Tensor decomposition is an important research area with numerous applications in data mining and computational neuroscience. An important class of tensor decomposition is sum-of-squares (SOS) tensor decomposition. SOS tensor decomposition has a close connection with SOS polynomials, and SOS polynomials are very important in polynomial theory and polynomial optimization. In this paper, we give a detailed survey on recent advances of high-order SOS tensors and their applications. It first shows that several classes of symmetric structured tensors available in the literature have SOS decomposition in the even order symmetric case. Then, the SOS-rank for tensors with SOS decomposition and the SOS-width for SOS tensor cones are established. Further, a sharper explicit upper bound of the SOS-rank for tensors with bounded exponent is provided, and the exact SOS-width for the cone consists of all such tensors with SOS decomposition is identified. Some potential research directions in the future are also listed in this paper.

This paper is devoted to the study on the spectrum of Hermitizable tridiagonal matrices. As an illustration of the application of the author’s recent results on Hermitizable matrices, an explicit criterion for discrete spectrum of the matrices is presented, with a slight and technical restriction. The problem is well known, but from the author’s knowledge, it has been largely opened for quite a long time. It is important in various application, in quantum mechanics for instance. The main tool to solve the problem is the isospectral technique developed a few years ago. Two alternative constructions of the isospectral operator are presented; they are helpful in theoretical analysis and in numerical computations, respectively. Some illustrated examples are included.

We provide the H^{2}-regularity result of the solution ψ and its first- order time derivative ψ_{t} and the second-order time derivative ψ_{tt} for the complex Ginzburg-Landau equation with the Dirichlet or Neumann boundary conditions. The analysis shows that these regularity results are uniform when t tends to ∞ and 0 and are dependent of the powers of ε^{−1}.

Consider the generalized dispersive equation defined by $\{\begin{array}{cc}{\mathrm{i}\partial}_{t}u+\phi \sqrt{-\mathrm{\Delta}})u=0,& (x,t)\in {\mathbb{R}}^{n}\times \mathbb{R},\\ u(x,0)=f\left(x\right),& f\in \mathit{\varphi}\left({\mathbb{R}}^{n}\right),\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}(*)$where $\phi (\sqrt{-\mathrm{\Delta}})$ is a pseudo-differential operator with symbol $\phi (\left|\xi \right|)$. In the present paper, assuming that $\phi $ satisfies suitable growth conditions and the initial data in ${H}^{s}\left({\mathbb{R}}^{n}\right)$, we bound the Hausdorff dimension of the sets on which the pointwise convergence of solutions to the dispersive equations (*) fails. These upper bounds of Hausdorff dimension shall be obtained via the Kolmogorov-Seliverstov-Plessner method.

Suppose that g(f) are bi-parameter Littlewood-Paley square functions which were introduced by H. Martikainen. It is known that the ${L}^{\mathrm{2}}({\mathbb{R}}^{n}\times {\mathbb{R}}^{m})$ boundedness and the ${H}^{\mathrm{1}}({\mathbb{R}}^{n}\times {\mathbb{R}}^{m})-{L}^{\mathrm{1}}({\mathbb{R}}^{n}\times {\mathbb{R}}^{m})$ boundedness of g(f) have been proved by H. Martikainen and by Z. Li and Q. Xue, respectively. In this paper, we apply the vector-valued theory, the atomic decomposition of product Hardy spaces, and Journe's covering lemma to show that g(f) are bounded from ${H}^{p}({\mathbb{R}}^{n}\times {\mathbb{R}}^{m})$ to ${L}^{p}({\mathbb{R}}^{n}\times {\mathbb{R}}^{m})$ with p smaller than 1.

Let R be an arbitrary associated ring. For an integer N≥2 and a self-orthogonal subcategory $\mathcal{W}$ of R-modules, we study the notion of Cartan-Eilenberg $\mathcal{W}$N-complexes. We show that an N-complex X is Cartan-Eilenberg $\mathcal{W}$ if and only if $X\cong X\text{'}\oplus X\text{'}\text{'}$ in which $X\text{'}$ is a $\mathcal{W}$N-complex and $X\text{'}\text{'}$ is a graded R-module with ${{\displaystyle X}}_{n}^{\text{'}\text{'}}\in \mathcal{W}$ for all $n\in \mathbb{Z}$. As applications of the result, we obtain some characterizations of Cartan-Eilenberg projective and injective N-complexes, establish Cartan and Eilenberg balance of N-complexes, and give some examples for some fixed integers N to illustrate our main results.

This paper develops the Bernstein tensor concentration inequality for random tensors of general order, based on the use of Einstein products for tensors. This establishes a strong link between these and matrices, which in turn allows exploitation of existing results for the latter. An interesting application to sample estimators of high-order moments is presented as an illustration.

We introduce and study property T and strong property T for unital *-homomorphisms between two unital C^{*}-algebras. We also consider the relations between property T and invariant subspaces for some canonical unital *-representations. As a corollary, we show that when G is a discrete group, G is nite if and only if G is amenable and the inclusion map i :${C}_{r}^{*}\left(G\right)\to \beta \left({l}^{2}\left(G\right)\right)$ has property T: We also give some new equivalent forms of property T for countable discrete groups and strong property T for unital C^{*}-algebras.

The π_{2}-diffeomorphism finiteness result of F. Fang-X. Rong and A. Petrunin-W. Tuschmann (independently) asserts that the diffeomorphic types of compact n-manifolds M with vanishing first and second homotopy groups can be bounded above in terms of n; and upper bounds on the absolute value of sectional curvature and diameter of M: In this paper, we will generalize this π_{2}-diffeomorphism finiteness by removing the condition that π_{1}(M) = 0 and asserting the diffeomorphism finiteness on the Riemannian universal cover of M:

We consider the simple restricted modules for special contact Lie superalgebras of odd type over an algebraically closed field of characteristic p>3: We give a suffcient and necessary condition in terms of typical or atypical weights for restricted Kac modules to be simple. In the process, we also determine the socle for each restricted Kac module and the length for each simple restricted module.