一、Basic Course Information
Course No.:202032023007
Course name：Modern Engineering Mathematics
Class hours and allocation:total class hours: 48, class teaching 40 hours, class discussion 8 hours
Credit:3
Start semester:Fall semester
Applicable subject:science and engineering
Object-oriented course: PhD students
Assessment method:closed-book exam
Teaching method:classroom teaching
Course starter:College of Mathematical Sciences
二、Advanced Courses
Matrix theory; mathematical analysis; linear algebra.
三、Course objectives and basic requirements
"Modern Engineering Mathematics" is a degree course for doctoral students of various engineering majors. It mainly adopts extremely powerful expression forms and highly abstract views, methods, and theories to clarify the collection structure and properties of space. Its contents include topology and linear space theory. Possibility to connect various practical things extensively, the theory and methods of "modern engineering mathematics" can be widely applied to all branches of engineering technology, and many disciplines in the natural sciences and other scientific fields such as circuit networks, theoretical physics, computers, cybernetics, etc. have a wide range of applications. This course simplifies the content of the original course and the intersection of undergraduates and postgraduates and the relatively backward content, and increases the basic theory of mathematics necessary for modern science. It changes the teaching theory that emphasizes theory rather than practice, and improves the starting point and strengthens the foundation. "Division of levels and rationalization of relationships", considering both the order of knowledge structure and the interrelationship of knowledge modules, and at the same time combining ideological and political cases into the teaching of matrix theory courses, on the one hand, it can enhance students' understanding of what they have learned, On the other hand, it can also guide students to apply the knowledge they have learned, especially some examples of specific technical problems in military operations. It can not only reflect the application of theory, but also provide students with patriotism education in national defense awareness. Of military problems solved with mathematical knowledge. Through the teaching of example questions, students' strong patriotism is stimulated. At present, the United States and its allies often carry out targeted military exercises in the South China Sea and the Pacific Ocean, which seriously threatens the sovereignty of China's territorial waters, which can only be solved scientifically. Many similar problems will make them understand the profound meaning of loving the motherland.
Therefore, learning this course not only enables students to master basic theories and methods, but also lays a good foundation for students' future development, and can enhance students' abstract thinking and logical thinking ability and the methods, and methods of analyzing and solving problems.
Basic requirements:
(1) Students are required to master the theory of using point set analysis to study mathematics, the specific content includes: Set, Mapping,
(2) Students are required to grasp the important concepts and properties of topological space, continuous mapping basic concepts, product space, quotient space and so on.
(3) Students are required to master the basic concepts of linear space and its dual space, and master several special linear space corresponding matrix theories.
四、Teaching content
Part 1: Point set topology
Chapter 1Set Theory and Equivalence
Set theory, relationship and mapping, identity relationship and equivalence relationship, equivalence class and quotient set
Chapter 2Metric Space and Continuous Mapping
The concept of metric space and examples of commonly used metrics, other concepts in metric space, spherical neighborhood and open set of metric space, continuous mapping in metric space.
Chapter 3Topological Space and Continuous Mapping
Topological spaces and topological subspaces, product spaces and quotient spaces, open sets of topological spaces and other subsets, axioms of open and closed sets of topological spaces, sequences and other concepts in topological spaces. Continuous mapping and homeomorphic mapping in topological space, topological invariance and homeomorphism between topological spaces. The judgment and application of homeomorphism.
Chapter 4Connectivity of Topological Spaces
Connectivity and connected subsets of topological spaces, the nature of connectivity, the application of connectivity in Euclidean space, the heritability and multiplicativeness of connectivity, the property of connectivity that remains constant under continuous mapping, and connectivity Application in political cases. The concept and nature of road connectivity and connected branches, the relationship between road connectivity and connectivity, and some applications of road connectivity.
Chapter 5Countability of Topological Spaces
The first countability and the second countability of topological space, the separability and lindeluff of topological space, the correlation between countability, the heritability of countability, multiplicatability and under continuous mapping Maintain immutability and application of countability in national defense science and life.
Chapter 6Separation of Topological Space
Ti(i=0,1,2,3,4) The concept and nature of space, the concept and nature of regular space and normal space, the concept and nature of completely regular space, the heredity of separation, the multiplicativeness and the continuous The inevitable connection between the nature under the mapping and the separation. Separated ideological and political cases
Chapter 7Compactness of Topological Spaces
Topological space's compactness and compaction subsets, properties of compactness, heredity of compactness, multiplicatability and properties under continuous mapping, compactness and separation, countable compaction of topological space The relationship between sex, sequence compactness and column compactness, several kinds of compactness, and the application of compactness in digital watermarking.
Part 2: Introduction to Linear Space
Chapter 8Linear Space and Linear Mapping
Concepts and properties of linear spaces and linear subspaces, common linear spaces, coordinates of bases and elements of finite dimensional linear spaces, basis transformations and coordinate transformations, transition matrix concepts and methods, linear mapping and linear transformation concepts, linear functions Application in actual cases of military defense.
Chapter 9Inner Product Space and Bilinear
Euclidean space and unitary space, unitary transformation, method to determine whether a space is chieftain space, similarities and differences between chieftain space and real inner product space; the concept of normal matrix and determination theorem and properties of standard orthogonal basis, bilinear and metric matrix. The application of multi-linear, unitary space in practical cases
Chapter 10Duality Space
The concept of dual space and dual base, the method of finding dual base, the relationship between inner product space and its dual base. The application of duality space in ideological and political cases.
五、Course main reference materials, reference literature and learning resources
(1) Textbooks, monographs and references
1. Xiong Jincheng. Point Set Topology Lecture (Fourth Edition). Higher Education Press, 2001.
2. Ye Mingxun. Introduction to Linear Space. Wuhan University Press, 2002.
3. Lin Mang. Linear Space and Matrix Theory. Harbin Engineering University Press, 2015.
Syllabus writer:
The signature of the teaching leader of the department:
Date:
Ideological and political educationcases
Teaching content |
Basic elements or cases |
Application of Topological Insulators in the Field of Military Information Technology |
Leading the future is a long-term fundamental task for the development and innovation of national defense science and technology. Once major breakthroughs are made in basic science and cutting-edge technologies, new scientific and technological revolutions will often be catalyzed, and even human society will be transformed. The discovery of topology theory has become the focus of the scientific community for the relationship between the "star material" in the field of physics-graphene, and other "super materials". The principle of topology is a basic theory in the field of physics, and "super materials" such as graphene are based on specific materials under certain principles. In other words, one of them is to answer "why?" and "what is?" these basic theoretical questions related to the cause; the other is to solve the practical technical problems related to the method of "how?" Due to its outstanding advantages such as hard materials, high light transmittance, and low energy loss, graphene makes people look forward to its future application prospects in the military field, and has gradually become a key breakthrough in the development of military technology in major powers. Therefore, if we want to insist on leading the future and seize the strategic commanding heights of technological revolution and military competition, we must stand at the forefront of science and technology. Only in this way can the army stand in the forefront of world military development. |
Dual Space Supports Theoretical Mechanics |
Secure communication has always been an important subject of military research, and the application of cryptography is the basis of secure communication. How to use passwords to protect military privacy is also a focus topic in various countries. In previous cryptography research, bilinear pairing functions (Weil pairing and Tate pairing) are usually used in cryptography. By using the pairing function, the discrete logarithm problem on some elliptic curves can be reduced to the discrete logarithm problem on the finite field. Protocols such as encryption, signature, and key agreement based on bilinear pairing functions. These breakthrough works have opened up new ideas for the construction of cryptographic protocols. Due to the characteristics of bilinear pairing functions, this case focuses on the application of bilinear pairing functions in constructing new cryptographic protocols. |
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