Programmable sufficient conditions for the strong ellipticity of partially symmetric tensors

M-eigenvalues of fourth-order partially symmetric tensors play important roles in the nonlinear elastic material analysis and the entanglement problem of quantum physics. In this paper, we introduce M-identity tensor and establish two M-eigenvalue inclusion intervals with n parameters for fourth-order partially symmetric tensors, which are sharper than some existing results. Numerical examples are proposed to verify the efficiency of the obtained results. As applications, we provide some checkable sufficient conditions for the positive definiteness and establish bound estimations for the M-spectral radius of fourth-order partially symmetric nonnegative tensors.

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New M-eigenvalue intervals and application to the strong ellipticity of fourth-order partially symmetric tensors

Journal of Industrial & Management Optimization ◽

10.3934/jimo.2020139 ◽

2017 ◽

Vol 13 (5) ◽

pp. 0-0

Author(s):

Haitao Che ◽

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Haibin Chen ◽

Guanglu Zhou ◽

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...

Keyword(s):

Fourth Order ◽

Strong Ellipticity ◽

Symmetric Tensors ◽

Eigenvalue Intervals ◽

Partially Symmetric

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Prescribed Schouten Tensor in Locally Conformally Flat Manifolds

Results in Mathematics ◽

10.1007/s00025-019-1086-8 ◽

2019 ◽

Vol 74 (4) ◽

Author(s):

Marcos Tulio Carvalho ◽

Mauricio Pieterzack ◽

Romildo Pina

Keyword(s):

Sufficient Conditions ◽

Conformally Flat ◽

Symmetric Tensors ◽

Conformally Flat Manifolds ◽

Dimensional Group ◽

Schouten Tensor ◽

Locally Conformally Flat ◽

Flat Manifolds ◽

Necessary And Sufficient ◽

Complete Metrics

Abstract We consider the pseudo-Euclidean space $$({\mathbb {R}}^n,g)$$(Rn,g), with $$n \ge 3$$n≥3 and $$g_{ij} = \delta _{ij} \varepsilon _{i}$$gij=δijεi, where $$\varepsilon _{i} = \pm 1$$εi=±1, with at least one positive $$\varepsilon _{i}$$εi and non-diagonal symmetric tensors $$T = \sum \nolimits _{i,j}f_{ij}(x) dx_i \otimes dx_{j} $$T=∑i,jfij(x)dxi⊗dxj. Assuming that the solutions are invariant by the action of a translation $$(n-1)$$(n-1)- dimensional group, we find the necessary and sufficient conditions for the existence of a metric $$\bar{g}$$g¯ conformal to g, such that the Schouten tensor $$\bar{g}$$g¯, is equal to T. From the obtained results, we show that for certain functions h, defined in $$\mathbb {R}^{n}$$Rn, there exist complete metrics $$\bar{g}$$g¯, conformal to the Euclidean metric g, whose curvature $$\sigma _{2}(\bar{g}) = h$$σ2(g¯)=h.

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Analytical expressions of copositivity for fourth-order symmetric tensors

Analysis and Applications ◽

10.1142/s0219530520500049 ◽

2020 ◽

pp. 1-22 ◽

Cited By ~ 1

Author(s):

Yisheng Song ◽

Liqun Qi

Keyword(s):

Lower Bound ◽

Fourth Order ◽

Scalar Potential ◽

Sufficient Conditions ◽

Scalar Fields ◽

Positive Definiteness ◽

Necessary Condition ◽

Symmetric Tensors ◽

Scalar Potentials ◽

Sufficient And Necessary Condition

In particle physics, scalar potentials have to be bounded from below in order for the physics to make sense. The precise expressions of checking lower bound of scalar potentials are essential, which is an analytical expression of checking copositivity and positive definiteness of tensors given by such scalar potentials. Because the tensors given by general scalar potential are fourth-order and symmetric, our work mainly focuses on finding precise expressions to test copositivity and positive definiteness of fourth-order tensors in this paper. First of all, an analytically sufficient and necessary condition of positive definiteness is provided for fourth-order 2-dimensional symmetric tensors. For fourth-order 3-dimensional symmetric tensors, we give two analytically sufficient conditions of (strictly) copositivity by using proof technique of reducing orders or dimensions of such a tensor. Furthermore, an analytically sufficient and necessary condition of copositivity is showed for fourth-order 2-dimensional symmetric tensors. We also give several distinctly analytically sufficient conditions of (strict) copositivity for fourth-order 2-dimensional symmetric tensors. Finally, these results may be applied to check lower bound of scalar potentials, and to present analytical vacuum stability conditions for potentials of two real scalar fields and the Higgs boson.

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Characterization of the symmetry class of an elasticity tensor using polynomial covariants

Mathematics and Mechanics of Solids ◽

10.1177/10812865211010885 ◽

2021 ◽

pp. 108128652110108

Author(s):

Marc Olive ◽

Boris Kolev ◽

Rodrigue Desmorat ◽

Boris Desmorat

Keyword(s):

Fourth Order ◽

Sufficient Conditions ◽

Cross Product ◽

Elasticity Tensor ◽

Theory Of Elasticity ◽

Symmetry Class ◽

Symmetric Tensors ◽

Minimal Set ◽

Algebraic Sets ◽

Symmetry Classes

We formulate effective necessary and sufficient conditions to identify the symmetry class of an elasticity tensor, a fourth-order tensor which is the cornerstone of the theory of elasticity and a toy model for linear constitutive laws in physics. The novelty is that these conditions are written using polynomial covariants. As a corollary, we deduce that the symmetry classes are affine algebraic sets, a result which seems to be new. Meanwhile, we have been lead to produce a minimal set of 70 generators for the covariant algebra of a fourth-order harmonic tensor and introduce an original generalized cross-product on totally symmetric tensors. Finally, using these tensorial covariants, we produce a new minimal set of 294 generators for the invariant algebra of the elasticity tensor.

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