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Bieske T: Elementary Real Analysis

Elementary Real Analysis

A Practical Introduction

The Analyst's Gambit: A Second Course in Functional Analysis is a textbook written to serve a graduate course in Functional Analysis. It provides a sequel to the author's other volume, A First Course in Functional Analysis, but it is not necessary to have read one in order to make use of the other. As a graduate text, the reader is assumed to have taken undergraduate courses in set theory, calculus, metric spaces and topology, complex analysis, measure theory (or, alternatively, have enough mathematical maturity to carry on without having seen every particular fact that is used).A particular strength of the book is that it includes numerous applications. Besides being engaging and interesting in their own right, these applications also illustrate how functional analysis is used in other parts of mathematics. The applications to problems from varied fields (PDEs, Fourier series, group theory, neural networks, topology, etc.) constitute a gratifying external motivation for studying functional analysis. There are also applications of the material to functional analytic problems (Lomonosov's invariant subspace theorem, the spectral theorem, Stone's theorem), showcasing the power of the results as well as the elegance and unity of the theory.Features¿ Rich variety of exercises¿ Can be used as the primary textbook for a graduate course in functional analysis¿ Emphasis on substantial and modern applicationsOrr Moshe Shalit is a professor of mathematics at the Technion Israel Institute of Technology, where he teaches and conducts research in operator theory, operator algebras, functional analysis and function theory. His first book, A First Course in Functional Analysis, was published by Chapman &amp; Hall / CRC in 2017.

A Second Course in Functional Analysis

The Analyst’s Gambit

Cambridge Monographs on Applied and Computational Mathematics

Cubature rules are indispensable tools in scientific computing and applied sciences whenever evaluating or discretizing integrals is needed. This monograph is the first comprehensive resource devoted to cubature rules in English since Stroud's classic 1971 book, and the first book about minimal cubature rules. The book explores the subject's theoretical side, which intersects with many branches of mathematics. Minimal cubature rules are intimately connected with common zeros of orthogonal polynomials, which can be described via the polynomial ideals and varieties. Many prominent or practical cubature rules are invariant under a finite group, and some involve symmetric functions and the discrete Fourier transform. Based on state-of-the-art research, the book systematically studies Gauss and minimal cubature rules, and includes a chapter on the practical aspects of construction cubature rules on triangles and simplexes. This comprehensive guide is ideal for researchers and advanced graduate students across the computational and applied mathematics community.

Theory and Practice

Minimal Cubature Rules

Reliability, Maintenance, and Safety Engineering

Nearly all our safety data collection and reporting systems are backward-looking: incident reports; dashboards; compliance monitoring systems, and so on. This book shows how we can use safety data in a forward-looking, predictive sense. It contains real use cases where organizations have reduced incidents by employing predictive analytics.

Reducing Risk through Modeling and Machine Learning

Predictive Safety Analytics

Encyclopedia of Mathematics and its Applications

Single and Multiple Number Series

Elements in the Philosophy of Mathematics

This Element introduces a young field, the 'philosophy of mathematical practice'. We first offer a general characterisation of the approach to the philosophy of mathematics that takes mathematical practice seriously and contrast it with 'mathematical philosophy'. The latter is traced back to Bertrand Russell and the orientation referred to as 'scientific philosophy' that was active between 1850 and 1930. To give a better sense of the field, the Element further contains two examples of topics studied, that of mathematical structuralism and visual thinking in mathematics. These are in part presented from a methodological point of view, focussing on mathematics as an activity and questions related to how mathematics develops. In addition, the Element contains several examples from mathematics, both historical and contemporary , to illustrate and support the philosophical points.

The Element introduces the philosophy of mathematical practice, which takes mathematical practice and contrasts it with mathematical philosophy, a term traced back to Bertrand Russell's scientific philosophy. It discusses mathematical structuralism and visual thinking, focusing on methodological aspects and historical and contemporary examples.

Introducing the Philosophy of Mathematical Practice

Number Systems: A Path into Rigorous Mathematics aims to introduce number systems to an undergraduate audience in a way that emphasises the importance of rigour, and with a focus on providing detailed but accessible explanations of theorems and their proofs.The book continually seeks to build upon students' intuitive ideas of how numbers and arithmetic work, and to guide them towards the means to embed this natural understanding into a more structured framework of understanding. The author's motivation for writing this book is that most previous texts, which have complete coverage of the subject, have not provided the level of explanation needed for first-year students. On the other hand, those that do give good explanations tend to focus broadly on Foundations or Analysis and provide incomplete coverage of Number Systems.Features<ul> <li>Approachable for first year undergraduates, but still of interest to more advanced students and postgraduates</li> <li>Does not merely present definitions, theorems and proofs, but also motivates them in terms of intuitive knowledge and discusses methods of proof</li> <li>Draws attention to connections with other areas of mathematics</li> <li>Plenty of exercises for students, both straightforward problems and more in-depth investigations</li> <li>Introduces many concepts that are required in more advanced topics in mathematics</li> </ul>New to the second edition<ul> <li>Complete solutions to all exercises, and hints for the in-depth investigations</li> <li>Extensive changes to chapters 4 and 5, including defining integral domains as distinct from commutative rings, a more complete discussion of irreducibles, primes and unique factorisation, and more topics in elementary number theory</li> <li>A completely revised chapter 8, giving a more coherent account of quadratic rings and their unique (or non-unique) factorisation properties</li> <li>A thorough correction of typos and errors across all chapters</li> <li>Updates to the bibliography</li> </ul>

A Path into Rigorous Mathematics

Number Systems

Textbooks in Mathematics

This popular textbook provides a detailed examination of the central assertions of measure theory in n-dimensional Euclidean space. The book emphasizes the roles of Hausdorff measure and capacity in characterizing the fine properties of sets and functions.

This popular textbook provides a detailed examination of the central assertions of measure theory in n-dimensional Euclidean space. The book emphasizes the roles of Hausdorff measure and capacity in characterizing the fine properties of sets and functions.

Measure Theory and Fine Properties of Functions

Wiley Series in Probability and Statistics

Timely update of a popular edition on permutation testing with numerous case studies included throughout The newly revised and updated Second Edition of Permutation Tests for Complex Data describes permutation tests from the point of view of experimental design, with methodological details and illustrating the process of devising an appropriate permutation test through case studies. In addition to the text, this book includes two open source packages for permutation tests in Python and R which include a comprehensive code base to implement common permutation tests as well as code to implement each of the book's case studies. The focus of this book is the permutation approach to a variety of univariate and multivariate problems of hypothesis testing in a typical nonparametric framework. The book examines the most up-to-date methodologies of univariate and multivariate permutation testing, includes real case studies from both experimental and observational studies, and presents and discusses solutions to the most important and frequently encountered real problems in multivariate analyses. Written by two highly qualified authors in the field of nonparametrics and applied statistics, Permutation Tests for Complex Data includes information on sample topics including: <ul> <li>Theory of one-dimensional and multi-dimensional permutation tests, covering test statistics, arguments for selecting permutation tests, and examples of one-sample and multi-sample problems</li> <li>Multiplicity control and closed testing, covering raw and adjusted p-values, the MinP Bonferroni-Holm procedure, and weighted methods for controlling FWE and FDR</li> <li>Multivariate categorical variables, covering stochastic ordering, tests on moments for ordered variables, and heterogeneity comparisons</li> <li>NPC tests for survival analysis and shape analysis, covering analysis of PERC data and analysis with correlated landmarks</li> </ul> Presenting a thorough overview of permutation testing with both formal description and proofs, Permutation Tests for Complex Data is an excellent introduction to permutation tests for graduate-level statistics or data science courses and will be ideal as a handbook for researchers hoping to use the open source code.

Theory, Applications and Software

Permutation Tests for Complex Data

Dynamical Zeta Functions Nielsen Theory & Reide

Generalized Quantum Calculus with Applications is devoted to the qualitative theory of general quantum calculus and its applications to general quantum differential equations and inequalities. The book is aimed at upper-level undergraduate students and beginning graduate students in a range of interdisciplinary courses including physical sciences and engineering, from quantum mechanics to differential equations, with pedagogically organized chapters that each concludes with a section of practical problems. Generalized quantum calculus includes a generalization of the q-quantum calculus and the time scale calculus. There are many open problems and difficulties in q-quantum calculus and time-scale calculus, and this book explores how to use the generalized quantum operators to solve difficulties arising in q-quantum calculus and time-scale calculus, including but not limited to generalized quantum integration, generalized quantum chain rules, and generalized quantum Taylor formula. Since generalized quantum calculus includes the q-quantum and time-scale calculus, this book can be utilized by a wide audience of researchers and students. This text is one of few foundational books on generalized quantum calculus and can be used for future discoveries in the area of integral transforms, variational calculus, integral equations, and inequalities in the language of generalized quantum calculus. This book also offers detailed proofs, exercises, and examples to aid instructors, researchers, and users in their studies.

Generalized Quantum Calculus with Applications

Series In Computational Methods

This unique compendium deals with the differentiation and integration of vector functions. It examines critical effects and extracts important features using powerful tools of differentiation and integration. Techniques and codes for computing the divergence, curl, and gradients of a given field function, which reveal the mathematical behavior of the vector field, are discussed. Green's theorem, Stokes's theorem, and Gauss's formula, along with their novel extensions, are presented in detail with applications such as the smoothed gradient method.Written in Jupyter notebook format, the book offers a unified environment for theory description, code execution, and real-time interaction, making it ideal for reading, practicing, and further exploration.

Vector Calculus: Formulations, Applications And Python Codes

AK Peters/CRC Recreational Mathematics Series

Overlooked and Omitted Topics in Mathematics

I Wish They'd Taught Me That

Elementary Real Analysis

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