Topic
Product (mathematics)
About: Product (mathematics) is a research topic. Over the lifetime, 44382 publications have been published within this topic receiving 377809 citations.
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TL;DR: In this paper, the problem of finding sets of distinct positive rational numbers such that the product of any two is one less than a rational square was posed by Diophantus, and some sets of six such numbers are presented and the computational algorithm used to find them is described.
Abstract: A famous problem posed by Diophantus was to find sets of distinct positive rational numbers such that the product of any two is one less than a rational square. Some sets of six such numbers are presented and the computational algorithm used to find them is described. A classification of quadruples and quintuples with examples and statistics is also given.
61 citations
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26 Dec 1996
TL;DR: In this article, a product supporting and feeding system is provided in which a plurality of tracks is mounted in spaced relationship from each other and equipped with springloaded sliders adapted to move the product along the tracks.
Abstract: A product supporting and feeding system is provided in which a plurality of tracks is mounted in spaced relationship from each other and equipped with spring-loaded sliders adapted to move the product along the tracks.
61 citations
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27 Mar 2012
TL;DR: This work proposes a new approach to extract and use so-called product codes to identify products and distinguish them from similar product variations and shows that the UPC information in product offers is often error-prone and can lead to insufficient match decisions.
Abstract: Product matching is a challenging variation of entity resolution to identify representations and offers referring to the same product. Product matching is highly difficult due to the broad spectrum of products, many similar but different products, frequently missing or wrong values, and the textual nature of product titles and descriptions. We propose the use of tailored approaches for product matching based on a preprocessing of product offers to extract and clean new attributes usable for matching. In particular, we propose a new approach to extract and use so-called product codes to identify products and distinguish them from similar product variations. We evaluate the effectiveness of the proposed approaches with challenging real-life datasets with product offers from online shops. We also show that the UPC information in product offers is often error-prone and can lead to insufficient match decisions.
61 citations
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01 May 2019TL;DR: The first polynomial-time algorithm for robust regression in the list-decodable setting where an adversary can corrupt a greater than 1/2 fraction of examples was given in this paper.
Abstract: We give the first polynomial-time algorithm for robust regression in the list-decodable setting where an adversary can corrupt a greater than 1/2 fraction of examples. For any \alpha < 1, our algorithm takes as input a sample {(x_i,y_i)}_{i \leq n} of n linear equations where \alpha n of the equations satisfy y_i = \langle x_i,\ell^*\rangle +\zeta for some small noise \zeta and (1-\alpha) n of the equations are {\em arbitrarily} chosen. It outputs a list L of size O(1/\alpha) - a fixed constant - that contains an \ell that is close to \ell^*. Our algorithm succeeds whenever the inliers are chosen from a certifiably anti-concentrated distribution D. In particular, this gives a (d/\alpha)^{O(1/\alpha^8)} time algorithm to find a O(1/\alpha) size list when the inlier distribution is a standard Gaussian. For discrete product distributions that are anti-concentrated only in regular directions, we give an algorithm that achieves similar guarantee under the promise that \ell^* has all coordinates of the same magnitude. To complement our result, we prove that the anti-concentration assumption on the inliers is information-theoretically necessary. To solve the problem we introduce a new framework for list-decodable learning that strengthens the ``identifiability to algorithms'' paradigm based on the sum-of-squares method.
61 citations
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TL;DR: In this paper, the authors introduce the concept of the inward product of two matrices, which is defined as an operation of internal composition in matrix spaces and present a wide prospect of applications, as well as evident connections with other mathematical structures like tagged sets and vector semispaces, useful in theoretical chemistry applications.
Abstract: A special matrix product, the inward product, of two matrices is defined as an operation of internal composition in matrix spaces. A generalisation and an extension of the inward product make it a very flexible algorithmic tool. This product, known from a long time, also called Hadamard or Schur product, presents characteristic properties, which made such an operation interesting enough due to its potential use in many fields, such as quantum chemistry, where matrix manipulation is a common trait. Here, besides the introduction to the main features of inward product, it is shown that there is a wide prospect of applications, as well as evident connections with other mathematical structures like tagged sets and vector semispaces, useful in theoretical chemistry applications. Approximate least-squares solutions forced to belong to a vector semispace are also discussed. This constrained least-squares procedure furnishes a new theoretical basis to the foundation of QSAR or QSPR in the framework of quantum similarity and other approaches.
61 citations