We show that for @math and for measurable subsets of the unit square with Lebesgue measure @math there exist bi-Lipschitz maps with bounded Lipschitz constant (uniformly over all such sets) which are identity on the boundary and increases the Lebesgue measure of the set to at least @math .
The question considered in this paper has connections to several different problems in analysis, an interested reader is referred to @cite_5 and the references therein for more details.
@cite_5
2082729696 2115083313 2100452521 1551374365
This paper reports on a novel technique for literature indexing and searching in a mechanized library system. The notion of relevance is taken as the key concept in the theory of information retrieval and a comparative concept of relevance is explicated in terms of the theory of probability. The resulting technique called “Probabilistic Indexing,” allows a computing machine, given a request for information, to make a statistical inference and derive a number (called the “relevance number”) for each document, which is a measure of the probability that the document will satisfy the given request. The result of a search is an ordered list of those documents which satisfy the request ranked according to their probable relevance. The paper goes on to show that whereas in a conventional library system the cross-referencing (“see” and “see also”) is based solely on the “semantical closeness” between index terms, statistical measures of closeness between index terms can be defined and computed. Thus, given an arbitrary request consisting of one (or many) index term(s), a machine can elaborate on it to increase the probability of selecting relevant documents that would not otherwise have been selected. Finally, the paper suggests an interpretation of the whole library problem as one where the request is considered as a clue on the basis of which the library system makes a concatenated statistical inference in order to provide as an output an ordered list of those documents which most probably satisfy the information needs of the user. Information extraction in specialized texts raises different problems related to the kind of searched information. In this paper, we are interested in relation identification between some concepts in medical reports, task that was evaluated in the i2b2 2010 challenge. As relations are expressed in natural language with a great variety of forms, we proceeded to sentence analysis by extracting features that enable all together to identify a relation and we modeled this task as a multi-class classification based on an SVM, each type of relation representing a class. We will present the selection of the features used by our system and an error analysis. This approach allowed us to obtain an F-measure of 0.70, classifying the system among the best systems. Link analysis typically focuses on a single type of connection, e.g., two journal papers are linked because they are written by the same author. However, often we want to analyze data that has multiple linkages between objects, e.g., two papers may have the same keywords and one may cite the other. The goal of this paper is to show that multilinear algebra provides a tool for multilink analysis. We analyze five years of publication data from journals published by the Society for Industrial and Applied Mathematics. We explore how papers can be grouped in the context of multiple link types using a tensor to represent all the links between them. A PARAFAC decomposition on the resulting tensor yields information similar to the SVD decomposition of a standard adjacency matrix. We show how the PARAFAC decomposition can be used to understand the structure of the document space and define paper-paper similarities based on multiple linkages. Examples are presented where the decomposed tensor data is used to find papers similar to a body of work (e.g., related by topic or similar to a particular author's papers), find related authors using linkages other than explicit co-authorship or citations, distinguish between papers written bymore » different authors with the same name, and predict the journal in which a paper was published.« less We consider the problem of answering queries from databases that may be incomplete. A database is incomplete if some tuples may be missing from some relations, and only a part of each relation is known to be complete. This problem arises in several contexts. For example, systems that provide access to multiple heterogeneous information sources often encounter incomplete sources. The question we address is to determine whether the answer to a specific given query is complete even when the database is incomplete. We present a novel sound and complete algorithm for the answer-completeness problem by relating it to the problem of independence of queries from updates. We also show an important case of the independence problem (and therefore ofthe answer-completeness problem) that can be decided in polynomial time, whereas the best known algorithm for this case is exponential. This case involves updates that are described using a conjunction of comparison predicates. We also describe an algorithm that determines whether the answer to the query is complete in the current state of the database. Finally, we show that our ‘treatment extends naturally to partiallyincorrect databases. Permission to copy without fee all or part of this material is granted provided that the copies aTe not made OT distributed for direct commercial advantage, the VLDB copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Very Large Data Base Endowment. To copy otherwise, OT to republish, requirea a fee and or special permission from the Endowment. Proceedings of the 22nd VLDB Conference Mumbai(Bombay), India, 1996