ISBN: 0817639586 9780817639587 3764339586 9783764339586: OCLC Number: 36498785: Description: vi, 361 pages: illustrations; 25 cm. Contents: Blowup of small data solutions for a class of quasilinear wave equations in two dimensions: an outline of the proof / Serge Alinhac --Concentration effects in critical nonlinear wave equation and scattering theory / Hajer Bahouri and Patrick Gerard. Get this from a library! Geometrical Optics and Related Topics. [F Colombini; Nicolas Lerner] -- This book contains fourteen research papers which are expanded versions of conferences given at a meeting held in September 1996 in Cortona, Italy. The topics include blowup questions for quasilinear. Avijit Lahiri, in Basic Optics, 2016. Abstract. Ray optics, or geometrical optics, is based on the short-wavelength approximation of electromagnetic theory.It is defined in terms of a package of rules the rules of geometrical optics that can be arrived at from the Maxwell equations in a consistent approximation scheme, referred to as the eikonal approximation, which is briefly outlined in. This is known as the dispersion equation. For frequencies above resonance σ 0 2 - σ 2 < 0, the oscillators are about 180 o with the driving electric field see the discussion of phase-lag, and n < 1.When the frequency is below resonance, σ 0 2 - σ 2 > 0, the oscillation is in phase with the applied electric field and n > 1. In general, index of refraction increases with frequency. Michael E. Taylor Department of Mathematics University of North Carolina Chapel Hill, NC 27599 USA met@math. ISSN 0066-5452 ISBN 978-1-4419-7054-1 e-ISBN 978-1-4419-7055-8.
We are excited to announce that Fixed Point Theory and Applications will be relaunched as Fixed Point Theory and Algorithms for Sciences and Engineering in 2021.The journal is open for submissions and celebrates its relaunch with a Topical Collection on Optimization and Real World Applications. This relaunch marks a shift towards a broadened scope with a clear emphasis on applications. and to introduce those working in partial diﬀerential equations to some fas-cinating applications containing many unresolved nonlinear problems arising in geometry. My intention is that after reading these notes someone will feel that they can cope with current research articles. In fact, the quite sketchy. Weak approximation for semilinear stochastic evolution equations. In Stochastic Analysis and Related Topics VIII. Progress in Probability 53 111–128. Birkhäuser, Basel. Numerical Solution of Stochastic Differential Equations. Applications of Mathematics New York 23. Semilinear geometric optics with boundary amplification Coulombel. The nonlinear geometric optics presents an introduction to methods developed within the last twenty years including a rethinking of the linear case. Much of the material has grown out of courses that I have taught. The crucial step was a series of eight lectures on nonlinear geometric optics at the Institute for Advanced Study Park City.
Geometrical optics, or ray optics, is a model of optics that describes light propagation in terms of rays.The ray in geometric optics is an abstraction useful for approximating the paths along which light propagates under certain circumstances. The simplifying assumptions of geometrical optics include that light rays: propagate in straight-line paths as they travel in a homogeneous medium. D. Del Santo, V. Georgiev and E. Mitidieri, Global existence of the solutions and formation of singularities for a class of hyperbolic systems, In: ``Geometric Optics and Related Topics'' Eds. F. Colombini & N. Lerner, Progress in Nonlinear Differential Equations and Their Applications, Vol. 32, pp. 117-140, Birkhäuser, Boston, 1997. 8.
R. B. Melrose, Geometric optics and the bottom of the spectrum, In F. Colombini and N. Lerner, editors, Geometrical optics and related topics, volume 32 of Progress in nonlinear differential equations and their applications, Birkhäuser, Basel - Boston - Berlin 1997. 33. R. B. Thin Lenses When the size of the physical and optical objects of a system are much larger than the wavelength of the light or as λ→ 0, we are in the realm of geometrical optics. Optical systems in which the wave nature of light must be taken into account interference, diffraction are called physical optics. Of course, every real system experiences diffraction effects, so geometric. Received May 2017 Revised October 2017 Published December 2017. Fund Project: The author is supported by NSF grant DMS 1361754. Lectures on nonlinear optics. This note covers the following topics: Introduction to nonlinear optics, The nonlinear susceptibilities and their symmetries, Quasi-monochromatic fields and the degeneracy factor in nonlinear optics, Quantum mechanics: Formulation of linear optical interactions, Formulation of nonlinear optical interactions, Linking the microcscopic to the macroscopi and Spatial. Lectures on nonlinear optics. This note covers the following topics: Introduction to nonlinear optics, The nonlinear susceptibilities and their symmetries, Quasi-monochromatic fields and the degeneracy factor in nonlinear optics, Quantum mechanics: Formulation of linear optical interactions, Formulation of nonlinear optical interactions.
Oct 10, 2000 · The existence of the critical curve for p-q systems for nonlinear wave equations was already established by D. Del Santo, V. Georgiev, and E. Mitidieri [1997, Global existence of the solutions and formation of singularities for a class of hyperbolic systems, in “Geometric Optics and Related Topics” F. Colombini and N. Lerner, Eds., Progress in Nonlinear Differential Equations and. This book is not a textbook, but rather a coherent collection of papers from the field of partial differential equations. Nevertheless we believe that it may very well serve as a good introduction into some topics of this classical field of analysis which, despite of its long history, is highly modem and well prospering. Further, some lower and upper bounds of the lifespan of classical solutions are found too. This work is an extension of work [Geometric Optics and Related Topics, Progress in Nonlinear Differential. These models are usually formulated as nonlinear partial differential equations and are rather difficult to make use of. With the help of Lie group analysis it is possible to describe some important properties of these models and to obtain interesting reductions in a clear and understandable algorithmic way. Del Santo et al. [“Geometric Optics and Related Topics” F. Colombini and N. Lerner, Eds., Progress in Nonlinear Differential Equations and Their Applications, Vol..
Oct 22, 2016 · The Monge-Ampère Equation Progress in Nonlinear Differential Equations and Their Applications Book 89 - Kindle edition by Gutiérrez, Cristian E. Download it once and read it on your Kindle device, PC, phones or tablets. Use features like bookmarks, note taking and highlighting while reading The Monge-Ampère Equation Progress in Nonlinear Differential Equations and Their Applications. Optics is the cornerstone of photonics systems and applications. Geometrical optics, or ray optics, is to study the geometry of paths of lights and their imagery through optical systems. Light will be treated as a form of energy which travels in straight lines called rays. Topics in this volume include stationary and time-dependent surface PDEs for geometric flows, large deformations of nonlinearly geometric plates and rods, level set and phase field methods and applications, free boundary problems, discrete Riemannian calculus and morphing, fully nonlinear PDEs including Monge-Ampere equations, and PDE constrained optimization. Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.
Bibliographic notes, updated and expanded from the first edition, are included at the end of every chapter for further reading on Monge-Ampère-type equations and their diverse applications in the areas of differential geometry, the calculus of variations, optimization problems, optimal mass transport, and geometric optics. Differential Geometry and its Applications publishes original research papers and survey papers in differential geometry and in all interdisciplinary areas in mathematics which use differential geometric methods and investigate geometrical structures. The following main areas are covered: differential equations on manifolds, global analysis. What distinguishes differential geometry in the last half of the twentieth century from its earlier history is the use of nonlinear partial differential equations in the study of curved manifolds, submanifolds, mapping problems, and function theory on manifolds, among other topics. The differential equations appear as tools and as objects of study, with analytic and geometric advances fueling. Among the topics carefully presented in the book are nonlinear geometric optics, the asymptotic analysis of short wavelength solutions, and nonlinear interaction of such waves. Studied in detail are the damping of waves, resonance, dispersive decay, and solutions to the compressible Euler equations with dense oscillations created by resonant. Motivation. I have recently started thinking about the interrelations among algebraic geometry and nonlinear PDEs. It is well known that the methods and ideas of algebraic geometry have lead to a number of important achievements in the study of PDEs, suffice it to mention the construction of finite-gap solutions to integrable PDEs see e.g. this book and the geometric approach to PDEs.
Del Santo et al. [“Geometric Optics and Related Topics” F. Colombini and N. Lerner, Eds., Progress in Nonlinear Differential Equations and Their Applications, Vol. 32, pp. 117–140, Birkhauser, Boston, 1997] have studied the existence and nonexistence of global classical solutions of the Cauchy problem except for the critical case. Topics include simple harmonic oscillation including damping and resonance, coupled oscillators and normal modes, nonlinear oscillations, basic wave properties, traveling and standing waves, waves in media, dispersion relations, derivation of wave equations, geometric optics, diffraction and interference, Fourier analysis. 3 hours lecture. The school consisted of four courses: Extremal problems for quasiconformal mappings in space by Luca Capogna, Fully nonlinear equations in geometry by Pengfei Guan, Monge-Ampere type equations and geometric optics by Cristian E. Gutiérrez, and On the Levi Monge Ampere equation. We establish the geometric origin of the nonlinear heat equation with arctangential nonlinearity: ∂ t D = Δ arctan D by deriving it, together and in duality with the mean curvature flow equation, from the minimal surface equation in Minkowski space-time, through a suitable quadratic change of time. After examining various properties of the arctangential heat equation in particular. A homogenization approach for the motion of motor proteins with S. Mirrahimi, NODEA Nonlinear Differential Equations Appl. 20 2013, no. 1, 129?47. Stochastic averaging lemmas for kinetic equations with P.-L. Lions and B. Perthame, Seminaire Equations Aux Derivee Partielles Ecole Polytechnique 2011-2012, no. 1.
This course will review key attributes of optical materials that allow them to be used in a range of applications. Physical properties and their structural origin will be used as a means to predict performance and limitations of these materials as used in devices and components in optical systems. transforms, ordinary differential equations. Differential equations theory and applications. First-order linear and nonlinear differential equations with analytic and numerical techniques. Higher-order linear differential equations and complex algebra. Phase trajectory and stability analysis. Systems of linear differential equations. The topics covered belong to the general areas of the theory of function spaces, partial differential equations of elliptic, parabolic, and dissipative types, geometric optics, free boundary problems, and ergodic theory, and the emphasis is on a host of new concepts, methods, and results. Singular nonlinear boundary value problems for second order ordinary differential equations, with J. A. Gatica and P. Waltman, Journal of Differential Equations, vol.79, no.1 1989, 62-78. On reconstructing a reflecting surface from the scattering data in the geometric optics approximation, Journal of Inverse problems, 5 1989, 51-65. One of the challenges of teaching a truly "reformed," context-based course in calculus or differential equations is finding interesting applications which lead naturally to the consideration of traditional topics. In this note, we outline a multi-part project for investigating two simple problems in geometric optics.
Partial differential equationsPDE are significantly more difficult than ODE, and we won’t talk about it at this moment. Order. Order of a differential equations is the order of the highest derivative in the equation. Order 1: Order 2: Degree. The degree of a differential equation is the degree of the highest derivative in the equation. Seminar on current research topics in dynamic systems modeling, control, and applications. Topics selected from process control, differential games, nonlinear estimation, adaptive filtering, industrial and aerospace applications, etc. Letter grading. M272A. Nonlinear Dynamic Systems. The Painlevé equations were discovered as nonlinear ordinary differential equations that define new special functions, and their importance has long been recognized. Since the 1990s, there have been many studies on various generalizations of the Painlevé equations such as discretizations, higher dimensional analogues, quantizations, and so on. Partial Differential Equations; Laser Dynamics; Numerical Mathematics and Scientific Computing; Nonlinear Optimization and Inverse Problems; Interacting Random Systems; Stochastic Algorithms and Nonparametric Statistics; Thermodynamic Modeling and Analysis of Phase Transitions; Nonsmooth Variational Problems and Operator Equations; Flexible.
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