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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.


Papers
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Journal ArticleDOI
TL;DR: In this article, Chen and Garay showed that pointwise semi-slant submanifolds of Kaehler manifolds have warped product warping functions, and established an inequality for the squared norm of the second fundamental form in terms of the warping function.
Abstract: It is known that there exist no warped product semi-slant submanifolds in Kaehler manifolds \\cite{Sahin}. Recently, Chen and Garay studied pointwise-slant submanifolds of almost Hermitian manifolds in \\cite{CG} and obtained many new results for such submanifolds. In this paper, we first introduce pointwise semi-slant submanifolds of Kaehler manifolds and then we show that there exists non-trivial warped product pointwise semi-slant submanifolds of Kaehler manifold by giving an example, contrary to the semi-slant case. We present a characterization theorem and establish an inequality for the squared norm of the second fundamental form in terms of the warping function for such warped product submanifolds in Kaehler manifolds. The equality case is also considered.

66 citations

Patent
22 May 2002
TL;DR: In this paper, product information is directly collected from a product provided with a label or tag that can be electronically scanned using, for example, optical scanning technology or radio-frequency scanning technology.
Abstract: In connection with a sales transaction, product information is directly collected from a product provided with a label or tag that can be electronically scanned using, for example, optical scanning technology or radio-frequency scanning technology. The product information is electronically provided to a third party for the purpose of a post sale activity such as recording of the sale of the product or product warranty registration.

66 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that a manifold admits a Riemannian metric with harmonic forms whose product is not harmonic if and only if it is not a rational homology sphere.
Abstract: We prove that manifolds admitting a Riemannian metric for which products of harmonic forms are harmonic satisfy strong topological restrictions, some of which are akin to properties of flat manifolds. Others are more subtle and are related to symplectic geometry and Seiberg-Witten theory. We also prove that a manifold admits a metric with harmonic forms whose product is not harmonic if and only if it is not a rational homology sphere.

66 citations

Journal ArticleDOI
TL;DR: A fast iterative algorithm, called Tubal-AltMin, that is inspired by a similar approach for low-rank matrix completion is proposed that improves the recovery error by several orders of magnitude and is faster than TNN-ADMM by a factor of 5.
Abstract: The low-tubal-rank tensor model has been recently proposed for real-world multidimensional data. In this paper, we study the low-tubal-rank tensor completion problem, i.e., to recover a third-order tensor by observing a subset of its elements selected uniformly at random. We propose a fast iterative algorithm, called Tubal-AltMin , that is inspired by a similar approach for low-rank matrix completion. The unknown low-tubal-rank tensor is represented as the product of two much smaller tensors with the low-tubal-rank property being automatically incorporated, and Tubal-AltMin alternates between estimating those two tensors using tensor least squares minimization. First, we note that tensor least squares minimization is different from its matrix counterpart and nontrivial as the circular convolution operator of the low-tubal-rank tensor model is intertwined with the sub-sampling operator. Secondly, the theoretical performance guarantee is challenging since Tubal-AltMin is iterative and nonconvex. We prove that 1) Tubal-AltMin generates a best rank- $r$ approximate up to any predefined accuracy $\epsilon $ at an exponential rate, and 2) for an $n \times n \times k$ tensor $\mathcal {M}$ with tubal-rank $r \ll n$ , the required sampling complexity is $O((nr^{2}k ||\mathcal {M}||_{F}^{2} \log ^{3}~n) / \overline {\sigma }_{rk}^{2})$ , where $\overline {\sigma }_{rk}$ is the $rk$ -th singular value of the block diagonal matrix representation of $\mathcal {M}$ in the frequency domain, and the computational complexity is $O(n^{2}r^{2}k^{3} \log n \log (n/\epsilon))$ . Finally, on both synthetic data and real-world video data, evaluation results show that compared with tensor-nuclear norm minimization using alternating direction method of multipliers (TNN-ADMM), Tubal-AltMin-Simple (a simplified implementation of Tubal-AltMin) improves the recovery error by several orders of magnitude. In experiments, Tubal-AltMin-Simple is faster than TNN-ADMM by a factor of 5 for a $200 \times 200 \times 20$ tensor.

66 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