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 4¤ëºtÁ¿ PDE Seminar, ¤¤¥¡¬ã¨s°|¼Æ¾Ç¬ã¨s©Ò ¤ý®üÀÜ±Ð±Â Thursday, April 13, 11:00¡X16:00 ¼Æ¾Ç¨t3¼Ó·|Ä³«Ç Title: Milne Problem for Boltzmann Equation with Soft Potentials Colloquium, ¦¨¥\¤j¾Ç²Î­p¨t ³¯·ç±l±Ð±Â Thursday, April 13, 16:10¡X17:00 ¼Æ¾Ç¨t3174 Title: Sequential Designs Based on Bayesian Uncertainty Quantification in Sparse Representation Surrogate ModelingAbstract: A numerical method, called OBSM, was recently proposed which employs overcomplete basis functions to achieve sparse representations. While the method can handle non-stationary response without the need of inverting large covariance matrices, it lacks the capability to quantify uncertainty in predictions. We address this issue by proposing a Bayesian approach which first imposes a normal prior on the large space of linear coefficients, then applies the MCMC algorithm to generate posterior samples for predictions. From these samples, Bayesian credible intervals can then be obtained to assess prediction uncertainty. A key application for the proposed method is the efficient construction of sequential designs. Several sequential design procedures with different infill criteria are proposed based on the generated posterior samples. Numerical studies show that the proposed schemes are capable of solving problems of positive point identification, optimization, and surrogate fitting. Colloquium, °ª¶¯¤j¾ÇÀ³¥Î¼Æ¾Ç¨t ³¢©¨©Ó±Ð±Â Thursday, April 20, 15:10¡X16:00 ¼Æ¾Ç¨t3174 Title: Continuation Methods for Computing Z-/H-eigenpairs of Nonnegative TensorsAbstract: In this talk, a homotopy continuation method for the computation of nonnegative Z-/H-eigenpairs of a nonnegative tensor is presented. We show that the homotopy continuation method is guaranteed to compute a nonnegative eigenpair. Additionally, using degree analysis, we show that the number of positive Z-eigenpairs of an irreducible nonnegative tensor is odd. A novel homotopy continuation method is proposed to compute an odd number of positive Z-eigenpairs, and some numerical results are presented. Geometry Seminar, Tufts University §ùªZ«G±Ð±Â Wednesday, April 26, 15:10¡X16:00 ¼Æ¾Ç¨t3¼Ó·|Ä³«Ç Title: The classical Lefschetz fixed point theorem states that the number of fixed points, counted with multiplicity $\pm 1$, of a smooth map $f$ from a manifold $M$ to itself can be calculated as the alternating sum $\sum (-1)^q {\mathrm tr} f^*|_{H^q(M;\mathbb Q)}$ of the trace of the induced homomorphism in cohomology. In 1964, at a conference in Woods Hole, Shimura conjectured a Lefschetz fixed point theorem for a holomorphic map, which Atiyah and Bott proved and generalized into a fixed point theorem for elliptic complexes. However, in Shimura's recollection, he had conjectured more than the holomorphic Lefschetz fixed point theorem. He said he had made a conjecture for a holomorphic correspondence, but he could not remember what it said nor did he have any notes. This talk is an exploration of Shimura's forgotten conjecture, first a proof for a smooth correspondence, and then a conjectural statement for a holomorphic correspondence. Colloquium, ¥xÆW¤j¾Ç¼Æ¾Ç¨t ¤ý®¶¨k±Ð±Â Thursday, April 27, 15:10¡X16:00 ¼Æ¾Ç¨t3174 Title: Global Identification of a Singular Potential in the Plane by Boundary Measurements Abstract: In this talk, I would like to discuss the global uniqueness of determining a singular potential in the plane by the full Cauchy data on the boundary. We the Schrodinger equation with potentials in space, where p>4/3. We will make use of an approach introduced by Bukhgeim for solving inverse boundary values in the plane. This approach consists of two key ingredients: complex geometrical optics solutions with quadratic phases and the method of stationary phase. The talk is based on a recent joint work with E. Blasten and L. Tzou.

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 °ê¥ß¦¨¥\¤j¾Ç¼Æ¾Ç¨t 70101 ¥x«n¥«¤j¾Ç¸ô¤@¸¹ ¹q¸Ü¡J(06) 2757575 Âà 65100   ¶Ç¯u¡J(06) 2743191 em65100[at]email.ncku.edu.tw