
Title: Algebraic
Properties and Preconditioning Methods for Block TwobyTwo Matrices
Zhongzhi Bai Chinese Academy of Sciences
Abstract: Large sparse systems of linear equations of block twobytwo
coefficient matrices have plentiful application backgrounds in scientific computing
and engineering applications. The standard and the generalized saddle point problems
are their typical special cases. We discuss algebraic properties of the block
twobytwo matrix, including nonsingularity, blocktriangular factorizations,
and eigenvalue estimates; provide algorithmic constructions of the block
preconditioners and eigenproperties of the preconditioned matrices; and
introduce effective approximations to the Schur complements. In addition, we
address the augmented Lagrangian approach for designing the blockdiagonal
and blocktriangular preconditioning matrices for application problems such
as the discrete NavierStokes equations.

Title: Incremental
Linear Discriminant Analysis
Delin Chu National University of Singapore
Abstract: It has been a challenge problem to develop fast and efficient
incremental linear discriminant analysis (LDA) algorithms although several
incremental LDA algorithms have been proposed in the past. For this purpose,
we conduct a new study on LDA and develop a new and efficient incremental LDA
algorithm. We first propose a new batch LDA algorithm called LDA/QR which
only depends on the data matrix and the sizes of data classes. LDA/QR is
obtained by computing the economic QR factorization of the data matrix
followed by solving a lower triangular linear system. Hence, LDA/QR is a
simple and fast LDA algorithm. The relationship between LDA/QR and
Uncorrelated LDA (ULDA) is also revealed. Based on LDA/QR, we develop a new
incremental LDA algorithm called ILDA/QR which is the exact incremental
version of LDA/QR.
The main features of our incremental LDA algorithm ILDA/QR include: (i) it
can easily handle not only the case that only one new sample is inserted but
also the case that a chunk of new samples are added; (ii) it has pleasant
computational complexity and space complexity; and (iii) it is very fast and
always achieves comparative classification accuracy compared with ULDA
algorithm and existing incremental LDA algorithms. Numerical experiments
using some real world data demonstrate that our ILDA/QR is very efficient and
competitive with the stateoftheart incremental LDA algorithms in terms of
classification accuracy, computational complexity and space complexity.

Title: Graphs associated with
matrices over a finite field
Liping Huang Changsha University of Science & Technology
Abstract: Let \mathbb{F} be a finite field. Let \mathbb{F}^{m\times n} be
the set of m\times n matrices over \mathbb{F}, and {\mathcal K}_n the
set of all n\times n alternate matrices over \mathbb{F}, respectively. Consider
a graph G=(V,\sim), where vertex set V=\mathbb{F} ^{m\times n} (resp.
{\mathcal K}_n), and A\sim B \Leftrightarrow rank (AB)=1 (resp. 2) for all
A,B\in V. We introduce some graph properties of graphs associated with
m\times n matrices (resp. alternate matrices) over a finite field, they are:
graph homomorphism, independence number, chromatic number, to distinguish
Hamiltonian graph, to distinguish perfect graph, etc.

Title: Minimum rank solutions to the matrix approximation problems in
spectral norm
Musheng Wei Shanghai Normal University
Abstract: In
this paper, we discuss the following two minimum rank matrix approximation
problems in spectral norm: \min\limits_X \rank(X), subject to \ABXC\_2=\theta,
where \theta \triangleq \min \limits_{Y} \ABYC\_2; \min\limits_X\rank(X),
subject to \ABXC\_2<\xi, where \xi >\theta . We solve
the first problem by applying the normpreserving dilation theorem and the
restricted singular value decomposition (RSVD) to characterize three
expressions of the minimum rank, and derive general forms of minimum rank
solutions. We also characterize the expressions of the minimum rank of the
second problem, which is a generalization of the result in the literature.

Title: Recursion Theorem: from Liar’s
paradox to computer virus
Guohua Wu Nanyang Technological University
Abstract: Kleene’s
Recursion Theorem is one of the most fundamental theorems in computability
theory, which says that no computable function can be fixedpoint free. In
this talk, I will give an analysis of Kleene’s recursion theorem, from
selfreference point of view, and show how people develop theoretical
foundation of computer virology with recursion theorem as a basis.

Title: List decoding of Gabidulin
codes
Chaoping Xing Nangyang Technological University
Abstract: Gabidulin codes have found various applications in reliable
communication of messages in linear network coding, crisscross
errorcorrection in magnetic tape recording or memory chip arrays, spacetime
coding in wireless communications, public key cryptosystems. Despite this
interest and many results paralleling ReedSolomon codes, an algorithm for
list decoding Gabidulin codes beyond half the distance has remained elusive.
In this talk, we present a Monte Carlo construction of a Gabidulin code that
is efficiently list decodable up to the Singleton bound.

Title: Galois Theory, Geometry and
Coding
Jietai Yu University of Hongkong
Abstract: In this talk we will explore the close connection among the
classical Galois theory, geometry, camplex analysis and matrix theory via
symmetry and representation, and possible applications in coding theory.

Title: Some
efficient algorithms for skew polynomials
Yang Zhang University of Manitoba
Abstract: In this talk, we will outline some
efficient algorithms for factoring skew polynomials and computing normal
forms of matrices over skew polynomials, and describe some applications in
cryptography, coding and control theory.

Title:
DNAlike structure of nonlinear functions
Zhenyue Zhang Zhejiang University
Abstract: The importance of DNA to living organisms and its general
structure of a double helix joined by base pairings whose sequences encode
the genetic information are well known facts. Demonstrated in this article,
via elementary calculus, is that there is a remarkably similar structure
underneath all smooth nonlinear functions. The idea is based on generalizing
the notion of gradient adaption from a scalar field to vector fields —
singular vectors of the Jacobian of any given function form a moving frame
pointing in directions along which the function changes most critically.
Trajectories of these singular vectors, referred to as singular curves,
unveil some interesting, perplexing, innate dynamics per the given function.
At points where two or more singular values coalesce, curious and complex
behavior occurs, manifesting specific landmarks for the underlying function.
This talk reports for the first time such a dynamical system. For
demonstration purpose only, the current study focuses primarily on the case
of twovariable functions which, commonly used to characterize parametric
surfaces, already exhibits some intriguing patterns. It is shown that,
analogous to the double helix structure of DNA, two strands and eight base
pairings could encode properties of an arbitrary surface.

