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This textbook provides an accessible and visually intuitive introduction to Fourier series, which is one of the most powerful mathematical tools used to understand patterns, signals, and data. Equally well suited as a classroom textbook or a self-study resource, it is designed to be accessible to students encountering the subject for the first time, while also offering the rigor and breadth needed to support their scientific development in later years. The text balances depth, clarity, and readability, integrating strong visual intuition with fully rigorous theoretical development. Concepts are introduced in a way that highlights the geometric and intuitive structures underlying Fourier analysis, while maintaining the mathematical precision expected at higher levels of study. The exposition is carefully structured to guide readers from basic principles to more advanced results without sacrificing accessibility or conceptual insight.The book is also particularly distinctive in its coherent and unified presentation of both theory and real-world applications, weaving significant applications from engineering, physics, data analysis, and differential equations seamlessly into the theoretical narrative. The result is a text that not only teaches Fourier analysis in a rigorous and accessible way but also demonstrates its power and versatility across a wide range of scientific and technological fields.

This textbook provides an accessible and visually intuitive introduction to Fourier series, which is one of the most powerful mathematical tools used to understand patterns, signals, and data. Equally well suited as a classroom textbook or a self-study resource, it is designed to be accessible to students encountering the subject for the first time, while also offering the rigor and breadth needed to support their scientific development in later years. The text balances depth, clarity, and readability, integrating strong visual intuition with fully rigorous theoretical development. Concepts are introduced in a way that highlights the geometric and intuitive structures underlying Fourier analysis, while maintaining the mathematical precision expected at higher levels of study. The exposition is carefully structured to guide readers from basic principles to more advanced results without sacrificing accessibility or conceptual insight.The book is also particularly distinctive in its coherent and unified presentation of both theory and real-world applications, weaving significant applications from engineering, physics, data analysis, and differential equations seamlessly into the theoretical narrative. The result is a text that not only teaches Fourier analysis in a rigorous and accessible way but also demonstrates its power and versatility across a wide range of scientific and technological fields.

This volume collects contributions from the speakers at an INdAM Intensive period held at the University of Bari in 2017. The contributions cover several aspects of partial differential equations whose development in recent years has experienced major breakthroughs in terms of both theory and applications. The topics covered include nonlocal equations, elliptic equations and systems, fully nonlinear equations, nonlinear parabolic equations, overdetermined boundary value problems, maximum principles, geometric analysis, control theory, mean field games, and bio-mathematics. The authors are trailblazers in these topics and present their work in a way that is exhaustive and clearly accessible to PhD students and early career researcher. As such, the book offers an excellent introduction to a variety of fundamental topics of contemporary investigation and inspires novel and high-quality research.

The topics covered include nonlocal equations, elliptic equations and systems, fully nonlinear equations, nonlinear parabolic equations, overdetermined boundary value problems, maximum principles, geometric analysis, control theory, mean field games, and bio-mathematics.

This monograph introduces a new mathematical model in population dynamics that describes two species sharing the same environmental resources in a situation of open hostility.  Its main feature is the expansion of the family of Lotka-Volterra systems by introducing a new term that defines aggression.  Because the model is flexible, it can be applied to various scenarios in the context of human populations, such as strategy games, competition in the marketplace, and civil wars.Drawing from a variety of methodologies within dynamical systems, ODEs, and mathematical biology, the authors' approach focuses on the dynamical properties of the system. This is accomplished by detecting and describing all possible equilibria, and analyzing the strategies that may lead to the victory of the aggressive population. Techniques typical of two-dimensional dynamical systems are used, such as asymptotic behaviors regulated by the Poincaré–Bendixson Theorem.A New Lotka-Volterra Model of Competition With Strategic Aggression will appeal to researchers and students studying population dynamics and dynamical systems, particularly those interested in the cross section between mathematics and ecology.

This book presents in a detailed and self-contained way a new and important density result in the analysis of fractional partial differential equations, while also covering several fundamental facts about space- and time-fractional equations.

These lecture notes are devoted to the analysis of a nonlocal equation in the whole of Euclidean space. In studying this equation, all the necessary material is introduced in the most self-contained way possible, giving precise references to the literature when necessary.The results presented are original, but no particular prerequisite or knowledge of the previous literature is needed to read this text. The work is accessible to a wide audience and can also serve as introductory research material on the topic of nonlocal nonlinear equations.

This is a textbook that covers several selected topics in the theory of elliptic partial differential equations which can be used in an advanced undergraduate or graduate course.The book considers many important issues such as existence, regularity, qualitative properties, and all the classical topics useful in the wide world of partial differential equations. It also includes applications with interesting examples.The structure of the book is flexible enough to allow different chapters to be taught independently.The book is friendly, welcoming, and written for a newcomer to the subject.It is essentially self-contained, making it easy to read, and all the concepts are fully explained from scratch, combining intuition and rigor, and therefore it can also be read independently by students, with limited or no supervision.

This is a textbook that covers several selected topics in the theory of elliptic partial differential equations which can be used in an advanced undergraduate or graduate course.The book considers many important issues such as existence, regularity, qualitative properties, and all the classical topics useful in the wide world of partial differential equations. It also includes applications with interesting examples.The structure of the book is flexible enough to allow different chapters to be taught independently.The book is friendly, welcoming, and written for a newcomer to the subject.It is essentially self-contained, making it easy to read, and all the concepts are fully explained from scratch, combining intuition and rigor, and therefore it can also be read independently by students, with limited or no supervision.

This book offers a captivating journey through the world of elliptic partial differential equations (PDEs) and their surprising appearances across science, engineering, and even everyday life. Blending mathematical rigor with an accessible and engaging style, it introduces the Laplace operator and elliptic PDEs through diverse real-world phenomena, from physics and biology to artificial intelligence and image processing.Readers will explore fundamental topics such as diffusion, wave propagation, and fluid dynamics, alongside unexpected applications — including pattern formation in nature, traffic jams, quantum mechanics, and even the mathematics behind fighting pandemics and cancer. With a focus on both intuition and formalism, the book provides a wealth of examples, historical insights, and thought-provoking problems that make the subject come alive.Awarded the prestigious Book Prize of the Unione Matematica Italiana, this work has been praised as "a remarkable journey through delightful applications of elliptic PDEs, all while maintaining the highest standards of mathematical rigor." Whether you are a student, researcher, or simply curious about how mathematics shapes the world around us, this book will challenge and inspire you.