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Revealing the Hidden Mathematical Beauties of the Cayley-Hamilton Method

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American Journal of Computational Mathematics 2024年第2期14卷 257-263页

作者： Haiduke Sarafian The Pennsylvania State University University College York PA USA

The inversion of a non-singular square matrix applying a Computer algebra System (CAS) is straightforward. The CASs make the numeric computation efficient but mock the mathematical characteristics. The algorithms conducive to the output are sealed and inaccessible. In practice, other than the CPU timing, the applied inversion method is irrelevant. This research-oriented article discusses one such process, the Cayley-Hamilton (C.H.) [1]. Pursuing the process symbolically reveals its unpublished hidden mathematical characteristics even in the original article [1]. This article expands the general vision of the original named method without altering its practical applications. We have used the famous CAS Mathematica [2]. We have briefed the theory behind the method and applied it to different-sized symbolic and numeric matrices. The results are compared to the named CAS’s sealed, packaged library commands. The codes are given, and the algorithms are unsealed.

关键词： Cayley-Hamilton Method Matrix Inversion linear algebra Computer algebra System Mathematica

Applying Neural Network in Classifying Parkinson’s Disease

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Journal of Computer and Communications 2020年第10期8卷 19-23页

作者： Yuan-Liang (Tom) Liao Fenglinlvzhou F01-10D Beijing China

This project uses knowledge of neural network to analyze if the person under study is analyzed to be Parkinson disease patient or not. Binary classification is constructed based on the multi-feature database. A decision boundary is clearly plotted to separate patient with and without Parkinson disease. Results show that over 80% accuracy could be obtained with the preliminary results. Future efforts could be performed to construct more complicated neural network to improve the accuracy.

关键词： Machine Learning linear algebra Error Analysis Neural Network

Why Are There as Many Elements in the Cantor Set as There Are Real Numbers?

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Open Journal of Applied Sciences 2023年第11期13卷 2183-2185页

作者： Wenbing Wu Xiaojian Yuan School of Big Data Fuzhou University of Foreign Studies and Trade Fuzhou China

There are many important concepts in linear algebra, such as linear correlation and linear independence, eigenvalues and eigenvectors, and so on. The article provides a graphical explanation of how to distinguish between the concepts of linear correlation and linear independence. The conclusion points out that linear independence means that there are no two (base) vectors with the same direction in a vector graph;otherwise, it is a linear correlation.

关键词： Cantor Ternary Set linear Independence Vector linear algebra

Use Elementary Column Operations to Calculate the Basis of the Null Space of a Matrix

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Journal of Mathematics and System Science 2013年第3期3卷 162-165页

作者： Chen Bihong Shen Da Xin Cun Rooml 7-201 ShenZhen 518033 GuangDong China

This paper gives and proofs a theorem, for any matrix A, do elementary column operations, change it to a matrix which is partitioned to two blocks which left one is column full rank and right one is zero matrix. That is, use a invertible matrix P to let AP = （B,O）, O is zero matrix with n-r columns, r and n is rank and column number of A, so the P＇s right n-r columns is just the basis of the null space of the matrix A. On the basis of the theorem, lots of problems of linear algebra can be resolved and lots of theorems can be proofed by elementary column operations. Perhaps the textbooks used in universities will have a lot of change with the result of the paper. This result is first found by author in 2010.12.8 in http：//***/***/default/releasepaper/content/201012-232, but is not formal published.

关键词： linear algebra null space systems of linear equations.

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通信密钥分散管理的两个新方案

哈尔滨工业大学学报 1987年第1期 128-129+125页

作者：刘锐曹珍富哈尔滨工业大学应用数学系

In 1976, Diffie and Hellman advanced a public key cryptosystem structured by the "trap-door one-way function". The system requires that each user has two cryptography keys—an encryption key E and a decryption key D, where D is needed to keep secret strictly. In 1979, Shamir presented a polynomial interpolation to divide D into n sub-cryptography keys D;, D;,…, D;, and they were distributed n people to keep with. It requires that arbitrary k people can recover D, but arbitrary k-1 people cannot. Later, Mceliece and Lu Tiecheng et al also put forward some other system by using RS or CR（Chinese remaining theorem） codes

关键词： Cryptography Key management linear algebra set theory.

Analysis of the Impact of Inductive Reasoning on the Mathematical Thinking Style of Deaf and Hard-of-Hearing Students

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Journal of Contemporary Educational Research 2024年第11期8卷 113-122页

作者： Yan Cui Zhili Ge Zhaosong Zhu Lin Xiang Wuxia Yan School of Mathematics and Information Science Nanjing Normal University of Special EducationNanjing 210038Jiangsu ProvinceChina

In this paper,we combine the teaching and learning situation of deaf and hard-of-hearing students in the linear algebra course of the Computer Science and Technology major at the Nanjing Normal University of Special *** on the cognitive style of deaf and hard-of-hearing students,we apply example induction,exhaustive induction,and mathematical induction to the teaching of linear algebra by utilizing specific course *** aim is to design comprehensive teaching that caters to the cognitive style characteristics of deaf and hard-of-hearing students,strengthen their mathematical thinking styles such as quantitative thinking,algorithmic thinking,symbolic thinking,visual thinking,logical thinking,and creative thinking,and enhance the effectiveness of classroom teaching and learning outcomes in linear algebra for deaf and hard-of-hearing students.

关键词： Cognitive style Mathematical thinking style Deaf university students Inductive reasoning linear algebra