave equation model with Markovian switching to represent complex systems with state‐jumping phenomena better, and the well‐posedness of the model is proved. In addition, a numerical method with non‐uniform grids is also proposed for the proposed model to simulate the data in realistic situations, which is based on the use of finite element discretization in space and central difference discretization in time. Finally, we conduct several experiments to analyze the errors and stability of the proposed model and the traditional model. The results show that the Markov switching model proposed in this paper has a smaller error than the traditional models while ensuring stability and can more accurately simulate the state jump phenomena of real‐world systems.

P-adic Brownian motion is a scaling limit

A p-adic Brownian motion is a continuous time stochastic process in a p-adic state space that has a Vladimirov operator as its infinitesimal generator. The current work shows that any such process is the scaling limit of a discrete time random walk on a discrete group. Earlier work required the exponent of the Vladimirov operator to be in (1,∞) , and the convergence was the weak convergence of probability measures on the Skorohod space of paths on a compact time interval. The current approach simplifies the earlier approach, allows for any positive exponent, eliminates the restriction to compact time intervals, and establishes some moment estimates for the discrete time processes that are of independent interest.

Discrete time crystal phase of higher dimensional integrable models

This paper investigates the possibility of generating Floquet-time crystals in higher dimensions (d≥2) through the time-periodic driving of integrable free-fermionic models. The realization leads to rigid time-crystal phases that are ideally resistant to thermalization and decoherence. By utilizing spin-orbit coupling, we are able to realize a robust time-crystal phase that can be detected using novel techniques. Moreover, we discuss the significance of studying the highly persistent subharmonic responses and their implementation in a Kitaev spin liquid, which contributes to our understanding of time translational symmetry breaking and its practical implications.

Multicriteria optimization of the composition, thermodynamic and strength properties of fly-ash as an additive in metakaolin-based geopolymer composites

This paper presents the construction of intelligent systems for selecting the optimum concentration of geopolymer matrix components based on ranking optimality criteria. A peculiarity of the methodology is replacing discrete time intervals with a sequence of states. Markov chains represent a synthetic property accumulating heterogeneous factors. The computational basis for the calculations was the digitization of experimental data on the strength properties of fly ashes collected from thermal power plants in the Czech Republic and used as additives in geopolymers. A database and a conceptual model of priority ranking have been developed, that are suitable for determining the structure of relations of the main factors. Computational results are presented by studying geopolymer matrix structure formation kinetics under changing component concentrations in real- time. Multicriteria optimization results for fly-ash as an additive on metakaolin-based geopolymer composites show that the optimal composition of the geopolymer matrix within the selected variation range includes 100 g metakaolin, 90 g potassium activator, 8 g silica fume, 2 g basalt fibers and 50 g fly ash by ratio weight. This ratio gives the best mechanical, thermal, and technological properties.

Statistical design and analysis of controlled human malaria infection trials

BackgroundMalaria is a potentially life-threatening disease caused by Plasmodium protozoa transmitted by infected Anopheles mosquitoes. Controlled human malaria infection (CHMI) trials are used to assess the efficacy of interventions for malaria elimination. The operating characteristics of statistical methods for assessing the ability of interventions to protect individuals from malaria is uncertain in small CHMI studies. This paper presents simulation studies comparing the performance of a variety of statistical methods for assessing efficacy of intervention in CHMI trials.MethodsTwo types of CHMI designs were investigated: the commonly used single high-dose design (SHD) and the repeated low-dose design (RLD), motivated by simian immunodeficiency virus (SIV) challenge studies. In the context of SHD, the primary efficacy endpoint is typically time to infection. Using a continuous time survival model, five statistical tests for assessing the extent to which an intervention confers partial or full protection under single dose CHMI designs were evaluated. For RLD, the primary efficacy endpoint is typically the binary infection status after a specific number of challenges. A discrete time survival model was used to study the characteristics of RLD versus SHD challenge studies.ResultsIn a SHD study with the continuous time survival model, log-rank test and t-test are the most powerful and provide more interpretable results than Wilcoxon rank-sum tests and Lachenbruch tests, while the likelihood ratio test is uniformly most powerful but requires knowledge of the underlying probability model. In the discrete time survival model setting, SHDs are more powerful for assessing the efficacy of an intervention to prevent infection than RLDs. However, additional information can be inferred from RLD challenge designs, particularly using a likelihood ratio test.ConclusionsDifferent statistical methods can be used to analyze controlled human malaria infection (CHMI) experiments, and the choice of method depends on the specific characteristics of the experiment, such as the sample size allocation between the control and intervention groups, and the nature of the intervention. The simulation results provide guidance for the trade off in statistical power when choosing between different statistical methods and study designs.

Examining the performance of Laguerre-based and nonlinear predictive control models for wheeled mobile robots when facing obstacles

This focuses on the problem of controlling the formation of a team of rotating robots that are not homonomic and are not able to interact in a barrier environment. The virtual structure formation control strategy is obtained here to calculate each path of the robot separately while the reference path of the virtual center of the formation is generated by artificial potential fields. In this presentation, we will introduce a new control algorithm that utilizes Model predictive control and nonlinear system dynamics. Three traditional model prediction control controllers (MPC): logger-based MPC, nonlinear MPC, and traditional MPC are used to apply the control algorithm for the nonlinear system. Model predictive control is used to calculate the torques required to track the path by using the dynamic model of the Rotary moving robot. A proposed control rule is the most effective approach to solving the formation and tracking problems. Model Predictive Control will be introduced through the introduction of basic ideas and terms. Traditional MPC, logger-based MPC, and non-linear MPC are among the three MPC methods to be offered. The discrete time display of the power plant under control is taken into account by developing MPC methods for linear or nonlinear systems for practical implementation. The model predictor controller optimizes a cost function to calculate an optimal control sequence at any sampling point. While ignoring the rest, the system only achieves the first control action of this sequence. During subsequent sampling, the optimization problem is resolvable with the use of updated process measurements and a different horizon. The cost function formula is determined by the controller’s purpose. An error function often plays a role in determining the objective by revealing the difference between the desired and actual response. The shape of the formation can be preserved by the law of control by utilizing information from other robots, which can be assured through a new cost function. In the current conditions of disruption, the proposed controllers can also achieve the goal of tracking the path and maintaining formation, according to the results.

Second-order, Positive, and Unconditional Energy Dissipative Scheme for Modified Poisson–Nernst–Planck Equations

First-order energy dissipative schemes in time are available in literature for the Poisson–Nernst–Planck (PNP) equations, but second-order ones are still in lack. This work proposes novel second-order discretization in time and finite volume discretization in space for modified PNP equations that incorporate effects arising from ionic steric interactions and dielectric inhomogeneity. A multislope method on unstructured meshes is proposed to reconstruct positive, accurate approximations of mobilities on faces of control volumes. Numerical analysis proves that the proposed numerical schemes are able to unconditionally ensure the existence of positive numerical solutions, original energy dissipation, mass conservation, and preservation of steady states at discrete level. Extensive numerical simulations are conducted to demonstrate numerical accuracy and performance in preserving properties of physical significance. Applications in ion permeation through a 3D nanopore show that the modified PNP model, equipped with the proposed schemes, has promising applications in the investigation of ion selectivity and rectification. The proposed second-order discretization can be extended to design temporal second-order schemes with original energy dissipation for a type of gradient flow problems with entropy.

Tunable subdiffusion in the Caputo fractional standard map

The Caputo fractional standard map (C-fSM) is a two-dimensional nonlinear map with memory given in action–angle variables (I,θ). It is parameterized by K and α∈(1,2] which control the strength of nonlinearity and the fractional order of the Caputo derivative, respectively. In this work we perform a scaling study of the average squared action I2 along strongly chaotic orbits, i.e. when K≫1. We numerically prove that I2∝nμ with 0≤μ(α)≤1, for large enough discrete times n. That is, we demonstrate that the C-fSM displays subdiffusion for 1<α<2. Specifically, we show that diffusion is suppressed for α→1 since μ(1)=0, while standard diffusion is recovered for α=2 where μ(2)=1. We describe our numerical results with a phenomenological analytical estimation.

The [formula omitted]-dimensional bootstrap percolation models with axial neighbourhoods

Fix positive integers d,r and a1≤a2≤⋯≤ad. For large L, each site of {1,…,L}d⊂Zd can be at state 0 or 1 (infected), and its neighbourhood consists of the ak nearest neighbours in the ±ek-directions for each k∈{1,2,…,d}. The state will evolve in discrete time as follows: At time 0, vertices are independently 1 with some probability p. We infect any vertex v∈{1,…,L}d at state 0 already having r infected neighbours, and infected sites remain infected forever.In this paper we study the critical length for percolation, defined by Lc(Nra1,…,ad,p)=min{L∈N:Pp({1,…,L}dis eventually infected)≥1/2}. We determine the (d−1)-times iterated logarithm of Lc(Nra1,…,ad,p) up to a constant factor, for all d-tuples (a1,…,ad) and all r∈{a2+⋯+ad+1,…,a1+a2+⋯+ad}.We conjecture that we can reduce the problem of determining this threshold for all d≥3 and all r∈{ad+1,…,a1+a2+⋯+ad}, to that of determining the threshold for all d≥3 and r∈{ad+1,…,ad−1+ad} only.

Dynamic rebalancing optimization for bike-sharing systems: A modeling framework and empirical comparison

Station-based Bike-sharing systems have been implemented in multiple major cities, offering a low-cost and environmentally friendly transportation alternative. As a remedy to unbalanced stations, operators typically rebalance bikes by trucks. The resulting dynamic planning has received significant attention from the Operations Research community. Due to its modeling flexibility, mixed-integer programming remains a popular choice. However, the complex planning problem requires significant simplifications to obtain a computationally tractable model. As a result, existing models have used a large variety of modeling assumptions and techniques regarding decision variables and constraints. Unfortunately, the impact of such assumptions on the solutions’ performance in practice remains generally unexplored.In this paper, we first systematically survey the literature on rebalancing problems and their modeling assumptions. We then propose a general mixed-integer programming model for multi-period rebalancing problems that can be easily adapted to different assumptions, including trip modeling, time discretization, trip distribution, and event sequences. We develop an instance generator to synthesize realistic station networks and customer trips, as well as a realistic fine-grained simulator to evaluate the operational performance of rebalancing strategies. Finally, extensive numerical experiments are carried out, both on the synthetic and real-world data, to analyze the effectiveness of various modeling assumptions and techniques. Based on our results, we identify the assumptions that empirically provide the most effective rebalancing strategies in practice. Specifically, a set of specific trip distribution constraints and event sequences ignored in the previous literature seem to provide particularly good results.

Lifetime prevalence, risk, and treatment of mood and anxiety disorders in Qatar's national mental health study.

To estimate lifetime prevalence, risk, and treatment for mental disorders and their correlates in Qatar's general population for the first time. We conducted a national phone survey of 5,195 Qatari and Arab residents in Qatar (2019-2022) using the Composite International Diagnostic Interview Version 3.3 and estimated lifetime mood and anxiety defined diagnoses. Survival-based discrete time models, lifetime morbid risk, and treatment projections were estimated. Lifetime prevalence of any disorder was 28.0% and was associated with younger cohorts, females, and migrants, but lower formal education. Treatment contact in the year of disorder onset were 13.5%. The median delay in receiving treatment was 5years (IQR=2-13). Lifetime treatment among those with a lifetime disorder were 59.9% for non-healthcare and 63.5% for healthcare; it was 68.1% for any anxiety and 80.1% for any mood disorder after 50years of onset.Younger cohorts and later age of onset were significantly predictors of treatment. Lifetime prevalence of mental disorders in Qatar is comparable to other countries. Treatment is significantly delayed and delivered largely in non-healthcare sectors thus the need for increased literacy of mental illness to reduce stigma and improve earlier help-seeking in healthcare settings.