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The ordinal numbers
 Journal of Formalized Mathematics
, 1989
"... Summary. We present the choice function rule in the beginning of the article. In the main part of the article we formalize the base of cardinal theory. In the first section we introduce the concept of cardinal numbers and order relations between them. We present here CantorBernstein theorem and oth ..."
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Cited by 731 (68 self)
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Summary. We present the choice function rule in the beginning of the article. In the main part of the article we formalize the base of cardinal theory. In the first section we introduce the concept of cardinal numbers and order relations between them. We present here CantorBernstein theorem
Number Theory
"... This set of notes on number theory was originally written in 1995 for students at the IMO level. It covers the basic background material that an IMO student should be familiar with. This text is meant to be a reference, and not a replacement but rather a supplement to a number theory textbook; sever ..."
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Cited by 3 (0 self)
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This set of notes on number theory was originally written in 1995 for students at the IMO level. It covers the basic background material that an IMO student should be familiar with. This text is meant to be a reference, and not a replacement but rather a supplement to a number theory textbook
Determining the Number of Factors in Approximate Factor Models
, 2000
"... In this paper we develop some statistical theory for factor models of large dimensions. The focus is the determination of the number of factors, which is an unresolved issue in the rapidly growing literature on multifactor models. We propose a panel Cp criterion and show that the number of factors c ..."
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Cited by 561 (30 self)
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In this paper we develop some statistical theory for factor models of large dimensions. The focus is the determination of the number of factors, which is an unresolved issue in the rapidly growing literature on multifactor models. We propose a panel Cp criterion and show that the number of factors
The Theory of Economic Regulation
 The Bell Journal of Economics and Management Science
, 1971
"... The potential usex of public resources and powers to improve the economic stuius of economic groups (such as industries and occupations) are analyzed to provide a scheme of the demand for regulation. The characteristics of the political process which allow relatively small groups to obtain such regu ..."
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Cited by 1101 (1 self)
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such regulation is then sketched lo provide elemenls of a theory of supply of regulation. A variety of empirical evidence and illustration is also presented. W The state—the machinery and power of the state—is a potential resource or threat to every industry in the society. With its power to prohibit or compel
A Theory of the Learnable
, 1984
"... Humans appear to be able to learn new concepts without needing to be programmed explicitly in any conventional sense. In this paper we regard learning as the phenomenon of knowledge acquisition in the absence of explicit programming. We give a precise methodology for studying this phenomenon from ..."
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Cited by 1985 (15 self)
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a computational viewpoint. It consists of choosing an appropriate information gathering mechanism, the learning protocol, and exploring the class of concepts that can be learnt using it in a reasonable (polynomial) number of steps. We find that inherent algorithmic complexity appears to set serious
A theory of communicating sequential processes
, 1984
"... A mathematical model for communicating sequential processes is given, and a number of its interesting and useful properties are stated and proved. The possibilities of nondetermimsm are fully taken into account. ..."
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Cited by 4185 (17 self)
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A mathematical model for communicating sequential processes is given, and a number of its interesting and useful properties are stated and proved. The possibilities of nondetermimsm are fully taken into account.
Stochastic Perturbation Theory
, 1988
"... . In this paper classical matrix perturbation theory is approached from a probabilistic point of view. The perturbed quantity is approximated by a firstorder perturbation expansion, in which the perturbation is assumed to be random. This permits the computation of statistics estimating the variatio ..."
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Cited by 907 (36 self)
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the variation in the perturbed quantity. Up to the higherorder terms that are ignored in the expansion, these statistics tend to be more realistic than perturbation bounds obtained in terms of norms. The technique is applied to a number of problems in matrix perturbation theory, including least squares
Lightness and Retinex Theory
 JOURNAL OF THE OPTICAL SOCIETY OF AMERICA
, 1971
"... Sensations of color show a strong correlation with reflectance, even though the amount of visible light reaching the eye depends on the product of reflectance and illumination. The visual system must achieve this remarkable result by a scheme that does not measure flux. Such a scheme is described as ..."
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Cited by 497 (6 self)
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as the basis of retinex theory. This theory assumes that there are three independent cone systems, each starting with a set of receptors peaking, respectively, in the long, middle, and shortwavelength regions of the visible spectrum. Each system forms a separate image of the world in terms of lightness
Estimating the number of clusters in a dataset via the Gap statistic
, 2000
"... We propose a method (the \Gap statistic") for estimating the number of clusters (groups) in a set of data. The technique uses the output of any clustering algorithm (e.g. kmeans or hierarchical), comparing the change in within cluster dispersion to that expected under an appropriate reference ..."
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Cited by 502 (1 self)
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null distribution. Some theory is developed for the proposal and a simulation study that shows that the Gap statistic usually outperforms other methods that have been proposed in the literature. We also briey explore application of the same technique to the problem for estimating the number of linear
Results 1  10
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48,484