Hardback
ISBN 9781032588100
236 Pages 15 B/W Illustrations
April 14, 2026 by Chapman & Hall
Differential Modules over Differential Rings provides an introduction and reference for researchers in commutative and differential algebra and could be used as the basis for a graduate course or seminar. The book is best suited to an audience for whom the terminology of rings, modules, homomorphisms, and categories is already familiar. Although the topic is rooted in differential algebra, and the book should be of interest to workers in that area, no particular prior knowledge of differential algebra is assumed. When it is necessary to use specialized results from differential algebra, especially Picard—Vessiot theory, the necessary definitions and theorems are supplied.
Collects the basic definitions and results about differential modules in one convenient reference with uniform notation
Accessible to readers who don’t have extensive specialized knowledge of differential algebra or commutative ring theory
The first book of its kind dedicated exclusively to the topic in this generality
Presents new formulations of previously published work as well as new results not previously published
1. Differential Rings
2. Differential Modules over Differential Fields
3. Differential Rings over Differential Fields
4. Differential Projective Modules
5. K-Theory of Differential Modules
Hardback
ISBN 9781041208990
326 Pages 8 B/W Illustrations
May 29, 2026 by Chapman & Hall
This book offers a rigorous, comprehensive, and modern presentation of the most traditional concepts in measure theory and integration. Building on the classical foundations, it introduces the theory with full generality and meticulous attention to detail, following the stylistic tradition first introduced by Nicolas Bourbaki. The book is designed for graduate students and young researchers seeking a thorough exposition of the theory in an abstract setting, complete proofs, and the strategies underlying them, fostering good mathematical habits in logical reasoning and clarity of deduction.
Beyond standard treatments, the book features several distinctive elements: Some classical results, such as Radon-Nikodým theorem, and Lebesgue and Hahn decompositions, have been presented with original proofs, aimed to clarifying the logic behind the results; some topics that are often overlooked, such as kernels, uniform integrability, the Vitali-Hahn-Saks and Dunford-Pettis theorems are developed in full in dedicated chapters, and a full account of the disintegration of measures is developed. The book also pays special attention to modern applications, including the construction of product measures for an arbitrary family of measures, by exploiting the properties of kernels, a full account of Daniell’s and Carathéodory’s methods for constructing and extending measures, and a thorough coverage of the theory of convergence, and showing two paramount applications of the theory to the presentation of the Lebesgue measure and the family of Hausdorff measures.
The book is largely self-contained, with supplementary sections on topology and differential calculus, and an appendix on filters and ultrafilters also included to help the reader to fully understand the notion of convergence with respect to a filter.
I. The Foundation of Measure Theory
II. Integration
III. Construction and Extension of Measures
IV. Kernels and Products of Measures
V. Riesz Spaces and Signed Measures
VI. The Lp Spaces
VII. Measures on a Topological Space
VIII. Convergence and Uniform Integrability
IX. Weak Convergence of Probability Measures
X. Disintegration of Measures
XI. Lebesgue Measure
XII. Hausdorff Measures
Hardback
ISBN 9781041120674
304 Pages 16 Color & 1 B/W Illustrations
February 19, 2026 by CRC Press
The book explores the theory and applications of fractional differential equations with variable order. It delves into the mathematical foundations, including notations, definitions, and key theorems, while also addressing practical applications in various fields. The book will guide readers from basic concepts to advanced topics, making it suitable for both researchers and graduate students in mathematics, engineering, and applied sciences. Each chapter includes illustrative examples and numerical applications to reinforce theoretical concepts. The book also emphasizes stability analysis and the use of fixed-point theorems in solving fractional differential equations.
Preface
1. Introduction
2. Preliminary
Notations and Definitions
Fractional Calculus
Measure of Noncompactness
Some Fixed Point Theorems
The Ulam Type Stability
3. Initial and Terminal Value Problems of R-Liouville Fractional Differential Equations of Variable Order
Initial and Terminal Value Issues for Variable-order Nonlinear Fractional Differential Equations using Kuratowski MNC Method.
Implicit Initial and Terminal Value Problems of R-Liouville Fractional Differential Equations of Variable Order.
Existence, Uniqueness, and Stability of Solutions to Variable Fractional Order Initial and Terminal Value Problems.
Initial and Terminal Value Problem for Nonlinear Fractional Differential Equations of Variable Order with Finite Delay Via Kuratowski’s Measure of Noncompactness.
Variable-Order Implicit Fractional Differential Equations based on the Kuratowski MNC Technique.
4. Problems for Caputo’s Fractional Differential Equations with Variable Order
Initial and Terminal Value Problems for Variable-Order Caputo Fractional Differential Equations via Kuratowski’s Measure of Noncompactness.
Fractional Variable Order Differential Equations with Impulses.
Existence and Stability of Solutions for a Class of Nonlinear Fractional Integro-Differential Equations of Variable Order.
Initial and Terminal Value Problem of a Fractional Thermostat Model of Variable Order via Piecewise Constant Function.
5. Initial and Terminal Value Problems of Hadamard Fractional Differential Equations of Variable Order
Qualitative Study on Solutions of a Hadamard Variable-Order Initial and Terminal Value Problem via Ulam-Hyers-ℜassias
Stability.
Darbo Fixed Point Criterion for Solutions of a Nonlinear Variable-Order Hadamard Problem and Ulam-Hyers-ℜassias
Stability.
Initial and Terminal Value Problems for Hadamard Fractional Differential Equations of Variable Order via the Kuratowski
Measure of Noncompactness Technique.
Multiterm Initial and Terminal Value Problems of Hadamard Fractional Differential Equations of Variable Order.
6. Problems for Caputo-Hadamard Fractional Differential Equations with Variable Order
Multiterm Impulsive Caputo–Hadamard-Type Differential Equations of Fractional Variable Order.
On the Caputo-Hadamard Fractional IVP with Variable Order using the Upper–Lower Solutions Technique.
Stability of an Initial and Terminal Value Problem Involving the Riemann–Liouville Fractional Derivative in the Sense of
Atangana–Baleanu of Variable Order.
7. Conclusion.
References
Index
A publication of the Société Mathématique de France
Hardcover
Hardcover ISBN: 978-2-37905-221-7
Product Code: COSP/32
Cours Spécialisés Volume: 32;
2026; 526 pp
MSC: Primary 35; 37; 53
The statistical properties of hyperbolic dynamical systems—such as ergodicity and mixing—can be studied through spectral theory, in particular via anisotropic Sobolev spaces of distributions. In settings where the geodesic flow exhibits hyperbolic features, rigidity phenomena in Riemannian geometry—showing that certain spectral or geometric invariants determine the underlying geometry—can likewise be addressed using microlocal analysis.
This book offers a comprehensive introduction to microlocal analysis and its applications to hyperbolic dynamics and Riemannian rigidity. It is intended for graduate students and researchers seeking to familiarize themselves with these powerful techniques.
A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.
Graduate students and researchers looking for a comprehensive introduction to microlocal analysis and its applications to hyperbolic dynamics and Riemannian rigidity.
This volume
of Surveys in Differential Geometry is the first of the two volumes dedicated to Professor Shing-Tung Yau’s 75th birthday. It consists of articles by speakers at the special conference “Current Developments in Mathematics and Physics” held at Tsinghua University, April 3-6 2024, as well as invited contributions.
Publisher
International Press of Boston, Inc.
Pages 322
Publish Date ISBN-13 Medium Binding Size Publish Status
2026-01-30 978-1-57146-600-6 Print Hardcover 7” x 10” In Print
ISBN: 978-1-836-69091-7
January 2026
288 pages
Complex Manifolds and Geometric Algebraic Analysis is intended for graduate students in mathematics, physics, and beyond.
The book is divided into ten chapters. Chapter 1 deals with the properties of holomorphic functions of several complex variables. Chapter 2 introduces tools for studying complex manifolds and analytic varieties, whilst Chapter 3 covers the foundational material from sheaves and cohomology. Chapter 4 concerns the study of divisors and line bundles on complex manifolds, and Chapter 5 is devoted to some fundamental theorems. Chapter 6 covers definitions and examples of abelian varieties, whilst Chapter 7 studies theta functions on complex projective tori. Lastly, the aim of Chapter 8 is to discuss an interesting interaction between complex algebraic geometry and dynamical systems.
This book is supplemented with two appendices, one on Riemann surfaces and algebraic curves and the other covering elliptic functions and elliptic integrals. Additionally, various examples, exercises, and problems with solutions are provided throughout the book.
Preface ix
Chapter 1 Holomorphic Functions of Several Complex Variables 1
1.1.Notations, definitions and properties 1
1.2.Cauchy integral formula and applications 7
1.3 Power series in several variables 9
1.4 Various fundamental properties 20
1.5 Inversemapping and implicit function theorems 28
1.6.Exercises 32
Chapter 2 Complex Manifolds and Analytic Varieties 37
2.1 Preliminaries 37
2.2.Examples of complexmanifolds 42
2.2.1 Cn and open subsets 42
2.2.2 Complex algebraic curves or compact Riemann surfaces 43
2.2.3 Spheres 43
2.2.4 Projective spaces 47
2.2.5.Grassmannians 50
2.2.6.Tori 51
2.3 Tangent spaces and tangent bundles 52
2.4.Constant rank theorem 53
2.5 Submanifolds, subvarieties and examples 55
2.6.Exercises 58
Chapter 3 Sheaves and Cohomology 65
3.1 Sheaves 65
3.2 Cech cohomology 69
3.3 De Rham cohomology 71
3.4 Dolbeault cohomology 72
3.5.Connections and curvature 74
3.6 Curvature form, first Chern class of line bundles 76
3.7.ThePoincaré dual 77
3.8.Exercises 78
Chapter 4 Divisors and Line Bundles 79
4.1.Divisors 79
4.2 Line bundles 81
4.3 Sections of line bundles 83
4.4.Tori andRiemann form 89
4.5 Line bundles on complex tori 91
4.6.Exercises 97
Chapter 5 Some Fundamental Theorems 101
5.1 Preliminaries and various notions 101
5.1.1 Projectivevariety 101
5.1.2 Tangent and cotangent bundles 103
5.1.3 Dolbeault cohomology group 103
5.1.4 FirstChern class 106
5.1.5 Hodge forms, hermitian metrics, harmonic space, Hodge star operator and Hodge theorem 107
5.1.6.Kählermetric,Kähler form,Kählermanifold 109
5.2.TheKodaira–Nakano vanishing theorem 111
5.3 The Lefschetz theorem on hyperplane sections 113
5.4 The Lefschetz theorem on (1, 1)-classes 114
5.5.TheKodaira embedding theorem 117
5.6.Exercises 126
Chapter 6 Abelian Varieties 133
6.1.Definitions and examples 133
6.2.Riemann conditions, polarization andRiemann form 138
6.3 Isogenies and reducibleAbelian varieties 141
6.4.DualAbelian varieties 142
6.5 Prymvarieties 145
6.6 Projectively normal embedding 154
6.7 Number of even and odd sections of a line bundle 155
6.8.Exercises 156
Contents vii
Chapter 7 Theta Functions and Complex Projective Tori 159
7.1.Meromorphic functions and theta functions 159
7.2.Lefschetz theorem 173
7.3.Exercises 179
Chapter 8 Algebraically Completely Integrable Systems 183
8.1 Preliminaries 183
8.2 The Hénon–Heiles system 190
8.3.Kowalewski’s spinning top 204
8.4 Kirchhoff’s equations of motion of a solid in an ideal fluid 213
8.5.Exercises 216
Chapter 9 Appendix: Riemann Surfaces and Algebraic Curves 227
Chapter 10 Appendix: Elliptic Functions and Elliptic Integrals 233
10.1 Elliptic functions 233
10.2.Weierstrass functions 235
10.3 Elliptic integrals and Jacobi elliptic functions 240
10.4 Application: simple pendulum 244
References 247
Index 251