Language: English
Published/Copyright: 2025
Derivations on von Neumann algebras are well understood and are always inner, meaning that they act as commutators with a fixed element from the algebra itself. The purpose of this book is to provide a complete description of derivations on algebras of operators affiliated with a von Neumann algebra. The book is designed to serve as an introductory graduate level to various measurable operators affiliated with a von Neumann algebras and their properties. These classes of operators form their respective algebras and the problem of describing derivations on these algebras was raised by Ayupov, and later by Kadison and Liu. A principal aim of the book is to fully resolve the Ayupov-Kadison-Liu problem by proving a necessary and sufficient condition of the existence of non-inner derivation of algebras of measurable operators. It turns out that only for a finite type I von Neumann algebra M may there exist a non-inner derivation on the algebra of operators affiliated with M. In particular, it is established that the classical derivation d/dt of functions of real variables can be extended up to a derivation on the algebra of all measurable functions. This resolves a long-standing problem in classical analysis.
Detailed description of topological, order-theoretic, and analytical aspects of algebras of measurable operators
Complete description of derivations on these algebras
Existence of nontrivial derivations on algebras of measurable functions
Introduction 1
1 Preliminaries on the theory of von Neumann algebras and general theory of linear operators in a Hilbert space 7
2 Classes of unbounded operators 87
3 Properties of locally measurable operators 124
4 Topologies on algebras of unbounded operators 181
5 Properties of derivations on algebras of locally measurable operators 231
6 Derivations on the algebras of measurable operators affiliated with a type Ifin von Neumann algebra 276
7 Complete description of derivations on algebras of locally measurable operators 339
Bibliography 393
Subject index 401
Notation index
Language: English
Published/Copyright: 2025
This book delves into the recent findings and research methods in the existence and regularity theory for Non-Newtonian Fluids with Variable Power-Law. The aim of this book is not only to introduce recent results and research methods in the existence and regularity theory, such as higher integrability, higher differentiability, and Holder continuity for flows of non-Newtonian fluids with variable power-laws, but also to summarize much of the existing literature concerning these topics. While this book mainly focuses on steady-state flows of non-Newtonian fluids, the methods and ideas presented in this book can be applied to unsteady flows (as discussed in Chapter 7) and other related problems such as complex non-Newtonian fluids, plasticity, elasticity, p(x)-Laplacian type systems, and so on.
The book is intended for researchers and graduate students in the field of mathematical fluid mechanics and partial differential equations with variable exponents. It is expected to contribute to the advancement of mathematics and its applications.
A comprehensive review of mathematical theory for non-Newtonian fluids.
In-depth discussions on unifying and systematic theory about regularity.
Concerning boundary value problems, which is rather realistic for real world applications
Frontmatter
Publicly Available I
Preface
Publicly Available V
Contents
Publicly Available IX
1 Motivation and preliminaries 1
2 Existence and higher integrability for steady flows of non-Newtonian fluids 56
3 Higher differentiability for steady flows of non-Newtonian fluids 79
4 Partial regularity for steady flows of non-Newtonian fluids in 3D 134
5 C1,α-regularity for steady flows of non-Newtonian fluids in 2D 199
6 Calderon?Zygmund estimate for steady flows of non-Newtonian fluids 219
7 Holder continuity for 2D unsteady flows of non-Newtonian fluids 242
A Standard notations 273
Bibliography 277
Index
Language: English
Published/Copyright: 2025
This book explores robust control strategies to manage the inherent uncertainties and maintain
the admissibility and performance of fractional-order singular systems. It covers essential topics such as
system admissibility, robust stabilization, H∞ control, positive real control, fault detection, delay systems,
and provides a comprehensive framework for both the theoretical analysis and practical implementation of robust control methods.
Gives a systematic presentation of the theory of fractional-order singular systems and its research methods.
Fills the vacuum in the literature regarding robust control problems and stabilisation.
Frontmatter
Publicly Available I
Preface
Publicly Available VII
Contents
Publicly Available IX
Abbreviations and Notations
Publicly Available XIII
1 Introduction and overview 1
2 Admissibility 13
3 Robust admissibility and robust stabilization 50
4 Robust H∞ control 74
5 Robust positive real control 105
6 Fault detection 134
7 Delay systems 154
8 Discrete systems 187
Bibliography 207
Index
*
Language: English
Published/Copyright: 2025
This book is a comprehensive guide to the space-time algebra of sixteen-component values "sedeons". This algebra is designed to provide a compact representation of equations that describe various physical systems. The book considers the symmetry of physical quantities concerning the operations of spatial and temporal inversion. This approach allows the formulation of a wide class of mathematical physics equations within a unified framework and enables the generalization of these equations for essential problems in electrodynamics, hydrodynamics, plasma physics, field theory, and quantum mechanics. In particular, it is shown that the broken symmetry between electricity and magnetism in electrodynamics equations is a result of choosing an asymmetric representation of these phenomena. The sedeonic algebra enables the formulation of Maxwell-like equations for the fields with a nonzero mass of quantum, which facilitates the calculation of energy for baryon-baryon interaction and the semi-classical interpretation of this interaction. It also allows one to generalize the hydrodynamics equations for the case of vortex turbulent flows and for a hydrodynamic two-fluid model of electron-ion plasma.
This book provides original space-time algebra of 16-component sedeons.
It contains novel results on electrodynamics, hydrodynamics, and plasma physics.
The topics of field theory and quantum mechanics are also included.
Frontmatter
Publicly Available I
Foreword
V. L. Mironov and S. V. Mironov
Publicly Available VII
Introduction
Publicly Available IX
Contents
Publicly Available XI
1 Quaternions, vectors, matrices 1
2 Algebra of sedeons 10
3 Sedeons in relativistic physics 18
4 Equations for electromagnetic field 23
5 Equations for weak gravitational field 40
6 First-order wave equation for massless field 48
7 Sedeonic equations for fields with non-zero mass of quantum 55
8 Symmetric form of equations for massive and massless fields 72
9 Equations of relativistic quantum mechanics 86
10 Sedeonic field equations for dyons 96
11 Sedeonic equations of hydrodynamics 113
12 Hydrodynamic model of plasma 140
13 Hydrodynamic model of electron vortex fluid in solids 153
14 Sedeonic generalization of telegraph equations 166
A Matrix representation of sedeons 175
B Space-time sedenions 179
Bibliography 189
Index
Language: English
Published/Copyright: 2025
This project is a second edition of the textbook: Deformation Theory of Discontinuous Groups (De Gruyter 2022). It is devoted to studying various geometric and topological concepts related to the deformation and moduli spaces of discontinuous group actions, and building some interrelationships between these concepts. It presents full proofs of recent results, computes fundamental examples, and serves as an introduction and reference for students and researchers in Lie theory, discontinuous groups, and deformation spaces. A part of the first edition, the setting of affine actions is introduced and new ideas and methods are developed with full proofs. The setting of compact extensions is also re-written with new approaches and proofs. It also contains the most recent developments of the theory, extending from basic concepts to a comprehensive exposition, and highlighting the newest approaches and methods in deformation theory. It also includes the most recent solutions to many open questions over the last decades and brings related newest research results in this area. For specialists and beginners in deformation theory, the settings of Heisenberg and Threadlike cases are differently re-written with full details and proofs.
The second edition contains
The most recent developments in the theory of discontinuous groups
Develops full proofs of recent results and the most known examples
Unique in the literature discussing discontinuous groups in the solvable setting
Frontmatter
Publicly Available I
Preface
Ali Baklouti
Publicly Available V
Contents
Publicly Available IX
1 Structure theory and first basic concepts 1
2 Proper actions on solvable homogeneous spaces
3 Proper action for compact extensions 75
4 Proper action of affine discontinuous groups 96
5 Deformation and moduli spaces 119
6 Local rigidity and stability 190
7 Deforming in extensions of Heisenberg groups 254
8 Discontinuous actions in the threadlike setting 344
Bibliography 423
Index
Language: English
Published/Copyright: 2025
There are very few systematic books on the dynamics of entire functions. Unfortunately, reading these books is often difficult for non-specialists since their proofs are not clearly written, and readers struggle to understand the arguments fully. This book is a comprehensive introduction to the iteration theory of entire complex functions. It is intended to introduce the reader to the key topics in the field and to form a basis for further study. In general, the proofs are more detailed; therefore, the book will also help non-specialist mathematicians become acquainted with complex dynamics. In no sense is this manuscript a complete account of the subject. Nevertheless, the book may also be helpful to young researchers in this field before they tackle more specific works. The book deals with three possible aspects: theory, practice, and computer graphics. In Appendix C, we explained the necessary rudiments of MATLAB RGB images to create computer graphics of different sets considered in the book, such as the sets of Julia and Mandelbrot. In this Appendix, a gallery is also included where beautiful and spectacular images are shown. The author has obtained all these images using MATLAB, most of which are revealed here for the first time.
The book covers theory, practice, and computer graphics (using Matlab).
It is suitable for non-specialist mathematicians and a potential postgraduate textbook.
Frontmatter
Publicly Available I
Preface
Publicly Available VII
Contents
Publicly Available IX
List of Special Symbols
Publicly Available XIII
1 Introduction 1
2 Normal families 17
3 Value distribution theory 23
4 Periodic points 40
5 Local conjugacy 57
6 The Fatou and Julia sets 77
7 Fatou components 98
8 Singular values 115
9 The exponential family 129
10 The parameter plane 147
11 Convergence and dynamics 157
A Univalent functions in the disc 169
B The Newton method 173
C Computer graphics of complex dynamics 179
D Hints to selected exercises 193
Bibliography 199
Index 203