Mathematics: Calculus and Combinatorial Mathematics in Abstract Algebra

Course 13: Mathematics: Calculus and Combinatorial Mathematics in Abstract Algebra

I. Course Description

Abstract algebra, calculus, and combinatorial mathematics are important and widely used branches in mathematics. Abstract algebra exploring the general properties of algebraic structures, such as groups, rings, and domains, provides rich tools of abstract thinking for the mathematical field. Calculus studies the concept of change and accumulation, which provides a basis for the modeling and analysis of natural and scientific phenomena. Combinatorial mathematics focuses on the permutation, combination and counting of discrete objects, and has important applications in information science and cryptography.

The combination of abstract algebra with calculus and combinatorial mathematics not only provides possibilities for the further study of mathematics itself, but also opens up new ways for solving practical problems and the application of mathematics in other disciplines. For example, in cryptography, abstract algebra can be used to design powerful encryption algorithms, while calculus helps to analyze the trends in data flow, and combinatorial mathematics can be applied to generate passwords and keys. Therefore, a deep exploration of the intersection of abstract algebra, calculus and combinatorial mathematics will bring rich possibilities for mathematical research and practical application.

Through this course, students will have a comprehensive grasp of the core concepts of combinatorial mathematics, calculus and probability theory, and can flexibly apply them to the business and scientific fields. The course aims to develop students mathematical modeling and problem solving skills to analyze practical business and scientific challenges using tools such as power sum, permutation, and probability distribution. Advanced mathematical tools such as gamma functions and elliptic integration to solve complex problems are applied in statistics, stochastic processes and machine learning. Through the comprehensive curriculum, students will be able to understand the associations between different branches of mathematics, provide innovative mathematical solutions to practical problems in interdisciplinary fields.

II. Professor Introduction

Dan Ciubotaru – Tenured Professor at Oxford University

Dan Ciubotaru Professor He completed his PhD in Cornell University, and began decades of research and teaching. Dan Ciubotaru Professor He also has teaching experience in top universities in China, Britain and the United States. At present, he is a tenured professor in the Department of Mathematics at the University of Oxford, and as a mathematics tutor at Somerville College, Oxford, responsible for pre-exam tutoring and pure mathematical theory tutoring. Since 2014, the Professor has been in charge of the Admissions Officer at Somerville College, Oxford. Professor Dan Ciubotarus research field is representation representation theory, a field of symmetry, who is also interested in the units of reduced Lie groups and Heckes algebra in the framework of local Langlands correspondence. The professors research was recently funded by New Vision from the UK Research Centre for Engineering and Physics (EPSRC).

III. Syllabus

1. Combinatorial mathematics
2. Infinity series and generating function
3. Zeta function with the Bernoulli polynomials
4. calculus
5. Probability integral, Γ (1 / 2), double integral
6. Gamma functions: functional equations, beta functions, elliptic integrals, and applications
7. Discrete probability, conditional probability, and combined application
8. High-level topics of calculus and probability
9. 9, an extension of the concept of calculus in the gamma function
10. Application of combined mathematics in machine learning