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Introduction to Abelian Varieties
This book presents an elementary and self-contained approach to Abelian varieties, a subject that plays a central role in algebraic and analytic geometry, number theory, and complex analysis. The book is based on notes from a course given at Concordia University and would be useful for independent study or as a textbook for graduate courses in complex analysis, Riemann surfaces, number theory, or analytic geometry. Murty works mostly over the complex numbers, discussing the theorem of Abel-Jacobi and Lefschetz's theorem on projective embeddings. After presenting some examples, Murty touches on Abelian varieties over number fields, as well as the conjecture of Tate (Faltings's theorem) and its relation to Mordell's conjecture. References are provided to guide the reader in further study.
Advanced Machine Intelligence and Signal Processing
This book covers the latest advancements in the areas of machine learning, computer vision, pattern recognition, computational learning theory, big data analytics, network intelligence, signal processing, and their applications in real world. The topics covered in machine learning involve feature extraction, variants of support vector machine (SVM), extreme learning machine (ELM), artificial neural network (ANN), and other areas in machine learning. The mathematical analysis of computer vision and pattern recognition involves the use of geometric techniques, scene understanding and modeling from video, 3D object recognition, localization and tracking, medical image analysis, and so on. Compu...
An Introduction to the Circle Method
The circle method, pioneered by Ramanujan and Hardy in the early 20th century, has over the past 100 years become part of the standard tool chest of analytic number theory. Its scope of applications is ever-expanding, and the subject continues to see important breakthroughs. This book provides an introduction to the circle method that is accessible to undergraduate students with no background in number theory. The authors' goal is to show the students the elegance of the circle method and at the same time give a complete solution of the famous Waring problem as an illustration of the method. The first half of this book is a curated introduction to elementary number theory with an emphasis on topics needed for the second half. The second half showcases the two most “classic” applications of the circle method, to Waring's problem (following Hardy–Littlewood–Hua) and to Goldbach's conjectures (following Vinogradov, with improvements by Vaughan). This text is suitable for a one-semester undergraduate course or for independent study and will be a great entry point into this fascinating area of research.
Seminar on Fermat's Last Theorem
The most significant recent development in number theory is the work of Andrew Wiles on modular elliptic curves. Besides implying Fermat's Last Theorem, his work establishes a new reciprocity law. Reciprocity laws lie at the heart of number theory. Wiles' work draws on many of the tools of modern number theory and the purpose of this volume is to introduce readers to some of this background material. Based on a seminar held during 1993-1994 at the Fields Institute for Research in Mathematical Sciences, this book contains articles on elliptic curves, modular forms and modular curves, Serre's conjectures, Ribet's theorem, deformations of Galois representations, Euler systems, and annihilators of Selmer groups. All of the authors are well known in their field and have made significant contributions to the general area of elliptic curves, Galois representations, and modular forms. Features: Brings together a unique collection of number theoretic tools. Makes accessible the tools needed to understand one of the biggest breakthroughs in mathematics. Provides numerous references for further study.
Geometry, Algebra, Number Theory, and Their Information Technology Applications
This volume contains proceedings of two conferences held in Toronto (Canada) and Kozhikode (India) in 2016 in honor of the 60th birthday of Professor Kumar Murty. The meetings were focused on several aspects of number theory: The theory of automorphic forms and their associated L-functions Arithmetic geometry, with special emphasis on algebraic cycles, Shimura varieties, and explicit methods in the theory of abelian varieties The emerging applications of number theory in information technology Kumar Murty has been a substantial influence in these topics, and the two conferences were aimed at honoring his many contributions to number theory, arithmetic geometry, and information technology.
Human Heredity
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Palaeoproterozoic of India
The Indian shield represents a vast repository of the Palaeoproterozoic geological record. Built over the four large amalgamated Archaean nuclei (Dharwar, Bastar, Singhbhum and Aravalli-Bundelkhand) the major and minor Palaeoproterozoic sedimentary basins and supracrustal sequences in India are comparable in scale, and perhaps also in development, to those of North America, Africa, Australia and Brazil. The deformation of these supracrustal sequences, attendant metamorphism and emplacement of plutonic bodies hold important clues to their connection with major orogenies. Research in these areas has led to investigations into global correlation, which in turn has had a direct bearing on refining models of Palaeoproterozoic supercontinent assembly and break-up. This book covers various aspects of regional geology as well as broader issues of the Indian Palaeoproterozoic geology and its global context. It is an outcome of the UNESCO-IGCP 509 Palaeoproterozoic Supercontinents and Global Evolution research project.
Mining Intelligence and Knowledge Exploration
This book constitutes the proceedings of the Second International Conference on Mining Intelligence and Knowledge Exploration, MIKE 2014, held in Cork, Ireland, in December 2014. The 40 papers presented were carefully reviewed and selected from 69 submissions. The papers cover topics such as information retrieval, feature selection, classification, clustering, image processing, network security, speech processing, machine learning, recommender systems, natural language processing, language, cognition and computation, and business intelligence.
The Mathematical Legacy of Srinivasa Ramanujan
Srinivasa Ramanujan was a mathematician brilliant beyond comparison who inspired many great mathematicians. There is extensive literature available on the work of Ramanujan. But what is missing in the literature is an analysis that would place his mathematics in context and interpret it in terms of modern developments. The 12 lectures by Hardy, delivered in 1936, served this purpose at the time they were given. This book presents Ramanujan’s essential mathematical contributions and gives an informal account of some of the major developments that emanated from his work in the 20th and 21st centuries. It contends that his work still has an impact on many different fields of mathematical research. This book examines some of these themes in the landscape of 21st-century mathematics. These essays, based on the lectures given by the authors focus on a subset of Ramanujan’s significant papers and show how these papers shaped the course of modern mathematics.
