Infinite History Research Articles

A Stability Result for a Timoshenko System with Infinite History and Distributed Delay Term

This manuscript is mainly focusing on a general stability of solution for one-dimensional Timoshenko system with infinite history and distributed delay term regardless also of the speeds of wave propagation. We prove our result by using the energy method combined with some properties of convex functions.

The generalized Khasminskii-type conditions in establishing existence, uniqueness and moment estimates of solution to neutral stochastic functional differential equations

The main objective of this paper involving neutral stochastic functional differential equations (NSFDEs), with finite or infinite history dependence, is to prove the existence and uniqueness of their global solutions by imposing conditions that hold for highly non-linear NSFDEs? coefficients. For that purpose, Yamada-Watanabe condition or the local Lipschitz condition for the drift and diffusion coefficients are imposed, together with contractivity condition for the neutral term. Also, instead of the linear growth condition for the drift and diffusion coefficients of the equations, generalized Khasminskii-type conditions are applied. The proof of the existence and uniqueness of the solution also leads us to estimates of the moments to the solution. Consequently, we discuss some asymptotic properties of the solution in terms of the generalized Lyapunov exponent. Additionally, we consider a class of neutral stochastic differential equations with state-dependent delay, as a special case of NSFDEs. The theoretical results are illustrated with two examples.

Emma ROTHSCHILD, An Infinite History. The Story of a Family in France over Three Centuries

Emma ROTHSCHILD, An Infinite History. The Story of a Family in France over Three Centuries

A Storied History

This is a book of stories—ninety-eight stories, to be precise—as Emma Rothschild is careful to tell us on the first page. Enlarging the scope of what will follow, she adds that they are stories about the “three or four thousand people who lived in agitated times” in the provincial city of Angoulême in the eighteenth and nineteenth centuries. But even this more expansive description is misleading. An Infinite History: The Story of a Family in France over Three Centuries contains multitudes. It is both many stories and many kinds of histories. It is a history of an extended family and those associated with it across time; or rather, as Rothschild insists, “a history with a family at its center, and not a family history” (197). It is a history, again she insists, “not of the economy but of economic lives”—lovingly and meticulously recovered lives forged in the face of hardship,...

General stability result for a porous thermoelastic system with infinite history and microtemperatures effects

In this paper, we consider a one‐dimensional porous thermoelastic system with microtemperatures effects and past history term acting only on the porous equation. We show that the system is well‐posed in the sens of semigroup and based on the energy method we establish a general decay result for the solutions of the system under an appropriate assumptions on the kernel. Furthermore, our result does not depend on any relationship between the coefficients of the system. The result of this paper improves some results in the literature.

An Infinite History: The Story of a Family in France over Three Centuries by Emma Rothschild

<i>An Infinite History: The Story of a Family in France over Three Centuries</i> by Emma Rothschild

New general decay result of the laminated beam system with infinite history

This paper is concerned with the asymptotic behavior of the solution of a laminated Timoshenko beam system with viscoelastic damping. We extend the work known for this system with finite memory to the case of infinite memory. We use minimal and general conditions on the relaxation function and establish explicit energy decay formula, which gives the best decay rates expected under this level of generality. We assume that the relaxation function g satisfies, for some nonnegative functions ξ and H, g′(t)≤−ξ(t)H(g(t)), ∀t≥0. Our decay results generalize and improve many earlier results in the literature. Moreover, we remove some assumptions on the boundedness of initial data used in many earlier papers in the literature.

General decay rate for a Moore–Gibson–Thompson equation with infinite history

In previous work (Alves et al. in Z Angew Math Phys 69:106, 2018), by using the linear semigroup theory, Alves et al. investigated the existence and exponential stability results for a Moore–Gibson–Thompson model encompassing memory of type 1, 2 or 3 in a history space framework. In this paper, we continue to consider the similar problem with type 1 and establish explicit and general decay results of energy for system in both the subcritical and critical cases, by introducing suitable energy and perturbed Lyapunov functionals and following convex functions ideas presented in Guesmia (J Math Anal Appl 382:748–760, 2011). Our results allow a much larger class of the convolution kernels which improves the earlier related results.

Viscoelastic versus frictional dissipation in a variable coefficients plate system with time-varying delay

In this paper, we are concerned with variable coefficients plate system subjected to three partially distributed feedbacks: time-varying delay, frictional and viscoelastic dissipations. This work is devoted to, without any prior quantification of both decay rate of relaxation function and growth rate of frictional dissipation near the origin, establish a general decay result which corresponds to a certainly stable ODE. Our result extends the decay result obtained for some kind of problems with finite history to problem with infinite history. Moreover, this paper allows a wider class of kernels of infinite history, and the usual exponential and polynomial decay rates are only special cases. The proof is based on the multiplier method and some techniques about convex functionals.

Asymptotic behavior for a past history viscoelastic problem with acoustic boundary conditions

ABSTRACTIn this paper, we consider a viscoelastic equation of variable coefficients in the presence of infinite memory (past history) with nonlinear damping term and nonlinear delay term in the boundary feedback and acoustic boundary conditions. Under suitable assumptions, two arbitrary decay results of the energy solution are established via suitable Lyapunov functionals and some properties of the convex functions. The first stability result is given with a relation between the damping term and relaxation function and the second result is given without imposing any restrictive growth assumption on the damping term and the kernel function g. Our result extends the decay result obtained for problems with finite history to those with infinite history.

General Decay for a Viscoelastic Equation of Variable Coefficients in the Presence of Past History with Delay Term in the Boundary Feedback and Acoustic Boundary Conditions

In this paper, we consider a viscoelastic equation of variable coefficients in the presence of infinite memory (past history) with nonlinear damping term and nonlinear delay term in the boundary feedback and acoustic boundary conditions. Under suitable assumptions, two arbitrary decay results of the energy solution are established via suitable Lyapunov functionals and some properties of the convex functions. The first stability result is given with relation between the damping term and relaxation function. The second result is given without imposing any restrictive growth assumption on the damping term and the kernel function g$g$. Our result extends the decay result obtained for problems with finite history to those with infinite history.

Playful palette

We present Playful Palette, a color picker interface for digital paint programs that derives intuition from oil paint and watercolor palettes, but extends them with digital features. A Playful Palette is a set of blobs of color that blend together to create gradients and gamuts. They can be directly manipulated to explore arrangements and harmonies. All edits are non-destructive, and an infinite history allows previous palettes to be revisited and modified, recoloring the painting. The Playful Palette design is motivated by a pilot study of how artists use paint palettes, and we evaluate the final design with a set of traditional and digital media painters to demonstrate that Playful Palette is effective both at enabling artists' color tasks, and at amplifying their creativity.

A general decay result for a memory-type Timoshenko-thermoelasticity system with second sound

In this paper we consider a system of Timoshenko-thermoelasticity with second sound in the presence of an infinite history term. We prove, under suitable conditions on the coefficients and the relaxation function, a general decay result from which the exponential and polynomial decay estimates are only special cases. We also establish a lack of exponential decay result and discuss the optimality for some situation.

Stability of a von Karman equation with infinite memory

In this paper, we consider a von Karman equation with infinite memory. For von Karman equations with finite memory, there is a lot of literature concerning on existence of the solutions, decay of the energy, and existence of the attractors. However, there are few results on existence and energy decay rate of the solutions for von Karman equations with infinite memory. The main goal of the present paper is to generalize previous results by treating infinite history instead of finite history.

States of Linear Systems, Automata, and Marriage : A Tribute to Rudolf Kalman [Historical Perspectives

During my time as a Ph.D. student in mathematics at the Massachusetts Institute of Technology (MIT), around 1962 or 1963, the authors of a book on control theory [1] introduced me to two researchers at MIT's Lincoln Laboratory, Michael Athanassiades and Peter Falb, who were writing a book on mathematical control theory. They, in turn, provided me with a draft of their book in progress and introduced me to the work of Rudolf Kalman. I read their draft with great interest not only for its intrinsic merit but also because I had been inspired by Norbert Wiener's Cybernetics and studied the Wiener-Hopf filter (and been a student of Wiener's at MIT for a while). Moreover, Kalman's stress on current state rather than the infinite past history of the system whose behavior was being extrapolated chimed with another of my interests, namely automata theory. There the concept of state was paramount-but in the world of sets and functions rather than vector spaces and linear transformations.

A general decay result of a viscoelastic equation with infinite history and nonlinear damping

In this paper, we consider a viscoelastic equation with a nonlinear frictional damping and in the presence of an infinite-memory term. We prove an explicit and general decay result using some properties of the convex functions. Our approach allows a wider class of kernels, from which those of exponential decay type are only special cases.

A new approach to the stability of an abstract system in the presence of infinite history

In this paper, we consider the following problem{utt(t)+Au(t)−∫0+∞g(s)Au(t−s)ds=0,∀t>0u(−t)=u0(t),∀t⩾0ut(0)=u1, where A is a self-adjoint positive definite operator and g is a positive nonincreasing function. We adopt the method introduced in [19], for finite history, with some modifications imposed by the nature of our problem, to establish a general decay result which depends only on the behavior of the relaxation function. Our result extends the decay result obtained for problems with finite history to those with infinite history. In addition, it improves, in some cases, some decay results obtained earlier in [15].

Two-Person Zero-Sum Stochastic Games with Semicontinuous Payoff

Consider a two-person, zero-sum stochastic game with Borel state space S, finite action sets A,B, and Borel measurable law of motion q. Suppose the payoff is a bounded function f of the infinite history of states and actions that is measurable for the product of the Borel σ-field for S and the σ-fields of all subsets for A and B, and is lower semicontinuous for the product of the discrete topologies on the coordinate spaces. Then the game has a value and player II has a subgame perfect optimal strategy.

Discussion: Bootstrap methods for dependent data: A review

This review of bootstrap methods for time series is most welcome especially coming from two key figures in the development of thesemethods.Wewould like to complement their exposition by focusing on some further issues of current interest. Block bootstrap methods for time series data have been most intensively studied under the assumption of stationarity and mixing. An important example is the stationary bootstrap of Politis and Romano (1994). As detailed in the review, this method is a variation of the standard block bootstrap that manages to create bootstrap series that are strictly stationary. This property not only replicates the strict stationarity of the original time series, but it also makes the method easy to analyze theoretically. However, some rather involved Mean Square Error calculations of Lahiri (1999) seemed to imply that the stationary bootstrap has poor asymptotic efficiency relative to the usual block bootstrap in the sample mean case. As a result, practitioners have been understandably reluctant in endorsing the stationary bootstrap after the publication of Lahiri’s (1999) paper. It was quite a surprise when Nordman (2009) discovered an error in Lahiri’s (1999) calculations; as it turns out, the stationary bootstrap has identical asymptotic accuracy as the block bootstrapwith non-overlapping blocks; the latter iswellknown to be only slightly less efficient than the full-overlap block bootstrap. These new findings are expected to rekindle the interests of researchers since the stationarity of bootstrap sample paths is a very powerful tool. For example, functional data are of current interest in the 21st century; see the review by McMurry and Politis (2011). It suffices to note that Politis and Romano (1994b) were able to prove the asymptotic validity of the stationary bootstrap for the sample mean of functional data almost twenty years ago. Such a result does not seem to be available for the regular block bootstrap even to date. The assumptions of strict stationarity and strong mixing can be too strong for many applications in economics and finance. First, for many of these applications, the time span is rather long, making it unlikely that stationarity holds. A more appropriate assumption is to allow for some time series heterogeneity (for instance, in the form of time varying moments). Similarly, mixing is too strong a dependence condition to be broadly applicable. Andrews (1984) gives an example of a simple AR(1) model that fails to be strong mixing. More generally, although measurable functions of mixing processes are themselves mixing when the function depends on a finite number of lagged values of the mixing process, this is not necessarily the case when the observed time series depends on the infinite history of an underlying mixing process. For instance, popular ARCH (Engle, 1982) and GARCH (Bollerslev, 1986) processes, which are a function of the entire past history of a fundamental driving innovation, are known to be mixing only when we assume these innovations to be i.i.d. (see e.g. Carrasco & Chen, 2002, and Basrak, Davis & Mikosch, 2002).

A stochastic Ramsey theorem

We establish a stochastic extension of Ramsey's theorem. Any Markov chain generates a filtration relative to which one may define a notion of stopping times. A stochastic colouring is any k-valued ( k < ∞ ) colour function defined on all pairs consisting of a bounded stopping time and a finite partial history of the chain truncated before this stopping time. For any bounded stopping time θ and any infinite history ω of the Markov chain, let ω | θ denote the finite partial history up to and including the time θ ( ω ) . Given k = 2 , for every ϵ > 0 , we prove that there is an increasing sequence θ 1 < θ 2 < ⋯ of bounded stopping times having the property that, with probability greater than 1 − ϵ , the history ω is such that the values assigned to all pairs ( ω | θ i , θ j ) , with i < j , are the same. Just as with the classical Ramsey theorem, we also obtain an analogous finitary stochastic Ramsey theorem. Furthermore, with appropriate finiteness assumptions, the time one must wait for the last stopping time (in the finitary case) is uniformly bounded, independently of the probability transitions. We generalise the results to any finite number k of colours.