In recent years, the classical theory of stochastic integration and stochastic differential equations has been extended to a non-commutative set-up to develop models for quantum noises. The author, a specialist of classical stochastic calculus and martingale theory, tries to provide an introduction to this rapidly expanding field in a way which should be accessible to probabilists familiar with the Ito integral. It can also, on the other hand, provide a means of access to the methods of stochastic calculus for physicists familiar with Fock space analysis. For this second edition, the author has added about 30 pages of new material, mostly on quantum stochastic integrals.
Projective Geometry - Volume I
Publisher: Potter Press | 2007-03-15 | ISBN:1406747173 | Pages:352 | DJVU | 4 MB
PEOJECTIVE GEOMETRY BY OSWALD VEBLEN PROFESSOR OP MATHEMATICS, PRINCETON UNIVERSITY AND JOHN WESLEY YOUNG PROFESSOR OF MATHEMATICS, UNIVERSITY OF KANSAS VOLUME I GINN AND COMPANY BOSTON NEW YORK CHICAGO LONDON PREFACE Geometry, which had been for centuries the most perfect example of a deductive science, during the creative period of the nineteenth century outgrew its old logical forms. The most recent period has however brought a clearer understanding of the logical foundations of mathematics and thus has made it possible for the exposition of geometry to resume the purely deductive form. But the treatment in the books which have hitherto appeared makes the work of lay ing the foundations seem so formidable as either to require for itself a separate treatise, or to be passed over without attention to more than the outlines. This is partly due to the fact that in giving the complete foundation for ordinary real or complex geometry, it is necessary to make a study of linear order and continuity, a study which is not only extremely delicate, but whose methods are those of the theory of functions of a real variable rather than of elemen tary geometry. The present work, which is to consist of two volumes and is in tended to be available as a text in courses offered in American uni versities to upper-class and graduate students, seeks to avoid this difficulty by deferring the study of order and continuity to the sec ond volume. The more elementary part of the subject rests on a very simple set of assumptions which characterize what may be called general protective geometry. It will be found that the theorems selected on this basis of logical simplicity are also elemen tary in the sense of being easily comprehended and often used. Even the limited space devoted in this volume to the foundations may seem a drawback from the pedagogical point of view of some mathematicians. To this we can only reply that, in our opinion, an adequate knowledge of geometry cannot be obtained without attention to the foundations. We believe, moreover, that the abstract treatment is peculiarly desirable in protective geometry, because it is through the latter that the other geometric disciplines are most readily coordinated. Read More »
The Universe in a Handkerchief: Lewis Carroll’s Mathematical Recreations, Games, Puzzles, and Word Plays
Martin Gardner, “The Universe in a Handkerchief: Lewis Carroll’s Mathematical Recreations, Games, Puzzles, and Word Plays”
Publisher: Springer | Pages: 158 | ISBN:0387256415 | PDF | 12.3 MB
This book contains scores of intriguing puzzles and paradoxes from Lewis Carroll, the author of Alice in Wonderland, whose interests ranged from inventing new games like Arithmetical Croquet to important problems in symbolic logic and propositional calculus. Written by Carroll expert and well-known mathematics author Martin Gardner, this tour through Carroll’s inventions is both fun and informative.
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A logic machine is a device, electrical or mechanical, designed specifically for solving problems in formal logic. A logic
diagram is a geometrical method for doing the same thing. The two fields are closely intertwined, and this book is the first attempt in any language to trace their curious, fascinating histories.
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Richard Beals, “Analysis: An Introduction”
Cambridge University Press (September 13, 2004) | ISBN: 0521600472 | 272 pages | PDF | 1,8 Mb
“Analysis: An Introduction is most appropriate for a undergraduate who has already grappled with the main ideas from real analysis, and who is looking for a succinct, well-written treatise that connects these concepts to some of their most powerful applications. Beals’ book has the potential to serve this audience very well indeed.”
MAA Reviews, Christopher Hammond, Connecticut College
This self-contained text, suitable for advanced undergraduates, provides an extensive introduction to mathematical analysis, from the fundamentals to more advanced material. It begins with the properties of the real numbers and continues with a rigorous treatment of sequences, series, metric spaces, and calculus in one variable. Further subjects include Lebesgue measure and integration on the line, Fourier analysis, and differential equations. In addition to this core material, the book includes a number of interesting applications of the subject matter to areas both within and outside the field of mathematics. The aim throughout is to strike a balance between being too austere or too sketchy, and being so detailed as to obscure the essential ideas. A large number of examples and 500 exercises allow the reader to test understanding, practise mathematical exposition and provide a window into further topics.
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W.J. Wickless, ” A First Graduate Course in Abstract Algebra”
CRC; 1 edition (March 28, 2004) | ISBN: 0824756274 | 250 pages | Djvu | 3,1 Mb
This is a very useful text on abstract algebra at the beginning graduate levelthe notions of tensor product and projectivity of modules is introduced early and serve in several places to simplify proofsnumerous worked out examples shed light on the abstract theory and help to understand what is going on.
- Monatshefte für Mathematik
Since abstract algebra is so important to the study of advanced mathematics, it is critical that students have a firm grasp of its principles and underlying theories before moving on to further study. To accomplish this, they require a concise, accessible, user-friendly textbook that is both challenging and stimulating. A First Graduate Course in Abstract Algebra is just such a textbook. Divided into two sections, this book covers both the standard topics (groups, modules, rings, and vector spaces) associated with abstract algebra and more advanced topics such as Galois fields, noncommutative rings, group extensions, and Abelian groups. The author includes review material where needed instead of in a single chapter, giving convenient access with minimal page turning. He also provides ample examples, exercises, and problem sets to reinforce the material. This book illustrates the theory of finitely generated modules over principal ideal domains, discusses tensor products, and demonstrates the development of determinants. It also covers Sylow theory and Jordan canonical form. A First Graduate Course in Abstract Algebra is ideal for a two-semester course, providing enough examples, problems, and exercises for a deep understanding. Each of the final three chapters is logically independent and can be covered in any order, perfect for a customized syllabus.
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H. Ted Davis , Kendall T. Thomson, “Linear Algebra and Linear Operators in Engineering: with Applications in Mathematica”
Academic Press; 1st edition (January 15, 2000) | ISBN:012206349X | 547 pages | PDF | 15,4 Mb
Designed for advanced engineering, physical science, and applied mathematics students, this innovative textbook is an introduction to both the theory and practical application of linear algebra and functional analysis. The book is self-contained, beginning with elementary principles, basic concepts, and definitions. The important theorems of the subject are covered and effective application tools are developed, working up to a thorough treatment of eigenanalysis and the spectral resolution theorem. Building on a fundamental understanding of finite vector spaces, infinite dimensional Hilbert spaces are introduced from analogy. Wherever possible, theorems and definitions from matrix theory are called upon to drive the analogy home. The result is a clear and intuitive segue to functional analysis, culminating in a practical introduction to the functional theory of integral and differential operators. Numerous examples, problems, and illustrations highlight applications from all over engineering and the physical sciences. Also included are several numerical applications, complete with Mathematica solutions and code, giving the student a “hands-on” introduction to numerical analysis. Linear Algebra and Linear Operators in Engineering is ideally suited as the main text of an introductory graduate course, and is a fine instrument for self-study or as a general reference for those applying mathematics.
· Contains numerous Mathematica examples complete with full code and solutions
· Provides complete numerical algorithms for solving linear and nonlinear problems
· Spans elementary notions to the functional theory of linear integral and differential equations
· Includes over 130 examples, illustrations, and exercises and over 220 problems ranging from basic concepts to challenging applications
· Presents real-life applications from chemical, mechanical, and electrical engineering and the physical sciences
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COMPLEX NUMBERS AND VECTORS IN THE SECONDARY CURRICULUM INTRODUCTION Complex numbers and vectors are both important areas of study within the senior secondary mathematics curriculum. The rationale for the inclusion of complex numbers in the curriculum is often related to: arguments for completeness of algebraic analysis of polynomial functions and the solution of related equations consideration of certain types of transformations of the (complex) plane, in particular those involving combinations of dilations and rotations, as well as some curves and regions in the complex plane Students typically encounter complex numbers in the guise of some kind of special number that enables one to extend certain algebraic manipulations on quadratic functions of a real variable with real coefficients to ensure that the rule of any quadratic function q]g x = ax + bx + c can be expressed as a product of two linear factors, and the equations q]g x = 0 always has two (not necessarily distinct) roots. The best problems can appear deceptively simple, but can create the need to explore new ways of thinking about our world and how we choose to describe it. A small problem may allow an insight. Student Activity 2.1 can be used to illustrate this point to students.
Éric Charpentier, Annick Lesne , Nikolaï K. Nikolski, “Kolmogorov’s Heritage in Mathematics”
Springer; 1 edition (October 11, 2007) | ISBN: 3540363491 | 318 pages | 4,4 Mb
A.N. Kolmogorov (b. Tambov 1903, d. Moscow 1987) was one of the most brilliant mathematicians that the world has ever known. Incredibly deep and creative, he was able to approach each subject with a completely new point of view: in a few magnificent pages, which are models of shrewdness and imagination, and which astounded his contemporaries, he changed drastically the landscape of the subject.
Most mathematicians prove what they can, Kolmogorov was of those who prove what they want. For this book several world experts were asked to present one part of the mathematical heritage left to us by Kolmogorov.
Each chapter treats one of Kolmogorov’s research themes, or a subject that was invented as a consequence of his discoveries. His contributions are presented, his methods, the perspectives he opened to us, the way in which this research has evolved up to now, along with examples of recent applications and a presentation of the current prospects.
This book can be read by anyone with a master’s (even a bachelor’s) degree in mathematics, computer science or physics, or more generally by anyone who likes mathematical ideas. Rather than present detailed proofs, the main ideas are described. A bibliography is provided for those who wish to understand the technical details.
One can see that sometimes very simple reasoning (with the right interpretation and tools) can lead in a few lines to very substantial results.
Which is the smallest integer that can be expressed as a sum of consecutive integers in a given number of ways?
Alternating iterations of the Smarandache function and the Euler phi-function respectively the sum of divisors function. Some light is thrown on loops and invariants resulting from these iterations. An important question is resolved with the amazing involvement of the famous Fermat numbers.
A particularly interesting subject is the Smarandache partial perfect additive sequence, it has a simple definition and a strange behaviour.
Smarandache general continued fractions are treated in great detail and proof is given for the convergence under specified conditions.
Smarandache k-k additive relationships as well as subtractive relationships are treated with some observations on the occurrence of prime twins.
A substantial part is devoted to concatenation and deconcatenation problems. Some divisibilty properties of very large numbers is studied. In particular some questions raised on the Smarandache deconstructive sequence are resolved.