to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References.

Quantum Inverse Scattering Method and Correlation Functions

Preface.- Preface to the 2nd edition.- Notational conventions.- 1 Preliminary general material.- I Steady-state problems.- 2 Pseudomonotone or weakly continuous mappings.- 3 Accretive mappings.- 4 Potential problems: smooth case.- 5 Nonsmooth problems variational inequalities.- 6. Systems of equations: particular examples.- II Evolution problems.- 7 Special auxiliary tools.- 8 Evolution by pseudomonotone or weakly continuous mappings.- 9 Evolution governed by accretive mappings.- 10 Evolution governed by certain set-valued mappings.- 11 Doubly-nonlinear problems.- 12 Systems of equations: particular examples.- References.- Index.

/pdf/nonlinear-partial-differential-equations-with-applications-204hhsvanv.pdf

Nonlinear partial differential equations with applications

Ferroelectric and ferroelastic switching cause ferroelectric ceramics to depolarize and deform when subjected to excessive electric field or stress. Switching is the source of the classic butterfly shaped strain vs electric field curves and the corresponding electric displacement vs electric field loops [1]. It is also the source of a stress—strain curve with linear elastic behavior at low stress, non-linear switching strain at intermediate stress, and linear elastic behavior at high stress [2, 3]. In this work, ceramic lead lanthanum zirconate titanate (PLZT) is polarized by loading with a strong electric field. The resulting strain and polarization hysteresis loops are recorded. The polarized sample is then loaded with compressive stress parallel to the polarization and the stress vs strain curve is recorded. The experimental results are modeled with a computer simulation of the ceramic microstructure. The polarization and strain for an individual grain are predicted from the imposed electric field and stress through a Preisach hysteresis model. The response of the bulk ceramic to applied loads is predicted by averaging the response of individual grains that are considered to be statistically random in orientation. The observed strain and electric displacement hysteresis loops and the nonlinear stress—strain curve for the polycrystalline ceramic are reproduced by the simulation.

Ferroelectric/ferroelastic interactions and a polarization switching model

1. Some Mathematical Tools.- 1.1 Measure and Integration.- 1.2 Function Spaces.- 1.3 Nonlinear Equations.- 1.4 Ordinary Differential Equations.- 2. Hysteresis Operators.- 2.1 Basic Examples.- 2.2 General Hysteresis Operators.- 2.3 The Play Operator.- 2.4 Hysteresis Operators of Preisach Type.- 2.5 Hysteresis Potentials and Energy Dissipation.- 2.6 Hysteresis Counting and Damage.- 2.7 Characterization of Preisach Type Operators.- 2.8 Hysteresis Loops in the Prandtl Model.- 2.9 Hysteresis Loops in the Preisach Model.- 2.10 Composition of Preisach Type Operators.- 2.11 Inverse and Implicit Hysteresis Operators.- 2.12 Hysteresis Count and Damage, Part II.- 3. Hysteresis and Differential Equations.- 3.1 Hysteresis in Ordinary Differential Equations.- 3.2 Auxiliary Imbedding Results.- 3.3 The Heat Equation with Hysteresis.- 3.4 A Convexity Inequality.- 3.5 The Wave Equation with Hysteresis.- 4. Phase Transitions and Hysteresis.- 4.1 Thermodynamic Notions and Relations.- 4.2 Phase Transitions and Order Parameters.- 4.3 Landau and Devonshire Free Energies.- 4.4 Ginzburg Theory and Phase Field Models.- 5. Hysteresis Effects in Shape Memory Alloys.- 5.1 Phenomenology and Falk's Model.- 5.2 Well-Posedness for Falk's Model.- 5.3 Numerical Approximation.- 5.4 Complementary Remarks.- 6. Phase Field Models With Non-Conserving Kinetics.- 6.1 Auxiliary Results from Linear Elliptic and Parabolic Theory.- 6.2 Well-Posedness of the Caginalp Model.- 6.3 Well-Posedness of the Penrose-Fife Model.- 6.4 Complementary Remarks.- 7. Phase Field Models With Conserved Order Parameters.- 7.1 Well-Posedness of the Caginalp Model.- 7.2 Well-Posedness of the Penrose-Fife Model.- 8. Phase Transitions in Eutectoid Carbon Steels.- 8.1 Phenomenology of the Phase Transitions.- 8.2 The Mathematical Model.- 8.3 Well-Posedness of the Model.- 8.4 The Jominy Test: A Numerical Study.

Hysteresis and Phase Transitions

In this technical note, we analyze the error behavior of the discrete-time extended Kalman filter for nonlinear systems with intermittent observations. Modelling the arrival of the observations as a random process, we show that, given a certain regularity of the system, the estimation error remains bounded if the noise covariance and the initial estimation error are small enough. We also study the effect of different measurement models on the bounds for the error covariance matrices.

Stochastic Stability of the Extended Kalman Filter With Intermittent Observations

In this paper, Pontryagin's principle is proved for a fairly general problem of optimal control of populations with continuous time and age variable. As a consequence, maximum principles are developed for an optimal harvesting problem and a problem of optimal birth control.

Pontryagin's principle for control problems in age-dependent population dynamics.

This paper is concerned with an optimal control problem for a system
of ordinary differential equations with rate independent hysteresis
modelled as a rate independent evolution variational inequality
with a closed convex constraint $Z\subset \mathbb{R}^m$.
We prove existence of optimal solutions as well as necessary optimality
conditions of first order. In particular, under certain regularity
assumptions we completely characterize the jump behaviour of the adjoint.

Optimal control of ODE systems involving a rate independent variational inequality

As shown by Krasnosel" skii, the classical Preisach model allows to construct a hysteresis operator Wbetween spaces of real functions of time. This construction, via the definition of a measure mu in the so-called Preisach plane, is recalled. Characterizations in terms of mu are given for several mapping and continuity properties of W in various function spaces, for the invertibility of W and for the corresponding mapping and continuity properties of the inverse.

https://kluedo.ub.uni-kl.de/files/663/gruen_30.pdf

Martin Brokate

Papers

Hysteresis and Phase Transitions

Stochastic Stability of the Extended Kalman Filter With Intermittent Observations

Pontryagin's principle for control problems in age-dependent population dynamics.

Optimal control of ODE systems involving a rate independent variational inequality

Properties of the Preisach model for hysteresis