DSpace Community: KAIST Dept. of Mathematical SciencesKAIST Dept. of Mathematical Scienceshttp://hdl.handle.net/10203/5272021-07-26T05:00:44Z2021-07-26T05:00:44ZDegree 3 unramified cohomology of classifying spaces for exceptional groupsBaek, Sanghoonhttp://hdl.handle.net/10203/2854032021-06-30T01:11:40Z2021-10-01T00:00:00ZTitle: Degree 3 unramified cohomology of classifying spaces for exceptional groups
Authors: Baek, Sanghoon
Abstract: Let G be a reductive group defined over an algebraically closed field of characteristic 0 such that the Dynkin diagram of G is the disjoint union of diagrams of types G(2), F-4, E-6, E-7, E-8. We show that the degree 3 unramified cohomology of the classifying space of G is trivial. In particular, combined with articles by Merkurjev [11] and the author [1], this completes the computations of degree 3 unramified cohomology and reductive invariants for all split semisimple groups of a homogeneous Dynkin type. (C) 2021 Elsevier B.V. All rights reserved.2021-10-01T00:00:00ZComponentwise linearity of projective varieties with almost maximal degreeCuong, Doan TrungKwak, Sijonghttp://hdl.handle.net/10203/2828692021-06-30T01:11:45Z2021-09-01T00:00:00ZTitle: Componentwise linearity of projective varieties with almost maximal degree
Authors: Cuong, Doan Trung; Kwak, Sijong
Abstract: The degree of a projective subscheme has an upper bound deg(X) <= ((e+r)(e)) in terms of the codimension eand the reduction number r. It was proved in [3] that deg(X) = ((e r)(e)) if and only if Xis arithmetically Cohen-Macaulay and has an (r+ 1)-linear resolution. Moreover, if the degree of a projective variety Xsatisfies deg(X) = ((e+r)(e)) - 1, then the Betti table is described with some constraints. In this paper, we build on this work to show that most of such varieties are componentwise linear and the componentwise linearity is particularly suitable for understanding their Betti tables. As an application, the graded Betti numbers of those varieties with componentwise linear resolutions are computed. (C) 2021 Elsevier B.V. All rights reserved.2021-09-01T00:00:00ZDiscontinuous bubble immersed finite element method for Poisson-Boltzmann-Nernst-Planck modelKwon, InKwak, Do YoungJo, Gwanghyunhttp://hdl.handle.net/10203/2855582021-06-30T01:11:59Z2021-08-01T00:00:00ZTitle: Discontinuous bubble immersed finite element method for Poisson-Boltzmann-Nernst-Planck model
Authors: Kwon, In; Kwak, Do Young; Jo, Gwanghyun
Abstract: We develop a numerical scheme for Poisson-Boltzmann-Nernst-Planck (PBNP) model. We adopt Gummel's method to treat the nonlinearity of PBNP where Poisson-Boltzmannequation and Nernst-Planckequation are iteratively solved, and then the idea of discontinuous bubble (DB) to solve the Poisson-Boltzmannequation is exploited [6]. First, we regularize the solution of Poisson-Boltzmannequation to remove the singularity. Next, we introduce the DB function as in [6] to treat the nonhomogeneous jump conditions of the regularized solution. Then, we discretize the discontinuous bubble and the bilinear form of Poisson-Boltzmannequation and solve the discretized linear problem by the immersed finite element method. Once Poisson-Boltzmannequation is solved, we apply the control volume method to solve Nernst-Planckequation via an upwinding concept. This process is repeated by updating the previous approximation until the total residual of the system decreases below some tolerance. We provide our numericalexperiments. We observe optimal convergence rates for the concentration variable in all examples having analytic solutions. We observe that our scheme reflects well without oscillations the effect on the distribution of electrons caused by locating the singular charge close to the interface. (C) 2021 Elsevier Inc. All rights reserved.2021-08-01T00:00:00ZTranslation partitions of unity, symmetry properties, and Gabor framesChristensen, OleGoh, Say SongKim, Hong OhKim, Rae Younghttp://hdl.handle.net/10203/2865492021-07-15T08:30:07Z2021-08-01T00:00:00ZTitle: Translation partitions of unity, symmetry properties, and Gabor frames
Authors: Christensen, Ole; Goh, Say Song; Kim, Hong Oh; Kim, Rae Young
Abstract: We consider the general question of constructing a partition of unity formed by translates of a compactly supported function g : DOUBLE-STRUCK CAPITAL Rd -> Double-struck capital C. In particular, we prove that such functions have a special structure that simplifies the construction of partitions of unity with specific properties. We also prove that it is possible to modify the function g in such a way that it becomes symmetric with respect to a given symmetry group on DOUBLE-STRUCK CAPITAL Zd. The results are illustrated with constructions of dual pairs of Gabor frames for L2(DOUBLE-STRUCK CAPITAL Rd). In addition, we obtain general approaches to construct dual Gabor frames whose window functions are symmetric with respect to an arbitrary symmetry group. Through sampling and periodization, these dual Gabor frames for L-2(DOUBLE-STRUCK CAPITAL Rd) lead to dual pairs of discrete Gabor frames for l(2)(DOUBLE-STRUCK CAPITAL Zd) and finite Gabor frames for periodic sequences on DOUBLE-STRUCK CAPITAL Zd.2021-08-01T00:00:00Z