scholarly journals Integrity basis for a second-order and a fourth-order tensor

1982 ◽  
Vol 5 (1) ◽  
pp. 87-96 ◽  
Author(s):  
Josef Betten
Keyword(s):  
Fourth Order ◽  
Second Order ◽  
Tensor Function ◽  
Order Tensor ◽  
Isotropic Tensor ◽  
Anisotropic Properties ◽  
Isotropic Tensor Function ◽  
Fourth Order Tensor

In this paper a scalar-valued isotropic tensor function is considered, the variables of which are constitutive tensors of orders two and four, for instance, characterizing the anisotropic properties of a material. Therefore, the system of irreducible invariants of a fourth-order tensor is constructed. Furthermore, the joint or simultaneous invariants of a second-order and a fourth-order tensor are found. In a similar way one can construct an integrity basis for a tensor of order greater than four, as shown in the paper, for instance, for a tensor of order six.

scholarly journals Variance Measures for Symmetric Positive (Semi-) Definite Tensors in Two Dimensions

2021 ◽  
pp. 3-22
Author(s):  
Magnus Herberthson ◽  
Evren Özarslan ◽  
Carl-Fredrik Westin
Keyword(s):  
Tensor Product ◽  
Fourth Order ◽  
Equivalence Problem ◽  
Elasticity Tensor ◽  
Two Dimensions ◽  
Second Order ◽  
Two Dimensional ◽  
Order Tensor ◽  
Symmetry Properties ◽  
Fourth Order Tensor

AbstractCalculating the variance of a family of tensors, each represented by a symmetric positive semi-definite second order tensor/matrix, involves the formation of a fourth order tensor $$R_{abcd}$$ R abcd . To form this tensor, the tensor product of each second order tensor with itself is formed, and these products are then summed, giving the tensor $$R_{abcd}$$ R abcd the same symmetry properties as the elasticity tensor in continuum mechanics. This tensor has been studied with respect to many properties: representations, invariants, decomposition, the equivalence problem et cetera. In this paper we focus on the two-dimensional case where we give a set of invariants which ensures equivalence of two such fourth order tensors $$R_{abcd}$$ R abcd and $$\widetilde{R}_{abcd}$$ R ~ abcd . In terms of components, such an equivalence means that components $$R_{ijkl}$$ R ijkl of the first tensor will transform into the components $$\widetilde{R}_{ijkl}$$ R ~ ijkl of the second tensor for some change of the coordinate system.

scholarly journals A Unified Fourth-Order Tensor-Based Smart Community System

Sensors ◽  
2020 ◽  
Vol 20 (21) ◽  
pp. 5990
Author(s):  
Chang Liu ◽  
Huaiyu Wu ◽  
Junyuan Wang ◽  
Mingkai Wang
Keyword(s):  
Human Activity ◽  
Fourth Order ◽  
Intelligent Systems ◽  
High Energy ◽  
Tensor Model ◽  
Order Tensor ◽  
Community System ◽  
Smart Community ◽  
Smart Communities ◽  
Fourth Order Tensor

Empowered by the ubiquitous sensing capabilities of Internet of Things (IoT) technologies, smart communities could benefit our daily life in many aspects. Various smart community studies and practices have been conducted, especially in China thanks to the government’s support. However, most intelligent systems are designed and built individually by different manufacturers in diverging platforms with different functionalities. Therefore, multiple individual systems must be deployed in a smart community to have a set of functions, which could lead to hardware waste, high energy consumption and high deployment cost. More importantly, current smart community systems mainly focus on the technologies involved, while the effects of human activity are neglected. In this paper, a fourth-order tensor model representing object, time, location and human activity is proposed for human-centered smart communities, based on which a unified smart community system is designed. Thanks to the powerful data management abilities of a high-order tensor, multiple functions can be integrated into our system. In addition, since the tensor model embeds human activity information, complex functions could be implemented by exploring the effects of human activity. Two exemplary applications are presented to demonstrate the flexibility of the proposed unified fourth-order tensor-based smart community system.

scholarly journals Multi-view foreground segmentation via fourth order tensor learning

2013 ◽  
Vol 7 (3) ◽  
pp. 885-906 ◽  
Author(s):  
Michael K. Ng ◽  
◽  
Chi-Pan Tam ◽  
Fan Wang ◽  
Keyword(s):  
Fourth Order ◽  
Foreground Segmentation ◽  
Order Tensor ◽  
Fourth Order Tensor
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