DSpace Community: KAIST Dept. of Mathematical Sciences
http://hdl.handle.net/10203/527
KAIST Dept. of Mathematical SciencesSun, 11 Oct 2020 22:25:36 GMT2020-10-11T22:25:36Zl-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, Wansuhttp://hdl.handle.net/10203/251897The 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.Tue, 01 Dec 2020 00:00:00 GMThttp://hdl.handle.net/10203/2764032020-12-01T00:00:00ZBranch-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.Tue, 01 Dec 2020 00:00:00 GMThttp://hdl.handle.net/10203/2764022020-12-01T00:00:00ZPricing of American lookback spread options
http://hdl.handle.net/10203/276063
Title: Pricing of American lookback spread options
Authors: Woo, Min Hyeok; Choe, Geon Ho
Abstract: We find the closed form formula for the price of the perpetual American lookback spread option, whose payoff is the difference of the running maximum and minimum prices of a single asset. We solve an optimal stopping problem related to both maximum and minimum. We show that the spread option is equivalent to some fixed strike options on some domains, find the exact form of the optimal stopping region, and obtain the solution of the resulting partial differential equations. The value function is not differentiable. However, we prove the verification theorem due to the monotonicity of the maximum and minimum processes.Thu, 01 Oct 2020 00:00:00 GMThttp://hdl.handle.net/10203/2760632020-10-01T00:00:00Z