Discrete-time Equation Research Articles

Exact Discretization of an Economic Accelerator and Multiplier with Memory

Fractional differential equations of macroeconomics, which allow us to take into account power-law memory effects, are considered. We describe an economic accelerator and multiplier with fading memory in the framework of discrete-time and continuous-time approaches. A relationship of the continuous- and discrete-time fractional-order equations is considered. We propose equations of the accelerator and multiplier for economic processes with power-law memory. Exact discrete analogs of these equations are suggested by using the exact fractional differences of integer and non-integer orders. Exact correspondence between the equations with finite differences and differential equations lies not so much in the limiting condition, when the step of discretization tends to zero, as in the fact that mathematical operations, which are used in these equations, satisfy in many cases the same mathematical laws.

Investigation of to the Effect of Bedrock Stiffness on Seismic Behaviour of Roller Compacted Concrete Dam

In this research, the effect of bedrock stiffness on seismic performance of roller compacted concrete (RCC) dam is evaluated using probabilistic analysis. Due to the geometry and behavior of RCC dams, a two-dimensional modeling was selected for system. Ansys software is used for modeling and analysis of dam-reservoir- foundation system. Newmark implicit time integration scheme is developed to solve the time-discretized equations which are an unconditionally stable method. The Watana dam, due to San Fernando earthquake has been selected as a case study. In order to propagate the parametric sensitivity to the seismic performance of the system, Monte Carlo simulation with Latin hypercube sampling (LHS) method is used as a probabilistic method and uncertainty analysis. The sensitivity of responses under seismic loading is reliably examined utilizing different values of ratio of bedrock stiffness to body concrete stiffness as random inputs. Consider to obtained results, it is revealed that the bedrock stiffness how can effect on seismic behavior of concrete gravity dams due to earthquake. Regarding the safety of dams due to compressive stresses, various ways have been assessed to investigate the induced tensile stress in the heel and the results have been investigated. Finally, appropriate range of the ratio of bedrock stiffness to concrete stiffness of dam body is presented to assess the safety design.

Abstract 4542: A mathematical model for immuno-chemotherapy

Abstract Chemotherapy and immunotherapy have sometimes been considered two mutually exclusive approaches. On the one hand, chemotherapy has long been considered as a modestly effective treatment (outside pediatric oncology), with a short-lived effect caused by drug resistance, a poor tolerance profile and a detrimental effect on the immune system. On the other hand, the oncologist continues using chemotherapies for many reasons: if the control of a large tumor mass has to be obtained urgently irrespective of the side effects on the immune system, if immunotherapy has failed or has not been validated as a first line of treatment, when no targeted therapy can apply, etc. However, the issue of modifying the long-validated chemotherapy protocols to make them a good fit for an adjuvant (or concomitant) immunotherapy has not been addressed yet. The practice of gluing together an old Maximum Tolerated Dose (MTD) protocol with a new immunotherapy protocol is often the standard practice, even though some re-design would be needed to mitigate the massive and prolonged immunosuppression caused by chemotherapy protocols. This lack of flexibility in the chemotherapy regimen could seriously jeopardize the effects of adjuvant or concomitant immunotherapy. Fortunately, chemotherapy and immunotherapy may not be incompatible per se: there are accumulating evidence that low-dose chemotherapies with little or no break-period (the so-called metronomic regimen) might have an immunological mode of action (in addition to the classical explanation of its anti-angiogenic effect and cytotoxicity). Furthermore, even with MTD protocols, experienced oncologists can improve the clinical outcome with the rationale use of pro-immunogenic drugs. Nevertheless, the best way to combine chemotherapy with immunotherapy is unknown and to find it in a trial and error manner is unrealistic (high number of possibilities). This is why we present a pharmacodynamic model of chemotherapy in combination with immune checkpoint blockers, in order to provide a practical tool for the design and the management of clinical trials. Based on simple equations, this model does not predict a priori the best protocol, but helps the analysis of clinical data and the subsequent identification of optimal combinations. The mathematical model is based on discrete-time equations of the tumor mass dynamics, under the retro-control of a specific immune response partly antagonized by chemotherapy, but also stimulated by the release of tumor antigens. An original dynamic model of lymphopenia is introduced to describe the immunosuppressive consequences of chemotherapy. The effects of immune checkpoint blockers are described in coherence with a previous publication in Cancer Research by the same authors. Acquired drug resistance and tumor angiogenesis are also modeled. Examples of practical applications are provided by showing how the model can describe various pre-clinical and clinical data. Citation Format: Raphael Serre, Dominique Barbolosi. A mathematical model for immuno-chemotherapy [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2017; 2017 Apr 1-5; Washington, DC. Philadelphia (PA): AACR; Cancer Res 2017;77(13 Suppl):Abstract nr 4542. doi:10.1158/1538-7445.AM2017-4542

A note on the necessity of filtering mechanism for polynomial observability of time-discrete wave equation

The problem of uniform polynomial observability was recently analyzed. It is shown that, when the continuous model is uniformly polynomially observable, it is sufficient to filter initial data to derive uniform polynomial observability inequalities for suitable time-discretization schemes. In this note, we prove that a filtering mechanism of high frequency modes is necessary to obtain uniform polynomial observability.
 More precisely, we give a counterexample which proves that this latter fails without filtering the initial data for time semi-discrete approximations of the wave equation.

Relevant parameters in models of cell division control.

A recent burst of dynamic single-cell data makes it possible to characterize the stochastic dynamics of cell division control in bacteria. Different models were used to propose specific mechanisms, but the links between them are poorly explored. The lack of comparative studies makes it difficult to appreciate how well any particular mechanism is supported by the data. Here, we describe a simple and generic framework in which two common formalisms can be used interchangeably: (i) a continuous-time division process described by a hazard function and (ii) a discrete-time equation describing cell size across generations (where the unit of time is a cell cycle). In our framework, this second process is a discrete-time Langevin equation with simple physical analogues. By perturbative expansion around the mean initial size (or interdivision time), we show how this framework describes a wide range of division control mechanisms, including combinations of time and size control, as well as the constant added size mechanism recently found to capture several aspects of the cell division behavior of different bacteria. As we show by analytical estimates and numerical simulations, the available data are described precisely by the first-order approximation of this expansion, i.e., by a "linear response" regime for the correction of size fluctuations. Hence, a single dimensionless parameter defines the strength and action of the division control against cell-to-cell variability (quantified by a single "noise" parameter). However, the same strength of linear response may emerge from several mechanisms, which are distinguished only by higher-order terms in the perturbative expansion. Our analytical estimate of the sample size needed to distinguish between second-order effects shows that this value is close to but larger than the values of the current datasets. These results provide a unified framework for future studies and clarify the relevant parameters at play in the control of cell division.

Direct Stator Flux Vector Control Strategy for IPMSM using a Full-order State Observer

A direct stator flux vector control scheme in discrete-time domain is proposed in this paper for the interior permanent magnet synchronous motor (IPMSM) drive to remove the proportional-integral (PI) controller from the direct torque control (DTC) scheme applied to IPMSM and to obtain faster dynamic response and lower torque ripple output. The output of speed outer loop is used as the desired torque angle instead of the desired torque in the proposed scheme. The desired stator flux vector in dq coordinate is calculated with a given amplitude. The state-space equations in discrete-time for IPMSM are established, the actual stator flux vector is estimated in deadbeat manner by a full-order state observer, and then the closed-loop control is achieved by the pole placement. The stator flux error vector is utilized to calculate the reference stator voltage vector. Extracting the angle position and amplitude from the estimated stator flux vector and estimating the output torque are eliminated for the direct feedback control of the stator flux vector. The proposed scheme is comparatively investigated with a PI-SVM DTC scheme by experiment results. Experimental results show the feasibility and advantages of the proposed control scheme.

Метод багатокритеріального розв’язання конфліктної ситуації між двома повітряними суднами у тривимірному просторі на основі динамічного програмування

Мета: Глобальні тенденції зростання інтенсивності повітряного руху обумовлюють збільшення кількості конфліктних ситуацій між повітряними судами. Актуальною проблемою є розробка нових методів розв’язання конфліктних ситуацій, які повинні забезпечувати синтез безконфліктних траєкторій у тривимірному просторі у відповідності до різних критеріїв ефективності польотів. Методи: Розроблено метод багатокритеріального розв’язання конфліктної ситуації між двома повітряними суднами із застосуванням маневрування зміною курсу, швидкості та висоти польоту. Описаний метод на основі динамічного програмування забезпечує синтез оптимальної безконфліктної траєкторії відповідно до критеріїв регулярності, економічності польотів та складності маневрування. Наведено рівняння багатокритеріального динамічного програмування для визначення множини Парето-оптимальних оцінок безконфліктних траєкторій у неперервній та дискретній формі. Синтез Парето-оптимальних безконфліктних траєкторій здійснюється із застосуванням прямої процедури дискретного багатокритеріального динамічного програмування. Моделювання траєкторій польоту виконується із використанням спеціальної моделі керованого руху повітряного судна. Вибір оптимальної безконфліктної траєкторії з множини Парето-оптимальних виконується із застосуванням згортки критеріїв оптимальності. В рамках методу визначено наступні процедури: прогнозування порушень мінімумів ешелонування; дискретизації станів та керувань, інтерполяції оцінок ефективності траєкторій за встановленими критеріями оптимальності. Результати: Дослідження запропонованого методу виконано шляхом комп’ютерного моделювання, результати якого показали, що розрахована оптимальна безконфліктна траєкторія забезпечує усунення конфліктної ситуації та відповідає встановленим критеріям оптимальності. Обговорення: Основними перевагами методу є: застосування маневрів по зміні курсу, швидкості та висоти польоту для усунення конфлікту; багатокритеріальна оптимізація безконфліктних траєкторій; застосування динамічного програмування, що підвищує обчислювальну ефективність. Запропонований метод може бути використаний при розробці засобів розв’язання конфліктних ситуацій для автоматизованих систем управління повітряним рухом.

Explicit iterative algorithms for solving coupled discrete‐time Lyapunov matrix equations

In this study, the authors aim to study explicit iterative algorithms for solving coupled discrete-time Lyapunov matrix equations. First, an explicit iterative algorithm based on fixed point theory of dynamic equations is presented via adding a tuning parameter. Second, a necessary and sufficient condition is provided for the convergence of the proposed algorithm. Moreover, the optimal value of the tuning parameter is derived for the fastest convergence of the algorithm. Third, by using the latest updated information, a modified version of the presented explicit iterative algorithm is also established with a necessary and sufficient condition being provided to guarantee the convergence of the modified algorithm. Finally, a numerical example is given to demonstrate the effectiveness of the proposed algorithms.

Approximation of a physiologically structured population model with seasonal reproduction by a stage-structured biomass model

Seasonal reproduction causes, due to the periodic inflow of young small individuals in the population, seasonal fluctuations in population size distributions. Seasonal reproduction furthermore implies that the energetic body condition of reproducing individuals varies over time. Through these mechanisms, seasonal reproduction likely affects population and community dynamics. While seasonal reproduction is often incorporated in population models using discrete time equations, these are not suitable for size-structured populations in which individuals grow continuously between reproductive events. Size-structured population models that consider seasonal reproduction, an explicit growing season and individual-level energetic processes exist in the form of physiologically structured population models. However, modeling large species ensembles with these models is virtually impossible. In this study, we therefore develop a simpler model framework by approximating a cohort-based size-structured population model with seasonal reproduction to a stage-structured biomass model of four ODEs. The model translates individual-level assumptions about food ingestion, bioenergetics, growth, investment in reproduction, storage of reproductive energy, and seasonal reproduction in stage-based processes at the population level. Numerical analysis of the two models shows similar values for the average biomass of juveniles, adults, and resource unless large-amplitude cycles with a single cohort dominating the population occur. The model framework can be extended by adding species or multiple juvenile and/or adult stages. This opens up possibilities to investigate population dynamics of interacting species while incorporating ontogenetic development and complex life histories in combination with seasonal reproduction.

Novel Parametric Circuit Modeling for Li-Ion Batteries

Because of their simplicity and dynamic response, current pulse series are often used to extract parameters for equivalent electrical circuit modeling of Li-ion batteries. These models are then applied for performance simulation, state estimation, and thermal analysis in electric vehicles. However, these methods have two problems: The assumption of linear dependence of the matrix columns and negative parameters estimated from discrete-time equations and least-squares methods. In this paper, continuous-time equations are exploited to construct a linearly independent data matrix and parameterize the circuit model by the combination of non-negative least squares and genetic algorithm, which constrains the model parameters to be positive. Trigonometric functions are then developed to fit the parameter curves. The developed model parameterization methodology was applied and assessed by a standard driving cycle.

Risk Measures and Financial Innovation with Backward Stochastic Difference Equations

Economic agents are exposed to the uncertain outcomes of future events. By enabling the exchange of securities, financial markets allow agents to reallocate their exposures in more efficient and mutually convenient arrangements to reduce perceived risks. The complexity and changing nature of the world in which agents operate make all markets incomplete, which means that there is scope for further risk reduction by the introduction of new market securities -- financial innovation -- to cover previously un-priced risks. This paper provides a convenient mathematical framework to solve the problem of dynamic equilibrium pricing and optimal design of new financial securities. Our mathematical tool for studying trading market equilibria is a novel theory of backward stochastic difference equations (BSΔEs) in discrete time, which we develop in analogy to the currently incomplete theory of backward stochastic differential equations (BSDEs) in continuous time. The new tool is used to define a family of dynamic risk measures which is used to solve the problems of optimal trading for a single economic agent, equilibrium pricing of new securities amongst multiple agents, and the optimal design of new securities. Our approach allows the characterization of unique dynamic trading equilibria, optimal instrument design, inter-agent risk transfer and the implications for real-life financial market structures which elude the continuous time BSDE theory. A simple intuitive numerical example is presented to illustrate the concepts introduced.

An Ensemble Kushner-Stratonovich-Poisson Filter for Recursive Estimation in Nonlinear Dynamical Systems

We propose a Monte Carlo filter for recursive estimation of diffusive processes that modulate the instantaneous rates of Poisson measurements. A key aspect is the additive update, through a gain-like correction term, empirically approximated from the innovation integral in the time-discretized Kushner-Stratonovich equation. The additive filter-update scheme eliminates the problem of particle collapse encountered in many conventional particle filters. Through a few numerical demonstrations, the versatility of the proposed filter is brought forth.

SIBC incorporated in conformal FDTD to efficiently simulate the thin conductive layer of periodic structure

In this study, surface impedance boundary condition (SIBC) incorporated in conformal finite-difference time-domain (CFDTD) to efficiently simulate the thin conductive layer of periodic structure is proposed. The method is based on the vector fitting (VF) technique to approximate the frequency-dependent surface impedance of the thin conductive layer by a series of partial fractions in terms of real or complex conjugate pole-residue pairs. A discrete-time equation of SIBC is then derived and incorporated in CFDTD update equations by Laplace transform and time central difference methods. The proposed method takes the following advantages, (i) the time step is no longer bounded by the fine resolution of the thin layer; (ii) The complicated recursive convolution approach is avoided, and the approximation of the frequency-dependent surface impedance of the thin conductive layer can be easy to handle with the VF method. This method is numerically verified for the time- and frequency-dependent scattering characteristics of several periodic structures behaved under normal and oblique incident waves.

A NEW VERSION OF THE SMITH METHOD FOR SOLVING SYLVESTER EQUATION AND DISCRETE-TIME SYLVESTER EQUATION

Recently, Xue etc. (37) discussed the Smith method for solving Sylvester equation AX + XB = C, where one of the matrices A and B is at least a nonsingular M-matrix and the other is an (singular or nonsingular) M-matrix. Furthermore, in order to nd the minimal non-negative solution of a certain class of non-symmetric algebraic Riccati equations, Gao and Bai (17) considered a doubling iteration scheme to inexactly solve the Sylvester equations. This paper discusses the iterative error of the standard Smith method used in (17) and presents the prior estimations of the accurate solution X for the Sylvester equation. Furthermore, we give a new version of the Smith method for solving discrete-time Sylvester equation or Stein equation AXB + X = C, while the new version of the Smith method can also be used to solve Sylvester equation AX + XB = C, where both A and B are positive denite. We also study the convergence rate of the new Smith method. At last, numerical examples are given to illustrate the eectiveness of our methods.

Existence and Boundedness of Solutions for Nonlinear Volterra Difference Equations in Banach Spaces

We consider a class of nonlinear discrete-time Volterra equations in Banach spaces. Estimates for the norm of operator-valued functions and the resolvents of quasi-nilpotent operators are used to find sufficient conditions that all solutions of such equations are elements of an appropriate Banach space. These estimates give us explicit boundedness conditions. The boundedness of solutions to Volterra equations with infinite delay is also investigated.

Emergy assessment of the benefits of closed-loop recycling accounting for material losses

Emergy analysis is applied to closed-loop recycling processes. The emergy balance is written under the form of a discrete-time equation following a comparable approach to the Lagrangian particle tracking in fluid mechanics. Two material losses were taken into consideration, the first one at the level of treatment (collection, dismantling, recycling etc.) and the second one at the transformation level. As expected, the emergy of a product increases with the number of cycles (after the transformation processes). To assess the environmental stress at each recycling stage, relative emergy of the product ΔEp(n) was used after discussing the limitation of others solutions like the environmental loading ratio (ELR) or the recycle benefit ratio (RBR) which are not suitable for use in the cases of multiple recycling analysis. The values obtained showed that the ΔEp(n) has an increasing trend for successive recycles. Additionally, this approach allowed the study of different cases of theoretical closed-loop recycling and an application on aluminum. The results indicated that the several cycles in the first stage of recycling have a significant impact, and an asymptotic behavior of ΔEp(n) can be noted. In any cases recycling remains a better option compared to raw material use to compensate the material losses. Furthermore, the presented equations illustrate the impact of the material recycle rate and the percentage of losses during the both recycling and transformation processes.

Optimal control of integrodifference equations in a pest-pathogen system

We develop the theory of optimal control for a system of integrodifference equations modelling a pest-pathogen system. Integrodifference equations incorporate continuous space into a system of discrete time equations. We design an objective functional to minimize the damaged cost generated by an invasive species and the cost of controlling the population with a pathogen. Existence, characterization, and uniqueness results for the optimal control and corresponding states have been completed. We use a forward-backward sweep numerical method to implement our optimization which produces spatio-temporal control strategies for the gypsy moth case study.

Constant pressure and temperature discrete-time Langevin molecular dynamics.

We present a new and improved method for simultaneous control of temperature and pressure in molecular dynamics simulations with periodic boundary conditions. The thermostat-barostat equations are built on our previously developed stochastic thermostat, which has been shown to provide correct statistical configurational sampling for any time step that yields stable trajectories. Here, we extend the method and develop a set of discrete-time equations of motion for both particle dynamics and system volume in order to seek pressure control that is insensitive to the choice of the numerical time step. The resulting method is simple, practical, and efficient. The method is demonstrated through direct numerical simulations of two characteristic model systems-a one-dimensional particle chain for which exact statistical results can be obtained and used as benchmarks, and a three-dimensional system of Lennard-Jones interacting particles simulated in both solid and liquid phases. The results, which are compared against the method of Kolb and Dünweg [J. Chem. Phys. 111, 4453 (1999)], show that the new method behaves according to the objective, namely that acquired statistical averages and fluctuations of configurational measures are accurate and robust against the chosen time step applied to the simulation.

Design of a simplified adaptive fuzzy backstepping control for uncertain discrete-time nonlinear systems

This paper presents a simplified adaptive fuzzy backstepping control for uncertain discrete-time nonlinear systems. It is assumed that the systems are described by a discrete-time equation with nonlinear uncertainties to be viewed as the modelling errors and the unknown external disturbances, and the states are observed with measurement noises. To design the simplified adaptive fuzzy backstepping control, the modelling errors are approximated by using the fuzzy inference approach based on the extended single-input rule modules, and the estimates for the unmeasurable states and the adjustable parameters are derived by using the weighted and its simplified weighted least squares estimators. It is proved that the states are ultimately bounded, and the estimation errors remain in the vicinity of zero. The effectiveness of the proposed approach is indicated through the simulation experiment of a simple numerical system.

Asymptotic description of stochastic neural networks. I. Existence of a large deviation principle

We study the asymptotic law of a network of interacting neurons when the number of neurons becomes infinite. The dynamics of the neurons is described by a set of stochastic differential equations in discrete time. The neurons interact through the synaptic weights that are Gaussian correlated random variables. We describe the asymptotic law of the network when the number of neurons goes to infinity. Unlike previous works which made the biologically unrealistic assumption that the weights were i.i.d. random variables, we assume that they are correlated. We introduce the process-level empirical measure of the trajectories of the solutions into the equations of the finite network of neurons and the averaged law (with respect to the synaptic weights) of the trajectories of the solutions into the equations of the network of neurons. The result (Theorem 3.1 below) is that the image law through the empirical measure satisfies a large deviation principle with a good rate function. We provide an analytical expression of this rate function in terms of the spectral representation of certain Gaussian processes.