Nonlinear partial differential equation is an important branch of modern mathematics. It is used to describe the theoretical and practical application problems in mechanics, control process, ecological and economic system, chemical circulation system and epidemiology. The nonlinear partial differential equation is used to describe the above problems, which fully considers the influence of space, time and time delay, so it can reflect the reality more accurately. This direction mainly studies nonlinear partial differential equation, H- semi-variational inequality, differential equation theory of optimal control system and its application in power system.

1. Research on nonlinear partial differential equations: We mainly study the existence, uniqueness (and multiplicity) and stability of solutions of partial differential equations; Existence and asymptotic behavior of global solutions (including periodic solutions and almost periodic solutions) for initial value problems and initial boundary value problems of partial differential equations; The existence of equilibrium solution, especially the bifurcation structure of equilibrium solution when the problem depends on some parameters, and the stability of equilibrium solution; Numerical solutions of nonlinear equations.

2. Research on H- semi-variational inequality: The generalized degree theory of multivalued (S) and pseudo-monotone mappings with maximal monotone operator perturbation, generalized fixed point index theory and nonlinear evolution H-semi-variational inequality theory with nonconvex non-differentiable functional are established to study nonlinear partial differential equations with discontinuous terms.

3. The differential equation theory of optimal control system and its application in power system: mainly study the theory and application of control system related to power production. Firstly, the optimal control system described by abstract nonlinear evolution equations in Banach space is studied. Non-smooth analysis is introduced to study the differential equation of optimal control system, and variational inequality theory is used to study multi-valued problems and numerical calculation. The theoretical results obtained are applied to many optimal control problems of power system (such as identification of transfer function of excitation regulator in power system, mathematical model of Newton optimal power flow, etc.). ).

(B) the characteristics of research direction

1. Variational inequality theory is closely related to the convexity of energy functional. Due to the needs of modern science and technology, especially the study of free boundary and solid mechanics, traditional methods can not solve this kind of problem. People study H- semi-variational inequality, which involves many important topics that need to be solved and developed in many fields such as modern analysis and application, partial differential equations and scientific calculation.

2. This research is an interdisciplinary research topic of modern mathematics and power production, which has very important theoretical guiding significance and practical application value for power production and management, and provides some important theoretical basis for the design, analysis and calculation of control system. In the research field of applied mathematics, this topic belongs to the frontier research work at home and abroad.

(3) Breakthroughs that can be achieved

1. deeply study the influence of space, time and time delay on the properties of solutions, such as the existence of static solutions, periodic solutions, the existence of solutions and asymptotic behavior; Seek their breakthrough in nonlinear partial differential equations with discontinuous terms.

2. Seek and find new methods to deal with nonmonotonic and nonconvex nondifferentiable energy functional (such as establishing convergence criteria of Ishikawa iterative sequence), establish G- convergence criteria of evolution equation, seek feasible smoothing methods of operator equation, and create new prior estimation methods.

3. Applying the theory of modern mathematics, the differential equation of most control system is studied, which provides some important theoretical basis and methods for the design, analysis and calculation of control system. (A) the main research content

Topology is an important and relatively young branch of mathematics, which can be divided into general topology, algebraic topology and differential topology. Since the late 1950s, the development of topology has promoted the development of mathematics and other disciplines. This direction mainly studies the singularity theory in topology, the properties of topological space and its mapping, some topics and applications in bifurcation theory.

Singularity theory is an important branch of differential topology. Singularity theory was put forward by R.Thom, a famous French mathematician, in the 20th century. Through the outstanding work of mather, Arnold and other mathematicians, great achievements have been made. In the application of geometry, the application of geometric differential equations and their geometric solutions, and the application of singularity theory and contact geometry in the study of partial differential equations have all achieved very important results.

We are committed to the research of these new topics, and have done a lot of work in the singularity classification of geometric solutions of first-order partial differential equations, the properties of singular solutions and the realization of geometric solutions. As the first and second main members, he has participated in two national natural science foundation projects, presided over the provincial natural science foundation project 1, presided over the key fund project of the provincial education department 1, and presided over the international small academic activities 1. It has also achieved some achievements that have reached the international advanced or domestic leading level. Because of these studies, we have been invited to attend international academic conferences many times. Won the second prize of scientific and technological progress in Hunan Province. We will continue our research in this field.

Golubitsky and others introduced the application of singularity theory to study the bifurcation of differential equations in 1979. In recent years, a large number of theoretical and applied research results have appeared at home and abroad. From the beginning, we followed the research frontier closely, studied several kinds of nonlinear boundary value problems with singularity theory, obtained some results about the existence of bifurcation solutions, and were invited to attend international academic conferences to give a report. There is still a lot of work to be done in this respect, especially in combination with the study of power system stability.

3. The properties of topological spaces and their mappings are one of the important branches of general topology research, which mainly studies the structure of topological spaces and the related properties of mappings between topological spaces. In recent years, we mainly study some properties of mapped images in metric spaces. And has made some remarkable achievements, published in important foreign academic journals or about to be published.

(B) the characteristics of research direction

Usually symmetry is not considered in the study of Legendre singularity in singularity theory, but we combine the theory of equivariant singularity with the study of Legendre singularity. In the research and classification of partial differential equations and their geometric solutions, we focus on the classification of more general equations and try to further study the properties of the geometric solutions after classification, which has not been done in previous studies. Especially in recent ten years, singularity theory has been applied to the geometric theory of partial differential equations. Usually, first-order equations are studied, and the future development is bound to be the trend of second-order partial differential equations. Therefore, the research direction has innovative significance and characteristics in research methods and objects.

Our research needs to combine modern topology, differential equations with geometry and algebra, and also needs to be calculated or verified by computer, which embodies the comprehensive trend of different branches of modern mathematics research and the unity of mathematics, so it has the nature of interdisciplinary research.

In addition, the application of topology theory in computer graphics and images has not started for a long time in the world, and it is still in its infancy. We can expect breakthroughs and innovations in methods and theories.

(3) Possible breakthroughs

In the research and classification of partial differential equations and their geometric solutions, we focus on the classification of more general equations and try to further study the properties of the geometric solutions after classification.

3. Using singularity theory to study nonlinear boundary value problems, and strive to achieve results on boundary bifurcation problems.

3. The research results of topological space and its mapping properties are used to study the application of computer graphics and images and electric traffic engineering. (A) the main research content

In today's scientific and engineering calculation, there are many problems such as nonlinear optimization, equation solving, least squares and eigenvalue calculation. How to design an efficient calculation method for these problems with the help of modern calculation tools and apply it to some practical problems is our main research content. Our research work will focus on the following aspects:

1. optimization calculation method and its application: study the numerical solution method of constrained nonlinear smooth and non-smooth equations, the efficient algorithm of constrained optimization problems, and analyze the properties and practical calculation performance of the established numerical method theoretically. Because the security and stability of power system can be described by nonlinear equations and optimization models, we will use new mathematical numerical methods to analyze the security and stability of power system to meet the needs of market-oriented reform of power system.

2. The application of numerical linear algebra (also called matrix calculation) is the core of scientific and engineering calculation, which mainly involves three major problems: linear algebraic equations, linear least squares and eigenvalue problems. Our research work will focus on the parallel algorithm of large-scale linear equations, the preprocessing method of ill-conditioned equations, the eigenvalue of structural matrix and the fast algorithm of least squares problem.

3. Constraint matrix equation problem: Constraint matrix equation problem includes matrix inverse eigenvalue problem, matrix least squares problem, matrix expansion problem and its best approximation problem. We will study the solvability, properties of solutions, numerical methods of constraint matrix equations and their applications in structural design, dynamic system model updating and many other engineering practices.

(B) the characteristics of research direction

1. In the research of optimization calculation method, we all consider the constraints, which not only gives the problem a general structure, but also conforms to the application background. In addition, the application analysis of power system security and stability is of great practical significance to promote the current power industry reform.

2. The research content of matrix calculation is closely related to many engineering problems, especially in signal processing, large-scale problems, ill-conditioned problems and structural matrix problems are often encountered. Therefore, our research has important theoretical and application value.

3. The research of constrained matrix equation not only uses the matrix partition, decomposition and order reduction technology of matrix theory, but also puts forward new matrix and matrix theory.

(3) Possible breakthroughs

1. Establish a numerical method for superlinear convergence of constrained nonsmooth equations; According to the decoupling method, an efficient and theoretically guaranteed algorithm for large-scale constrained nonlinear optimization problems is established. The online analysis of available transfer capability and congestion management in the safe and stable operation of power system is realized by using new mathematical methods.

2. In view of some special matrix calculation problems that often appear in process application, effective fast algorithms are designed and analyzed theoretically, thus forming high-level academic achievements.

3. Problems related to the solutions of matrix equations or new types of matrix equations under the constraints of new matrix sets; A new efficient numerical method is proposed. Some practical engineering problems are solved by using the existing constrained matrix equation theory.

(4) Introduction of major academic leaders

Tong: Professor, Ph.D., mainly engaged in the research of nonlinear equations, numerical methods of nonlinear optimization problems and power system security and stability. He presided over or participated in many research projects such as National Natural Science Foundation of China, Natural Science Foundation of Hunan Province, Outstanding Youth of Hunan Provincial Department of Education, and participated in the work of National 973 Project "Study on Several Major Issues of Catastrophe Prevention and Economic Operation of China Large Power System", and published more than 30 papers in important journals in recent 6 years. (1) Main contents

It has a solid research foundation in Markov process, stochastic analysis, mathematical finance, applied mathematical statistics and other fields, and has achieved a large number of important research results that are quite influential at home and abroad. In particular, Professor Qiu and his research group have made scientific research in the fields of two-parameter Markov processes, Markov chains and bifurcation processes in random environments and related functional equations. And in the research of IC card operating system and IC card application integration technology, it has achieved fruitful application results in the fields of human resource management, power load forecasting, traffic stochastic model, financial risk model and so on. Our research work will mainly focus on the following aspects:

1. Research on Markov chain theory in random environment: Markov chain in random environment is a hot spot in the study of contemporary random processes, and has achieved rich results, but these works need to be deepened and expanded. In this respect, we mainly study its general theories, such as irreducibility, recursion, instantaneity and the properties of its corresponding chain, large deviation theory, ergodic theory and related open problems. The properties of specific processes such as branching process, random walk, single chain and super process in random environment. Our research in this field will further improve the whole theoretical system of Markov process in random environment.

2. Theoretical study of two-parameter Markov process: Two-parameter Markov process is another hot spot in contemporary stochastic process research, and has achieved rich results, but at present, the research progress is slow, especially the study of sample orbit properties of two-parameter Markov process. The main reason is that the time parameters of the process are not completely orderly, and the methods used in the study of single-parameter Markov processes, such as first arrival time and infinitesimal operator, can no longer be used for reference, and new concepts and methods need to be introduced, but there is still no breakthrough in this respect.

3. Applied research: The research group has successfully applied probability statistics to short-term, medium-term and long-term power load forecasting of Guangxi Electric Power Bureau and its subordinate Guilin Electric Power Bureau, and achieved good economic and social benefits. We will sum up our experience and continue to do this applied research well. In addition, we are currently conducting research on the application of probability statistics in human resource management, image processing and finance. (A) the main research content

This direction mainly studies the geometric function theory in real complex analysis, the value distribution theory of meromorphic functions and some topics and applications in harmonic analysis.

1. Geometric function theory is a classical research field, which has attracted the attention of many mathematicians. Since 1970s and 1980s, with the application of convolution theory, differential subordination, fractional calculus operator and extreme point and support point theory, the research of geometric function theory has been revitalized. We are committed to the research of these new topics, and have done a lot of work in convolution operator, differential subordination, the combination of fractional calculus operator and univalent function theory, and achieved many important achievements, and won the first prize of outstanding natural science papers in Hunan Province. We will continue to explore this aspect, and have done some work in extending the relevant conclusions to quasi-* *-shaped mappings and functions with multiple complex variables.

The value distribution theory of meromorphic functions has been a hot issue in complex analysis since it was founded in the 1920s. Especially in recent twenty years, the uniqueness theory of meromorphic functions and the complex oscillation theory of differential equations have attracted the attention of many mathematicians. From the beginning, we followed the research frontier closely. At present, we have made a breakthrough in the research on the four-valued problem of meromorphic functions, and made some tentative work on the combination of the uniqueness of meromorphic functions and the complex oscillation of differential equations.

3. Harmonic analysis is one of the main branches of analytical mathematics. It mainly uses analytical tools to study the structure of function space and the boundedness of integral operators in function space, and commutators are one of the important operators. Commutators can be used to describe some function spaces and have many important applications in differential equation theory. Therefore, it has become a very active and hot research topic in recent years to study the boundedness of multilinear operators related to various integral operators (nontrivial generalization of commutators) in various function spaces. We mainly study the weighted boundedness of multilinear operators and the boundedness of multilinear operators in Hardy space and Herz space. , and achieved some remarkable results, published many papers in important academic journals at home and abroad.

Application of complex analysis theory in traffic and power engineering. The problem of pavement temperature field is studied by complex analysis theory, and the problem of temperature stress distribution in elastomer is solved. As a sub-project, the "Seventh Five-Year Plan" key project won the first prize of scientific and technological progress of the Ministry of Communications. We will continue to carry out research work in this field.

(B) the characteristics of research direction

1. Combining geometric function theory with differential equations and special functions, and combining * * shape mapping with quasi * * shape mapping can break through some technical difficulties, thus obtaining some classical results and new results more effectively and creating some new methods.

Combining the uniqueness theory of meromorphic functions with the study of complex oscillation of differential equations, it is possible to obtain some new results of complex oscillation theory of differential equations.

3. The boundedness of multilinear operators is a new research topic in harmonic analysis.

4. Pay attention to the exploration and research on the mutual connection and penetration of the above branches, so as to engage in the research of related topics from a higher angle, thus making breakthroughs and innovations in methods and theories.

(3) Possible breakthroughs

1. Deepen the theory and application of the research on the combination of differential subordination and univalent function, thus solving several difficult problems in univalent function theory.

The uniqueness theory of meromorphic functions is applied to the study of complex vibration theory of differential equations, and new results of its vibration properties are obtained.

3. The boundedness results of some multilinear operators on some function spaces are obtained. (A) the main research content

Algebra is an important basic branch of mathematics. Traditional algebra includes group theory, ring theory, module theory, field theory, linear algebra and multiple linear algebra (including matrix theory), finite-dimensional algebra, homology algebra, category and so on. At present, the development of algebra has several characteristics: first, it intersects with other branches of mathematics, such as geometry and number theory, resulting in the mainstream direction of current mathematics, such as algebraic geometry, arithmetic geometry and algebraic number theory, as well as matrix theory and combinatorics, resulting in combinatorial matrix theory. Second, the intersection of algebra with computational science and computer science has produced new directions such as computational algebra, mathematical mechanization, algebraic cryptography and algebraic automata. With the development of computing science, matrix theory is still in the development stage, showing its vitality. The third is that some old important branches of algebra are separated from algebra to form new mathematical branches, such as Lie groups and Lie algebras, algebraic K theory. And some old branches of algebra (such as ring theory) are no longer hot spots.

1. Matrix geometry and its application: At present, there are three main aspects in the development of matrix geometry: one is to extend the research of matrix geometry to rings with zero factors; The second is to simplify the conditions in the basic theorem of matrix geometry or find other equivalent conditions to find a simple proof in special circumstances; The third is to expand the research scope of matrix geometry to preserve other geometric invariants and infinite dimensional operator algebras. In recent years, our research has focused on matrix geometry and operator preservation over rings.

2. Matrix theory over rings and its application: quaternion and quaternion matrix theory have good applications in physics, mechanics, computer science and engineering technology, which have attracted the attention of engineering and technical circles at home and abroad. Matrix equation plays an important role in many practical problems (such as cybernetics and stability theory), and it is also a long-term research hotspot. We will study some important unsolved problems in matrix theory over rings and quaternion matrix theory, solve the theory of constrained matrix equations, and discuss their applications in practical problems.

3. Group theory and its application: Group theory is the foundation of algebra and the basic tool of physics. Typical group is a very important group type. We will study some important problems of typical groups over rings, and describe the structure of groups with arithmetic conditions (such as the order of groups and elements, the number of characteristic indexes, the length of yokes, etc.). ) and classify them. In this paper, the finite subgroups of general linear groups over number fields or integer rings are studied, and the structure of groups is characterized and classified by using some arithmetic conditions of groups.

4.Clifford Algebra, Hopf Algebra and Their Applications: At present, Clifford Algebra and Hopf Algebra have become popular tools in physics. Two-dimensional Clifford algebra is quaternion. We study some important problems of Clifford algebra and Hopf algebra, and discuss their applications in practical problems.

5. Application of algebra in computer science and information science: With the deepening and rapid development of informatization and Internet, information security is becoming more and more important, and protecting online information security is an extremely important new topic. Mainly using encryption technology and number recognition, in fact, it is mathematical technology, mainly using algebra, combinatorial mathematics and number theory. Image compression is a difficult and extremely important problem in information processing, and we have a good foundation in algebra.

(B) the characteristics of research direction

1. Matrix geometry is a mathematical research field initiated by mathematician Hua and inherited and developed by Chinese mathematician Academician Wan Zhexian. Belonging to the category of algebraic geometry, it has China characteristics. At present, our research in this field is at the first-class level in China.

2. With the development of computer science, the theory of matrix over rings has become an important mathematical tool and one of the important research directions of algebra in the future.

With the rapid development of Internet, information security becomes more and more important. In recent years, algebraic automata is a research direction where computer science and algebra intersect. Therefore, their basic theoretical research is particularly important.

(3) Breakthroughs that can be achieved

Continue to maintain the first-class level of matrix geometry and matrix theory research in China, and develop new research directions such as group theory, Clifford algebra, Hopf algebra, algebraic automata and algebraic cryptography. According to the actual situation of our college, we will strive to make some academic achievements in these new directions.