I.The Basic Information of the Course(Time New Rome, 12 points, bold font)
Course Number: 202012420016
The EnglishName of theCourse：Combinatorics in Tensor Analysis
The ChineseName of theCourse：张量分析中的组合学
In-classHours andAllocation(Sample: total class hours:32, classroom teaching: 20 classhours, classroom discussion: 6 classhours, 6 classhours on computer, etc.)
Credit(s):2
Semester:2
AppliedDiscipline(Professional Degree Category)：Mathematics
CourseObjectOriented：Academic DoctorandAcademicMaster
EvaluationMode:Process Evaluation
TeachingMethod:Blended Teaching
CourseOpening Department: College of Mathematics Sciences
Notes:Filling explanation to applied discipline: the general courses like foreign languages, ideological and political theory course may fill in “AllDisciplines”,Mathematics general course may fill inScience andEngineeringDiscipline, and the other courses fill in “Name ofApplicableDiscipline” according to actual teaching objects, and the number of applicable discipline may be more than one.
II.Prerequisite Course( Time New Rome 12 points bold font)
Matrix Analysis, Advanced Algebra
III.The Objectives and Requirements of the Course(Time New Rome 12 points, total words: about 200 words)
Combinatorics in Tensor Analysis mainly introduces the research methods and ideas of combinatorics, which is based on multiple linear algebra and hypergraphs, and combined with the spectrum of tensors and applications spectrum and its applications. Through this course, one can learn the basic knowledge and application principle of tensor analysis and combinatorial mathematics, and master the international frontier issues and development trends in this direction. And through the case heuristic teaching, one has the ability to combine tensor analysis and combinatorial mathematics to solve problems. The course cultivates the scientific research quality, innovation ability of postgraduate students, and the ability to track the research dynamics independently, read quickly and master the main content of the literature.
IV.The Content of the Course(Time New Rome, 12 points, bold font, 1000-2000 words)
This course focuses on concept of tensors, eigenvalues, nonnegative tensor theory, hypergraph theory and their applications in artificial intelligence, from the perspective of the development of matrices to tensors. Emphasis is laid on the combinatorial methods and ideas in tensor spectral problems, included the directed graph of tensors, the existence set of tensor eigenvalues, the spectrum of hypergraphs and the structure of hypergraphs. And combined with domestic and foreign frontier issues, This course introduces the applications of tensor and combinatorial mathematics in engineering problems, which arouses students' interests.
There are 5 chapters in this course. The first chapterintroducesthebasic knowledge, included the concept of tensors, tensor operations, several eigenvalues of tensors based on different research problems, several types inclusion sets of tensor eigenvalues. Chapter 2 introduces the nonnegative tensor theory included that the Perron-Frobenius theory on the nonnegative tensor, irreducible tensor and the weakly irreducible tensor. Chapter 3 introduces the directed graph representation of tensors, the inclusion set of tensor eigenvalues by the directed graph tool and other directed graph methods in tensor analysis. Chapter 4 introduces the adjacency tensor, Laplacian tensor and signless Laplacian tensor corresponding to hypergraphs. Chapter is is about the application of hypergraphs in practical problems included the spectral clustering via the graph partition model and the image matching problem. In the course ideological and political aspects, through the introduction of Chinese mathematicians, to overcome the difficult course of academic cutting-edge problems, and Chinese mathematicians have made abundant achievements in tensor theory and combinatorial mathematics in recent years, which carry forward the spirit of patriotism, remain true to our mission of pursuing truth in scientific research, and keep in mind the mission of rejuvenating the country through scientific innovation.
The teaching content of this course is arranged as follows:
Chapter 1 Tensor
1.1 Products of tensors
1.2 Eigenvalues of tensors
1.3Inclusion sets of tensor eigenvalues
Chapter 2 Spectrum of nonnegative tensors
2.1 Irreducible tensors
2.2 Perron-Frobenius theory
2.3 Primitive tensors
Chapter 3 Tensor and direct graph
3.1 Direct graphs for tensors
3.2 Inclusion sets for tensor eigenvalues via direct graph
Chapter 4Spectrum of hypergraphs
4.1 Tensors corresponding to hypergraphs
4.2 Spectrum for hypergraph via tensors
4.3Characteristic polynomial for hypergraphs
Chapter 5 Applications for the spectrum of tensors
5.1Image matching
5.2 Partitioning of hypergraphs
V. Reference Books, Reference Literatures, and Reference Materials
A. Text Books, Monographs and References
1. Qi L, Luo Z. Tensor Analysis: Spectral Theory and Special Tensors. Philadelphia (PA):Society of industrial and applied mathematics, 2017
2. QiL, ChenH, Chen Y. TensorEigenvaluesand TheirApplications. Springer, 2018
3. WangJ. Foundation for the Spectrum of Hypergraphs.Higher Education Press, 2006.
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