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This book provides an introduction to information theory, focussing on Shannon¿s three foundational theorems of 1948¿1949. Shannon¿s first two theorems, based on the notion of entropy in probability theory, specify the extent to which a message can be compressed for fast transmission and how to erase errors associated with poor transmission. The third theorem, using Fourier theory, ensures that a signal can be reconstructed from a sufficiently fine sampling of it. These three theorems constitute the roadmap of the book. The first chapter studies the entropy of a discrete random variable and related notions. The second chapter, on compression and error correcting, introduces the concept of coding, proves the existence of optimal codes and good codes (Shannon's first theorem), and shows how information can be transmitted in the presence of noise (Shannon's second theorem). The third chapter proves the sampling theorem (Shannon's third theorem) and looks at its connections with other results, such as the Poisson summation formula. Finally, there is a discussion of the uncertainty principle in information theory.Featuring a good supply of exercises (with solutions), and an introductory chapter covering the prerequisites, this text stems out lectures given to mathematics/computer science students at the beginning graduate level.

This book presents in a compact form the program carried out in introductory statistics courses and discusses some essential topics for research activity, such as Monte Carlo simulation techniques, methods of statistical inference, best fit and analysis of laboratory data. All themes are developed starting from fundamentals, highlighting their applicative aspects, up to the detailed description of several cases particularly relevant for technical and scientific research. The text is dedicated to university students in scientific fields and to all researchers who have to solve practical problems by applying data analysis and simulation procedures. The R software is adopted throughout the book, with a rich library of original programs accessible to the readers through a website.

The book is intended as an advanced undergraduate or first-year graduate course for students from various disciplines, including applied mathematics, physics and engineering. It has evolved from courses offered on partial differential equations (PDEs) over the last several years at the Politecnico di Milano. These courses had a twofold purpose: on the one hand, to teach students to appreciate the interplay between theory and modeling in problems arising in the applied sciences, and on the other to provide them with a solid theoretical background in numerical methods, such as finite elements. Accordingly, this textbook is divided into two parts. The first part, chapters 2 to 5, is more elementary in nature and focuses on developing and studying basic problems from the macro-areas of diffusion, propagation and transport, waves and vibrations. In turn the second part, chapters 6 to 11, concentrates on the development of Hilbert spaces methods for the variational formulation and the analysis of (mainly) linear boundary and initial-boundary value problems.

This book serves as a concise textbook for students in an advanced undergraduate or first-year graduate course in various disciplines such as applied mathematics, control, and engineering, who want to understand the modern standard of numerical methods of ordinary and delay differential equations. Experts in the same fields can also learn about the recent developments in numerical analysis of such differential systems. Ordinary differential equations (ODEs) provide a strong mathematical tool to express a wide variety of phenomena in science and engineering. Along with its own significance, one of the powerful directions toward which ODEs extend is to incorporate an unknown function with delayed argument. This is called delay differential equations (DDEs), which often appear in mathematical modelling of biology, demography, epidemiology, and control theory. In some cases, the solution of a differential equation can be obtained by algebraic combinations of known mathematical functions. In many practical cases, however, such a solution is quite difficult or unavailable, and numerical approximations are called for. Modern development of computers accelerates the situation and, moreover, launches more possibilities of numerical means. Henceforth, the knowledge and expertise of the numerical solution of differential equations becomes a requirement in broad areas of science and engineering.One might think that a well-organized software package such as MATLAB serves much the same solution. In a sense, this is true; but it must be kept in mind that blind employment of software packages misleads the user. The gist of numerical solution of differential equations still must be learned. The present book is intended to provide the essence of numerical solutions of ordinary differential equations as well as of delay differential equations. Particularly, the authors noted that there are still few concise textbooks of delay differential equations, and then they set about filling the gap through descriptions as transparent as possible. Major algorithms of numerical solution are clearly described in this book. The stability of solutions of ODEs and DDEs is crucial as well. The book introduces the asymptotic stability of analytical and numerical solutions and provides a practical way to analyze their stability by employing a theory of complex functions.

This self-contained and comprehensive textbook of algebraic number theory is useful for advanced undergraduate and graduate students of mathematics. The book discusses proofs of almost all basic significant theorems of algebraic number theory including Dedekind¿s theorem on splitting of primes, Dirichlet¿s unit theorem, Minkowski¿s convex body theorem, Dedekind¿s discriminant theorem, Hermite¿s theorem on discriminant, Dirichlet¿s class number formula, and Dirichlet¿s theorem on primes in arithmetic progressions. A few research problems arising out of these results are mentioned together with the progress made in the direction of each problem.Following the classical approach of Dedekind¿s theory of ideals, the book aims at arousing the reader¿s interest in the current research being held in the subject area. It not only proves basic results but pairs them with recent developments, making the book relevant and thought-provoking. Historical notes are given at various places. Featured with numerous related exercises and examples, this book is of significant value to students and researchers associated with the field. The book also is suitable for independent study. The only prerequisite is basic knowledge of abstract algebra and elementary number theory.

The prerequisites consist of the knowledge of basics in affine and projective geometry, basic algebraic concepts regarding rings, modules, fields, linear algebra, basic notions in the theory of categories, and some elementary point-set topology.This book can be used as a textbook for an undergraduate course in algebraic geometry.

This textbook is addressed to PhD or senior undergraduate students in mathematics, with interests in analysis, calculus of variations, probability and optimal transport.

The exposition is particularly suitable for students of Mathematics, Physics, Engineering and Statistics, besides providing the foundation essential for the study of Probability Theory and many branches of Applied Mathematics, including the Analysis of Financial Markets and other areas of Financial Engineering.

In this book we describe the magic world of mathematical models: starting from real-life problems, we formulate them in terms of equations, transform equations into algorithms and algorithms into programs to be executed on computers.A broad variety of examples and exercises illustrate that properly designed models can, e.g.: predict the way the number of dolphins in the Aeolian Sea will change as food availability and fishing activity vary; describe the blood flow in a capillary network; calculate the PageRank of websites.This book also includes a chapter with an elementary introduction to Octave, an open-source programming language widely used in the scientific community. Octave functions and scripts for dealing with the problems presented in the text can be downloaded from https://paola-gervasio.unibs.it/quarteroni-gervasioThis book is addressed to any student interested in learning how to construct and apply mathematical models.

Then, after a self-contained presentation of degree theory for continuous self-maps of the circumference, we study the global theory of plane curves, introducing winding and rotation numbers, and proving the Jordan curve theorem for curves of class C2, and Hopf theorem on the rotation number of closed simple curves.

Stats for social sciences is frequently the subject of textbooks, while maths for social sciences is often neglected: monographs on specific themes (like, for instance, social choice systems or game theory applications) are available, but they do not adequately cover the topic in general.

This book provides a basic, initial resource, introducing science and engineering students to the field of optimization. It covers three main areas: mathematical programming, calculus of variations and optimal control, highlighting the ideas and concepts and offering insights into the importance of optimality conditions in each area.

As a limit theory of quantum mechanics, classical dynamics comprises a large variety of phenomena, from computable (integrable) to chaotic (mixing) behavior.

This book provides a basic introduction to reduced basis (RB) methods for problems involving the repeated solution of partial differential equations (PDEs) arising from engineering and applied sciences, such as PDEs depending on several parameters and PDE-constrained optimization.

This book, intended as a practical working guide for calculus students, includes 450 exercises.

Finite volume methods are used in numerous applications and by a broad multidisciplinary scientific community. The book communicates this important tool to students, researchers in training and academics involved in the training of students in different science and technology fields.

This primer covers a remarkably wide range of topics in discrete calculus, from combinatorics to recurrence relations. With examples and exercises throughout, the text's clarity and rigor are informed by extensive teaching experience and academic discussion.

Accordingly, this textbook is not meant to cover the whole range of this high-performance technical programming environment, but to motivate first- and second-year undergraduate students in mathematics and computer science to learn Matlab by studying representative problems, developing algorithms and programming them in Matlab.

This is an introductory textbook on general and algebraic topology, aimed at anyone with a basic knowledge of calculus and linear algebra. complete metric spaces and function spaces; It is suitable for anyone needing a basic, comprehensive introduction to general and algebraic topology and its applications.

The second edition has been revised to correct minor errata, and features a number of carefully selected new exercises, together with more detailed explanations of some of the topics. A complete Solutions Manual, containing solutions to all the exercises published in the book, is available.

This textbook presents problems and exercises at various levels of difficulty in the following areas: Classical Methods in PDEs (diffusion, waves, transport, potential equations); Variational Formulation of Elliptic Problems; and Weak Formulation for Parabolic Problems and for the Wave Equation.

This textbook will present, in a rigorous way, the basic theory of the discrete-time and the continuous-time Markov chains, along with many examples and solved problems. This textbook is intended for a basic course on stochastic processes at an advanced undergraduate level and the background needed will be a first course in probability theory.

This book is an introduction to mathematical biology for students with no experience in biology, but who have some mathematical background.

The book represents a basic support for a master course in electromagnetism oriented to numerical simulation.

The book collects over 120 exercises on different subjects of Mathematical Finance, including Option Pricing, Risk Theory, and Interest Rate Models.

This book is designed as an advanced undergraduate or a first-year graduate course for students from various disciplines like applied mathematics, physics, engineering.