Format: Hardback, 576 pages, height x width: 235x155 mm, 3 Illustrations, color;
11 Illustrations, black and white; X, 576 p. 14 illus., 3 illus. in color., 1 Hardback
Pub. Date: 14-Sep-2025
ISBN-13: 9783031966712
This monograph presents the theory of the Mellin transform and the resulting Mellin analysis in a rigorous and unified manner. Often dismissed as a subordinate topic within Fourier and Laplace transform, it is instead demonstrated here that the theory is completely independent, can be studied within a self-contained framework, and exhibits some typical characteristics.
In addition to highlighting the foundations of the theory, the book addresses applications to certain partial differential equations, sampling theory and numerical quadrature. These applications provide methods which are in turn of interest in various areas of mathematics, science, and engineering.
Each chapter is enriched by numerous references to further literature and potential research directions. Researchers working in this field will gain new insights and appreciate the deserved attention for this underrated topic in harmonic analysis.
1. Preliminaries.-
2. Polar-Analytic Functions.-
3. The Foundation of Mellin Analysis.-
4. The Mellin Convolution.-
5. The Finite Mellin transform and the MellinFourier Series.-
6. The Mellin Transform in Xc.-
7. The Mellin Transform in Spaces Xp/c for 1 < p 2.-
8. Mellin Bandlimited Functions and PaleyWiener Theorems in Mellin Setting.-
9. Mellin Transforms and Fractional Analysis.-
10. The Mellin Distance.-
11. Mellin Transform Methods for Partial Differential Equations.-
12. Exponential Sampling Theory.-
13. Generalized Exponential Sampling Theory.-
14. Applications to Quadrature over the Positive Real Axis.
Format: Hardback, 205 pages, height x width: 235x155 mm, 36 Illustrations, color;
55 Illustrations, black and white; X, 205 p. 91 illus., 36 illus. in color., 1 Hardback
Series: Mathematics of Data 3
Pub. Date: 24-Aug-2025
ISBN-13: 9783031979729
This open access book provides a robust exposition of the mathematical foundations of data representation, focusing on two essential pillars of dimensionality reduction methods, namely geometry in general and Riemannian geometry in particular, and category theory.
Presenting a list of examples consisting of both geometric objects and empirical datasets, this book provides insights into the different effects of dimensionality reduction techniques on data representation and visualization, with the aim of guiding the reader in understanding the expected results specific to each method in such scenarios.
As a showcase, the dimensionality reduction method of Uniform Manifold Approximation and Projection (UMAP) has been used in this book, as it is built on theoretical foundations from all the areas we want to highlight here. Thus, this book also aims to systematically present the details of constructing a metric representation of a locally distorted metric space, which is essentially the problem that UMAP is trying to address, from a more general perspective.
Explaining how UMAP fits into this broader framework, while critically evaluating the underlying ideas, this book finally introduces an alternative algorithm to UMAP. This algorithm, called IsUMap, retains many of the positive features of UMAP, while improving on some of its drawbacks.
Chapter 1. Introduction.
Chapter 2. Illustrating UMAP on some simple data sets.
Chapter 3. Metrics and Riemannian manifolds.
Chapter 4. Merging fuzzy simplicial sets and metric spaces: A category theoretical approach.-
Chapter 5. UMAP.
Chapter 6. IsUMap: An alternative to the UMAP embedding.
Format: Hardback, 557 pages, height x width: 235x155 mm, 1 Illustrations, color; 5 Illustrations, black and white; X, 557 p. 6 illus., 1 illus. in color., 1 Hardback
Series: Birkhauser Advanced Texts / Basler Lehrbucher
Pub. Date: 19-Aug-2025
ISBN-13: 9783031965050
The purpose of this book is to provide an invitation to the beautiful and important subject of ergodic theorems, both classical and modern, which lies at the intersection of many fundamental mathematical disciplines: dynamical systems, probability theory, topology, algebra, number theory, analysis and functional analysis. The book is suitable for undergraduate and graduate students as well as non-specialists with basic knowledge of functional analysis, topology and measure theory.
Starting from classical ergodic theorems due to von Neumann and Birkhoff, the state-of-the-art of modern ergodic theorems such as subsequential, multiple and weighted ergodic theorems are presented. In particular, two deep connections between ergodic theorems and number theory are discussed: Furstenbergs famous proof of Szemeredis theorem on existence of arithmetic progressions in large sets of integers, and the Sarnak conjecture on the random behavior of the Mobius function.
An extensive list of references to other literature for readers wishing to deepen their knowledge is provided.
Chapter 1. Preliminaries.- Part I. The Fundamentals.
Chapter 2. Measure-preserving systems.
Chapter 3. Minimality and ergodicity.
Chapter 4. Some Fourier analysis.
Chapter 5. The spectral theorem.
Chapter 6. Decompositions in Hilbert spaces.- Part II. Classical Ergodic Theorems.-
Chapter 7. Classical ergodic theorems and more.
Chapter 8. Factors of measure-preserving systems.
Chapter 9. First applications of ergodic theorems.
Chapter 10. Equidistribution.
Chapter 11. Groups, semigroups and homogeneous spaces.- Part III. More Ergodic Theorems.
Chapter 12. Subsequential ergodic theorems.
Chapter 13. Multiple recurrence.
Chapter 14. Nilsystems.
Chapter 15. GowersHostKra seminorms and multiple convergence.
Chapter 16. Weighted ergodic theorems.
Chapter 17. Sarnaks conjecture.
Format: Hardback, 294 pages, height x width: 235x155 mm, XIX, 294 p., 1 Hardback
Series: Springer Monographs in Mathematics
Pub. Date: 26-Aug-2025
ISBN-13: 9789819684717
The theme of this book is to establish a link between gauge theory and L2-cohomology theory. Although both theories focus on differential topology, they have been developed rather independently. One of the main reasons lies in the differing characteristics of these theories. This book introduces an integrated theory that bridges these subjects. One goal of the book is to propose differential-topological conjectures that are covering versions of the so-called 10/8-theorem. We include various pieces of evidence to support them. This book is almost self-contained and is accessible not only to graduate students in differential geometry but also to both the experts in L2-cohomology theory and gauge theory. This unique and fundamental book contains numerous unsolved problems, suggesting future directions of topology of smooth 4-manifolds by using various analytic methods.
After the introduction (Chap. 1), Chap. 2 gives a quick overview of the historical progress of differential topology. Chap. 3 covers the basic subjects of spin geometry. Chaps 4 and 5 deal with the foundations of the SeibergWitten and the BauerFuruta theories. In Chaps 6 and 7, we present the basic theory of L2-cohomology, L2-Betti numbers, amenability, and residual finiteness of discrete groups.
In Chap. 8, we treat the Singer conjecture and describe the solution to the conjecture for Kahler hyperbolic manifolds. We then describe various variations of Furuta's 10/8-inequalities and how the aspherical 10/8-inequalities conjecture is induced. We provide the evidence by examining various classes of 4-manifolds, such as aspherical surface bundles and complex surfaces.
Chapter 1 Introduction.
Chapter 2 A glimpse of progress of dierential topology.
Chapter 3 Spin geometry.
Chapter 4 SeibergWitten theory.-
Chapter 5 BauerFuruta theory.-
Chapter 6 ??^?? cohomology.-
Chapter 7 ??^2-Betti number and von Neumann trace.
Chapter 8 Aspherical 10/8 -inequality and Singers conjecture.- Solutions.-
References.- Index.
Format: Hardback, 545 pages, height x width: 235x155 mm, 1 Illustrations, color; IX, 545 p. 1 illus. in color., 1 Hardback
Series: SISSA Springer Series 6
Pub. Date: 08-Sep-2025
ISBN-13: 9783031929977
This book is unique in the current scenario because it provides a complete treatment of modern Brouwer and of Leray-Schauder degree theories, including detailed proofs and selected applications to differential equations. In addition, new results and open problems appear throughout. It aims to introduce Ph.D. students to research on nonlinear boundary value problems for ordinary, elliptic, and evolution equations using the topological methods of nonlinear functional analysis. It is suitable for self-study, as all proofs are detailed, with prerequisites provided when necessary. It is based on personal notes compiled over more than forty years of teaching.
The theory.- Applications to the topology of RN.- Applications to fixed
point theory.- Applications to ODEs.