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Isomorphism

About: Isomorphism is a research topic. Over the lifetime, 9263 publications have been published within this topic receiving 119508 citations.


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Book
16 Dec 1981
TL;DR: The first part of the text as discussed by the authors provides an introduction to ergodic theory suitable for readers knowing basic measure theory, including recurrence properties, mixing properties, the Birkhoff Ergodic theorem, isomorphism, and entropy theory.
Abstract: This text provides an introduction to ergodic theory suitable for readers knowing basic measure theory. The mathematical prerequisites are summarized in Chapter 0. It is hoped the reader will be ready to tackle research papers after reading the book. The first part of the text is concerned with measure-preserving transformations of probability spaces; recurrence properties, mixing properties, the Birkhoff ergodic theorem, isomorphism and spectral isomorphism, and entropy theory are discussed. Some examples are described and are studied in detail when new properties are presented. The second part of the text focuses on the ergodic theory of continuous transformations of compact metrizable spaces. The family of invariant probability measures for such a transformation is studied and related to properties of the transformation such as topological traitivity, minimality, the size of the non-wandering set, and existence of periodic points. Topological entropy is introduced and related to measure-theoretic entropy. Topological pressure and equilibrium states are discussed, and a proof is given of the variational principle that relates pressure to measure-theoretic entropies. Several examples are studied in detail. The final chapter outlines significant results and some applications of ergodic theory to other branches of mathematics.

3,550 citations

Posted Content
TL;DR: In this article, the authors explore why organizations tend to be increasingly and inevitably homogeneous in their forms and practices, and suggest that organizational fields are structured into an organizational field by powerful forces that lead them to become similar.
Abstract: Instead of examining why organizations are dissimilar, this study explores why organizations tend to be increasingly and inevitably homogenous in their forms and practices. Organizations in a similar line of work are structured into an organizational field by powerful forces that lead them to become similar. Rather than the causes of rationalization and bureaucratization suggested by Max Weber, including competition and the need for efficiency, institutional similarity is due to the structuration of organizational fields, a process caused largely by the state and the professions, which are the great rationalizers of the late 20th century. In highly structured organizational fields, rational efforts of individuals aggregately lead to structural, cultural, and output homogeneity. Homogenization is best captured by the concept of isomorphism, the process whereby one element in a population resembles others that confront the same environmental conditions. The two types of isomorphism are competitive and institutional. Three processes lead to organizational similarity: (1) coercive isomorphism stemming from political influence and the problem of legitimacy; (2) mimetic isomorphism resulting from uniform responses to uncertainty; and (3) normative isomorphism associated with professionalism. While these isomorphic processes improve organizational transactions, they do not necessarily increase internal efficiency. Twelve hypotheses are offered for further research about which organizational fields will be most homogenous. These hypotheses relate the impact of resource centralization and dependency, goal ambiguity and technical uncertainty, and professionalism and structuration on isomorphic change. Finally, useful implications of the study for theories of organizations and social change are offered. (TNM)

2,879 citations

Journal ArticleDOI
TL;DR: A new algorithm is introduced that attains efficiency by inferentially eliminating successor nodes in the tree search by means of a brute-force tree-search enumeration procedure and a parallel asynchronous logic-in-memory implementation of a vital part of the algorithm is described.
Abstract: Subgraph isomorphism can be determined by means of a brute-force tree-search enumeration procedure. In this paper a new algorithm is introduced that attains efficiency by inferentially eliminating successor nodes in the tree search. To assess the time actually taken by the new algorithm, subgraph isomorphism, clique detection, graph isomorphism, and directed graph isomorphism experiments have been carried out with random and with various nonrandom graphs. A parallel asynchronous logic-in-memory implementation of a vital part of the algorithm is also described, although this hardware has not actually been built. The hardware implementation would allow very rapid determination of isomorphism.

2,319 citations

Journal ArticleDOI
TL;DR: In this article, the authors define a theory of higher Chow groups CH*(X, n), n 2 0, so as to obtain isomorphisms, which is a formal consequence of the existence of a I-structure on G,(X).

700 citations

Journal ArticleDOI
TL;DR: In this paper, the authors present a generalisation of K-theory to non-compact spaces, namely equivariant Ktheory on G-spaces, which is a generalization of the notion of vector-bundles.
Abstract: The purpose of this thesis is to present a fairly complete account of equivariant K-theory on compact spaces. Equivariant K-theory is a generalisation of K-theory, a rather well-known cohomology theory arising from consideration of the vector-bundles on a space. Equivariant K-theory, or KG-theory, is defined not on a space but on G-spaces, i.e. pairs (X,α), where X is a space and α is an action of a fixed group G on X, and it arises from consideration of G-vector-bundles on X, i.e. vector-bundles on whose total space G acts in a suitable way (of 3.1). In this thesis G will always be a compact group. But KG-theory does not appear in the first three chapters, which are introductory. Chapter 1 consists of preliminary discussions of little relevance to the sequel, but which permit me to make a few propositions in the later chapters shorter or more elegant. It was intended to be amusing, and the reader may prefer to omit it. Chapter 2 is devoted to the representation-theory of compact groups. When X is a point a G-vector-bundle on X is just a representation-module for G, so the representation-ring, or character-ring, R(G) plays a fundamental role in KG-theory. In chapter 2 I investigate its algebraic structure, and in particular when G is a compact Lie group I determine completely its prime ideals. To do this I have to discuss first the space of conjugacy-classes of a compact Lie group, and outline an induced-representation construction for obtaining finite-dimensional modules for G from modules for suitable subgroups not of finite index. Chapter 3 is a rather full collection of technical results concerning G-vector-bundles: they are all essentially well-known, but have not been stated in the equivariant case. Chapter 4 presents basic equivariant K-theory. I show that it can be defined in three ways: by G-vector-bundles, by complexes of G-vector-bundles, and by Fredholm complexes of infinite-dimensional G-vector-bundles. This chapter also treats the continuity of KG with respect to inverse limits of G-spaces, the Thorn homomorphism for a G-vector-bundle and the periodicity-isomorphism, and the question of extending KG to non-compact spaces. In chapter 5 I obtain for KG(X) a filtration and spectral sequence generalising those of [6], but without dissecting the space X. My method is based on a Cech approach: for each open covering of X I construct an auxiliary space homotopy-equivalent to X which has the natural filtration that X lacks. Also in chapter 5 I prove the localisation-theorem (5.3), which, together with the theory of chapter 6, is one of the most important tools in applied KG-theory. KG(X) is a module over the character-ring R(G), so one can localise it at the prime ideals of R(G), which I have determined in 2.5. The simplest and most important case of the localisation-theorem states that, if β is the prime ideal of characters of G vanishing at a conjugacy-class γ, and if Xγ is the part of X where elements in γ have fixed-points, then the natural restriction-map KG(X) r KG(Xγ) induces an isomorphism when localised at β. In chapter 6 I show how to associate to certain maps f : X r Y of (G-spaces a homomorphism f! : KG(X) r KG(Y). It is the analogue of the Gysin homomorphism in ordinary cohomology-theory; but it can also be regarded as a generalisation of the induced-representation construction of 2.4. In the important special case when f is a fibration whose fibre is a rational algebraic variety I prove that f! is left-inverse to the natural map f! : KG(Y) r KG(X); and I apply that to obtain the general Thom isomorphism theorem. Finally in chapter 7 I prove the theorem towards which my thesis was originally directed. Just as a G-module defines a vector-bundle on the classifying-space BG for G (of [1]), so a G~vector-bundle on X defines a vector-bundle on the space XG fibred over BG with fibre X. Thus one gets a homomorphism α : KG(X) r K(XG). I prove that if KG(X) and K(XG) are given suitable topologies then in certain circumstances K(XG)is complete and α induces an isomorphism of the completion of KG(X) with K(XG). This generalises the theorem of Atiyah-Hirsebruch that R(G)^ ≅ K(BG).

625 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202214
2021518
2020522
2019492
2018487
2017444