ADVANCED TEXTBOOKS IN MATHEMATICS – serie

"The book is mainly addressed to the non-expert reader, in that it assumes only a little background in complex analysis and algebraic geometry, but no previous knowledge in transcendental number theory is required. The technical language is introduced smoothly, and illustrative examples are provided where appropriate … The book is carefully written, and the relevant literature is provided in the list of references. "Mathematical Reviews ClippingsThis book gives an introduction to some central results in transcendental number theory with application to periods and special values of modular and hypergeometric functions. It also includes related results on Calabi–Yau manifolds. Most of the material is based on the author's own research and appears for the first time in book form. It is presented with minimal of technical language and no background in number theory is needed. In addition, except the last chapter, all chapters include exercises suitable for graduate students. It is a nice book for graduate students and researchers interested in transcendence.

This book provides an in-depth and rigorous study of the Wigner transform and its variants. They are presented first within a context of a general mathematical framework, and then through applications to quantum mechanics. The Wigner transform was introduced by Eugene Wigner in 1932 as a probability quasi-distribution which allows expression of quantum mechanical expectation values in the same form as the averages of classical statistical mechanics. It is also used in signal processing as a transform in time-frequency analysis, closely related to the windowed Gabor transform.Written for advanced-level students and professors in mathematics and mathematical physics, it is designed as a complete textbook course providing analysis on the most important research on the subject to date. Due to the advanced nature of the content, it is also suitable for research mathematicians, engineers and chemists active in the field.

This book provides an in-depth and rigorous study of the Wigner transform and its variants. They are presented first within a context of a general mathematical framework, and then through applications to quantum mechanics. The Wigner transform was introduced by Eugene Wigner in 1932 as a probability quasi-distribution which allows expression of quantum mechanical expectation values in the same form as the averages of classical statistical mechanics. It is also used in signal processing as a transform in time-frequency analysis, closely related to the windowed Gabor transform.Written for advanced-level students and professors in mathematics and mathematical physics, it is designed as a complete textbook course providing analysis on the most important research on the subject to date. Due to the advanced nature of the content, it is also suitable for research mathematicians, engineers and chemists active in the field.

Mathematics of Planet Earth (MPE) was started and continues to be consolidated as a collaboration of mathematical science organisations around the world. These organisations work together to tackle global environmental, social and economic problems using mathematics.This textbook introduces the fundamental topics of MPE to advanced undergraduate and graduate students in mathematics, physics and engineering while explaining their modern usages and operational connections. In particular, it discusses the links between partial differential equations, data assimilation, dynamical systems, mathematical modelling and numerical simulations and applies them to insightful examples.The text also complements advanced courses in geophysical fluid dynamics (GFD) for meteorology, atmospheric science and oceanography. It links the fundamental scientific topics of GFD with their potential usage in applications of climate change and weather variability. The immediacy of examples provides an excellent introduction for experienced researchers interested in learning the scope and primary concepts of MPE.

Mathematics of Planet Earth (MPE) was started and continues to be consolidated as a collaboration of mathematical science organisations around the world. These organisations work together to tackle global environmental, social and economic problems using mathematics.This textbook introduces the fundamental topics of MPE to advanced undergraduate and graduate students in mathematics, physics and engineering while explaining their modern usages and operational connections. In particular, it discusses the links between partial differential equations, data assimilation, dynamical systems, mathematical modelling and numerical simulations and applies them to insightful examples.The text also complements advanced courses in geophysical fluid dynamics (GFD) for meteorology, atmospheric science and oceanography. It links the fundamental scientific topics of GFD with their potential usage in applications of climate change and weather variability. The immediacy of examples provides an excellent introduction for experienced researchers interested in learning the scope and primary concepts of MPE.

The book contains self-contained descriptions of existing models, accompanied by critical analyses of their properties both from a theoretical and practical standpoint. It aims to develop 'modeling skills' within the readers, giving them the ability to develop their own models and improve existing ones. Written in connection with a full, open source Python Library, this project also enables readers to run the simulations discussed within the text.

"I very much enjoyed reading this book … Each chapter comes with well thought-out exercises, solutions to which are given at the end of the chapter. Conformal Maps and Geometry presents key topics in geometric function theory and the theory of univalent functions, and also prepares the reader to progress to study the SLE. It succeeds admirably on both counts."MathSciNetGeometric function theory is one of the most interesting parts of complex analysis, an area that has become increasingly relevant as a key feature in the theory of Schramm–Loewner evolution.Though Riemann mapping theorem is frequently explored, there are few texts that discuss general theory of univalent maps, conformal invariants, and Loewner evolution. This textbook provides an accessible foundation of the theory of conformal maps and their connections with geometry.It offers a unique view of the field, as it is one of the first to discuss general theory of univalent maps at a graduate level, while introducing more complex theories of conformal invariants and extremal lengths. Conformal Maps and Geometry is an ideal resource for graduate courses in Complex Analysis or as an analytic prerequisite to study the theory of Schramm–Loewner evolution.

In today's world, we are increasingly exposed to the words 'machine learning' (ML), a term which sounds like a panacea designed to cure all problems ranging from image recognition to machine language translation. Over the past few years, ML has gradually permeated the financial sector, reshaping the landscape of quantitative finance as we know it.An Introduction to Machine Learning in Quantitative Finance aims to demystify ML by uncovering its underlying mathematics and showing how to apply ML methods to real-world financial data. In this book the authorsFeatured with the balance of mathematical theorems and practical code examples of ML, this book will help you acquire an in-depth understanding of ML algorithms as well as hands-on experience. After reading An Introduction to Machine Learning in Quantitative Finance, ML tools will not be a black box to you anymore, and you will feel confident in successfully applying what you have learnt to empirical financial data!The Python codes contained within An Introduction to Machine Learning in Quantitative Finance have been made publicly available on the author's GitHub: https://github.com/deepintomlf/mlfbook.git

In today's world, we are increasingly exposed to the words 'machine learning' (ML), a term which sounds like a panacea designed to cure all problems ranging from image recognition to machine language translation. Over the past few years, ML has gradually permeated the financial sector, reshaping the landscape of quantitative finance as we know it.An Introduction to Machine Learning in Quantitative Finance aims to demystify ML by uncovering its underlying mathematics and showing how to apply ML methods to real-world financial data. In this book the authorsFeatured with the balance of mathematical theorems and practical code examples of ML, this book will help you acquire an in-depth understanding of ML algorithms as well as hands-on experience. After reading An Introduction to Machine Learning in Quantitative Finance, ML tools will not be a black box to you anymore, and you will feel confident in successfully applying what you have learnt to empirical financial data!The Python codes contained within An Introduction to Machine Learning in Quantitative Finance have been made publicly available on the author's GitHub: https://github.com/deepintomlf/mlfbook.git

The quest for the optimal is ubiquitous in nature and human behavior. The field of mathematical optimization has a long history and remains active today, particularly in the development of machine learning.Classical and Modern Optimization presents a self-contained overview of classical and modern ideas and methods in approaching optimization problems. The approach is rich and flexible enough to address smooth and non-smooth, convex and non-convex, finite or infinite-dimensional, static or dynamic situations. The first chapters of the book are devoted to the classical toolbox: topology and functional analysis, differential calculus, convex analysis and necessary conditions for differentiable constrained optimization. The remaining chapters are dedicated to more specialized topics and applications.Valuable to a wide audience, including students in mathematics, engineers, data scientists or economists, Classical and Modern Optimization contains more than 200 exercises to assist with self-study or for anyone teaching a third- or fourth-year optimization class.

The quest for the optimal is ubiquitous in nature and human behavior. The field of mathematical optimization has a long history and remains active today, particularly in the development of machine learning.Classical and Modern Optimization presents a self-contained overview of classical and modern ideas and methods in approaching optimization problems. The approach is rich and flexible enough to address smooth and non-smooth, convex and non-convex, finite or infinite-dimensional, static or dynamic situations. The first chapters of the book are devoted to the classical toolbox: topology and functional analysis, differential calculus, convex analysis and necessary conditions for differentiable constrained optimization. The remaining chapters are dedicated to more specialized topics and applications.Valuable to a wide audience, including students in mathematics, engineers, data scientists or economists, Classical and Modern Optimization contains more than 200 exercises to assist with self-study or for anyone teaching a third- or fourth-year optimization class.

"I very much enjoyed reading this book … Each chapter comes with well thought-out exercises, solutions to which are given at the end of the chapter. Conformal Maps and Geometry presents key topics in geometric function theory and the theory of univalent functions, and also prepares the reader to progress to study the SLE. It succeeds admirably on both counts."MathSciNetGeometric function theory is one of the most interesting parts of complex analysis, an area that has become increasingly relevant as a key feature in the theory of Schramm–Loewner evolution.Though Riemann mapping theorem is frequently explored, there are few texts that discuss general theory of univalent maps, conformal invariants, and Loewner evolution. This textbook provides an accessible foundation of the theory of conformal maps and their connections with geometry.It offers a unique view of the field, as it is one of the first to discuss general theory of univalent maps at a graduate level, while introducing more complex theories of conformal invariants and extremal lengths. Conformal Maps and Geometry is an ideal resource for graduate courses in Complex Analysis or as an analytic prerequisite to study the theory of Schramm–Loewner evolution.

This book sets out to demonstrate, interpret, and analyse the geometrical structures underlying classical mechanics. Through exploring the applications of these structures in the context of dissipative, autonomous, and nonautonomous conservative dynamical systems, a series of insightful exercises are developed in order to both consolidate and clarify the theoretical concepts introduced.Geometry of Mechanics provides an informative exploration of the classic geometrical structures, including the symplectic structure and its application for constructing Lagrangian and Hamiltonian formalisms of autonomous regular systems, the cosymplectic structure (and others) for nonautonomous (time-dependent) systems, and the Riemannian structure underlying systems of "mechanical" type. Lesser-known frameworks are also investigated, such as the (Skinner–Rusk) unified Lagrangian–Hamiltonian formalism, a new geometric setting for the Hamilton–Jacobi equation, and the latest treatment of nonconservative systems by contact geometry. Although the main focus of this exposition is on the regular Lagrangian and Hamiltonian systems, singular systems are also briefly explained where applicable.Each chapter concludes with a set of problems, some of which are intended to be solved solely by the application of results presented, while others contain instructive results complementary to those presented in the chapters, complete with comments, suggestions, and recommendations. Interested readers will also find an extensive and up-to-date bibliography, including a great number of works produced in recent decades related to all topics explored.

This book sets out to demonstrate, interpret, and analyse the geometrical structures underlying classical mechanics. Through exploring the applications of these structures in the context of dissipative, autonomous, and nonautonomous conservative dynamical systems, a series of insightful exercises are developed in order to both consolidate and clarify the theoretical concepts introduced.Geometry of Mechanics provides an informative exploration of the classic geometrical structures, including the symplectic structure and its application for constructing Lagrangian and Hamiltonian formalisms of autonomous regular systems, the cosymplectic structure (and others) for nonautonomous (time-dependent) systems, and the Riemannian structure underlying systems of "mechanical" type. Lesser-known frameworks are also investigated, such as the (Skinner–Rusk) unified Lagrangian–Hamiltonian formalism, a new geometric setting for the Hamilton–Jacobi equation, and the latest treatment of nonconservative systems by contact geometry. Although the main focus of this exposition is on the regular Lagrangian and Hamiltonian systems, singular systems are also briefly explained where applicable.Each chapter concludes with a set of problems, some of which are intended to be solved solely by the application of results presented, while others contain instructive results complementary to those presented in the chapters, complete with comments, suggestions, and recommendations. Interested readers will also find an extensive and up-to-date bibliography, including a great number of works produced in recent decades related to all topics explored.