Sharp upper bounds on the maximum $M$-eigenvalue of fourth-order partially symmetric nonnegative tensors

<p style='text-indent:20px;'><inline-formula><tex-math id="M1">\begin{document}$ M $\end{document}</tex-math></inline-formula>-eigenvalues of partially symmetric nonnegative tensors play important roles in the nonlinear elastic material analysis and the entanglement problem of quantum physics. In this paper, we establish two upper bounds for the maximum <inline-formula><tex-math id="M2">\begin{document}$ M $\end{document}</tex-math></inline-formula>-eigenvalue of partially symmetric nonnegative tensors, which improve some existing results. Numerical examples are proposed to verify the efficiency of the obtained results.</p>

Properties and calculation for C-eigenvalues of a piezoelectric-type tensor

<p style='text-indent:20px;'>This paper mainly considers the <i>C</i>-eigenvalues of a piezoelectric-type tensor. For this, we first discuss its relationship with <inline-formula><tex-math id="M1">\begin{document}$ l^{k, s} $\end{document}</tex-math></inline-formula>-singular values of a partially symmetric rectangular tensor, and then present three types of <i>C</i>-eigenvalue inclusion intervals which can be used to locate all <i>C</i>-eigenvalues of a piezoelectric-type tensor and can provide an upper and a lower bound for the largest <i>C</i>-eigenvalue of a piezoelectric-type tensor. Finally, we present an alternative method to compute all <i>C</i>-eigenpairs of a piezoelectric-type tensor.</p>

A Flower-Shape Geometry and Nonlinear Problems on Strip-Like Domains

In the present paper, we show how to define suitable subgroups of the orthogonal group $${O}(d-m)$$ O ( d - m ) related to the unbounded part of a strip-like domain $$\omega \times {\mathbb {R}}^{d-m}$$ ω × R d - m with $$d\ge m+2$$ d ≥ m + 2 , in order to get “mutually disjoint” nontrivial subspaces of partially symmetric functions of $$H^1_0(\omega \times {\mathbb {R}}^{d-m})$$ H 0 1 ( ω × R d - m ) which are compactly embedded in the associated Lebesgue spaces. As an application of the introduced geometrical structure, we prove (existence and) multiplicity results for semilinear elliptic problems set in a strip-like domain, in the presence of a nonlinearity which either satisfies the classical Ambrosetti–Rabinowitz condition or has a sublinear growth at infinity. The main theorems of this paper may be seen as an extension of existence and multiplicity results, already appeared in the literature, for nonlinear problems set in the entire space $${\mathbb {R}}^d$$ R d , as for instance, the ones due to Bartsch and Willem. The techniques used here are new.

On the Non-Uniqueness of the Sets Computing a Partially Symmetric Rank at Most Three

We describe all partially symmetric tensors which have rank two in more than one way and gives many examples, perhaps all, for rank three partially symmetric tensors.

An SDP Method for Copositivity of Partially Symmetric Tensors

In this paper, we consider the problem of detecting the copositivity of partially symmetric rectangular tensors. We first propose a semidefinite relaxation algorithm for detecting the copositivity of partially symmetric rectangular tensors. Then, the convergence of the proposed algorithm is given, and it shows that we can always catch the copositivity of given partially symmetric tensors. Several preliminary numerical results confirm our theoretical findings.

M-Eigenvalues-Based Sufficient Conditions for the Positive Definiteness of Fourth-Order 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.