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Event(s) on February 2008

  • Monday, 18th February, 2008

    Title: On Acute and Nonobtuse Simplicial Partitions
    Speaker: Prof. Michal Krizek, Institute of Mathematics, Czech Academy of Sciences, Czech Republic
    Time/Place: 11:30  -  12:30
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    Abstract: We survey some results on acute and nonobtuse simplicies and associated spatial partitions. These partitions are relevant in numerical mathematics, including piecewise polynomial approximation theory and the finite element method. Special attention is paid to a basic type of nonobtuse simplicies called path-simplicies, the generalization of right triangles to higher dimensions. In addition to applications in numerical mathematics, we give examples of the appearance of acute and nonobtuse simplices in other areas of mathematics.

  • Tuesday, 19th February, 2008

    Title: The Asymptotic Existence of Resolvable Graph Designs
    Speaker: Prof. Alan Ling, College of Engineering and Mathematical Sciences, The University of Vermont, USA
    Time/Place: 11:30  -  12:30
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    Abstract: Let $v ge k$ and $lam$ be positive integers. A {em block design} BD$(v,k,lam)$ is a collection $cA$ of $k$-subsets of a $v$-set $X$ in which every unordered pair of elements from $X$ is contained in exactly $lam$ elements of $cA$. More generally, for a fixed simple graph $G$, a {em graph design} GD$(v,G,lam)$ is a collection $cA$ of graphs isomorphic to $G$ with vertices in $X$ such that every unordered pair of elements from $X$ is an edge of exactly $lam$ elements of $cA$. A famous result of Wilson says that for a fixed $G$ and $lam$, there exists a GD$(v,G,lam)$ for all sufficiently large $v$ satisfying certain necessary conditions. A block (graph) design as above is said to be {em resolvable} if $cA$ can be partitioned into partitions of (graphs whose vertex sets partition) $X$. Lu has shown asymptotic existence of resolvable BD$(v,k,lam)$, yet for over twenty years the analogous problem for resolvable GD$(v,k,lam)$ has remained open. In this talk, we settle asymptotic existence of resolvable graph designs. (a joint work with Peter Dukes)

  • Wednesday, 20th February, 2008

    Title: Adaptive meshless RBF collocation method for solving modified Helmholtz equations
    Speaker: Mr. Ting On Kwok, Department of Mathematics, Hong Kong Baptist University, HKSAR, China
    Time/Place: 16:30  -  17:30
    Abstract: In this talk, we proposed a least-squares based Kansa's method for solving the modified Helmholtz equations. In the theoretical part, we proved the convergence of the proposed method providing that the collocation points are sufficiently dense. For numerical implementation, a subspace selection process for the trial space (the so-called greedy algorithm) is employed. With the proven theories, the resultant overdetermined systems are solved by the standard least-squares method. Numerical experiments are provided To conclude the work.

  • Friday, 22nd February, 2008

    Title: Advance Hybrid High Order Methods for Multi-scales Problems in Shocked Flow
    Speaker: Prof. Wai Sun Don, The Division of Applied Mathematics, Brown University, USA
    Time/Place: 10:30  -  11:30
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    Abstract: Classical high order methods such as spectral methods and classical WENO methods are reaching their limits in handling shocked flows with both large and small scales structures, such as the classical Richtmyer-Meshkov instability and shock-particle laden flow, in an efficient and accuracy manner due to some of their inherent weakness. Spectral methods, though highly efficient and accuracy for smooth problems, suffer the Gibbs oscillations when discontinuities formed in the solution in a nonlinear hyperbolic conservation laws. High order WENO methods, on the other hand, are capable of capturing shocks in an essentially non-oscillatory manner due to its non-linear adaptive stencils switching and lowering of the order at the non-smooth stencils. However, the reconstruction procedure is an expensive undertaking and quite dissipative in nature. In this talk, I will present the work on the hybridization of these two methodologically different methods in a spatially and temporary adaptive multi-domain framework yielding a scheme which can capture shock and resolving smooth small scales structures in an efficient manner. The adaptively is based on the high order multi-resolution analysis. Examples of the standard Riemann IVP problems, Mach 3 shock-density wave interaction and two dimensional Shock-vortex interaction and Richtmyer-Meshkov instability at high Mach numbers will be shown.

  • Friday, 22nd February, 2008

    Title: What Mathematics is Hidden Behind the Astronomical Clock of Prague?
    Speaker: Dr. Alena Solcova, Department of Mathematics, Czech Technical University
    Time/Place: 16:30  -  17:30
    University Chapel
    Abstract: We present several geometrical and algebraical theorems that have a close connection with the astronomical clock (horologe) of Prague. In particular, we show that there is a remarkable relationship between the triangular numbers and the bellworks of the astronomical clock. The dial-plate is an astrolabe controlled by a clockwork mechanism. It represents a stereographic projection of the celestial sphere from its North Pole onto the tangent plane passing through the South Pole.

  • Wednesday, 27th February, 2008

    Title: Nonlinear Problems in Analysis of Krylov Subspace Methods
    Speaker: Prof. Zdenek Strakovs, Institute of Computer Science, Czech Academy of Sciences and Faculty of Mathematics and Physics, Charles University, Prague
    Time/Place: 11:30  -  12:30
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    Abstract: Consider a system of linear algebraic equations Ax = b where A is an n by n real matrix and b a real vector of length n. Unlike in the linear iterative methods based on the idea of splitting of A, the Krylov subspace methods, which are used in computational kernels of various optimization techniques, look for some optimal approximate solution xn in the subspaces Kn (A, b) = span {b, Ab, …, An-1b}, n = 1, 2,… (here we assume, with no loss of generality, x0 = 0). As a consequence, though the problem Ax = b is linear, Krylov subspace methods are not. Their convergence behaviour cannot be viewed as an (unimportant) initial transient stage followed by the subsequent convergence stage. Apart from very simple, and from the point of view of Krylov subspace methods uninteresting cases, it cannot be meaningfully characterized by an asymptotic rate of convergence. In Krylov subspace methods such as the conjugate gradient method (CG) or the generalized minimal residual method (GMRES), the optimality at each step over Krylov subspaces of increasing dimensionality makes any linearized description inadequate. CG applied to Ax = b with a symmetric positive definite A can be viewed as a method for numerical minimization the quadratic functional 1/2 (Ax, x) - (b,x). In order to reveal its nonlinear character, we consider CG a matrix formulation of the Gauss-Christoffel quadrature, and show that it essentially solves the classical Stieltjes moment problem. Moreover, though the CG behaviour is fully determined by the spectral decomposition of the problem, the relationship between convergence and spectral information is nothing but simple. We will explain several phenomena where an intuitive commonly used argumentation can lead to wrong conclusions, which can be found in the literature. We also show that rounding error analysis of CG brings fundamental understanding of seemingly unrelated problems in convergence analysis and in theory of the Gauss-Christoffel quadrature. In remaining time we demonstrate that in the unsymmetric case the spectral information is not generally sufficient for description of behaviour of Krylov subspace methods. In particular, given an arbitrary prescribed convergence history of GMRES and an arbitrary prescribed spectrum of the system matrix, there is always a system Ax = b such that GMRES follows the prescribed convergence while A has the prescribed spectrum.

  • Thursday, 28th February, 2008

    Title: Model-less Pose Tracking
    Speaker: Dr. Eric Yu, Department of Computer Science and Engineering, The Chinese University of Hong Kong, Hong Kong
    Time/Place: 11:30  -  12:30
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    Abstract: Acquiring 3-D motion of a camera from image sequences is one of the key components in a wide range of applications such as human computer interaction. Given the 3-D structure, the problem of camera motion recovery can be solved using the model-based approaches, which are well-known and have good performance under a controlled environment. If prior information on the scene is not available, traditional Structure from Motion (SFM) algorithms, which simultaneously estimate the scene structure and pose information, are required. The research presented in this talk belongs to a different category: Motion from Motion (MFM), in which the main concern is the camera position and orientation. To be more precise, MFM algorithms have the capability of estimating 3-D camera motion directly from 2-D image motion without the explicit reconstruction of the scene structure, even though the 3-D model structure is not known in prior. As keeping track of the structural information is no longer required, putting these types of algorithms into real applications is relatively easy and convenient. We have recently proposed a high-speed recursive MFM approach based on the trifocal tensor. It is demonstrated in the experiment that our algorithm is efficient, stable and accurate compared to existing methods. Furthermore, the proposed algorithm has been applied to applications such as mixed reality, virtual reality, robotics and super-resolution to show its performance in real situations.