A New Approach to Equations with Memory

In this work, we present a novel approach to the mathematical analysis of equations with memory based on the notion of a state, namely, the initial configuration of the system which can be unambiguously determined by the knowledge of the future dynamics. As a model, we discuss the abstract version of an equation arising from linear viscoelasticity. It is worth mentioning that our approach goes back to the heuristic derivation of the state framework, devised by L.Deseri, M.Fabrizio and M.J.Golden in "The concept of minimal state in viscoelasticity: new free energies and applications to PDEs", Arch. Ration. Mech. Anal., vol. 181 (2006) pp.43-96. Starting from their physical motivations, we develop a suitable functional formulation which, as far as we know, is completely new.Comment: 39 pages, no figur

Global attractors for nonlinear viscoelastic equations with memory

We study the asymptotic properties of the semigroup S(t) arising from a nonlinear viscoelastic equation with hereditary memory on a bounded three-dimensional domain written in the past history framework of Dafermos. We establish the existence of the global attractor of optimal regularity for S(t) for a wide class of nonlinearities as well as within the most general condition on the memory kernel

hepaccelerate: Fast Analysis of Columnar Collider Data

At HEP experiments, processing terabytes of structured numerical event data to a few statistical summaries is a common task. This step involves selecting events and objects within the event, reconstructing high-level variables, evaluating multivariate classifiers with up to hundreds of variations and creating thousands of low-dimensional histograms. Currently, this is done using multi-step workflows and batch jobs. Based on the CMS search for H(μμ), we demonstrate that it is possible to carry out significant parts of a real collider analysis at a rate of up to a million events per second on a single multicore server with optional GPU acceleration. This is achieved by representing HEP event data as memory-mappable sparse arrays, and by expressing common analysis operations as kernels that can be parallelized across the data using multithreading. We find that only a small number of relatively simple kernels are needed to implement significant parts of this Higgs analysis. Therefore, analysis of real collider datasets of billions events could be done within minutes to a few hours using simple multithreaded codes, reducing the need for managing distributed workflows in the exploratory phase. This approach could speed up the cycle for delivering physics results at HEP experiments. We release the hepaccelerate prototype library as a demonstrator of such accelerated computational kernels. We look forward to discussion, further development and use of efficient and easy-to-use software for terabyte-scale high-level data analysis in the physical sciences

A reaction-diffusion equation with memory

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Uniform attractors for a phase-field model with memory and quadratic nonlinearity

A phase-field system with memory which describes the evolution of both the temperature variation $\theta$ and the phase variable $\chi$ is considered. This thermodynamically consistent model is based on a linear heat conduction law of Coleman-Gurtin type. Moreover, the internal energy linearly depends both on the present value of $\theta$ and on its past history, while the dependence on $\chi$ is represented through a function with quadratic nonlinearity. A Cauchy-Neumann initial and boundary value problem associated with the evolution system is then formulated in a history space setting. This problem is shown to generate a non-autonomous dynamical system which possesses a uniform attractor. In the autonomous case, the attractor has finite Hausdorff and fractal dimensions whenever the internal energy linearly depends on $\chi$

Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent

In this paper the long time behaviour of the solutions of 3-D strongly damped wave equation is studied. It is shown that the semigroup generated by this equation possesses a global attractor in H_{0}^{1}(\Omega)\times L_{2}(\Omega) and then it is proved that this global attractor is a bounded subset of H^{2}(\Omega)\times H^{2}(\Omega) and also a global attractor in H^{2}(\Omega)\cap H_{0}^{1}(\Omega)\times H_{0}^{1}(\Omega)

Robust exponential attractors for a family of nonconserved phase-field systems with memory

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Delayed benign surgery during the COVID-19 pandemic. The other side of the coin

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