Format: Paperback / softback, 154 pages, height x width: 235x155 mm, 37 Illustrations, color; 34 Illustrations,
black and white; VI, 154 p. 71 illus., 37 illus. in color.
Series: Frontiers in Applied Dynamical Systems: Reviews and Tutorials
Pub. Date: 28-Jul-2022
ISBN-13: 9783031047893
This monograph uses braids to explore dynamics on surfaces, with an eye towards applications to mixing in fluids. The text uses the particular example of taffy pulling devices to represent pseudo-Anosov maps in practice. In addition, its final chapters also briefly discuss current applications in the emerging field of analyzing braids created from trajectory data. While written with beginning graduate students, advanced undergraduates, or practicing applied mathematicians in mind, the book is also suitable for pure mathematicians seeking real-world examples. Readers can benefit from some knowledge of homotopy and homology groups, but these concepts are briefly reviewed. Some familiarity with Matlab is also helpful for the computational examples.
Introduction.- Topological dynamics on the torus.- Stretching with three rods.- Braids.- The Thurston-Nielsen classification.- Topological entropy.- Train tracks.- Dynnikov coordinates.- The braidlab library.- Braids and data analysis.- References.- Appendix: Derivation of Dynnikov update rules.
Format: Hardback, 244 pages, height x width: 235x155 mm, 80 Tables, color; 54 Illustrations,
color; 62 Illustrations, black and white; XIV, 244 p. 116 illus., 54 illus. in color., 1 Hardback
Series: Graduate Texts in Physics
Pub. Date: 05-Aug-2022
ISBN-13: 9783031012488
This book presents a clear and concise introduction to the field of nonlinear dynamics and chaos, suitable for graduate students in mathematics, physics, chemistry, engineering, and in natural sciences in general. This second edition includes additional material and in particular a new chapter on dissipative nonlinear systems. The book provides a thorough and modern introduction to the concepts of dynamical systems' theory combining in a comprehensive way classical and quantum mechanical description. It is based on lectures on classical and quantum chaos held by the author at Heidelberg and Parma University. The book contains exercises and worked examples, which make it ideal for an introductory course for students as well as for researchers starting to work in the field.
Introduction.- Fundamental terminology.- Complexity.- Classical versus quantum dynamics.- Problems.- References.- Dynamical systems.- Evolution law.- One-dimensional maps.- Problems.- References.- Nonlinear Hamiltonian systems.- Integrable examples.- Hamiltonian formalism.- Important techniques in the Hamiltonian formalism.- Integrable systems.- Non-integrable systems.- Perturbation of low-dimensional systems.- Canonical perturbation theory.- Transition to chaos in Hamiltonian systems.- Criteria for local and global chaos.- Appendix.- Problems.- References.- Dissipative systems - Introduction - Fixed points - Fixed point scenarios in two dimensional systems - Damped Oscillators - Harmonic oscillator - Nonlinear oscillators - Nonlinear damping - Poincare-Bendixson Theorem - Damped forced oscillators - Driven one-Dimensional harmonic oscillator - Duffing oscillator - Lorenz model for turbulence - Fractals - Simple examples - Box-counting dimension - Examples from nature - Bifurcation scenarios - Examples of pitchfork bifurcations - Tangent bifurcations - Transcritical bifurcations - Higher-order bifurcations - Hopft bifurcations - Intermittency - Coupled Oscillators - Synchronisation - Kuramoto model - Increasing complexity - Problems - References. - Aspects of quantum chaos.- Introductory remarks on quantum mechanics.- Semiclassical quantization of integrable systems.- Semiclassical description of non-integrable systems.- Wave functions in phase space.- Anderson and dynamical localization.- Universal level statistics.- Concluding remarks.- Appendix.- Problems.- References.- Index.
Format: Paperback / softback, 120 pages, height x width: 235x155 mm, X, 120 p., 1 Paperback / softback
Series: Lecture Notes in Mathematics 2306
Pub. Date: 22-Aug-2022
ISBN-13: 9783031055126
This book develops tools to handle C*-algebras arising as completions of convolution algebras of sections of line bundles over possibly non-Hausdorff groupoids. A fundamental result of Gelfand describes commutative C*-algebras as continuous functions on locally compact Hausdorff spaces.
Kumjian, and later Renault, showed that Gelfand's result can be extended to include non-commutative C*-algebras containing a commutative C*-algebra. In their setting, the C*-algebras in question may be described as the completion of convolution algebras of functions on twisted Hausdorff groupoids with respect to a certain norm. However, there are many natural settings in which the Kumjian?Renault theory does not apply, in part because the groupoids which arise are not Hausdorff. In fact, non-Hausdorff groupoids have been a source of surprising counterexamples and technical difficulties for decades. Including numerous illustrative examples, this book extends the Kumjian?Renault theory to a much broader class of C*-algebras.
This work will be of interest to researchers and graduate students in the area of groupoid C*-algebras, the interface between dynamical systems and C*-algebras, and related fields.
Introduction.- Part I Inclusions.- Local modules.- Regular ideals and the localizing projection.- Regular inclusions.- Invariant ideals.- Extended multiplication for normalizers.- Regularity of maximal ideals in regular inclusions.- Extension of pure states, relative free points and smooth normalizers.- Free points.- Fourier coefficients.- Opaque and gray ideals.- Topologically free inclusions.- Pseudo-expectations.- Part II Groupoids.- E#tale groupoids.- Twists and line bundles.- The C*-algebra of a twisted groupoid.- Topologically free groupoids.- The essential groupoid C*-algebra.- Kwasniewski and Meyer's version of the essential groupoid C*-algebra.- The relative Weyl groupoid.- Fell bundles over inverse semigroups.- Topological freeness of the Weyl groupoid and the main Theorem.- Semi-masas.- Canonical states.- Part III Examples and Open Questions.- Example: non-smooth normalizers.- Example: periodic functions on the interval.- Example: the gray ideal of twisted groupoid C*-algebras.- Some open questions.- Part IV Appendix.- Isotropy projection.- Symbol index.- Concept index.- References.
Format: Paperback / softback, 240 pages, height x width: 235x155 mm, 3 Tables, color;
3 Illustrations, color; X, 240 p. 3 illus. in color.,
Series: Lecture Notes in Mathematics 2315
Pub. Date: 06-Aug-2022
ISBN-13: 9783031052958
T his book provides an introduction to recent developments in the theory of generalized harmonic analysis and its applications. It is well known that convolutions, differential operators and diffusion processes are interconnected: the ordinary convolution commutes with the Laplacian, and the law of Brownian motion has a convolution semigroup property with respect to the ordinary convolution. Seeking to generalize this useful connection, and also motivated by its probabilistic applications, the book focuses on the following question: given a diffusion process Xt on a metric space E, can we construct a convolution-like operator * on the space of probability measures on E with respect to which the law of Xt has the *-convolution semigroup property? A detailed analysis highlights the connection between the construction of convolution-like structures and disciplines such as stochastic processes, ordinary and partial differential equations, spectral theory, special functions and integral transforms.
The book will be valuable for graduate students and researchers interested in the intersections between harmonic analysis, probability theory and differential equations.
1.Introduction.-
2. Preliminaries.-
3. The Whittaker convolution.-
4. Generalized convolutions for Sturm-Liouville operators.-
5. Convolution-like structures on multidimensional spaces.- A. Some open problems.- References.- Index.
Format: Hardback, 646 pages, height x width: 235x155 mm, X, 646 p., 1 Hardback
Series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 74
Pub. Date: 19-Aug-2022
ISBN-13: 9783031067525
This monograph provides a comprehensive introduction to the theory of complex normal surface singularities, with a special emphasis on connections to low-dimensional topology. In this way, it unites the analytic approach with the more recent topological one, combining their tools and methods. In the first chapters, the book sets out the foundations of the theory of normal surface singularities. This includes a comprehensive presentation of the properties of the link (as an oriented 3-manifold) and of the invariants associated with a resolution, combined with the structure and special properties of the line bundles defined on a resolution. A recurring theme is the comparison of analytic and topological invariants. For example, the Poincare series of the divisorial filtration is compared to a topological zeta function associated with the resolution graph, and the sheaf cohomologies of the line bundles are compared to the Seiberg-Witten invariants of the link. Equivariant Ehrhart theory is introduced to establish surgery-additivity formulae of these invariants, as well as for the regularization procedures of multivariable series. In addition to recent research, the book also provides expositions of more classical subjects such as the classification of plane and cuspidal curves, Milnor fibrations and smoothing invariants, the local divisor class group, and the Hilbert-Samuel function. It contains a large number of examples of key families of germs: rational, elliptic, weighted homogeneous, superisolated and splice-quotient. It provides concrete computations of the topological invariants of their links (Casson(-Walker) and Seiberg-Witten invariants, Turaev torsion) and of the analytic invariants (geometric genus, Hilbert function of the divisorial filtration, and the analytic semigroup associated with the resolution). The book culminates in a discussion of the topological and analytic lattice cohomologies (as categorifications of the Seiberg-Witten invariant and of the geometric genus respectively) and of the graded roots. Several open problems and conjectures are also formulated. Normal Surface Singularities provides researchers in algebraic and differential geometry, singularity theory, complex analysis, and low-dimensional topology with an invaluable reference on this rich topic, offering a unified presentation of the major results and approaches.
1 Introduction.- 2 Resolution of Surface Singularities.- 3 The Link.- 4 Coverings.- 5 Examples.- 6 Invariants Associated With a Resolution.- 7 The Artin-Laufer Program.- 8 Multivariable Divisorial Filtration.- 9 Topological Invariants. The Seiberg-Witten Invariant.- 10 Ehrhart Theory and the Seiberg-Witten Invariant.- 11 Lattice Cohomology.- 12 Appendix. Complex Analytic Spaces.- References.- Index.