Application of Time Frequency Transformation in Medical and Biological Engineering
Prof. Yih-Nen Jeng ( Department of Aeronautics and Astronautics, National Cheng Kung University )2008 - 04 - 11 (Fri.)
13:30 - 14:30
405, Mathematics Research Center Building (ori. New Math. Bldg.)The talk will introduce how to remove the non-sinusoidal part of a continuous data string. The procedure to obtain a Fourier sine or cosine spectrum via a monotonic cubic interpolation will be shown too. Next, the Fourier sine spectrogram, which is a time frequency transformation generated from the Fourier sine spectrum, is theoretically compared with the existing short time Fourier transform and continuous wavelet transform. Finally, both the spectrum and spectrogram are employed to examine ECG and ABP data of several typical cases. Meanwhile, a sonocardiography system to sense wrist pulse is employed to take data and then analyzed by the Fourier sine spectrogram.

Prof. Yih-Nen Jeng  ( Department of Aeronautics and Astronautics, National Cheng Kung University )

Average performance of the sparsest approximation using a dictionary
Prof. Mila Nikolova ( Centre de Mathématiques et de Leurs Applications, ENS Cachan )2008 - 04 - 11 (Fri.)
10:30 - 11:20
405, Mathematics Research Center Building (ori. New Math. Bldg.)We consider the minimization of the number of non-zero coefficients (the ℓ0 norm) of the representation of a data set in terms of a dictionary under a fidelity constraint. (Both the dictionary and the norm defining the constraint are arbitrary.) This (non-convex) optimization problem naturally leads to the sparsest representations, compared with other objectives. Our goal is to measure the sets of data yielding a K-sparse or sparser solution - i.e. involving at most K non-zero components. Data are assumed uniformly distributed on a domain defined by a norm. Relevant bounds on the Lebesgue measure of these sets are derived. They naturally lead to bound the probability of getting a K-sparse (or sparser) solution. We specify these results in the case of the Euclidian norm and an arbitrary dictionary. Then we specialize the problem to orthogonal bases (e.g. splines or wavelets) and focus on popular choices for the two norms. We get sharper bounds that are easy to compute. These results have a considerable theoretical and practical importance.

Mila Nikolova 2008-04-11
10:30:00 - 11:20:00 Average performance of the sparsest approximation using a dictionary405 , Mathematics Research Center Building (ori. New Math. Bldg.)

Yih-Nen Jeng 2008-04-11
13:30:00 - 14:30:00 TBA405 , Mathematics Research Center Building (ori. New Math. Bldg.)

Chern-Simons-Higgs Models in Superconductivity (Ⅲ)
Prof. Chang-Shou Lin ( Department of Mathematics, National Taiwan University )2008 - 04 - 10 (Thu.)
11:10 - 12:20
308, Mathematics Research Center Building (ori. New Math. Bldg.)

Comparison theorems in Rimannian geometry
Mr. Sheng Fu Chio ( Department of Mathematics, National Taiwan University )2008 - 04 - 10 (Thu.)
19:00 - 20:30
405, Mathematics Research Center Building (ori. New Math. Bldg.)

Application of Braid Group Theory to PDE
Prof. Chin-Chin Wu ( Department of Applied Mathematics, National Chiao Tung University )2008 - 04 - 10 (Thu.)
14:00 - 15:00
308, Mathematics Research Center Building (ori. New Math. Bldg.)

Stability of spacelike hypersurfaces in foliated spacetims
Mr. Ye-Kai Wang ( Department of Mathematics, National Taiwan University )2008 - 04 - 10 (Thu.)
13:00 - 14:00
103, Old Mathematics Building

Mr. Ye-Kai Wang  ( Department of Mathematics, National Taiwan University )

Ye-Kai Wang 2008-04-10
13:00:00 - 14:00:00 Stability of spacelike hypersurfaces in foliated spacetimes103 , Old Mathematics Building

Physics and mathematics of dispersive shocks: An introduction
Prof. Anatoly Kamchatnov ( Institute of Spectroscopy, Russian Academy of Science )2008 - 04 - 09 (Wed.)
15:30 - 17:20
405, Mathematics Research Center Building (ori. New Math. Bldg.)In this lecture, we shall apply the Whitham modulation theory developed earlier for the KdV equation to description of dispersive shocks. Gurevich-Pitaevskii problem is formulated. Its solution is given for a dispersive shock formed due to decay of an initial step-like distribution.

The Cauchy problem of the Ward equation I
Prof. Derchyi Wu ( Academia Sinica )2008 - 04 - 08 (Tue.)
14:00 - 15:00
405, Mathematics Research Center Building (ori. New Math. Bldg.)

Prof. Derchyi Wu  ( Academia Sinica )

Ricci flow
Prof. Chang-Shou Lin ( Department of Mathematics, National Taiwan University )2008 - 04 - 08 (Tue.)
09:10 - 12:10
405, Mathematics Research Center Building (ori. New Math. Bldg.)

Derchyi Wu 2008-04-08
14:00:00 - 15:00:00 The Cauchy problem of the Ward equation (I)405 , Mathematics Research Center Building (ori. New Math. Bldg.)

Comparison Theorems in Riemannian Geometry
Mr. Jui-En Chang ( Department of Mathematics, National Taiwan University )2008 - 04 - 03 (Thu.)
19:00 - 20:30
405, Mathematics Research Center Building (ori. New Math. Bldg.)

Ricci Flow
Prof. Yu-Chao Tu ( University of Utah, USA )2008 - 04 - 03 (Thu.)
13:00 - 14:00
103, Old Mathematics Building

Physics and mathematics of dispersive shocks: An introduction
Prof. Anatoly Kamchatnov ( Institute of Spectroscopy, Russian Academy of Science )2008 - 04 - 02 (Wed.)
15:30 - 17:20
405, Mathematics Research Center Building (ori. New Math. Bldg.)In this lecture, we shall introduce basic notions of Whitham modulation theory. The method will be illustrated by a simple example of derivation of Whitham equations for a modulated periodic wave which evolution is governed by linear Klein-Gordon equation. After that we shall proceed to derivation of Whitham equations for the KdV case.

Black holes and compact Einstein manifolds
Prof. Yoshitake Hashimoto ( Osaka City University, Japan )2008 - 04 - 01 (Tue.)
11:10 - 12:00
308, Mathematics Research Center Building (ori. New Math. Bldg.)This is a joint work with M. Sakaguchi and Y.Yasui. We construct an infinite family of inhomogeneous Einstein metrics on a 3-sphere bundle on a 2-sphere. They appear as limits of Kerr-de Sitter balck holes. Our construction is a higher dimesional version of the method of D. Page, by which he constructed an inhomogeneous compact Einstein 4-manifold.

Differential Geometry of Lagrangian submanifolds and related variational problems
Prof. Yoshihiro Ohnita ( Osaka City University, Japan & Osaka City University Advanced Mathematical Institute, Japan )2008 - 04 - 01 (Tue.)
09:00 - 09:50
308, Mathematics Research Center Building (ori. New Math. Bldg.)In this talk I will give a survey of my recent works and their environs on Lagrangian submanifolds in specific K\"ahler manifolds such as complex projective spaces, complex space forms and Hermitian symmmetric spaces and so on. Especially I shall discuss (1) the Hamiltonian stability problem of Lagrangian submanifolds, (2) the classification problems of homogeneous Lagrangian submanifolds, (3) the relationship between certain minimal Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres. This talk is mainly based on joint works with Amartuvshin Amarzaya(Natonal Univ. Mongolia), Hui Ma (Tsinghua Univ., Peking).

Surfaces of prescribed $p$-mean curvature in the Heisenberg group
Prof. Jih-Hsin Cheng ( Academia Sinica )2008 - 04 - 01 (Tue.)
14:00 - 14:50
308, Mathematics Research Center Building (ori. New Math. Bldg.)I will report on the study of the prescribed $p$% -mean curvature equation in recent years. We interpreted the $p$-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a characteristic curve (in dimension $2$), and as a quantity in terms of calibration geometry. As a differential equation, the prescribed $p$-mean curvature equation is degenerate (hyperbolic and elliptic) in dimension 2 while subelliptic in the nonsingular domain for higher dimensions. Also in dimension 2 with zero $p$-mean curvature, it is a special type of scalar conservation law in the variable of the so-called characteristic angle. We analyzed the singular set and formulated an extension theorem. This allowed us to classify the entire solutions to this equation and to solve a Bernstein-type problem. As a geometric application, we proved the nonexistence of $C^{2}$ smooth hyperbolic surfaces having bounded $p$-mean curvature, immersed in a pseudohermitian 3-manifold. From the variational formulation of the equation, we studied the Dirichlet problem by proving the existence and the uniqueness of the ($p$-)minimizers. An intriguing point is that, in dimension 2, a $C^{2}$ smooth solution from the PDE viewpoint may not be a minimizer. As an application to neuro-biology, we could use such a $p$-minimal surface theory as an image completion model for missing visual data in the first layer of the visual cortex. About the regularity of a $C^{1}$ smooth surface with continuous prescribed $p$-mean curvature, we proved that the horizontal normal gains one more derivative by invoking the jump formulas along characteristic curves. Near a singular (or shock) curve we depicted the Rankine-Hugoniot jump condition as characteristic incident and reflected angles being equal. In the last section we reviewed the second variation formula for a $C^{2}$ smooth $p$-minimal surface in an arbitrary $3$-dimensional pseudohermitian manifold.

Special solutions to Lagrangian mean curvature flow
Prof. Yng-Ing Lee ( Department of Mathematics, National Taiwan University )2008 - 04 - 01 (Tue.)
10:00 - 10:50
308, Mathematics Research Center Building (ori. New Math. Bldg.)I have been interested in and working on the existence of special Lagrangian/Lagrangian minimal. There are several different approaches we have taken and obtained various results. In recent years, we took the approach of mean curvature flow. Singularities may occur in the flow. They usually model on self-similar solutions and translating solutions. Hence it is very important to study these solutions in studying the singularities. We have constructed many such solutions in a joint work with Joyce, and Tsui. Among our examples, there are three types that are of particular interests. One family is Lagrangian self-expanders that are asymptotic to pair of Lagrangian planes. The Lagrangian angle can be made arbitrarily small, and these examples can serve as local model for surgery. Another family is translating solutions with arbitrarily small Lagrangian angle. Our example is rather surprising and indicates that the flow does not behave well even if under the almost calibrated assumption, which is different from the expectation in the past. Some other families of our examples give external Brakke solutions.

The Li-Yau Eigenvalue Estimates and CR Obata Theorem in Pseudohermitian Manifolds "
Prof. Shu-Cheng Chang ( Department of Mathematics, National Taiwan University )2008 - 04 - 01 (Tue.)
15:00 - 15:50
308, Mathematics Research Center Building (ori. New Math. Bldg.)In this talk, based on a jointed work with Hung-Lin Chiu, we first have the CR analogue of Li-Yau's eigenvalue estimate on the lower bound estimate of psitive first eigenvalue of the sub-Laplacian in pseudohermitian $(2n+1)$-manifolds. Second, we prove the CR Obata's theorem on a closed pseudohermitian manifold with vanishing torsion. The key step is a discovery of CR analogue of Bochner formula which involving the CR Paneitz operator and nonnegativity of CR Paneitz operator for any closed pseudohermitian $(2n+1)$-manifold with $n>1$.

Prof. Ping Lin  ( University of Dundee, UK )

Weakly reflective orbits and tangentially degenerate orbits of $s$-representations
Prof. Takashi Sakai ( Tokyo Metropolitan University, Japan & Osaka City University Advanced Mathematical Institute, Japan )2008 - 04 - 01 (Tue.)
16:10 - 17:00
308, Mathematics Research Center Building (ori. New Math. Bldg.)This talk is based on a joint work with Osamu Ikawa and Hiroyuki Tasaki. A linear isotropy representation of a Riemannian symmetric space is called an $s$-representation. In this talk we will study some geometric properties of orbits of $s$-representations as submanifolds in the Euclidean sphere. We introduce the notion of a weakly reflective submanifold, which is an austere submanifold with a certain global condition, and study its fundamental properties. Using these, we give the classification of weakly reflective orbits and austere orbits of $s$-representations. We observe that some wealky reflective orbits have a remarkable property, namely, they are tangentially degenerate. Furthermore these orbits provide many new examples of tangentially degenerate submanifolds which satisfy the Ferus equality.

Yoshihiro Ohnita 
2008-04-01 
09:00 - 09:50
308 , Mathematics Research Center Building (ori. New Math. Bldg.)

Yng-Ing Lee 
2008-04-01 
10:00- 10:50
308 , Mathematics Research Center Building (ori. New Math. Bldg.)

Yoshitake Hashimoto 2008-04-01 
11:10 - 12:00 308 , Mathematics Research Center Building (ori. New Math. Bldg.)

Jih-Hsin Cheng 2008-04-01 
14:00 - 14:50 308 , Mathematics Research Center Building (ori. New Math. Bldg.)