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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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Patent
08 Oct 2010
TL;DR: In this paper, the authors propose a method to generate data describing a product from a pre-existing product database using a taxonomy and an ontology, which includes the steps of comparing a digitized query image of the product to digitized pre-existing product images in a product database, and then extracting the node in the taxonomy, the attribute data, or the attribute value data linked to the previous image retrieved earlier.
Abstract: Disclosed is a method to generate data describing a product. The method includes the steps of comparing a digitized query image of the product to digitized pre-existing product images in a pre-existing product database. The pre-existing product database is organized using a taxonomy and an ontology. The pre-existing product images are linked to a corresponding node in the taxonomy and are also linked to attribute data and attribute value data in the ontology. At least one pre-existing product image is then retrieved that most closely matches the query image based on at least one matching criterion selected in whole or in part by a user. From the pre-existing product database is extracted the node in the taxonomy, the attribute data, or the attribute value data linked to the pre-existing product image retrieved earlier. In this fashion, product data relevant to the item depicted in the query image can be generated automatically from existing product data.

70 citations

Journal ArticleDOI
TL;DR: This paper describes a series of embedding theorems, which show that any distributive category has a full faithful embedding into a recognizable distributivecategory, and which can be "solidified" faithfully to produce an extensive distributive categories.
Abstract: Distributive category theory is the study of categories with two monoidal structures, one of which “distributes” over the other in some manner When these are the product and coproduct, this distribution is taken to be the lawwhich asserts that the obvious canonical map has an inverse A distributive category is here taken to mean a category with finite products and binary coproducts such that this law is satisfiedIn any distributive category the coproduct of the final object with itself, 1 + 1, forms a boolean algebra Thus, maps into 1 + 1 provide a boolean logic: if each such map recognizes a unique subobject, the category is a recognizable distributive category If, furthermore, the category is such that these recognizers classify detachable subobjects (coproduct embeddings), it is an extensive distributive categoryExtensive distributive categories can be approached in various ways For example, recognizable distributive categories, in which coproducts are disjoint or all preinitials are isomorphic, are extensive Also, a category X having finite products and binary coproducts satisfying the slice equation (due to Schanuel and Lawvere) is extensive This paper describes a series of embedding theorems Any distributive category has a full faithful embedding into a recognizable distributive category Any recognizable distributive category can be "solidified" faithfully to produce an extensive distributive category Any extensive distributive category can be embedded into a toposA peculiar source of extensive distributive categories is the coproduct completion of categories with familial finite products In particular, this includes the coproduct completion of cartesian categories, which is serendipitously, therefore, also the distributive completion Familial distributive categories can be characterized as distributive categories for which every object has a finite decomposition into indecomposables

70 citations

Journal ArticleDOI
TL;DR: Describing circuits for computation of a large class of algebraic functions on polynomials, power series, and integers, for which, it has been a long standing open problem to compute in depth less than $\Omega (\log n)^2 $.
Abstract: This paper describes circuits for computation of a large class of algebraic functions on polynomials, power series, and integers, for which, it has been a long standing open problem to compute in depth less than $\Omega (\log n)^2 $.Algebraic circuits assume unit cost for elemental addition and multiplication. This paper describes $O(\log n)$ depth algebraic circuits which given as input the coefficients of n degree polynomials (over an appropriate ring), compute the product of $n^{O(1)} $ polynomials, the symmetric functions, as well as division and interpolation of real polynomials. Also described are $O(\log n)$ depth algebraic circuits which are given as input the first n coefficients of a power series (over an appropriate ring) compute the product of $n^{O(1)} $ power series, as well as division, reciprocal and reversion of real power series.Furthermore this paper describes boolean circuits of depth $O(\log n(\log \log n))$ which, given n-bit binary numbers, compute the product of n numbers and integ...

70 citations

Proceedings ArticleDOI
04 May 1998
TL;DR: The ISPR (Information System for Product Recovery) is proposed, whose key component is an electronic device, the so-called EDL (Electronic Data Log), which is integrated in a product to record and store data strongly correlated with the degradation of components during the use stage of a product.
Abstract: Information on product properties and the history of product use are essential for higher levels of product recovery like the reuse of components or remanufacturing. These product recovery options are economically more attractive than materials recycling. In this paper, the ISPR (Information System for Product Recovery) is proposed. Its key component is an electronic device, the so-called EDL (Electronic Data Log), which is integrated in a product to record and store data strongly correlated with the degradation of components during the use stage of a product. The data recorded and processed during the use stage are retrieved and analyzed by the ISPR when the product is returned. The objectives for the development of the EDL, its implementation and its economical efficiency are discussed in detail.

70 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20251
20244
20239,015
202219,171
20212,013
20202,263