Subjects: Mathematics >> Control and Optimization. submitted time 2023-12-04
Abstract： In this paper, we mainly dicuss the analytic conditions of copositivity of a class of 4th order 3-dimensional symmetric tensors. For a 4th order 3-dimensional symmetric tensor with its entries $1$ or $-1$, an analytic necessary and sufficient condition is given for its strict copositivity with the help of the properties of strictly semi-positive tensors. And by means of usual maxi-min theory, a necessary and sufficient condition is established for copositivity of such a tensor also. Applying these conclusions to a general 4th order 3-dimensional symmetric tensor, the analytic conditions are successfully obtained for verifying the (strict) copositivity, and these conditions can be very easily parsed and validated.
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Awaiting Review
Subjects: Mathematics >> Control and Optimization. submitted time 2017-07-25
Abstract：In this paper, one of our main purposes is to prove the boundedness of solution set of tensor complementarity problem with B tensor such that the specific bounds only depend on the structural properties of tensor. To achieve this purpose, firstly, we present that each B tensor is strictly semi-positive and each B$_0$ tensor is semi-positive. Subsequencely, the strictly lower and upper bounds of different operator norms are given for two positively homogeneous operators defined by B tensor. Finally, with the help of the upper bounds of different operator norms, we show the strcitly lower bound of solution set of tensor complementarity problem with B tensor. Furthermore, the upper bounds of spectral radius and $E$-spectral radius of B (B$_0$) tensor are obtained, respectively, which achieves our another objective. In particular, such the upper bounds only depend on the principal diagonal entries of tensors.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Mathematics （General） submitted time 2017-07-03
Abstract：In this paper, we introduce the concept of an m-order n-dimensional generalized Hilbert tensor \mathcal{H}_{n}=(\mathcal{H}_{i_{1}i_{2}\cdots i_{m}}),
\mathcal{H}_{i_{1}i_{2}\cdots i_{m}}=\frac{1}{i_{1}+i_{2}+\cdots i_{m}-m+a}, a\in \mathbb{R}\setminus\mathbb{Z}^-;\ i_{1},i_{2},\cdots,i_{m}=1,2,\cdots,n,
and show that its H-spectral radius and its Z-spectral radius are smaller than or equal to M(a)n^{m-1} and M(a)n^{\frac{m}{2}}, respectively, here M(a) is a constant only dependent on a. Moreover, both infinite and finite dimensional generalized Hilbert tensors are positive definite for a\geq1. For an m-order infinite dimensional generalized Hilbert tensor $\mathcal{H}_{\infty} with a>0, we prove that \mathcal{H}_{\infty} defines a bounded and positively (m-1)-homogeneous operator from l^{1} into l^{p}\ (1
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Mathematics （General） submitted time 2016-11-09
Abstract：In this paper, the $m-$order infinite dimensional Hilbert tensor (hypermatrix) is intrduced to define an $(m-1)$-homogeneous operator on the spaces of analytic functions, which is called Hilbert tensor operator. The boundedness of Hilbert tensor operator is presented on Bergman spaces $A^p$ ($p>2(m-1)$). On the base of the boundedness, two positively homogeneous operators are introduced to the spaces of analytic functions, and hence the upper bounds of norm of such two operators are found on Bergman spaces $A^p$ ($p>2(m-1)$). In particular, the norms of such two operators on Bergman spaces $A^{4(m-1)}$ are smaller than or equal to $\pi$ and $\pi^\frac1{m-1}$, respectively.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Mathematics （General） submitted time 2016-07-05
Abstract：In this paper, the Pazy's Fixed Point Theorems of monotone $\alpha-$nonexpansive mapping $T$ are proved in a uniformly convex Banach space $E$ with the partial order ``$\leq$". That is, we obtain that the fixed point set of $T$ with respect to the partial order ``$\leq$" is nonempty whenever the Picard iteration $\{T^nx_0\}$ is bounded for some comparable initial point $x_0$ and its image $Tx_0$. When restricting the demain of $T$ to the cone $P$, a monotone $\alpha-$nonexpansive mapping $T$ has at least a fixed point if and only if the Picard iteration $\{T^n0\}$ is bounbed. Furthermore, with the help of the properties of the normal cone $P$, the weakly and strongly convergent theorems of the Picard iteration $\{T^nx_0\}$ are showed for finding a fixed point of $T$ with respect to the partial order ``$\leq$" in uniformly convex ordered Banach space.
Peer Review Status:Awaiting Review
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