# Introduction to Probability For example , the probability that there are no fixed points , when n = 10 , 20 , or 30 is estimated to be between .35 and .37 . We shall see later ( see Example 3.12 ) that for n > 10 the exact probabilities Pn ( 0 ) are , to six ...

Publisher: American Mathematical Soc.

ISBN: 0821807498

Category: Mathematics

Page: 510

View: 617

This text is designed for an introductory probability course at the university level for undergraduates in mathematics, the physical and social sciences, engineering, and computer science. It presents a thorough treatment of probability ideas and techniques necessary for a firm understanding of the subject.
Categories: Mathematics

# Dynamical Systems parameter and the eigenvalue associated with the system linearization at the fixed point changes discontinuously from a to b at μ = 0. Scenario A2: (Persistence of unstable fixed point) or No Attractor → No Attractor in [Yuan (1997), ...

Publisher: CRC Press

ISBN: 9780429647420

Category: Mathematics

Page: 400

View: 283

Chaos is the idea that a system will produce very different long-term behaviors when the initial conditions are perturbed only slightly. Chaos is used for novel, time- or energy-critical interdisciplinary applications. Examples include high-performance circuits and devices, liquid mixing, chemical reactions, biological systems, crisis management, secure information processing, and critical decision-making in politics, economics, as well as military applications, etc. This book presents the latest investigations in the theory of chaotic systems and their dynamics. The book covers some theoretical aspects of the subject arising in the study of both discrete and continuous-time chaotic dynamical systems. This book presents the state-of-the-art of the more advanced studies of chaotic dynamical systems.
Categories: Mathematics

# Handbook of Topological Fixed Point Theory However, f may have extra points only on T, and furthermore the fixed points on T can be determined by the differentials Df(z), 2 € S, explicitly. In particular, if S contains no fixed points of f, or if Df(z) does not have positive ...

Author: Robert F. Brown

Publisher: Springer Science & Business Media

ISBN: 9781402032226

Category: Mathematics

Page: 972

View: 233

This book is the first in the world literature presenting all new trends in topological fixed point theory. Until now all books connected to the topological fixed point theory were devoted only to some parts of this theory. This book will be especially useful for post-graduate students and researchers interested in the fixed point theory, particularly in topological methods in nonlinear analysis, differential equations and dynamical systems. The content is also likely to stimulate the interest of mathematical economists, population dynamics experts as well as theoretical physicists exploring the topological dynamics.
Categories: Mathematics

# Fixed Points and Economic Equilibria The continuous mapping restricted on |K| to itself, however, never has a fixed point since by the property of class B ... of fφ (a certain kind of linear approximation of φ) is 0 for sufficiently fine K as long as φ has no fixed point.

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Publisher:

ISBN: 9789814469180

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# Homotopy Methods in Topological Fixed and Periodic Points Theory Let B be an open ball in the Euclidean space E and let f : cl B – E be a map with no fixed point on the boundary . Then ind ( f ) = 0 implies a homotopy ft : cl B → E from fo = f to a fixed point free map . Moreover , the homotopy ft ...

Author: Jerzy Jezierski

Publisher: Springer Science & Business Media

ISBN: 1402039301

Category: Mathematics

Page: 320

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The notion of a ?xed point plays a crucial role in numerous branches of mat- maticsand its applications. Informationabout the existence of such pointsis often the crucial argument in solving a problem. In particular, topological methods of ?xed point theory have been an increasing focus of interest over the last century. These topological methods of ?xed point theory are divided, roughly speaking, into two types. The ?rst type includes such as the Banach Contraction Principle where the assumptions on the space can be very mild but a small change of the map can remove the ?xed point. The second type, on the other hand, such as the Brouwer and Lefschetz Fixed Point Theorems, give the existence of a ?xed point not only for a given map but also for any its deformations. This book is an exposition of a part of the topological ?xed and periodic point theory, of this second type, based on the notions of Lefschetz and Nielsen numbers. Since both notions are homotopyinvariants, the deformationis used as an essential method, and the assertions of theorems typically state the existence of ?xed or periodic points for every map of the whole homotopy class, we refer to them as homotopy methods of the topological ?xed and periodic point theory.
Categories: Mathematics

# Fixed Point Theory (H. 3) COROLIARY (Eace is ion) If K, is a subcomplex of K such that f has no fixed points on |K| \ (IK, \ k, ), then ind (K, f, L) = ind (Kı, f, L). Pyroo Evident from (11.1) and (4.2). (H. H. ) PROPOSITTON (Homotopy) If h; ...

Publisher: Springer

ISBN: 9783540386001

Category: Mathematics

Page: 518

View: 823

Categories: Mathematics

# A Walk Through Combinatorics In a more mathematical formulation: how many permutations of the set [n] have no fixed points, that is, have the element i in the ith position for no il Such permutations are called derangements.

Author: Mikl¢s B¢na

Publisher: World Scientific

ISBN: 9789812568854

Category: Mathematics

Page: 469

View: 379

This is a textbook for an introductory combinatorics course that can take up one or two semesters. An extensive list of problems, ranging from routine exercises to research questions, is included. In each section, there are also exercises that contain material not explicitly discussed in the preceding text, so as to provide instructors with extra choices if they want to shift the emphasis of their course. Just as with the first edition, the new edition walks the reader through the classic parts of combinatorial enumeration and graph theory, while also discussing some recent progress in the area: on the one hand, providing material that will help students learn the basic techniques, and on the other hand, showing that some questions at the forefront of research are comprehensible and accessible for the talented and hard-working undergraduate. The basic topics discussed are: the twelvefold way, cycles in permutations, the formula of inclusion and exclusion, the notion of graphs and trees, matchings and Eulerian and Hamiltonian cycles. The selected advanced topics are: Ramsey theory, pattern avoidance, the probabilistic method, partially ordered sets, and algorithms and complexity. As the goal of the book is to encourage students to learn more combinatorics, every effort has been made to provide them with a not only useful, but also enjoyable and engaging reading.
Categories: Mathematics

# Lectures on Algebraic Topology §"-*S" has no fixed point then deg(/)=(- l)n+1. If /: §"-*§" has no antipodal point (fx+-x) then deg(/)= + l. In particular, every map f : S2*-»§2* has a fixed point or an antipodal point. Proof. If /has no fixed point then dtx = (l ...

Author: Albrecht Dold

Publisher: Springer Science & Business Media

ISBN: 9783540586609

Category: Mathematics

Page: 379

View: 648

Springer is reissuing a selected few highly successful books in a new, inexpensive softcover edition to make them easily accessible to younger generations of students and researchers. Springer-Verlag began publishing books in higher mathematics in 1920. This is a reprint of the Second Edition.
Categories: Mathematics

# Probability and Statistics with Integrated Software Routines A mapping with no fixed points is called a derangement. (Matching Problem) a) How many ways can the integers 1, 2, and 3 be uniquely mapped onto themselves? b) Compute the probability of at least one of the integers being a fixed point ...

Author: Ronald Deep