Department Seminars & Colloquia
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Abstract: Let $Omegasubsetmathbb{R}^n$ be a smooth bounded domain,
$a_1,a_2,dots,a_{i_0}inOmega$ and
$widehat{Omega}=Omegasetminus{a_1,a_2,dots,a_{i_0}}$ and
$widehat{R^n}=mathbb{R}^nsetminus{a_1,a_2,dots,a_{i_0}}$. We
prove the existence of solution $u$ of the fast diffusion equation
$u_t=Delta u^m$, $u>0$, in $widehat{Omega}times (0,infty)$
($widehat{R^n}times (0,infty)$ respectively) which satisfies
$u(x,t)toinfty$ as $xto a_i$ for any $t>0$ and $i=1,cdots,i_0$,
when $0<m<frac{n-2}{n}$, $ngeq 3$, and the initial value satisfies
$0le u_0in L^p_{loc}(2{Omega}setminus{a_1,cdots,a_{i_0}})$
($u_0in L^p_{loc}(widehat{R^n})$ respectively) for some constant
$p>frac{n(1-m)}{2}$ and $u_0(x)ge lambda_i|x-a_i|^{-gamma_i}$ for
$xapprox a_i$ and some constants
$gamma_i>frac{2}{1-m},lambda_i>0$, for all $i=1,2,dots,i_0$. We
also find the blow-up rate of such solutions near the blow-up points
$a_1,a_2,dots,a_{i_0}$, and obtain the asymptotic large time
behaviour of such singular solutions. More precisely we prove that if
$u_0gemu_0$ on $widehat{Omega}$ ($widehat{R^n}$, respectively)
for some constant $mu_0>0$ and $gamma_1>frac{n-2}{m}$, then the
singular solution $u$ converges locally uniformly on every compact
subset of $widehat{Omega}$ (or $widehat{R^n}$ respectively) to
infinity as $ttoinfty$. If $u_0gemu_0$ on $widehat{Omega}$
($widehat{R^n}$, respectively) for some constant $mu_0>0$ and
satisfies $lambda_i|x-a_i|^{-gamma_i}le u_0(x)le
lambda_i'|x-a_i|^{-gamma_i'}$ for $xapprox a_i$ and some constants
$frac{2}{1-m}<gamma_ilegamma_i'<frac{n-2}{m}$, $lambda_i>0$,
$lambda_i'>0$, $i=1,2,dots,i_0$, we prove that $u$ converges in
$C^2(K)$ for any compact subset $K$ of
$2{Omega}setminus{a_1,a_2,dots,a_{i_0}}$ (or $widehat{R^n}$
respectively) to a harmonic function as $ttoinfty$. This is joint work with Sunghoon Kim.
I. Blockchain이란?
분산원장 기반기술인 블록체인의 개념을 알아보고, 암호화폐를 중심으로 한 Public Blockchain과
기업이 활용할 수 있는 IT 인프라 중심의 Private Blockchain의 특징을 소개
하고;
II. Enterprise Blockchain(Private Blockchain)
Hyperledger Fabric 1.0의 주요 특징 및 구성을 살펴보고, 기술적 장점을 소개
하고;
III. Use Cases
국내외 금융권, 비금융권 기업들에서 시도되고 활용되고 있는 Blockchain Use Case 내용
을 소개할 예정입니다.
The phenomena in the Lotka-Volterra three-species competition-diffusion system are rich and complicated. In the joint work with Hung, Mimura, Tohma and Ueyama, we combined the method of exact/semi-exact solutions and the numerical approach to investigate the wave behaviors of this system. In this talk, we will explain our research and show that the exact/semi-exact solutions can provide very interesting information for the study of new dynamical patterns as well as the study of competitor-mediated coexistence in situations where one exotic competing species invades a system that already contains two strongly competing species. Also, some further applications of exact/semi-exact solutions will be discussed.
In this talk, we present our work of finite volume method for stochastic partial differential equations, both from the viewpoint of theoretical study and numerical simulations. In joint work with T. Funaki and D. Hilhorst [1, 2], we consider a first-order conservation law involving a Q-Brownian motion. We prove that the discrete solution converges along a subsequence in the sense of Young measures to a measure-valued entropy solution as the maximum diameter of the volume elements and the time step tend to zero. We then present the Kato's inequality and as a corollary we deduce the uniqueness of the measure-valued entropy solution as well as the uniqueness of the weak entropy solution. For the numerical simulations, we show results of stochastic Burgers equation by Monte-Carlo method [1]. And some recent simulations of phase-field model, namely Cahn-Hilliard equation and Swift-Hohenberg equation.
[1] T. Funaki, Y. Gao and D. Hilhorst, Convergence of a finite volume scheme for a stochastic conservation law involving a Q-Brownian motion, Accepted for publication by DCDS-B, AIMS, hal-01404119.
[2] T. Funaki, Y. Gao and D. Hilhorst, Uniqueness of the entropy solution of a stochastic conservation law with a Q-Brownian motion, in preparation.
Using elliptic regularity results, we construct for every starting point, weak solutions to SDEs in R^d with Sobolev diffusion and locally integrable drift coefficient up to their explosion times. Subsequently, we develop non-explosion criteria which allow for linear growth, singularities of the drift coefficient inside an arbitrarily large compact set, and an interplay between the drift and the diffusion coefficient. Moreover, we show strict irreducibility of the solution, which by construction is a strong Markov process with continuous sample paths on the one-point compactification of R^d. Joint work with Haesung Lee
We study localization occurring during high-speed shear deformations of metals leading to the formation of shear bands. The localization instability results from the competition among Hadamard instability (caused by softening response) and the stabilizing effects of strain-rate hardening. We consider a hyperbolic-parabolic system that expresses the above mechanism and construct self-similar solutions of localizing type that arise as the outcome of the above competition.
The existence of self-similar solutions is turned, via a series of transformations, into a problem of constructing a heteroclinic orbit for an induced dynamical system. The dynamical system is four-dimensional but has a fast-slow structure with respect to a small parameter capturing the strength of strain-rate hardening. Geometric singular perturbation theory is applied to construct the heteroclinic orbit as a transversal intersection of two invariant manifolds in the phase space. The recognized orbit is numerically captured as well. This is to numerically capture a saddle-saddle connection in the phase space. Method of continuation via software package AUTO is employed in the computation.
E6-1, ROOM 3433
Discrete Math
O-joung Kwon (Technische Universität Berlin, Berlin, Germany)
Erdős-Pósa property of chordless cycles and its applications
E6-1, ROOM 3433
Discrete Math
A chordless cycle in a graph G is an induced subgraph of G which is a cycle of length at least four. We prove that the Erdős-Pósa property holds for chordless cycles, which resolves the major open question concerning the Erdős-Pósa property. Our proof for chordless cycles is constructive: in polynomial time, one can find either k+1 vertex-disjoint chordless cycles, or c k^2 log k vertices hitting every chordless cycle for some constant c. It immediately implies an approximation algorithm of factor O(OPT log OPT) for Chordal Vertex Deletion. We complement our main result by showing that chordless cycles of length at least ℓ for any fixed ℓ≥ 5 do not have the Erdős-Pósa property.
Abstract: We discuss optimal transport maps between non-convex domains.
In the convex cases, an optimal map is the gradient of a potential function. Moreover, the potential function is a solution to a Monge-Ampere equation.
In the general non-convex case, the map is a diffeomorphism between subsets of given domain, and the complements of the subsets are measure zero.
Then, the complements can be considered as free boundaries of the potential function. In this talk, we discuss how to apply the well-developed theory in free boundary problems for Monge-Ampere equations to this optimal transport maps.
E6-1, ROOM 3433
Discrete Math
Joonkyung Lee (University of Oxford, Oxford, UK)
Counting tree-like graphs in locally dense graphs
E6-1, ROOM 3433
Discrete Math
We prove that a class of graphs obtained by gluing complete multipartite graphs in a tree-like way satisfies a conjecture of Kohayakawa, Nagle, Rödl, and Schacht on random-like counts for small graphs in locally dense graphs. This implies an approximate version of the conjecture for graphs with bounded tree-width. We also prove an analogous result for odd cycles instead of complete multipartite graphs.
The proof uses a general information theoretic method to prove graph homomorphism inequalities for tree-like structured graphs, which may be of independent interest.
Okounkov bodies have become a very interesting and useful tool to understand the positivity of divisors. Although the Okounkov body carries rich positivity data of a divisor, it only provides information near a single point. In this talk, we introduce a new convex body of a divisor that is effective in handling the positivity theory in a multi-point setting. We study its various properties, and observe local positivity data via this convex body.
I will explain our recent progress on the construction of exceptional vector bundles on surfaces when they admit Q-Gorestein degenerations to singularities of class T_d. This is a generalization of the result of Hacking who has studied the case d=1. We give the construction of block(=completely orthogonal exceptional collection) of length d when d>1. If the underlying spaces are del Pezzo surfaces, then our construction explains the paralleism between toric degenerations and three block collections in derived categories.