An Introduction to Inverse Limits with Set-valued Functions PDF Download
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Author: W.T. Ingram
Publisher: Springer Science & Business Media
ISBN: 146144487X
Category : Mathematics
Languages : en
Pages : 93
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Book Description
Inverse limits with set-valued functions are quickly becoming a popular topic of research due to their potential applications in dynamical systems and economics. This brief provides a concise introduction dedicated specifically to such inverse limits. The theory is presented along with detailed examples which form the distinguishing feature of this work. The major differences between the theory of inverse limits with mappings and the theory with set-valued functions are featured prominently in this book in a positive light. The reader is assumed to have taken a senior level course in analysis and a basic course in topology. Advanced undergraduate and graduate students, and researchers working in this area will find this brief useful.
Author: W.T. Ingram
Publisher: Springer Science & Business Media
ISBN: 146144487X
Category : Mathematics
Languages : en
Pages : 93
Get Book Here
Book Description
Inverse limits with set-valued functions are quickly becoming a popular topic of research due to their potential applications in dynamical systems and economics. This brief provides a concise introduction dedicated specifically to such inverse limits. The theory is presented along with detailed examples which form the distinguishing feature of this work. The major differences between the theory of inverse limits with mappings and the theory with set-valued functions are featured prominently in this book in a positive light. The reader is assumed to have taken a senior level course in analysis and a basic course in topology. Advanced undergraduate and graduate students, and researchers working in this area will find this brief useful.
Author:
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
Author: Alexander Nelson Cornelius
Publisher:
ISBN:
Category : Funtions, Continuous
Languages : en
Pages : 69
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Book Description
Much is known about inverse limits of compact spaces with continuous bonding maps. When the requirement that the bonding maps be continuous functions is relaxed, to allow for upper semi-continuous set-valued functions, many of the standard tools used to study inverse limit spaces are no longer available. Also, the structure of the inverse limit becomes more complicated. In this dissertation, the structure of compact subsets of the inverse limit is studied. Necessary and sufficient conditions are given for a compact subset of an inverse limit to be the inverse limit of its projections. Weak crossovers are introduced and used to describe the relationship between a set and the inverse limit of its projections. In the process of examining weak crossovers, tools with potentially broader applications are developed. When set-valued bonding maps are continuous and can be written as a union of continuous functions, sufficient conditions to make the inverse limit a Hausdorff continuum are given. It is also shown that, with another assumption, the inverse limit is in fact decomposable. When the bonding map can be written as a union of set-valued functions between compact subsets of the factor space, a description of the inverse limit is given. This uses a generalization of the itinerary of a point.
Author: Christopher David Jacobsen
Publisher:
ISBN:
Category : Continuum (Mathematics)
Languages : en
Pages : 60
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Book Description
"While there is a wealth of information pertaining to inverse limits with single valued bonding maps, comparatively little is known about inverse limits with upper semi-continuous set valued bonding functions. In order to add somewhat to the communal knowledge on the subject, this paper provides an example of an inverse limit with a single upper semi-continuous set valued bonding function. It is then shown that the space is a continuum, and its structure is examined via its arc components and through various of its properties, such as dimension and decomposability"--Abstract, leaf iii.
Author: W.T. Ingram
Publisher: Springer Science & Business Media
ISBN: 146141797X
Category : Mathematics
Languages : en
Pages : 229
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Book Description
Inverse limits provide a powerful tool for constructing complicated spaces from simple ones. They also turn the study of a dynamical system consisting of a space and a self-map into a study of a (likely more complicated) space and a self-homeomorphism. In four chapters along with an appendix containing background material the authors develop the theory of inverse limits. The book begins with an introduction through inverse limits on [0,1] before moving to a general treatment of the subject. Special topics in continuum theory complete the book. Although it is not a book on dynamics, the influence of dynamics can be seen throughout; for instance, it includes studies of inverse limits with maps from families of maps that are of interest to dynamicists such as the logistic and the tent families. This book will serve as a useful reference to graduate students and researchers in continuum theory and dynamical systems. Researchers working in applied areas who are discovering inverse limits in their work will also benefit from this book.
Author: Michael Lockyer
Publisher:
ISBN:
Category : Continuum (Mathematics)
Languages : en
Pages : 111
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Book Description
Generalised inverse limits are a new topic of study in the area of continuum theory. Although similarly defined to the more traditional inverse limits of continuous functions on continua, they have a much richer structure. It was realised early on that many of the theorems relating to the inverse limits of continuous functions do not carry over to generalised inverse limits, which use set valued functions. Much of the research in generalised inverse limits to this point has been to attempt to characterise their structure. This thesis contains three main chapters, each of which is based on a topic in generalised inverse limits research. In the first of these we explore the structure of a particular generalised inverse limit known as K(0,1). This is the inverse limit of a generalised tent map. The inverse limits of tent maps have played an important role in continuum theory in the past, and with the introduction of generalised inverse limits we can include generalised tent maps that are not continuous functions. The only such generalised tent map whose inverse limit does not have a very simple structure is K(0,1). In this chapter we prove a number of topological properties of K(0,1), and give an embedding of K(0,1)) into R3. For the second topic we characterise a certain kind of disconnection in generalised inverse limits over Hausdorff continua. This generalises a result by Greenwood and Kennedy for generalised inverse limits over intervals. Connectedness is a property that has attracted much interest in generalised inverse limits, as these are not necessarily connected, unlike the inverse limits of continuous functions, which are. In the final chapter we prove a result relating to path connectedness in generalised inverse limits. Path connectedness in generalised inverse limits is in some ways a very different property to connectedness. For example, a generalised inverse limit may not be path connected even though all its finite approximants are path connected. This cannot happen if we replace the words "path connected" with "connected" in the previous sentence. The result proved in this chapter links the path connectedness of a generalised inverse limit with path connected properties of its finite approximants.
Author: Sergio Macías
Publisher: Springer
ISBN: 3319909029
Category : Mathematics
Languages : en
Pages : 451
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Book Description
This book is a significant companion text to the existing literature on continuum theory. It opens with background information of continuum theory, so often missing from the preceding publications, and then explores the following topics: inverse limits, the Jones set function T, homogenous continua, and n-fold hyperspaces. In this new edition of the book, the author builds on the aforementioned topics, including the unprecedented presentation of n-fold hyperspace suspensions and induced maps on n-fold hyperspaces. The first edition of the book has had a remarkable impact on the continuum theory community. After twelve years, this updated version will also prove to be an excellent resource within the field of topology.
Author: Rafal K. Goebel
Publisher: SIAM
ISBN: 1611977983
Category : Mathematics
Languages : en
Pages : 234
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Book Description
Set-valued analysis, convex analysis, and nonsmooth analysis are relatively modern branches of mathematical analysis that have become increasingly relevant in current control theory and control engineering literature. This book serves as a broad introduction to analytical tools in these fields and to their applications in dynamical and control systems and is the first to cover these topics with this scope and at this level. Both continuous-time and discrete-time mutlivalued dynamics, modeled by differential and difference inclusions, are considered. Set-Valued, Convex, and Nonsmooth Analysis in Dynamics and Control: An Introduction is aimed at graduate students in control engineering and applied mathematics and researchers in control engineering who have no prior exposure to set-valued, convex, and nonsmooth analysis. The book will also be of interest to advanced undergraduate mathematics students and mathematicians with no prior exposure to the topic. The expected mathematical background is a course on nonlinear differential equations / dynamical systems and a course on real analysis. Knowledge of some control theory is helpful, but not essential.
Author: Shoumei Li
Publisher: Springer Science & Business Media
ISBN: 9781402009181
Category : Mathematics
Languages : en
Pages : 414
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Book Description
This book presents a clear, systematic treatment of convergence theorems of set-valued random variables (random sets) and fuzzy set-valued random variables (random fuzzy sets). Topics such as strong laws of large numbers and central limit theorems, including new results in connection with the theory of empirical processes are covered. The author's own recent developments on martingale convergence theorems and their applications to data processing are also included. The mathematical foundations along with a clear explanation such as Hölmander's embedding theorem, notions of various convergence of sets and fuzzy sets, Aumann integrals, conditional expectations, selection theorems, measurability and integrability arguments for both set-valued and fuzzy set-valued random variables and newly obtained optimizations techniques based on invariant properties are also given.
Author: Asen L. Dontchev
Publisher: Springer
ISBN: 149391037X
Category : Mathematics
Languages : en
Pages : 495
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Book Description
The implicit function theorem is one of the most important theorems in analysis and its many variants are basic tools in partial differential equations and numerical analysis. This second edition of Implicit Functions and Solution Mappings presents an updated and more complete picture of the field by including solutions of problems that have been solved since the first edition was published, and places old and new results in a broader perspective. The purpose of this self-contained work is to provide a reference on the topic and to provide a unified collection of a number of results which are currently scattered throughout the literature. Updates to this edition include new sections in almost all chapters, new exercises and examples, updated commentaries to chapters and an enlarged index and references section.
