Alexander G. Ramm

Bio: Alexander G. Ramm is an academic researcher from Kansas State University. The author has contributed to research in topics: Scattering & Inverse scattering problem. The author has an hindex of 36, co-authored 660 publications receiving 8033 citations. Previous affiliations of Alexander G. Ramm include Technion – Israel Institute of Technology & Centre national de la recherche scientifique.

Scattering by obstacles

Abstract: I. Scattering by an Obstacle.- 1. Statement of the Problem. Basic Integral Equations.- 2. Existence and Uniqueness of the Solution to the Scattering Problem.- 3. Eigenfunction Expansion Theorem.- 4. Properties of the Scattering Amplitude.- 5. The S-Matrix and Wave Operators.- 6. Inequalities for Solutions to Helmholtz's Equation for Large frequencies.- 7. Representations of solutions to Helmholtz's Equation.- II. The Inverse Scattering (Diffraction) Problem.- 1. Statement of the Problem and Uniqueness Theorems.- 2. Reconstruction of Obstacles from the Scattering Data at High Frequencies.- 3. Stability of the Surface with Respect to Small Perturbations of the Data.- III. Time Dependent Problem.- 1. Statement of the Problems.- 2. The Limiting Amplitude Principle (Abstract Results).- 3. The Limiting Amplitude Principle for the Laplacian in Exterior Domains.- 4. Decay of Energy.- 5. Singularity and Eigenmode Expansion Methods.- IV. T-Matrix Scheme and Other Numerical Schemes.- 1. Statement of the Problem.- 2. Justification of the T-Matrix Scheme.- 3. Numerical Results.- 4. Other Schemes.- V. Scattering by Small Bodies.- 1. Scattering by a Single Small Body.- 2. Scattering by Many Small Bodies.- 3. Electromagnetic Wave Scattering by Small Bodies.- 4. Behavior of the Solutions to Exterior Boundary Value Problems at Low Frequencies.- VI. Some Inverse Scattering Problems of Geophysics.- 1. Inverse Scattering for Geophysical Problems.- 2. Two Parameter Inversion.- 3. An Inversion Formula in Scattering Theory.- 4. A Model Inverse Problem of Induction Logging.- VII. Scattering by Obstacles with Infinite Boundaries.- 1. Statement of the Problem.- 2. Spectral Properties of the Laplacians.- 3. Spectral Properties of the Dirichlet Laplacian in Semi-Infinite Tubes.- 4. Absence of Positive Eigenvalues for the Dirichlet Laplacian Under Local Assumptions at Infinity.- 5. The Limiting Absorption Principle and Compact Perturbations of the Boundary.- 6. Eigenfunction Expansions in Canonical Domains.- Appendix 1. Summary of some Results in Potential Theory and Embedding Theorems.- Appendix 2. Summary of some Results in Operator Theory.- Appendix 4. Stable Numerical Differentiation.- Appendix 5. Limit of the Spectra of the Interior Neumann Problems when a Solid Domain Shrinks to a Plane One.- Appendix 6. Construction of a Surface from its Principal Curvatures.- Appendix 7. Resonances.- Research Problems.- Bibliographical Notes.- List of Symbols.

The radon transform and local tomography

Abstract: Introduction Brief Description of New Results and the Aims of the Book Review of Some Applications of the Radon Transform Properties of the Radon Transform and Inversion Formulas Definitions and Properties of the Radon Transform and Related Transforms Inversion Formulas for R Singular Value Decomposition of the Radon Transform Estimates in Sobolev Spaces Inversion Formulas for the Backprojection Operator Inversion Formulas for X-Ray Transform Uniqueness Theorems for the Radon and X-Ray Transforms Attenuated and Exponential Radon Transforms Convergence Properties of the Inversion Formulas on Various Classes of Functions Range Theorems and Reconstruction Algorithms Range Functions for R on Smooth Functions Range Functions for R on Sobolev Spaces Range Theorems for R* Range Theorem for X-Ray Transform Numerical Solution of the Equation Rf = g with Noisy Data Filtered Backprojection Algorithm Other Reconstruction Algorithms Singularities of the Radon Transform Introduction Singular Support of the Radon Transform The Relation Between S and S (WE NEED A "HAT" OVER THE LAST S. See hard copy of toc for details) The Envelopes and the Duality Law Asymptotics of Rf Near S Singularities of the Radon Transform: An Alternative Approach Asymptotics of the Fourier Transform Wave Front Sets Singularities of X-Ray Transform Stable Calculation of the Legendre Transform Local Tomography Introduction A Family of Local Tomography Functions Optimization of Noise Stability Algorithm for Finding Values of Jumps of a Function Using Local Tomography Numerical Implementation Local Tomography for the Exponential Radon Transform Local Tomography for the Generalized Radon Transform Local Tomography for the Limited-Angle Data Asymptotics of Pseudodifferential Operators, Acting on a Piecewise-Smooth Function, f, Near the Singular Support of f Pseudolocal Tomography Introduction Definition of a Pseudolocal Tomography Function Investigation of the Convergence frc(x) Ae f(x) as r Ae 0 More Results on Functions frc, fr, and on convergence frc Ae f A Family of Pseudolocal Tomography Functions Numerical Implementation of Pseudolocal Tomography Pseudolocal Tomography for the Exponential Radon Transform Geometric Tomography Basic Idea Description of the Algorithm and Numerical Experiments Inversion of Incomplete Tomographic Data Inversion of Incomplete Fourier Transform Data Filtered Backprojection Method for Inversion of the Limited-Angle Tomographic Data The Extrapolation Problem The Davison-Grunbaum Algorithm Inversion of Cone-Beam Data Inversion of the Complete Cone-Beam Data Inversion of Incomplete Cone-Beam Data An Exact Algorithm for the Cone-Beam Circle Geometry g-Ray Tomography Radon Transform of Distributions Main Definitions Properties of the Test Function Spaces Examples Range Theorem for the Radon Transform on e' A Definition Based on Spherical Harmonics Expansion When Does the Radon Transform on Distributions Coincide with the Classical Radon Transform? The Dual Radon Transform on Distributions Abel-Type Integral Equation The Classical Abel Equation Abel-Type Equations Reduction of the Equation to a More Stable One Finding Locations and Values of Jumps of the Solution to the Abel Equation Multidimensional Algorithm for Finding Discontinuities of Signals from Noisy Discrete Data Introduction Edge Detection Algorithm Thin Line Detection Algorithm Generalization of the Algorithms Justification of the Edge Detection Algorithm Justification of the Algorithm for Thin Line Detection Justification of the General Scheme Numerical Experiments Proof of Auxiliary Results Test of Randomness and Its Applications Introduction Consistency of Rank Test Against Change Points (Change Surfaces) Alternative Consistency of Rank Test Against Trend in Location Auxiliary Results Abstract and Functional Spaces Distribution Theory Pseudodifferential and Fourier Integral Operators Special Functions Asymptotic Expansions Linear Equations in Banach Spaces Ill-Posed Problems Examples of Regularization of Ill-Posed Problems Radon Transform and PDE Statistics Research Problems Bibliographical Notes References Index List of Notations

Multidimensional inverse scattering problems

Abstract: Summary form only given, as follows. An overview of the author's results is given. The inverse problems for obstacle, geophysical and potential scattering are considered. The basic method for proving uniqueness theorems in one- and multi-dimensional inverse problems is discussed and illustrated by numerous examples. The method is based on property C for pairs of differential operators. Property C stands for completeness of the sets of products of solutions to homogeneous differential equations. To prove a uniqueness theorem in the inverse scattering problem one assumes that there are two operators which generate the same scattering data. This assumption allows one to derive an orthogonality relation from which, via property C, the uniqueness theorem follows. New results are discussed. These include property C for ordinary differential equations, inversion of I-function (impedance function), inversion of incomplete scattering data (for example, the phase shift of s-wave without the knowledge of bound states and norming constants but assuming a priori that the potential has compact support, etc). Analytical solution of the ground-penetrating radar problem is outlined. Open problems are formulated.

Dynamical Systems Method for Solving Operator Equations

Abstract: Preface Contents 1. Introduction 2. Ill-posed problems 3. DSM for well-posed problems 4. DSM and linear ill-posed problems 5. Some inequalities 6. DSM for monotone operators 7. DSM for general nonlinear operator equations 8 DSM for operators satisfying a spectral assumption 9. DSM in Banach spaces 10. DSM and Newton-type methods without inversion of the derivative 11. DSM and unbounded operators 12. DSM and nonsmooth operators 13. DSM as a theoretical tool 14. DSM and iterative methods 15. Numerical problems arising in applications 16. Auxiliary results from analysis Bibliographical notes Bibliography Index

The collected works

Abstract: This is a review of Collected Works of John Tate. Parts I, II, edited by Barry Mazur and Jean-Pierre Serre. American Mathematical Society, Providence, Rhode Island, 2016. For several decades it has been clear to the friends and colleagues of John Tate that a “Collected Works” was merited. The award of the Abel Prize to Tate in 2010 added impetus, and finally, in Tate’s ninety-second year we have these two magnificent volumes, edited by Barry Mazur and Jean-Pierre Serre. Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate. For an overview of Tate’s work, the editors refer the reader to [4]. Before discussing the volumes, I describe some of Tate’s work. 1. Hecke L-series and Tate’s thesis Like many budding number theorists, Tate’s favorite theorem when young was Gauss’s law of quadratic reciprocity. When he arrived at Princeton as a graduate student in 1946, he was fortunate to find there the person, Emil Artin, who had discovered the most general reciprocity law, so solving Hilbert’s ninth problem. By 1920, the German school of algebraic number theorists (Hilbert, Weber, . . .) together with its brilliant student Takagi had succeeded in classifying the abelian extensions of a number field K: to each group I of ideal classes in K, there is attached an extension L of K (the class field of I); the group I determines the arithmetic of the extension L/K, and the Galois group of L/K is isomorphic to I. Artin’s contribution was to prove (in 1927) that there is a natural isomorphism from I to the Galois group of L/K. When the base field contains an appropriate root of 1, Artin’s isomorphism gives a reciprocity law, and all possible reciprocity laws arise this way. In the 1930s, Chevalley reworked abelian class field theory. In particular, he replaced “ideals” with his “idèles” which greatly clarified the relation between the local and global aspects of the theory. For his thesis, Artin suggested that Tate do the same for Hecke L-series. When Hecke proved that the abelian L-functions of number fields (generalizations of Dirichlet’s L-functions) have an analytic continuation throughout the plane with a functional equation of the expected type, he saw that his methods applied even to a new kind of L-function, now named after him. Once Tate had developed his harmonic analysis of local fields and of the idèle group, he was able prove analytic continuation and functional equations for all the relevant L-series without Hecke’s complicated theta-formulas. Received by the editors September 5, 2016. 2010 Mathematics Subject Classification. Primary 01A75, 11-06, 14-06. c ©2017 American Mathematical Society