Format: Paperback / softback, 133 pages, height x width: 235x155 mm, weight: 232 g, IX, 133 p.,
Series: Sources and Studies in the History of Mathematics and Physical Sciences
Pub. Date: 17-Dec-2022
ISBN-13: 9783030872984
Paris of the year 1900 left two landmarks: the Tour Eiffel, and David Hilbert's celebrated list of twenty-four mathematical problems presented at a conference opening the new century. Kurt Goedel, a logical icon of that time, showed Hilbert's ideal of complete axiomatization of mathematics to be unattainable. The result, of 1931, is called Goedel's incompleteness theorem. Goedel then went on to attack Hilbert's first and second Paris problems, namely Cantor's continuum problem about the type of infinity of the real numbers, and the freedom from contradiction of the theory of real numbers. By 1963, it became clear that Hilbert's first question could not be answered by any known means, half of the credit of this seeming faux pas going to Goedel. The second is a problem still wide open. Goedel worked on it for years, with no definitive results; The best he could offer was a start with the arithmetic of the entire numbers. This book, Goedel's lectures at the famous Princeton Institute for Advanced Study in 1941, shows how far he had come with Hilbert's second problem, namely to a theory of computable functionals of finite type and a proof of the consistency of ordinary arithmetic. It offers indispensable reading for logicians, mathematicians, and computer scientists interested in foundational questions. It will form a basis for further investigations into Goedel's vast Nachlass of unpublished notes on how to extend the results of his lectures to the theory of real numbers. The book also gives insights into the conceptual and formal work that is needed for the solution of profound scientific questions, by one of the central figures of 20th century science and philosophy.
Goedel's Functional Interpretation in Context.- Part I: Axiomatic Intuitionist Logic.- Part II: The Functional Interpretation.- References.- Name Index.
Format: Hardback, 425 pages, height x width: 235x155 mm,
weight: 816 g, 1 Illustrations, black and white; X, 425 p. 1 illus.
Pub. Date: 02-Jan-2023
ISBN-13: 9783031238161
This book introduces the reader to quantum groups, focusing on the simplest ones, namely the closed subgroups of the free unitary group. Although such quantum groups are quite easy to understand mathematically, interesting examples abound, including all classical Lie groups, their free versions, half-liberations, other intermediate liberations, anticommutation twists, the duals of finitely generated discrete groups, quantum permutation groups, quantum reflection groups, quantum symmetry groups of finite graphs, and more. The book is written in textbook style, with its contents roughly covering a one-year graduate course. Besides exercises, the author has included many remarks, comments and pieces of advice with the lone reader in mind. The prerequisites are basic algebra, analysis and probability, and a certain familiarity with complex analysis and measure theory. Organized in four parts, the book begins with the foundations of the theory, due to Woronowicz, comprising axioms, Haar measure, Peter-Weyl theory, Tannakian duality and basic Brauer theorems. The core of the book, its second and third parts, focus on the main examples, first in the continuous case, and then in the discrete case. The fourth and last part is an introduction to selected research topics, such as toral subgroups, homogeneous spaces and matrix models. Introduction to Quantum Groups offers a compelling introduction to quantum groups, from the simplest examples to research level topics.
Part I. Quantum groups.
Chapter 1. Quantum spaces.
Chapter 2. Quantum groups.
Chapter 3. Representation theory.
Chapter 4. Tannakian duality.- Part II. Quantum rotations.
Chapter 5. Free rotations.
Chapter 6. Unitary groups.
Chapter 7. Easiness, twisting.
Chapter 8. Probabilistic aspects.- Part III. Quantum permutations.
Chapter 9. Quantum permutations.
Chapter 10. Quantum reflections.
Chapter 11. Classification results.
Chapter 12. The standard cube.- Part IV. Advanced topics.
Chapter 13. Toral subgroups.-
Chapter 14. Amenability, growth.
Chapter 15. Homogeneous spaces.
Chapter 16. Modelling questions.- Bibliography.- Index.
Format: Paperback / softback, 333 pages, height x width: 235x155 mm, weight: 539 g, XVII, 333 p
Series: Indian Statistical Institute Series
Pub. Date: 19-Dec-2022
ISBN-13: 9789811678837
This book discusses the theory of wavelets on local fields of positive characteristic. The discussion starts with a thorough introduction to topological groups and local fields. It then provides a proof of the existence and uniqueness of Haar measures on locally compact groups. It later gives several examples of locally compact groups and describes their Haar measures. The book focuses on multiresolution analysis and wavelets on a local field of positive characteristic. It provides characterizations of various functions associated with wavelet analysis such as scaling functions, wavelets, MRA-wavelets and low-pass filters. Many other concepts which are discussed in details are biorthogonal wavelets, wavelet packets, affine and quasi-affine frames, MSF multiwavelets, multiwavelet sets, generalized scaling sets, scaling sets, unconditional basis properties of wavelets and shift invariant spaces.
Local Fields.- Multiresolution Analysis on Local Fields.- Affine, Quasi-Affine and Co-Affine Frames.- Characterizations in Wavelet Analysis.- Biorthogonal Wavelets.- Wavelet Packets and Frame Packets.- Wavelets as Unconditional Bases.- Shift-Invariant Spaces and Wavelets.
Format: Paperback / softback, 448 pages, height x width: 235x155 mm, 63 Illustrations, black and white; XII, 448 p. 63 illus
Pub. Date: 10-Mar-2023
ISBN-13: 9783030885779
The Mathematics of Errors presents an original, rigorous and systematic approach to the calculus of errors, targeted at both the engineer and the mathematician. Starting from Gauss's original point of view, the book begins as an introduction suitable for graduate students, leading to recent developments in stochastic analysis and Malliavin calculus, including contributions by the author. Later chapters, aimed at a more mature audience, require some familiarity with stochastic calculus and Dirichlet forms. Sensitivity analysis, in particular, plays an important role in the book. Detailed applications in a range of fields, such as engineering, robotics, statistics, financial mathematics, climate science, or quantum mechanics are discussed through concrete examples. Throughout the book, error analysis is presented in a progressive manner, motivated by examples and appealing to the reader's intuition. By formalizing the intuitive concept of error and richly illustrating its scope for application, this book provides readers with a blueprint to apply advanced mathematics in practical settings. As such, it will be of immediate interest to engineers and scientists, whilst providing mathematicians with an original presentation. Nicolas Bouleau has directed the mathematics center of the Ecole des Ponts ParisTech for more than ten years. He is known for his theory of error propagation in complex models. After a degree in engineering and architecture, he decided to pursue a career in mathematics under the influence of Laurent Schwartz. He has also written on the production of knowledge, sustainable economics and mathematical models in finance. Nicolas Bouleau is a recipient of the Prix Montyon from the French Academy of Sciences.
I. Error computations a la Gauss.-
1. The different approaches.-
2. Finite dimensional examples.-
3. An intuitive introduction to error structures.-
4. Weakly and strongly random errors.- II. Probabilistic and Functional Models.-
5. Strongly continuous semi-groups and Dirichlet forms.-
6. Error structures.-
7. Images and products of error structures.-
8. The gradient and the sharp and other calculation tools.-
9. Error structures on fundamental spaces.- III. The subtleness of the notion of bias.-
10. Approximation and bias operators.-
11. Computations and simulation methods.- IV. Error structures and its applications.-
12. Statistical identification of error structures.-
13. The instantaneous structure of a stochastic process.-
14. Models inspired by finance.-
15. Examples in Physics.-
16. The principle of arbitrary functions and error structures.- V. Historical elements and research themes.-
17. Error calculations from Gauss and Laplace.-
18. Extensions and open questions.- Hints for exercises.- Pedagogical References.- Chronological Bibliography.
Format: Hardback, 368 pages, height x width: 235x155 mm, 20 Tables, color; 20 Illustrations, color; 128 Illustrations,
black and white; VIII, 368 p. 148 illus., 20 illus. in color. With online files/update.
Pub. Date: 06-Apr-2023
ISBN-13: 9783031227561
This text describes a comprehensive adjoint sensitivity analysis methodology (C-ASAM), developed by the author, enabling the efficient and exact computation of arbitrarily high-order functional derivatives of model responses to model parameters in large-scale systems. The model's responses can be either scalar-valued functionals of the model's parameters and state variables (as customarily encountered, e.g., in optimization problems) or general function-valued responses, which are often of interest but are currently not amenable to efficient sensitivity analysis. The C-ASAM framework is set in linearly increasing Hilbert spaces, each of state-function-dimensionality, as opposed to exponentially increasing parameter-dimensional spaces, thereby breaking the so-called "curse of dimensionality" in sensitivity and uncertainty analysis. The C-ASAM applies to any model; the larger the number of model parameters, the more efficient the C-ASAM becomes for computing arbitrarily high-order response sensitivities. The text includes illustrative paradigm problems which are fully worked-out to enable the thorough understanding of the C-ASAM's principles and their practical application. The book will be helpful to those working in the fields of sensitivity analysis, uncertainty quantification, model validation, optimization, data assimilation, model calibration, sensor fusion, reduced-order modelling, inverse problems and predictive modelling. It serves as a textbook or as supplementary reading for graduate course on these topics, in academic departments in the natural, biological, and physical sciences and engineering. This Volume Three, the third of three, covers systems that are nonlinear in the state variables, model parameters and associated responses. The selected illustrative paradigm problems share these general characteristics. A separate Volume One covers systems that are linear in the state variables.
Part A: Function-Valued Responses
Chapter 1: The First- and Second-Order Comprehensive Adjoint Sensitivity Analysis Methodologies for Nonlinear Systems with Function-Valued Responses 1.1. The First-Order Comprehensive Selective Adjoint Sensitivity Analysis Methodology (CS-ASAM-1) for Nonlinear Systems with Function-Valued Responses 1.1.1: CS-ASAM-1 Methodology: Finite-Dimensional (Algebraic) Systems 1.1.2: CS-ASAM-1 Methodology: Infinite-Dimensional (Operator) Systems 1.2. The Second-Order Comprehensive Selective Adjoint Sensitivity Analysis Methodology (CS-ASAM-2) for Nonlinear Systems with Function-Valued Responses 1.2.1: CS-ASAM-2 Methodology: Finite-Dimensional (Algebraic) Systems 1.2.2: CS-ASAM-2 Methodology: Infinite-Dimensional (Operator) Systems 1.3. The First-Order Comprehensive Kernel Adjoint Sensitivity Analysis Methodology (CK-ASAM-1) for Linear Systems with Function-Valued Responses 1.3.1: CK-ASAM-1 Methodology: Finite-Dimensional (Algebraic) Systems 1.3.2: CK-ASAM-1 Methodology: Infinite-Dimensional (Operator) Systems 1.4. The Second-Order Comprehensive Kernel Adjoint Sensitivity Analysis Methodology (CK-ASAM-2) for Linear Systems with Function-Valued Responses 1.4.1: CK-ASAM-2 Methodology: Finite-Dimensional (Algebraic) Systems 1.4.2: CK-ASAM-2 Methodology: Infinite-Dimensional (Operator) Systems
Chapter 2: The Third-Order Comprehensive Adjoint Sensitivity Analysis Methodology (C-ASAM-3) for Nonlinear Systems with Function-Valued Responses 2.1. The Third-Order Comprehensive Selective Adjoint Sensitivity Analysis Methodology (CS-ASAM-3) for Nonlinear Systems with Function-Valued Responses 2.1.1: CS-ASAM-3 Methodology: Finite-Dimensional (Algebraic) Systems 2.1.2: CS-ASAM-3 Methodology: Infinite-Dimensional (Operator) Systems 2.2. The Third-Order Comprehensive Kernel Adjoint Sensitivity Analysis Methodology (CK-ASAM-3) for Linear Systems with Function-Valued Responses 2.2.1: CK-ASAM-3 Methodology: Finite-Dimensional (Algebraic) Systems 2.2.2: CK-ASAM-3 Methodology: Infinite-Dimensional (Operator) Systems
Chapter 3: The Fourth-Order Comprehensive Adjoint Sensitivity Analysis Methodology (C-ASAM-4) for Nonlinear Systems with Function-Valued Responses 3.1. The Fourth-Order Comprehensive Selective Adjoint Sensitivity Analysis Methodology (CS-ASAM-4) for Nonlinear Systems with Function-Valued Responses 3.1.1: CS-ASAM-4 Methodology: Finite-Dimensional (Algebraic) Systems 3.1.2: CS-ASAM-4 Methodology: Infinite-Dimensional (Operator) Systems 3.2. The Fourth-Order Comprehensive Kernel Adjoint Sensitivity Analysis Methodology (CK-ASAM-4) for Linear Systems with Function-Valued Responses 3.2.1: CK-ASAM-4 Methodology: Finite-Dimensional (Algebraic) Systems 3.2.2: CK-ASAM-4 Methodology: Infinite-Dimensional (Operator) Systems
Chapter 4: The Nth-Order Adjoint Sensitivity Analysis Methodology (C-ASAM-N) for Nonlinear Systems with Function-Valued Responses 4.1. The Arbitrarily-High Nth-Order Comprehensive Selective Adjoint Sensitivity Analysis Methodology (CS-ASAM-N) for Nonlinear Systems with Function-Valued Responses 4.1.1: CS-ASAM-N Methodology: Finite-Dimensional (Algebraic) Systems 4.1.2: CS-ASAM-N Methodology: Infinite-Dimensional (Operator) Systems< 4.2. The Arbitrarily-High Nth-Order Comprehensive Kernel Adjoint Sensitivity Analysis Methodology (CK-ASAM-N) for Nonlinear Systems with Function-Valued Responses 4.2.1: CK-ASAM-N Methodology: Finite-Dimensional (Algebraic) Systems 4.2.2: CK-ASAM-N Methodology: Infinite-Dimensional (Operator) Systems Part B: Scalar-Valued Responses
Chapter 5: The Fourth-Order Comprehensive Adjoint Sensitivity Analysis Methodology (C-ASAM-4) for Nonlinear Systems with Scalar-Valued Responses 5.1. The Fourth-Order Comprehensive Selective Adjoint Sensitivity Analysis Methodology (CS-ASAM-4) for Nonlinear Systems with Scalar-Valued Responses 5.1.1: CS-ASAM-4 Methodology: Finite-Dimensional (Algebraic) Systems 5.1.2: CS-ASAM-4 Methodology: Infinite-Dimensional (Operator) Systems 5.2. The Fourth-Order Comprehensive Kernel Adjoint Sensitivity Analysis Methodology (CK-ASAM-4) for Nonlinear Systems with Scalar-Valued Responses 5.2.1: CK-ASAM-4 Methodology: Finite-Dimensional (Algebraic) Systems 5.2.2: CK-ASAM-4 Methodology: Infinite-Dimensional (Operator) Systems
Chapter 6: The Nth-Order Adjoint Sensitivity Analysis Methodology (C-ASAM-N) for Nonlinear Systems with Scalar-Valued Responses 6.1. The Arbitrarily-High Nth-Order Comprehensive Selective Adjoint Sensitivity Analysis Methodology (CS-ASAM-N) for Nonlinear Systems with Scalar-Valued Responses 6.1.1: CS-ASAM-N Methodology: Finite-Dimensional (Algebraic) Systems 6.1.2: CS-ASAM-N Methodology: Infinite-Dimensional (Operator) Systems 6.2. The Arbitrarily-High Nth-Order Comprehensive Kernel Adjoint Sensitivity Analysis Methodology (CK-ASAM-N) for Nonlinear Systems with Scalar-Valued Responses 6.2.1: CS-ASAM-N Methodology: Finite-Dimensional (Algebraic) Systems 6.2.2: CS-ASAM-N Methodology: Infinite-Dimensional (Operator) Systems
Chapter 7: Applications of C-ASAM to Uncertainty Analysis