Fading Memory Research Articles

Dynamics of Ebola Disease in the Framework of Different Fractional Derivatives.

In recent years the world has witnessed the arrival of deadly infectious diseases that have taken many lives across the globe. To fight back these diseases or control their spread, mankind relies on modeling and medicine to control, cure, and predict the behavior of such problems. In the case of Ebola, we observe spread that follows a fading memory process and also shows crossover behavior. Therefore, to capture this kind of spread one needs to use differential operators that posses crossover properties and fading memory. We analyze the Ebola disease model by considering three differential operators, that is the Caputo, Caputo–Fabrizio, and the Atangana–Baleanu operators. We present brief detail and some mathematical analysis for each operator applied to the Ebola model. We present a numerical approach for the solution of each operator. Further, numerical results for each operator with various values of the fractional order parameter are presented. A comparison of the suggested operators on the Ebola disease model in the form of graphics is presented. We show that by decreasing the value of the fractional order parameter , the number of individuals infected by Ebola decreases efficiently and conclude that for disease elimination, the Atangana–Baleanu operator is more useful than the other two.

Reservoir Computing Universality With Stochastic Inputs.

The universal approximation properties with respect to L p -type criteria of three important families of reservoir computers with stochastic discrete-time semi-infinite inputs are shown. First, it is proven that linear reservoir systems with either polynomial or neural network readout maps are universal. More importantly, it is proven that the same property holds for two families with linear readouts, namely, trigonometric state-affine systems and echo state networks, which are the most widely used reservoir systems in applications. The linearity in the readouts is a key feature in supervised machine learning applications. It guarantees that these systems can be used in high-dimensional situations and in the presence of large data sets. The L p criteria used in this paper allow the formulation of universality results that do not necessarily impose almost sure uniform boundedness in the inputs or the fading memory property in the filter that needs to be approximated.

A new fractional analysis on the polluted lakes system

Abstract In this paper, we use Atangana–Baleanu derivative which is defined with the Mittag–Leffler function and has all the properties of a classical fractional derivative for solving the system of fractional differential equations. The classical model of polluted lakes system is modified by using the concept of fractional differentiation with nonsingular and nonlocal fading memory. The new numerical scheme recommended by Toufik and Atangana is used to analyze the modified model of polluted lakes system. Some numerical illustrations are presented to show the effect of the new fractional differentiation.

Inclusion of fading memory to Banister model of changes in physical condition

We introduced the fading memory effect to the model portraying the prediction in physical condition. The classical model is known as the Banister model. We presented the existence and uniqueness conditions of the exact solutions of this model using three different memory including the bad memory induces by the power law and the good memories induced by exponential decay law and the Mittag-Leffler law. We derived the exact solutions using the Laplace transform for the non-delay version.

Dynamic Keynesian Model of Economic Growth with Memory and Lag

A mathematical model of economic growth with fading memory and continuous distribution of delay time is suggested. This model can be considered as a generalization of the standard Keynesian macroeconomic model. To take into account the memory and gamma-distributed lag we use the Abel-type integral and integro-differential operators with the confluent hypergeometric Kummer function in the kernel. These operators allow us to propose an economic accelerator, in which the memory and lag are taken into account. The fractional differential equation, which describes the dynamics of national income in this generalized model, is suggested. The solution of this fractional differential equation is obtained in the form of series of the confluent hypergeometric Kummer functions. The asymptotic behavior of national income, which is described by this solution, is considered.

Phillips model with exponentially distributed lag and power-law memory

In this paper, we propose two generalizations of the Phillips model of multiplier–accelerator by taking into account memory with power-law fading. In the first generalization we consider the model, where we replace the exponential weighting function by the power-law memory function. In this model we consider two power-law fading memories, one on the side of the accelerator (induced investment responding to changes in output with memory-fading parameter α) and the other on the supply side (output responding to demand with memory-fading parameter β). To describe power-law memory we use the fractional derivatives in the accelerator equation and the fractional integral in multiplier equation. The solution of the model fractional differential equation is suggested. In the second generalization of the Phillips model of multiplier–accelerator we consider the power-law memory in addition to the continuously (exponentially) distributed lag. Equation, which describes generalized Phillips model of multiplier–accelerator with distributed lag and power-law memory, is solved using Laplace method.

Reinforcement with Fading Memories

We study the effect of imperfect memory on decision making in the context of a stochastic sequential action-reward problem. An agent chooses a sequence of actions which generate discrete rewards at different rates. She is allowed to make new choices at rate $\beta$, while past rewards disappear from her memory at rate $\mu$. We focus on a family of decision rules where the agent makes a new choice by randomly selecting an action with a probability approximately proportional to the amount of past rewards associated with each action in her memory. We provide closed-form formulae for the agent's steady-state choice distribution in the regime where the memory span is large ($\mu \to 0$), and show that the agent's success critically depends on how quickly she updates her choices relative to the speed of memory decay. If $\beta \gg \mu$, the agent almost always chooses the best action, i.e., the one with the highest reward rate. Conversely, if $\beta \ll \mu$, the agent chooses an action with a probability roughly proportional to its reward rate.

Harrod–Domar Growth Model with Memory and Distributed Lag

In this paper, we propose a macroeconomic growth model, in which we take into account memory with power-law fading and gamma distributed lag. This model is a generalization of the standard Harrod–Domar growth model. Fractional differential equations of this generalized model with memory and lag are suggested. For these equations, we obtain solutions, which describe the macroeconomic growth of national income with fading memory and distributed time-delay. The asymptotic behavior of these solutions is described.

Verbal Working Memory Processes in Students With Mild and Borderline Intellectual Disabilities: Differential Developmental Trajectories for Rehearsal and Redintegration.

In verbal working memory, two processes serve to retain a fading memory trace: subvocal rehearsal and lexical redintegration. While recent studies on students with mild and borderline intellectual disabilities (MBID) have yielded mixed results on rehearsal, redintegration has not been researched in MBID, yet. Furthermore, most studies have used a group-matched design which, due to methodological constraints, can only distinguish between two different development patterns. Thus, we study both rehearsal and redintegration in students with MBID using developmental trajectories that have greater potential for identifying differential developmental patterns than traditional group-matching approaches. We investigate whether three aspects in working memory develop differently in students with MBID in comparison to typically developing students: (a) the general capacity of the phonological loop, and the effectiveness of (b) rehearsal, and (c) redintegration. We use three different developmental indicators to compare trajectories: chronological age, cognitive capacity, and vocabulary size. N = 210 students (87 students with MBID, 123 typically developing students) completed working memory span tasks with short and long (1- vs. 3-syllable) real words and pseudowords. The effect for word length (short vs. long) measures rehearsal, and the lexicality effect (real words vs. pseudowords) measures redintegration. Results show that developmental trajectories reveal an intercept difference but no slowed rate in rehearsal, and no impairment in redintegration. However, concerning the developmental relation between redintegration and vocabulary size, students with MBID reveal a differential pattern as redintegration appears higher for students with small vocabulary size, but unexpectedly decreases as vocabulary size increases. We conclude that students with MBID show a delayed onset in the development of capacity of the phonological loop and rehearsal and that they do not catch up in their development. Redintegration does not seem to be impaired in relation to age and cognitive capacity. However, the differential relation of redintegration with vocabulary size calls for further research. While impaired subvocal rehearsal appears to be connected to the developmental problems of students with MBID, lexical redintegration seems to be intact in relation to chronological age and cognitive capacity, making it a possible area of strength.

Discrete-Time Direct Model Reference Adaptive Control Application in a High-Precision Inertially Stabilized Platform

This paper studies the practical use of discrete-time direct adaptive control in a high-precision inertially stabilized platform's turbulence isolation system for the purpose of enhancing isolation performance. Under low-frequency and low-velocity environments, the velocity-stabilized loop of the platform shows severe nonlinear characteristic; therefore, its isolation performance is limited. In previous research, we constructed a nonlinear model of the velocity-stabilized loop by using an improved Stribeck friction model, and also designed a feed-forward compensation strategy. Both have obtained outstanding performance during practical experiments. However, errors still exist as disturbance of the unmodeled part and environmental change causes the system's parameter to vary. To solve this, a novel discrete-time direct model reference adaptive control based on nonlinear friction compensation is introduced to the original proportional-integral control system. An improved projection algorithm and a recursive least-square algorithm with fading memory are respectively used to design the adaptive law. By using a turbulence observer to provide a reference signal, both types of controls are applied to the carrier turbulence isolation system. Results of the practical experiments prove that model reference adaptive control can further enhance the system's isolation ability.

What happens after consumption Role of autobiographical memory experience, nostalgia and word of mouth behaviour

The objective of the research work is to widen the possibilities of autobiographical memory experience in the service marketing literature. The focus is on the role of autobiographical memory experience in post consumption stage. By theory driven approach, the study explores the relationship with overall customer experience on autobiographical memory experience, nostalgia and word of mouth behaviour. Very little research has been done on autobiographical memory's impact on information processing. A study on 355 number of theme park consumers revealed that the core customer experience would influence the autobiographical memory experience, nostalgia and word of mouth behaviour. This work initially gives emphasis on the complexity of services, consumption experience stages and how the post consumption stage is being formed, further it explores on the role of behavioural intentions. Due to the complexity of services and the fading memory of unpleasant experiences managers and researchers must understand the influencers of autobiographical memory experience, since it is a very important factor, which impacts evangelising nature of customers.

Response functions in linear viscoelastic constitutive equations and related fractional operators

This study addresses the stress–strain relaxation functions of solid polymers in the framework of the linear viscoelasticity with aim to establish the adequate fractional operators emerging from the hereditary integrals. The analysis encompasses power-law and non-power-law materials, thus allowing to see the origins of application of the tools of the classical fractional calculus with singular memory kernels and the ideas leading towards fractional operators with non-singular (regular) kernels. A step ahead in modelling with hereditary integrals is the decomposition of non-power-law relaxation curves by Prony series, thus obtaining discrete relaxation kernels with a finite number of terms. This approach allows for seeing the physical background of the newly defined Caputo–Fabrizio time fractional derivative and demonstrates how other constitutive equations could be modified with non-singular fading memories. The non-power-law relaxation curves also allow for approximations by the Mittag–Leffler function of one parameter that leads reasonably into stress–strain hereditary integrals in terms of Atangana–Baleanu fractional derivative of Caputo sense. The main outcomes of the analysis done are the demonstrated distinguishes between the relaxation curve behaviours of different materials and are therefore the adequate modelling with suitable fractional operators.

Self-organization with memory

In this paper we propose a generalization of method of adiabatic elimination of fast relaxing variables, which is a special form of the slaving principle, to the case of processes with power-law memory. Simple models to describe the distinctive features of the self-organized processes with power-law memory are considered. We have proved that when the memory fading parameters change, the processes with memory can significantly change its rates (by many orders of magnitude) and change the characteristic times both upward and downward. Therefore the parameters of memory fading can be considered as control parameters that change of characteristic time of processes. Some principles to describe specific features of self-organization with memory are proposed.

Real-Time Weighted Data Fusion Algorithm for Temperature Detection Based on Small-Range Sensor Network.

Biological oxidation pretreatment, which can improve the yield of gold, is the main gold extraction technology for disposing refractory gold ore with high arsenic and sulfur. The temperature of the oxidation tank influences the oxidation efficiency between the ore pulp and bacteria, including the yield of gold. Therefore, measurement has consistently been an important subject for researchers. As an effective data processing method, data fusion has been used extensively in many fields of industrial production. However, the interference of equipment or external factors such as the diurnal temperature difference or powerful wind may constantly increase measurement errors and damage certain sensors, which may transmit error data. These problems can be solved by following a pretreatment process. First, we establish a heat transfer mechanism model. Second, we design a small-range sensor network for the pretreatment process and present a layered fusion structure of sharing sensors using a multi-connected fusion structure. Third, we introduce the idea of iterative operation in data processing. In addition, we use prior data for predicting state values twice in order to improve the effectiveness of extended Kalman filtering in one time step. This study also proposes multi-fading factors on the basis of a weighted fading memory index to adjust the prediction error covariance. Finally, the state estimation accuracy of each sensor can be used as a weighting principle for the predictive confidence of each sensor by adding a weighting factor. In this study, the performance of the proposed method is verified by simulation and compared with the traditional single-sensor method. Actual industrial measurement data are processed by the proposed method for the equipment experiment. The performance index of the simulation and the experiment shows that the proposed method has a higher global accuracy than the traditional single-sensor method. Simulation results show that the accuracy of the proposed method has a 55% improvement upon that of the traditional single-sensor method, on average. In the equipment experiment, the accuracy of the industrial measurement improved by 37% when using the proposed method.

Rapid time series prediction with a hardware-based reservoir computer.

Reservoir computing is a neural network approach for processing time-dependent signals that has seen rapid development in recent years. Physical implementations of the technique using optical reservoirs have demonstrated remarkable accuracy and processing speed at benchmark tasks. However, these approaches require an electronic output layer to maintain high performance, which limits their use in tasks such as time-series prediction, where the output is fed back into the reservoir. We present here a reservoir computing scheme that has rapid processing speed both by the reservoir and the output layer. The reservoir is realized by an autonomous, time-delay, Boolean network configured on a field-programmable gate array. We investigate the dynamical properties of the network and observe the fading memory property that is critical for successful reservoir computing. We demonstrate the utility of the technique by training a reservoir to learn the short- and long-term behavior of a chaotic system. We find accuracy comparable to state-of-the-art software approaches of a similar network size, but with a superior real-time prediction rate up to 160 MHz.

Dynamics of predator–prey system with fading memory

A predator–prey model with fading memory for general Holling type functional response is proposed. The fading memory term is used with the hypothesis that the predator’s growth rate at present depends on the recent past quantities of prey. The effects of predator harvesting is also considered in the model. The model is analysed theoretically as well as numerically. Two parameter bifurcation analysis are done and the existence of Hopf point bifurcation is identified. Both supercritical and subcritical Hopf bifurcations are obtained with the variation of system parameters. Maximum sustainable yield with respect to harvesting effort is also determined. It is to be observed that the system dynamics is very rich in presence of fading memory. The obtained results may be useful in the field of agriculture and fishery.

Nostalgia film tourism and its potential for destination development

ABSTRACTCinematic nostalgia can influence visitors in choosing destinations where particular films have been made or shot. Focusing on the case of Hong Kong this study investigated the incidence of nostalgia amongst Taiwanese filmgoers, and whether this generates feelings of familiarity that induce subsequent travel intentions. As an input to potentially wide-ranging destination experiences, it is found that nostalgia film tourism provides tourists with psychological benefits and constitutes a destination experience that is responsive to the ambient surroundings. The researchers identified five film nostalgia domains, namely: memories of backdrops, stories and movie stars, mimicking, envy, and culture and history. Memories of backdrops and mimicking were found to be the best predictors of perceived familiarity with the films’ origin territory and of future travel intentions. Given the continuing preference for “blockbusters” amongst contemporary filmgoers and the faddishness of audience preferences, there is some risk that fading memories may cause film destinations to lose their appeal. Film nostalgia nevertheless provides a potential medium to maintain the profile of destinations which have served as film locations.

Nonlinear dynamics of infectious diseases via information-induced vaccination and saturated treatment

The present study of a nonlinear compartmental SVIR model accounts for the dynamics of infectious diseases in population. In this model, the effect of information on vaccination coverage is quantified when medical resources are limited during the epidemic outbreak. Model analysis is performed and the global asymptotic stability of the disease free equilibrium is established when treatment is available for all infective individuals. However, the existence of multiple endemic equilibria is observed due to saturation in medical treatment and information-induced vaccination coverage. If the basic reproduction number is greater than unity, a unique endemic equilibrium for the model is obtained under some parametric conditions. Using the geometric approach, the global asymptotic stability of this endemic equilibrium is established under a parametric constraint. Further, when medical resources are limited, the existence Hopf bifurcation is shown analytically, which infers the oscillatory persistence of disease within the population. Occurrence of Hopf–Hopf bifurcation is also investigated numerically. At first Hopf bifurcation threshold, an endemic equilibrium loses its stability and produces oscillations, and further, it regains its stability at another Hopf bifurcation threshold when oscillations disappear. Hence, multiple stability switches are observed due to the saturated treatment and information-induced vaccination. The effect of the fading of memory also plays a crucial role and gives rise to complex dynamical behaviour and also affects the stability properties. Numerically, different types of backward bifurcation are also examined. Extensive numerical experimentations are also carried out to further explore the model system.

Chaos in a 5-D hyperchaotic system with four wings in the light of non-local and non-singular fractional derivatives

A new 5-D hyperchaotic system with four wings is studied in the light of the newly introduced operator by Atangana and Baleanu with non-local and non-singular fading memory. The basic properties and stability analysis are studied. Picard–Lindelof method is used to examine the existence and uniqueness of solutions of the new 5-D hyperchaotic system with four wings. The numerical simulation results depict a new chaotic behaviours with the ABC numerical scheme.

A fractional model of vertical transmission and cure of vector-borne diseases pertaining to the Atangana–Baleanu fractional derivatives

The model of transmission dynamics of vector-borne diseases with vertical transmission and cure within a targeted population is extended to the concept of fractional differentiation and integration with non-local and non-singular fading memory introduced. The effect of vertical transmission and cure rate on the basic reproduction number is shown. The Atangana–Baleanu fractional operator in caputo sense (ABC) with non-singular and non-local kernels is used to study the model. The existence and uniqueness of solutions are investigated using the Picard–Lindel method. Ultimately, for illustrating the acquired results, we perform some numerical simulations and show graphically to observe the impact of the arbitrary order derivative. It is expected that the proposed model will show better approximation than the classical model established before.