https://doi.org/10.1142/12244 | December 2021
Pages: 400
ISBN: 978-981-123-575-7 (hardcover)
ISBN: 978-981-123-625-9 (softcover)
This volume contains techniques of integration which are not found in standard calculus and advanced calculus books.
It can be considered as a map to explore many classical approaches to evaluate integrals. It is intended for students
and professionals who need to solve integrals or like to solve integrals and yearn to learn more about the various
methods they could apply. Undergraduate and graduate students whose studies include mathematical analysis or
mathematical physics will strongly benefit from this material. Mathematicians involved in research and teaching in
areas related to calculus, advanced calculus and real analysis will find it invaluable.
The volume contains numerous solved examples and problems for the reader. These examples can be used in classwork
or for home assignments, as well as a supplement to student projects and student research.
Special Substitutions
Solving Integrals by Differentiation with Respect to a Parameter
Solving Logarithmic Integrals by Using Fourier Series
Evaluating Integrals by Laplace and Fourier Transforms Integrals Related to Riemann's Zeta Function
Various Techniques
List of Solved Integrals
Graduate and undergraduate students, professors and researchers in mathematics related to calculus, advanced calculus,
mathematical analysis, real analysis, and mathematical physics; physics, and engineering.
https://doi.org/10.1142/12478 | February 2022
Pages: 730
ISBN: 978-981-124-410-0 (hardcover)
This book presents a powerful way to study Einstein's special theory of relativity and its underlying hyperbolic geometry
in which analogies with classical results form the right tool. The premise of analogy as a study strategy is to make
the unfamiliar familiar. Accordingly, this book introduces the notion of vectors into analytic hyperbolic geometry,
where they are called gyrovectors. Gyrovectors turn out to be equivalence classes that add according to the
gyroparallelogram law just as vectors are equivalence classes that add according to the parallelogram law. In the
gyrolanguage of this book, accordingly, one prefixes a gyro to a classical term to mean the analogous term in hyperbolic
geometry. As an example, the relativistic gyrotrigonometry of Einstein's special relativity is developed and employed
to the study of the stellar aberration phenomenon in astronomy.
Furthermore, the book presents, for the first time, the relativistic center of mass of an isolated system of noninteracting
particles that coincided at some initial time t = 0. It turns out that the invariant mass of the relativistic center of mass
of an expanding system (like galaxies) exceeds the sum of the masses of its constituent particles. This excess of mass
suggests a viable mechanism for the formation of dark matter in the universe, which has not been detected but is needed
to gravitationally "glue" each galaxy in the universe. The discovery of the relativistic center of mass in this book
thus demonstrates once again the usefulness of the study of Einstein's special theory of relativity in terms of its
underlying hyperbolic geometry.
Gyrogroups
Gyrocommutative Gyrogroups
Gyrogroup Extension
Gyrovectors and Cogyrovectors
Gyrovector Spaces
Rudiments of Differential Geometry
Gyrotrigonometry
Bloch Gyrovector of Quantum Information and Computation
Special Theory of Relativity: The Analytic Hyperbolic Geometric Viewpoint
Relativistic Gyrotrigonometry
Stellar and Particle Aberration
Enriched Special Relativity Theory: Special Relativity of Signature (m,n)
The book is aimed at a large audience. It includes both elementary and advanced topics, and is structured so it can be enjoyed equally
by undergraduates, graduate students, researchers and academics in geometry, algebra, mathematical physics, theoretical physics and astronomy.
https://doi.org/10.1142/12687 | December 2021
Pages: 488
ISBN: 978-981-125-092-7 (hardcover)
This monograph provides a coherent development of operads, infinity operads, and monoidal categories, equipped with
equivariant structures encoded by an action operad. A group operad is a planar operad with an action operad equivariant
structure. In the first three parts of this monograph, we establish a foundation for group operads and for their higher
coherent analogues called infinity group operads. Examples include planar, symmetric, braided, ribbon, and cactus operads,
and their infinity analogues. For example, with the tools developed here, we observe that the coherent ribbon nerve of
the universal cover of the framed little 2-disc operad is an infinity ribbon operad.
In Part 4 we define general monoidal categories equipped with an action operad equivariant structure and provide
a unifying treatment of coherence and strictification for them. Examples of such monoidal categories include symmetric,
braided, ribbon, and coboundary monoidal categories, which naturally arise in the representation theory of quantum groups
and of coboundary Hopf algebras and in the theory of crystals of finite dimensional complex reductive Lie algebras.
Operads with Group Equivariance:
Introduction
Planar Operads
Symmetric Operads
Group Operads
Braided Operads
Ribbon Operads
Cactus Operads
Constructions of Group Operads:
Naturality
Group Operads as Algebras
Group Operads with Varying Colors
Boardman-Vogt Construction for Group Operads
Infinity Group Operads:
Category of Group Trees
Contractibility of Group Tree Category
Generalized Reedy Structure
Realization-Nerve Adjunction for Group Operads
Nerve Theorem for Group Operads
Coherent Realization-Nerve and Infinity Group Operads
Coherence for Monoidal Categories with Group Equivariance:
Monoidal Categories
G-Monoidal Categories
Coherence for G-Monoidal Categories
Braided and Symmetric Monoidal Categories
Ribbon Monoidal Categories
Coboundary Monoidal Categories
Graduate students and researchers with an interest in operads and categories.
Series on Advances in Mathematics for Applied Sciences
https://doi.org/10.1142/12345 | March 2022
Pages: 250
ISBN: 978-981-123-943-4 (hardcover)
The aim of this book is to provide a basic and self-contained introduction to the ideas underpinning fractal analysis.
The book illustrates some important applications issued from real data sets, real physical and natural phenomena
as well as real applications in different fields, and consequently, presents to the readers the opportunity to implement
fractal analysis in their specialties according to the step-by-step guide found in the book.
Besides advanced undergraduate students, graduate students and senior researchers, this book may also serve
scientists and research workers from industrial settings, where fractals and multifractals are required for modeling
real-world phenomena and data, such as finance, medicine, engineering, transport, images, signals, among others.
For the theorists, rigorous mathematical developments are established with necessary prerequisites that make
the book self-containing. For the practitioner often interested in model building and analysis, we provide
the cornerstone ideas.
Introduction
Basics of Measure Theory
Martingales with Discrete Time
Hausdorff Measure and Dimension
Capacity Dimension of Sets
Packing Measure and Dimension
Multifractal Analysis of Measures
Extensions to Multifractal Cases
Bibliography
Index
Young researchers at master's level in sciences; researchers in PhD studies in pure a
nd applied mathematical/physical sciences; and researchers at advanced levels are provided the necessary
tools that allow them to understand and adapt fractal analysis to their needs such as supervision and development
of research projects. Advanced undergraduate students will gain a clear idea on what fractal analysis is,
that will guide them to decide their course on their future scientific research areas.
It is applicable to the industrial and professional sectors such as ready-made constructions, town planning,
such as construction and the development of plans for urban areas (fractal cities); for economists to develop good
forecasting models and for researchers in nanofractal materials, geology, geosciences, biology, among others.
https://doi.org/10.1142/12539 | March 2022
Pages: 250
ISBN: 978-981-124-666-1 (hardcover)
ISBN: 978-981-124-755-2 (softcover)
This book is a textbook for a semester-long or year-long introductory course in abstract algebra at the upper
undergraduate or beginning graduate level.
It treats set theory, group theory, ring and ideal theory, and field theory (including Galois theory), and culminates
with a treatment of Dedekind rings, including rings of algebraic integers.
In addition to treating standard topics, it contains material not often dealt with in books at this level. It provides
a fresh perspective on the subjects it covers, with, in particular, distinctive treatments of factorization theory in integral domains and of Galois theory.
As an introduction, it presupposes no prior knowledge of abstract algebra, but provides a well-motivated, clear,
and rigorous treatment of the subject, illustrated by many examples. Written with an eye toward number theory, it contains numerous applications to number theory (including proofs of Fermat's theorem on sums of two squares and of the Law of Quadratic Reciprocity) and serves as an excellent basis for further study in algebra in general and number theory in particular.
Each of its chapters concludes with a variety of exercises ranging from the straightforward to the challenging
in order to reinforce students' knowledge of the subject. Some of these are particular examples that illustrate
the theory while others are general results that develop the theory further.
Set Theory
Group Theory
Ring Theory
Field Theory
Rings of Algebraic Integers and Dedekind Rings
Advanced undergraduate and beginning graduate students in mathematics, suitable for introductory abstract algebra course in general, and particularly suitable for such a course with an orientation toward number theory.