Finding the minimal H-eigenvalue of tensors is an important topic in tensor computation and numerical multilinear algebra. This paper is devoted to a sum-of-squares (SOS) algorithm for computing the minimal H-eigenvalues of tensors with some sign structures called extended essentially nonnegative tensors (EEN-tensors), which includes nonnegative tensors as a subclass. In the even-order symmetric case, we first discuss the positive semi-definiteness of EEN-tensors, and show that a positive semi-definite EEN-tensor is a nonnegative tensor or an M-tensor or the sum of a nonnegative tensor and an M-tensor, then we establish a checkable sufficient condition for the SOS decomposition of EEN-tensors. Finally, we present an efficient algorithm to compute the minimal H-eigenvalues of even-order symmetric EEN-tensors based on the SOS decomposition. Numerical experiments are given to show the efficiency of the proposed algorithm.

Let m, ${m}^{\prime}$, n be positive integers such that $m\ne {m}^{\prime}$. Let $A$ be an mth order n-dimensional tensor, and let $B$ be an ${m}^{\prime}$th order n-dimensional tensor. λ ∈ $?$ is called a $B$-eigenvalue of $A$ if $A{x}^{m-\mathrm{1}}=\lambda B{x}^{{m}^{\prime}-\mathrm{1}}$ and $B{x}^{{m}^{\prime}}=\mathrm{1}$ for some x ∈${?}^{n}\backslash \{\mathrm{0}\}$. In this paper, we propose a linear homotopy method for solving this eigenproblem. We prove that the method finds all isolated $B$-eigenpairs. Moreover, it is easy to implement. Numerical results are provided to show the efficiency of the proposed method.

We treat even-order tensors with Einstein product as linear operators from tensor space to tensor space, define the null spaces and the ranges of tensors, and study their relationship. We extend the fundamental theorem of linear algebra for matrix spaces to tensor spaces. Using the new relationship, we characterize the least-squares ($M$) solutions to a multilinear system and establish the relationship between the minimum-norm ($N$) leastsquares ($M$) solution of a multilinear system and the weighted Moore-Penrose inverse of its coefficient tensor. We also investigate a class of even-order tensors induced by matrices and obtain some interesting properties.

An algorithm is presented for decomposing a symmetric tensor into a sum of rank-1 symmetric tensors. For a given tensor, by using apolarity, catalecticant matrices and the condition that the mapping matrices are commutative, the rank of the tensor can be obtained by iteration. Then we can find the generating polynomials under a selected basis set. The decomposition can be constructed by the solutions of generating polynomials under the condition that the solutions are all distinct which can be guaranteed by the commutative property of the matrices. Numerical examples demonstrate the efficiency and accuracy of the proposed method.

Let $A$ be an mth order n-dimensional tensor, where m, nare some positive integers and N:= m(n−1).Then $A$ is called a Hankel tensor associated with a vector $v\in {?}^{N+\mathrm{1}}$ if ${A}_{\sigma}={v}_{k}$ for each k= 0, 1, …,Nwhenever σ= (i_{1}, …,i_{m}) satisfies i_{1} +…+i_{m} = m+k.We introduce the elementary Hankel tensors which are some special Hankel tensors, and present all the eigenvalues of the elementary Hankel tensors for k= 0, 1, 2. We also show that a convolution can be expressed as the product of some third-order elementary Hankel tensors, and a Hankel tensor can be decomposed as a convolution of two Vandermonde matrices following the definition of the convolution of tensors. Finally, we use the properties of the convolution to characterize Hankel tensors and (0,1) Hankel tensors.

We give a further study on B-tensors and introduce doubly B-tensors that contain B-tensors. We show that they have similar properties, including their decompositions and strong relationship with strictly (doubly) diagonally dominated tensors. As an application, the properties of B-tensors are used to localize real eigenvalues of some tensors, which would be very useful in verifying the positive semi-definiteness of a tensor.

A supertree is a connected and acyclic hypergraph. For a hypergraph H,the maximal modulus of the eigenvalues of its adjacency tensor is called the spectral radius of H.By applying the operation of moving edges on hypergraphs and the weighted incidence matrix method, we determine the ninth and the tenth k-uniform supertrees with the largest spectral radii among all k-uniform supertrees on nvertices, which extends the known result.

We consider approximation algorithms for nonnegative polynomial optimization problems over unit spheres. These optimization problems have wide applications e.g., in signal and image processing, high order statistics, and computer vision. Since these problems are NP-hard, we are interested in studying on approximation algorithms. In particular, we propose some polynomial-time approximation algorithms with new approximation bounds. In addition, based on these approximation algorithms, some efficient algorithms are presented and numerical results are reported to show the efficiency of our proposed algorithms.

We consider an M^{X}=M=c queue with catastrophes and state-dependent control at idle time. Properties of the queues which terminate when the servers become idle are first studied. Recurrence, equilibrium distribution, and equilibrium queue-size structure are studied for the case of resurrection and no catastrophes. All of these properties and the first effective catastrophe occurrence time are then investigated for the case of resurrection and catastrophes. In particular, we obtain the Laplace transform of the transition probability for the absorbing M^{X}=M=c queue.

A method based on higher-order partial differential equation (PDE) numerical scheme are proposed to obtain the transition cumulative distribution function (CDF) of the diffusion process (numerical differentiation of the transition CDF follows the transition probability density function (PDF)), where a transformation is applied to the Kolmogorov PDEs first, then a new type of PDEs with step function initial conditions and 0, 1 boundary conditions can be obtained. The new PDEs are solved by a fourth-order compact difference scheme and a compact difference scheme with extrapolation algorithm. After extrapolation, the compact difference scheme is extended to a scheme with sixth-order accuracy in space, where the convergence is proved. The results of the numerical tests show that the CDF approach based on the compact difference scheme to be more accurate than the other estimation methods considered; however, the CDF approach is not time-consuming. Moreover, the CDF approach is used to fit monthly data of the Federal funds rate between 1983 and 2000 by CKLS model.

Let $d\ge \mathrm{3}$ be an integer, and set $r={\mathrm{2}}^{d-\mathrm{1}}+\mathrm{1}?\mathrm{for}?\mathrm{3}\le d\le \mathrm{4},$$r={\displaystyle \frac{\mathrm{17}}{\mathrm{32}}}?{\mathrm{2}}^{d}+\mathrm{1}?\mathrm{for}?5\le d\le \mathrm{6},$$r={d}^{\mathrm{2}}+d+\mathrm{1}?\mathrm{for}?\mathrm{7}\le d\le \mathrm{8},$ and $r={d}^{\mathrm{2}}+d+\mathrm{2}?\mathrm{for}d\ge \mathrm{9},$ respectively. Suppose that ${{\displaystyle \Phi}}_{i}(x,y)\in ?\left|x,y\right|(\mathrm{1}\le i\le r)$ are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ_{1}, λ_{2}, . . . , λ_{r} are nonzero real numbers with λ_{1}/λ_{2} irrational, and λ_{1}Φ_{1} (x_{1}, y_{1}) + λ_{2}Φ_{2} (x_{2}, y_{2}) + · · · + λ_{r}Φ_{r} (x_{r}, y_{r}) is indefinite. Then for any given real η and σ with 0<σ<22−d, it is proved that the inequality has infinitely many solutions in integers x_{1}, x_{2}, . . . , x_{r}, y_{1}, y_{2}, . . . , y_{r}. This result constitutes an improvement upon that of B. Q. Xue.

For a holomorphic function f defined on a strongly pseudo-convex domain in ${?}^{n}$ such that it has only isolated critical points, we define a twisted Cauchy-Riemann operator ${{\displaystyle \stackrel{\u203e}{\partial}}}_{\tau f}:\stackrel{\u203e}{\partial}+\tau \partial f\wedge $. We will give an asymptotic estimate of the corresponding harmonic forms as τ tends to infinity. This asymptotic estimate is used to recover the residue pairing of the singularity defined by f.

Linear mixed effects models with general skew normal-symmetric (SNS) error are considered and several properties of the SNS distributions are obtained. Under the SNS settings, ANOVA-type estimates of variance components in the model are unbiased, the ANOVA-type F-tests are exact F-tests in SNS setting, and the exact confidence intervals for fixed effects are constructed. Also the power of ANOVA-type F-tests for components are free of the skewing function if the random effects normally distributed. For illustration of the main results, simulation studies on the robustness of the models are given by comparisons of multivariate skew-normal, multivariate skew normal-Laplace, multivariate skew normal-uniform, multivariate skew normal-symmetric, and multivariate normal distributed errors. A real example is provided for the illustration of the proposed method.

H is called an ${{\displaystyle \mathcal{M}}}_{p}$-embedded subgroup of G, if there exists a pnilpotent subgroup B of G such that H_{p} ∈ Syl_{p} (B) and B is ${{\displaystyle \mathcal{M}}}_{p}$-supplemented in G. In this paper, by considering prime divisor 3, 5, or 7, we use ${{\displaystyle \mathcal{M}}}_{p}$-embedded property of primary subgroups to investigate the solvability of finite groups. The main result is follows. Let E be a normal subgroup of G, and let P be a Sylow 5-subgroup of E. Suppose that $\mathrm{1}\langle d\le \left|P\right|$ and d divides |P|. If every subgroup H of P with $\left|H\right|=d$ is ${{\displaystyle \mathcal{M}}}_{5}$-embedded in G, then every composition factor of E satisfies one of the following conditions: (1) I/C is cyclic of order 5, (2) I/C is 5'-group, (3) $I/C\cong {A}_{5}$