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Argument Principle

Mathematics document containing theorems and formulas.

Excerpt: Argument principle. Let C be a simple closed curve, and suppose f is analytic on C. Suppose moreover that the only singularities of f inside C are poles. If fYz 0 for all zOC, then @ = fYC is a closed curve which does not pass through the origin. If L is a complex...

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Applications to the Behavioral and Managerial Sciences

Mathematics document containing theorems and formulas.

Excerpt: Communication And Soclometric Matrices. Matrices having only the entries 0 and 1 are useful in the analysis of graphs and networks. We shall not attempt to give a complete treatment of the subject here, but merely illustrate some of its more interesting applications. A communication network consists of a set of people, A,, A,, . . . ,A,, such that between some pairs of persons there is a communication link. Such a link may be either one-way or two-way. A two-way...

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Applications

Mathematics document containing theorems and formulas.

Excerpt: This chapter presents some applications of manifold theory and tensor analysis to physics and engineering. Our selection is a of limited scope and depth, with the intention of providing an introduction to the techniques. There are many other applications of the ideas of this book as well. We list below a few selected references for further reading in the same spirit: // 1. Arnol?d [1982], Abraham and Marsden [1978], Chernoff and Marsden [1974],Weinstein [1977], ...

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Appendix B : Lawton's Proof of Theorem 9. 6

Mathematics document containing theorems and formulas.

Excerpt: In this appendix we present Lawton's proof of Theorem 9.4. The proof follows Lawton [17] closely, and uses standard results on topological entropy (from Chapters 6 and 7). Let G be a compact abelian group, with T an automorphism of G. If H G is a T{invariant closed subgroup, then denote by TH the automorphism of H obtained by restricting T to H. Definition B.1. The dual group of G, bG, is said to be nitely generated under bT if there are elements 1; n in bG such...

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Appendix a : Weil's Proof of Theorem 8. 1

Mathematics document containing theorems and formulas.

Excerpt: Let P denote the set of all rational primes. For each p 2 P, the field of p{adic numbers, Qp, is defined to be the completion of the rationals in the p{adic valuation, jrjp = p ordp(r); where ordp(r) is the highest power of p dividing r. Extend the de nition of j jp by setting j0jp = 0. Write p = 1 for the usual valuation on Q, jrj1 = jrj. Then Q1 = R. For each p 2 f1g [ P, the field Qp is a non{discrete locally compact eld, with topology given by the metric dp(...

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An Introduction to Lie Groups

Mathematics document containing theorems and formulas.

Excerpt: Rigid Body. Consider a free rigid body rotating about its center of mass, taken to be the origin. ?Free? means that there are no external forces, and ?rigid? means that the distance between any two points of the body is unchanged during the motion. Consider a point X of the body at time t = 0, and denote its position at time t by f(X, t). Rigidity of the body and the assumption of a smooth motion imply that f(X, t) = A(t)X, where A(t) is a proper rotation, that ...

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All Games Bright and Beautiful

Mathematics document containing theorems and formulas.

Excerpt: What is a combinatorial game? The usual definition is a game in which (i) there are two players moving alternately; (ii) there are no chance devices and both players have perfect information; (iii) the rules are such that the game must eventually end; and (iv) there are no draws, and the winner is determined by who moves last. In this section, two master expositors lead us through many examples of such games, and introduce the theory that has been developed to d...

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A Little about Representations

Mathematics document containing theorems and formulas.

Excerpt: There is still a great deal to uncover about the structure of simple Lie algebras but it is worthwhile to make a slight detour to discuss something about representations This will lead to some useful relations for the adjoint representation.

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A Geometrical Vector in Three Dimensional Space

Mathematics document containing theorems and formulas.

Excerpt: A geometrical vector in three dimensional space can be represented by a column vector whose entries are the x y and z components of the vector A rotation of the vector can be represented by a three by three matrix In particular a rotation by about the z axis is given by...

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Take-Away Games

By: Michael Zieve

Mathematics document containing theorems and formulas.

Excerpt: Several works, including [Berlekamp et al. 1982; Epp and Ferguson 1980; Schwenk 1970], have studied take-away games where the players alternately remove a positive number of counters from a single pile, the player removing the last counter being the winner. On his initial move, the first player may remove any number of counters, so long as he leaves at least one. On each subsequent move, a player may remove at most f(n) counters, where n is the number of counter...

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Scientific Computing with Maple Programming

By: Zhonggang Zeng

Mathematics document containing theorems and formulas.

Excerpt: Maple as a calculator Maple can perform many of mathematical with one-line commands.

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Littlewood Richardson Semigroups

By: Andrei Zelevinsky

Mathematics document containing theorems and formulas.

Excerpt: This article is based on my talk at the workshop on Representation Theory and Symmetric Functions, MSRI, April 14, 1997. I thank the organizers (Sergey Fomin, Curtis Greene, Phil Hanlon and Sheila Sundaram) for bringing together a group of outstanding combinatorialists and for giving me a chance to bring to their attention some of the problems that I find very exciting and beautiful.

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The Research Mathematician as Storyteller

By: William Yslas

Mathematics document containing theorems and formulas.

Excerpt: The Southwest Regional Institute in the Mathematical Sciences (SWRIMS) was funded by the National Science Foundation (NSF) as an e ort to integrate research and education. Part of the NSF's vision is that the researcher should be simultaneously involved in research activities and educational initiatives. One of the purposes of SWRIMS was to bring into focus what these educational initiatives might be, and to answer the following questions: How can a researcher u...

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On the Education of Mathematics Majors

By: Hung-Hsi Wu

Mathematics document containing theorems and formulas.

Excerpt: Re-examination of Standard Upper-Division Courses: Upper division courses in college are where math majors learn real mathematics. For the first time they get to examine the foundations of algebra, geometry and analysis, come face-to-face with the deductive nature of mathematics on a consistent basis and, most importantly, learn to do serious theorem-proving. For reasons not unlike these, most mathematicians enjoy teaching these courses more than others.

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The Gamesman's Toolkit

By: David Wolfe

Mathematics document containing theorems and formulas.

Excerpt: The Gamesman's Toolkit, written in C and running under UNIX, implements virtually all of the material in finite combinatorial game theory. The toolkit can be used in one of two ways: as a games calculator, by the gamesman who wishes to do standard algebraic manipulations on games; and as a programming toolkit, to analyze a particular game, for instance. A broad base of game-theoretic functions is provided, along with a parser and output routines. Either way, the...

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Model Comparison

By: William of Ockham

Mathematics document containing theorems and formulas.

Excerpt: We have seen in some detail how to conduct inferences { test hypotheses, estimate parameters, predict future observations { within the context of a preassigned model, representing some working hypothesis about the phenomenon being observed. But a scientist must be concerned also with a bigger problem; how to decide between different models when both seem able to account for the facts. Indeed, the progress of science requires comparison of different conceivable m...

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Algorithms and Complexity

By: Herbert S. Wilf

Mathematics document containing theorems and formulas.

Excerpt: Internet Edition, Summer, 1994 This edition of Algorithms and Complexity may be taken at no charge by all interested persons...

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Algorithms and Complexity

By: Herbert S. Wilf

Mathematics document containing theorems and formulas.

Excerpt: Internet Edition, Summer, 1994 This edition of Algorithms and Complexity. It may be taken at no charge by all interested persons...

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Algorithms and Complexity

By: Herbert S. Wilf

Mathematics document containing theorems and formulas.

Excerpt: Internet Edition, Summer, 1994 This edition of Algorithms and Complexity. It may be taken at no charge by all interested persons...

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Algorithms and Complexity

By: Herbert S. Wilf

Mathematics document containing theorems and formulas.

Excerpt: This edition of Algorithms and Complexity is the file pub/wilf/AlgComp.ps.Z. It may be taken at no charge by all interested persons...

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