setting, difficult to come by. A good solution probably should not involve the notion of homogeneity (see Supplement 5) since, in applications, the spaces constructed which are to be checked are obviously manifolds at some points, so that recognizing homogeneity is precisely the difficulty. Finally, a satisfactory solution should allow one to solve problems of independent interest. Before spring of this year (1977) no conjectured characterization of topological manifolds seemed to have a clear cut advantage over any other. But the situation has changed rapidly so that we can make the following conjecture with some confidence. 1.3. CONJECTURE. A topological «-manifold may be characterized as a generalized «-manifold satisfying a minimal amount of general position. (Definitions follow.) Prerequisites for understanding the conjecture in particular and the paper in general include a knowledge of basic homology theory (as presented, for example, in [33]) and basic PL topology (as presented, for example, in [57]). A good introduction to the particular point of view that we shall pursue (concerning tameness and wildness) appears in [19]. Nevertheless, even without those prerequisites the reader will probably understand and enjoy some of the historical material in §§2 and 3 and the discussion of Antoine's necklace in §§5 and 6. 1.4. DEFINITION. A generalized «-manifold M is a Euclidean neighborhood retract (ENR) (=>= retract of an open subset of some Euclidean space E) with the local homology groups at each point of Euclidean «-space E: H+(M, M ~{x};Z) s H*(E9 E n (0};Z) (for each x E M). The probable appropriate general position condition for « > 5 is the disjoint disk property. 1.4'. DEFINITION. A space M satisfies the disjoint disk property if arbitrary maps ƒ, g: B-+ M from the 2-dimensional disk B into M can be approximated by maps/ , g': B -» M with ƒ'(B) n g\\B) = 0 . The conjecture, as completed by Definitions 1.4 and 1.4', was proved during the spring of this year (1977) for a large class of generalized manifolds by J. W. Cannon [22] and Cannon, J. L. Bryant, and R. C. Lacher [23] (see Supplement 4). The fertile source of generalized manifolds supplied by cell-like upper semicontinuous decompositions of manifolds was then, for « > 5, completely mastered by R. D. Edwards [36] (see Supplement 4); his result confirmed the conjecture for all cell-like decompositions of «-manifolds, « > 5. An infinite dimensional analogue of the conjecture for Q manifolds was proved early in the year by H. Torunczyk [65] (see Supplement 3). An easy consequence of the work, and one of its great motivations, is the famous double suspension conjecture: 1.5. THEOREM. The double suspension S//\" of any homology «-sphere is homeomorphic with the (« + 2)-sphere S*. (A homology «-sphere H is an n-manifold satisfying H^(H;Z) s H+(S;Z); the kth suspension of a space is the join of that space with the (k — 1)-sphere S~; see Definition 1.6 for the definition of a sphere.)

/pdf/the-recognition-problem-what-is-a-topological-manifold-1juzfzmflw.pdf

The recognition problem: What is a topological manifold?

Current best-match ranking (BMR) systems perform well but cannot handle word mismatch between a query and a document. The best known alternative ranking method, hierarchical clustering-based ranking (HCR), seems to be more robust than BMR with respect to this problem, but it is hampered by theoretical and practical limitations. We present an approach to document ranking that explicitly addresses the word mismatch problem by exploiting interdocument similarity information in a novel way. Document ranking is seen as a query-document transformation driven by a conceptual representation of the whole document collection, into which the query is merged. Our approach is based on the theory of concept (or Galois) lattices, which, we argue, provides a powerful, well-founded, and computationally-tractable framework to model the space in which documents and query are represented and to compute such a transformation. We compared information retrieval using concept lattice-based ranking (CLR) to BMR and HCR. The results showed that HCR was outperformed by CLR as well as by BMR, and suggested that, of the two best methods, BMR achieved better performance than CLR on the whole document set, whereas CLR compared more favorably when only the first retrieved documents were used for evaluation. We also evaluated the three methods' specific ability to rank documents that did not match the query, in which case the superiority of CLR over BMR and HCR (and that of HCR over BMR) was apparent.

/pdf/order-theoretical-ranking-1d3nav7321.pdf

Order-theoretical ranking

Publisher Summary This chapter provides an overview of strange attractors in fluid dynamics. Strange attractors are relevant to the transition from regular to chaotic flow in some parametric regimes, whether or not the chaotic flow is called turbulent, weakly turbulent, or simply aperiodic. The term strange attractor for chaotic solutions of dissipative systems with regular forcing was introduced by Ruelle and Takens (1971), who discovered the phenomenon and suggested its relevance to turbulence. Ruelle and Takens conjectured that a strange attractor would appear at the third bifurcation in the Landau sequence so that at most two incommensurate frequencies could appear prior to the transition to chaotic flow. This scenario, with some variation in the number of incommensurate frequencies that appear prior to chaos, has been observed in both Rayleigh-Benard convection and circular Couette flow. The chapter describes concepts related to Lorenz's convection model. It provides details about the Howard–Malkus–Welander convection model. Mathematical routes to turbulence are also elaborated in the chapter.

Strange attractors in fluid dynamics

We show that if the similarity function of a retrieval system leads to a (pseudo-) metric, the retrieval, the similarity and the Everett-Cater metric topology coincide and are generally different from the discrete topology. This is the case if we represent documents by lists and use the Jaccard similarity measure. The corresponding metric is then the Marczewski-Steinhaus metric. We further study the special case of a one-element query space consisting of a single-item query.

Jaccard similarity leads to the Marczewski-Steinhaus topology for information retrieval

A unified mathematical definition for the classical (vector and probabilistic) models of information retrieval is given. Also, a mathematical structure (Diophantine set) behind relevance feedback is identified.

A unified mathematical definition of classical information retrieval

Document-retrieval systems can be classified on the basis of the topology each system induces on its set of possible documents. The induced topology measures closeness among documents, with two documents considered close if a retrieval which includes one document is likely to also include the other. The induced topology may be distinct from the usual topology which the set of documents inherits from its indexing system. Under a reasonably general definition of document-retrieval systems, the existence of a query which returns one of two preselected documents, but not the other, is a necessary and sufficient condition for the document-retrieval system to induce not only a topology, but also a metric on its set of documents. the topological structure and induced metric of the following document-retrieval systems are analyzed: the vector-space model, the fuzzy-set model, the Extended-Boolean model, a probabilistic model, and the TIRS model

Topology of document retrieval systems

Publisher Summary This chapter presents shrinking countable decompositions of E 3 into points and tame cells. It presents theorem that states that if G is an upper semi-continuous decomposition of E 2 such that no decomposition element seperates E 2 , then, E 2 / G = E 2 . If G is an upper semi-continuous decomposition of E 3 such that each point of E 3 / G has arbitrarily small neighborhoods U such that FR U is a 2-sphere missing π (G * ), then E 3 / G = E 3 . A decomposition H of E 3 is called a closed sub-decomposition of G if the nondegenerate elements of H form a closed subset of the nondegenerate elements of G. It follows immediately that H is an upper semi-continuous.

Shrinking countable decompositions of e3 into points and tame cells

During the past two decades a new perspective known as nonlinear dynamics has caught the imagination of researchers in a number of disciplines of the physical sciences. Recently, some of the assumptions of this theory have begun to be applied to the social sciences, including economics, political science and public administration. We believe that the way the current presidential nomination system aggregates individual preferences into a social choice exhibits behavior consistent with nonlinear dynamics. We argue that the nomination system is characterized by two nonlinear processes: sensitive dependence and nonmonotonic behavior.



In this paper, we present a nonlinear dynamical perspective for understanding presidential nomination processes. In particular, we address the question of whether very modest changes, in parameters can produce major changes in the outcome of the campaign. We suggest that resource acquisition, resource allocations, vote outcomes, and media verdicts interact with each other in a complex and dynamic fashion that has dramatic effects upon the allocation of delegates, and hence, the final nomination outcome.

An application of nonlinear dynamics to the presidential nomination process

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Daniel M. Everett

Author Tools

Chat about Author

Papers

Topology of document retrieval systems

Shrinking countable decompositions of e3 into points and tame cells

An application of nonlinear dynamics to the presidential nomination process