Multi-layer self-organizing polynomial neural networks and their development with the use of genetic algorithms

In this paper, we introduce a new architecture of genetic algorithms (GA)-based self-organizing polynomial neural networks (SOPNN) and discuss a comprehensive design methodology. Let us recall that the design of the “conventional” PNNs uses an extended group method of data handling (GMDH) and exploits polynomials (such as linear, quadratic, and modified quadratic functions) as well as considers a fixed number of input nodes (as being selected in advance by a network designer) at polynomial neurons (or nodes) located in each layer. The proposed GA-based SOPNN gives rise to a structurally optimized structure and comes with a substantial level of flexibility in comparison to the one we encounter in conventional PNNs. The design procedure applied in the construction of each layer of a PNN deals with its structural optimization involving the selection of preferred nodes (or PNs) with specific local characteristics (such as the number of input variables, the order of the polynomial, and a collection of the specific subset of input variables) and addresses specific aspects of parametric optimization. An aggregate performance index with a weighting factor is proposed in order to achieve a sound balance between approximation and generalization (predictive) abilities of the network. To evaluate the performance of the GA-based SOPNN, the model is experimented with using chaotic time series data. A comparative analysis reveals that the proposed GA-based SOPNN exhibits higher accuracy and superb predictive capability in comparison to some previous models available in the literature.     

素数在域Q(u~[1/(l~n)])上素理想分解

设F=Q(u~[1/(l~n)]),其中l是奇素数.本文给出了奇素数p在整数环OF中的素理想分解形式.     

韦达定理及其推广应用

韦达定理是初等数学中的重要内容,它是揭示一元二次方程根与系数关系的重要定理。利用多项式理论将其推广到一元n次方程中,并介绍其简单应用。     

E-characteristic polynomials of real rectangular tensor

By the resultant theory, the E-characteristic polynomial of a real rectangular tensor is defined. It is proved that an E-singular value of a real rectangular tensor is always a root of the E-characteristic polynomial. The definition of the regularity of square tensors is generalized to the rectangular tensors, and in the regular case, a root of the E-characteristic polynomial of a special rectangular tensor is an E-singular value of the rectangular tensor. Moreover, the best rank-one approximation of a real partially symmetric rectangular tensor is investigated.     

经管类专业本科生定量分析能力现状及影响因素

通过对经管类专业本科生定量分析能力的现状及影响因素的量化研究,结果表明：定量分析能力可分为三个模块：初级能力、中级能力、高级能力;现实状况是学生对初、中级能力的掌握较好,对高级能力的掌握较差。其次,对学生在掌握定量分析能力程度上,验证了男生比女生有优势,理科生比文科生有优势。第三,将学生掌握定量分析能力的影响因素归纳为4个方面,有3个方面通过了显著性检验,其中“学习方法”、“认识作用”是最重要的影响因素。     

对古塔变形趋势的若干思考

通过建立数学模型来分析古塔的总体变形状况,就是为了科学地对古塔变形趋势有所预测,能有效地采取措施对古塔进行维护。文章采用均值法确定古塔各层中心位置;通过大量的分析假设,运用MATLAB软件结合曲线拟合的最小二乘法,对所测数据进行多项式拟合,列出各层中心点所在曲线的参数方程,求得曲线的曲率和挠率,分析古塔的弯曲、扭曲情况;根据离散的数据图形,比较直观地预测它的变形趋势,有的放矢地采取维护方法。     

（3,n）型多项式函数方程xf^2（x）＋xg^2（x）=h^2（x）的解

讨论在复数域上,当f（x）与g（x）的次数都等于3,并且g（x）的次数不超过3时,多项式函数方程xf（x）＋xg^2（x）=h^2（x）的解的情况,得到部分结果.主要结果为：如果h（x）的次数等于1,那么这个函数方程无解;如果h（x）的次数等于2,那么这个函数方程一共有8组解;如果h（x）的次数等于3,那么h（x）的1次项系数等于零时,这个函数方程一共有24组解;当h（x）的2次项系数等于零时,但1次项系数不等于零时,这个函数方程一共有36组解.     

广义高阶Bernoulli多项式的一些恒等式及其应用

Bernoulli多项式及其多种推广形式在组合数学、解析数论等领域中起着十分重要的作用。广义Bernoulli多项式Bn,χ(x)与Euler多项式、Dirichlet级数有密切的联系。应用绝对收敛Laurent级数的卷积公式,给出了广义高阶Bernoulli多项式的一些表达式和一个推论。     

代数学基本定理的推论及其应用

用代数学基本定理的推论讨论多项式和矩阵问题,给出了方阵乘积的伴随矩阵与参与乘积矩阵的伴随矩阵关系一个新的证明,得到了实对称矩阵正交相似关系的一个新结果.     

对称反循环矩阵求逆的探讨

根据对称反循环矩阵的性质,利用生成多项式和特征多项式,采用行初等变换的方法,给出了求对称反循环矩阵的逆的一种方法,有一定的实用性。