Department Seminars & Colloquia
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Modeling of debris flow is of great importance in various research areas, and there have been several debris flow models have been proposed in the last couple of decades. However, most of them do not consider the erosional effect. Here, we discuss a mathematical approach to model a debris flow with erosional effect. The (energetic) variational approach is applied to derive the resulting system of partial differential equation (PDEs) with the erosional effect. Since the erosional effect plays a key role on the interface between flow and the base, it is crucial to find kinematic boundary condition on the interface to elucidate erodible debris or granular flows. We first model an erosional potential energy with power law inspired by the frictional potential. In order to find proper kinematic boundary conditions on the surface and interface, the modeled erosional potential is incorporated with Bernoulli's equation of velocity potential, and Luke's variational principle is used. Then we employ a shallow-water assumption to derive the system of PDEs describing debris or granular flow with erodible base. In order to ensure stable finite volume discretization for shallow water type equation, the hydrostatic reconstruction method and implicit treatment of additional source terms is implemented. In order to verify the resulting mathematical system with the erodible effects, we simulated three reference experiments. The derived mathematical system properly describes erosional effect of granular flows and the simulated results are agreed well with experimental data. One of 2011 Umyeon Mt. debris flows is also simulated by the derived model.
The main aim of this talk is to design efficient and novel numerical algorithms for highly oscillatory dynamical systems with multiple time scales. Classical numerical methods for such problems need temporal resolution to resolve the finest scale and become very inefficient when the longer time intervals are of interest. In order to accelerate computations and improve the long time accuracy of numerical schemes, we take advantage of various multiscale structures established from a separation of time scales. The framework of the heterogeneous multiscale method (HMM) will be considered as a general strategy both for the design and for the analysis of multiscale methods.
E6-1, ROOM 1409
Discrete Math
Boris Aronov (Department of Computer Science and Engineering, Po)
The complexity of unions of shapes
E6-1, ROOM 1409
Discrete Math
Over the years, the following class of problems has been studied quite a lot: Given a class of simply-shaped objects in the plane (disks, unit disks, squares, axis-aligned squares, isosceles triangles, shapes definable with a small number of polynomial equations and inequalities), how complicated can be the union of N shapes from the class? There are several different ways in which one can measure this (combinatorial) complexity. Two popular measures are the number of connected components of the complement, and the number of places where two object boundaries intersect on the boundary of the union (so-called “vertices” of the union).
It is easy to see that, if each object is “simple,” the union of N objects cannot be larger than O(N^2) and a matching construction is easy. Are there classes of objects for which this quantity is near-linear in N? (Yes, there are: disks, axis-aligned squares, and more.) The quest for such classes, over the years, motivation for the problem, generalizations to higher dimensions, and other puzzles will constitute the content of this talk.
If I ever get to it, the latest and most amazing result in this area is joint work with Mark de Berg, Esther Ezra, and Micha Sharir. It is quite technical and I will not be able to say much about this during the talk, but if anyone is interested, I can provide lots of technical details on request. An overview of the subject will be mostly based on a survey of Agarwal, Pach, and Sharir.
In this talk, we will discuss about the following question posed by S. Tolman.
Question : If a smooth compact sympletic manifold (M, ω) admits a Hamiltonian torus action with only isolated fixed points, then is the sequence of even Betti numbers of M unimodal?
R. Stanley proved that the above question is “yes” when our torus has a full dimension, i.e. (M, ω) is a toric variety. One of the main purpose of this talk is to show that the answer is also “yes” if dimM ≤ 8.
Secondly, we will see that we can deal with the above question using graph theory. Let’s assume that T be a compact torus of dimension k and (M, ω) be a 2n-dimensional closed Hamiltonian T -manifold with some nice condition.
Then a 1-skeleton of the corresponding moment polytope, which is called a GKM graph, is an n-valent graph embedded in R^k satisfying several nice properties. In this case, we will see that our question is equivalent to the followings.
Question (2nd version) : If we have a GKM graph Γ, let ξ∈R^k be any vector such that ξ is not orthogonal to any edge of Γ. Then ξ gives an orientation overrightarrow{e} for each edge e in Γ such that <overrightarrow{e},ξ> 0. Let b_r(Γ, ξ) be the number of vectices with index r. Then is the sequence b_0, b_1,... unimodal?
자연과학동 Room 1409
Discrete Math
Eunjung Kim (CNRS, LAMSADE, Paris, France)
On subexponential and FPT-time inapproximability
자연과학동 Room 1409
Discrete Math
Fixed-parameter algorithms, approximation algorithms and moderately exponential algorithms are three major approaches to algorithms design. While each of them being very active in its own, there is an increasing attention to the connection between these different frameworks. In particular, whether Independent Set would be better approximable once allowed with subexponential-time or FPT-time is a central question. Recently, several independent results appeared regarding this question, implying negative answer toward the conjecture. They state that, for every 0<r<1, there is no r-approximation which runs in better than certain subexponential-function time. We outline the results in these papers and overview the important concepts and techniques used to obtain such results.
The research on flips is a crucial part to study minimal model program (MMP). To understand flips, it is worthwhile to get some numerical invariants of flips to understand it better. The minimal log discrepancy (mld) is one of the important invariant to give a geometrical information about flips, and it appear naturally in global contexts. There have been several conjectures on mld related with the termination of flips. I will introduce basic concepts of mld and talk about some relations between mld and other invariants with some examples of flips.
In this talk I will discuss the spreading properties of solutions of a prey-predator type reaction-diffusion system. This system belongs to the class of reaction-diffusion systems for which the comparison principle does not hold. For such class of systems, little has been know about the spreading properties of the solutions. Here, by a spreading property, we mean the way the solution propagates when starting from compactly supported initial data. We show that propagation of both the prey and the predator occur with a definite spreading speed. Furthermore, quite intriguingly, the spreading speed of the prey and that of the predator are different in some situations. This is joint work with Arnaud Ducrot and Thomas Giletti.
The classical p-typical Witt vectors were contrived by Teichm?ller and Witt to build unramified extensions of the field of p-adic numbers from their residue fields in a functorial way. Dress and Siebeneicher introduced a fascinating generalization of them called "Witt-Burnside rings" in a group-theoretical way. In this talk, we will briefly review the basic theory of Witt vectors and Witt-Burnside rings. Recent developments in this area, in particular, some open problems concerned with Witt vector construction will be also dealt with.
The rational homology groups of the matching complexes are closely related to the syzygies of the Veronese embeddings. In this talk, I will give a proof of the shellability of certain skeleta of matching complexes, thus proving that the coordinate rings of the Veronese varieties satisfy property $N_{2d-2}$. Using duality and explicit computation of homology groups of matching complexes, we will deduce the Ottaviani-Paoletti conjecture for fourth Veronese embeddings.