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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 . And some recent simulations of phase-field model, namely Cahn-Hilliard equation and Swift-Hohenberg equation.
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
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.
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.
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.