Introduction to Analysis


Author: Edward Gaughan
Publisher: American Mathematical Soc.
ISBN: 9780821847879
Category: Mathematics
Page: 240
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Introduction to Analysis is designed to bridge the gap between the intuitive calculus usually offered at the undergraduate level and the sophisticated analysis courses the student encounters at the graduate level. In this book the student is given the vocabulary and facts necessary for further study in analysis. The course for which it is designed is usually offered at the junior level, and it is assumed that the student has little or no previous experience with proofs in analysis. A considerable amount of time is spent motivating the theorems and proofs and developing the reader's intuition. Of course, that intuition must be tempered with the realization that rigorous proofs are required for theorems. The topics are quite standard: convergence of sequences, limits of functions, continuity, differentiation, the Riemann integral, infinite series, power series, and convergence of sequences of functions. Many examples are given to illustrate the theory, and exercises at the end of each chapter are keyed to each section. Also, at the end of each section, one finds several Projects. The purpose of a Project is to give the reader a substantial mathematical problem and the necessary guidance to solve that problem. A Project is distinguished from an exercise in that the solution of a Project is a multi-step process requiring assistance for the beginner student.

An Introduction to Complex Analysis and Geometry


Author: John P. D'Angelo
Publisher: American Mathematical Soc.
ISBN: 0821852744
Category: Mathematics
Page: 163
View: 1104

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An Introduction to Complex Analysis and Geometry provides the reader with a deep appreciation of complex analysis and how this subject fits into mathematics. The book developed from courses given in the Campus Honors Program at the University of Illinois Urbana-Champaign. These courses aimed to share with students the way many mathematics and physics problems magically simplify when viewed from the perspective of complex analysis. The book begins at an elementary level but also contains advanced material. The first four chapters provide an introduction to complex analysis with many elementary and unusual applications. Chapters 5 through 7 develop the Cauchy theory and include some striking applications to calculus. Chapter 8 glimpses several appealing topics, simultaneously unifying the book and opening the door to further study. The 280 exercises range from simple computations to difficult problems. Their variety makes the book especially attractive. A reader of the first four chapters will be able to apply complex numbers in many elementary contexts. A reader of the full book will know basic one complex variable theory and will have seen it integrated into mathematics as a whole. Research mathematicians will discover several novel perspectives.

Einführung in die Mechanik und Symmetrie

Eine grundlegende Darstellung klassischer mechanischer Systeme
Author: Jerrold E. Marsden,Tudor S. Ratiu
Publisher: Springer-Verlag
ISBN: 3642568599
Category: Mathematics
Page: 598
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Symmetrie spielt in der Mechanik eine große Rolle. Dieses Buch beschreibt die Entwicklung zugrunde liegender Theorien. Besonderes Gewicht wird der Symmetrie beigemessen. Ursache hierfür sind Entwicklungen im Bereich dynamischer Systeme, der Einsatz geometrischer Verfahren und neue Anwendungen. Dieses Lehrbuch stellt Grundlagen bereit und beschreibt zahlreiche spezifische Anwendungen. Interessant für Physiker und Ingenieure. Ausgewählte Beispiele, Anwendungen, aktuelle Verfahren/Techniken veranschaulichen die Theorie.

Advanced Calculus

An Introduction to Modern Analysis
Author: Voxman
Publisher: Routledge
ISBN: 1351468677
Category: Mathematics
Page: 696
View: 6400

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Advanced Calculus: An Introduction to Modem Analysis, an advanced undergraduate textbook,provides mathematics majors, as well as students who need mathematics in their field of study,with an introduction to the theory and applications of elementary analysis. The text presents, inan accessible form, a carefully maintained balance between abstract concepts and applied results ofsignificance that serves to bridge the gap between the two- or three-cemester calculus sequence andsenior/graduate level courses in the theory and appplications of ordinary and partial differentialequations, complex variables, numerical methods, and measure and integration theory.The book focuses on topological concepts, such as compactness, connectedness, and metric spaces,and topics from analysis including Fourier series, numerical analysis, complex integration, generalizedfunctions, and Fourier and Laplace transforms. Applications from genetics, spring systems,enzyme transfer, and a thorough introduction to the classical vibrating string, heat transfer, andbrachistochrone problems illustrate this book's usefulness to the non-mathematics major. Extensiveproblem sets found throughout the book test the student's understanding of the topics andhelp develop the student's ability to handle more abstract mathematical ideas.Advanced Calculus: An Introduction to Modem Analysis is intended for junior- and senior-levelundergraduate students in mathematics, biology, engineering, physics, and other related disciplines.An excellent textbook for a one-year course in advanced calculus, the methods employed in thistext will increase students' mathematical maturity and prepare them solidly for senior/graduatelevel topics. The wealth of materials in the text allows the instructor to select topics that are ofspecial interest to the student. A two- or three?ll?lester calculus sequence is required for successfuluse of this book.

Number Systems: An Introduction to Algebra and Analysis


Author: Sergei Ovchinnikov
Publisher: American Mathematical Soc.
ISBN: 147042018X
Category: Mathematics
Page: 144
View: 4533

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This book offers a rigorous and coherent introduction to the five basic number systems of mathematics, namely natural numbers, integers, rational numbers, real numbers, and complex numbers. It is a subject that many mathematicians believe should be learned by any student of mathematics including future teachers. The book starts with the development of Peano arithmetic in the first chapter which includes mathematical induction and elements of recursion theory. It proceeds to an examination of integers that also covers rings and ordered integral domains. The presentation of rational numbers includes material on ordered fields and convergence of sequences in these fields. Cauchy and Dedekind completeness properties of the field of real numbers are established, together with some properties of real continuous functions. An elementary proof of the Fundamental Theorem of Algebra is the highest point of the chapter on complex numbers. The great merit of the book lies in its extensive list of exercises following each chapter. These exercises are designed to assist the instructor and to enhance the learning experience of the students.

Spaces: An Introduction to Real Analysis


Author: Tom L. Lindstrøm
Publisher: American Mathematical Soc.
ISBN: 1470440628
Category: Functional analysis
Page: 369
View: 955

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Spaces is a modern introduction to real analysis at the advanced undergraduate level. It is forward-looking in the sense that it first and foremost aims to provide students with the concepts and techniques they need in order to follow more advanced courses in mathematical analysis and neighboring fields. The only prerequisites are a solid understanding of calculus and linear algebra. Two introductory chapters will help students with the transition from computation-based calculus to theory-based analysis. The main topics covered are metric spaces, spaces of continuous functions, normed spaces, differentiation in normed spaces, measure and integration theory, and Fourier series. Although some of the topics are more advanced than what is usually found in books of this level, care is taken to present the material in a way that is suitable for the intended audience: concepts are carefully introduced and motivated, and proofs are presented in full detail. Applications to differential equations and Fourier analysis are used to illustrate the power of the theory, and exercises of all levels from routine to real challenges help students develop their skills and understanding. The text has been tested in classes at the University of Oslo over a number of years.

Finite-Elemente-Methoden


Author: Klaus-Jürgen Bathe
Publisher: DrMaster Publications
ISBN: 9783540668060
Category: Technology & Engineering
Page: 1253
View: 7781

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Dieses Lehr- und Handbuch behandelt sowohl die elementaren Konzepte als auch die fortgeschrittenen und zukunftsweisenden linearen und nichtlinearen FE-Methoden in Statik, Dynamik, Festkörper- und Fluidmechanik. Es wird sowohl der physikalische als auch der mathematische Hintergrund der Prozeduren ausführlich und verständlich beschrieben. Das Werk enthält eine Vielzahl von ausgearbeiteten Beispielen, Rechnerübungen und Programmlisten. Als Übersetzung eines erfolgreichen amerikanischen Lehrbuchs hat es sich in zwei Auflagen auch bei den deutschsprachigen Ingenieuren etabliert. Die umfangreichen Änderungen gegenüber der Vorauflage innerhalb aller Kapitel - vor allem aber der fortgeschrittenen - spiegeln die rasche Entwicklung innerhalb des letzten Jahrzehnts auf diesem Gebiet wieder. TOC:Eine Einführung in den Gebrauch von Finite-Elemente-Verfahren.-Vektoren, Matrizen und Tensoren.-Einige Grundbegriffe ingenieurwissenschaftlicher Berechnungen.-Formulierung der Methode der finiten Elemente.-Formulierung und Berechnung von isoparametrischen Finite-Elemente-Matrizen.-Nichtlineare Finite-Elemente-Berechnungen in der Festkörper- und Strukturmechanik.-Finite-Elemente-Berechnungen von Wärmeübertragungs- und Feldproblemen.-Lösung von Gleichgewichtsbeziehungen in statischen Berechnungen.-Lösung von Bewegungsgleichungen in kinetischen Berechnungen.-Vorbemerkungen zur Lösung von Eigenproblemen.-Lösungsverfahren für Eigenprobleme.-Implementierung der Finite-Elemente-Methode.

Introduction to Visual Computing

Core Concepts in Computer Vision, Graphics, and Image Processing
Author: Aditi Majumder,M. Gopi
Publisher: CRC Press
ISBN: 1315355523
Category: Computers
Page: 376
View: 2582

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Introduction to Visual Computing: Core Concepts in Computer Vision, Graphics, and Image Processing covers the fundamental concepts of visual computing. Whereas past books have treated these concepts within the context of specific fields such as computer graphics, computer vision or image processing, this book offers a unified view of these core concepts, thereby providing a unified treatment of computational and mathematical methods for creating, capturing, analyzing and manipulating visual data (e.g. 2D images, 3D models). Fundamentals covered in the book include convolution, Fourier transform, filters, geometric transformations, epipolar geometry, 3D reconstruction, color and the image synthesis pipeline. The book is organized in four parts. The first part provides an exposure to different kinds of visual data (e.g. 2D images, videos and 3D geometry) and the core mathematical techniques that are required for their processing (e.g. interpolation and linear regression.) The second part of the book on Image Based Visual Computing deals with several fundamental techniques to process 2D images (e.g. convolution, spectral analysis and feature detection) and corresponds to the low level retinal image processing that happens in the eye in the human visual system pathway. The next part of the book on Geometric Visual Computing deals with the fundamental techniques used to combine the geometric information from multiple eyes creating a 3D interpretation of the object and world around us (e.g. transformations, projective and epipolar geometry, and 3D reconstruction). This corresponds to the higher level processing that happens in the brain combining information from both the eyes thereby helping us to navigate through the 3D world around us. The last two parts of the book cover Radiometric Visual Computing and Visual Content Synthesis. These parts focus on the fundamental techniques for processing information arising from the interaction of light with objects around us, as well as the fundamentals of creating virtual computer generated worlds that mimic all the processing presented in the prior sections. The book is written for a 16 week long semester course and can be used for both undergraduate and graduate teaching, as well as a reference for professionals.

Analysis I


Author: Martin Barner
Publisher: Walter de Gruyter
ISBN: 3110809133
Category: Mathematics
Page: 544
View: 9667

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Measure and Integral

An Introduction to Real Analysis, Second Edition
Author: Richard L. Wheeden
Publisher: CRC Press
ISBN: 1498702902
Category: Mathematics
Page: 532
View: 7336

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Now considered a classic text on the topic, Measure and Integral: An Introduction to Real Analysis provides an introduction to real analysis by first developing the theory of measure and integration in the simple setting of Euclidean space, and then presenting a more general treatment based on abstract notions characterized by axioms and with less geometric content. Published nearly forty years after the first edition, this long-awaited Second Edition also: Studies the Fourier transform of functions in the spaces L1, L2, and Lp, 1 p Shows the Hilbert transform to be a bounded operator on L2, as an application of the L2 theory of the Fourier transform in the one-dimensional case Covers fractional integration and some topics related to mean oscillation properties of functions, such as the classes of Hölder continuous functions and the space of functions of bounded mean oscillation Derives a subrepresentation formula, which in higher dimensions plays a role roughly similar to the one played by the fundamental theorem of calculus in one dimension Extends the subrepresentation formula derived for smooth functions to functions with a weak gradient Applies the norm estimates derived for fractional integral operators to obtain local and global first-order Poincaré–Sobolev inequalities, including endpoint cases Proves the existence of a tangent plane to the graph of a Lipschitz function of several variables Includes many new exercises not present in the first edition This widely used and highly respected text for upper-division undergraduate and first-year graduate students of mathematics, statistics, probability, or engineering is revised for a new generation of students and instructors. The book also serves as a handy reference for professional mathematicians.

Reelle und Komplexe Analysis


Author: Walter Rudin
Publisher: Walter de Gruyter
ISBN: 9783486591866
Category: Analysis - Lehrbuch
Page: 499
View: 1044

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Besonderen Wert legt Rudin darauf, dem Leser die Zusammenhänge unterschiedlicher Bereiche der Analysis zu vermitteln und so die Grundlage für ein umfassenderes Verständnis zu schaffen. Das Werk zeichnet sich durch seine wissenschaftliche Prägnanz und Genauigkeit aus und hat damit die Entwicklung der modernen Analysis in nachhaltiger Art und Weise beeinflusst. Der "Baby-Rudin" gehört weltweit zu den beliebtesten Lehrbüchern der Analysis und ist in 13 Sprachen übersetzt. 1993 wurde es mit dem renommierten Steele Prize for Mathematical Exposition der American Mathematical Society ausgezeichnet. Übersetzt von Uwe Krieg.

Mathematical Analysis

A Concise Introduction
Author: Bernd S. W. Schröder
Publisher: John Wiley & Sons
ISBN: 9780470226766
Category: Mathematics
Page: 584
View: 8936

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A self-contained introduction to the fundamentals of mathematical analysis Mathematical Analysis: A Concise Introduction presents the foundations of analysis and illustrates its role in mathematics. By focusing on the essentials, reinforcing learning through exercises, and featuring a unique "learn by doing" approach, the book develops the reader's proof writing skills and establishes fundamental comprehension of analysis that is essential for further exploration of pure and applied mathematics. This book is directly applicable to areas such as differential equations, probability theory, numerical analysis, differential geometry, and functional analysis. Mathematical Analysis is composed of three parts: ?Part One presents the analysis of functions of one variable, including sequences, continuity, differentiation, Riemann integration, series, and the Lebesgue integral. A detailed explanation of proof writing is provided with specific attention devoted to standard proof techniques. To facilitate an efficient transition to more abstract settings, the results for single variable functions are proved using methods that translate to metric spaces. ?Part Two explores the more abstract counterparts of the concepts outlined earlier in the text. The reader is introduced to the fundamental spaces of analysis, including Lp spaces, and the book successfully details how appropriate definitions of integration, continuity, and differentiation lead to a powerful and widely applicable foundation for further study of applied mathematics. The interrelation between measure theory, topology, and differentiation is then examined in the proof of the Multidimensional Substitution Formula. Further areas of coverage in this section include manifolds, Stokes' Theorem, Hilbert spaces, the convergence of Fourier series, and Riesz' Representation Theorem. ?Part Three provides an overview of the motivations for analysis as well as its applications in various subjects. A special focus on ordinary and partial differential equations presents some theoretical and practical challenges that exist in these areas. Topical coverage includes Navier-Stokes equations and the finite element method. Mathematical Analysis: A Concise Introduction includes an extensive index and over 900 exercises ranging in level of difficulty, from conceptual questions and adaptations of proofs to proofs with and without hints. These opportunities for reinforcement, along with the overall concise and well-organized treatment of analysis, make this book essential for readers in upper-undergraduate or beginning graduate mathematics courses who would like to build a solid foundation in analysis for further work in all analysis-based branches of mathematics.

Lineare Funktionalanalysis

Eine anwendungsorientierte Einführung
Author: Hans Wilhelm Alt
Publisher: Springer-Verlag
ISBN: 3642222617
Category: Mathematics
Page: 449
View: 6279

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Die lineare Funktionalanalysis ist ein Teilgebiet der Mathematik, das Algebra mit Topologie und Analysis verbindet. Das Buch führt in das Fachgebiet ein, dabei bezieht es sich auf Anwendungen in Mathematik und Physik. Neben den vollständigen Beweisen aller mathematischen Sätze enthält der Band zahlreiche Aufgaben, meist mit Lösungen. Für die Neuauflage wurden die Inhalte komplett überarbeitet. Das Standardwerk auf dem Gebiet der Funktionalanalysis richtet sich insbesondere an Leser mit Interesse an Anwendungen auf Differentialgleichungen.

Anschauliche Geometrie


Author: David Hilbert,Stefan Cohn-Vossen
Publisher: Springer-Verlag
ISBN: 3662366851
Category: Mathematics
Page: 312
View: 1630

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Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

Diskrete Mathematik


Author: Martin Aigner
Publisher: Springer-Verlag
ISBN: 3322854965
Category: Mathematics
Page: 318
View: 3008

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Vor 50 Jahren gab es den Begriff "Diskrete Mathematik" nicht, und er ist auch heute im deutschen Sprachraum keineswegs gebrauchlich. Vorlesungen dazu werden nicht iiberall und schon gar nicht mit einem einheitlichen Themenkatalog angeboten (im Gegensatz zum Beispiel zu den USA, wo sie seit langem einen festen Platz haben). Die Mathematiker verstehen unter Diskreter Mathematik meist Kombinatorik oder Graphentheorie, die Informatiker Diskrete Strukturen oder Boolesche Algebren. Das Hauptanliegen dieses Buches ist daher, solch einen Themenkatalog zu prasentieren, der alle Grundlagen fiir ein weiterfiihrendes Studium enthalt. Die Diskrete Mathematik beschaftigt sich vor allem mit endlichen Mengen. Was kann man in endlichen Mengen studieren? Ais allererstes kann man sie abzahlen, dies ist das klassische Thema der Kombinatorik - in Teil I werden wir die wich tigsten Ideen und Methoden zur Abzahlung kennenlernen. Auf endlichen Mengen ist je nach Aufgabenstellung meist eine einfache Struktur in Form von Relationen gegeben, von denen die anwendungsreichsten die Graphen sind. Diese Aspekte fas sen wir in Teil II unter dem Titel Graphen uncl Algorithmen zusammen. Und schlieBlich existiert auf endlichen Mengen oft eine algebraische Struktur (oder man kann eine solche auf natiirliche Weise erklaren). Algebraische Systeme sind der Inhalt von Teil III. Diese drei Gesichtspunkte bilden den roten Faden des Buches. Ein weiterer Aspekt, der die Darstellung durchgehend pragt, betrifft den Begriff der Optimierung.

The Tools of Mathematical Reasoning


Author: Tamara J. Lakins
Publisher: American Mathematical Soc.
ISBN: 1470428997
Category: General -- Instructional exposition (textbooks, tutorial papers, etc.)
Page: 217
View: 2400

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This accessible textbook gives beginning undergraduate mathematics students a first exposure to introductory logic, proofs, sets, functions, number theory, relations, finite and infinite sets, and the foundations of analysis. The book provides students with a quick path to writing proofs and a practical collection of tools that they can use in later mathematics courses such as abstract algebra and analysis. The importance of the logical structure of a mathematical statement as a framework for finding a proof of that statement, and the proper use of variables, is an early and consistent theme used throughout the book.

Spectral Synthesis


Author: John J. Benedetto
Publisher: Springer-Verlag
ISBN: 3322966615
Category: Technology & Engineering
Page: 281
View: 3586

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Naive Mengenlehre


Author: Paul R. Halmos
Publisher: Vandenhoeck & Ruprecht
ISBN: 9783525405277
Category: Arithmetic
Page: 132
View: 1249

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Vektoranalysis


Author: Klaus Jänich
Publisher: Springer-Verlag
ISBN: 3662107503
Category: Mathematics
Page: 277
View: 9311

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Die Vektoranalysis handelt, in klassischer Darstellung, von Vektorfeldern, den Operatoren Gradient, Divergenz und Rotation, von Linien-, Flächen- und Volumenintegralen und von den Integralsätzen von Gauß, Stokes und Green. In moderner Fassung ist es der Cartansche Kalkül mit dem Satz von Stokes. Das vorliegende Buch vertritt grundsätzlich die moderne Herangehensweise, geht aber auch sorgfältig auf die klassische Notation und Auffassung ein. Das Buch richtet sich an Mathematik- und Physikstudenten ab dem zweiten Studienjahr, die mit den Grundbegriffen der Differential- und Integralrechnung in einer und mehreren Variablen sowie der Topologie vertraut sind. Der sehr persönliche Stil des Autors und die aus anderen Büchern bereits bekannten Lernhilfen, wie: viele Figuren, mehr als 50 kommentierte Übungsaufgaben, über 100 Tests mit Antworten, machen auch diesen Text zum Selbststudium hervorragend geeignet.