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53,798
The fundamental properties of natural numbers
- Journal of Formalized Mathematics
, 1989
"... Summary. Some fundamental properties of addition, multiplication, order relations, exact division, the remainder, divisibility, the least common multiple, the greatest common divisor are presented. A proof of Euclid algorithm is also given. MML Identifier:NAT_1. WWW:http://mizar.org/JFM/Vol1/nat_1.h ..."
Abstract
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Cited by 688 (73 self)
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.html The articles [4], [6], [1], [2], [5], and [3] provide the notation and terminology for this paper. A natural number is an element of N. For simplicity, we use the following convention: x is a real number, k, l, m, n are natural numbers, h, i, j are natural numbers, and X is a subset of R
THE NATURAL NUMBERS
"... We are ready for the official introduction of the natural numbers. I ask you to suspend belief in the truth of the things you already knowabout the natural numbers. I insist you keep every bit of your knowledge- just stop taking things for granted. “The Natural Numbers ” is nowa name only, to be use ..."
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We are ready for the official introduction of the natural numbers. I ask you to suspend belief in the truth of the things you already knowabout the natural numbers. I insist you keep every bit of your knowledge- just stop taking things for granted. “The Natural Numbers ” is nowa name only
Basic objects in natural categories
- COGNITIVE PSYCHOLOGY
, 1976
"... Categorizations which humans make of the concrete world are not arbitrary but highly determined. In taxonomies of concrete objects, there is one level of abstraction at which the most basic category cuts are made. Basic categories are those which carry the most information, possess the highest categ ..."
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Cited by 892 (1 self)
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significant numbers of attributes in common, (b) have motor programs which are similar to one another, (c) have similar shapes, and (d) can be identified from averaged shapes of members of the class. The eight experiments of Part II explore implications of the structure of categories. Basic objects are shown
The lexical nature of syntactic ambiguity resolution
- Psychological Review
, 1994
"... Ambiguity resolution is a central problem in language comprehension. Lexical and syntactic ambiguities are standardly assumed to involve different types of knowledge representations and be resolved by different mechanisms. An alternative account is provided in which both types of ambiguity derive fr ..."
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Cited by 557 (24 self)
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from aspects of lexical representation and are resolved by the same processing mechanisms. Reinterpreting syntactic ambiguity resolution as a form of lexical ambiguity resolution obviates the need for special parsing principles to account for syntactic interpretation preferences, reconciles a number
Relations between the statistics of natural images and the response properties of cortical cells
- J. Opt. Soc. Am. A
, 1987
"... The relative efficiency of any particular image-coding scheme should be defined only in relation to the class of images that the code is likely to encounter. To understand the representation of images by the mammalian visual system, it might therefore be useful to consider the statistics of images f ..."
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Cited by 831 (18 self)
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relatively high signal-to-noise ratio and permits information to be transmitted with only a subset of the total number of cells. These results support Barlow's theory that the goal of natural vision is to represent the information in the natural environment with minimal redundancy.
Transformation-Based Error-Driven Learning and Natural Language Processing: A Case Study in Part-of-Speech Tagging
- Computational Linguistics
, 1995
"... this paper, we will describe a simple rule-based approach to automated learning of linguistic knowledge. This approach has been shown for a number of tasks to capture information in a clearer and more direct fashion without a compromise in performance. We present a detailed case study of this learni ..."
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Cited by 924 (8 self)
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this paper, we will describe a simple rule-based approach to automated learning of linguistic knowledge. This approach has been shown for a number of tasks to capture information in a clearer and more direct fashion without a compromise in performance. We present a detailed case study
1 Infinite and natural numbers
, 2002
"... The infinite numbers of the set M of finite and infinite natural numbers are defined starting from the sequence 0Φ, where 0 is the first natural number, Φ a succession of symbols S and xS is the successor of the natural number x. The concept of limit of the natural number n, when n tends to infinite ..."
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The infinite numbers of the set M of finite and infinite natural numbers are defined starting from the sequence 0Φ, where 0 is the first natural number, Φ a succession of symbols S and xS is the successor of the natural number x. The concept of limit of the natural number n, when n tends
Difficulties of the set of natural numbers
"... In this article some difficulties are deduced from the set of natural numbers. The demonstrated difficulties suggest that if the set of natural numbers exists it would conflict with the axiom of regularity. As a result, we have the conclusion that the class of natural numbers is not a set but a prop ..."
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In this article some difficulties are deduced from the set of natural numbers. The demonstrated difficulties suggest that if the set of natural numbers exists it would conflict with the axiom of regularity. As a result, we have the conclusion that the class of natural numbers is not a set but a
Weighted Voting for Replicated Data
, 1979
"... In a new algorithm for maintaining replicated data, every copy of a replicated file is assigned some number of votes. Every transaction collects a read quorum of r votes to read a file, and a write quorum of w votes to write a file, such that r+w is greater than the total number number of votes assi ..."
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Cited by 598 (0 self)
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In a new algorithm for maintaining replicated data, every copy of a replicated file is assigned some number of votes. Every transaction collects a read quorum of r votes to read a file, and a write quorum of w votes to write a file, such that r+w is greater than the total number number of votes
Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm
- Machine Learning
, 1988
"... learning Boolean functions, linear-threshold algorithms Abstract. Valiant (1984) and others have studied the problem of learning various classes of Boolean functions from examples. Here we discuss incremental learning of these functions. We consider a setting in which the learner responds to each ex ..."
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Cited by 773 (5 self)
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example according to a current hypothesis. Then the learner updates the hypothesis, if necessary, based on the correct classification of the example. One natural measure of the quality of learning in this setting is the number of mistakes the learner makes. For suitable classes of functions, learning
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