학과 세미나 및 콜로퀴엄
In this talk, a dynamic algorithm to predict the pricing of bond options using actual data set of gilts will be considered. The pre-measured interest rate traces lose the meaning of interest rate prediction because the actual distribution is not equal to the normal distribution which is used for bond option models in general. Thus, to obtain the interest rate dynamics, we need the cumulative distribution function (CDF) of actual data of gilts using numerical methods. In addition, Pricing models of Bond Option with jumps will be discussed.
Latent Dirichlet Allocation (LDA) and Hierarchical Dirichlet Processes
(HDP) have become popular models for discovering latent semantics from
text corpora. I will first start this talk with what LDA and HDP are and
how they are used in common text analysis tasks. Then, I will present
three recent papers from our research group that extend LDA and HDP to
analyze three different text corpora: online reviews, news articles, and
conference proceedings. With the online reviews, we propose a variant of
LDA called Aspect and Sentiment Unification Model (ASUM) to analyze topics
and sentiments jointly in an unsupervised fashion. With the news articles,
we use LDA to generate topic chains to model temporal patterns of similar
topics. With the conference proceedings, we propose a variant of HDP
called distant dependent Chinese Restaurant Franchise (ddCRF) to discover
how new topics emerge through time.
5시부터 피자 제공
Random walks in random scenery (RWRS) are processes where a random
walker collects a random reward (or scenery) at each site it visits.
If it visits a site multiple times, it collects the same reward many
times thus leading to correlations in this partial sum process. Cohen
and Samorodnitsky (2006) studied a certain
renormalization of RWRS and proposed self-similar, symmetric
alpha-stable processes, which generalize fractional Brownian motion,
renormalization of RWRS and proposed self-similar, symmetric
alpha-stable processes, which generalize fractional Brownian motion,
as their scaling limits. The limiting processes can be written as
stochastic integrals with random kernels, and the processes have
self-similarity exponents H>1/alpha.
We consider a modification in which a sign associated to the reward
(scenery) alternates upon successive visits. The resulting process is
what we call a random walk at random time. We will discuss weak
convergence of
the discrete processes to their scaling limits, and in particular,
show that the alternating scenery leads to limiting processes which
convergence of
the discrete processes to their scaling limits, and in particular,
show that the alternating scenery leads to limiting processes which
are also stochastic integrals with quite simple random kernels of
indicator type. Our results complement the above results in that the
processes have self-similarity exponents H<1/alpha.
In 2007, families of new examples of homogeneous domains were presented
by K.H. Lee - homogeneous in the sense that the action by their J - holomor-
phic automorphism group is transitive. To the surprise of many experts, such
examples are almost-complex but not complex. Such examples were formerly
expected to be non-existent, or if they existed, they should be sporadic and
very rare, such as nitely many. But not only are there in nitely many such
examples, but they form a continuous family when the dimension (of the man-
ifold) is 6 or higher. So such homogeneous almost-complex (but not complex),
which we call the models, have emerged as a research object that requires careful
investigation.
In this talk, we introduce the uniformization theorem in Riemann surface and
the di erential geometry of the standard unit ball. In this viewpoint, we discuss
the di erential geometric characterization of the models which are mentioned
above.
2000년대 이후 우리 사회에는 무선 통신을 위한 인프라가 구축되었으며 많은 무선 통신 기기가 보급되었다. 그 이면에는 이를 뒷받침할만한 통신 기술의 발전이 있었으며 시스템의 성능을 분석하는데 있어서 수학이 큰 역할을 하였다. 이번 발표에서는 우선 시스템의 성능을 분석하는 대표적인 방법인 큐잉 이론에 대하여 알아본다. 큐잉 이론은 전화 교환망에 대한 성능 분석을 시작으로 현재 존재하는 거의 모든 통신 시스템의 성능을 분석하는데에 쓰여 왔다. 또한, 큐잉 이론을 이용하여 무선 통신에서 가장 간단한 형태인 일대일 통신의 성능을 큐잉 이론을 적용하여 분석하는 방법에 대하여 소개하고 차세대 통신 기술로써 현재 많은 연구가 되고 있는 인지무선통신에 대해서 간단히 소개한다.
E6-1 Room 1409
Discrete Math
Xuding Zhu (Zhejiang Normal University)
Fractional colouring of product graphs
E6-1 Room 1409
Discrete Math
Given two graphs $G$ and $H$, the categorical product $G \times H$ has vertex set $V(G) \times V(H)$, and two vertices $(x,y)$ and $(x',y')$ are adjacent if $xx' \in E(G)$ and $yy' \in E(H)$. The famous Hedetniemi-Lovasz Conjecture asserts that teh chromatic number of $G \times H$ equals the minimum of $\chi(G)$ and $\chi(H)$. In this talk, I will sketch a proof of the fractional version of the conjecture, which says that the fractional chromatic number of $G \times H$ equals to the minimum of the fractional chromatic numbers of $G$ and $H$. This result is then used to prove a conjecture of Burr-Erdos-Lovasz on the chromatic Ramsey number of graphs.
In 1970s, B. C. Berndt has found a more general class of Eisenstein series and has computed
transformation formulas for them. He also has converted these transformation formulas into
transformation formulas for a large class of functions containing the classical Dedekind
eta function. By this process, he has established a number of interesting infinite series identities,
many of which are stated in the Notebooks of Ramanujan or are proved by many authors.
In this talk, we introduce his work and, continuing his work, we prove three modular transformation
formulas. Furthermore we give more general infinite series identities regarding hyperbolic
functions and trigonometric functions.
Vertex colouring of graphs is an active research area in graph theory. It has wide applications to practical problems and also has many theoretically deep results and challenging problems. Circular colouring of graphs is a refinement of the conventional vertex colouring of graphs. It has atttract a lot of attention in the past twenty years, and becomes an important branch of chromatic graph theory, with many exciting results and new techniques. This talk gives a survey on research in this area.
The pseudo-spectral (or spectral collocation) method is a numerical approximation method based on spectral basis functions for differential operators on a discretized domain specified with boundary conditions. Considered as the generalized discretized Fourier transformation, this method provides spectral accuracy with respect to
the number of discretization in approximating differentiable functions, therefore it is a powerful tool in numerical approximation for smooth enough solutions to PDE in accuracy and efficiency. In this talk, some applications of this method are presented, particularly highlighting how to properly treat multi-dimensional, unbounded,
singular properties in solving some nonlinear PDE that naturally arise in the course of mathematical modeling in physical science and engineering.
Algebraic quantum field theory aims at providing a mathematical description of the structure of quantum field theories. In this talk I will introduce the basic notions of quantum field theory and QFT algebra, sketch some relevant classical ideas - such as homological and homotopy algebras, basic deformation theory, etc, for a general audience.
BK21 연수연구원 세미나 (2011.4.14(목), 16:00~17:30, 자연과학동 3433호)
1. 권오상
-제목 : Existence of solutions for elliptic partial differential equations
-Abstract
: In this talk, I will briefly review the basic concepts of the calculus of variations and Lyapunov-Schmidt reduction method, and prove the existence of solutions for some elliptic partial differential equations using Lyapunov-Schmidt reduction method.
2. 박혜인
-제목 : Heat kernel estimates for Parabolic operators
-Abstract
: We study the relation between parabolic operators and markov processes. Also we establish the heat kernel estimates for the parabolic operators.
3. 정재환
-제목 : Number of zeros and Heat equation in the real line
-Abstract
: For a solution to the one dimensional heat equation, number of zeros of the solution does not increase with time; we will discuss the nonincreasing property and its applications. Long time asymptotics of zeros will be also mentioned.
4. 곽민석
-제목 : Optimal Portfolio, Consumption and Retirement Decision under a Preference Change
-Abstract
: We consider an optimal portfolio, consumption and retirement decision problem in which an economic agent can determine the discretionary stopping time as a retirement time.
It is assumed that the agent's coefficient of relative risk aversion becomes higher after retirement.
Under a CRRA utility function, we obtain the optimal policies in closed-forms using martingale methods and variational inequality methods.
We analyze the properties of the optimal policies with numerical examples.
5. 조윤형
-제목 : TBA
One version of Kakeya's problem asks for the smallest convex garden in which a ladder of length one can be rotated fully. The answer to this question was shown to be the equilateral triangle of height one by Pal in 1920.
We consider the following generalization: Given a (not necessarily finite) family F of line segments in the plane, what is the smallest convex figure K such that every segment in F can be translated to lie in K?
We show that as in the classic case, the answer can always be chosen to be a triangle. We also give an O(n log n) time algorithm to compute such an optimal triangle if F consists of n line segments.
Joint work with Sang Won Bae, Hee-Kap Ahn, Joachim Gudmundsson, Takeshi Tokuyama, and Antoine Vigneron.
Based on various experiences of speaker as teaching assistants, in this talk, we first describe a 'good/bad‘ teaching assistant in Mathematics for students, professors, Head TAs, and oneself. After reviewing some specific cases, in order to be a good TA, let us consider what to do or what not to do.
In this talk, I will explain how to construct the quantum invariants for classical singularity theory via constructing the virtual cycle on moduli spaces of $W$-curves. Those invariants are closely related to the study of mirror symmetry and integrable hierarchy.
The purpose of this talk is to discover how Koreans worked hard to learn modern western mathematics and science in late 19th century by themselves. We will see many hundred years old Korean mathematics books which were written in Korean. We will talk aboutfrontiers of Korean modern mathematics between 1894 to 1945 such as LEE Sang-Seol, YU Il-Sun, CHOE Gyu-Dong, REE Im-Hak and CHOE Yun-Sik. It will give us a fine view of Korean efforts in Western modernity for mathematics in Korea that was not known before.
BK21 연수연구원 세미나 (2011.4.6(수), 16:00~17:30, 자연과학동 3433호)
1. 신동화
-제목 : Class number one problem
-Abstract
: We revisit the class number one problem done by Heegner and Stark, and present a new proof by making use of Shimura's reciprocity law. 2. 최영준
-제목 : The differential geometric models in almost complex manifolds and the negatively curved Kähler manifolds
-Abstract
: In this talk, we introduce the differential geometric Models in Almost Complex Manifolds and discuss their differential geometric characterization theorem. We also introduce the classical problem related the uniformization theorem, in particular the Negatively curved Kähler manifold.
3. 박선정
-제목 : Topological classification of quasitoric manifolds
-Abstract
: A quasitoric manifold is a smooth manifold with locally standard torus action whose orbit space is a simple convex polytope. In this talk, I will explain a cohomological rigidity problem and give some affirmative results, that is, some quasitoric manifolds are classified by their integral cohomology rings up to homeomorphism
Although singularities of degree 1 or -1 appear in the energy minimizers for harmonic maps or Oseen-Frank energy, there are many situations where singulairties of degree 1/2 or -1/2 have been observed. But there is no mathematical frame work to prove such singularities. In a joint with P. Bauman and D. Phillips, we use Landau-de Gennes Energy to prove that such singularities do exist. In the proof, we employ a famous well-known theory developed by Bethuel, Brezis, and Helein.
The study of singularities (such as ADE type singularities) has a long history in mathematics, and has also appeared in physics. In this talk, I will explain the basic concepts in quantum singularity theory. For example, $W$-curves, Witten equation, moduli spaces and the relation with $r$-spin curve theory, and integrable hierarchy.
20세기에야 비로소 독립된 분야로 확립된 게임이론은, 수학에 그 큰 빚을 지고 있다. 경제주체들의 행동을 설명하고자 했던 이 수학적 시도는, 불규칙적이고 혼란스러워 보이는 인간행동에 수학적 법칙과 질서를 부여한다.
이 세미나에서는, 누구에게나 친숙한 Nash Equilibrium을 출발점으로 게임이론의 중심화두 중 하나인 "균형"에 대해 살펴보고자 한다. 특히, Axiomatic Bargaining Game Theory의 흥미로운 예들을 통해, 수학적 균형점이 가지는 사회과학적 함의에 대해 함께 생각해 보자.
A classical problem regarding pointwise behavior of the free Schr\"odinger equation is to determine the optimal exponent $s$ for which
$\lim_{t\to 0} e^{it\Delta}f(x)=f(x) \text{ a.e. }x\in \mathbb R^d,$ whenever $f\in H^s(\mathbb R^d)$. This problem was settled for $d=1$ but still remains open for the higher dimensions even though some progresses have been made for $d=2$. In this talk we consider the convergence problem for Schr\"odinger equations with quadratic potentials, which include the hermite Schr\"odinger equation. This problem is also closely related to pointwise convergence along variable curves. We will show that these problems are more or less equivalent to that of free Schr\odinger equation and discuss on equivalence of related time-space estimates. Part of this talk is based on a joint work with Keith Rogers.
We introduce knots and links in dimension three and their deformation in dimension four which was first introduced by Fox and Milnor. We discuss its key role in understanding the topology of 4-dimensional manifolds and the interplay with 3-dimensional topology. We discuss briefly recent developments, including joint work with Kent Orr which presents new L2-theoretic methods for amenable groups.
Keeping the previous notation, these satisfy the stronger condition n>=2c+1.
Here we will talk about special cases of the (in)famous Hartshorne Conjecture.
Lecture III: Defective manifolds. There are two kinds of such: secant
defective and dual defective. We shall investigate both and see that they
are actually related.
The weakly over-penalized symmetric interior penalty (WOPSIP) method, which belongs to the family of discontinuous Galerkin methods, was introduced for second order elliptic problems by Brenner et al. in 2008. We will discuss a preconditioner for the WOPSIP method that is based on balancing domain decomposition by constraints (BDDC). Theoretical results on the condition number estimate of the preconditioned system will be presented along with numerical results. This is joint work with Susanne C. Brenner and Li-yeng Sung.
Matrix analogues of embedded surfaces, as well as discrete versions of curvature and the Euler characteristic will be presented, and the new notions, related to a novel algebraic formulation of classical differential geometry in terms of Poisson/Nambu brackets, will be illustrated by several examples.
This paper presents a mathematical analysis of the time harmonic
electrical complex potential in an electrical conducting object. In fact, there is little previous work on the complex elliptic partial differential equation (PDE) and it is difficult for us to sketch distributions of complex potential and current density for a given admittivity distribution inside the conducting object. In order to develop admittivity imaging reconstruction methods, we need to study the complex elliptic PDE and its solution. In this paper, we will introduce auxiliary real potentials which enable us to predict distributions of the complex potential and its current density. Above all, in this paper, we are interested in the interrelation between angular frequency $\om$, conductivity and permittivity distributions $\sigma$, $\ep$ inside the conducting object, and the complex potential. Although auxiliary potentials are quite useful for understanding the behavior of the complex potential, it is still difficult to see the influence of $\om$, $\sigma$, and $\ep$ to the complex potential. So we will try to investigate relations between the complex potential and $u_{\sigma}$, $u_{\ep}$, where $u_{\sigma}$ and $u_{\ep}$ are well-known solutions to the standard elliptic PDE with coefficient $\sigma$ and $\epsilon$, respectively. In particular, we will observe that the imaginary $h_{\gamma^\om}$ is closely related with the difference $u_\sigma - u_\ep$. Finally, as an application, this observation will be applied to multi-frequency Trans-Admittance Scanner in order to understand the interpretation of the frequency difference methods for breast cancer diagnosis. It is shown that the weighted difference method proposed in \cite{klswz} is actually based on the difference Neumann data $\f{\p u_\sigma}{\p \n} - \f{\p u_\ep}{\p \n}$.
Interactions between mathematics and physics are incrasing in the application of mathematics to problems in physics as well as in solving mathematical problems inspierd by physics. In this talk, mathematical frameworks to describe quantum mechanics will be introduced. Some known results and heuristic arguments to understand many-particle systems will also be explained.
An $L(j,k)$ labeling of a graph is a vertex labeling such that the difference of the labels of any two adjacent vertices is at least $j$ and that of any two vertices of distance 2 is at least $k$. The minimum of the spans of all $L(j,k)$-labelings of $G$ is denoted by $\lambda_k^j(G)$. Recently Haque and Jha \cite{HJ} proved if $G$ is a direct product of complete graphs, then $\lambda_k^j(G)$ coincide with the trivial lower bound $(N-1)k$ where $N$ is the order of $G$ when $\frac{j}{k}$ is within a certain bound.
In this paper, we suggest a new labeling method of such a graph $G$. With this method, we extend the range of $\frac{j}{k}$ such that $\lambda_k^j(G)=(N-1)k$ holds. Moreover, we obtain an upper bound of $\lambda_k^j(G)$ for the remaining cases.
In this talk we discuss the key ideas on -theory on stochastic parabolic equations and systems in an informal manner. We use only simple equations and systems to avoid technical complexity. We talk over the followings:
● A heat equation with stochastic force and its solution.
● Estimation of the solution using BDG inequality and a generalized Littlewood- Paley inequality.
● Half space domain and weights.
● A warning on systems.
● Stochastic parabolic systems.
E6-1 #1409
Discrete Math
김석진 (건국대)
Decomposition of Sparse Graphs into Forests and a Graph with Bounded Degree
E6-1 #1409
Discrete Math
Say that a graph with maximum degree at most d is d-bounded. For d>k, we prove a sharp sparseness condition for decomposability into k forests and a d-bounded graph. Consequences ar e that every graph with fractional arboricity at most k+ d/(k+d+1) has such a decomposition, and (for k=1) every graph with maximum average degree less than 2+2d/(d+2) decomposes into a forest and a d-bounded graph. When d=k+1, and when k=1 and d≤6, the d-bounded graph in the decomposition can also be required to be a forest. When k=1 and d≤2, the d-bounded forest can also be required to have at most d edges in each component.
This is joint work with A.V. Kostochka, D.B. West, H. Wu, and X. Zhu.
A time harmonic scattering problem of electromagnetic waves from a two-dimensional open cavity is considered. A variational formulation reduces the scattering problem into a bounded domain(the cavity) problem. In this talk, a stability of the solution isestablished for the bounded domain problem in the energy space.
Moreover, the stability estimate provides the explicit dependence on the high wave number.
Introduction of the "Basic Notions in Physics" seminar (hosted by Prof. Wei-Dong Ruan):
Mathematics and physics are very much intertwined, and development in mathematics has been greatly impacted by ideas from physics in the recent decades. This seminar aims at giving math students (graduate and advanced undergraduate students) and faculty an introduction to basic concepts and ideas in physics, in languages they can understand. Hopefully it will help mathematicians better understand some of the development in mathematics brought by physics ideas. We are very glad to have Prof. Stewart, an expert in theoretical physics, to give talks in the seminar.
A page in Ramanujan's lost notebook contains two identities for trigonometric sums in terms of doubly infinite series of Bessel functions. One is related to the famous "circle problem'' and the other to the equally famous "divisor problem." These relations are discussed as well as various attempts to prove the identities. Our methods also yield new identities for certain trigonometric sums, for which analogues of the circle and divisor problems are proposed. The research to be described is joint work with Sun Kim and Alexandru Zaharescu.
We establish sharp two-sided estimates for the Dirichlet heat kernel of time-dependent parabolic operators with singular drifts in a $C^{1,\alpha}$-domain in $\R^d$, where $d\ge 1$ and $\alpha \in (0, 1]$.
Our operator is $\partial_t - L - \mu \cdot \nabla_x$, where $L$ is a time-dependent uniformly elliptic divergent
operator with Dini continuous coefficients
and $\mu$ is a signed measure on $(0,\infty)\times \R^d$ belonging to the parabolic Kato class.
Along the way, a gradient estimate is
also established. In a probabilistic counterpart, we construct a Markov process corresponding to the operator $\partial_t - L - \mu \cdot \nabla_x$ and obtain two-sided estimates for the transition density of the Markov process.