June 2025
Pages: 192
ISBN: 978-981-98-0668-3 (hardcover)
Non Additive Geometry introduces a groundbreaking approach to arithmetic geometry, replacing traditional structure of commutative rings with Props and Bioperads ? algebraic systems that can handle matrix multiplication and block direct sums. These structures allow for a deeper exploration of algebraic geometry, where addition no longer holds as a universal operation, particularly at the critical "Real prime."
The book presents an innovative and comprehensive study of this new geometric framework, discussing its implications for arithmetic geometry and its potential applications in physics. Chapters explore topics such as generalized schemes, sheaves, ideals and primes, localization, and higher K-theory, following Grothendieck's pioneering methods while extending them to accommodate the needs of arithmetic. The text also addresses future applications, leaving room for readers to explore new directions and potential breakthroughs.
This monograph is essential reading for advanced graduate students, researchers, and professionals in mathematics and theoretical physics interested in the foundations of arithmetic geometry, the role of Props and Bioperads, and their applications to broaden our concept of geometry, and therefore have new geometrical objects, such as the Arithmetical Surface Spec(? ? ?), the product of the primes Spec(?) with themselves.
Props
Associating Props with Rings
Bio(perads)
Commutativity for Bios
Ideals and Primes
The Spectrum
Localization
Sheaves
Generalized Schemes
Pro-Schemes
Valuations and Beta Integrals
Categorical Objects and A-Modules
Derivations and Differential
Simplicial Objects and the Cotangent Complex
Properties of Generalized Schemes
Higher K-Theory
The Witt Ring
Modules Over the Sphere Spectrum
Researchers, graduate students, and professionals in the fields of algebraic geometry, commutative algebra, category theory, and K-theory.
June 2025
Pages: 316
ISBN: 978-981-98-1219-6 (hardcover)
This comprehensive volume offers an in-depth exploration of advanced integration theories, extending beyond classical methods to unify and expand the field. Building on the foundational work of Jaroslav Kurzweil and Ralph Henstock, the book delves into the Henstock?Kurzweil and McShane gauge integrals, presenting a more intuitive and versatile alternative to the traditional Lebesgue integral. By bridging gaps in existing literature, the authors provide a rigorous treatment of integration on metric measure spaces, exploring critical concepts such as completeness, compactness, and Cousin's lemma.
The book systematically introduces advanced topics, including the variational Henstock integral in locally convex spaces, the Riemann?Lebesgue integral for vector-valued functions, and generalizations of the Sugeno integral. Further chapters explore convergence in Banach spaces on time scales, set-valued integrals, and applications to harmonic analysis and partial differential equations, including solutions to the heat equation in distribution spaces.
Notably, the text presents innovative approaches like the symmetric Laplace integral and the q-Homotopy Analysis Method for solving nonlinear integral equations, offering practical tools for modern analysis. Unified integral representations for generalized Mittag-Leffler functions further highlight the book's engagement with special functions.
Ideal for researchers and advanced students in mathematical analysis, this book seamlessly integrates classical theories with modern advancements, offering both theoretical insights and practical applications across mathematics, physics, and engineering.
Preface
Gauge Integrals on Metric Measure Spaces (S P S Kainth and N Singh)
Variational Henstock Integral and its Variational Measure in Locally Convex Space (S Bhatnagar)
Variational Version of Henstock type Integral and Application in Harmonic Analysis (V Skvortsov)
A Survey on the Riemann-Lebesgue Integrability in Non-additive Setting (A Croitoru, A Gavrilu?, A Iosif and A R Sambucini)
Some Nonlinear Integrals of Vector Multifunctions with Respect to a Submeasure (C Stamate and A Croitoru)
Convergence of Riemann Integrable Functions over Banach Spaces on Time Scales (H Bharali, V Sekhose and H Kalita)
Comparative Results among Different Types of Generalized Integrals (H Kalita and A Croitoru)
The Heat Equation with the Lp Primitive Integral (E Talvila)
On the Symmetric Laplace Integral and Its Application to Trigonometric Series (S Mahanta)
Finite and Infinite Integral Formulae Associated with the Family of Incomplete I-Functions (S Bhatter, Nishant, S D Purohit)
Homotopy Analysis Method for Solving Nonlinear Fredholm Integral Equations of Second Kind (S Paul and S Koley)
L1-space of Vector Measures with Density Defined on Â-rings (C Avalos-Ramos)
More on Unified Approach to Integration (M A Robdera)
Some Unified Integral Representations of the Four-parameter Mittag-Leffler Functions (A Pal and K Kumari)
Author Index
Primarily for researchers in Integration Theory and postgraduate students in Real and Functional Analysis.
Also suitable for advanced undergraduates preparing for graduate studies in mathematics.
September 2025
Pages: 300
ISBN: 978-981-98-1514-2 (hardcover)
ISBN: 978-981-98-1578-4 (softcover)
This book presents 125 unusual geometric relationships, offering clear proofs that will captivate readers of all backgrounds. Geometry is one of the most visually compelling and surprising branches of mathematics, full of unexpected relationships that are easy to see but astonishing to uncover. From the concurrency of triangle lines ? medians, altitudes, and bisectors ? to the mysterious parallelogram hidden in any quadrilateral, the discoveries explored in this volume reveal the intuitive and exciting nature of geometry.
To ensure all readers can fully appreciate these ideas, the opening chapters serve as a refresher on key high school geometry concepts, such as tangents and trigonometry, as well as powerful but often-overlooked theorems such as those by Ptolemy, Ceva, and Apollonius. With these tools in hand, you'll be ready to explore remarkable relationships that will deepen your appreciation for geometry. Whether you're a student, teacher, or math enthusiast, this collection will inspire fresh curiosity and excitement. Prepare to be amazed!
Introduction
The Basic Elements of Geometry
Spectacular Named Geometric Relationships
Applications of Some Named Geometric Relationships
Properties of Components of Triangles
Unusual 'Geometrick' Experiences
'Geometrick' Proofs
This book is intended for the general readership who would like to pursue mathematics
(particularly geometry) in a motivating fashion beyond the high school curriculum.
November 2025
Pages: 650
ISBN: 978-981-98-1154-0 (hardcover)
ISBN: 978-981-98-1240-0 (softcover)
The theory of ordinary differential equations is addressed in detail in this textbook, and is split into three sections: linear equations and systems, the general theory of nonlinear systems, and the theory of dynamical systems. These topics can be taken together or studied independently.
the theory of linear equation and systems with holomorphic coefficients; Kneser's theorem on the complexity of the set of solutions in the absence of uniqueness; the method of sub- and supersolutions for cooperative systems; and a detailed construction of the global bifurcation diagrams for some parametric classes of one-dimensional boundary value problems, which are pivotal for applications of the theory.
This is a self-contained, rigorous treatment of ordinary differential equations that is complemented by a variety of illustrating examples of the theory in practice. Many of these examples are related to models in Physics and Applied Sciences, making them suitable for students in Physics, Chemistry, Engineering, Mathematical Biology, Economics and Ecology as well as in Mathematics. Each chapter contains exercises to test students' understanding of the topic and concludes with some historical notes and further discussions.
Linear Systems:
First Order Linear Systems
First Order Linear Systems with Constant Coefficients
First Order Linear Systems with Holomorphic Coefficients
Nonlinear Systems:
An Introduction to Nonlinear Differential Equations
Cauchy?Lipschitz Theory
High Order Nonlinear Equations
Peano Theory
Method of Sub- and Supersolutions
Dynamical Systems:
Some Paradigmatic Dynamical Systems
Newtonian Planar Conservative Systems
Non-Conservative Systems
This book is targeted at undergraduate and graduate students in Mathematics, and professors in this subject. I
t would also be suitable for undergraduate and graduate students and professors in Physics, Chemistry, Engineering, Mathematical Biology, Economics and Ecology.