The Book of Numbers

Conway, John H.; Guy, Richard K.

doi:10.1007/978-1-4612-4072-3

Cited by 459 publications

(373 citation statements)

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“…Suppose for every natural number k, ({a} n∈N )(α)) κ∈N (n) = a · (n + 1). The theorem is a consequence of (12).…”

Section: Sequencesmentioning

confidence: 93%

“…It is defined in Section 6 quite generally as the sum Another important examples of transcendental numbers are e and π [7], [13], [6]. At the end, we show that the construction of an arbitrary Lioville constant satisfies the properties of a Liouville number [12], [1]. We show additionally, that the set of all Liouville numbers is infinite, opening the next item from Abad and Abad's list of "Top 100 Theorems".…”

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confidence: 85%

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Introduction to Liouville Numbers

Grabowski

Korniłowicz

2017

Formalized Mathematics

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Summary. The article defines Liouville numbers, originally introduced byJoseph Liouville in 1844 [17] as an example of an object which can be approximated "quite closely" by a sequence of rational numbers. A real number x is a Liouville number iff for every positive integer n, there exist integers p and q such that q > 1 andIt is easy to show that all Liouville numbers are irrational. Liouville constant, which is also defined formally, is the first transcendental (not algebraic) number. It is defined in Section 6 quite generally as the sumfor a finite sequence {a k } k∈N and b ∈ N. Based on this definition, we also introduced the so-called Liouville number as

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“…Suppose for every natural number k, ({a} n∈N )(α)) κ∈N (n) = a · (n + 1). The theorem is a consequence of (12).…”

Section: Sequencesmentioning

confidence: 93%

mentioning

confidence: 85%

Introduction to Liouville Numbers

Grabowski

Korniłowicz

2017

Formalized Mathematics

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“…This methodology will be used in Section 4 to study Turing machines and to obtain some more accurate results with respect to those obtainable by using the traditional framework [5,43]. In order to start, let us remind that numerous trials have been done during the centuries to evolve existing numeral systems in such a way that numerals representing infinite and infinitesimal numbers could be included in them (see [3,4,6,18,19,25,29,46]). Since new numeral systems appear very rarely, in each concrete historical period their significance for Mathematics is very often underestimated (especially by pure mathematicians).…”

Section: The Grossone Methodsologymentioning

confidence: 99%

Single-tape and multi-tape Turing machines through the lens of the Grossone methodology

Sergeyev

Garro

2013

J Supercomput

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The paper investigates how the mathematical languages used to describe and to observe automatic computations influence the accuracy of the obtained results. In particular, we focus our attention on Single and Multi-tape Turing machines which are described and observed through the lens of a new mathematical language which is strongly based on three methodological ideas borrowed from Physics and applied to Mathematics, namely: the distinction between the object (we speak here about a mathematical object) of an observation and the instrument used for this observation; interrelations holding between the object and the tool used for the observation; the accuracy of the observation determined by the tool. Results of the observation executed by the traditional and new languages are compared and discussed.

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“…Here, P is called the partition function, which outputs the number of different ways one can write an integer number as a sum of positive integers [58]. In this case, a modified version is proposed of the partition function, P(n, N), which is the number of different ways one can write n as a sum of only up to N positive integers.…”

Section: (B) Optimization Search Algorithmmentioning

confidence: 99%

Shelf life modelling for first-expired-first-out warehouse management

Hertog

Uysal

Carthy

et al. 2014

Phil. Trans. R. Soc. A.

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In the supply chain of perishable food products, large losses are incurred between farm and fork. Given the limited land resources and an ever-growing population, the food supply chain is faced with the challenge of increasing its handling efficiency and minimizing post-harvest food losses. Huge value can be added by optimizing warehouse management systems, taking into account the estimated remaining shelf life of the product, and matching it to the requirements of the subsequent part of the handling chain. This contribution focuses on how model approaches estimating quality changes and remaining shelf life can be combined in optimizing first-expired-first-out cold chain management strategies for perishable products. To this end, shelf-life-related performance indicators are used to introduce remaining shelf life and product quality in the cost function when optimizing the supply chain. A combinatorial exhaustive-search algorithm is shown to be feasible as the complexity of the optimization problem is sufficiently low for the size and properties of a typical commercial cold chain. The estimated shelf life distances for a particular batch can thus be taken as a guide to optimize logistics.

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The Book of Numbers

Cited by 459 publications

References 0 publications

Introduction to Liouville Numbers

Introduction to Liouville Numbers

Single-tape and multi-tape Turing machines through the lens of the Grossone methodology

Shelf life modelling for first-expired-first-out warehouse management

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