Algebraic Theory of Measure and Integration

Author: Constantin Carathéodory

Publisher: American Mathematical Soc.

ISBN: 0821852736

Category: Mathematics

Page: 378

View: 3678

DOWNLOAD NOW »

In this book, the famous Greek mathematician Constantin Caratheodory presents his generalization of the classical theory of measure and integration. The main component of this generalization is the replacement of the Boolean algebra of subsets of the base space $X$ with an arbitrary Boolean algebra. The author develops measure theory associated with an arbitrary Boolean algebra. Then, using an appropriate generalization of the notion of a function on $X$, he introduces the generalized procedure of integration on $X$ and studies the main properties of this generalized integration. These include analogs of the Radon-Nikodym theorem, of the Lebesque convergence theorem, and of other classical theorems of real analysis.

Generalized Calculus with Applications to Matter and Forces

Author: Luis Manuel Braga de Costa Campos

Publisher: CRC Press

ISBN: 1420071157

Category: Mathematics

Page: 885

View: 7306

DOWNLOAD NOW »

Combining mathematical theory, physical principles, and engineering problems, Generalized Calculus with Applications to Matter and Forces examines generalized functions, including the Heaviside unit jump and the Dirac unit impulse and its derivatives of all orders, in one and several dimensions. The text introduces the two main approaches to generalized functions: (1) as a nonuniform limit of a family of ordinary functions, and (2) as a functional over a set of test functions from which properties are inherited. The second approach is developed more extensively to encompass multidimensional generalized functions whose arguments are ordinary functions of several variables. As part of a series of books for engineers and scientists exploring advanced mathematics, Generalized Calculus with Applications to Matter and Forces presents generalized functions from an applied point of view, tackling problem classes such as: Gauss and Stokes’ theorems in the differential geometry, tensor calculus, and theory of potential fields Self-adjoint and non-self-adjoint problems for linear differential equations and nonlinear problems with large deformations Multipolar expansions and Green’s functions for elastic strings and bars, potential and rotational flow, electro- and magnetostatics, and more This third volume in the series Mathematics and Physics for Science and Technology is designed to complete the theory of functions and its application to potential fields, relating generalized functions to broader follow-on topics like differential equations. Featuring step-by-step examples with interpretations of results and discussions of assumptions and their consequences, Generalized Calculus with Applications to Matter and Forces enables readers to construct mathematical–physical models suited to new observations or novel engineering devices.

Measure and Integration

Author: Sterling K. Berberian

Publisher: American Mathematical Soc.

ISBN: 9780821853283

Category: Mathematics

Page: 312

View: 2588

DOWNLOAD NOW »

This highly flexible text is organized into two parts: Part I is suitable for a one-semester course at the first-year graduate level, and the book as a whole is suitable for a full-year course. Part I treats the theory of measure and integration over abstract measure spaces. Prerequisites are a familiarity with epsilon-delta arguments and with the language of naive set theory (union, intersection, function). The fundamental theorems of the subject are derived from first principles, with details in full. Highlights include convergence theorems (monotone, dominated), completeness of classical function spaces (Riesz-Fischer theorem), product measures (Fubini's theorem), and signed measures (Radon-Nikodym theorem). Part II is more specialized; it includes regular measures on locally compact spaces, the Riesz-Markoff theorem on the measure-theoretic representation of positive linear forms, and Haar measure on a locally compact group. The group algebra of a locally compact group is constructed in the last chapter, by an especially transparent method that minimizes measure-theoretic difficulties. Prerequisites for Part II include Part I plus a course in general topology. To quote from the Preface: ``Finally, I am under no illusions as to originality, for the subject of measure theory is an old one which has been worked over by many experts. My contribution can only be in selection, arrangement, and emphasis. I am deeply indebted to Paul R. Halmos, from whose textbook I first studied measure theory; I hope that these pages may reflect their debt to his book without seeming to be almost everywhere equal to it.''

Constantin Carathéodory

Mathematics and Politics in Turbulent Times

Author: Maria Georgiadou

Publisher: Springer Science & Business Media

ISBN: 3642185622

Category: Mathematics

Page: 651

View: 4868

DOWNLOAD NOW »

With breathtaking detail, Maria Georgiadou sheds light on the work and life of Constantin Carathéodory, who until now has been ignored by historians. In her thought-provoking book, Georgiadou maps out the mathematician’s oeuvre, life and turbulent historical surroundings. Descending from the Greek élite of Constantinople, Carathéodory graduated from the military school of Brussels, became engineer at the Assiout dam in Egypt and finally dedicated a lifetime to mathematics and education. He significantly contributed to: calculus of variations, the theory of point set measure, the theory of functions of a real variable, pdes, and complex function theory. An exciting and well-written biography, once started, difficult to put down.

Integration and Modern Analysis

Author: John J. Benedetto,Wojciech Czaja

Publisher: Springer Science & Business Media

ISBN: 9780817646561

Category: Mathematics

Page: 575

View: 9754

DOWNLOAD NOW »

This textbook and treatise begins with classical real variables, develops the Lebesgue theory abstractly and for Euclidean space, and analyzes the structure of measures. The authors' vision of modern real analysis is seen in their fascinating historical commentary and perspectives with other fields. There are comprehensive treatments of the role of absolute continuity, the evolution of the Riesz representation theorem to Radon measures and distribution theory, weak convergence of measures and the Dieudonné–Grothendieck theorem, modern differentiation theory, fractals and self-similarity, rearrangements and maximal functions, and surface and Hausdorff measures. There are hundreds of illuminating exercises, and extensive, focused appendices on functional and Fourier analysis. The presentation is ideal for the classroom, self-study, or professional reference.

Advanced Real Analysis

Author: Anthony W. Knapp

Publisher: Springer Science & Business Media

ISBN: 9780817644420

Category: Mathematics

Page: 466

View: 8347

DOWNLOAD NOW »

* Presents a comprehensive treatment with a global view of the subject * Rich in examples, problems with hints, and solutions, the book makes a welcome addition to the library of every mathematician