Over the past decade, the randomized singular value decomposition (RSVD)
algorithm has proven to be an efficient, reliable alternative to classical
algorithms for computing low-rank approximations in a number of applications.
However, in cases where no information is available on the singular value
decay of the data matrix or the data matrix is known to be close to full-rank,
the RSVD is ineffective. In recent years, there has been great interest in
randomized algorithms for computing full factorizations that excel in this
regime.  In this talk, we will give a brief overview of some key ideas in
randomized numerical linear algebra and introduce a new randomized algorithm for
computing a full, rank-revealing URV factorization.

We introduce and analyze a discontinuous Galerkin method for the Oseen equations in two dimension spaces. The boundary conditions are mixed and they are assumed to be of three different types:
the vorticity  and the normal component of the velocity are given on a first part of the boundary, the pressure and the tangential component of the velocity are given on a second part of the boundary and the Dirichlet condition is given on the remainder part . We establish a priori error estimates in the energy norm for the velocity and in the L2 norm for the pressure. An a posteriori error estimate is also carried out yielding optimal convergence rate. The analysis is based on rewriting the method in a non-consistent manner using lifting operators in the spirit of Arnold, Brezzi, Cockburn and Marini.

We consider numerical methods for solving  time dependent partial differential equations with convection-diffusion terms and anti-diffusive fractional operator of order $\alpha \in (1,2)$. These equations are motivated by two distinct applications: a dune morphodynamics model and a signal filtering method. 
We propose numerical schemes based on local discontinuous Galerkin methods to approximate the solutions of these equations. Numerical stability and convergence of these schemes are investigated. 
Finally numerical experiments are given to illustrate qualitative behaviors of solutions for both applications and to confirme the convergence results. 

Several scholars of evolutionary biology have suggested that functional redundancy (also known as "biological degener-
acy") is important for robustness of biological networks. Structural redundancy indicates the existence of structurally
similar subsystems that can perform the same function. Functional redundancy indicates the existence of structurally
di erent subsystems that can perform the same function. For networks with Ornstein--Uhlenbeck dynamics, Tononi et al.
[Proc. Natl. Acad. Sci. U.S.A. 96, 3257{3262 (1999)] proposed measures of structural and functional redundancy that are
based on mutual information between subnetworks. For a network of n vertices, an exact computation of these quantities
requires O(n!) time. We derive expansions for these measures that one can compute in O(n3) time. We use the expan-
sions to compare the contributions of di erent types of motifs to a network's functional redundancy.

In this talk, we will discuss some recent progress on the study of six-dimensional S-matrices as well as their applications. Six-dimensional theories we are interested include the world-volume theories of single probe M5-brane and D5-brane, as well as 6D super Yang-Mills and supergravity. We will present twistor-string-like formulas for all these theories, analogue to that of Witten’s twistor string formulation for 4D N=4 SYM. 
As the applications, from the 6D results we also deduce new formulas for scattering amplitudes of theories in lower dimensions, such as SYM and supergravity in five dimensions, and 4D N=4 SYM on the Columbo branch. 
 

It is an old problem in number theory to count number fields of a fixed degree and having a fixed Galois group for its Galois closure, ordered by their absolute discriminant, say. In this talk, I shall discuss some background of this problem, and then report a recent work with Stanley Xiao. In our paper, we considered quartic $D_4$-fields whose ring of integers has a certain nice algebraic property, and we counted such fields by their conductor.