Softcover ISBN: 978-1-4704-7332-7
Product Code: MAWRLD/31
Mathematical WorldVolume: 31;
2023; 185 pp
MSC: Primary 11; 12; 13;
This book is a concrete introduction to abstract algebra and number theory. Starting from the basics, it develops the rich parallels between the integers and polynomials, covering topics such as Unique Factorization, arithmetic over quadratic number fields, the RSA encryption scheme, and finite fields.
In addition to introducing students to the rigorous foundations of mathematical proofs, the authors cover several specialized topics, giving proofs of the Fundamental Theorem of Algebra, the transcendentality of e
, and Quadratic Reciprocity Law. The book is aimed at incoming undergraduate students with a strong passion for mathematics.
High school teachers and undergraduate students interested in algebra.
Preface
Chapter 1. The Integers
1.1. Basic Properties
1.2. Well Ordering Principle
1.3. Primes
1.4. Many Primes
1.5. Euclidean Algorithm
1.6. Factoring Integers
1.7. Irrational Numbers
1.8. Unique Factorization in More General Rings
Notes on Chapter 1
Chapter 2. Modular Arithmetic
2.1. Linear Equations
2.2. Congruences
2.3. The Ring \bZ_{??}
2.4. Equivalence Relations
2.5. Chinese Remainder Theorem
2.6. Congruence Equations
2.7. Fermat’s Little Theorem
2.8. Euler’s Theorem
2.9. More on Euler’s Phi Function
2.10. Primitive Roots
Notes on Chapter 2
Chapter 3. Diophantine Equations and Quadratic Number Domains
3.1. Pythagorean Triples
3.2. Fermat’s Equation for ??=4
3.3. Quadratic Number Domains
3.4. Pell’s Equation
3.5. The Gaussian Integers
3.6. Quadratic Reciprocity
Notes on Chapter 3
Chapter 4. Codes and Factoring
4.1. Codes
4.2. The Rivest-Shamir-Adelman Scheme
4.3. Primality Testing
4.4. Factoring Algorithms
Notes on Chapter 4
Chapter 5. Real and Complex Numbers
5.1. Real Numbers
5.2. Complex Numbers
5.3. Polar Form
5.4. The Exponential Function
5.5. Fundamental Theorem of Algebra
5.6. Real Polynomials
Notes on Chapter 5
Chapter 6. The Ring of Polynomials
6.1. Preliminaries on Polynomials
6.2. Unique Factorization for Polynomials
6.3. Irreducible Polynomials in \bZ[??]
6.4. Eisenstein’s Criterion
6.5. Factoring Modulo Primes
6.6. Algebraic Numbers
6.7. Transcendental Numbers
6.8. Sturm’s Algorithm
6.9. Symmetric Functions
6.10. Cubic Polynomials
Notes on Chapter 6
Chapter 7. Finite Fields
7.1. Arithmetic Modulo a Polynomial
7.2. An Eight-Element Field
7.3. Fermat’s Little Theorem for Finite Fields
7.4. Characteristic
7.5. Algebraic Elements
7.6. Finite Fields
7.7. Automorphisms of \bF_{??^{??}}
7.8. Irreducible polynomials of all degrees
7.9. Factoring Algorithms for Polynomials
7.10. Factoring Rational Polynomials
Notes on Chapter 7
Bibliography
Index
Hardcover ISBN: 978-1-4704-7525-3
Expected availability date: January 09, 2024
Graduate Studies in Mathematics Volume: 237;
2023; 325 pp
MSC: Primary 35; Secondary 47; 58; 65;
It is remarkable that various distinct physical phenomena, such as wave propagation, heat diffusion, electron movement in quantum mechanics, oscillations of fluid in a container, can be described using the same differential operator, the Laplacian. Spectral data (i.e., eigenvalues and eigenfunctions) of the Laplacian depend in a subtle way on the geometry of the underlying object, e.g., a Euclidean domain or a Riemannian manifold, on which the operator is defined. This dependence, or, rather, the interplay between the geometry and the spectrum, is the main subject of spectral geometry. Its roots can be traced to Ernst Chladni's experiments with vibrating plates, Lord Rayleigh's theory of sound, and Mark Kac's celebrated question “Can one hear the shape of a drum?” In the second half of the twentieth century spectral geometry emerged as a separate branch of geometric analysis. Nowadays it is a rapidly developing area of mathematics, with close connections to other fields, such as differential geometry, mathematical physics, partial differential equations, number theory, dynamical systems, and numerical analysis.
This book can be used for a graduate or an advanced undergraduate course on spectral geometry, starting from the basics but at the same time covering some of the exciting recent developments which can be explained without too many prerequisites.
Graduate students and researchers interested in differential geometry and Laplace operators.
Strings, drums, and the Laplacian
The spectral theorems
Variational principles and applications
Nodal geometry of eigenfunctions
Eigenvalue inequalities
Heat equation, spectral invariants, and isospectrality
The Steklov problem and the Dirichlet-to-Neumann map
A short tutorial on numerical spectral geometry
Background definitions and notation
Fractional Differential Equations: Theoretical Aspects and Applicationspresents the latest mathematical and conceptual developments in the field of Fractional Calculus and explores the scope of applications in research science and computational modeling. The book delves into these methods and applied computational modelling techniques, including analysis of equations involving fractional derivatives, fractional derivatives and the wave equation, analysis of FDE on groups, direct and inverse problems, functional inequalities, and computational methods for FDEs in physics and engineering. Other modeling techniques and applications explored include general fractional derivatives involving the special functions in analysis and fractional derivatives with respect to other functions. Fractional Calculus, the field of mathematics dealing with operators of differentiation and integration of arbitrary real or even complex order, extends many of the modelling capabilities of conventional calculus and integer-order differential equations and finds its application in various scientific areas, such as physics, mechanics, engineering, economics, finance, biology, and chemistry, among others.
Provides the most recent and up-to-date developments in the theory and scientific applications Fractional Differential Equations
Includes transportable computer source codes for readers in MATLAB, with code descriptions as it relates to the mathematical modelling and applications
Provides readers with a comprehensive foundational reference for this key topic in computational modeling, which is a mathematical underpinning for most areas of scientific and engineering research
Copyright 2024
Hardback
ISBN 9781032575032
268 Pages
October 30, 2023
Multiplicative Partial Differential Equations presents an introduction to the theory of multiplicative partial differential equations (MPDEs). It is suitable for all types of basic courses on MPDEs. The authors' aim is to present a clear and well-organized treatment of the concepts behind the development of mathematics and solution techniques. The text is presented in a highly readable, mathematically solid format. Many practical problems are illustrated, displaying a wide variety of solution techniques.
Includes new classification and canonical forms of second-order MPDEs
Proposes the latest techniques in solving the multiplicative wave equation such as the method of separation of variables and the energy method
Useful in allowing for the basic properties of multiplicative elliptic problems, fundamental solutions, multiplicative integral representation of multiplicative harmonic functions, meant-value formulas, strong principle of maximum, multiplicative Poisson equation, multiplicative Green functions, method of separation of variables, and theorems of Liouville and Harnack
1. 1. General Introduction
2. 2. Classification of Second Order Multiplicative Partial Differential Equations
3. 3. Classification and Canonical Forms
4. 4. The Multiplicative Wave Equation
5. 5. The Heat Equation
6. 6. The Laplace Equation
7. 7. The Cauchy-Kovalevskaya Theorem
Copyright 2024
Hardback
ISBN 9781032397818
458 Pages 56 Color Illustrations
December 12, 2023 by Chapman & Hall
Spatio-Temporal Methods in Environmental Epidemiology with R, like its First Edition, explores the interface between environmental epidemiology and spatio-temporal modeling. It links recent developments in spatio-temporal theory with epidemiological applications. Drawing on real-life problems, it shows how recent advances in methodology can assess the health risks associated with environmental hazards. The book's clear guidelines enable the implementation of the methodology and estimation of risks in practice.
New additions to the Second Edition include: a thorough exploration of the underlying concepts behind knowledge discovery through data; a new chapter on extracting information from data using R and the tidyverse; additional material on methods for Bayesian computation, including the use of NIMBLE and Stan; new methods for performing spatio-temporal analysis and an updated chapter containing further topics. Throughout the book there are new examples, and the presentation of R code for examples has been extended. Along with these additions, the book now has a GitHub site (https://spacetime-environ.github.io/stepi2) that contains data, code and further worked examples.
? Explores the interface between environmental epidemiology and spatio--temporal modeling
? Incorporates examples that show how spatio-temporal methodology can inform societal concerns about the effects of environmental hazards on health
? Uses a Bayesian foundation on which to build an integrated approach to spatio-temporal modeling and environmental epidemiology
? Discusses data analysis and topics such as data visualization, mapping, wrangling and analysis
? Shows how to design networks for monitoring hazardous environmental processes and the ill effects of preferential sampling
? Through the listing and application of code, shows the power of R, tidyverse, NIMBLE and Stan and other modern tools in performing complex data analysis and modeling
Representing a continuing important direction in environmental epidemiology, this book ? in full color throughout ? underscores the increasing need to consider dependencies in both space and time when modeling epidemiological data. Readers will learn how to identify and model patterns in spatio-temporal data and how to exploit dependencies over space and time to reduce bias and inefficiency when estimating risks to health.
1. An overview of spatio-temporal epidemiology and knowledge discovery
2. An introduction to modelling health risks and impacts
3. The importance of uncertainty: assessment and quantification
4. Extracting information from data
5. Embracing uncertainty: the Bayesian approach
6. Approaches to Bayesian computation
7. Strategies for modelling
8. The challenges of working with real-world data
9. Spatial modelling: areal data
10. Spatial modelling: point-referenced data
11. Modelling temporal data: time series analysis and forecasting
12. Bringing it all together: modelling exposures over space and time
13. Causality: issues and challenges
14. The quality of data: the importance of network design
15. Further topics in spatio-temporal modelling
In the series De Gruyter Textbook
This book is intended to be used as a rather informal, and surely not complete, textbook on the subjects indicated in the title. It collects my Lecture Notes held during three academic years at the University of Siena for a one semester course on "Basic Mathematical Physics", and is organized as a short presentation of few important points on the arguments indicated in the title.
It aims at completing the students' basic knowledge on Ordinary Differential Equations (ODE) - dealing in particular with those of higher order - and at providing an elementary presentation of the Partial Differential Equations (PDE) of Mathematical Physics, by means of the classical methods of separation of variables and Fourier series. For a reasonable and consistent discussion of the latter argument, some elementary results on Hilbert spaces and series expansion in othonormal vectors are treated with some detail in Chapter 2.
Prerequisites for a satisfactory reading of the present Notes are not only a course of Calculus for functions of one or several variables, but also a course in Mathematical Analysis where - among others - some basic knowledge of the topology of normed spaces is supposed to be included. For the reader's convenience some notions in this context are explicitly recalled here and there, and in particular as an Appendix in Section 1.4. An excellent reference for this general background material is W. Rudin's classic Principles of Mathematical Analysis. On the other hand, a complete discussion of the results on ODE and PDE that are here just sketched are to be found in other books, specifically and more deeply devoted to these subjects, some of which are listed in the Bibliography.
In conclusion and in brief, my hope is that the present Notes can serve as a second quick reading on the theme of ODE, and as a first introductory reading on Fourier series, Hilbert spaces, and PDE
A short introduction to the 3 subjects in the title
Convenient for students in Maths, Physics, Engineering or Biology
Appropriate for an one-semester course
Raffaele Chiappinelli obtained his Laurea in Physics at the University of Naples, Italy (1974) and then his Ph.D. at the University of Sussex, UK (1986) under the advice of D.E. Edmunds. He has been Associate Professor of Mathematical Analysis for more than 30 years, first at the University of Calabria (1986-1994) and then at the University of Siena (1994-2018), where he is remained in activity as a Senior Professor. His research interests focus on Eigenvalue problems and spectral theory for nonlinear operators acting in Banach spaces and on the related Bifurcation theory, with applications to ordinary and partial differential equations. He is author of more than 40 publication.