相关论文: Long and Short Memory in Economics: Fractional-Ord…

This work introduces and analyzes a finite element scheme for evolution problems involving fractional-in-time and in-space differentiation operators up to order two. The left-sided fractional-order derivative in time we consider is employed…

The mathematical model of a linear system with the short memory about own stochastic behavior is proposed. It is assumed that the system is under a continual influence of independent stochastic impulses. In a short memory approximation the…

Derivatives and integrals of non-integer order may have a wide application in describing complex properties of materials including long-term memory, non-locality of power-law type and fractality. In this paper we consider extensions of…

The steady state of a Langevin equation with short ranged memory and coloured noise is analyzed. When the fluctuation-dissipation theorem of second kind is not satisfied, the dynamics is irreversible, i.e. detailed balance is violated. We…

Memory effects are a key feature in the description of the dynamical systems governed by the generalized Langevin equation, which presents an exact reformulation of the equation of motion. A simple measure for the estimation of memory…

Time non-locality, or memory, is a non-equilibrium property shared by all physical systems. Here, we show that memory is sufficient to induce a phase of spatial long-range order (LRO) even if the system's primary dynamical variables are…

We study the generalized dynamic factor model in a long-memory setting. Unlike most recent work, which assumes a finite-dimensional factor space and short memory, our framework allows the factor space to be infinite-dimensional and the…

This chapter presents some numerical methods to solve problems in the fractional calculus of variations and fractional optimal control. Although there are plenty of methods available in the literature, we concentrate mainly on approximating…

We propose to describe the dynamics of phase transitions in terms of a non-stationary Generalized Langevin Equation for the order parameter. By construction, this equation is non-local in time, i.e.~it involves memory effects whose…

Fractional difference operators possess nonlocal structure which largely affects and complicates the qualitative analysis of fractional difference equations. In this article, we discuss the effect of this memory property on asymptotic…

This paper, that will appear in the Notices of the AMS, begins with a brief historical account of the beginnings of fractional calculus and the crucial roles played by Leibniz and Fourier. Fourier's definition of fractional derivative is…

In this work we propose a new class of long-memory models with time-varying fractional parameter. In particular, the dynamics of the long-memory coefficient, $d$, is specified through a stochastic recurrence equation driven by the score of…

In this review, we present some fundamental classical and quantum phenomena in view of time fractional formalism. Time fractional formalism is a very useful tool in describing systems with memory and delay. We hope that this study can…

In this work, a second-order approximation of the fractional substantial derivative is presented by considering a modified shifted substantial Gr\"{u}nwald formula and its asymptotic expansion. Moreover, the proposed approximation is…

The concept of memory is of central importance for characterizing complex systems and phenomena. Presence of long-term memories indicates how their dynamics can be less sensitive to initial conditions compared to the chaotic cases. On the…

We derive a moment formula for generalized fractional polynomial processes, i.e., for polynomial-preserving Markov processes time-changed by an inverse L\'evy-subordinator. If the time change is inverse $\alpha$-stable, the time-derivative…

We introduce an extended version of the fractional Ornstein-Uhlenbeck (FOU) process where the integrand is replaced by the exponential of an independent L\'evy process. We call the process the generalized fractional Ornstein-Uhlenbeck…

Derivatives of fractional order are introduced in different ways: as left-inverse of the fractional integral or by generalizing the limit of the difference quotient defining integer-order derivatives. Although the two approaches lead (under…

Memristors have emerged as ideal components for modeling synaptic connections in neural networks due to their ability to emulate synaptic plasticity and memory effects. Discrete models of memristor-coupled neurons are crucial for…

Memory and forgetting constitute two sides of the same coin, and although the first has been rigorously investigated, the latter is often overlooked. A number of experiments under the realm of psychology and experimental neuroscience have…