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Methods Of Modern Mathematical Physics

There have been significant recent advances in realizing band structures with geometrical and topological features in experiments on cold atomic gases. This review summarizes these developments, beginning with a summary of the key concepts of geometry and topology for Bloch bands. Descriptions are given of the different methods that have been used to generate these novel band structures for cold atoms and of the physical observables that have allowed their characterization. The focus is on the physical principles that underlie the different experimental approaches, providing a conceptual framework within which to view these developments. Also described is how specific experimental implementations can influence physical properties. Moving beyond single-particle effects, descriptions are given of the forms of interparticle interactions that emerge when atoms are subjected to these energy bands and of some of the many-body phases that may be sought in future experiments.

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Topological bands for ultracold atoms.

More than 100 offshore mass-movement deposits have been studied in Holocene and Pleistocene sediments. The processes can be divided into three main types: slides/slumps, plastic flows, and turbidity currents, of which 13 main varieties have been recognized. The three types are differentiated mainly by motion, architecture, and shape of failure surface. For slides, the morphology of deposits can usually be linked to a process, but for plastic flows and turbidity currents, information about the motion is mainly provided by the sedimentary record. A static classification based on these features is given, and is related to a dynamic classification system to try to underline the morphological transformation of an offshore event from initiation to deposition.

Classification of offshore mass movements

R Leis 1986 Chichester: John Wiley viii + 266 pp price £2450 ISBN 0 471 90863 0 This book is an account of some recent developments in the theory of partial differential equations for readers thoroughly familiar with functional analysis It deals with various types of problem for the wave equation, Maxwell's equations, Schrodinger's equation, the plate equation etc, but dissipative systems are hardly discussed

https://iopscience.iop.org/article/10.1088/0031-9112/37/10/027/pdf

Initial Boundary Value Problems in Mathematical Physics

Singularity formation for Prandtl’s equations

In this paper we derive a reaction-diffusion-chemotaxis model for the dynamics of multiple sclerosis. We focus on the early inflammatory phase of the disease characterized by activated local microglia, with the recruitment of a systemically activated immune response, and by oligodendrocyte apoptosis. The model consists of three equations describing the evolution of macrophages, cytokine and apoptotic oligodendrocytes. The main driving mechanism is the chemotactic motion of macrophages in response to a chemical gradient provided by the cytokines. Our model generalizes the system proposed by Calvez and Khonsari (Math Comput Model 47(7–8):726–742, 2008) and Khonsari and Calvez (PLos ONE 2(1):e150, 2007) to describe Balo’s sclerosis, a rare and aggressive form of multiple sclerosis. We use a combination of analytical and numerical approaches to show the formation of different demyelinating patterns. In particular, a Turing instability analysis demonstrates the existence of a threshold value for the chemotactic coefficient above which stationary structures develop. In the case of subcritical transition to the patterned state, the numerical investigations performed on a 1-dimensional domain show the existence, far from the bifurcation, of complex spatio-temporal dynamics coexisting with the Turing pattern. On a 2-dimensional domain the proposed model supports the emergence of different demyelination patterns: localized areas of apoptotic oligodendrocytes, which closely fit existing MRI findings on the active MS lesion during acute relapses; concentric rings, typical of Balo’s sclerosis; small clusters of activated microglia in absence of oligodendrocytes apoptosis, observed in the pathology of preactive lesions.

Demyelination patterns in a mathematical model of multiple sclerosis

A phenomenological operator description of dynamics of crowds: Escape strategies

(H, ρ)-induced dynamics and the quantum game of life

We discuss the role of pseudofermions in the analysis of some two-dimensional models, recently introduced in connection with non-self-adjoint Hamiltonians. Among other aspects, we discuss the appearance of exceptional points in connection with the validity of the extended anticommutation rules which define the pseudofermionic structure.

Francesco Gargano

Papers

Singularity formation for Prandtl’s equations

Demyelination patterns in a mathematical model of multiple sclerosis

A phenomenological operator description of dynamics of crowds: Escape strategies

(H, ρ)-induced dynamics and the quantum game of life

Model pseudofermionic systems: Connections with exceptional points