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Provides a student's first encounter with the concepts of measure theory and functional analysis. This book reflects the belief that difficult concepts should be introduced in their simplest and most concrete forms. It is suitable for an advanced undergraduate course or for the start of a graduate course.
Book Synopsis A (Terse) Introduction to Lebesgue Integration by : John M. Franks
Download or read book A (Terse) Introduction to Lebesgue Integration written by John M. Franks and published by American Mathematical Soc.. This book was released on 2009 with total page 219 pages. Available in PDF, EPUB and Kindle. Book excerpt: Provides a student's first encounter with the concepts of measure theory and functional analysis. This book reflects the belief that difficult concepts should be introduced in their simplest and most concrete forms. It is suitable for an advanced undergraduate course or for the start of a graduate course.
A User-Friendly Introduction to Lebesgue Measure and Integration provides a bridge between an undergraduate course in Real Analysis and a first graduate-level course in Measure Theory and Integration. The main goal of this book is to prepare students for what they may encounter in graduate school, but will be useful for many beginning graduate students as well. The book starts with the fundamentals of measure theory that are gently approached through the very concrete example of Lebesgue measure. With this approach, Lebesgue integration becomes a natural extension of Riemann integration. Next, -spaces are defined. Then the book turns to a discussion of limits, the basic idea covered in a first analysis course. The book also discusses in detail such questions as: When does a sequence of Lebesgue integrable functions converge to a Lebesgue integrable function? What does that say about the sequence of integrals? Another core idea from a first analysis course is completeness. Are these -spaces complete? What exactly does that mean in this setting? This book concludes with a brief overview of General Measures. An appendix contains suggested projects suitable for end-of-course papers or presentations. The book is written in a very reader-friendly manner, which makes it appropriate for students of varying degrees of preparation, and the only prerequisite is an undergraduate course in Real Analysis.
Book Synopsis A User-Friendly Introduction to Lebesgue Measure and Integration by : Gail S. Nelson
Download or read book A User-Friendly Introduction to Lebesgue Measure and Integration written by Gail S. Nelson and published by American Mathematical Soc.. This book was released on 2015-11-09 with total page 221 pages. Available in PDF, EPUB and Kindle. Book excerpt: A User-Friendly Introduction to Lebesgue Measure and Integration provides a bridge between an undergraduate course in Real Analysis and a first graduate-level course in Measure Theory and Integration. The main goal of this book is to prepare students for what they may encounter in graduate school, but will be useful for many beginning graduate students as well. The book starts with the fundamentals of measure theory that are gently approached through the very concrete example of Lebesgue measure. With this approach, Lebesgue integration becomes a natural extension of Riemann integration. Next, -spaces are defined. Then the book turns to a discussion of limits, the basic idea covered in a first analysis course. The book also discusses in detail such questions as: When does a sequence of Lebesgue integrable functions converge to a Lebesgue integrable function? What does that say about the sequence of integrals? Another core idea from a first analysis course is completeness. Are these -spaces complete? What exactly does that mean in this setting? This book concludes with a brief overview of General Measures. An appendix contains suggested projects suitable for end-of-course papers or presentations. The book is written in a very reader-friendly manner, which makes it appropriate for students of varying degrees of preparation, and the only prerequisite is an undergraduate course in Real Analysis.
The Lebesgue integral is now standard for both applications and advanced mathematics. This books starts with a review of the familiar calculus integral and then constructs the Lebesgue integral from the ground up using the same ideas. A Primer of Lebesgue Integration has been used successfully both in the classroom and for individual study. Bear presents a clear and simple introduction for those intent on further study in higher mathematics. Additionally, this book serves as a refresher providing new insight for those in the field. The author writes with an engaging, commonsense style that appeals to readers at all levels.
Book Synopsis A Primer of Lebesgue Integration by : H. S. Bear
Download or read book A Primer of Lebesgue Integration written by H. S. Bear and published by Academic Press. This book was released on 2002 with total page 184 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Lebesgue integral is now standard for both applications and advanced mathematics. This books starts with a review of the familiar calculus integral and then constructs the Lebesgue integral from the ground up using the same ideas. A Primer of Lebesgue Integration has been used successfully both in the classroom and for individual study. Bear presents a clear and simple introduction for those intent on further study in higher mathematics. Additionally, this book serves as a refresher providing new insight for those in the field. The author writes with an engaging, commonsense style that appeals to readers at all levels.
Ein lebendig und präzise geschriebenes Lehrbuch zu Lebesgue-Maß und Lebesgue-Integration, das sehr ins Detail geht und viele historische Aspekte anspricht. Reich ausgestattet mit Illustrationen, Übungsaufgaben und zahlreichen Beispielen, legt dieses Buch die Grundlage für eine Vorlesung über abstrakte Maße und Integrationstheorie. (11/97)
Book Synopsis Lebesgue Measure and Integration by : Frank Burk
Download or read book Lebesgue Measure and Integration written by Frank Burk and published by John Wiley & Sons. This book was released on 1997-11-03 with total page 316 pages. Available in PDF, EPUB and Kindle. Book excerpt: Ein lebendig und präzise geschriebenes Lehrbuch zu Lebesgue-Maß und Lebesgue-Integration, das sehr ins Detail geht und viele historische Aspekte anspricht. Reich ausgestattet mit Illustrationen, Übungsaufgaben und zahlreichen Beispielen, legt dieses Buch die Grundlage für eine Vorlesung über abstrakte Maße und Integrationstheorie. (11/97)
The Stieltjes Integral provides a detailed, rigorous treatment of the Stieltjes integral. This integral is a generalization of the Riemann and Darboux integrals of calculus and undergraduate analysis, and can serve as a bridge between classical and modern analysis. It has applications in many areas, including number theory, statistics, physics, and finance. It begins with the Darboux integral, builds the theory of functions of bounded variation, and then develops the Stieltjes integral. It culminates with a proof of the Riesz representation theorem as an application of the Stieltjes integral. For much of the 20th century the Stjeltjes integral was a standard part of the undergraduate or beginning graduate student sequence in analysis. However, the typical mathematics curriculum has changed at many institutions, and the Stieltjes integral has become less common in undergraduate textbooks and analysis courses. This book seeks to address this by offering an accessible treatment of the subject to students who have had a one semester course in analysis. This book is suitable for a second semester course in analysis, and also for independent study or as the foundation for a senior thesis or Masters project. Features: Written to be rigorous without sacrificing readability. Accessible to undergraduate students who have taken a one-semester course on real analysis. Contains a large number of exercises from routine to challenging.
Book Synopsis The Stieltjes Integral by : Gregory Convertito
Download or read book The Stieltjes Integral written by Gregory Convertito and published by CRC Press. This book was released on 2023-02-28 with total page 250 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Stieltjes Integral provides a detailed, rigorous treatment of the Stieltjes integral. This integral is a generalization of the Riemann and Darboux integrals of calculus and undergraduate analysis, and can serve as a bridge between classical and modern analysis. It has applications in many areas, including number theory, statistics, physics, and finance. It begins with the Darboux integral, builds the theory of functions of bounded variation, and then develops the Stieltjes integral. It culminates with a proof of the Riesz representation theorem as an application of the Stieltjes integral. For much of the 20th century the Stjeltjes integral was a standard part of the undergraduate or beginning graduate student sequence in analysis. However, the typical mathematics curriculum has changed at many institutions, and the Stieltjes integral has become less common in undergraduate textbooks and analysis courses. This book seeks to address this by offering an accessible treatment of the subject to students who have had a one semester course in analysis. This book is suitable for a second semester course in analysis, and also for independent study or as the foundation for a senior thesis or Masters project. Features: Written to be rigorous without sacrificing readability. Accessible to undergraduate students who have taken a one-semester course on real analysis. Contains a large number of exercises from routine to challenging.
Very roughly speaking, representation theory studies symmetry in linear spaces. It is a beautiful mathematical subject which has many applications, ranging from number theory and combinatorics to geometry, probability theory, quantum mechanics, and quantum field theory. The goal of this book is to give a ``holistic'' introduction to representation theory, presenting it as a unified subject which studies representations of associative algebras and treating the representation theories of groups, Lie algebras, and quivers as special cases. Using this approach, the book covers a number of standard topics in the representation theories of these structures. Theoretical material in the book is supplemented by many problems and exercises which touch upon a lot of additional topics; the more difficult exercises are provided with hints. The book is designed as a textbook for advanced undergraduate and beginning graduate students. It should be accessible to students with a strong background in linear algebra and a basic knowledge of abstract algebra.
Book Synopsis Introduction to Representation Theory by : Pavel I. Etingof
Download or read book Introduction to Representation Theory written by Pavel I. Etingof and published by American Mathematical Soc.. This book was released on 2011 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: Very roughly speaking, representation theory studies symmetry in linear spaces. It is a beautiful mathematical subject which has many applications, ranging from number theory and combinatorics to geometry, probability theory, quantum mechanics, and quantum field theory. The goal of this book is to give a ``holistic'' introduction to representation theory, presenting it as a unified subject which studies representations of associative algebras and treating the representation theories of groups, Lie algebras, and quivers as special cases. Using this approach, the book covers a number of standard topics in the representation theories of these structures. Theoretical material in the book is supplemented by many problems and exercises which touch upon a lot of additional topics; the more difficult exercises are provided with hints. The book is designed as a textbook for advanced undergraduate and beginning graduate students. It should be accessible to students with a strong background in linear algebra and a basic knowledge of abstract algebra.
The calculus of variations is used to find functions that optimize quantities expressed in terms of integrals. Optimal control theory seeks to find functions that minimize cost integrals for systems described by differential equations. This book is an introduction to both the classical theory of the calculus of variations and the more modern developments of optimal control theory from the perspective of an applied mathematician. It focuses on understanding concepts and how to apply them. The range of potential applications is broad: the calculus of variations and optimal control theory have been widely used in numerous ways in biology, criminology, economics, engineering, finance, management science, and physics. Applications described in this book include cancer chemotherapy, navigational control, and renewable resource harvesting. The prerequisites for the book are modest: the standard calculus sequence, a first course on ordinary differential equations, and some facility with the use of mathematical software. It is suitable for an undergraduate or beginning graduate course, or for self study. It provides excellent preparation for more advanced books and courses on the calculus of variations and optimal control theory.
Book Synopsis A Primer on the Calculus of Variations and Optimal Control Theory by : Mike Mesterton-Gibbons
Download or read book A Primer on the Calculus of Variations and Optimal Control Theory written by Mike Mesterton-Gibbons and published by American Mathematical Soc.. This book was released on 2009 with total page 274 pages. Available in PDF, EPUB and Kindle. Book excerpt: The calculus of variations is used to find functions that optimize quantities expressed in terms of integrals. Optimal control theory seeks to find functions that minimize cost integrals for systems described by differential equations. This book is an introduction to both the classical theory of the calculus of variations and the more modern developments of optimal control theory from the perspective of an applied mathematician. It focuses on understanding concepts and how to apply them. The range of potential applications is broad: the calculus of variations and optimal control theory have been widely used in numerous ways in biology, criminology, economics, engineering, finance, management science, and physics. Applications described in this book include cancer chemotherapy, navigational control, and renewable resource harvesting. The prerequisites for the book are modest: the standard calculus sequence, a first course on ordinary differential equations, and some facility with the use of mathematical software. It is suitable for an undergraduate or beginning graduate course, or for self study. It provides excellent preparation for more advanced books and courses on the calculus of variations and optimal control theory.
Both fractal geometry and dynamical systems have a long history of development and have provided fertile ground for many great mathematicians and much deep and important mathematics. These two areas interact with each other and with the theory of chaos in a fundamental way: many dynamical systems (even some very simple ones) produce fractal sets, which are in turn a source of irregular 'chaotic' motions in the system. This book is an introduction to these two fields, with an emphasis on the relationship between them. The first half of the book introduces some of the key ideas in fractal geometry and dimension theory - Cantor sets, Hausdorff dimension, box dimension - using dynamical notions whenever possible, particularly one-dimensional Markov maps and symbolic dynamics. Various techniques for computing Hausdorff dimension are shown, leading to a discussion of Bernoulli and Markov measures and of the relationship between dimension, entropy, and Lyapunov exponents. In the second half of the book some examples of dynamical systems are considered and various phenomena of chaotic behaviour are discussed, including bifurcations, hyperbolicity, attractors, horseshoes, and intermittent and persistent chaos. These phenomena are naturally revealed in the course of our study of two real models from science - the FitzHugh - Nagumo model and the Lorenz system of differential equations. This book is accessible to undergraduate students and requires only standard knowledge in calculus, linear algebra, and differential equations. Elements of point set topology and measure theory are introduced as needed. This book is a result of the MASS course in analysis at Penn State University in the fall semester of 2008.
Book Synopsis Lectures on Fractal Geometry and Dynamical Systems by : Ya. B. Pesin
Download or read book Lectures on Fractal Geometry and Dynamical Systems written by Ya. B. Pesin and published by American Mathematical Soc.. This book was released on 2009 with total page 334 pages. Available in PDF, EPUB and Kindle. Book excerpt: Both fractal geometry and dynamical systems have a long history of development and have provided fertile ground for many great mathematicians and much deep and important mathematics. These two areas interact with each other and with the theory of chaos in a fundamental way: many dynamical systems (even some very simple ones) produce fractal sets, which are in turn a source of irregular 'chaotic' motions in the system. This book is an introduction to these two fields, with an emphasis on the relationship between them. The first half of the book introduces some of the key ideas in fractal geometry and dimension theory - Cantor sets, Hausdorff dimension, box dimension - using dynamical notions whenever possible, particularly one-dimensional Markov maps and symbolic dynamics. Various techniques for computing Hausdorff dimension are shown, leading to a discussion of Bernoulli and Markov measures and of the relationship between dimension, entropy, and Lyapunov exponents. In the second half of the book some examples of dynamical systems are considered and various phenomena of chaotic behaviour are discussed, including bifurcations, hyperbolicity, attractors, horseshoes, and intermittent and persistent chaos. These phenomena are naturally revealed in the course of our study of two real models from science - the FitzHugh - Nagumo model and the Lorenz system of differential equations. This book is accessible to undergraduate students and requires only standard knowledge in calculus, linear algebra, and differential equations. Elements of point set topology and measure theory are introduced as needed. This book is a result of the MASS course in analysis at Penn State University in the fall semester of 2008.
What can we compute--even with unlimited resources? Is everything within reach? Or are computations necessarily drastically limited, not just in practice, but theoretically? These questions are at the heart of computability theory. The goal of this book is to give the reader a firm grounding in the fundamentals of computability theory and an overview of currently active areas of research, such as reverse mathematics and algorithmic randomness. Turing machines and partial recursive functions are explored in detail, and vital tools and concepts including coding, uniformity, and diagonalization are described explicitly. From there the material continues with universal machines, the halting problem, parametrization and the recursion theorem, and thence to computability for sets, enumerability, and Turing reduction and degrees. A few more advanced topics round out the book before the chapter on areas of research. The text is designed to be self-contained, with an entire chapter of preliminary material including relations, recursion, induction, and logical and set notation and operators. That background, along with ample explanation, examples, exercises, and suggestions for further reading, make this book ideal for independent study or courses with few prerequisites.
Book Synopsis Computability Theory by : Rebecca Weber
Download or read book Computability Theory written by Rebecca Weber and published by American Mathematical Soc.. This book was released on 2012 with total page 218 pages. Available in PDF, EPUB and Kindle. Book excerpt: What can we compute--even with unlimited resources? Is everything within reach? Or are computations necessarily drastically limited, not just in practice, but theoretically? These questions are at the heart of computability theory. The goal of this book is to give the reader a firm grounding in the fundamentals of computability theory and an overview of currently active areas of research, such as reverse mathematics and algorithmic randomness. Turing machines and partial recursive functions are explored in detail, and vital tools and concepts including coding, uniformity, and diagonalization are described explicitly. From there the material continues with universal machines, the halting problem, parametrization and the recursion theorem, and thence to computability for sets, enumerability, and Turing reduction and degrees. A few more advanced topics round out the book before the chapter on areas of research. The text is designed to be self-contained, with an entire chapter of preliminary material including relations, recursion, induction, and logical and set notation and operators. That background, along with ample explanation, examples, exercises, and suggestions for further reading, make this book ideal for independent study or courses with few prerequisites.
Introduces the reader to the techniques, ideas, and consequences related to the Erdős problem. The authors introduce these concepts in a concrete and elementary way that allows a wide audience to absorb the content and appreciate its far-reaching implications. In the process, the reader is familiarized with a wide range of techniques from several areas of mathematics and can appreciate the power of the resulting symbiosis.
Book Synopsis The Erdos Distance Problem by : Julia Garibaldi
Download or read book The Erdos Distance Problem written by Julia Garibaldi and published by American Mathematical Soc.. This book was released on 2011 with total page 166 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduces the reader to the techniques, ideas, and consequences related to the Erdős problem. The authors introduce these concepts in a concrete and elementary way that allows a wide audience to absorb the content and appreciate its far-reaching implications. In the process, the reader is familiarized with a wide range of techniques from several areas of mathematics and can appreciate the power of the resulting symbiosis.
