Superpixel-level hyperspectral image classification is a representative spec-tral-spatial classification method. Compared with the pixel-wise classification method, it has obvious advantages in classification accuracy and efficiency. However, the main disadvantage of superpixel-level classification algorithms is that the classification results depend heavily on the segmentation scale of superpixels. Existing literature shows that the optimal segmentation scale of superpixels is usually an experimental result, and it is difficult to be specified in advance. To weaken this dependency, a superpixel-level hyperspectral image classification algorithm based on superpixel merging is proposed in this work. Local modularity function is first used to merge the sparse weighted superpixel graph constructed. By the newly defined mapping, each superpixel is represented as a sample. Then popular KNN method is adopted to classify the merged image at the superpixel level. The superpixel merging enhances the role of spatial information in classification, effectively weakens the dependence of classification results on the segmentation scale of superpixels, and improves the classification accuracy. To evaluate the effectiveness of the method, the proposed algorithm is compared with some competitive hyperspectral image classification methods on four publicly real hyperspectral datasets. The experimental and comparative results show that the proposed method not only effectively reduces the influence of superpixel segmentation scale on the classification results, but also has obvious advantages both in classification accuracy and computational efficiency.

This paper deals with the Zero-Hopf bifurcation in high dimensional polynomial differential systems. First, we reduce the problem of bifurcation analysis to an algebraic problem, and we give a method for determining the bifurcation set of the Zero-Hopf bifurcation points of differential systems by using symbolic algorithm for solving semi-algebraic systems. Then, based on the second order averaging method, the algorithmic framework of the Zero-Hopf bifurcation analysis of differential systems is derived, and the limit cycle bifurcation problem is studied through specific examples by using the methods of symbolic computation, and some new results are obtained. Finally, we propose several related research problems.

Multidimensional systems are often described by polynomial matrices, and problems on the equivalence of multidimensional systems in system theory are often transformed into problems on the equivalence of polynomial matrices. In this paper, we mainly study the equivalence of two kinds of multivariate polynomial matrices, and obtain the discriminant conditions for the equivalence of these matrices and their Smith forms, respectively. The conditions are easily verified, and an example is also used to illustrate these in the paper.

Different from the known methods of order theory and topology theory on partial order and $T_0$ topology, this paper presents a method to find all partial orders as well as $T_0$ topologies on the finite set $[n]=\{1,2,\cdots,n\}$ through solving a polynomial set over the finite field $\mathbb{F}_2$, and illustrates the correspondence between zeros of the polynomial set and partial orders as well as $T_0$ topologies by examples. Based on Gr$\ddot{\text{o}}$bner bases theory, a symbolic computation method for computing the number of partial orders and the number of $T_0$ topologies on $[n]$ is obtained. Some examples are given to illustrate our method using Maple.

For the integer polynomials $\sum_{i=0}^m a_i\,x^i$ of degree $m$, where $a_m=1$, this paper presents a fixed-point iterative algorithm: %and the corresponding fixed point iterative algorithm $$\left\{ \begin{array}{ll} u_1=\tilde{u}_1, & \\ u_2=\tilde{u}_2, & \\ \quad\,\,\,\vdots& \\ u_{m-1}=\tilde{u}_{m-1}, & \\ \displaystyle{u_n=-\Big{(}a_{m-1}+\frac{a_{m-2}}{u_{n-1}}+\frac{a_{m-3}}{u_{n-1}u_{n-2}}+\cdots+\frac{a_{0}}{u_{n-1}u_{n-2}\cdots u_{n-(m-1)}}\Big{)}\,\,(n\geq m).} & %\displaystyle{u_n=-\frac{a_{m-1}}{a_m}-\frac{a_{m-2}}{a_mu_{n-1}}-\frac{a_{m-3}}{a_mu_{n-1}u_{n-2}}-\cdots-\frac{a_{0}}{a_mu_{n-1}u_{n-2}\cdots u_{n-(m-1)}}\,\,(n\geq m).} & \end{array} \right. $$ 1) Obviously, if the iteration has a rational limit value, then the value is a zero of the polynomial, so that the polynomial is reducible over $\mathbb{Q}$. 2) This iteration does not need to choose the initial values: If the polynomial has $m$ rational zeros with different absolute values, then for any $m-1$ non-zero rational initial values $u_i\,(1\leq i\leq m-1)$, the iteration approaches to one of the zeros. Therefore, the polynomial is reducible. 3) Assuming that $\{\zeta_i\,\big{|}\, |\zeta_1|\geq|\zeta_{2}|\geq\cdots\geq|\zeta_m|,\,\zeta_i\in\mathcal{C},\,1\leq i\leq m\}$ are the distinct zeros of the above polynomial, there exist $m$ complex numbers $\{\beta_i\,|\, \beta_i\in\mathcal{C},\,1\leq i\leq m\}$ such that $u_n$ can be expressed in the following form \begin{equation*}u_n=\frac{\beta_1\zeta_1^{n+1}+\beta_2\zeta_2^{n+1}+\cdots+\beta_m\zeta_m^{n+1}}{\beta_1\zeta_1^n+\beta_2\zeta_2^n+\cdots+\beta_m\zeta_m^n}%\,\,(\beta_i\in\mathcal{R}) .\end{equation*} Among the $m$ elements of the vector $\beta$, let $\beta_l$ be the first non-zero element and $\beta_k$ the first non-zero element after it, that is, $\{\beta_i\,|\, \beta_i\in\mathcal{C},\,1\leq i\leq m\}= \{\!\!\underbrace{0,0,\cdots,0}_{\mbox{All are zero.}},\beta_l(\neq0), \underbrace{0,0,\cdots,0}_{\mbox{ All are zero.}},\beta_k(\neq0),\cdots,\beta_m\}.$ In this case, if $|\zeta_l|>|\zeta_k|$, then the iteration converges to $\zeta_l$. Therefore, if $\zeta_l\in\mathcal{Q}$, then the polynomial is reducible.

This paper presented a method of algebraic representation of the Ramsey Theorem and gave an implementation for the mechanical proofs to the theorem for two classic cases, i.e., $R(3,3)=6$ and $R(3,4)=9$ using symbolic computation. A divide-and-conquer method was also suggested for tackling with complicated cases including $R(3,5)=14$ and $R(3,3,3)=17$. Different from the existed computer aided algorithms, the proposed method can generate mechanical proof of Ramsey theorem through polynomial computation.

Interpolation strategies have shown to be very effective in the reconstruction black-box functions, in particular in polynomial algebra, for sparse multivariate polynomials. In addition, sparse multivariate interpolation algorithms with polynomial time complexity have been widely studied and applied. Recently, Huang(2021) presented a sparse polynomial interpolation algorithm based on diversification. The algorithm costs $O(nT \log^2 q + nT\sqrt D\log q)$ bit operations and it is the first one to achieve the complexity of fractional power about $D$, while keeping linear in $n$, $T$ over finite fields $\mathbb{F}_q$. Since Huang's algorithm is probabilistic with correct interpolates $f$ with probability at lest $\frac{3}{4}$, in order to improve the success probabilistic of interpolation, three failure cases of Huang's algorithm are analyzed, and the corresponding modification schemes are given. Based on revised strategies, a high probability sparse interpolation algorithm based on diversified polynomial is designed. Extensive numerical experiments show the effectiveness of the algorithm.