Part of Cambridge Monographs on Applied and Computational Mathematics
Not yet published - available from March 2022
FORMAT: Hardback ISBN: 9781107072633
Quasi-interpolation is one of the most useful and often applied methods for the approximation of functions and data in mathematics and applications. Its advantages are manifold: quasi-interpolants are able to approximate in any number of dimensions, they are efficient and relatively easy to formulate for scattered and meshed nodes and for any number of data. This book provides an introduction into the field for graduate students and researchers, outlining all the mathematical background and methods of implementation. The mathematical analysis of quasi-interpolation is given in three directions, namely on the basis (spline spaces, radial basis functions) from which the approximation is taken, on the form and computation of the quasi-interpolants (point evaluations, averages, least squares), and on the mathematical properties (existence, locality, convergence questions, precision). Learn which type of quasi-interpolation to use in different contexts and how to optimise its features to suit applications in physics and engineering.
Provides an in-depth summary of approximations using quasi-interpolation
Explains the advantages of several different approaches to quasi-interpolation, including different convergence properties, smoothness and precision of approximants
Offers a large range of examples and practical applications of quasi-interpolants, including scattered data, uniform data, solving PDEs and data compression
1. Introduction
2. Generalities on quasi-interpolation
3. Univariate RBF quasi-interpolants
4. Spline quasi-interpolants
5. Quasi-interpolants for periodic functions
6. Multivariate spline quasi-interpolants
7. Multivariate quasi-interpolants: construction in n dimensions
8. Quasi-interpolation on the sphere
9. Other quasi-interpolants and wavelets
10. Special cases and applications
References
Index.
Part of Institute of Mathematical Statistics Monographs
Not yet published - available from May 2022
FORMAT: Hardback ISBN: 9781108480291
Stable Levy processes lie at the intersection of Levy processes and self-similar Markov processes. Processes in the latter class enjoy a Lamperti-type representation as the space-time path transformation of so-called Markov additive processes (MAPs). This completely new mathematical treatment takes advantage of the fact that the underlying MAP for stable processes can be explicitly described in one dimension and semi-explicitly described in higher dimensions, and uses this approach to catalogue a large number of explicit results describing the path fluctuations of stable Levy processes in one and higher dimensions. Written for graduate students and researchers in the field, this book systemically establishes many classical results as well as presenting many recent results appearing in the last decade, including previously unpublished material. Topics explored include first hitting laws for a variety of sets, path conditionings, law-preserving path transformations, the distribution of extremal points, growth envelopes and winding behaviour.
This self-contained reference catalogues a wide range of results from the literature, including worked computations and new proofs
Documents the last 15 years of development, presented by authors who helped clarify the theory through their research
Written in a friendly, accessible style
Part of Cambridge Monographs on Applied and Computational Mathematics
Not yet published - available from April 2022
FORMAT: HardbackISBN: 9781316518878
Recovering the phase of the Fourier transform is a ubiquitous problem in imaging applications from astronomy to nanoscale X-ray diffraction imaging. Despite the efforts of a multitude of scientists, from astronomers to mathematicians, there is, as yet, no satisfactory theoretical or algorithmic solution to this class of problems. Written for mathematicians, physicists and engineers working in image analysis and reconstruction, this book introduces a conceptual, geometric framework for the analysis of these problems, leading to a deeper understanding of the essential, algorithmically independent, difficulty of their solutions. Using this framework, the book studies standard algorithms and a range of theoretical issues in phase retrieval and provides several new algorithms and approaches to this problem with the potential to improve the reconstructed images. The book is lavishly illustrated with the results of numerous numerical experiments that motivate the theoretical development and place it in the context of practical applications.
Features a careful analysis of the class of maps used in most algorithms, called hybrid iterative maps, including a complete description of the geometry underlying this class of maps that reveals many surprising properties
Equips the reader with tools to easily see when an algorithm provides reliable phase information for particular frequencies
Contains extensive background material on the mathematics employed in the book, making it accessible to a wide range of technically sophisticated physicists and engineers with an interest in phase retrieval
Includes nearly 200 color illustrations, including numerical examples whose results are displayed graphically
Part I. Theoretical Foundations:
1. The geometry near an intersection
2. Well posedness
3. Uniqueness and the non-negativity constraint
4. Some preliminary conclusions
Part II. Analysis of Algorithms for Phase Retrieval:
6. Introduction to Part II
7. Algorithms for Phase Retrieval
8. Discrete classical phase retrieval
9. The non-negativity constraint
10. Asymptotics of hybrid iterative maps
Part III. Further Properties of Hybrid Iterative Algorithms and Suggestions for Improvement:
11. Introduction to Part III
12. Statistics of algorithms
13. Suggestions for improvements
14. Concluding Remarks
15. Notational conventions.
Not yet published - available from May 2022
FORMAT: Hardback ISBN: 9781108423564
Fay and Brittain present statistical hypothesis testing and compatible confidence intervals, focusing on application and proper interpretation. The emphasis is on equipping applied statisticians with enough tools - and advice on choosing among them - to find reasonable methods for almost any problem and enough theory to tackle new problems by modifying existing methods. After covering the basic mathematical theory and scientific principles, tests and confidence intervals are developed for specific types of data. Essential methods for applications are covered, such as general procedures for creating tests (e.g., likelihood ratio, bootstrap, permutation, testing from models), adjustments for multiple testing, clustering, stratification, causality, censoring, missing data, group sequential tests, and non-inferiority tests. New methods developed by the authors are included throughout, such as melded confidence intervals for comparing two samples and confidence intervals associated with Wilcoxon-Mann-Whitney tests and Kaplan-Meier estimates. Examples, exercises, and the R package asht support practical use.
Encapsulates 60 years of experience with consequential applications in a unified presentation of the most useful methods and how to evaluate and modify them
Presents ideas of causal inference used in clinical trials for decades together with a modern account of causal ideas using potential outcomes
Provides R package asht with functions for immediate application of both new and classical methods
Includes 95 end-of-chapter exercises, and over 100 examples that illustrate key concepts
1. Introduction
2. Theory of tests, p-values, and confidence intervals
3. From scientific theory to statistical hypothesis test
4. One sample studies with binary responses
5. One sample studies with ordinal or numeric responses
6. Paired data
7. Two sample studies with binary responses
8. Assumptions and hypothesis tests
9. Two sample studies with ordinal or numeric responses
10. General methods for creating decision rules
11. K-Sample studies and trend tests
12. Clustering and stratification
13. Multiplicity in testing
14. Testing from models
15. Causality
16. Censoring
17. Missing data
18. Group sequential and related adaptive methods
19. Testing fit, equivalence, and non-inferiority
20. Power and sample size.
Not yet published - available from May 2022
FORMAT: Hardback ISBN: 9781108831086
Group theory, originating from algebraic structures in mathematics, has long been a powerful tool in many areas of physics, chemistry and other applied sciences, but it has seldom been covered in a manner accessible to undergraduates. This book from renowned educator Robert Kolenkow introduces group theory and its applications starting with simple ideas of symmetry, through quantum numbers, and working up to particle physics. It features clear explanations, accompanying problems and exercises, and numerous worked examples from experimental research in the physical sciences. Beginning with key concepts and necessary theorems, topics are introduced systematically including: molecular vibrations and lattice symmetries; matrix mechanics; wave mechanics; rotation and quantum angular momentum; atomic structure; and finally particle physics. This comprehensive primer on group theory is ideal for advanced undergraduate topics courses, reading groups, or self-study, and it will help prepare graduate students for higher-level courses.
Uniquely suitable for advanced undergraduate level, as most group theory texts and courses begin at graduate level, enabling students to appreciate why group theory is central to modern physics
A wide range of examples clarify the use of group theory, including selected topics not often covered at this level, e.g. matrix mechanics, entanglement, EPR paradox, and photon correlation
Avoid formal math-heavy proofs in the main text, but important results are provided in the Appendices
1. Fundamental concepts
2. Matrix representations of discrete groups
3. Molecular vibrations
4. Crystalline solids
5. Bohr's quantum theory and matrix mechanics
6. Wave mechanics, measurement, and entanglement
7. Rotation
8. Quantum angular momentum
9. The structure of atoms
10. Particle physics. Appendices.