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

Calculating 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.

A Unified Fourth-Order Tensor-Based Smart Community System

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.

The memory tesseract: Mathematical equivalence between composite and separate storage memory models

Computational memory models can explain the behaviour of human memory in diverse experimental paradigms. But research has produced a profusion of competing models, and, as different models focus on different phenomena, there is no best model. However, by examining commonalities among models, we can move towards theoretical unification. Computational memory models can be grouped into composite and separate storage models. We prove that MINERVA 2, a separate storage model of long-term memory, is mathematically equivalent to composite storage memory implemented as a fourth order tensor, and approximately equivalent to a fourth-order tensor compressed into a holographic vector. Building of these demonstrations, we show that MINERVA 2 and related separate storage models can be implemented in neurons. Our work clarifies the relationship between composite and separate storage models of memory, and thereby moves memory models a step closer to theoretical unification.

On inelasticity of damaged quasi rate independent anisotropic materials

This paper deals with a body that has a random 3D-distribution of two phase inclusions: spheroidal mutually parallel voids, and differently oriented reinforcing parallel stiff spheroidal short fibers. By the effective field approach the effective stiffness fourth order tensor is formulated and found numerically. Simultaneous and sequential embeddings of inclusions are compared. Damage evolution is described by a modified Vakulenko approach to the endochronic thermodynamics. A brief account of the problem of effective elastic symmetry is considered. The results of the theory are applied to the damage-elasto-viscoplastic strain of a reactor stainless steel AISI 316H.