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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01 Nov 2004
TL;DR: In this article, a finite dimensional complex Hilbert space of dimension d is associated with a finite level quantum system for i = 1, 2,...,k and k = 2.
Abstract: LetH ibe a finite dimensional complex Hilbert space of dimensiond
i associated with a finite level quantum system Ai for i = 1, 2, ...,k. A subspaceS ⊂
$${\mathcal{H}} = {\mathcal{H}}_{A_1 A_2 ...A_k } = {\mathcal{H}}_1 \otimes {\mathcal{H}}_2 \otimes \cdots \otimes {\mathcal{H}}_k $$
is said to becompletely entangled if it has no non-zero product vector of the formu
1⊗u
2 ⊗ ... ⊗u
k with ui inH
i for each i. Using the methods of elementary linear algebra and the intersection theorem for projective varieties in basic algebraic geometry we prove that
$$\mathop {max}\limits_{S \in \varepsilon } dim S = d_1 d_2 ...d_k - (d_1 + \cdots + d_k ) + k - 1$$
where e is the collection of all completely entangled subspaces. When
$${\mathcal{H}} = {\mathcal{H}}_2 $$
andk = 2 an explicit orthonormal basis of a maximal completely entangled subspace of
$${\mathcal{H}}_1 \otimes {\mathcal{H}}_2 $$
is given. We also introduce a more delicate notion of aperfectly entangled subspace for a multipartite quantum system, construct an example using the theory of stabilizer quantum codes and pose a problem.
95 citations
TL;DR: In this paper, conformal and related changes of the product metric on the product of two almost contact metric manifolds are studied, and it is shown that if one factor is Sasakian, the other is not, but locally the second factor is of the type studied by Kenmotsu.
Abstract: This paper studies conformal and related changes of the product metric on the product of two almost contact metric manifolds. It is shown that if one factor is Sasakian, the other is not, but that locally the second factor is of the type studied by Kenmotsu. The results are more general and given in terms of trans-Sasakian, a-Sasakian and s-Kenmotsu structures.
95 citations
Patent•
30 Nov 2015
TL;DR: In this paper, a method includes receiving an image of a product, obtaining content relevant to using the product, and displaying the content in an augmented reality view of the product by overlaying the content on the image on a display device.
Abstract: A method includes receiving an image of a product, obtaining content relevant to using the product, and displaying the content in an augmented reality view of the product by overlaying the content on the image of the product on a display device.
95 citations
04 Nov 2003
TL;DR: A UML profile for software product lines is proposed that includes stereotypes, tagged values, and structural constraints and it makes possible to define PL models with variabilities.
Abstract: This paper proposes a UML profile for software product lines. This profile includes stereotypes, tagged values, and structural constraints and it makes possible to define PL models with variabilities. Product derivation consists in generating product models from PL models. The derivation should preserve and ensure a set of constraints which are specified using the OCL.
95 citations
Book•
18 Dec 2012TL;DR: Decomposition into 3-balls, Ideal Polyhedra, and I-bundles and essential product disks are presented.
Abstract: 1 Introduction.- 2 Decomposition into 3-balls.- 3 Ideal Polyhedra.- 4 I-bundles and essential product disks.- 5 Guts and fibers.- 6 Recognizing essential product disks.- 7 Diagrams without non-prime arcs.- 8 Montesinos links.- 9 Applications.- 10 Discussion and questions.
95 citations