1st ed. 2018, VIII, 140 p.
Softcover
ISBN 978-981-13-0055-4
Series : SpringerBriefs in Mathematical Physics
Restricts readers' attention to the best-known example of mirror symmetry: a
quintic hypersurface in CP^4
Explains mirror symmetry from the point of view of a researcher involved in
physics and mathematics
Provides a detailed exposition of the procedure of computation in the last two
chapters
This book furnishes a brief introduction to classical mirror symmetry, a term that denotes the
process of computing Gromov?Witten invariants of a Calabi?Yau threefold by using the Picard?
Fuchs differential equation of period integrals of its mirror Calabi?Yau threefold. The book
concentrates on the best-known example, the quintic hypersurface in 4-dimensional projective
space, and its mirror manifold. First, there is a brief review of the process of discovery of mirror
symmetry and the striking result proposed in the celebrated paper by Candelas and his
collaborators. Next, some elementary results of complex manifolds and Chern classes needed
for study of mirror symmetry are explained. Then the topological sigma models, the A-model
and the B-model, are introduced. The classical mirror symmetry hypothesis is explained as the
equivalence between the correlation function of the A-model of a quintic hyper-surface and
that of the B-model of its mirror manifold. On the B-model side, the process of construction of
a pair of mirror Calabi?Yau threefold using toric geometry is briefly explained. Also given are
detailed explanations of the derivation of the Picard?Fuchs differential equation of the period
integrals and on the process of deriving the instanton expansion of the A-model Yukawa
coupling based on the mirror symmetry hypothesis. On the A-model side, the moduli space of
degree d quasimaps from CP^1 with two marked points to CP^4 is introduced, with
reconstruction of the period integrals used in the B-model side as generating functions of the
intersection numbers of the moduli space. Lastly, a mathematical justification for the process of
the B-model computation from the point of view of the geometry of the moduli space of
quasimaps is given. The style of description is between that of mathematics and physics, with
the assumption that readers have standard graduate student backgrounds in both disciplines.
Due 2018-06-14
2nd ed. 2018, XXV, 679 p.
Hardcover
ISBN 978-3-319-91040-6
Series : Undergraduate Texts in Mathematics
Develops a strong conceptual grounding for applying linear algebra in
numerous modern applications
Weaves the theory of linear algebra with applications across engineering,
science, computing, data analysis, and beyond
Provides an ideal preparation for future study in applied differential equations
Presents material in engaging and informative full color
This textbook develops the essential tools of linear algebra, with the goal of imparting
technique alongside contextual understanding. Applications go hand-in-hand with theory, each
reinforcing and explaining the other. This approach encourages students to develop not only
the technical proficiency needed to go on to further study, but an appreciation for when, why,
and how the tools of linear algebra can be used across modern applied mathematics.Providing
an extensive treatment of essential topics such as Gaussian elimination, inner products and
norms, and eigenvalues and singular values, this text can be used for an in-depth first course,
or an application-driven second course in linear algebra. In this second edition, applications
have been updated and expanded to include numerical methods, dynamical systems, data
analysis, and signal processing, while the pedagogical flow of the core material has been
improved. Throughout, the text emphasizes the conceptual connections between each
application and the underlying linear algebraic techniques, thereby enabling students not only
to learn how to apply the mathematical tools in routine contexts, but also to understand what
is required to adapt to unusual or emerging problems. No previous knowledge of linear
algebra is needed to approach this text, with single-variable calculus as the only formal
prerequisite. However, the reader will need to draw upon some mathematical maturity to
engage in the increasing abstraction inherent to the subject. Once equipped with the main
tools and concepts from this book, students will be prepared for further study in differential
equations, numerical analysis, data science and statistics, and a broad range of applications.
Due 2018-06-13
1st ed. 2018, XIX, 378 p. 6 illus.
Hardcover
ISBN 978-981-13-0145-2
Series : Developments in Mathematics
Describes the developments in the research directions and the types of the
generalized inverses since mid-1970s
Includes the fundamentals as well as advanced topics
Ready to be adopted as a textbook or reference book for graduate courses
This book begins with the fundamentals of the generalized inverses, then moves to more
advanced topics. It presents a theoretical study of the generalization of Cramer's rule,
determinant representations of the generalized inverses, reverse order law of the generalized
inverses of a matrix product, structures of the generalized inverses of structured matrices,
parallel computation of the generalized inverses, perturbation analysis of the generalized
inverses, an algorithmic study of the computational methods for the full-rank factorization of a
generalized inverse, generalized singular value decomposition, imbedding method, finite
method, generalized inverses of polynomial matrices, and generalized inverses of linear
operators. This book is intended for researchers, postdocs, and graduate students in the area
of the generalized inverses with an undergraduate-level understanding of linear algebra.
Due 2018-06-25
1st ed. 2018, XVII, 318 p.
Hardcover
ISBN 978-3-319-78947-7
Series : Developments in Mathematics
Provides a categorical approach to quantales and applications
Develops the theory of modules on unital quantales
Includes exercises and bibliographical notes
This monograph provides a modern introduction to the theory of quantales. First coined by C.J.
Mulvey in 1986, quantales have since developed into a significant topic at the crossroads of
algebra and logic, of notable interest to theoretical computer science. This book recasts the
subject within the powerful framework of categorical algebra, showcasing its versatility through
applications to C*- and MV-algebras, fuzzy sets and automata. With exercises and historical
remarks at the end of each chapter, this self-contained book provides readers with a valuable
source of references and hints for future research. This book will appeal to researchers across
mathematics and computer science with an interest in category theory, lattice theory, and manyvalued
logic.
Due 2018-06-19
1st ed. 2018, XIV, 124 p. 34 illus., 32 illus. in color.
Softcover
ISBN 978-3-319-78809-8
Series Lecture Notes in Mathematics
Provides the first systematic treatment of rotation sets
The abstract treatment is augmented by concrete examples of applications in
polynomial dynamics
The clear and detailed exposition is accompanied by numerous illustrations,
making it accessible to graduate students
This monograph examines rotation sets under the multiplication by d (mod 1) map and their
relation to degree d polynomial maps of the complex plane. These sets are higher-degree
analogs of the corresponding sets under the angle-doubling map of the circle, which played a
key role in Douady and Hubbard's work on the quadratic family and the Mandelbrot set.
Presenting the first systematic study of rotation sets, treating both rational and irrational cases
in a unified fashion, the text includes several new results on their structure, their gap
dynamics, maximal and minimal sets, rigidity, and continuous dependence on parameters. This
abstract material is supplemented by concrete examples which explain how rotation sets arise
in the dynamical plane of complex polynomial maps and how suitable parameter spaces of
such polynomials provide a complete catalog of all such sets of a given degree. As a main
illustration, the link between rotation sets of degree 3 and one-dimensional families of cubic
polynomials with a persistent indifferent fixed point is outlined. The monograph will benefit
graduate students as well as researchers in the area of holomorphic dynamics and related
fields.