Handbook of Calculus of Variations for Absolute Beginners
The book aims at endowing any student with a survival toolkit to start safely diving into the realm of Calculus of Variations. In summary, the latter is a part of mathematical analysis devoted to minimization/maximization problems. A great effort has been made to present the themes and methods considered in the book in the simplest possible way: the reader will not find here general statements or proofs based on general abstract theories. In contrast, the main focus of the book is on introducing some key concepts "from scratch", by means of simple and meaningful explicit examples (including for instance, the classical isoperimetric and brachistocrone problems, as well as the boundary value problem for harmonic functions). In particular, the book is mainly (but not exclusively) designed to smoothly introduce the reader to the so-called Direct Method of the Calculus of Variations, which is a central concept in the field. Accordingly, a good part of the book is devoted to discussing spaces of weakly differentiable functions (i.e., Sobolev and Lipschitz functions), which are essential tools of the Direct Method. A long list of problems will guide the student through the study of the subject. Almost all the problems come with their fully detailed solutions. The book is complemented by four appendices, which contribute to making it self-contained, as well as to deepening the study of certain parts. Despite being designed for students, even the researchers in the field could find a reading of the book profitable, at least for certain parts concerning the properties of Sobolev spaces, functional inequalities of the Sobolev-Poincar矇 type, tricks to handle nonlinear elliptic PDEs, and a gentle introduction to some techniques of modern regularity theory for elliptic PDEs.
Advances in Fuzzy MCDM, Hybrid Methods, Fuzzy Number Ranking and Their Applications
This Special Issue of Axioms entitled "Advances in Fuzzy MCDM, Hybrid Methods, Fuzzy Number Ranking, and Their Applications" consists of a collection of ten papers written by eminent mathematicians and experts in their fields, covering numerous different areas of fuzzy MCDM, hybrid methods, ranking methods, and/or their applications. The objective of this Special Issue is to provide a platform for researchers to publish their recent work, delve deeper into various problems, and solve them mathematically.
Exponential Sums, Hypergeometric Sheaves, and Monodromy Groups
An examination of some of the remarkable connections between group theory and arithmetic algebraic geometry over finite fields Exponential sums have been of great interest ever since Gauss, and their importance in analytic number theory goes back a century to Kloosterman. Grothendieck's creation of the machinery of l-adic cohomology led to the understanding that families of exponential sums give rise to local systems, while Deligne, who gave his general equidistribution theorem after proving the Riemann hypothesis part of the Weil conjectures, established the importance of the monodromy groups of these local systems. Deligne's theorem shows that the monodromy group of the local system incarnating a given family of exponential sums determines key statistical properties of the family of exponential sums in question. Despite the apparent simplicity of this relation of monodromy groups to statistical properties, the actual determination of the monodromy group in any particular situation is highly nontrivial and leads to many interesting questions. This book is devoted to the determination of the monodromy groups attached to various explicit families of exponential sums, especially those attached to hypergeometric sheaves, arguably the simplest local systems on G_m, and to some simple (in the sense of simple to write down) one-parameter families of one-variable sums. These last families turn out to have surprising connections to hypergeometric sheaves. One of the main technical advances of this book is to bring to bear a group-theoretic condition (S+), which, when it applies, implies very strong structural constraints on the monodromy group, and to show that (S+) does indeed apply to the monodromy groups of most hypergeometric sheaves.
Variance-Constrained Filtering for Stochastic Complex Systems
Metrical and Ergodic Theory of Continued Fraction Algorithms
This monograph presents the work of the authors in metrical theory of continued fractions in the last two decades. The monograph cuts a particular path through this extensive theory and describes the theory in its current form for three families of continued fractions, namely, θ-continued fractions, N-continued fractions, and generalized R矇nyi continued fractions. The book systematically lays out the required preliminaries, making the book easy to read. This monograph provides a solid introduction into the theory of continued fractions. The book is intended for researchers in metrical theory, as well as advanced graduate students and mathematicians interested in this field.
Generalization of Some Inequalities for Rational Functions with Prescribed Poles
Elements of Algebraic Topology
This classic text appears here in a new edition for the first time in four decades. The new edition, with the aid of two new authors, brings it up to date for a new generation of mathematicians and mathematics students.Elements of Algebraic Topology provides the most concrete approach to the subject. With coverage of homology and cohomology theory, universal coefficient theorems, Kunneth theorem, duality in manifolds, and applications to classical theorems of point-set topology, this book is perfect for communicating complex topics and the fun nature of algebraic topology for beginners.This edition retains the essential features of the original book. Most of the notation and terminology is the same. There are some useful additions. There is a new introduction to homotopy theory. A new Index of Notation is included. Many new exercises are added.Algebraic topology is a cornerstone of modern mathematics. Every working mathematician should have at least an acquaintance with the subject. This book, which is based largely on the theory of triangulations, provides such an introduction. It should be accessible to a broad cross-section of the profession--both students and senior mathematicians. Students should have some familiarity with general topology.Algebraic topology is a cornerstone of modern mathematics. Every working mathematician should have at least an acquaintance with the subject. This book, which is based largely on the theory of triangulations, provides such an introduction. It should be accessible to a broad cross-section of the profession--both students and senior mathematicians. Students should have some familiarity with general topology.
Game-Based Learning, Gamification in Education and Serious Games 2023
In a world where video games are more than just entertainment, Current Advances in Serious Games, Game-based Learning and Gamification in Education explores the transformative power of games in shaping learning, awareness, and engagement. This collection highlights groundbreaking research on game-based learning, digital intelligence, and adaptive gamification. Featuring insights from leading experts, it examines how serious games combat misinformation, enhance media literacy, promote cultural heritage, and personalize educational experiences. Whether you are an educator, researcher, developer, or game enthusiast, this book provides a compelling look at how games are revolutionizing education, training, and social awareness. Dive into the cutting-edge world where learning meets play!
Introduction to Linear Algebra with Earth Science Applications
Deep Learning model for prediction and prevention of COVID-19 outbreak
Surveys in Combinatorics 2024
This volume contains nine survey articles by the invited speakers of the 30th British Combinatorial Conference, held at Queen Mary University of London in July 2024. Each article provides an overview of recent developments in a current hot research topic in combinatorics. Topics covered include: Latin squares, Erdős covering systems, finite field models, sublinear expanders, cluster expansion, the slice rank polynomial method, and oriented trees and paths in digraphs. The authors are among the world's foremost researchers on their respective topics but their surveys are accessible to nonspecialist readers: they are written clearly with little prior knowledge assumed and with pointers to the wider literature. Taken together these surveys give a snapshot of the research frontier in contemporary combinatorics, helping researchers and graduate students in mathematics and theoretical computer science to keep abreast of the latest developments in the field.
