DSpace Community: KAIST Dept. of Mathematical SciencesKAIST Dept. of Mathematical Scienceshttp://hdl.handle.net/10203/5272024-05-19T05:04:58Z2024-05-19T05:04:58ZPrime vertex-minors of a prime graphKim, DonggyuOum, Sang-ilhttp://hdl.handle.net/10203/3168562023-12-26T01:00:18Z2024-05-01T00:00:00ZTitle: Prime vertex-minors of a prime graph
Authors: Kim, Donggyu; Oum, Sang-il
Abstract: A graph is prime if it does not admit a partition (A, B) of its vertex set such that min{vertical bar A vertical bar, vertical bar B vertical bar} >= 2 and the rank of the AxB submatrix of its adjacency matrix is at most 1. A vertex v of a graph is non-essential if at least two of the three kinds of vertex-minor reductions at v result in prime graphs. In 1994, Allys proved that every prime graph with at least four vertices has a non-essential vertex unless it is locally equivalent to a cycle graph. We prove that every prime graph with at least four vertices has at least two non-essential vertices unless it is locally equivalent to a cycle graph. As a corollary, we show that for a prime graph G with at least six vertices and a vertex x, there is a vertex v not equal x such that G \ v or G * v \ v is prime, unless x is adjacent to all other vertices and G is isomorphic to a particular graph on odd number of vertices. Furthermore, we show that a prime graph with at least four vertices has at least three non-essential vertices, unless it is locally equivalent to a graph consisting of at least two internally-disjoint paths between two fixed distinct vertices having no common neighbors. We also prove analogous results for pivot-minors.2024-05-01T00:00:00ZGrowth of torsion subgroups of elliptic curves over number fields without rationally defined CMIm, Bo-HaeKim, Hansolhttp://hdl.handle.net/10203/3180192024-02-13T07:00:14Z2024-05-01T00:00:00ZTitle: Growth of torsion subgroups of elliptic curves over number fields without rationally defined CM
Authors: Im, Bo-Hae; Kim, Hansol2024-05-01T00:00:00ZGrowth of torsion groups of elliptic curves over number fields without rationally defined CMIm, Bo-HaeKim, HanSolhttp://hdl.handle.net/10203/3179852024-02-14T09:00:21Z2024-05-01T00:00:00ZTitle: Growth of torsion groups of elliptic curves over number fields without rationally defined CM
Authors: Im, Bo-Hae; Kim, HanSol
Abstract: For a quadratic field K without rationally defined complex multiplication, we prove that there exists of a prime pK depending only on K such that if d is a positive integer whose minimal prime divisor is greater than pK, then for any extension L/K of degree d and any elliptic curve E/K, we have E (L)tors = E (K)tors. By not assuming the GRH, this is a generalization of the results by Genao, and Gonalez-Jimenez and Najman. (c) 2023 Elsevier Inc. All rights reserved.2024-05-01T00:00:00ZTHE MODIFIED SCATTERING FOR DIRAC EQUATIONS OF SCATTERING-CRITICAL NONLINEARITYCho, YonggeunKwon, SoonsikLee, KiyeonYang, Changhunhttp://hdl.handle.net/10203/3145332023-11-14T01:00:12Z2024-03-01T00:00:00ZTitle: THE MODIFIED SCATTERING FOR DIRAC EQUATIONS OF SCATTERING-CRITICAL NONLINEARITY
Authors: Cho, Yonggeun; Kwon, Soonsik; Lee, Kiyeon; Yang, Changhun
Abstract: In this paper, we consider the Maxwell-Dirac system in 3 dimension under zero magnetic field. We prove the global well-posedness and modified scattering for small solutions in the weighted Sobolev class. Imposing the Lorenz gauge condition, (and taking the Dirac projection operator), it becomes a system of Dirac equations with Hartree type non -linearity with a long-range potential as |x|-1. We perform the weighted energy estimates. In this procedure, we have to deal with various resonance functions that stem from the Dirac projections. We use the space-time resonance argument of Germain-Masmoudi-Shatah ([14, 15, 16]), as well as the spinorial null-structure. On the way, we recognize a long-range interaction which is responsible for a logarithmic phase correction in the modified scattering statement. This result was obtained by Cloos in his dissertation [9], via a different technique (see Remark 1.2).2024-03-01T00:00:00Z