Geometry of the Phase Retrieval Problem

Geometry of the Phase Retrieval Problem
Author: Alexander H. Barnett
Publisher: Cambridge University Press
Total Pages: 321
Release: 2022-05-05
Genre: Mathematics
ISBN: 1316518876

This book provides a theoretical foundation and conceptual framework for the problem of recovering the phase of the Fourier transform.

Nanoscale Photonic Imaging

Nanoscale Photonic Imaging
Author: Tim Salditt
Publisher: Springer Nature
Total Pages: 634
Release: 2020-06-09
Genre: Science
ISBN: 3030344134

This open access book, edited and authored by a team of world-leading researchers, provides a broad overview of advanced photonic methods for nanoscale visualization, as well as describing a range of fascinating in-depth studies. Introductory chapters cover the most relevant physics and basic methods that young researchers need to master in order to work effectively in the field of nanoscale photonic imaging, from physical first principles, to instrumentation, to mathematical foundations of imaging and data analysis. Subsequent chapters demonstrate how these cutting edge methods are applied to a variety of systems, including complex fluids and biomolecular systems, for visualizing their structure and dynamics, in space and on timescales extending over many orders of magnitude down to the femtosecond range. Progress in nanoscale photonic imaging in Göttingen has been the sum total of more than a decade of work by a wide range of scientists and mathematicians across disciplines, working together in a vibrant collaboration of a kind rarely matched. This volume presents the highlights of their research achievements and serves as a record of the unique and remarkable constellation of contributors, as well as looking ahead at the future prospects in this field. It will serve not only as a useful reference for experienced researchers but also as a valuable point of entry for newcomers.

Advancements in Complex Analysis

Advancements in Complex Analysis
Author: Daniel Breaz
Publisher: Springer Nature
Total Pages: 538
Release: 2020-05-12
Genre: Mathematics
ISBN: 3030401200

The contributions to this volume are devoted to a discussion of state-of-the-art research and treatment of problems of a wide spectrum of areas in complex analysis ranging from pure to applied and interdisciplinary mathematical research. Topics covered include: holomorphic approximation, hypercomplex analysis, special functions of complex variables, automorphic groups, zeros of the Riemann zeta function, Gaussian multiplicative chaos, non-constant frequency decompositions, minimal kernels, one-component inner functions, power moment problems, complex dynamics, biholomorphic cryptosystems, fermionic and bosonic operators. The book will appeal to graduate students and research mathematicians as well as to physicists, engineers, and scientists, whose work is related to the topics covered.

The Fractional Fourier Transform

The Fractional Fourier Transform
Author: Haldun M. Ozaktas
Publisher: John Wiley & Sons
Total Pages: 546
Release: 2001-02-08
Genre: Mathematics
ISBN:

The discovery of the Fractional Fourier Transform and its role in optics and data management provides an elegant mathematical framework within which to discuss diffraction and other fundamental aspects of optical systems. This book explains how the fractional Fourier transform has allowed the generalization of the Fourier transform and the notion of the frequency transform. It will serve as the standard reference on Fourier transforms for many years to come.

Coherent X-Ray Optics

Coherent X-Ray Optics
Author: David Paganin
Publisher: OUP Oxford
Total Pages: 424
Release: 2006-01-12
Genre: Science
ISBN: 019152431X

This book gives a thorough treatment of the rapidly-expanding field of coherent x-ray optics, which has recently experienced something of a renaissance with the availability of third-generation synchrotron sources. It is the first book of its kind. The author begins with a treatment of the fundamentals of x-ray diffraction for both coherent and partially coherent radiation, together with the interactions of x-rays with matter. X-ray sources, optics elements and detectors are then discussed, with an emphasis on their role in coherent x-ray optics. Various facets of coherent x-ray imaging are then discussed, including holography, interferometry, self imaging, phase contrast and phase retrieval. Lastly, the foundations of the new field of singular x-ray optics are examined. Most topics are developed from first principles, with numerous references given to the contemporary research literature. This book will be useful to x-ray physicists and students, together with optical physicists and engineers who wish to learn more about the fascinating subject of coherent x-ray optics.

Sparse Solutions of Underdetermined Linear Systems and Their Applications

Sparse Solutions of Underdetermined Linear Systems and Their Applications
Author: Ming-Jun Lai
Publisher: SIAM
Total Pages:
Release: 2021-06-25
Genre: Mathematics
ISBN: 1611976510

This textbook presents a special solution to underdetermined linear systems where the number of nonzero entries in the solution is very small compared to the total number of entries. This is called a sparse solution. Since underdetermined linear systems can be very different, the authors explain how to compute a sparse solution using many approaches. Sparse Solutions of Underdetermined Linear Systems and Their Applications contains 64 algorithms for finding sparse solutions of underdetermined linear systems and their applications for matrix completion, graph clustering, and phase retrieval and provides a detailed explanation of these algorithms including derivations and convergence analysis. Exercises for each chapter help readers understand the material. This textbook is appropriate for graduate students in math and applied math, computer science, statistics, data science, and engineering. Advisors and postdoctoral scholars will also find the book interesting and useful.

Foundations of Time-Frequency Analysis

Foundations of Time-Frequency Analysis
Author: Karlheinz Gröchenig
Publisher: Springer Science & Business Media
Total Pages: 367
Release: 2013-12-01
Genre: Technology & Engineering
ISBN: 1461200032

Time-frequency analysis is a modern branch of harmonic analysis. It com prises all those parts of mathematics and its applications that use the struc ture of translations and modulations (or time-frequency shifts) for the anal ysis of functions and operators. Time-frequency analysis is a form of local Fourier analysis that treats time and frequency simultaneously and sym metrically. My goal is a systematic exposition of the foundations of time-frequency analysis, whence the title of the book. The topics range from the elemen tary theory of the short-time Fourier transform and classical results about the Wigner distribution via the recent theory of Gabor frames to quantita tive methods in time-frequency analysis and the theory of pseudodifferential operators. This book is motivated by applications in signal analysis and quantum mechanics, but it is not about these applications. The main ori entation is toward the detailed mathematical investigation of the rich and elegant structures underlying time-frequency analysis. Time-frequency analysis originates in the early development of quantum mechanics by H. Weyl, E. Wigner, and J. von Neumann around 1930, and in the theoretical foundation of information theory and signal analysis by D.

High-Dimensional Probability

High-Dimensional Probability
Author: Roman Vershynin
Publisher: Cambridge University Press
Total Pages: 299
Release: 2018-09-27
Genre: Business & Economics
ISBN: 1108415199

An integrated package of powerful probabilistic tools and key applications in modern mathematical data science.

Introduction to Information Retrieval

Introduction to Information Retrieval
Author: Christopher D. Manning
Publisher: Cambridge University Press
Total Pages:
Release: 2008-07-07
Genre: Computers
ISBN: 1139472100

Class-tested and coherent, this textbook teaches classical and web information retrieval, including web search and the related areas of text classification and text clustering from basic concepts. It gives an up-to-date treatment of all aspects of the design and implementation of systems for gathering, indexing, and searching documents; methods for evaluating systems; and an introduction to the use of machine learning methods on text collections. All the important ideas are explained using examples and figures, making it perfect for introductory courses in information retrieval for advanced undergraduates and graduate students in computer science. Based on feedback from extensive classroom experience, the book has been carefully structured in order to make teaching more natural and effective. Slides and additional exercises (with solutions for lecturers) are also available through the book's supporting website to help course instructors prepare their lectures.

Dispersion, Complex Analysis and Optical Spectroscopy

Dispersion, Complex Analysis and Optical Spectroscopy
Author: Kai-Erik Peiponen
Publisher: Springer Science & Business Media
Total Pages: 152
Release: 1998-11-30
Genre: Technology & Engineering
ISBN: 9783540645221

This book is devoted to dispersion theory in linear and nonlinear optics. Dispersion relations and methods of analysis in optical spectroscopy are derived with the aid of complex analysis. The book introduces the mathematical basis and derivations of various dispersion relations that are used in optical spectroscopy. In addition, it presents the dispersion theory of the nonlinear optical processes which are essential in modern optical spectroscopy. The book includes new methods such as the maximum entropy model for wavelength-dependent spectra analysis.