Research Projects
- Deterministic and Stochastic models in Science, Medicine and Technology
- Các phương pháp đại số và tổ hợp cho đồ thị và siêu đồ thị;
- Some selected problems in algebra and number theory;
- Some
Deterministic and Stochastic models in Science, Medicine and Technology
Manager: Dr Pham Viet Hung
1. Research scope.
Study the existence of weak solution to a class of fractional PDEs. Study the large time behavior of solutions to the incompressible Navier-Stokes equations. Study the Boussinesq systems on the background of a line solitary wave. Study ergodic property of some stochastic iterated systems and applications in additive combinatorics. Study the affect of noise in synchronization for some simple models. Study the conjunction probability of smooth random fields. Study the statistical analysis in pesticides business. Study the least square method.
2. Literature review
In practice, to investigate the phenomenon in science, medicine and technology, ones usually model them by ordinary differential equations or partial differential equations. For instance, the SIRS models describe the evolution of epidemics, or the celebrated Navier-Stokes equations describe some fluid motions. Moreover, to consider the unknown environmental affects, some stochastic noises are added. Therefore, the study of the validity of the models and the properties of solutions such as existence, asymptotic behaviors, ergodicity , etc, plays an important role not only in theory but also in practice.
Fractional partial differential equations are generalizations of classical integer order partial differential equations. This kind of equations are considered to describe more precisely some phenomenon in nature, finance and biology. However, to give an exact analytic solution is challenging. Study the weak solution to these equations has been draw much of interest and needs to investigate more.
The incompressible Navier-Stokes equations and Boussinesq systems are the fundamental equation of hydraulics to model the isotropic fluids. Study the properties of the solution of these equations is always of central interest in Analysis.
The random dynamical systems have been investigated since 1905 by Einstein. The mathematicians realize that the effect of the noise is non-trivial, for example, noise can stabilize unstable equilibria and shift bifurcations. Study the ergodicity of random iterated systems and the affect of noise in synchronization deserve to be considered more.
Study of the conjunction probability of smooth Gaussian fields is motivated from the statistical test to analysis the similarity and difference in functional brain region between man and woman. Moreover, it is also an interesting theoretical question.
To analysis data, in statistics, many methods have been proposed such as: least square method, principal component analysis, regression, survival analysis, multilevel analysis,… To apply these methods in a particular problem to analysis the pesticides business is highly practical and interesting.
3. Approaches and Methods
Using the tools from Functional analysis, Harmonic analysis, PDEs, Stochastic analysis and Statistics.
Manager: Dr. Tran Nam Trung
1. The purpose of the project
Study the algebraic and combinatorial structures via analyzing the properties of graphs and hypergraphs and related topics.
2. Overview of research in the field of the project
The relationship between algebraic structures and combinatorial structures are studying very active and have many applications in many branches of mathematics.
One of the important problem is to understand the relationship between algebraic objects associated to graphs or hypergraphs, such as: edge ideals, cover ideals, Leavitt path algebra ... Recently, this topics continue to elude many researchers. We would like to study these objects in more detail and deeper on both algebraic and combinatorial aspects.
We also study the structure of dynamic systems on graphs and parallel dynamic systems.
3. Approaches, study methods
Establishing the relationship between the algebraic structures and the structures of associated graphs and hypergraphs. Applying the structure of graphs and related algebraic properties to study discrete dynamic systems
Using tools from Combinatoric Commutative Algebra to our objects based on underlying graphs or hypergraphs.
Extend some results from graphs to weighted graphs
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Some selected problems in algebra and number theory
Manager: Dr. Nguyen Bich Van
1. The purpose of the project:
We study some selected problems in algebra and number theory listed below.
a) We study applications of Nevanlinna theory for meromorphic functions. We concentrate on studying the problem of defining a unique meromorphic function under the condition of the derivative polynomial. We also study the distribution of values of derivative polynomials and extend Mue’s results for differential polynomials in a more general form. We present the relation between the counting function and the characteristic function in Nevanlinna theory.
b) The classic zeta function is an important object of mathematics during more 100 passed years. In the recent years, there are some interesting results about similar objects built from fields of positive characteristic instead of the field of rational numbers for the classic zeta function. We study advantages obtained in this research. Namely, we study the construction of the L-functions of Pellarin and some results about their special values.
c) We study the structure of the maximal pro-p quotient of an absolute Galois group.
d) We study higher secant varieties of some projective
e) We study fields of values of complex irreducible characters of finite (almost quasisimple) groups.
f) We study induced graded simple modules over Steinberg algebras
g) We study the depth functions and the regularity indices of ordinary powers and symbolic powers of ideals. We try to answer some open questions on the depth functión and regularity indices of powers of ideals of polynomial rings.
2. Overview of research in the field of the project
a) This research has attracted many mathematicians in the world. They obtained various important results.
b) L-functions over fields of positive characteristics have been studied by various approaches. In the topic of our project, there were constructions and results of Carlitz from 1930 to 1970 and then there were contributions of Goss, Thakur, Anderson in 1980’s and 1990’s.
In 2011, F. Pellarin constructed a class of L-functions and proved some their properties. This work attracted of many mathematicians in the world. In the recent years, there were various advantages in the theory of L-functions over fields of positive characteristic.
c) Absolute Galois groups and their maximal pro-p quotients are important objects in mathematics. Let F be a field and G(p) denote the maximal pro-p quotient of the absolute Galois group of F. The papers of Kawada, Demushkin, Serre and Labute allow us describe G(p) via generators and relations when F is a local field containing the p-th primitive root. These results showed that the relation r of G(p) has a special form. A recent result showed that when we deform the relation a little bit, we will obtain groups which are never isomorphic to the maximal pro-p quotient of the absolute Galois group of any field containg the p-th primitive root.
d) Secant varieties were studied by Italian algebraic geometers in the 19th century. Recently, algebraic geometers pay more and more attention on secant varieties. Secant varieties have applications in some pure mathematical fields such as algebraic statistics and in various fields related directly to our life such as computer science, biology ...In general it is very difficult to compute with secant varieties.
e) When we study the character tables of finite groups, we see that there is not any irreducible character of odd order which has the field containing Q and square roots of 2 or -2 as the field of values. On the other hand, the extension of Q containg the square root of -3 is the field of values of linear characters of a cyclic group of order 3. This observation shows that the field of values of the irreducible characters of a finite group has special, interesting properties which can be studied. Some results about these properties were obtained by I.M. Isaacs, M.W. Liebeck, G. Navarro, P.H. Tiep in 2019.
f) For an ample groupoid G and an element u in the unit space of G B. Steinberg constructed induction functor ($In{\rm{d}}_u$) and restriction functor $Res_u$ ) between the category of modules over the Steinberg algebra Steinberg $A_R(G)$ and the category of modules over the isotropy group algebra of u. We will prove a graded version of these functors and some related results, namely, we will show that every spectral graded simple module over the Steinberg algebra of a graded ample groupoid has the form $In{\rm{d}}_u (N)$, where u is an element in the unit space of this groupoid, N is a graded simple module over the isotropy group algebra of u. When we apply these results for Leavitt path algebras, we can show that all the graded simple modules over Leavitt path algebras constructed by Hazrat-Rangaswamy using E-algebraic branching systems are induced graded modules. In particular, we will prove that all the (non graded) Chen simple modules are induced simple modules.
g) In the late 1970’s Brodmann proved that the depth function of an ordinary power of an ideal of a local ring always converges. Since then there are various papers about algebraic properties and asympotic homology of powers of ideals. For example, Cutkosky-Herzog-Trung and Kodiyalam proved in 1999-2000 that the regularity index function of ordinary powers of a homogeneous ideal of a polynomial ring is a linear function when the powers are big enough. In this project, we concentrate on 2 recent open questions about the depth functions and the regularity index functions. The first question is the Herzog-Hibi’s conjecture, which states that every convergent nonnegative function is the depth function of some ideal of a polynomial ring. The second one is the question of Minh-T.N.Trung (2018): is the regularity index function of symbolic powers of a monomial squarefree ideal a linear function when the powers are big enough?
Herzog-Hibi’s conjecture was verified for nondecreasing convergent functions. The question of Minh-T.N.Trung has the positive answer for Stanley-Reisner ideals of matroid simplicial complexes.
3. Approaches, study methods
a) We use knowledges of complex analysis, number theory,...
b) We try to understand the construction, typical techniques in studying L-functions over fields of positive characteristic. We hope that in this process we can give some advantages in these techniques and we are interested in solving some problems arising from the theory of L-functions over fields of positive characteristic.
c) We use Galois theory and construction of Galois extensions.
d) We study some basic properties of higher secant varieties of some special varieties such as Veronese variety, Grassmanian variety, Serge variety, then we try to extend results for projective varieties.
e) We try to understand properties of fields of values of irreducible characters of (almost quasisimple) finite groups.
f) We use representation theory, category theory, theory of graded modules.
g) To work with the Herzog-Hibi’s conjecture we concentrate on the construction of nondecreasing convergent depth functions and the constructio of depth functions which have values 0,0,...,0,1,0,...i.e. they have value 0 everywhere except a position. If we can construct these functions, then we will be able to show that Herzog-Hibi’s conjecture hold true. Moreover, besides solving Herzog-Hibi’s conjecture, we can concentrate on studying characterization of the depth functions of squarefree monomial ideals. It is a much more difficult question, since examples of squarefree monomial ideals with increasing depth functions exist only in polynomial rings of dimension larger or equal to 10 (the well-known example of Kaiser-Stehlik-Skrekovski in 2014 exists in the dimension 12 ).
To answer the question of Minh-T.N. Trung we study a class of squarefree monomial ideals arising from graphs, i.e. covering ideals of graphs. We can show that the regularity index functions of powers of covering ideals are quasilinear with period 2. We will concentrate on the problem of existence of example of a regularity index function which is actually quasilinear with period 2. TheTakayama’s formula which expresses the relation between local cohomology of monomial ideals and reduced simplicial homology of simplicial complexes is a powerful tool in this research.
4. International cooperation (if any):
a) We work with Professor William Cherry (North Texas University) and Professor Julie Wang (Institute of Mathematics, Taiwan)
d) We cooperate with some groups of mathematicians in South Korea and the US.
g) We work with professor Hà Huy Tài (Tulane University, USA) and Dr. Seyed Amin Seyed Fakhari (Iran).
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1.Objectives
The objective of this project is
(i) to study some selected problems in optimization and control for dynamical systems described by differential-difference control equations, fractional-order and singular systems with delays: Stability, stabilization, H-infinity control, controllability.
(ii) to research and propose improved methods/algorithms for some opimization problems: Maximum principle in optimal control; minimax and equilibirum; Nonlinear Modokhovich analysis, Hybrid gradient methods, Convex optimization; Second order conditions for opimal control of some partial differential euqations.
2. Overview of research situation of the project
Dynamical systems modelled by differential-difference equations, fractional-order euqtions and partial differential equations have been appeared in many practical system applications such as power systems, economic systems, digital communication networks, urban traffic network, etc. ). These models are more general and provide an excellent instrument for the description of memory and hereditary processes. Qualitative problems of these systems have been widely studied during the past decades and have recieved a lot of excellent results by using various effective methods/algorithms as bounded variational approaches, positive measured functions on Borel σ-algebra methods, Riesz theorem on continuous functions, etc.
However, the obtained results were focused on specific dynamical systems. To the best of our knowledge, many qualitative problems for a more general class of dynamical systems decribed by differential-difference equations, fractional-order euqtions and partial differential equations, with delays, etc. are not fully investigated.
Research on optimal rpoblems, stability and control of such is one of the difficult problems in qualitative theory of dynamical systems. This problem were widely investigated for specific models of the systems, meanwhile the investigation for the models with more complicated structure on both the system and the delay has been still unsolved. That is why the problems we aim to solve in the project are new, actual and have scientific significance.
3. Approaches and methods
To solve the problems we use new mathematical tools in optimazation and control theory (nonlinear analysis, nonsmooth analysis), theory of differetial-difference equations, linear algebra, functional analysis, convex analysis, etc. Depending on the investigation of each objective system, new method and techniques will be employed for solving the problems.
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