In this paper, the authors present a method to express a univariate positive semi-definite polynomial into the sum of lader-like squares, i.e., squares of several polynomials which degrees are strictly decreasing. When the coefficients of the given polynomial are rational numbers, the coefficients of the degree-descending polynomials are also rational numbers. The authors have also extended the method to multi-variate polynomials. Namely, if a multi-variate polynomial has any sum of square representation, then a special representation with rational coefficients of this polynomial can be obtained using this method.

This paper discusses a polynomial-based barrier certificate construction method for verifying the safety of neural network controlled systems. First, the neural network model is abstracted using methods such as global sector constraints, local sector constraints, and overlay sector constraints to obtain corresponding semi-algebraic constraints. Then, using Positivstellenstz in computational real algebraic geometry, the barrier certificate conditions are transformed into corresponding sum-of-squares constraints, which are solved by using semi-definite programming. Finally, the effects of the above different neural network abstraction methods on the ability of constructing the barrier certificates of the neural network controlled systems are analyzed and compared through examples.

This paper mainly discusses the equivalence of two classes of bivariate polynomial matrices to their Smith forms. Some necessary and sufficient conditions for equivalence of these classes of matrices to their Smith forms are given. And an example is given to illustrate how to realize the equivalence reduction.

As presented by Curto, et al. (2013), neural rings and ideals serve as powerful algebraic constructs that facilitate the systematic organization and analysis of combinatorial data within neural codes. In this paper, we present the relationship between the Gröbner basis and canonical form of the neural ideal. Some new RF-relationships are given by analyzing the forms of elements in neural ideals.

第十三届中国数学会计算机数学大会是由中国数学会下属的计算机数学专业委员会主办的一系列学术盛会. 旨在为全国范围内从事计算机数学研究的科研人员提供一个汇总和交流国内外计算机数学最新研究成果的平台. 此次大会由大连理工大学数学科学学院、大连海事大学理学院以及中国科学院数学机械化重点实验室联合承办, 于2023年6月15日至18日在辽宁大连成功举办, 吸引了来自国内科研院所、高校和企业的200多名科研人员参与. 大会设立了涵盖计算几何、密码学、人工智能、组合数学、符号计算、符号数值混合计算、数学软件以及数学教育等八个专题. 特邀了来自中国科学院数学与系统科学研究院的闫振亚研究员与李子明研究员、香港理工大学的祁力群教授、复旦大学的高卫国教授做大会报告, 并邀请了各个专题的八位青年科研人员做青年邀请报告. 此外, 还有70多名与会人员做分组报告, 介绍了各自领域的最新成果.会议期间, 中国数学会计算机数学专业委员会还颁发了第四届``吴文俊计算机数学青年学者奖".
大会共收到约75篇稿件, 其中包括中英文长文40多篇.经过大会程序委员会及相关领域专家的认真评选, 选取了7篇中文长文组成专题, 涵盖了符号计算、符号数值混合计算以及人工智能等方向的最新研究成果. 在此, 我们要感谢会议程序委员会的各位委员和论文评审专家对评审工作的辛勤付出.同时, 特别感谢本次会议的组织委员会, 尤其是于波教授与董波教授及其团队, 为成功举办本次盛会所做的努力. 也感谢《系统科学与数学》杂志的各位同仁对本专题的大力协助.我们衷心希望本专题能为计算机数学的发展做出有力推动.

Extensive work has shown that the hyperspectral image classification method based on the fusion of spectral-spatial information can obtain satisfactory results. It is a challenging problem in the field of remote sensing that how to make full use of the spatial information contained in hyperspectral images and properly integrate it with spectral features to obtain better classification results. In this paper, a multi-scale hyperspectral image classification method based on siamese network architecture and graph convolution is proposed. First, the original hyperspectral image is divided into three images of the same size but with different spectral features. Then, three images are segmented separately into superpixels using different scales and merged them. The merged superpixels not only greatly reduce the size of the graph and improve the computational efficiency, but also further enhance the role of spatial information in classification. Next, based on the extended siamese network architecture, the main spectral features of the three images are extracted using graph convolution, respectively. Finally, the extracted spectral features are fused using the attention mechanism and input into the fully connected network for classification. Experimental and comparative results on two public datasets, Indian Pines and Salinas, show that the proposed method performs better than several competing methods in classification completion.

Singular two-point boundary value problems arise in a variety of applied mathematics and physics. It is a classical problem and many researchers have done a lot of research work on this issue. In this paper, we apply cubic B-spline to explore the numerical solutions of a class of singular two-point boundary value problems. The paper method is primarily based on the super convergence in approximating second-order derivative values at the knots by the combination of second-order derivative values of cubic B-spline. The paper proposed cubic B-spline possesses super convergence in approximating the first-order derivative/second-order derivative of given function and thus the approximation order of our method reaches fourth order. Some numerical experiments are provided to demonstrate the effectiveness of our method compared to the other existing methods.

It is well known that if a polynomial$f\in \mathbb{R}[x]$ is strictly positive on the unit box $I^n=[0,1]^n$, then $f$ can be written as a Bernstein expansion with strictly positive coefficients. However, the above conclusion no longer holds if $f$ has zero points on $I^{n}$. In this paper, we consider the case of $f$ with corner zero points (vertices of $I^n$). As a result, we provide a necessary and sufficient condition for the Bernstein expansion of $f$ with non-negative coefficients when the zeros are only at corner of $I^n$. Our method relies on constructing the$d$-multiple form whose terms are homogeneous, the problem is transformed into the verification of coefficients of a given $d$-multiple form.