DSpace Community: KAIST Dept. of Mathematical Sciences
http://hdl.handle.net/10203/527
KAIST Dept. of Mathematical Sciences2020-10-26T15:55:26Zl-adic étale cohomology of Shimura varieties of Hodge type with non-trivial coefficients
http://hdl.handle.net/10203/251897
Title: l-adic étale cohomology of Shimura varieties of Hodge type with non-trivial coefficients
Authors: Hamacher, Paul; Kim, WansuPROPAGATION OF THE MONO-KINETIC SOLUTION IN THE CUCKER-SMALE-TYPE KINETIC EQUATIONS
http://hdl.handle.net/10203/276896
Title: PROPAGATION OF THE MONO-KINETIC SOLUTION IN THE CUCKER-SMALE-TYPE KINETIC EQUATIONS
Authors: Kang, Moon-Jin; Kim, Jeongho
Abstract: In this paper, we study the propagation of the mono-kinetic distribution in the CuckerSmale-type kinetic equations. More precisely, if the initial distribution is a Dirac mass for the variables other than the spatial variable, then we prove that this "mono-kinetic" structure propagates in time. For that, we first obtain the stability estimate of measure-valued solutions to the kinetic equation, by which we ensure the uniqueness of the mono-kinetic solution in the class of measure-valued solutions with compact supports. We then show that the mono-kinetic distribution is a special measure-valued solution. The uniqueness of the measure-valued solution implies the desired propagation of mono-kinetic structure.The average cut-rank of graphs
http://hdl.handle.net/10203/276403
Title: The average cut-rank of graphs
Authors: Nguyen, Huy-Tung; Oum, Sang-il
Abstract: The cut-rank of a set X of vertices in a graph G is defined as the rank of the X x (V(G) \ X) matrix over the binary field whose (i, j)-entry is 1 if the vertex i in X is adjacent to the vertex j in V(G)\X and 0 otherwise. We introduce the graph parameter called the average cut-rank of a graph, defined as the expected value of the cut-rank of a random set of vertices. We show that this parameter does not increase when taking vertex-minors of graphs and a class of graphs has bounded average cut-rank if and only if it has bounded neighborhood diversity. This allows us to deduce that for each real alpha, the list of induced-subgraphminimal graphs having average cut-rank larger than (or at least) alpha is finite. We further refine this by providing an upper bound on the size of obstruction and a lower bound on the number of obstructions for average cut-rank at most (or smaller than) alpha for each real alpha >= 0. Finally, we describe explicitly all graphs of average cut-rank at most 3/2 and determine up to 3/2 all possible values that can be realized as the average cut-rank of some graph.2020-12-01T00:00:00ZBranch-depth: Generalizing tree-depth of graphs
http://hdl.handle.net/10203/276402
Title: Branch-depth: Generalizing tree-depth of graphs
Authors: DeVos, Matt; Kwon, O-joung; Oum, Sang-il
Abstract: We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs as follows. For a graph G = (V, E) and a subset A of E we let lambda(G)(A) be the number of vertices incident with an edge in A and an edge in E \ A. For a subset X of V, let rho(G)(X) be the rank of the adjacency matrix between X and V \ X over the binary field. We prove that a class of graphs has bounded tree-depth if and only if the corresponding class of functions lambda(G) has bounded branch depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions rho(G) has bounded branch-depth, which we call the rank-depth of graphs. Furthermore we investigate various potential generalizations of tree-depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi ordered by restriction.2020-12-01T00:00:00Z