KMS Meeting - Programs and Abstracts
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- Plenary Lecture
- ⋅ 17:20 − 18:10 Fokker-Planck equations for a free energy functional or Markov process on a graph (Shui-Nee Chow)
- Shui-Nee Chow, Georgia Institute of Technology
The classical Fokker-Planck equation is a linear parabolic equation which describes the time evolution of probability distribution of a stochastic process defined on an Euclidean space. Corresponding to a stochastic process, there often exists a free energy functional which is defined on the space of probability distributions and is a linear combination of a potential and an entropy. In recent years, it has been shown that Fokker-Planck equation is the gradient flow of the free energy functional defined on the Riemannian manifold of probability distributions whose inner product is generated by a 2-Wasserstein distance. In this paper, we consider analogous matters for a free energy functional or Markov process defined on a graph with a finite number of vertices and edges. If $N\ge 2$ is the number of vertices of the graph, we show that the corresponding Fokker-Planck equation is a system of $N$ nonlinear ordinary differential equations defined on a Riemannian manifold of probability distributions. However, in contrast to stochastic processes defined on Euclidean spaces, situation is more subtle for discrete spaces. We have different choices for inner products on the space of probability distributions resulting in different Fokker-Planck equations for the same process. It is shown that there is a strong connection but also substantial discrepancies between the systems of ordinary differential equations and the classical Fokker-Planck equation on Euclidean spaces. Furthermore, both systems of ordinary differential equations are gradient flows for the same free energy functional defined on the Riemannian manifolds of probability distributions with different metrics. Some examples will also be discussed.
2010 Mathematics Subject Classification: 37H10,60J27,60J60
Key Words and Phrases: Markov process, Fokker-Planck equation, Gibbs density, graph
- Special Lecture
- ⋅ 16:30 − 17:20 한국연구재단의 기초연구 지원정책 (Se-Jung Oh)
- 오세정(한국연구재단 이사장)
Se-Jung Oh, President of National Research Foundation of Korea
이 강연에서는 먼저 우리나라 R$\&$D의 성과와 당면문제, 특히 기초연구의 현주소를 살펴본 후, 이에 대응하는 정부의 기초 및 원천연구 지원정책 방향을 간단히 정리해 볼 것이다. 그리고 이에 따라 한국연구재단에서 앞으로 대학의 기초연구를 지원하기 위해서 펼칠 정책을 논의하려 한다.
2010 Mathematics Subject Classification:
Key Words and Phrases:
- Invited Lectures
- ⋅ 10:10 − 10:50 Dynamics of group action on non-positively curved spaces and its application (Seonhee Lim, Hee Oh)
- 임선희*(서울대), 오희(브라운대)
Seonhee Lim*, Seoul National University, Hee Oh, Brown University
We introduce several dynamical properties of group action on non-positively curved spaces, and relate them with topological and geometrical properties of the spaces. We will also show some examples of its applications to problems in number theory.
2010 Mathematics Subject Classification: 57Sxx
Key Words and Phrases: dynamics of Lie groups, ergodic theory, metric spaces of non-positive curvature
- ⋅ 10:10 − 10:50 Resolvent average on symmetric cones (Sangho Kum)
- 금상호(충북대)
Sangho Kum, Chungbuk National University
Recently Bauschke et al. introduced a very interesting and new notion of proximal average in the context of convex analysis, and studied this subject systemically from various viewpoints. In addition, this new concept was applied to positive semidefinite matrices under the name of resolvent average, and basic properties of the resolvent average are successfully established by themselves from a totally different view and techniques of convex analysis rather than the classical matrix analysis. Inspired by their works and the well-known fact that the convex cone of positive definite matrices is a typical example of a symmetric cone (self-dual homogeneous convex cone), we study the resolvent average on symmetric cones, and derive corresponding results in a different manner based on a purely Jordan algebraic technique. Some average for general convex functions is also introduced.
2010 Mathematics Subject Classification: 17C99, 47A10, 90C25
Key Words and Phrases: Resolvent average, symmetric cone, Hua's identity, logarithmic barrier, Bregman distance
- ⋅ 11:00 − 11:40 Symmetry of planar curves and capillary surface in a wedge in $\mathbb H^3$ (Sung-ho Park)
- 박성호(한국외대)
Sung-ho Park, Hankuk university of foreign studies
We give several conditions for a planar curve to be a circle in $\mathbb R^2$ and $\mathbb H^2$, and apply the result to derive a necessary condition for capillary surfaces in a wedge in $\mathbb H^3$.
2010 Mathematics Subject Classification: 53A10, 53A04
Key Words and Phrases: capillary surface, symmetry of planar curve
- ⋅ 11:00 − 11:40 On the quasi-regularity of non-sectorial Dirichlet forms by processes having the same polar sets (Lucian Beznea)
- Lucian Beznea, Institute of Mathematics of the Romanian Academy
We obtain a criterion for the quasi-regularity of generalized (non- sectorial) Dirichlet forms, which extends the result of P. J. Fitzsimmons on the quasi-regularity of (sectorial) semi-Dirichlet forms. Given the right (Markov) process associated to a semi-Dirichlet form, we present sufficient conditions for a second right process to be a standard one, having the same state space. The above mentioned quasi-regularity criterion is then an application. The conditions are expressed in terms of the associated capacities, nests of compacts, polar sets, and quasi-continuity. A second application is on the quasi-regularity of the generalized Dirichlet forms obtained by perturbing a semi-Dirichlet form with kernels. The talk is based on a joint work with Gerald Trutnau.
2010 Mathematics Subject Classification: 31C25, 31C15, 60J45, 47A55, 47D07, 60J35, 60J40
Key Words and Phrases: Dirichlet form, generalized Dirichlet form, quasi-regularity, standard process, capacity, quasi-continuity, polar set, right process, weak duality
- ⋅ 13:10 − 13:50 Witt vectors and Witt-Burnside rings (Young Tak Oh)
- 오영탁(서강대)
Young Tak Oh, Sogang University
The Witt-Burnside ring of a profinite group is a group-theoretical generalization of big Witt vectors. For instance, the classical $p$-typical Witt vectors of Teichmuller and Witt correspond to the Witt-Burnside ring of the profinite $p$-completion of the infinite cyclic group $C$ and the universal Witt vectors of Lang and Witt to that of the profinite completion of $C$. In this talk, we explain recent results on the structure of Witt-Burnside rings. A special emphasis will be put on the classification and decomposition. Connection to representation theory will be also discussed.
2010 Mathematics Subject Classification: 13F35
Key Words and Phrases: Witt vector, Witt-Burnside ring, Profinite group, Representaion
- ⋅ 13:10 − 13:50 Existence of topological solutions in the self-dual gauge field theories via super and subsolution method (Jongmin Han)
- 한종민(경희대)
Jongmin Han, Kyung Hee University
In this talk, we discuss how to find topological solutions in the self-dual gauge field theories via the method of super and subsolutions. In particluar, we propose a new method to find an explicit subsolution. As an application, we provide a new proof of the existence of topological solutions in two Higgs model. We also prove the existence of topological solutions in the Maxwell gauged $O(3)$ sigma model and the Chern-Simons gauged $O(3)$ sigma model.
2010 Mathematics Subject Classification: 81T13, 35B40
Key Words and Phrases: topological solutions, self-dual equations, method of super and subsolutions
- ⋅ 13:10 − 13:50 창의 중심의 미래형 수학과 교육과정 시안 개발의 방향과 내용 체계 (Sunwook Hwang)
- 황선욱(숭실대)
Sunwook Hwang, Soongsil University
TBA
2010 Mathematics Subject Classification:
Key Words and Phrases:
- ⋅ 14:00 − 14:40 Thue choice number of graphs (Xuding Zhu)
- Xuding Zhu, Zhejiang Normal University
A sequence of even length is a repetition if the first half is identical to the second half. A sequence is said to contain a repetition if it has a subsequence which is a repetition. A classical result of Thue says that there is an infinite sequence on $3$ symbols which contains no repetition. This result motivated many deep research and challenging problems. One graph concept related to this result is Thue-colouring. A Thue-colouring of a graph $G$ is a mapping which assigns to each vertex of $G$ a colour (a symbol) in such a way that the colour sequence of any path of $G$ contains no repetition. The Thue-chromatic number of a graph is the minimum number of colours needed in a Thue-colouring of $G$. Thue's result is equivalent to say that the infinite path has Thue-chromatic number $3$. It is also known that the Thue-chromatic number of any tree is at most $4$. Thue-choice number of a graph $G$ is the list version of its Thue-chromatic number, which is the minimum integer $k$ such that if each vertex of $G$ is given $k$-permissible colours, then there is a Thue-colouring of $G$ using a permissible colour for each vertex. This talk will survey some research related to Thue Theorem and will show that Thue-choice number of paths is at most $4$ and Thue choice number of trees are unbounded.
2010 Mathematics Subject Classification: 05Cxx
Key Words and Phrases: Thue-colouring, Thue-choice number, graph
- ⋅ 14:00 − 14:40 Non-localization phenomena of oscillatory integral operators (Chan Woo Yang)
- 양찬우(고려대)
Chan Woo Yang, Korea University
In this talk we consider oscillatory integral operators $T_{\lambda}$ of the form $$T_{\lambda}f(x)=\int_{\mathbb{R}} e^{i \lambda S(x,y)} f(y) dy,$$ where $S$ is a polynomial with two real variables $x$ and $y$. We call these operators non-localized oscillatory integral operators. Usually oscillatory integral operators are operators of the form $$T_{\lambda}f(x)=\int_{\mathbb{R}} e^{i \lambda S(x,y)} \chi (x,y) f(y) dy,$$ where $\chi$ is a smooth cut-off function supported in a small neighborhood of the origin in $\mathbb{R}^2$. Because of the presence of the cut-off function $\chi$ we call these operators localized oscillatory integral operators. Almost full picture of decay rate of operator norm of localized oscillatory integral operators have been drawn in terms of Newton polygon of the phase function $S$ in numerous articles. However non-localized oscillatory integral operators have not been fully understood. The aim of this talk is to discuss interesting phenomena of mapping properties of non-localized oscillatory integral operators due to the absense of the cut-off function $\chi$ in terms of `non-localized' Newton polygon in adapted coordinate system.
2010 Mathematics Subject Classification: 42-xx
Key Words and Phrases: non-localized, oscillatory integral operators, Newton polygon
- ⋅ 15:20 − 16:00 Poincare-Dulac normal form reduction for unconditional well-posedness of the periodic cubic (Soonsik Kwon, Tadahiro Oh)
- 권순식*(카이스트), 오충홍(프린스턴대)
Soonsik Kwon*, KAIST, Tadahiro Oh, Princeton University
We implement an infinite iteration scheme of Poincare-Dulac normal form reductions to establish an energy estimate on the one-dimensional cubic nonlinear Schrodinger equation. As a result, we prove its unconditional well-posedness in $L^2$. In the talk we explain how to adopt Poincare-Dulac normal form technique to PDE setting with multilinear estimates.
2010 Mathematics Subject Classification: 35Q55
Key Words and Phrases: NLS, normal form, well-posedness, uniqueness
- ⋅ 15:20 − 16:00 Efficient ID-based signatures and multisignatures based on the q-SDH problem (Kyung-Ah Shim)
- 심경아(국가수리과학연구소)
Kyung-Ah Shim, National Institute for Mathematical Sciences
Paterson and Schuldt proposed an ID-based signature scheme based on Waters' signature scheme, secure in the standard model under the Computational Diffie-Hellman assumption. They claimed that the scheme allows secure ID-based aggregate signature (multisignature) scheme. We show that the ID-based aggregate (multisignature) scheme based on Paterson and Schuldt's signature scheme is insecure against forgery attacks. We then propose a new ID-based signature scheme based on the q-SDH assumption which allows us to construct an efficient multisignature scheme secure in the standard model.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: ID-based system, digital signature, multisignature, bilinear pairing, q-Strong Diffie-Hellman problem
- Mathematical Logic and Foundations
- ⋅ 09:00 − 09:30 Computer verification of mathematical proofs (Joohee Jeong)
- 정주희(경북대)
Joohee Jeong, Kyungpook National University
It is well known that mathematical proofs can be formalized, in principle, so that the validity of the proofs can be rigorously verified by computers. The actual implementation of a proof verifier faces two challenges. One is the efficiency and the other is human readability. We investigate the strengths and weaknesses of the existing systems such as LPL(Language, Proof and Logic: CSLI) and Coq. Then we present Proofmood, a web-based proof verifier developed by the author.
2010 Mathematics Subject Classification: 03B35
Key Words and Phrases: formal proof, verification
- ⋅ 09:30 − 09:50 Consistency of ZFC$+\neg$ AC (Ji Hoon No)
- 노지훈(연세대)
Ji Hoon No, Yonsei University
We will show that $\neg$ axiom of choice is consistent with ZFC. For this work, we will study about forcing method. Then, for given countable transitive model M, we can define partial ordering, and find generic extension. Finally, we will find some generic extension. This model satisfy ZFC, but we can`t find well-ordering of $\wp(\omega)$. This means axiom of choice is failed in this model.
2010 Mathematics Subject Classification: 03E25
Key Words and Phrases: Axiom of choice, Forcing
- ⋅ 10:10 − 10:40 Amalgamation functors and homology groups in model theory (Byunghan Kim)
- 김병한(연세대)
Byunghan Kim, Yonsei University
We present definitions of homology groups $H_n$, $n\geq 0$, associated to a family of ``amalgamation functors''. We show that if the generalized amalgamation properties hold, then the homology groups are trivial. We compute the group $H_2$ for strong types in stable theories and show that any profinite abelian group can occur as the group $H_2$ in the model-theoretic context.
2010 Mathematics Subject Classification: 03C45, 55N35
Key Words and Phrases: amalgamation functors, homology groups
- ⋅ 10:40 − 11:10 Logic and software verification (Gyesik Lee)
- 이계식(한경대)
Gyesik Lee, Hankyong National University
Logical foundation for software verification
2010 Mathematics Subject Classification: 03F03
Key Words and Phrases: logic, software verification
- ⋅ 11:10 − 11:30 The unprovability of some combinatorial statement within a system satisfying the Peano axioms (Sup Gun Lee)
- 이섭건(연세대)
Sup Gun Lee, Yonsei University
Godel's first incompleteness theorem states that, in any consistent, axiomatizable theory that can encode the arithmetic of natural numbers, there must exist a statement that is true but unprovable. In this paper, we will illustrate the Godel's idea by showing that a given combinatorial statement is true and yet unprovable within a system satisfying the Peano axioms
2010 Mathematics Subject Classification: 03C99
Key Words and Phrases: Peano axioms, unprovability
- Representation Theory
- ⋅ 10:00 − 10:20 Poisson structures of hypersurface (Sei-Qwon Oh)
- 오세권(충남대)
Sei-Qwon Oh, Chungnam National University
Let $s,t$ be elements of the polynomial algebra $A=\Bbb C[x,y,z]$ such that $s$ and $t$ are coprime. Then $A$ is a Poisson algebra with qm-Poisson bracket determined by $s/t$. In this talk, Poisson spectrum and Poisson primitive spectrum of $A$ are given and the results are applied to hypersurfaces.
2010 Mathematics Subject Classification: 17B63, 16S36, 13N15
Key Words and Phrases: Poisson algebra, Poisson prime ideal
- ⋅ 10:20 − 10:40 $\lambda$-ring structure of the Green ring of a cyclic $p$-group (Sun-Young Nam, Young-Tak Oh)
- 남선영*(서강대), 오영탁(서강대)
Sun-Young Nam*, Sogang University, Young-Tak Oh, Sogang University
Let $\mathbb F_p$ be a field of order $p$, $G$ a cyclic group of order $p^k$, and $RS(p^k)$ the Green ring of $G$ over $\mathbb F_p$. This talk concerns the conjecture on the $\lambda$-ring structure of a certain quotient ring $RS(p^k)/I_{p^k}$ of $RS(p^k)$ when $k\ge 2$, which was originally due to Kouwenhoven.
2010 Mathematics Subject Classification: 20C05, 20C20
Key Words and Phrases: modular representation, Green ring, $\lambda$-ring, Adams operation
- ⋅ 10:40 − 11:00 Irreducible modules over Khovanov-Lauda-Rouquier algebras of classical type (Euiyong Park, Georgia Benkart, Seok-Jin Kang and Se-jin Oh)
- 박의용*(고등과학원), Georgia Benkart(Univ. of Wisconsin), 강석진(서울대), 오세진(서울대)
Euiyong Park*, Korea Institute for Advanced Study, Georgia Benkart, University of Wisconsin, Seok-Jin Kang, Seoul National University, Se-jin Oh, Seoul National University
In this talk, we give an explicit construction of irreducible modules over Khovanov-Lauda-Rouquier algebras and their cyclotomic quotients of classical type. This construction is compatible with Kashiwara operators.
2010 Mathematics Subject Classification: 16G99
Key Words and Phrases: Khovanov-Lauda-Rouquier algebras, crystal
- ⋅ 11:00 − 11:20 Crystal bases for the quantum queer superalgebra and semistandard decomposition tableaux (Myungho Kim, Dimitar Grantcharov, Ji Hye Jung, Seok-Jin Kang and Masaki Kashiwara)
- 김명호*(서울대), Dimitar Grantcharov(Univ. of Texas at Arlington), 정지혜(서울대), 강석진(서울대), Masaki Kashiwara(RIMS(교토대), 서울대)
Myungho Kim*, Seoul National University, Dimitar Grantcharov, University of Texas at Arlington, Ji Hye Jung, Seoul National University, Seok-Jin Kang, Seoul National University, Masaki Kashiwara, Research Institute for Mathematical Sciences(Kyoto University) / Seoul National University
In this talk, we give an explicit combinatorial realization of the crystal $B(\lambda)$ for an irreducible highest weight $U_q(\mathfrak{q}(n))$-module $V(\lambda)$ in terms of semistandard decomposition tableaux. We present an insertion scheme for semistandard decomposition tableaux and give algorithms of decomposing the tensor product of $\mathfrak{q}(n)$-crystals. Consequently, we obtain explicit combinatorial descriptions of the shifted Littlewood-Richardson coefficients.
2010 Mathematics Subject Classification: 17B37, 81R50
Key Words and Phrases: quantum queer superalgebras, crystal bases, odd Kashiwara operators, semistandard decomposition tableaux, shifted Littlewood-Richardson coefficient
- ⋅ 11:20 − 11:40 Realization and exoticity index of fusion systems (Sejong Park)
- 박세종(서강대)
Sejong Park, Sogang University
Fusion systems are categories which makes precise and extends the traditional notion of local structure of finite groups, providing a fertile ground for interaction among group theory, representation theory and algebraic topology. In this talk, we will give a brief overview of the theory of fusion systems, then discuss realization problem and exoticity index of fusion systems.
2010 Mathematics Subject Classification: 20C20
Key Words and Phrases: fusion systems
- Several Complex Variables
- ⋅ 09:00 − 09:30 Algebraic generalizations of the Newlander-Nirenberg theorem (Chong-Kyu Han, Hyeseon Kim)
- 한종규*(서울대), 김혜선(성균관대)
Chong-Kyu Han*, Seoul National University, Hyeseon Kim, SungKyunKwan University
Given an almost complex structure on a smooth manifold of dimension $2m$, the Nijenheus tensor, or equivalently the torsion tensor, measures the non-integrability of the almost complex structure. With respect to a basis of $(1,0)$-forms the torsion tensor is represented as a matrix $T$ of dimension $m\times m(m-1)/2$. If $T$ is identically zero then the structure is integrable. We discuss the rank of $T$ on various submanifolds and their implications on the partial integrability of the structure. In cases of perturbation of the standard complex structure, these rank conditions are given by a system of second order PDE on the perturbation.
2010 Mathematics Subject Classification: 32W05, 35N10
Key Words and Phrases: almost complex structure, Nijenhuis tensor, integrability, Newlander-Nirenberg theorem
- ⋅ 09:30 − 10:00 A characterization of harmonic Bergman spaces (Kye Sook Nam)
- 남계숙(서울대)
Kye Sook Nam, Seoul National University
In this talk, we give a characterization of weighted harmonic Bergman spaces on the real ball and the half-space by means of integrability conditions of double integrals. Our observations reveal an interesting half-space phenomenon caused by the unboundedness of the halfspace. The talk is based on a joint work with B.R. Choe.
2010 Mathematics Subject Classification: 31B05, 31B10, 30H20
Key Words and Phrases: weighted harmonic Bergman spaces, unit ball, upper half-space
- ⋅ 10:10 − 10:50 Recent progress in composition operators in several variables (Hyungwoon Koo)
- 구형운(고려대)
Hyungwoon Koo, Korea University
In this talk we survey recent progress on composition operators in several variables. Unlike the unit disc case, there are smooth holomorphic self-maps which induce unbounded composition operators on Hardy or Bergman spaces on the unit ball. In this talk we review one variable results and coresponding progress in several variables.
2010 Mathematics Subject Classification: 47B33
Key Words and Phrases: composition operators
- ⋅ 11:00 − 11:20 Function theory on Fock-Sobolev spaces (Hong Rae Cho)
- 조홍래(부산대)
Hong Rae Cho, Pusan National University
We consider the Fock-Sobolev space $F^{p,m}$ consisting of entire functions $f$ such that $f^{(m)}$, the $m$-th order derivative of $f$, is in the Fock space $F^p$. We show that an entire function $f$ is in $F^{p,m}$ if and only if the function $z^mf(z)$ is in $F^p$. We also characterize the Carleson measures for the spaces $F^{p,m}$, establish the boundedness of the weighted Fock projection on appropriate $L^p$ spaces, identify the Banach dual of $F^{p,m}$, and compute the complex interpolation space between two $F^{p,m}$ spaces.
2010 Mathematics Subject Classification: 30H20
Key Words and Phrases: Fock space, Fock-Sobolev space, Carleson measure
- ⋅ 11:20 − 11:40 Automorphism group of an almost CR manifold (Jae-Cheon Joo)
- 주재천(부산대)
Jae-Cheon Joo, Pusan National University
This talk concerns a recent progress on the characterization of strongly pseudoconvex almost CR manifolds by the action of the automorphism groups. I would also like to discuss about some interesting examples and related problems with this subject.
2010 Mathematics Subject Classification: 32V05
Key Words and Phrases: almost CR manifold
- Banach Spaces and Related Topics
- ⋅ 09:00 − 09:30 The dual space of $L(X,Y)$ endowed with the topology of uniform convergence on $p$-compact subsets of $X$ (Ju Myung Kim)
- 김주명(서울대)
Ju Myung Kim, Seoul National University
We establish a representation of the dual space of $L(X,Y)$, the space of bounded linear operators from a Banach space $X$ into a Banach space $Y$, endowed with the topology of uniform convergence on $p$-compact subsets of $X$.
2010 Mathematics Subject Classification: 46B28, 46B10
Key Words and Phrases: Banach space, $p$-compact set, operator, dual space
- ⋅ 09:30 − 10:00 Banach-Saks property and spreading model (Kyugeun Cho, Chongsung Lee)
- 조규근*(명지대), 이종성(인하대)
Kyugeun Cho*, MyongJi University, Chongsung Lee, Inha University
We study Banach-Saks property in Banach spaces using the Brunel-Sucheston's spreading model.
2010 Mathematics Subject Classification: 46B15
Key Words and Phrases: Banach-Saks property, spreading model
- ⋅ 10:10 − 10:40 Conditional integral transforms and conditional convolution products on a Banach space (Dong Hyun Cho)
- 조동현(경기대)
Dong Hyun Cho, kyonggi University
Let $C[0, t]$ denote the function space of all real-valued continuous paths on $[0, t]$.
Define $X_{n}: C[0, t]\to \mathbb R^{n+1}$ and $X_{n+1}: C[0, t]\to \mathbb R^{n+2}$ by
\begin{eqnarray*}
X_n(x) = (x(t_0),
x(t_1), \cdots, x(t_n))
\end{eqnarray*}
and
\begin{eqnarray*}
X_{n+1}(x) = (x(t_0), x(t_1), \cdots,x (t_n), x(t_{n+1})),
\end{eqnarray*}
where $0=t_0 < t_1 < \cdots < t_n<t_{n+1} =t$.
In the present talk,
using two simple formulas for the conditional expectations with the conditioning functions $X_n$ and $X_{n+1}$,
we evaluate the conditional analytic Fourier-Feynman transforms
and the conditional convolution products of the cylinder functions which have the form
\begin{eqnarray*}
f((v_1,x), \cdots, (v_r,x)) \text{ for } x\in C[0,t],
\end{eqnarray*}
where $\{v_1, \cdots, v_r\}$ is an orthonormal subset of $L_2[0,t]$ and $f\in L_1(\mathbb R^r)$.
We finally show that the conditional analytic Fourier-Feynman transforms of the conditional convolution products for the cylinder functions, can be expressed in terms of the products of the conditional analytic Fourier-Feynman transform of each function.
2010 Mathematics Subject Classification: 28C20
Key Words and Phrases: Fourier-Feynman transform, convolution, conditional Fourier-Feynman transform, conditonal convolution
- ⋅ 10:40 − 11:10 Weighted fourier algebra and complexification of compact Lie groups (Hun Hee Lee)
- 이훈희(충북대)
Hun Hee Lee, Chungbuk National University
In this talk we will introduce a class of commutative Banach algebra called Beurling-Fourier algebra, which is a weighted version of Fourier algebra of a compact group. Then, we will explain its connection to the complexification of compact Lie groups. Our primary examples of compact groups will be the torus and SU(2).
2010 Mathematics Subject Classification: 43A30
Key Words and Phrases: Fourier algebra, spectrum, complexification of Lie group
- ⋅ 11:10 − 11:40 The Bishop-Phelps-Bollob\'as theorem for operators from uniformly convex Banach spaces (Sun Kwang Kim, Yun Sung Choi)
- 김선광*(포항공대), 최윤성(포항공대)
Sun Kwang Kim*, Postech, Yun Sung Choi, Postech
In 1970, Bollob\'as sharpened the Bishop-Phelps theorem namely the Bishop-Phelps-Bollob\'as theorem. In this paper, we introduce a Approximate Optimization property for reflexive Banach spaces which is related with the Bishop-Phelps-Bollob\'as theorem and its application to the Bishop-Phelps-Bollob\'as property.
2010 Mathematics Subject Classification: 46b20
Key Words and Phrases: uniformly convex, Bishop Phelphs theorem, norm attaining
- p-Adic Analysis and Their Applications
- ⋅ 09:00 − 09:15 On periods of cusp forms for $SL_2(\mathbb Z)$ and $p$-adic Hecke series (Min-Soo Kim)
- 김민수(카이스트)
Min-Soo Kim, KAIST
Eichler discovered relations between periods of cusp forms, extended by Shimura. Manin made more explicit the connection of these relations with the coefficients in the $q$-expansion. Our first purpose is to give an exposition of a part of Manin's work. In addition, we introduce $p$-adic analogs of the Hecke series of parabolic forms in the manner of Iwasawa, Mazur and Manin.
2010 Mathematics Subject Classification: 11F67,11F85
Key Words and Phrases: cusp forms, $p$-adic Hecke series
- ⋅ 09:15 − 09:30 A numerical approach to $q$-Bernstein type polynomials (Young-Hee Kim)
- 김영희(광운대)
Young-Hee Kim, Kwangwoon University
Recently, various $q$-Bernstein type polynomials different from the $q$-Bernstein polynomials of Phillips [G. M. Phillips, Bernstein polynomials based on the $q$-integers, Ann. Numer. Math. {\bf 14} (1997), 511--518] has been studied by many authors. In this talk, we approach $q$-Bernstein type polynomials numerically and investigate shapes of these polynomials according to $q$. We also propose the $k$-dimensional generalization of $q$-Bernstein polynomials and discuss some results related to these polynomials.
2010 Mathematics Subject Classification: 41A10, 65D20
Key Words and Phrases: $q$-Bernstein polynomial
- ⋅ 09:30 − 09:45 Future internet and q-series (Daeyeoul Kim)
- 김대열(국가수리과학연구소)
Daeyeoul Kim, National Institute for Mathematical Sciences
In this talk, I will state future internet in our institute. And I shall explain an approach for future internet using q-series.
2010 Mathematics Subject Classification: 05A17
Key Words and Phrases: q-series, future internet
- ⋅ 09:45 − 10:00 A study on the eigenvalues of the $p$-adic $q$-transfer operator on the space of twisted $q$-Euler polynomials (WonJoo Kim, Lee-Chae Jang)
- 김원주*(건국대), 장이채(건국대)
WonJoo Kim*, Konkuk University, Lee-Chae Jang, Konkuk University
Let $T_p= \cup_{n\geq}C_{p^n}=\lim_{n\rightarrow\infty}C_{p^n}$ be the locally constant space, where $C_{p^n}=\{ w | w^{p^n}=1\}$ is the cyclic group of order of $p^n$. For $w\in T_p$, $\phi_w : \Bbb Z_p \rightarrow\Bbb C_p$ is the locally constant function $x \mapsto w^x$ (cf. [2-4]). In [14], Kim et al defined the twisted $q$-Euler polynomials by using $p$-adic $q$-integral on $\Bbb Z_p$ as follows: \begin{eqnarray} E_{n,w}^{(h,q)} (x)= \int_{\Bbb Z_p } q^{(h-1)y}w^y[x+y]_q^n d\mu_{-q} (y), \end{eqnarray} where $w\in T_p$ and $h\in \Bbb Z$ and $n\in \Bbb N$. The transfer operator is defined as an operator $L$ acting on the space of functions $\phi: X \rightarrow \Bbb C$ as $$ (L\phi)(x)=\sum_{y\in f^{-1}(x)} g(y)\phi(y) $$ where $g: X \longrightarrow \Bbb C$ is an auxiliary valuation function. In this paper we consider the $p$-adic $q$-transfer operator on the space of all twisted $q$-Euler polynomials and investigate the eigenvalues of the $p$-adic $q$-transfer operators related with twisted $q$-Euler polynomials.
2010 Mathematics Subject Classification: 11B68, 11S80,41A30
Key Words and Phrases: twisted $q$-Euler polynomials, $p$-adic $q$-integral, $q$-transfer operators
- ⋅ 10:10 − 10:25 A note on the twisted q-Euler numbers and polynomials with weight $\alpha$ (H.Y.Lee, N.S.Jung)
- 이희영*(한남대), 정남순(한남대)
H.Y.Lee*, Hannam University, N.S.Jung, Hannam University
In this paper we construct a new type of twisted $q$-Euler numbers $E_{n,q,w}^{(\alpha)}$ and polynomials $E_{n,q,w}^{(\alpha)}(x)$. Some interesting results and relationships are obtained.
2010 Mathematics Subject Classification: 11B68,11S40,11S80
Key Words and Phrases: Euler numbers and polynomials, $q$-Euler numbers and polynomials, weighted $q$-Euler numbers and polynomials
- ⋅ 10:25 − 10:40 Multivariate twisted $p$-adic q-integral on $\mathbb{Z}_p$ associated with twisted q-Bernoulli polynomials and numbers (Joung Hee Jin, Seog Hoon Rim, Eun Jung Moon and Sun Jung Lee)
- 진정희*(경북대), 임석훈(경북대), 문은정(경북대), 이선정(경북대)
Joung Hee Jin*, Kyungpook National University, Seog Hoon Rim, Kyungpook National University, Eun Jung Moon, Kyungpook National University, Sun Jung Lee, Kyungpook National University
Recently, many authors have studied twisted q-Bernoulli polynomials by using the $p$-adic invariant q-integral on $\mathbb{Z}_p$. In this paper, we define the twisted $p$-adic q-integral on $\mathbb{Z}_p$ and extend our result to the twisted q-Bernoulli polynomials and numbers. Finally, we derive some various identities related to the twisted q-Bernoulli polynomials.
2010 Mathematics Subject Classification: 11B68, 11S40, 11S80
Key Words and Phrases:
- ⋅ 10:40 − 10:55 Alon-Babai-Suzuki's conjecture on nonmodular (Kyung-won Hwang)
- 황경원(동아대)
Kyung-won Hwang, Donga-A University
In 1991, Alon-Babai-Suzuki made a conjecture. We prove that Alon-Babai-Suzuki's conjecture on nonmodular. Joint work with T. Kim, l. C. Jang, P. Kim, and G. Sohn.
2010 Mathematics Subject Classification: 58A
Key Words and Phrases: finite intersection, extremal set
- ⋅ 10:55 − 11:10 A systemic study of the generalized Barnes-type multiple twisted $q$-Euler polynomials of higher order (Lee Chae Jang)
- 장이채(건국대)
Lee Chae Jang, Konkuk University
Recently, many authors have studied the $q$-extension of various twisted Bernoulli or Euler polynomials and numbers (see[2,3,11,12,14-17]). In this paper, we try to consider the twisted $q$-extension of Barnes-type multiple Euler polynomials and numbers related to fermionic $p$-adic invariant integrals which were defined by Kim [4].
Note that
$$ \lim_{q \rightarrow 1} {\binom{n}{k}}_q =\binom{n}{k}=\frac{n!}{(n-k)!k!}=\frac{n(n-1)\cdots (n-k+1)}{k!}.$$
We say that $f$ is uniformly differentiable function at a point $a \in \Bbb Z_p$, and write $f\in UD(\Bbb Z_p)$, if the difference quotient $F_f(x,y)=\frac{f(x)-f(y)}{x-y}$ has a limit $f'(a)$ as $(x,y) \rightarrow(a,a)$.
For $f\in UD(\Bbb Z_p)$, the fermionic $p$-adic invariant integral on $\Bbb Z_p$ is defined as
$$
I_{q} (f) =\int_{\Bbb Z_p }f(x) d\mu_{q} (x) = \lim_{n\to
\infty} \dfrac{1}{[p^n ]_{-q}} \sum_{x=0}^{p^n -1} f(x) (-q)^x .
\leqno(1)$$
Thus we note that
$$
\lim_{q\rightarrow 1} I_{q} (f) = I_{1} (f) = \int_{\Bbb Z_p }f(x) d\mu_{1} (x).
\leqno(2)
$$
For $n \in \Bbb N$, let $f_n (x)= f(x+n)$. Then we have
$$ I_{1} (f_n) = (-1)^n I_{1}(f)+2\sum_{l=0}^{n-1} (-1)^{n-1-l}f(l).
\leqno(3)
$$
Using formula (3), we can derive the Euler polynomials, $E_n(x)$, namely
$$
I_{1}(e^{tx}) = \int_{\Bbb Z_p} e^{(x+y)t} d \mu_{1} (y) = \frac{2}{e^t+1}e^{xt}
=\sum_{n=0}^{\infty} E_n(x) \frac{t^n}{n!}.
\leqno(4)
$$
In this paper, we define Barnes-type multiple twisted $q$-Euler polynomials and numbers related
fermionic $p$-adic invariant integrals and discuss some properties of them. In particular,
we present a systemic study of the generalized Barnes-type multiple twisted $q$-Euler polynomials and numbers
of higher order and
give Barnes-type multiple twisted zeta functions which interpolate the generalized Barnes-type multiple twisted $q$-Euler polynomials.
2010 Mathematics Subject Classification: 11B68, 11S80
Key Words and Phrases: Euler polynomials and numbers, Fermionic $p$-adic invariant integrals, the Barnes-type multiple twisted $q$-Euler polynomials and numbers
- ⋅ 11:10 − 11:25 Dynamics of the zeros of Bernoulli polynomials (Cheon Seoung Ryoo)
- 유천성(한남대)
Cheon Seoung Ryoo, Hannam University
In this talk we observe the behavior of roots of the $(h,q)$-extension of Bernoulli polynomials $B_{n,q}^{(h)}(x)$. By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the $q$-extension of Bernoulli polynomials $B_{n,q}^{(h)}(x)$. The main purpose of this paper is also to investigate the zeros of the $(h,q)$-extension of Bernoulli polynomials $B_{n,q}^{(h)}(x)$. Furthermore, we give a table for the zeros of the $(h,q)$-extension of Bernoulli polynomials $B_{n,q}^{(h)}(x)$.
2010 Mathematics Subject Classification: 11B68, 11S40, 11S80
Key Words and Phrases: Bernoulli numbers, Bernoulli polynomials, q-extension of Bernoulli numbers, dynamics of the zeros
- ⋅ 11:25 − 11:40 The consideration on the connection and convergence among the content of mathematics learnt (Jong Jin Seo)
- 서종진(부경대)
Jong Jin Seo, Pukyong National University
In order for learners to learn a certain contents of mathematics, there are contents to be preceded. Here, let's consider two matters largely. One is: do they know the contents necessary for learning the contents of the mathematics given? Another is to evaluate what they know about the contents of mathematics learnt in connection with each other in what degree. Here, the evaluation is not that we simply evaluate what the learners know about the contents of the mathematics which they already learnt in what degree but it means detailed evaluation. In this study, we considered the method to measure what the learners knows about the contents of mathematics already learnt and in what degree they connect them. And based on the connection among the contents which learners already learnt we discussed what degree of convergence they had.
2010 Mathematics Subject Classification: 97D60
Key Words and Phrases: convergence, connection, mathematical concept, problem solving
- ⋅ 14:50 − 15:05 A new construction on the q-Bernoulli polynomials (Eun Jun Moon, Seog Hoon Rim, A. Bayad, Jeong Hee Jin and Sun Jung Lee)
- 문은정*(경북대), 임석훈(경북대), A. Bayad(Universite' d'Evry Val d'Essonne), 진정희(경북대), 이선정(경북대)
Eun Jun Moon*, Kyungpook National University, Seog Hoon Rim, Kyungpook National University, A. Bayad, Universite' d'Evry Val d'Essonne, Jeong Hee Jin, Kyungpook National University, Sun Jung Lee, Kyungpook National University
This paper performs a further investigation on the q-Bernoulli polynomials and numbers given by M. Acikgoz et al. (Advances in Difference Equations, vol. 2010, Article ID 951764, 9 pages), some incorrect properties are revised. It is pointed out that the definition concerning the q-Bernoulli polynomials and numbers is unreasonable. The purpose of this talk is to redefine the q-Bernoulli polynomials and numbers, and correct its wrong properties and rebuild its theorems.
2010 Mathematics Subject Classification: 11B68, 11S40, 11S80
Key Words and Phrases: q-Bernoulli polynomials
- ⋅ 15:05 − 15:20 On the fermionic $p$-adic integral representation of $q$-Bernstein polynomials associated with $q$-Genocchi numbers and polynomials (Sun Jung Lee, Seog Hoon Rim, Eun Jung Moon and Joung Hee Jin)
- 이선정*(경북대), 임석훈(경북대), 문은정(경북대), 진정희(경북대)
Sun Jung Lee*, Kyungpook National University, Seog Hoon Rim, Kyungpook National University, Eun Jung Moon, Kyungpook National University, Joung Hee Jin, Kyungpook National University
In this paper, we give some properties of several q-Bernstein polynomials to represent the fermionic $p$-adic Integral on ${\mathbb{Z}_p}$. From the properties, we construct some interesting identies of $q$-Bernstein polynomials associated with q-Genocchi numbers and polynomials.
2010 Mathematics Subject Classification: 11B68, 11S40, 11S80
Key Words and Phrases: q-Bernstein polynomials, $p$-Genocchi numbers and polynomials, $p$-adic q-integral.
- ⋅ 15:20 − 16:00 A study on fermionic $p$-adic integral representations of several $q$-Bernstein-type polynomials (Tae Kyun Kim)
- 김태균(광운대)
Tae Kyun Kim, Kwangwoon University
Recently, Simsek-Acikgoz and Kim-Jang-Yi have studied the $q$-extension of Bernstein polynomials. In this present paper, we suggest $q$-extensions of Bernstein polynomials of degree $n$ which differ from the $q$-Bernstein polynomials of Simsek-Acikgoz and Kim-Jang-Yi. Using these $q$-Bernstein polynomials, we derive fermionic $p$-adic integral representations of several $q$-Bernstein polynomials and $q$-Euler numbers.
2010 Mathematics Subject Classification: 41A25, 41A35
Key Words and Phrases: $q$-Bernstein polynomials, $p$-adic invariant $q$-inetgral, $p$-adic inetgral representations
- Graph Theory
- ⋅ 09:00 − 09:30 Rank-width and well-quasi-ordering of skew-symmetric or symmetric matrices (Sang-il Oum)
- 엄상일(카이스트)
Sang-il Oum, KAIST
We prove that every infinite sequence
of skew-symmetric or symmetric matrices $M_1$, $M_2$, $\ldots$
over a fixed finite field
must have a pair $M_i$, $M_j$ $(i<j)$ such
that $M_i$ is isomorphic to a principal submatrix of
the Schur complement of a nonsingular principal submatrix in $M_j$,
if those matrices have bounded rank-width.
This generalizes three theorems on
well-quasi-ordering of graphs or matroids admitting good tree-like decompositions;
(1) Robertson and Seymour's theorem for
graphs of bounded tree-width,
(2) Geelen, Gerards, and Whittle's theorem
for matroids representable over a fixed finite field having bounded
branch-width,
and (3) Oum's theorem for graphs of bounded rank-width with respect
to pivot-minors.
2010 Mathematics Subject Classification: 05C83
Key Words and Phrases: well-quasi-order, delta-matroid, rank-width, branch-width, principal pivot transformation, Schur complement
- ⋅ 09:30 − 10:00 Geometric distance-regular graphs with smallest eigenvalue $-3$ (Sejeong Bang)
- 방세정(영남대)
Sejeong Bang, Yeungnam University
A non-complete distance-regular graph $\Gamma$ is called geometric if there exists a set $\mathcal{C}$ of Delsarte cliques such that each edge of $\Gamma$ lies in a unique clique in $\mathcal{C}$. In this talk, we determine the non-complete distance-regular graphs satisfying $ \max \{3, \frac{8}{3}(a_1+1) \}< k < 4a_1+10-6c_2.$ To prove this result, we first show by considering non-existence of $4$-claws that any non-complete distance-regular graph satisfying $ \max \{3, \frac{8}{3}(a_1+1) \} < k < 4a_1+10-6c_2 $ is a geometric distance-regular graph with smallest eigenvalue $-3$. Moreover, we classify the geometric distance-regular graphs with smallest eigenvalue $-3$. As an application, $7$ feasible intersection arrays are ruled out.
2010 Mathematics Subject Classification: 05E30
Key Words and Phrases: geometric, n-claw, smallest eigenvalue, distance-regular graphs, feasible arrays
- ⋅ 10:10 − 10:40 The H-colouring dichotomy through a projective property (Mark Siggers)
- 시거스(경북데)
Mark Siggers, Kyungpook University
The H-colouring dichotomy of Hell and Nesetril, proved in 1990, is one of the most quoted results in the field of graph homomorphisms. It says that H-coloring, the the problem of deciding if a given graph G admits an homomorphism to the fixed graph H, is NP-complete if H contains an odd cycle, and otherwise polynomial time solvable. In this talk we present a short new proof of this result, recently published, using a new projective property defined for homomorphisms of powers of a graph G onto a graph H.
2010 Mathematics Subject Classification: 05C15
Key Words and Phrases: H-coloring, dichotomy, homomorphism
- ⋅ 10:40 − 11:10 On second Hamilton circuits in cubic graphs (Tommy Rene Jensen)
- 젠슨, 토미 르네(경북대)
Tommy Rene Jensen, Kyungpook National University
A classical theorem of Cedric Smith guarantees the existence of a second Hamilton circuit other than a given one in any hamiltonian cubic graph. It is an open problem in complexity theory whether the corresponding search problem is polynomially solvable. We observe that a search algorithm, implicit in Bill Tutte's nonconstructive proof of Smith's theorem, has exponential running time. We also mention two possible candidates for search algorithms with polynomial complexity.
2010 Mathematics Subject Classification: 05C85
Key Words and Phrases: Hamilton cycle, Smith theorem, complexity
- ⋅ 11:10 − 11:40 Recent progress of distance-regular graphs (Jack Koolen)
- 잭쿨렌(포항공대)
Jack Koolen, POSTECH
I will talk about recent progress of distance-regular graphs.
2010 Mathematics Subject Classification: 05E30
Key Words and Phrases: distance-regular graph
- 예비수학자 연구발표 I
- ⋅ 09:00 − 09:30 Restricted permutations with a certain pattern (Young Ja Park)
- 박영자(연세대)
Young Ja Park, Yonsei University
We study $132$ avoiding permutations that also avoids $(2r+1)(2r+2)\cdots 12$ but contains $(2r-1)(2r)\cdots 12$ pattern. Permutations which avoid $132$ and $3412$ simultaneously is studied too. We also show that the number of these sets are related to the Narayana number.
2010 Mathematics Subject Classification: 05A05
Key Words and Phrases: restricted permutation, narayana number
- ⋅ 09:30 − 10:00 Generalized dyck paths (Ji Sun Huh)
- 허지선(연세대)
Ji Sun Huh, Yonsei University
We study generalized Dyck paths with the different sizes of horizontal and vertical steps.
2010 Mathematics Subject Classification: 05C38
Key Words and Phrases: dyck path
- ⋅ 10:10 − 10:40 Any $3$-equivalenced association scheme is Frobenius (Jeong-Rye, Park, Mitsugu Hirasaka and Kyoung Tark Kim)
- 박정례*(부산대), 히라사카 미쯔구(부산대), 김경탁(부산대)
Jeong-Rye, Park*, Pusan National University, Mitsugu Hirasaka, Pusan National University, Kyoung Tark Kim, Pusan National University
Let $(\Omega,S)$ be an association scheme where $\Omega$ is a finite set and $S$ is a partition of $\Omega\times \Omega$. For a positive integer $k$ we say that $(\Omega,S)$ is $k$-equivalenced if each nonidentity element of $S$ has valency $k$. In this talk we focus on $3$-equivalenced association schemes and briefly show that they are Frobenius.
2010 Mathematics Subject Classification: 05E30
Key Words and Phrases: association scheme, Frobenius group, Frobenius scheme, 3-equivalenced
- ⋅ 10:40 − 11:10 Graphical union of posets and $P$-partition (Seoung Ji Hong, Seung Kyung Park)
- 홍성지*(연세대), 박승경(연세대)
Seoung Ji Hong*, Yonsei University, Seung Kyung Park, Yonsei University
Let $P$ and $Q$ be indexed posets whose index set is the natural numbers. We define the union of two posets $P\cup Q$ as the graphical union of $P$ and $Q$; i.e., the union of vertices and edges between $P$ and $Q$. It is known that if $P$ and $Q$ are disjoint the order polynomial of $P\cup Q$ is the Hadamard product of the two order polynomials of $P$ and $Q$. We extend to the multi-variable order polynomials and derive a similar result.
2010 Mathematics Subject Classification: 06A11
Key Words and Phrases: poset
- ⋅ 11:10 − 11:40 Regularized imaginary quadratic Doi-Naganuma lifting (Seok Ho Jin, Youngju Choie and Su Bong Lim)
- 진석호*(포항공대), 최영주(포항공대), 임수봉(포항공대)
Seok Ho Jin*, POSTECH, Youngju Choie, POSTECH, Su Bong Lim, POSTECH
We will study on the regularization of the Friedberg's imaginary quadratic Doi-Naganuma lifting.
2010 Mathematics Subject Classification: 11F37
Key Words and Phrases: theta lifting, regularization
- ⋅ 13:10 − 13:40 Infinitesimal rigidities of semi-Riemannian hypersurfaces in Lorentz spaces (Hye Lim Han, Ho Bum Kim and An Sook Shin)
- 한혜림*(연세대), 김호범(연세대), 신안숙(연세대)
Hye Lim Han*, Yonsei University, Ho Bum Kim, Yonsei University, An Sook Shin, Yonsei University
The main interests of this paper are concerning a rigidities of semi Riemannian surfaces in a Lorentz manifold, especially in the Minkowski 3-space. In particular, we show that planes in Lorentz manifolds is not infinitesimal rigid, and that the pseudoshperes are infinitesimal rigid. This paper is organized as follows. We begin by introducing some preliminaries about Lorentz geometry. And then, we present concepts of rigidities in various senses. At the end, we show the pseudoshperes are infinitesimal rigid with some formulas about conformal vector fields.
2010 Mathematics Subject Classification: 53Axx
Key Words and Phrases: rigidity, infinitesimal rigidity, semi Riemannian geometry, Lorentz manifold, umbilic hypersurface
- ⋅ 13:40 − 14:10 Perfect ideals of grade 3 defined by skew-symmetrizable matrices (Yong Sung Cho, Oh-Jin Kang and Hyoung June Ko)
- 조용성*(연세대), 강오진(인천대), 고형준(연세대)
Yong Sung Cho*, Yonsei University, Oh-Jin Kang, University of Incheon, Hyoung June Ko, Yonsei University
Brown gave a structure theorem for a class of perfect ideals of grade 3 with type 2 and $\lambda>0.$ We present a skew-symmetrizable matrix to describe a structure theorem for complete intersections of grade 4 in a noetherian local ring. We construct a class of perfect ideals $I$ of grade 3 with type 2 defined by a certain skew-symmetrizable matrix. We also construct a class of homogeneous perfect ideals $\tilde{I}$ of grade 3 with type 2 from these ideals and present the Hilbert function of the standard $k$-algebras $R/\tilde{I},$ where $R$ is the polynomial ring $R=k[x_0, x_1, \ldots, x_m]$ over a field $k$ with indeterminates $x_i$ and $\deg x_i=1.$
2010 Mathematics Subject Classification: 13C05
Key Words and Phrases: almost complete intersection of grade 3; perfect ideal of grade 3, minimal free resolution; linkage
- ⋅ 14:10 − 14:40 $\alpha$-completely positive maps on pro-$C^*$-algebras and their representations (Young Yi Kim, Jaeseong Heo and Un Cig Ji)
- 김영이*(충북대), 허재성(한양대), 지운식(충북대)
Young Yi Kim*, Chungbuk National University, Jaeseong Heo, Hanyang University, Un Cig Ji, Chungbuk National University
We study covariant KSGNS representation theorems for $\alpha$-completely positive maps on a pro-$C^*$-algebra and covariant representations of crossed products. Moreover, the Radon-Nikod\'{y}m type theorem of $\alpha$-completely positive maps on a pro-$C^*$-algebra is studied.
2010 Mathematics Subject Classification: Primary 60G20, 60G10, Secondary 60L25, 60G05.
Key Words and Phrases: pro-$C^*$-algebra, $\alpha$-completely positive map, $J$-representation, Krein module
- ⋅ 14:50 − 15:20 An upwind mixed element method for stress-velocity formulation for Oseen equations (Boyoon Seo, Eun-Jae Park)
- 서보윤*(연세대), 박은재(연세대)
Boyoon Seo*, Yonsei University, Eun-Jae Park, Yonsei University
We consider the following Oseen equations: \begin{eqnarray*} \begin{cases} ~\alpha\mathbf{u}-\nu\triangle\mathbf{u}+\mathbf{b}\cdot\nabla\mathbf{u}+\nabla p=\mathbf{f}~~\text{in} ~~\Omega\\ ~ \text{div}~\mathbf{u}=0~~\text{in}~~\Omega\\ ~\mathbf{u}=\mathbf{0}~~\text{on} ~~\partial\Omega,\end{cases} \end{eqnarray*} where $\mathbf{u}$ and $p$ are the velocity vector and pressure, respectively. A pseudostress-velocity formulation is considered for approximation in this talk. An upwind scheme is constructed for pseudostress-velocity formulation and convergence analysis is given. We prove first order convergence for quantities of interest using the lowest Raviart-Thomas space on rectangular mesh. Finally we give some results of numerical experiments for various cases to show the efficiency of this method.
2010 Mathematics Subject Classification: 65Gxx
Key Words and Phrases: upwind scheme, pseudostress-velocity formulation
- ⋅ 15:20 − 15:50 Nonconforming primal mixed finite element methods for elliptic equations (Sungmin Cho, Eun-Jae Park)
- 조성민*(연세대), 박은재(연세대)
Sungmin Cho*, Yonsei University, Eun-Jae Park, Yonsei University
In this talk, we propose and analyze a nonconforming primal mixed finite element for Stokes' and Oseen's equation. In the case of Stokes' equations we prove the optimal convergence order and in the case of Oseen's equations we use the nonconforming streamline diffusion method to resolve the convection dominated problems. To show its accuracy and efficiency, several numerical experiments are presented.
2010 Mathematics Subject Classification: 65Gxx
Key Words and Phrases: nonconforming primal mixed FEM, streamline, Stokes' equation, Oseen's equation
- 예비수학자 연구발표 II
- ⋅ 09:00 − 09:30 Properties of hyperholomorphic functions in Clifford analysis (Su Jin Lim)
- 임수진(부산대)
Su Jin Lim, Pusan National University
In this paper, we give definitions of hyperholomorphc funtions of quaternionic functions of quaternionic variables $z=x_{1}+ix_{2}+jx_{3}+kx_{4}$ by a pair $z=z_{1}+z_{2}j$ of two complex variables $z_{1}=x_{1}+ix_{2}$ and $z_{2}=x_{3}+ix_{4}$, and investigate properties of hyperholomorphic functions on quaternion analysis. Also, we define hyperholomorphic functions of octonionic functions of octonionic variables $z=\sum_{j=0}^{7}e_jx_j$ by $z=z_{1}+z_{2}e_{2}+z_{3}e_{4}+z_{4}e_{6}$ of four complex variables $z_{1}=x_{0}+e_{1}x_{1}$, $z_{2}=x_{2}+e_{1}x_{3}$,$z_{3}=x_{4}+e_{1}x_{5}$ and $z_{4}=x_{6}+e_{1}x_{7}$, and find the corresponding Cauchy-Riemann equations in octonion analysis.
2010 Mathematics Subject Classification: 30G35,32A99,11E88
Key Words and Phrases: hyperholomorphic, quaternion, octonion, Clifford analysis
- ⋅ 09:30 − 10:00 Generic diffeomorphism with shadowing property on transitive sets (Ju Mi Oh)
- 오주미(충남대)
Ju Mi Oh, Chungnam National University
Let $\Lambda$ be a transitive set. $f$ has the $C^1-$stably shadowing on $\Lambda$ is hyperbolic basic set. And $C^1$-generically, If $f|_\Lambda$ have shadowing property and isolated, then it is hyperbolic.
2010 Mathematics Subject Classification: 37A25, 37D20, 37C50
Key Words and Phrases: chain component, shadowing property, isolated, hyperbolic, basic set
- ⋅ 10:10 − 10:40 Homoclinic classes with shadowing property (Jiwon Ahn)
- 안지원(충남대)
Jiwon Ahn, Chungnam National University
We show that for $C^1$ generic diffeomorphisms, an isolated homoclinic class is shadowable if and only if homoclinic class is hyperbolic basic set.
2010 Mathematics Subject Classification: 37C20, 37C05, 37C29, 37D05
Key Words and Phrases: shadowing property, basic set, expansive, hyperbolic, generically, homoclinic class
- ⋅ 10:40 − 11:10 Discrete fractional calculus (Bowon Kang, Sung Kyu Choi and Namjip Koo)
- 강보원*(충남대), 최성규(충남대), 구남집(충남대)
Bowon Kang*, Chungnam National University, Sung Kyu Choi, Chungnam National University, Namjip Koo, Chungnam National University
In this talk we introduce the discrete fractional calculus: the fractional sum and fractional difference operators. Then we discuss some results on discrete fractional comparison principle.
2010 Mathematics Subject Classification: 39A12,26D15
Key Words and Phrases: discrete fractional calculus, discrete fractional comparison principle
- ⋅ 11:10 − 11:40 Stability of $J^*$-derivations on $J^*$-algebras (Sang Hoon Lee, Ick Soon Chang and Hark Mahn Kim)
- 이상훈*(충남대), 장익순(목원대), 김학만(충남대)
Sang Hoon Lee*, Chungnam National Unversity, Ick Soon Chang, Mokwon University, Hark Mahn Kim, Chungnam National University
In this note, we improve to establish the generalized stability and the superstability of $J^*$-derivations in $J^*$-algebras for the generalized Jensen-type functional equation $$rf(\frac{x+y}{r})+rf(\frac{x-y}{r})=2f(x).$$ Moreover, we also prove the generalized stability of $J^*$-derivations by using the fixed point alternative.
2010 Mathematics Subject Classification: 39B82
Key Words and Phrases: Hyers-Ulam-Rassias stability, $J^*$-algebra, generalized Jensen-type functional equation
- ⋅ 13:10 − 13:40 L-fuzzy topological spaces and L-fuzzy quasi-proximity spaces (Eun-Seok Kim, Seung-Ho Ahn and Dae Heui Park)
- 김은석*(전남대), 안승호(전남대), 박대희(전남대)
Eun-Seok Kim*, Chonnam National University, Seung-Ho Ahn, Chonnam National University, Dae Heui Park, Chonnam National University
This paper studies the relationship between L-fuzzy proximities and L-fuzzy topologies by topological fuzzy remote neighborhood systems. We will prove that the category of L-fuzzy topological spaces can be embedded in the category of L-fuzzy quasi-proximity spaces as a coreflective full subcategory.
2010 Mathematics Subject Classification: 54A40, 03E72
Key Words and Phrases: L-fuzzy topology, L-fuzzy uniformity, L-fuzzy proximity, fuzzy remote neighborhood systems
- ⋅ 13:40 − 14:10 Default modeling with multi-scale stochastic intensity (Sun-Hwa Cho, Jeong-Hoon Kim and Yong-Ki Ma)
- 조선화*(연세대), 김정훈(연세대), 마용기(연세대)
Sun-Hwa Cho*, Yonsei University, Jeong-Hoon Kim, Yonsei University, Yong-Ki Ma, Yonsei University
In this paper, we consider the intensity-based defaultable bonds where the volatility of the intensity process is driven by multi-time factors varying on fast and slow time scales. Corrections from the multi-scale stochastic volatility of the intensity process are computed and how these corrections can affect the term structure of interest rate derivatives is shown. Our results indicate that any additional slow or fast scale stochastic volatility of the intensity tends to make the yields of the defaultable bonds lower than the case of just constant volatility. Interestingly enough, the mixture of the fast and slow scales, however, neutralizes the effect produced by single (either fast or slow) scale volatility.
2010 Mathematics Subject Classification: 91G40
Key Words and Phrases: defaultable bond, stochastic intensity, stochastic volatility, multi-time scale, asymptotic analysis
- ⋅ 14:10 − 14:40 Pricing a perpetual American CatEPut option when the stock price is correlated with the catastrophe loss (Jerim Kim, Bara Kim and Hwa-Sung Kim)
- 김제림*(고려대), 김바라(고려대), 김화성(경희대)
Jerim Kim*, Korea University, Bara Kim, Korea University, Hwa-Sung Kim, Kyung Hee University
As the number of catastrophic events increases, insurance companies are required to hedge catastrophic loss through the catastrophe derivatives. A catastrophe equity put option (CatEPut option) is a catastrophe derivative that provides its holder with the right to sell the underlying stock at a specified price when a catastrophe loss exceeds a prespecified level. In practice CatEPut options are American-style and have a long maturity. This paper provides a pricing formula for a perpetual American CatEPut option that can approximate a relatively long-maturity American CatEPut option. In addition, it is important to consider the structure of the stock price reduction due to a catastrophe event so that the pricing model could fit realistic phenomena. To reflect a flexible correlation structure, this paper extends the existing one in that the log stock price can be moderately correlated with the catastrophic loss amounts using the Downton's bivariate exponential distribution.
2010 Mathematics Subject Classification: 91Gxx
Key Words and Phrases: catastrophe equity put, option pricing, perpetual put, bivatiate exponential distribution, correlated jump model
- ⋅ 14:50 − 15:20 Finite automata using proteins and position encoding (Sung Bong Kang)
- 강성봉(연세대)
Sung Bong Kang, Yonsei University
In this paper, we study finite automata by encoding orders in protein synthesis to solve a problem which determines whether an input-data has the specific sub-string. We propose basic algorithms of an implementation of finite state automata.
2010 Mathematics Subject Classification: 92-04
Key Words and Phrases: finite automaton, protein
- ⋅ 15:20 − 15:50 An autonomous computation by a protein synthesis mechanism (Su Hyung An, Seung Kyung Park)
- 안수형*(연세대), 박승경(연세대)
Su Hyung An*, Yonsei University, Seung Kyung Park, Yonsei University
In this paper, we present a construction of finite state automata to recognize the regular language of all strings that contain a specific substring and conduct autonomous computation by a protein-synthesis mechanism in biology.
2010 Mathematics Subject Classification: 92-04
Key Words and Phrases: finite automata, protein, RNA interference
- 예비수학자 연구발표 III
- ⋅ 09:00 - 09:30 Stability of derivations on Hilbert $C^{*}-$modules (Min Young Kim, Hark-Mahn Kim)
- 김민영*(충남대), 김학만(충남대)
Min Young Kim*, Chungnam University, Hark-Mahn Kim, Chungnam University
Consider the functional equation $\mathbf{\varepsilon}_{1}(f) = \mathbf{\varepsilon}_2(f) \ (\mathbf{\varepsilon})$ in a certain framework. We say a function $f_0$ is an approximate solution of $(\mathbf{\varepsilon})$ if $\mathbf{\varepsilon}_1(f_{0})$ and $\mathbf{\varepsilon}_2(f_{0})$ are close in some sense. The stability problem is whether or not there is an exact solution of $(\mathbf{\varepsilon})$ near $f_0$. In this paper, the stability of derivations on Hilbert $C^{*}$-modules is investigated in the spirit of Hyers-Ulam-Rassias.
2010 Mathematics Subject Classification: 39B82, 46L08, 46L57
Key Words and Phrases: stability, Hilbert $C^{*}$-module
- ⋅ 09:30 - 10:00 Weak law of large numbers for weighted sums in noncommutative probability space (Byoung Jin Choi, Un Cig Ji)
- 최병진*(충북대), 지운식(충북대)
Byoung Jin Choi*, Chungbuk National University, Un Cig Ji, Chungbuk National University
We study the weak law of large numbers for weighted sums of pairwise independent random variables in a noncommutative probability space of which the weights are in a von Neumann algebra. As applications, we study the weak law of large numbers for weighted sums of random variables in noncommutative Ciach space and the weak law of large numbers for weighted additive convolution sums, and, finally, we study the necessary and sufficient conditions for the weak law of large numbers for weighted free additive convolution sum.
2010 Mathematics Subject Classification: 46L53
Key Words and Phrases: noncommutative probability space, law of large numbers, weighted sum, convolution
- ⋅ 10:10 - 10:40 Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model (Sungeun Jung, Young-Pil Choi, Seung-Yeal Ha and Yongduck Kim)
- 정성은*(서울대), 최영필(서울대), 하승열(서울대), 김용덕(서울대)
Sungeun Jung*, Seoul national university, Young-Pil Choi, Seoul national university, Seung-Yeal Ha, Seoul national university, Yongduck Kim, Seoul national university
We discuss the asymptotic formation and nonlinear orbital stability of phase-locked states arising from the ensemble of non-identical Kuramoto oscillators. We provide an explicit lower bound for a coupling strength on the formation of phase-locked states, which only depends on the diameters of natural frequencies and initial phase configurations. We also show that the Kuramoto model is $\ell_1$-contractive in the sense that the $\ell_1$-distance along two smooth Kuramoto flows is less than or equal to that of initial configurations. In particular, when two initial configurations have the same averaged phases, the $\ell_1$-distance between them decays to zero exponentially fast. For the configurations with different phase averages, we use the method of average adjustment and translation-invariance of the Kuramoto model to show that one solution converges to the translation of the other solution exponentially fast. This establishes the orbital stability of the phase-locked states. In the formation process of phase-locked states, we also estimate the number of collisions between oscillators, and lower-upper bounds of transversal phase differences.
2010 Mathematics Subject Classification: 34D06
Key Words and Phrases: The kuramoto model, phase, frequency, complete synchronization, phase-locked state, $\ell_1$-contraction, orbital stability
- ⋅ 10:40 - 11:10 Asymptotic behavior of the nonlinear Vlasov equation with a self-consistent force (Sun-Ho Choi, Seung-Yeal Ha)
- 최선호*(서울대), 하승열(서울대)
Sun-Ho Choi*, Seoul National University, Seung-Yeal Ha, Seoul National University
We present a threshold phenomenon on the $L^1$-asymptotic completeness for the nonlinear Vlasov equation with a self-consistent force. For a long-ranged self-consistent force, we show that the nonlinear Vlasov equation has no $L^1$-asymptotic completeness, which means that the nonlinear Vlasov flow can not be approximated by the corresponding free flow in $L^1$-norm time-asymptotically. In contrast, for a short-ranged force, the nonlinear Vlasov flow can be approximated by the free flow time-asymptotically. Our result is the kinetic counterpart of the Glassey work on the nonexistence of $L^2$-asymptotic completeness to the Schr$\ddot{o}$dinger-Hartree equation in the quantum mechanics.
2010 Mathematics Subject Classification: 35B40, 35Q83
Key Words and Phrases: Asymptotic completeness, Vlasov equation, scattering
- 학부수학교육의 도구와 기법
- ⋅ 10:55 - 11:10 학부수학교재번역현황 (Seong-Deog Yang, Young Wook Kim and 이동선)
- 양성덕*(고려대), 김영욱(고려대), 이동선(고려대)
Seong-Deog Yang*, Korea University, Young Wook Kim, Korea University, 이동선, Korea University
이 발표는 영어로 쓰여진 수학전공도서가 일본어로 번역이 되어 있는 현황과 우리말로 번역이 되어 있는 현황을 비교조사한 결과를 다룬다.
2010 Mathematics Subject Classification:
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- ⋅ 11:10 - 11:40 우리말 수학전공도서 개발의 어려움 및 개선책에 관한 토론 (Seoung Dal Jung, Hyang Sook Kim, 오판근 and Seong-Deog Yang)
- 정승달*(제주대), 김향숙*(인제대), 오판근*(교우사), 양성덕*(고려대)
Seoung Dal Jung*, Jeju National University, Hyang Sook Kim, Inje University, 오판근, 교우사, Seong-Deog Yang, Korea University
이 토론에서는 현 상황에서 국문 수학전공도서를 개발하는데 어떠한 어려움이 있는지, 그런 어려움을 극복하기 위하여 어떤 공동노력이 필요한지 토론한다.
2010 Mathematics Subject Classification:
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- ⋅ 13:10 - 13:40 Development of smartphone apps for linear algebra with Sage (Kyung-Won Kim, Sang-Gu Lee)
- 김경원*(성균관대), 이상구*(성균관대)
Kyung-Won Kim*, Sungkyunkwan University, Sang-Gu Lee, Sungkyunkwan University
본 연구팀은 2009년 우리말 Sage 서버를 구축하고 모바일 환경에 맞는 기초 모듈을 개발하여, 새로운 IT 교육환경에 적합한 모바일 수학의 가능성을 제시하였다. 이를 이용하여 본 발표에서는 누구나 사용하기 쉬운 스마트폰을 이용한 선형대수학의 모바일 학습모델을 제시한다.
2010 Mathematics Subject Classification: 97Uxx
Key Words and Phrases: 선형대수학, linear algebra, 모바일, mobile, Sage
- ⋅ 13:40 - 14:10 Status of NIMSOFT as Advanced Toolkit for Scientific Computing and Collaborations (Taeyoung Ha, John J. Oh, Kiwan Jeon, SangHoon Oh and Myoungnyoun Kim)
- 하태영*(국가수리과학연구소), 오정근(국가수리과학연구소), 전기완(국가수리과학연구소), 오상훈(국가수리과학연구소), 김명년(국가수리과학연구소)
Taeyoung Ha*, National Institute for Mathematical Sciences, John J. Oh, National Institute for Mathematical Sciences, Kiwan Jeon, National Institute for Mathematical Sciences, SangHoon Oh, National Institute for Mathematical Sciences, Myoungnyoun Kim, National Institute for Mathematical Sciences
We present the status of NIMSOFT Project at NIMS so far. NIMSOFT is the abbreviation for ``Numerical, Integrated, and Multi-language Supporting Operational Functions and Toolkits". This is a sort of ``Integrated Development Environment (IDE)" designed for the purpose of scientific collaboration and high performance computing in various science fields. In this talk, we report the basic concept, architectures, and current status of development of NIMSOFT.
2010 Mathematics Subject Classification:
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- ⋅ 14:10 - 14:40 The advanced features and solutions in Mathematica (Jiwon Hwang)
- 황지원(다한테크)
Jiwon Hwang, Dahan Tech
Almost any workflow involves computing results, and that's what Mathematica does-from building a hedge fund trading website or publishing interactive engineering textbooks to developing embedded image recognition algorithms or teaching calculus. Mathematica is renowned as the world's ultimate application for computations. But it's much more. In this talk, I will show the new features in Mathematica 8, the latest version, with some examples. It introduced free-form language input form, integrated wavelet analysis, enhanced statistic capability and C code generation.
2010 Mathematics Subject Classification:
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- ⋅ 14:50 - 15:25 MathJax, 웹 수식 표현의 종결자 (남수진)
- 남수진((주)카카오)
남수진, (주)카카오
MathJax는 자바스크립트, CSS와 웹폰트로 구성된 웹 상에서의 수식 표현 도구로써 이를 사용하면 텍 또는 MathML 형식으로 간편하게 인터넷 웹 페이지에서 수식을 표현 할 수 있다. 최근에 유명한 여러 수학 관련 웹사이트에서 적극적으로 MathJax를 도입하기 시작했고, MathJax의 개발도 활발하여 최근의 버전은 1.1이다. 본 발표에서는 MathJax의 이용 방법 중에서 MathML보다는 수학자들에게 더욱 친숙한 텍에 초점을 맞추어 설명한다. 먼저 텍에서의 수식 조판 원리를 간단히 살펴보고, 기존의 웹에서의 수식 표현 방식과 비교하여 텍 방식을 이용한 MathJax의 수식 표현 방법이 얼마나 손쉽고 편리한 지를 설명한다. 그리고 기본적인 MathJax의 원리와 설치/이용 방법을 알아보고, MathJax의 응용으로 자신이 운영하는 홈페이지 또는 블로그와 인터넷 상의 이메일과 채팅에서 MathJax를 활용하는 방법을 소개하고, MathJax에서의 자주 사용하는 텍 매크로들을 정의하고 이용하는 방법을 알아본다. 마지막으로 MathJax를 이용한 간단한 자바스크립트 프로그램으로 MathJax의 우수성을 눈으로 직접 확인하고, MathJax를 잘 활용한 사이트들을 소개한다.
2010 Mathematics Subject Classification:
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- ⋅ 15:25 - 16:00 움직이는 수학 소프트웨어 GeoGebra의 소개와 활용 사례 (최경식)
- 최경식(서울대)
최경식, Seoul National University
본 강연은 움직이는(dynamic) 수학 소프트웨어인 GeoGebra에 대한 소개와 이를 활용한 다양한 사례를 제시하는 내용을 포함한다. GeoGebra는 2001년 오스트리아의 Markus Hohenwarter에 의하여 개발되었으며, 현재는 190여개국 이상의 수학교육 연구자, 교사, 학생에 의해서 사용되고 있는 ‘움직이는(dynamic) 수학 소프트웨어’이다. GeoGebra라는 소프트웨어의 이름은, 기하(Geometry)와 대수(Algebra)라는 이름의 합성어이며, 이에 걸맞게 DGS(Dynamic Geometry Software), CAS(Computer Algebra System), DHS(Data Handling Software)가 하나의 인터페이스 안에 결합되어 있다. GeoGebra의 철학은 ‘모두를 위한 움직이는 수학(Dynamic Mathematics for Everyone)’으로, 이러한 철학에 따라 교육을 위한 비영리 목적에 대하여는 무료로 제공되고 있으며, 수학에서 사용되는 기호 체계 외에 불필요한 사용자 인터페이스를 사용하지 않음으로, 수학 기호체계를 아는 사람이라면 누구나 쉽게 사용할 수 있다. 본 강연에서는 이러한 GeoGebra의 과거와 현재의 모습, 우리나라에서의 도입 과정 및 활용사례를 보여주고자 한다. 또한 GeoGebra의 발전 방향 및 이에 대한 프로토타입을 제시하여, 앞으로의 GeoGebra의 가능성을 모색한다.
2010 Mathematics Subject Classification:
Key Words and Phrases:
- Algebra
- ⋅ 09:00 − 09:30 Maximum gap in (inverse) cyclotomic polynomial (Eunjeong Lee, Hoon Hong, Hyang-Sook Lee and Cheol-Min Park)
- 이은정*(이화여대 수리과학연구소), 홍 훈(노스캐롤라이나 주립대), 이향숙(이화여대), 박철민(국가수리과학연구소)
Eunjeong Lee*, IMS, Ewha Womans University, Hoon Hong, Dept. of Math. NCSU, Hyang-Sook Lee, Dept. of Math. Ewha Womans University, Cheol-Min Park, National Institute for Mathematical Sciences
Let $g(f)$ denote the maximum of the differences (gaps) between two consecutive exponents occurring in a polynomial $f$. Let $\Phi_n$ denote the $n$-th cyclotomic polynomial and let $\Psi_n$ denote the $n$-th inverse cyclotomic polynomial. In this note, we study $g(\Phi_n)$ and $g(\Psi_n)$ where $n$ is a product of odd primes, say $p_1 < p_2 < p_3$, etc. It is trivial to determine $g(\Phi_{p_1})$, $g(\Psi_{p_1})$ and $g(\Psi_{p_1p_2})$. Hence the simplest non-trivial cases are $g(\Phi_{p_1p_2})$ and $g(\Psi_{p_1p_2p_3})$. We provide an exact expression for $g(\Phi_{p_1p_2}).$ We also provide an exact expression for $g(\Psi_{p_1p_2p_3})$ under a mild condition. The condition is almost always satisfied (only finite exceptions for each $p_1$). We also provide a lower bound and an upper bound for $g(\Psi_{p_1p_2p_3})$.
2010 Mathematics Subject Classification: 11Cxx
Key Words and Phrases: cyclomic polynomial, inverse cyclotomic polynomial
- ⋅ 09:30 − 10:00 Transcendence of the Iwasawa power series (Hae-Sang Sun)
- 선해상(충북대)
Hae-Sang Sun, Chungbuk National University
The Iwasawa power series is basically the Kubota-Leopoldt $p$-adic L-function. In this talk, we are going to explain the proof of transcendence of the Iwasawa power series over a rational function field of positive characteristic. This is an improvement of our previous work on the proof of transcendence under Borel's conjecture.
2010 Mathematics Subject Classification: 11S
Key Words and Phrases: Kubota-Leopoldt $p$-adic L-function, Iwasawa power series, transcendence, Dirichlet L-function
- ⋅ 10:10 − 10:40 Some module-theoretic characterizations of Krull domains (Hwankoo Kim)
- 김환구(호서대)
Hwankoo Kim, Hoseo University
In this talk, I present my recent works concerning some module-theoretic characterizations of Krull domains. More precisely, it is shown that for a completely integrally closed domain $R$, $R$ is a Krull domain if and only if $E(K/R)$, the injective envelope of $K/R$, is an injective cogenerator in the quotient category $\mathsf{Mod}(R)/\mathscr{T}_{\tau}(R)$. It is also shown that the following statements for an infra-Krull domain $R$ are equivalent: (1) $R$ is a Krull domain; (2) for any essentially finite $w$-module $M$ over $R$, the torsion submodule $t(M)$ of $M$ is a direct summand of $M$; (3) for any essentially finite $w$-module $M$ over $R$, $t(M) \cap \mathfrak{p}M = \mathfrak{p}t(M)$, for all maximal $w$-ideal $\mathfrak{p}$ of $R$; (4) $R$ satisfies the $w$-radical formula; (5) the $R$-module $R \oplus R$ satisfies the $w$-radical formula.
2010 Mathematics Subject Classification: 13F05, 13A15
Key Words and Phrases: Krull domain, infra-Krull domain, $w$-radical formula, strong Mori domain, injective cogenerator
- ⋅ 10:40 − 11:10 Linear operators that preserve term ranks of matrices over semirings (Seok-Zun Song, Kyung-Tae Kang)
- 송석준*(제주대), 강경태(제주대)
Seok-Zun Song*, Jeju National University, Kyung-Tae Kang, Jeju National University
The term rank of a matrix $A$ over a semiring $S$ is the least number of lines (rows or columns) needed to include all the nonzero entries in $A$. In this paper, we study linear operators that preserve term ranks of matrices over semiring $S$. We obtain that a linear operator $T$ on matrix space over $S$ preserves term rank if and only if $T$ preserves term ranks $1$ and $\alpha(\ge 2)$ if and only if $T$ preserves two consecutive term ranks in a restricted condition. Other characterizations of term-rank preservers are also given.
2010 Mathematics Subject Classification: 15A86, 15A03, 15A04
Key Words and Phrases: semiring, term rank, linear operator
- ⋅ 11:10 − 11:30 Automorphism group of the ternary self-dual code of length 8 (Hyun Jin Kim, June Bok Lee)
- 김현진*(이화여대), 이준복(연세대)
Hyun Jin Kim*, Ewha University, June Bok Lee, Yonsei University
The abstract structure of the automorphism group of the ternary self-dual code of length 8 is presented by convenient generators.
2010 Mathematics Subject Classification: 20H30
Key Words and Phrases: automorphism group, ternary code
- ⋅ 14:20 − 14:40 Weakly Krull domains and related domains of the form $A+B[\Gamma^*]$ (Jung Wook Lim, Byung Gyun Kang)
- 임정욱*(포항공대), 강병균(포항공대)
Jung Wook Lim*, POSTECH, Byung Gyun Kang, POSTECH
Let $A \subseteq B$ be an extension of integral domains and $\Gamma$ be a numerical semigroup with $\Gamma \subsetneq \mathbb{N}_0$. In this talk, we study when the ring $A+B[\Gamma^*]$ is a weakly Krull domain, an almost weakly factorial domain or a generalized weakly factorial domain. We also show that $A+B[\Gamma^*]$ is never a weakly factorial domain.
2010 Mathematics Subject Classification: 13A15, 13F05, 13F20, 13G05
Key Words and Phrases: numerical semigroup, $A+B[\Gamma^*]$, weakly krull domain
- ⋅ 14:50 − 15:10 Representations by $x_1^2+2x_2^2+x_3^2+x_4^2+x_1x_3+x_1x_4+x_2x_4$ (Ick Sun Eum, Dong Hwa Shin and Dong Sung Yoon)
- 엄익선*(카이스트), 신동화(카이스트), 윤동성(카이스트)
Ick Sun Eum*, KAIST, Dong Hwa Shin, KAIST, Dong Sung Yoon, KAIST
Let $r_Q(n)$ be the representation number of a nonnegative integer $n$ by the quaternary quadratic form $Q=x_1^2+2x_2^2+x_3^2+x_4^2+x_1x_3+x_1x_4+x_2x_4$. We prove the identity $r_Q(p^2n)=r_Q(p^2)r_Q(n)/r_Q(1)$ for any prime $p$ different from $13$ and any positive integer $n$ prime to $p$. And, we explicitly determine $r_Q(n^2)$ for any integer $n$, too.
2010 Mathematics Subject Classification: 11E25, 11F11, 11F25, 11M36
Key Words and Phrases: Eisenstein series, Hecke operators, modular forms, representations by quadratic forms
- ⋅ 15:10 − 15:30 Generation of class fields by Siegel-Ramachandra invariants (Dong Hwa Shin, Ja Kyung Koo)
- 신동화*(카이스트), 구자경(카이스트)
Dong Hwa Shin*, KAIST, Ja Kyung Koo, KAIST
We show that the Siegel-Ramachandra invariants could be primitive generators of the ray class fields over imaginary quadratic fields. By using Shimura's reciprocity law we give a modern explanation of the solution of class number one problem given by Heegner and Stark.
2010 Mathematics Subject Classification: 11G16, 11F03, 11G15, 11R29, 11R37
Key Words and Phrases: class field theory, complex multiplication, class numbers, elliptic and modular units, modular and automorphic
- ⋅ 15:30 − 15:50 The Riemann zeta function and the related polynomials with only zeros on the vertical line (Hana Kim, Gi-Sang Cheon)
- 김하나*(성균관대), 천기상(성균관대)
Hana Kim*, Sungkyunkwan University, Gi-Sang Cheon, Sungkyunkwan University
In this talk, we introduce a family of polynomials $p_n^{(m,\alpha,\beta)}(s)$ that satisfy some properties that the Riemann zeta function $\zeta(s)$ satisfies. This concept leads to a generalization of the Riemann's integral arising in the Riemann's second proof of the analytic continuation of $\zeta(s)$. One of main results shows that these polynomials satisfy the analog of Riemann hypothesis: all their zeros lie on the line of symmetry given by the functional equation. In addition, an interlacing property for their zeros, an explicit formula, a three term recurrence relation and a combinatorial interpretation for the polynomials are observed.
2010 Mathematics Subject Classification: 11M26, 05A15
Key Words and Phrases: Riemann zeta function, Riordan matrix, Sheffer sequence, Mellin transform
- Analysis I
- ⋅ 09:00 − 09:30 Critical point results for the semilinear biharmonic problem (Q-Heung Choi, Tacksun Jung)
- 최규흥*(인하대), 정택선(군산대)
Q-Heung Choi*, Inha University, Tacksun Jung, Kunsan National University
We get one theorem that there exists a unique solution of the fourth order semilinear elliptic Dirichlet boundary value problem when the number 0 and the coefficient of the semilinear part belongs to the open interval between two successive eigenvalues of the fourth order eigenvalue problem simultaneously. We prove this result by the contraction mapping principle method. We also get another theorem that there exists at least two solutions when there exist $n$ eigenvalues of the fourth order eigenvalue problem between the coefficient of the semilinear part and the number 0. We prove this result by the critical point theory and the variation linking method.
2010 Mathematics Subject Classification: 35J20, 35J25
Key Words and Phrases: fourth order elliptic boundary value problem, contraction mapping principle, critical point theory, variation linking method
- ⋅ 09:30 − 10:00 Leray-Schauder degree theory applied to the fourth order elliptic problem with variable coefficient semilinear term (Tacksun Jung, Q-Heung Choi)
- 정택선*(군산대), 최규흥(인하대)
Tacksun Jung*, Kunsan National University, Q-Heung Choi, Inha University
We obtain the multiple solutions for the fourth order elliptic problem with variable coefficient semilinear term. We show two theorems that there exist at least two solutions if the variable coefficient of the semilinear term crosses some number of the eigenvalues of the biharmonic eigenvalue problem or is less than all eigenvalues of the biharmonic eigenvalue problem. We obtain these multiplicity results by applying the Leray-Schauder degree theory.
2010 Mathematics Subject Classification: 35J20, 35J25, 35J20, 35Q72
Key Words and Phrases: fourth order elliptic equation, variable coefficient semilinear term, Dirichlet boundary condition, Leray-Schauder degree theory
- ⋅ 10:10 − 10:40 On local spectral properties of complex symmetric operators (Ji Eun Lee, Eungil Ko, Sungeun Jung and Mee-Jung Lee)
- 이지은*(이화여대), 고응일(이화여대), 정성은(이화여대), 이미정(이화여대)
Ji Eun Lee*, Ewha Womans University, Eungil Ko, Ewha Womans University, Sungeun Jung, Ewha Womans University, Mee-Jung Lee, Ewha Womans University
In this paper we study properties of complex symmetric operators. In particular, we prove that every complex symmetric operator having property ($\beta$) or ($\delta$) is decomposable. Moreover, we show that complex symmetric operator $T$ has Dunford's property ($C$) and it satisfies Weyl's theorem if and only if its adjoint does.
2010 Mathematics Subject Classification: 47A05, 47B20
Key Words and Phrases: complex symmetric operator, Dunford's property (C), property (beta), decomposable, Weyl's theorem
- ⋅ 10:40 − 11:10 Feynman's operational calculi for noncommuting operators (Byoung Soo Kim)
- 김병수(서울과학기술대)
Byoung Soo Kim, Seoul National University of Science and Technology
Feynman's 1951 paper on the operational calculus for noncommuting operators arouse out of his ingenious work on quantum electrodynamics and was inspired in part by his earlier work on the Feynman path integral. Indeed, Feynman thought of his operational calculus as a kind of generalized path integral. Much surprisingly varied work on the subject has been done since by mathematicians and physicists. Recently Jefferies and Johnson developed mathematical rigorous approach to Feynman's operational calculi. In this talk we give definitions and properties of Feynman's operational calculi initiated by Jefferies and Johnson. In particular, extraction of a linear factor and measure permutation formula for Feynman's operational calculi will be given.
2010 Mathematics Subject Classification: 47A60
Key Words and Phrases: Feynman's operational calculus, disentangling, extraction of a linear factor
- ⋅ 11:10 − 11:30 Properties of class $A$ operators (Sungeun Jung, Eungil Ko)
- 정성은*(이화여대), 고응일(이화여대)
Sungeun Jung*, Ewha Womans University, Eungil Ko, Ewha Womans University
In this paper we provide several properties of class $A$ operators. In particular, we show that the set of all class $A$ operators is norm closed and give conditions for class $A$ operators to have a nontrivial hyperinvariant subspace. We also study weakly supercyclic class $A$ operators.
2010 Mathematics Subject Classification: 47B20
Key Words and Phrases: class A, hyperinvariant subspace, weakly supercyclic
- ⋅ 14:50 − 15:20 Ordered orbits of the $\beta$-transformations, and a devil's slope (DoYong Kwon)
- 권도용(전남대)
DoYong Kwon, Chonnam National University
For $\beta>1$, let $T_\beta:[0,1]\rightarrow [0,1)$ be the $\beta$-transformation. We consider an invariant $T_\beta$-orbit contained in a closed interval with diameter $1/\beta$. Then such $T_\beta$-orbit has a unique rotation number. On the other hand, let $\beta>1$ be fixed. Then for each $\alpha>0$, there is at most one $T_\beta$-orbit such that it is contained in a closed interval with diameter $1/\beta$, and that its rotation number is equal to $\alpha$. We define a function $\Xi(\alpha,\beta)$ by the supremum of the $T_\beta$-orbit with rotation number $\alpha$, i.e., the maximum value in the $T_\beta$-orbit closure. This paper demonstrates that the function $\Xi$ enjoys devil's staircase-like properties. We also study the value of $\Xi$ in an arithmetical aspect.
2010 Mathematics Subject Classification: 26A30, 26A27, 11J82, 37B10, 68R15
Key Words and Phrases: $\beta$-shift, devil's staircase, Sturmian word, Christoffel word, irrationality measure, Liouville number
- Analysis II
- ⋅ 09:00 − 09:30 A study on the fuzzy stability of quadratic functional equations (Jung Rye Lee, Choonkil Park and Dong Yun Shin)
- 이정례*(대진대), 박춘길(한양대), 신동윤(서울시립대)
Jung Rye Lee*, Daejin University, Choonkil Park, Hanyang University, Dong Yun Shin, University of Seoul
Using the fixed point method, we prove the Hyers-Ulam stability of the following quadratic functional equations
\begin{eqnarray*}
&& cf\left(\sum_{i=1}^{n}x_{i}\right)+\sum_{j=2}^{n}
f\left(\sum_{i=1}^{n}x_{i} - (n+c-1)x_{j}\right) \\ &&
\qquad \quad =(n+c-
1)\left(f(x_{1})+c\sum_{i=2}^{n}f(x_{i})+\sum_{i<j, j=3}^{n}
\left(\sum_{i=2}^{n-1}f(x_{i} - x_{j})\right)\right), \\
&& f\left(\sum_{i=1}^n d_i x_i\right)+ \sum_{1 \le i < j \le n}
d_i d_j f(x_i-x_j) = \left(\sum_{i=1}^n d_i\right) \left( \sum_{i=1}^n d_i
f(x_i)\right)
\end{eqnarray*}
in fuzzy Banach spaces.
2010 Mathematics Subject Classification: 39B52
Key Words and Phrases: fuzzy Banach space, fixed poin, quadratic functional equation, Hyers-Ulam stability
- ⋅ 09:30 − 10:00 Stability of the Pexiderized Lobacevski equation (Gwang Hui Kim)
- 김광휘(강남대)
Gwang Hui Kim, Kangnam University
The aim of this paper is to investigate the solution and the superstability of the Pexiderized Lobacevski equation \begin{equation*} f\big(\frac{x+y}{2}\big)^2 = g(x)h(y), \end{equation*} where $f, g, h: G^{2} \to \mathbb{C}$ are unknown functions on an Abelian semigroup $(G,+)$. The obtained result is a generalization of P. G{\v a}vruta's result in 1994 and G. H. Kim's result in 2010, respectively.
2010 Mathematics Subject Classification: 39B82
Key Words and Phrases: Hyers-Ulam stability, superstability, Lobacevski equation, d'Alembert functional equation,, sine functional equation
- ⋅ 10:10 − 10:40 Stability of functional equations on restricted domains in group and their asymptotic behaviors (Jaeyoung Chung)
- 정재영(군산대)
Jaeyoung Chung, Kunsan National University
We consider the Hyers-Ulam stability problems for the Pexider equation, Cauchy equation, Jensen equation in general restricted domains in a group. The main purpose of this paper is to find the restricted domains such that the functional inequality satisfied in which domain extends to the inequality for whole domain and that the Hyers-Ulam stability theorem holds for the inequalities as it does when the inequality holds globally. We also consider a distributional version of the Hyers-Ulam stability of the Pexider equation in restricted domains and its asymptotic behaviors.
2010 Mathematics Subject Classification: 39B52
Key Words and Phrases: Hyers-Ulam stability,Pexider equation, Cauchy equation, asymptotic behavior
- ⋅ 10:40 − 11:10 Solutions of nonlinear discrete boundary value problems and its applications (Jea Hyun Park, Soon Yeoung Chung)
- 박재현*(이화여대 수리과학연구소), 정순영(서강대)
Jea Hyun Park*, Institute of Mathematical Science, Ewha Womans University, Soon Yeoung Chung, Songang University
In this talk, we discuss some conditions between $V$ and $f$ to exist solutions as the following equation:
\begin{equation*}\label{meq}
\left\{
\begin{array}{ll}
-\Delta_{p,\omega} u(x) = - V(x) |u(x)|^{q-2}u(x) + f(x, u(x)), & x \in S,\\
u(x) =0, & x \in \partial S,
\end{array}
\right.
\end{equation*}
where
$1<p<\infty$, $1<q<\infty$,
$\overline S = (S \cup \partial S, E)$ is a simple, connected and undirected graph,
$V$ is a function on $S$ and $f : S \times \mathbb{R} \to \mathbb{R}$ is a continuous function.
Moreover, we also give a bound of solutions. Finally, we give its applications.
2010 Mathematics Subject Classification: 39A20
Key Words and Phrases: $p$-Lapalcian, discrete $p$-Laplacian, graph $p$-Laplacian, discrete boundary value problems, difference equations
- ⋅ 11:10 − 11:40 On an inverse source problems for heat equation by using reciprocity gap method (Yun-Sung Chung, Soon-Yeong Chung)
- 정윤성*(서강대), 정순영(서강대)
Yun-Sung Chung*, Sogang University, Soon-Yeong Chung, Sogang University
We consider the problem of identification of a heat source in a bounded domain $\Omega\in {\mathbb R}^d$ for the case $d=1,2$ and $3.$ With the assumption that the point sources become constant for the time variable, we prove that they can be determined by measurements of the heat flux on $\partial\Omega\times (0,T).$ It will be shown that the reciprocity gap method, which has been widely used for solving inverse problems for elliptic equations, is still useful to solve the problems for parabolic equations.
2010 Mathematics Subject Classification: 35R30
Key Words and Phrases: Inverse source problem, heat equation
- ⋅ 14:50 − 15:20 Analysis of nematic liquid crystals with singularities of degree 1/2 (Jinhae Park, P.Bauman and D. Phillips)
- 박진해*(충남대), 바우만 필립스(퍼듀대)
Jinhae Park*, Chungnam National University, P.Bauman and D. Phillips, Purdue University
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.
2010 Mathematics Subject Classification: 49K20
Key Words and Phrases: singularity, liquid crystal, nematic,landau-de Gennes
- Analysis III
- ⋅ 09:00 − 09:20 Multiple solutions of discrete $p$-Laplacian equations (Heesoo Lee, Soon-Yeong Chung)
- 이희수*(서강대), 정순영(서강대)
Heesoo Lee*, Sogang University, Soon-Yeong Chung, Sogang University
We deal with the following discrete $p$-Laplacian equation; $$ -\Delta_{p,\omega}u+V|u|^{p-2}u=f $$ in $S$ and $u=0$ on $\partial S$. Under some assumptions such as an asymptotic behavior of the source to the origin and to infinity, we will find a solution which is a global maximizer or minimizer, and a second solution which is gained by the mountain pass theorem. Also We introduce some properties of such solutions.
2010 Mathematics Subject Classification: 35J25
Key Words and Phrases: discere $p$-Laplacian, the mountain pass theorem, elliptic discrete boundary value problem
- ⋅ 09:20 − 09:40 Rotfel'd type inequalities for norms (Eun-young Lee)
- 이은영(경북대)
Eun-young Lee, Kyungpook University
we consider some norm inequalities related to the Rotfel'd Trace Inequality $$ {\mathrm{Tr\,}} f(|A+B|) \le {\mathrm{Tr\,}} f(|A|)+f(|B|) $$ for concave functions $f:[0,\infty)\to [0,\infty)$ and arbitrary $n$-by-$n$ matrices. For instance we show that for a large class of non-negative concave functions $f(t)$ and for all symmetric norms we have $$ \| f(|A+B|) \| \le \sqrt{2} \| f(|A|)+f(|B|) \| $$ and we conjecture that this holds for all non-negative concave functions.
2010 Mathematics Subject Classification: 15A60, 47A30, 47A60
Key Words and Phrases: symmetric norms, operator inequalities, concave
- ⋅ 09:40 − 10:00 Global solution branches for $p(x)$-Laplace equation (Yun-Ho Kim)
- 김연호(상명대)
Yun-Ho Kim, Sangmyung University
We consider the following $p(x)$-Laplacian problem \begin{equation*} -\Delta_{p(x)}u=\mu|u|^{p(x)-2}u +f(\lambda,x,u,\nabla u)\quad \textmd{in } \Omega \end{equation*} subject to Dirichlet boundary conditions when $\mu$ is not an eigenvalue of $p(x)$-Laplacian. The aim of this talk is to study the structure of the set of solutions for the above equation. We first state some basic results for the generalized Lebesgue spaces and the generalized Lebesgue-Sobolev spaces. Based on these results, we present some properties of the corresponding integral operators. We will prove our main result on global bifurcation for the above problem, by using a bifurcation result in abstract setting which gives the existence of a branch of solutions. Finally, an example will be given to illustrate our bifurcation result.
2010 Mathematics Subject Classification: 35B32, 47J05, 46E35
Key Words and Phrases: bifurcation; nonlinear equation; $p(x)$-Laplacian; generalized Lebesgue-Sobolev spaces
- ⋅ 10:10 − 10:30 Generalization of the Littlewood-Paley inequality for the fractional Laplacian (Ildoo Kim, Kyeong-Hun Kim)
- 김일두*(고려대), 김경훈(고려대)
Ildoo Kim*, Korea University, Kyeong-Hun Kim, Korea University
Let $T_t$ denote the semigroup corresponding to the Laplacian. The classical Littlewood-Paley inequality says for any $p \in (1,p)$ and $f=f(x)$, $$ \int_{R^d} [\int^{\infty}_0 |DT_t f|^2 dt]^{p/2}dx\leq N(p) \|f\|^p_p. $$ Krylov extended this result by proving the parabolic version. In this talk, we introduce a parabolic version of the Littlewood-Paley inequality for the fractional Laplacian $(-\Delta)^{\alpha/2}, \alpha\in (0,2)$. That is, if $T^{\alpha}_t$ is the semigroup corresponding the fraction Laplacian and $f=f(t,x)$, $$ \int_{R^d}\int^b_a [\int^t_a|\partial^{\alpha/2}_xT^{\alpha}_{t-s}f|^2ds]^{p/2}dtdx\leq N(p)\int_{R^d}\int^b_a|f|^p dtdx. $$
2010 Mathematics Subject Classification: 42B25
Key Words and Phrases: Littlewood-Paley inequality
- ⋅ 10:30 − 10:50 The gauge-invariant ideals of labelled graph $C^*$-algebras (Sun Ho Kim, Ja A Jeong and Gi Hyun Park)
- 김선호*(서울대), 정자아(서울대), 박기현(한신대)
Sun Ho Kim*, Seoul National University, Ja A Jeong, Seoul National University, Gi Hyun Park, Hanshin University
In [3], Bates and Pask introduced a class of $C^*$-algebras $C^*(E,\mathcal{L},\mathcal{B})$ associated to labelled spaces $(E,\mathcal{L},\mathcal{B})$, where $\mathcal{L}$ is a labelling map assigning a label to each of the edges of the graph $E$ and $\mathcal{B}$ is an accommodating set consisting of vertex subsets satisfying certain conditions. This class includes all graph $C^*$-algebras $C^*(E)$ which has drawn great interest from operator algebraists for the past 15 years. Under some mild conditions, the simplicity together with some topological properties of $C^*(E,\mathcal{L},\mathcal{B})$ was studied in [4] when $\mathcal{B}$ is the smallest accommodating set $\mathcal{E}^{0,-}$, which extends the well known facts for the graph $C^*$-algebras ([1], [2], [6], [7]). We provide a sufficient and necessary condition of $(E,\mathcal{L}, \bar{\mathcal{E}})$ for $C^*(E,\mathcal{L}, \bar{\mathcal{E}})$ to be simple, where $\bar{\mathcal{E}}$ is an accommodating set such that $C^*(E,\mathcal{L}, \bar{\mathcal{E}})\cong C^*(E,\mathcal{L},\mathcal{E}^{0,-})$. We also show that for a labelled space $(E,\mathcal{L},\mathcal{B})$ there exists a one to one correspondence between the set of all hereditary saturated subsets of $\mathcal{B}$ and the gauge-invariant ideals of $C^*(E,\mathcal{L},\mathcal{B})$. This extends the similar results for graph $C^*$-algebras. For this, we introduce a notion of quotient labelled space $(E,\mathcal{L}, [\mathcal{B}]_{R})$ and their $C^*$-algebra $C^*(E,\mathcal{L}, [\mathcal{B}]_{R})$. Besides, the gauge-invariant uniqueness theorem for $C^*(E,\mathcal{L}, [\mathcal{B}]_{R})$ is obtained.
\begin{thebibliography}{9}
\bibitem{1} T. Bates, J. H. Hong, I. Raeburn, and W.Szymanski, {\it The ideal structure of the $C^*$-algebras of infinite graphs}, Illinois J. Math. {\bf 46} (2002), 1159--1176.
\bibitem{2} T. Bates, D. Pask, I. Raeburn, and W.Szymanski, {\it The $C^*$-algebras of row-finite graphs}, New York J. Math. {\bf 6} (2000), 307--324.
\bibitem{3} T. Bates and D. Pask, {\it $C^*$-algebras of labelled graphs}, J. Operator Theory. {\bf 57} (2007), 101--120.
\bibitem{4} T. Bates and D. Pask, {\it $C^*$-algebras of labelled graphs II -simplicity results}, Math. Scand. {\bf 104} (2009), no. 2, 249--274.
\bibitem{5} J. A Jeong and S. H. Kim, {\it On simple labelled graph $C^*$-algebras}, 2010, preprint (arXiv:1101.4739v1[math.OA]).
\bibitem{6} A. Kumjian, D. Pask, and I. Raeburn, {\it Cuntz-Krieger algebras of directed graphs}, Pacific J. Math. {\bf 184} (1998),161--174.
\bibitem{7} A. Kumjian, D. Pask, I. Raeburn, and J. Renault, {\it Graphs, groupoids, and Cuntz-Krieger algebras}, J. Funct. Anal. {\bf 144} (1997), 505--541.
\end{thebibliography}
2010 Mathematics Subject Classification: 46L05, 46L55
Key Words and Phrases: labelled graph $C^*$-algebra, gauge-invariant ideal
- ⋅ 10:50 − 11:10 Generalized analytic Fourier-Feynman transforms and convolutions on a Fresnel type class (Il Yong Lee, Seung Jun Chang)
- 이일용*(단국대), 장승준(단국대)
Il Yong Lee*, Dankook University, Seung Jun Chang, Dankook University
In this paper, we define an $L_p$ analytic generalized Fourier-Feynman
transform and a convolution product
of functionals in a Banach algebra
$\mathcal F (C_{a,b}[0,T])$ which is called the Fresnel
type class, and in more general class $\mathcal F_{A_1,A_2}$ of functionals
defined on general function space $C_{a,b}[0,T]$
rather than on classical Wiener space.
Also we obtain some relationships between the $L_p$ analytic generalized
Fourier-Feynman transform and convolution product for functionals
in $\mathcal F (C_{a,b}[0,T])$ and in $\mathcal F_{A_1,A_2}$.
2010 Mathematics Subject Classification: 60J25, 28C20
Key Words and Phrases: generalized Brownian motion process, generalized analytic Feynman integral,, generalized analytic Fourier-Feynman transform,, convolution product, Fresnel type class
- ⋅ 11:10 − 11:40 Remarks on uncertainty relation associated with a monotone pair skew information (Chul Ki Ko, Hyun Jae Yoo)
- 고철기*(연세대), 유현재(한경대)
Chul Ki Ko*, Yonsei University, Hyun Jae Yoo, Hankyong National University
The Heisenberg's uncertainty relation for any pair of observables $A$ and $B$ and a density matrix $\rho$ is expressed as $$ V_\rho(A)V_\rho(B)\ge\frac{1}{4}|\hbox{Tr}(\rho[A,B])|^2. $$ Here $V_\rho(A)=\hbox{Tr}(\rho A^2)-\hbox{Tr}(\rho A)^2$ is the variance of the observable $A$ for $\rho$ and the commutator $[A,B]=AB-BA$. This inequality was improved by many authors. Recently, using the Wigner-Yanase-Dyson skew information $I_{\rho,\alpha}(A)=\hbox{Tr}(\rho A^2)-\hbox{Tr}(\rho^\alpha A\rho^{1-\alpha}A)$, $\alpha\in[0.1],$ Yanagi gave the trace inequality $$ U_{\rho,\alpha}(A)U_{\rho,\alpha}(B)\ge\alpha(1-\alpha)|\hbox{Tr}(\rho[A,B])|^2$$ for any observable $A,B$, where $U_{\rho,\alpha}(A)=\sqrt{V_\rho(A)^2-(V_\rho(A)-I_{\rho,\alpha}(A))^2}$. When $\alpha=1/2$, the inequality reduces to the Luo's inequality. Furuichi introduced the monotone pair skew information as a generalization of the Wigner-Yanase-Dyson skew information as well as some other skew information. The aim of this talk is to derive a trace inequality leading to an uncertainty relation based on the monotone pair skew information. Our result extends a few number of known results on the uncertainty relations.
2010 Mathematics Subject Classification: 81P45,82B10
Key Words and Phrases: Wigner-Yanase skew information,, Wigner-Yanase-Dyson skew information,, monotone pair skew information,, uncertainty relation
- Geometry I
- ⋅ 09:00 − 09:30 The category of generalized complex structures (Hoil Kim, Kyurak Cheong)
- 김호일*(경북대), 정규락(한국고등연구소)
Hoil Kim*, Kyungpook National University, Kyurak Cheong, Korean Institute of advanced studies
We describe the category of generalized complex structures containing the category of complex and symplectic structures. The objects are Lie (bi-)algebroids or Lie groupoids and the morphisms are naturally defined. We relate this category to that of (infinite) stacks and topos.
2010 Mathematics Subject Classification: 14A
Key Words and Phrases: generalized complex structures, Lie algebroids, Lie groupids, stacks, topos
- ⋅ 09:30 − 10:00 Orthogonal and symplectic extensions of vector bundles over a Riemann surface (Insong Choe, George H. Hitching)
- 최인송*(건국대), 히칭(노르웨이)
Insong Choe*, Konkuk University, George H. Hitching, Hogskolen i Vestfold
An efficient way to classify the vector bundles over a Riemann surface is given by studying the subbundles. This provides us a natural stratification on the moduli of vector bundles, which have been understood extensively. In this talk, we present our recent results on the analogous problem on orthogonal and symplectic bundles.
2010 Mathematics Subject Classification: 14H60, 14N05
Key Words and Phrases: orthogonal/symplectic bundle over a Riemann surface, maximal isotropic subbundles, extension
- ⋅ 10:10 − 10:30 A curvature identity on a 4-dimensional Riemannian manifold and its applications (Yunhee Euh, JeongHyeung Park and Kouei Sekigawa)
- 어윤희*(성균관대), 박정형(성균관대), 세키가와(니가타대)
Yunhee Euh*, Sungkyunkwan University, JeongHyeung Park, Sungkyunkwan University, Kouei Sekigawa, Niigata University
We recall a curvature identity for 4-dimensional compact Riemannian manifolds as derived from the generalized Gauss-Bonnet formula. We extend this curvature identity to non-compact 4-dimensional Riemannian manifolds. We also give some applications of this curvature identity.
2010 Mathematics Subject Classification: 53B20, 53C20
Key Words and Phrases: generalized Gauss-Bonnet formula, 3-dimensional curvature identity, super-Einstein
- ⋅ 10:30 − 10:50 Almost Hermitian structures on the product of Sasakian spaces (Jungchan Lee, JeongHyeong Park and Kouei Sekigawa)
- 이정찬*(성균관대), 박정형(성균관대), 코에이 세끼가와(니가타대)
Jungchan Lee*, Sungkyunkwan University, JeongHyeong Park, Sungkyunkwan University, Kouei Sekigawa, Niigata University
\noindent Let $(M, g, \xi, \varphi, \eta)$ (resp. $(M', g', \xi', \varphi', \eta')$ ) be a ${2p+1}$-dimensional Sasakian manifold(resp. ${2q+1}$-dimensional Sasakian manifold). Let $\overline{M}=M\times M'$ be the product space of $M$ and $M'$. Then we define a 2-parameter family of almost Hermitian structures $(\bar{J}, \bar{g})$ on $\overline{M}$. We investigate the integrability and the curvature properties of the almost Hermitian structure $(\bar{J},\bar{g})$ on $\overline{M}$, as well as intermediate relations. We may note that the almost Hermitian manifold $\overline{M}= (\bar{M}, \bar{J},\bar{g})$ is a generalization of the Calabi-Eckmann manifold equipped with the Hermitian structure introduced by Tsukada\cite{T}. \begin{thebibliography}{99} \bibitem{T} K. Tsukada, {\it Eigenvalues of the Laplacian on Calabi-Eckmann manifolds}, J. Math. Soc. Japan {\bf 33} (1981), 673--691. \end{thebibliography}
2010 Mathematics Subject Classification: 53C15, 53C55
Key Words and Phrases: Calabi-Eckmann manifold, almost Hermitian manifold
- ⋅ 13:10 − 13:40 Ruled minimal submanifolds (Dong-Soo Kim)
- 김동수(전남대)
Dong-Soo Kim, Chonnam National University
In this article, we study minimal ruled submanifolds in Minkowski space $\Bbb L^m$ which include Riemannian and Lorentzian cases. As a result, we give a complete classification of minimal ruled submanifolds in Minkowski space $\Bbb L^m$ with arbitrary dimension and codimension. As an immediate byproduct, we achieve the classification of all minimal surfaces in Minkowski space $\Bbb L^m$ of arbitrary dimension.
2010 Mathematics Subject Classification: 53B25, 53C40
Key Words and Phrases: ruled submanifold, minimal submanifold, Minkowski space
- ⋅ 13:40 − 14:10 Scalar curvatures of certain homogeneous spaces (Joon-Sik Park)
- 박준식(부산외대)
Joon-Sik Park, Pusan University of Foreign Studies
In this paper, we study the scalar curvature of a certain compact Riemannian homogeneous space $(G/H,g)$ whose isotropy representation decomposes into two inequivalent irreducible summands. We obtain a necessary and sufficient condition for the scalar curvature of $(G/H,g)$ to be positive, negative and zero respectively.
2010 Mathematics Subject Classification: 53C07, 53A15
Key Words and Phrases: scalar curvature, homogeneous space, isotropy representation
- ⋅ 14:10 − 14:40 Transverse Killing forms on foliated manifolds (Seoung Dal Jung)
- 정승달(제주대)
Seoung Dal Jung, Jeju National University
On a Riemannian foliation $\mathcal F$ of codimension $q$, {\it transverse conformal Killing forms} are defined to be basic forms $\phi$ such that for any vector field $X$ normal to the foliation, \begin{align*} \nabla_X\phi ={1\over r+1}i(X)d\phi + {1\over q-r+1}X^*\wedge \delta_T\phi, \end{align*} where $r$ is the degree of the form $\phi$ and $X^*$ the dual 1-form of $X$. See Section 3 for the definition of $\delta_T$. The transverse conformal Killing forms with $\delta_T\phi=0$ are called {\it transverse Killing forms}. Transverse Killing forms (resp. transverse conformal Killing forms) are generalizations of transversal Killing fields (resp. transversal conformal Killing fields). In this talk, we study the transverse Killing forms on foliated manifolds.
2010 Mathematics Subject Classification: 53C12
Key Words and Phrases: foliation, transverse Killing form
- ⋅ 14:50 − 15:10 Bertrand curves in 3-dimensional space forms (Jin Ho Choi, Tae Ho Kang and Young Ho Kim)
- 최진호*(울산대), 강태호(울산대), 김영호(경북대)
Jin Ho Choi*, University of Ulsan, Tae Ho Kang, University of Ulsan, Young Ho Kim, Kyungpook National University
A helix in 3-dimensiona Euclidean space $\mathbb{E}^3$ is a space curve with the constant curvature and the constant torsion. It is well-known two generalizations of helices in $\mathbb{E}^3$ which are called by 'general helices' and 'Bertrand curves'. General helices and Bertrand curves are respectively characterized by \begin{equation*} a\kappa - \tau = 0 ~~ \text{ and } ~~ a\kappa + b\tau =1, \end{equation*} where $a\neq0$ and $b$ are constants and $kappa$ and $\tau$ denote the curvature and the tosion of a given curve, respectively. In this talk, we define Bertrand curves in a 3-dimensional Riemannian manifold and we show that above characterization of Bertrand curves in $\mathbb{E}^3$ is extended to that in 3-dimensional space forms
2010 Mathematics Subject Classification: 53A04, 53A35
Key Words and Phrases: Helix, Bertrand curve
- ⋅ 15:10 − 15:40 Spacelike surfaces with free boundary in the Lorentz-Minkowski space (Juncheol Pyo)
- 표준철(고등과학원)
Juncheol Pyo, Korea Institute for Advanced Study
We investigate a variational problem in the 3-dimensional Lorentz-Minkowski space $\Bbb L^3$ whose critical points are spacelike surfaces with constant mean curvature and making constant contact angle with a given support surface along its common boundary. We show that if the support surface is a pseudosphere, then the surface is a planar disc or a hyperbolic cap.
2010 Mathematics Subject Classification: 53A10
Key Words and Phrases: spacelike surface, constant mean curvature
- ⋅ 15:40 − 16:10 Applications of minimal surfaces to special Lagrangian surfaces (Hojoo Lee)
- 이호주(서울대)
Hojoo Lee, Seoul National University
We exploit the theory of minimal surfaces and maximal surfaces to study special Lagrangian surfaces. Lei Fu (Houston J. Math. 1998) and Yu Yuan (Invent. Math. 2002) proved that the potential functions of any entire special Lagrangian graphs in 4 dimensional Euclidean space endowed with the standard symplectic structure are quadratic polynomials or harmonic functions. First, we provide a simple proof of this Bernstein type theorem and show that the geometric correspondence we use in our proof is a particular case of the construction from minimal surfaces to special Lagrangian surfaces. Second, we construct the duality for special Lagrangian surfaces in 4 dimensional Euclidean space and pseudo-Euclidean space. This duality prescribes the corresponding Lagrangian angles. It is also a particular case of more general duality between minimal surfaces and maximal surfaces.
2010 Mathematics Subject Classification: 53A10, 53D12
Key Words and Phrases: special Lagrangian surfaces, Bernstein type theorem, mean curvature
- Geometry II
- ⋅ 09:00 − 09:30 H-slant submersions (Kwang-Soon Park)
- 박광순(서울대)
Kwang-Soon Park, Seoul National University
Given a $C^{\infty}-$submersion $F$ from a Riemannian manifold $(M,g_{M})$ onto a Riemannian manifold $(N,g_{N})$, there are several kinds of submersions according to the conditions on it: e.g. Riemannian submersion, slant submersion, almost Hermitian submersion, quaternionic submersion, etc. As we know, Riemannian submersions are related with physics and have their applications in the Yang-Mills theory, Kaluza-Klein theory, Supergravity and superstring theories, etc. And the quaternionic K\"{a}hler manifolds have applications in physics as the target spaces for nonlinear $\sigma-$models with supersymmetry. In this talk, we will give the definitions of the almost h-slant submersion and the h-slant submersion and then obtain some interesting properties about them. Furthermore, we will construct some examples for the almost h-slant submersions and the h-slant submersions.
2010 Mathematics Subject Classification: 53C15; 53C26; 53C43
Key Words and Phrases: slant submersion, K\"{a}hler manifold, hyperk\"{a}hler manifold
- ⋅ 09:30 − 10:00 A study on manifolds with harmonic curvature (Seongtag Kim)
- 김성택(인하대)
Seongtag Kim, Inha University
A Riemannian $n$-manifold $(M,g)$ is a manifold with harmonic curvature if the divergence of its curvature tensor $Riem$ vanishes, i.e.,
$$0 =(\delta Riem)_{jkl} =\nabla^i R_{ijkl}, \leqno(1) $$
which is equivalent to $$ 0= \nabla_k R_{jl}-\nabla_l R_{jk}.\leqno(2)
$$ By the Bianchi identity, the scalar curvature is constant. Einstein metrics, manifolds with parallel Ricci curvature have harmonic curvature. In the space of connections, the Yang-Mills connections correspond to metrics with harmonic curvature. A natural question is when a manifold of harmonic curvature must be Einstein or flat. We investigate this problem for some noncompact complete Riemannian
manifolds with positive Sobolev constant $Q(M, g)$, and obtain rigidity phenomena when the curvatures are close to those of the flat or Einstein space in $L_{n/2}$.
2010 Mathematics Subject Classification: 53C21
Key Words and Phrases: manifold with harmonic curvature, Einstein metric, rigidity
- ⋅ 10:10 − 10:40 Seiberg-Witten monopoles on noncompact complete 4-manifolds (Chanyoung Sung)
- 성찬영(건국대)
Chanyoung Sung, Konkuk University
We prove the $C^0$ bound of the spinor satisfying the Seiberg-Witten equations on noncompact complete Riemannian 4-manifolds satisfying certain lower bound on the Ricci curvature. This is an application of certain Liouville-type theorems.
2010 Mathematics Subject Classification: 31B05, 57R57, 53A30, 53C42
Key Words and Phrases: Liouville theorem, Seiberg-Witten equations
- ⋅ 13:10 − 13:40 Real hypersurfaces of Type $A$ in complex two-plane Grassmannians related to commuting shape operator (Imsoon Jeong, Young Jin Suh)
- 정임순*(경북대), 서영진(경북대)
Imsoon Jeong*, Kyungpook National University, Young Jin Suh, Kyungpook National University
In this talk, we introduce and give some characterizations of real hypersurfaces of Type $A$ in complex two-plane Grassmannians $G_{2}(\mathbb C)^{m+2}$, that is, a tube over a totally geodesic $G_{2}(\mathbb C)^{m+1}$ in $G_{2}(\mathbb C)^{m+2}$ with the commuting condition $A \phi_{1} \phi = \phi_{1} \phi A$ for the shape operator $A$, the structure tensors $\phi$ and $\phi_{1}$ and together with another geometric conditions.
2010 Mathematics Subject Classification: 53C40
Key Words and Phrases: real hypersurfaces, complex two-plane Grassmannians, Hopf hypersurface, commuting shape operator
- ⋅ 13:40 − 14:00 Real hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster $\xi$-parallel shape operator (Hyunjin Lee, Imsoon Jeong and Young Jin Suh)
- 이현진*(경북대), 정임순(경북대), 서영진(경북대)
Hyunjin Lee*, Kyungpook National University, Imsoon Jeong, Kyungpook National University, Young Jin Suh, Kyungpook National University
We have shown that there does not exist a hypersurface in $G_{2}(\mathbb C^{m+2})$ with parallel shape operator in the generalized Tanaka-Webster connection (to appear in Kodai Math. J. (2011)). In this talk, we consider the notion of $\xi$-parallel for a hypersurface $M$ in $G_{2}(\mathbb C^{m+2})$ and proved that $M$ is an open part of a tube around a totally geodesic $G_2({\mathbb C}^{m+1})$ in $G_2({\mathbb C}^{m+2})$.
2010 Mathematics Subject Classification: 53C40; 53C15
Key Words and Phrases: real hypersurfaces, complex two-plane Grassmannians, Hopf hypersurface, generalized Tanaka-Webster connection, $\xi$-parallel shape operator
- ⋅ 14:00 − 14:30 Isospectral Riemann surfaces (Hyunsuk Kang, Dennis Barden)
- 강현석*(고등과학원), 데니스 바든(캠브리지대)
Hyunsuk Kang*, Korea Institute for Advanced Study, Dennis Barden, University of Cambridge
Since the question `Can one hear the shape of a drum?' was raised by M. Kac in the 1960's, there has been extensive search of isospectral, i.e., having the same eigenvalue spectrum of the Laplacian on functions, non-isometric manifolds. It has been shown that there exist such Riemann surfaces of genus greater than three, and we show that it is also true for genus two and three. Thus we conclude that `one cannot hear the geometry of Riemann surfaces'. This is a joint work with Dennis Barden.
2010 Mathematics Subject Classification: 58J50
Key Words and Phrases: spectral geometry
- Topology I
- ⋅ 09:00 − 09:30 A complex surface of general type with $p_g=0$, $K^2=2$ and $H_1=\mathbb{Z}/4\mathbb{Z}$ (Heesang Park, Jongil Park and Dongsoo Shin)
- 박희상*(고등과학원), 박종일(서울대), 신동수(충남대)
Heesang Park*, Korea Institue for Advanced Study, Jongil Park, Seoul National University, Dongsoo Shin, Chungnam National University
We construct a new minimal complex surface of general type with $p_g=0$, $K^2=2$ and $H_1=\mathbb{Z}/4\mathbb{Z}$ (in fact $\pi_1^{\text{alg}}=\mathbb{Z}/4\mathbb{Z}$), which settles the existence question for numerical Campedelli surfaces with all possible algebraic fundamental groups. The main techniques involved in the construction are a rational blow-down surgery and a $\mathbb{Q}$-Gorenstein smoothing theory.
2010 Mathematics Subject Classification: 14J29
Key Words and Phrases: $\mathbb{Q}$-Gorenstein smoothing, rational blow-down surgery, surface of general type
- ⋅ 09:30 − 10:00 Weakly transitive sets with shadowing property (Manseob Lee)
- 이만섭(목원대)
Manseob Lee, Mokwon University
Let $M$ be a closed smooth manifold, and let $f$ be a diffeomorphism on $M.$ We show that if $f$ has the $C^1$-stably shadowing property on a weakly transitive set then the set is hyperbolic. And, $C^1$-generically, an isolated weakly transitive set is shadowable if and only if weakly transitive set is hyperbolic.
2010 Mathematics Subject Classification: 37B20, 37C29, 37D30, 37C25.
Key Words and Phrases: shadowing, hyperbolic, weakly transitive, Axiom A, basic set
- ⋅ 11:00 − 11:20 Maslov index via Chern-Weil theory (Hyung-Seok Shin, Cheol-hyun Cho)
- 신형석*(서울대), 조철현(서울대)
Hyung-Seok Shin*, Seoul National University, Cheol-hyun Cho, Seoul National University
We define the Maslov index via Chern-Weil theory. Our main application is defining a Maslov index for symplectic orbifold.
2010 Mathematics Subject Classification: 53D12
Key Words and Phrases: Maslov index, orbifold
- ⋅ 11:20 − 11:40 Continuation homomorphism in Rabinowitz Floer homology for symplectic deformations (Youngjin Bae, Urs Frauenfelder)
- 배영진*(서울대), Urs Frauenfelder(서울대)
Youngjin Bae*, Seoul National University, Urs Frauenfelder, Seoul National University
Will J. Merry computed Rabinowitz Floer homology above Mane's critical value in terms of loop space homology by establishing an Abbondandolo-Schwarz short exact sequence. The purpose of this talk is to provide an alternative proof of Merry's result. We construct a continuation homomorphism for symplectic deformations which enables us to reduce the computation to the untwisted case. Our construction takes advantage of a special version of the isoperimetric inequality which above Mane's critical value holds true.
2010 Mathematics Subject Classification: 53D40
Key Words and Phrases: Floer homology, Rabinowitz Floer homology, Mane's critical value, isoperimetric inequality
- ⋅ 13:10 − 13:40 Prime knots whose arc index is smaller than the crossing number (Hwa Jeong Lee, Gyo Taek Jin)
- 이화정*(카이스트), 진교택(카이스트)
Hwa Jeong Lee*, KAIST, Gyo Taek Jin, KAIST
It is known that the arc index of alternating knots is the minimal crossing number plus two and the arc index of nonalternating knots is less than or equal to the minimal crossing number. We study some cases when the arc index is strictly less than the minimal crossing number.
2010 Mathematics Subject Classification: 57M25
Key Words and Phrases: knot, arc index, crossing number
- ⋅ 13:40 − 14:00 The Gromov-Witten invariants of the symmetric product of symplectic manifolds (So Ra Yi)
- 이소라(이화여대)
So Ra Yi, Ewha Womans University
Let $X$ be the $n$-fold product space of a symplectic manifolds $M$. Using a symplectic form and a complex structure on $M$, we can consider $X$ as an almost complex and symplectic manifold, that is, $X$ has a symplectic form $\omega = \omega_1 + \cdots + \omega_n$, where $\omega_i = \omega_j$ is the symplectic form on $M$ for all $i,\; j = 1, \cdots, n$. Let $X'$ be the $n$-fold symmetric product of $M$, which is the orbit space $X' = X / S_n$ obtained by the standard action of the symmetric group $S_n$ of $n$ letters on $X$.\\ In this talk, we investigate the moduli space $\mathfrak{M}_{g,k}(X,A)$ of holomorphic curves $u : \Sigma \rightarrow X$ from a Riemann surface $\Sigma$ with $k$-marked points to the manifold $X$ representing the homology class $A$, the symmetric group $S_n$ action on the moduli space, and the moduli space $\mathfrak{M}_{g,k}(X',A')$ of holomorphic curves $u' : \Sigma \rightarrow X'$ on $X'$ where $A \in H_2(X, \mathbb{Z})$ is an $S_n$-invariant $2$-dimensional integral homology class of $X$ and $A'$ is a suitable image of a homology class in $X$ through the projection $\pi : X \longrightarrow X'$. Further, we research relations between $\mathfrak{M}_{g,k}(X,A)$ and $\mathfrak{M}_{g,k}(X',A')$.
2010 Mathematics Subject Classification: 57R17
Key Words and Phrases: symplectic manifold, symmetric product
- ⋅ 14:00 − 14:30 On Petrie's conjecture for torus manifolds (Suyoung Choi)
- 최수영(아주대)
Suyoung Choi, Ajou university
T. Petrie has shown that all homotopy equivalence between homotopy projective spaces admitting effective smooth half-dimensional compact torus actions should preserve their Pontrjagin classes. In this talk, we introduce some results on the invariance of Pontrjagin classes of torus manifolds, which can be regarded as a generalization of Petrie's theorem.
2010 Mathematics Subject Classification: 57R19
Key Words and Phrases: torus manifold, generalized Bott manifold, cohomological rigidity problem, homotopy Bott manifold, toric topology
- ⋅ 14:50 − 15:10 On coisotropic intersections (Jungsoo Kang)
- 강정수(서울대)
Jungsoo Kang, Seoul National University
In this talk, we introduce intersection problems and rigidity problems of coisotropic submanifolds. Taking advantage of Rabinowitz Floer theory, we prove some of them.
2010 Mathematics Subject Classification: 53D40, 37J10, 58J05
Key Words and Phrases: Coisotropic intersections, Rabinowitz Floer theory
- ⋅ 15:10 − 15:30 Quadrisecants of minimal polygonal figure eight knots (Seo Jung Park)
- 박서정(카이스트)
Seo Jung Park, KAIST
An n-secant line for a knot K is an oriented line whose intersection with K has at least n components. An n-secant is an ordered n-tuple of points in K which lie in order on an n-secant line. A 4-secant is called a quadrisecant. In this talk, we study quadrisecants of minimal polygonal figure eight konts.
2010 Mathematics Subject Classification: 54-XX
Key Words and Phrases: knot, quadrisecant
- ⋅ 15:30 − 16:00 Anosov theorem for coincidences on solvmanifolds (Jong Bum Lee)
- 이종범(서강대)
Jong Bum Lee, Sogang University
Suppose that $S$ and $S'$ are simply connected solvable Lie groups of type $(\mathrm{R})$ with the same dimension. We show that the Lefschetz coincidence numbers of maps $f, g : \Gamma\backslash{S}\to\Gamma'\backslash{S}'$ between special solvmanifolds can be computed algebraically as follows: $$ L(f, g) = \det(G_*-F_*), $$ where $F_*,G_*$ are the matrices, with respect to any preferred bases, of morphisms of Lie algebras induced by $f$ and $g$. This generalizes a recent result by S. W. Kim and J. B. Lee to special solvmanifolds of type $(\mathrm{R})$. Moreover, we can drop the dimension match condition imposed in the latter result.
2010 Mathematics Subject Classification: 55M20, 54H25, 57S30
Key Words and Phrases: Nielsen coincidence number, solvmanifold
- Topology II
- ⋅ 13:10 − 13:30 Orbifold morse homology (Hansol Hong, Cheol-Hyun Cho)
- 홍한솔*(서울대), 조철현(서울대)
Hansol Hong*, Seoul National University, Cheol-Hyun Cho, Seoul National University
Morse-Smale-Witten complex is a free module generated by critical points of a Morse function graded by their indices, and the differential on this complex is given by counting (signed) number of gradient lines between critical points of index difference one. We develop Morse-Smale-Witten complex for effective orbifolds $X$ where Morse functions on $X$ are given by invariant functions whose local lift is Morse. We also show that its homology is isomorphic to the singular homology of $|X|$, the underlying quotient space of $X$ (with certain assumptions).
2010 Mathematics Subject Classification: 57R18
Key Words and Phrases: orbifolds, morse theory
- ⋅ 13:30 − 13:50 $\mathbb{Q}$-trivial generalized Bott manifolds (Seonjeong Park, Dong Youp Suh)
- 박선정*(카이스트), 서동엽(카이스트)
Seonjeong Park*, KAIST, Dong Youp Suh, KAIST
A generalized Bott tower is a sequence of $\mathbb{C}P^{n_i}$-bundles for $i=1,\ldots,h$. The top manifold of the tower is called a generalized Bott manifold. When the cohomology ring of a generalized Bott manifold with $\mathbb{Q}$-coefficient is isomorphic to that of a product of complex projective spaces $\mathbb{C}P^{n_i}$, the generalized Bott manifold is said to be $\mathbb{Q}$-trivial. In this talk, we find the necessary and sufficient condition for a generalized Bott manifold to be $\mathbb{Q}$-trivial. In particular, every $\mathbb{Q}$-trivial generalized Bott manifold without $\mathbb{C}P^1$-fibration is diffeomorphic to a product of complex projective spaces.
2010 Mathematics Subject Classification: 57R19,57S25
Key Words and Phrases: generalized Bott manifold, cohomological rigidity, toric topology
- ⋅ 13:50 − 14:10 Classification of equivariant vector bundle over two-sphere (Min Kyu Kim)
- 김민규(경인교대)
Min Kyu Kim, Gyeongin National University of Education
We classify equivariant topological complex vector bundles over two-sphere under a compact Lie group (not necessarily effective) action. It is shown that nonequivariant Chern classes and isotropy representations at (at most) three points are sufficient to classify equivariant vector bundles except a few cases.
2010 Mathematics Subject Classification: 57S25
Key Words and Phrases: equivariant vector bundle, equivariant homotopy, representation
- ⋅ 14:20 − 14:40 Notes on Kontsevich-Soibelman's theorem about cyclic $A_{\infty}$ algebras (Sangwook Lee, Cheol-hyun Cho)
- 이상욱*(서울대), 조철현(서울대)
Sangwook Lee*, Seoul National University, Cheol-hyun Cho, Seoul National University
Kontsevich and Soibelman has proved a relation between a non-degenerate cyclic homology element of an $A_{\infty}$-algebra $A$ and its cyclic inner products on the minimal model of $A$. We find an explicit formula of this correspondence, in terms of the strong homotopy inner products and negative cyclic cohomology of $A$. We prove that an equivalence class of the induced strong homotopy inner product depends only on the given negative cyclic cohomology class. Also, we extend such a correspondence to the case of gapped filtered $A_{\infty}$-algebras.
2010 Mathematics Subject Classification: 57T99
Key Words and Phrases: $A_{\infty}$ algebras, strong homotopy inner product, negative cyclic cohomology
- Probability and Statistics
- ⋅ 10:10 − 10:40 Stability of a retrial queueing network with different classes of customers and restricted resource pooling (Bara Kim)
- 김바라(고려대)
Bara Kim, Korea University
We consider a retrial queueing network with different classes of customers and several servers. Each customer class is associated with a set of servers who can serve the class of customers. Customers of each class exogenously arrive according to a Poisson process. If an exogenously arriving customer finds upon his arrival any idle server who can serve the customer class, then he begins to receive a service by one of the available servers. Otherwise he joins the retrial group, and then tries his luck again after exponential time, the mean of which is determined by his customer class. Service times of each server are assumed to have general distribution. The retrial queueing network can be represented by a Markovprocess, with the number of customers of each class, and thecustomer class and the remaining service time of each busy server. Using the fluid limit technique, we find a necessary and sufficient condition for the positive Harris recurrence of the representing Markov process. This work is the first that applies the fluid limit technique to a model with retrial phenomena.
2010 Mathematics Subject Classification: 60K25
Key Words and Phrases: stochastic stability, fluid limit, retrial network, restricted resource pooling
- ⋅ 13:10 − 13:40 Some results on stochastic partial differential systems (Kyeong-Hun Kim)
- 김경훈(고려대)
Kyeong-Hun Kim, Korea University
Stochastic partial differential system is a parabolic system with some random noises. In this talk, we plan to give a brief survey on past results. Also some recent new works will be introduced. In particular, we will present an $L_p$-theory for stochastic systems defined on $C^1$ domains.
2010 Mathematics Subject Classification: 60H15
Key Words and Phrases: stochastic partial differential system
- ⋅ 13:40 − 14:10 On the discrete heat equation taking values on a simple tree (Kijung Lee, Carl Mueller)
- 이기정*(아주대), 칼 뮬러(로체스터대)
Kijung Lee*, Ajou University, Carl Mueller, University of Rochester, U.S.A.
In this talk we discuss a heat equation with ``values" in a simple tree, a non-smooth manifold. We suggest a solution motivated by the propabilistic representation of the solution on Euclidean space. The main obstacle is to define ``addition" on the tree so that the diffusion is well-described; we suggest a way to overcome this.
2010 Mathematics Subject Classification: 60H15
Key Words and Phrases: heat diffusion, simple tree valued function
- ⋅ 14:10 − 14:40 Two-sided Green function estimates for killed subordinate Brownian motions (Panki Kim)
- 김판기(서울대)
Panki Kim, Seoul National University
A subordinate Brownian motion is a Levy process which can obtained by replacing the time of Brownian motion by an independent increasing Levy process. In this talk, we consider a large class of subordinate Brownian motions without diffusion term. This class of processes include symmetric stable processes, relativistic stable processes, sums of independent symmetric stable processes, sums of independent elativistic stable processes, and much more. We give an explicit form of sharp two-sided estimates on the Green functions of these subordinate Brownian motions in bounded C1,1 open set. This is a joint work with Renming Song and Zoran Vondracek.
2010 Mathematics Subject Classification: 60J45
Key Words and Phrases: Green function, subordinate Brownian motion, symmetric stable process, boundary Harnack principle, fluctuation theory, regularly varying function
- ⋅ 14:50 − 15:10 A stochastic elasticity correction to the Cox-Ingersoll-Ross interest rate model (Yong-Ki Ma, Jeong-Hoon Kim)
- 마용기*(연세대), 김정훈(연세대)
Yong-Ki Ma*, Yonsei University, Jeong-Hoon Kim, Yonsei University
As known well, one-factor interest rate models tend to produce a certain undesirable gap between the theoretical yield curves and the observed yield curves. In this paper, we propose a new two-factor short-term interest rate model in such a way that the popular Cox-Ingersoll-Ross model is corrected by a function of a fast mean-reverting Ornstein-Uhlenbeck process. Correction effects are asymptotically analyzed and numerically computed for the term structure of interest rate derivatives.
2010 Mathematics Subject Classification: 91G30
Key Words and Phrases: Stochastic elasticity, term structure, Cox-Ingersoll-Ross model, Ornstein-Uhlenbeck process
- ⋅ 15:10 − 15:40 Financial data analysis with time-varying bivariate student distributions (Cheong Hee Ahn, Hi Jun Choe)
- 안청희*(아주대), 최희준(연세대)
Cheong Hee Ahn*, Ajou University, Hi Jun Choe, Yonsei University
We propose a time-varying bivariate student distribution of fitting returns between Kospi index and Dow Johns index.
2010 Mathematics Subject Classification: 91G70
Key Words and Phrases: time- varying bivariate student distribution, Fokker-Planck equation
- ⋅ 15:40 − 16:10 Exponential functionals of Wishart processes (Chulmin Kang, Wanmo Kang)
- 강철민*(카이스트), 강완모(카이스트)
Chulmin Kang*, KAIST, Wanmo Kang, KAIST
We characterize the exponential functionals of Wishart processes in terms of certain matrix Riccati integral equations with measure drift. The result of the paper generalizes the systematic exploitation of exponential functionals of squared Bessel processes in a natural way. As consequences of such characterization of exponential functionals, we found some formulae such as Laplace transforms, transition densities, and Pitman-Yor's formula for Wishart processes. Moreover, we also consider the exponential functionals of Wishart bridges. As an application of Wishart bridges, we propose an exact simulation method of multifactor volatility Heston model.
2010 Mathematics Subject Classification: 60J60
Key Words and Phrases: Wishart processes, matrix Riccati integral equations, exponential functionals
- Applied Mathematics I
- ⋅ 09:00 − 09:30 Identifiability problem of linearized sine-Gordon equation (Jun Hong Ha)
- 하준홍(한국기술교육대)
Jun Hong Ha, Korea University of Technology and Education
Consider the identification problem of parameters $q=(\alpha, \beta,\delta) \in P \subset R^n$ appearing in the one-dimensional sine-Gordon equation: \begin{eqnarray} y_{tt}+2\alpha y_{t}-\beta y_{xx}+ \gamma \sin y =f(t,x), \ \ (t,x) \in Q=(0,T) \times (0,1), \label{in1} \end{eqnarray} In [1], the parameter identification problem for (1) was studied from the theoretical and numerical perspectives. That is, they found $q^*$ satisfying \begin{eqnarray} J(q^*) = \inf_P J(q) = \|y(q)-z_d \|_{L^2(Q)}^2 \end{eqnarray} where $y(q)$ is the solution of (1)-(3) and $z_d$ is the target in $L^2(Q)$. Here we are interested in obtaining the information on $q^*=(\alpha^*, \beta^*,\delta^*)$ using not so many observed data on $(0,\pi)$. For it, we consider the linearized equation of (\ref{in1}) like \begin{eqnarray} y_{tt}+\alpha y_{t}-\beta y_{xx}+ \gamma y =f(t,x), \ \ (t,x) \in Q=(0,T) \times (0,1). \label{Lin1} \end{eqnarray} In this paper we consider an identifiability problem for (\ref{Lin1}). That is, if $y(t,x) = z_d(t,x)$ on a subset in $Q$, then $q=q^*$, where $z_d$ is the solution of (\ref{Lin1}) with $q=q^*$. [1] Junhong Ha and Semion Gutman, {\it Optimal parameters for a damped sine-Gordon equation}, J. Korean Math. Soc. {\bf 46} (2009), no. 5, 1105--1117.
2010 Mathematics Subject Classification: 35R30
Key Words and Phrases: identification, identifiability
- ⋅ 09:30 − 10:00 Curvature-incorporating interpolation for image zooming (Youngjoon Cha, Hakran Kim and Seongjai Kim)
- 차영준*(세종대), 김학란(미시시피주립대), 김성재(미시시피주립대)
Youngjoon Cha*, Sejong University, Hakran Kim, Mississippi State University, Seongjai Kim, Mississippi State University
This talk introduces a novel image zooming algorithm, called the curvature-incorporating method (CIM), which is partial differential equation-based and easy to implement. In order to minimize artifacts arising in image interpolation such as image blur and the checkerboard effect, the CIM first evaluates the curvature of the low resolution image. After interpolating the curvature to the high resolution image domain, the CIM constructs the high resolution image by solving a linearized curvature equation, incorporating the interpolated curvature as an explicit driving force. It has been numerically verified that the new zooming method can produce clear images of sharp edges which are already denoised and superior to those obtained from linear methods. Various results are given to prove effectiveness and reliability of the new method.
2010 Mathematics Subject Classification: 65M06, 68U10
Key Words and Phrases: curvature-incorporating method (CIM), interpolation, image zooming, super-resolution
- ⋅ 11:00 − 11:30 Alternating minimization algorithm for speckle reduction (Hyenkyun Woo, Sangwoon Yun)
- 우현균*(이화여대), 윤상운(고등과학원)
Hyenkyun Woo*, Ewha Womans University, Sangwoon Yun, Korea Institute for Advanced Study
In active imaging systems, such as synthetic aperture radar, the observed image is corrupted by the speckle. The variational models with the total variation (TV) regularization have attracted much interest in reducing the speckle due to the edge preserving feature of TV. Recently, several TV regularized convex variational model, such as the maximum a posteriori (MAP) model for log-transformed image and the I-divergence model, have been proposed. To solve those problems, the augmented Lagrangian with Gauss-Seidel method have been proposed recently. In this talk, we use alternating minimization algorithm to solve those problems without any inner iteration or inversion involving the Laplacian operator. The proposed method is very simple and highly parallelizable. Numerical results show that our proposed method outperforms the state-of-the-art algorithms for speckle reduction models.
2010 Mathematics Subject Classification: 49M37, 65K10
Key Words and Phrases: total variation, speckle, synthetic aperture radar, convex optimization, alternating minimization, denoising, multiplicative noise
- ⋅ 13:10 − 13:40 Optimal liquidation of a large block of stock with regime-switching (Dong-Hoon Shin)
- 신동훈(고려대)
Dong-Hoon Shin, Korea University
This work is concerned with an optimal selling rule for a large position of stock in a market. Selling a large block of stock in a short period typically depresses the market, which would result in a poor filling price. In addition, the large selling intensity makes the regime more likely to be poor state in the market. In this paper, regime switching and depressing terms associated with selling intensity are considered on a set of geometric Brownian models to capture movements of underlying asset. We also consider the liquidation strategy to sell much smaller number of shares in a long period. The goal is to maximize the overall return under state constraints. The corresponding value function with the selling strategy is shown to be a unique viscosity solution to the associated HJB equations. Optimal liquidation rules are characterized by a finite difference method. A numerical example is given to illustrate the result.
2010 Mathematics Subject Classification: 91G80
Key Words and Phrases: regime switching, optimal selling rule, optimal control, HJB equation
- ⋅ 13:40 − 14:10 Cubic helical splines with Frenet-frame continuity (Chang Yong Han, Song-Hwa Kwon)
- 한창용*(경희대), 권성화(가톨릭대)
Chang Yong Han*, Kyung Hee University, Song-Hwa Kwon, The Catholic University of Korea
A scheme of interpolating a sequence of points in space by a piecewise-cubic-helical spline with continuous Frenet frame is presented. Each cubic helical segment connecting adjacent data points is constructed such that its end tangents are symmetric with respect to the displacement vector of the end points. The existence condition of such segment is formulated in terms of the configuration involving the displacement vector and the Frenet frame at the initial point. A Frenet-frame-continuous spline is then obtained by matching the Frenet frames of sequentially constructed segments at their juncture points.
2010 Mathematics Subject Classification: 53A04
Key Words and Phrases: Pythagorean-hodograph, cubic helix, Frenet frame, geometric continuity, Hermite interpolation, spline, tangent indicatrix, quaternion
- ⋅ 14:10 − 14:40 Enzyme reactions on DNA and tangle analysis (Myeong-Ju Jeong)
- 정명주(카이스트 부설 한국과학영재학교)
Myeong-Ju Jeong, KSA of KAIST
We study topological geometry of a processive recombination along enzymatic reaction. We introduce an invariant of knots and links and compute the difference of values when an enzymatic reaction occurred. We find a necessary condition for a tangle surgery of the recombination and give a result to find the number of the reactions.
2010 Mathematics Subject Classification: 57M25
Key Words and Phrases: topology of DNA, knots, invariants of knots
- ⋅ 14:50 − 15:10 Copulas from Fokker-Planck equation (Beom Jin Kim, Cheonghee Ahn and Hi Jun Choe)
- 김범진*(연세대), 안청희(아주대), 최희준(연세대)
Beom Jin Kim*, Yonsei University, Cheonghee Ahn, Ajou University, Hi Jun Choe, Yonsei University
This paper develops a theoretical study addressing the dependence structure and provides a formula for a copula related to stochastic process arising in finance. The copula is a function that links univariate distributions to joint multivariate distribution. The tractability and important of copula lie on the method of inference function of margin(IFM) and our method is also in the spirit of IFM. We present a direct computation copula from the Fokker-Planck equation that can be used for computing the probability density for a stochastic process described by a stochastic differential equation (SDE). We find relation between copula and Fokker-Planck equation. We also present numerical results which illustrate a sensitive analysis of our scheme.
2010 Mathematics Subject Classification: 60Gxx
Key Words and Phrases: copulas, Fokker-Planck equation, marginal distribution functions, separable structure of stochastic
- ⋅ 15:10 − 15:30 Mean-reverting option price with Hull-White interest rate (Ji Hun Yoon, Jeong-Hoon Kim and Suk Hyun Yu)
- 윤지훈*(연세대), 김정훈(연세대), 유석현(삼성증권)
Ji Hun Yoon*, Yonsei University, Jeong-Hoon Kim, Yonsei University, Suk Hyun Yu, Samsung Security
In this paper we consider a multiscale extension of the well-known Black-Scholes option pricing model based on stochastic volatility driven by a fast mean-reverting Ornstein-Uhlenbeck process, a slowly varying diffusion process. Also, we introduce Hull-white interest rate model. We use singular and regular perturbation analysis to obtain an approximate price of European options under our multiscale formulation. We also provide a calibration method using observed option prices as represented by the so-called term structure of implied volatility. The resulting approximation is still independent of the particular details of the volatility model and gives more exibility in the parametrization of the implied volatility surface. In particular, the introduction of the slow factor and jump process factor gives a much better fit for options with longer maturities.
2010 Mathematics Subject Classification: 60Gxx
Key Words and Phrases: option price, mean-reversion, slow varying process, Hull-White interest rate process
- ⋅ 15:30 − 15:50 Necessary and sufficient conditions for global dynamics of a Rosenzweig-MacArthur model with prey immigration (Saito Yasuhisa and Jitsuro Sugie)
- Yasuhisa Saito(전남대), Jitsuro Sugie(Shimane Univ.)
Saito Yasuhisa, Chonnam National University, Jitsuro Sugie, Shimane University
Many natural predator and prey populations persist while their densities show sustained oscillations. Hence these populations must be regulated in such a way that the densities are kept away from the values where extinction is likely to occur. On the other hand, non-spatial simple predator-prey models show vigorous oscillations that can bring the populations to the brink of extinction or beyond. Predator-prey systems that are kept in the laboratory also tend to show fluctuations in densities that are severe enough to drive them to extinction. Since the amount of space that laboratory populations live in is small compared to those of natural populations, one is readily led to the hypothesis that spatial interactions must contribute to the regulation of natural predator-prey systems. In this paper, we construct a simplest type of spatially interacting populations by taking into account constant immigration of prey for a predator-prey model with a Holling type II functional response and derive necessary and sufficient conditions for both the uniqueness of limit cycles and the global asymptotic stability of a positive equilibrium. From these results, it is fully suggested (mathematically) that the prey immigration dampens the large fluctuations emerging in the predator-prey model and also stabilizes a positive equilibrium globally.
2010 Mathematics Subject Classification: 92B99
Key Words and Phrases: Rosenzweig-MacArthur model, prey immigration, predator-prey model
- Applied Mathematics II
- ⋅ 13:10 − 13:40 Recurrence relations for semilocal convergence of a family of methods in Banach spaces (Changbum Chun, Pantelimon Stanica and Beny Neta)
- 천창범*(성균관대), 펜트리몬 스탠시아(미국 해군대학원), 베니 네타(미국 해군대학원)
Changbum Chun*, Sungkyunkwan University, Pantelimon Stanica, Naval Postgraduate School, Beny Neta, Naval Postgraduate School
A family of iterative methods is studied for solving nonlinear equations $F(x)=0$ in Banach spaces. Results on convergence and uniqueness of solution are given, as well as error estimates by using recurrence relations. We give several applications to our method.
2010 Mathematics Subject Classification: 41A25, 65Q30, 65J15, 65H10, 65D99
Key Words and Phrases: nonlinear equations in Banach spaces, recurrence relations, semilocal convergence
- ⋅ 13:40 − 14:10 A quadrilateral Morley element for biharmonic equations (Chunjae Park, Dongwoo Sheen)
- 박춘재*(건국대), 신동우(서울대)
Chunjae Park*, Konkuk University, Dongwoo Sheen, Seoul National University
In this talk, we propose a Morley-type finite element for quadrilateral meshes to solve biharmonic problems. For each quadrilateral $Q$, the finite element space is defined by the span of $P_2(Q)$ plus two polynomials in $P_3(Q).$ Each of the cubic polynomials vanishes at a pair of opposite edges and the bimedian between them. We will prove the values at vertices and integrals of normal derivatives over edges are the degrees of freedom of the proposed element. An optimal order of convergence is analyzed and several numerical tests confirm it.
2010 Mathematics Subject Classification: 65N30
Key Words and Phrases: Morley, Biharmonic
- ⋅ 14:10 − 14:40 Stablility estimate for the scattering from an EM cavity (KiHyun Yun)
- 윤기현(한국외대)
KiHyun Yun, Hankuk University of Foreign Studies
We consider a time harmonic scattering problem of EM waves from an open cavity in two dimensions. A variational formulation reduces the scattering problem into a bounded domain problem. In this talk, a stability of the solution shall be established for the bounded domain problem in the energy space. Moreover, our result provides the explicit dependency on the wave number.
2010 Mathematics Subject Classification: 78M30, 35Q60, 35J57
Key Words and Phrases: EM cavity, stability, high frequency
- Mathematics for Information Sciences I
- ⋅ 09:00 − 09:30 Simple proofs for duality of generalized minimum poset weights and weight distributions of linear (Near-)MDS poset codes (Dae San Kim, Dong Chan Kim and Jong Yoon Hyun)
- 김대산*(서강대), 김동찬(서강대), 현종윤(이화여대)
Dae San Kim*, Sogang University, Dong Chan Kim, Sogang University, Jong Yoon Hyun, Ewha Womans University
In 1991, Wei introduced generalized minimum Hamming weights for linear codes and showed their monotonicity and duality. Recently, several authors extended these results to the case of generalized minimum poset weights by using different methods. Here, we would like to prove the duality by using matroid theory. This gives yet another and very simple proof of it. In particular, our argument will make it clear that the duality follows from the well-known relation between the rank function and the corank function of a matroid. In addition, we derive the weight distributions of linear MDS and Near-MDS poset codes in the same spirit.
2010 Mathematics Subject Classification: 05B30, 94B05, 94B60
Key Words and Phrases: duality, generalized minimum poset weight, weight distribution, matroid, Near-MDS poset code
- ⋅ 09:30 − 10:00 New metrics arising from matroids of linear codes (Dong-Chan Kim, Dae San Kim and Jong Yoon Hyun)
- 김동찬*(서강대, ETRI부설연구소), 김대산(서강대), 현종윤(이화여대)
Dong-Chan Kim*, Sogang University, The Attached Institute of ETRI, Dae San Kim, Sogang University, Jong Yoon Hyun, Ewha Womans University
In this paper, from each linear code in $\mathbb{F}_q^n$, we construct two metrics, the one called ``the metric associated to it'' and the other ``the symmetrized metric associated to it.'' They are respectively defined in terms of the rank function and the rank and corank functions of the matroid of the linear code. Such metrics constructed from linear codes give rise to abundant new metrics on the space $\mathbb{F}_q^n$. We investigate their properties and illustrate our results with some examples.
2010 Mathematics Subject Classification: 05B30, 94B05, 94B60
Key Words and Phrases: linear codes, matroid, rank function
- ⋅ 10:10 − 10:30 Quadratic generating functions and random walks (Sung-Tae Jin, Gi-Sang Cheon)
- 진성태*(성균관대), 천기상(성균관대)
Sung-Tae Jin*, Sungkyunkwan University, Gi-Sang Cheon, Sungkyunkwan University
Many familiar counting generating functions are algebraic of degree 2. Special well known cases include $C=1+zC^2, m=1+zm+z^2m^2$, and $r=1+zr+zr^2$ where $C, m$ and $r$ are the generating functions for the Catalan numbers, the Motzkin numbers, and the Schr\"oder numbers, respectively. In this talk we introduce general functional equations of degree 2 which include many familiar generating functions and their sequences. Further, we give their combinatorial interpretations by random walks on the lattice plane $\mathbb{Z \times Z}$. The concept of Riordan array leads to the functional equation of degree $r$.
2010 Mathematics Subject Classification: 05A15, 05C38
Key Words and Phrases: Riordan array, random walks
- ⋅ 10:30 − 11:00 Diffusion characters of graphs (Eun Youn Kim)
- 김은연(국가수리과학연구소)
Eun Youn Kim, Naional Institute for Mathematical Sciences
Combinatorial graphs and networks are used as mathematical models of a broad variety of systems including communications networks, brain connecttivity, food webs, or even to map out resource conflicts. The diffusion character matrix of a graph injects the vertices of a graph into Euclidean space so that Euclidean distances between vertices are closely tied to connectivity between those vertices in the graph. In this talk, diffusion characters and their associated matrices are defined, elementary properties are derived, and it is demonstrated that diffusion character matrices contain information not contained in the eigenvalues of the graph. Finally, the distance based on the diffusion character between vertices are defined and similarity of them and modularity of the graph are discussed.
2010 Mathematics Subject Classification: 05C50
Key Words and Phrases: matrix, graph, network, distance
- ⋅ 11:00 − 11:20 The competition number of a graph and the dimension of its hole space (Boram Park, Suh-Ryung Kim, Jung Yeun Lee and Yoshio Sano)
- 박보람*(서울대), 김서령(서울대), 이정연(국가수리과학연구소), 사노요시오(포항공대)
Boram Park*, Seoul National University, Suh-Ryung Kim, Seoul National University, Jung Yeun Lee, National Institute for Mathematical Sciences, Yoshio Sano, POSTECH
The competition graph of a digraph $D$ is a (simple undirected) graph which has the same vertex set as $D$ and has an edge between $x$ and $y$ if and only if there exists a vertex $v$ in $D$ such that $(x,v)$ and $(y,v)$ are arcs of $D$. For any graph $G$, $G$ together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number $k(G)$ of $G$ is the smallest number of such isolated vertices. In general, it is hard to compute the competition number $k(G)$ for a graph $G$ and it has been one of important research problems in the study of competition graphs to characterize a graph by its competition number. Recently, the relationship between the competition number and the number of holes of a graph is being studied. A hole of a graph is a cycle of length at least $4$ as an induced subgraph. In this paper, we propose the dimension of its hole space as an upper bound for the competition number of a graph and show that it is sharp for connected triangle-free graphs and present various classes graphs satisfying the inequality.
2010 Mathematics Subject Classification: 05C75
Key Words and Phrases: competition graph, competition number, cycle space, hole, hole space
- ⋅ 13:10 − 13:30 Spoiled child trees (Louis Shapiro)
- 루이스 샤피로(성균관대/ Howard University)
Louis Shapiro, SKKU and Howard University
The sequence starting 1, 2, 5, 15, 51, 188, ... is a close cousin of the Catalan numbers and shows up in a variety of settings. We will look at a few of these settings and show the Catalan connection. The most interesting new setting is that of spoiled child trees.
2010 Mathematics Subject Classification: 05A05
Key Words and Phrases: Catalan number, generating function, spoiled child tree
- ⋅ 13:30 − 13:50 $r$-Whitney numbers of the Dowling lattices (Ji-Hwan Jung, Gi-Sang Cheon)
- 정지환*(성균관대), 천기상(성균관대)
Ji-Hwan Jung*, Sungkyunkwan University, Gi-Sang Cheon, Sungkyunkwan University
Let $G$ be a finite group of order $m\ge1$. A Dowling lattice $Q_n(G)$ is the geometric lattice of rank $n$ over $G$. In this talk, we define the $r$-Whitney numbers of the first and second kind over $Q_n(G)$, respectively. This concept is a common generalization of the Whitney numbers and the $r$-Stirling numbers of both kinds. We give combinatorial interpretations for the $r$-Whitney numbers of the first and second kind respectively. In addition, we develop the $r$-Whitney Lah numbers from the classical Lah numbers. Finally, we define the the $r$-Dowling polynomials associated with the Dowling lattice and then we explore the interlacing property of their zeros.
2010 Mathematics Subject Classification: 05B35, 05A15
Key Words and Phrases: Dowling lattice, Whitney numbers, $r$-Stirling numbers, Lah numbers, Riordan matrix, Dowling polynomials
- ⋅ 14:50 − 15:20 Search for good linear codes using a simple extension algorithm (Sunghyu Han)
- 한성휴(한국기술교육대)
Sunghyu Han, Korea University of Technology and Education
Recently, there were several papers on finding good linear codes. Various methods were introduced to make good linear codes. In this paper, we give a simple iterative algorithm to make good linear codes. Using our algorithm we construct 86 codes which improve the minimum distance of previously best known linear codes for fixed parameters $n, k$.
2010 Mathematics Subject Classification: 94B05
Key Words and Phrases: best known linear codes, coding theory, extendable codes, linear codes
- ⋅ 15:20 − 15:50 MacWilliams duality and Gleason-type theorwm on self-dual bent functions (Jong Yoon Hyun, Heisook Lee and Yoonjin Lee)
- 현종윤*(이화여대 수리과학연구소), 이혜숙(이화여대), 이윤진(이화여대)
Jong Yoon Hyun*, Ewha Womans University, Heisook Lee, Ewha Womans University, Yoonjin Lee, Ewha Womans University
In this talk, we prove that the MacWilliams duality holds for bent functions, and by us- ing the MacWilliams duality, we prove the Gleason-type theorem on self-dual bent functions. As an application, we obtain a general explicit formula for an upper bound of the total number of self-dual bent functions.
2010 Mathematics Subject Classification: 94C10
Key Words and Phrases: self-dual bent functions
- Mathematics for Information Sciences II
- ⋅ 13:10 − 13:30 $-2-\sqrt2$ is a limit point for the smallest eigenvalues of regular graphs (Hyonju Yu)
- 유현주(포항공대)
Hyonju Yu, POSTECH
In this talk, we show that $-2-\sqrt2$ is a limit point for the smallest eigenvalues of regular graphs.
2010 Mathematics Subject Classification: 05E30
Key Words and Phrases:
- ⋅ 13:30 − 13:50 There are only finitely many distance-regular graphs with valency $\textbf{k}$ at least three, fixed ratio $\mathbf{\frac{k_2}{k}}$ and large diameter (Jongyook Park)
- 박종육(포항공대)
Jongyook Park, POSTECH
In this talk, we will show that for given positive integer $C$, there are only finitely many distance-regular graphs with valency $k$ at least three, diameter $D$ at least six and $\frac{k_2}{k}\leq C$. This extends a conjecture of Bannai and Ito.
2010 Mathematics Subject Classification: 05E30
Key Words and Phrases: distance-regular graphs
- ⋅ 14:50 − 15:20 Entringer families (Heesung Shin)
- 신희성(인하대)
Heesung Shin, Inha University
Andr\'e proved that the number of down-up permutations on $\{1, 2, ..., n\}$ is equal to the Euler number $E_n$. A refinement of Andr\'e's result was given by Entringer, who proved that counting down-up permutations according to the first element gives rise to Seidel's triangle $(E_{n,k})$ for computing the Euler numbers. Sequentially, using generating function method and induction, Poupard gave several further combinatorial interpretations for $E_{n,k}$ both in down-up permutations and increasing trees. Kuznetsov, Pak, and Postnikov have given more combinatorial interpretations of $E_{n,k}$ in the model of trees. The aim of this talk is to provide bijections between the different models for $E_{n,k}$ as well as some new interpretations. In particular, we give the first explicit one-to-one correspondence between Entringer's down-up permutation model and Poupard's increasing tree model.
2010 Mathematics Subject Classification: 05A05
Key Words and Phrases: bijection, Entringer, alternating permutation
- Mathematical Education
- ⋅ 14:00 − 14:30 A research on mathematician Sang-Seol Lee's contribution to the natural science in 19th century Korea (Jae Hwa Lee, Sang-Gu Lee, Yongkeun Son and Chae Sik Kim)
- 이재화*(성균관대), 이상구(성균관대), 손용근(성균관대), 김채식(성균관대)
Jae Hwa Lee*, Sungkyunkwan University, Sang-Gu Lee, Sungkyunkwan University, Yongkeun Son, Sungkyunkwan University, Chae Sik Kim, Sungkyunkwan University
Sang-Seol LEE wrote a manuscript HwaHakGyeMongCho with a brush in late 19th century. HwaHakGyeMongCho was transcribed from Science Primers: Chemistry (written by H. E. Roscoe), which is translated into Chinese by Joseph Edkins in 1886. LEE did not copy original writing exactly, but he understood the contents of each chapter and sections, then summarized it. In this paper, we discuss the significance of this book.
2010 Mathematics Subject Classification: 01A55, 01A90
Key Words and Phrases: Sang-Seol LEE, HwaHakGyeMongCho, science primers: chemistry, Henry Enfield Roscoe, Joseph Edkins, Robert Hart, 16 primers for western knowledge
- ⋅ 14:50 − 15:20 Effects of self-management skills on the mathematical academic achievement using hierarchical linear models (Sung Yeun Kim, Won Choi and In-Hyoun Kim)
- 김성연*(인천대), 최 원(인천대), 김인현(인천대)
Sung Yeun Kim*, The University of Incheon, Won Choi, The University of Incheon, In-Hyoun Kim, The University of Incheon
The purpose of this study was to examine effects of self-management skills on middle school students' achievements in mathematics using hierarchical linear growth models. This study used Korea Educational Longitudinal Study (KELS) from 2005 to 2007 which was composed of 6908 students that were obtained using a stratified cluster random sampling method from 703,914 middle school students in South Korea in 2005. Variables that were related to self-management skills were managing time, managing space and demanding social help. The main results of this study were as follows. First, among subscales of self-management skills, demanding social help skills was effective on enhancement of the mathematical academic achievement. Second, there was statistically significant difference in all subscales of self-management skills between high-achieving students who had academic achievement more than 70% and low-achieving students who had academic achievement less than 30%. In other words, high-achieving students had better self-management skills than low-achieving students.
2010 Mathematics Subject Classification: 97D60
Key Words and Phrases: self-management skills, hierarchical linear growth model, managing time, managing space, demanding social help
- Cryptography
- ⋅ 13:10 − 13:40 Pairing-friendly curves with minimal security loss (Cheol-Min Park, Hyang-Sook Lee)
- 박철민*(국가수리과학연구소), 이향숙(이화여대)
Cheol-Min Park*, National Institute for Mathematical Sciences, Hyang-Sook Lee, Ewha womans University
In ICISC 2007, Comuta et al. showed that among the methods for constructing pairing-friendly curves, those using cyclotomic polynomials (e.g., the BW method and the FST method) are affected by Cheon’s algorithm. This paper proposes a method for searching parameters of pairing-friendly elliptic curves that induces minimal security loss by Cheon’s algorithm. We also provide a sample set of parameters of BN-curves, FST-curves and KSS-curves for pairing-based cryptography.
2010 Mathematics Subject Classification: 11T71
Key Words and Phrases: cyclotomic polynomial, pairing-friendly elliptic curve, Cheon’s algorithm, strong diffie-hellman problem
- ⋅ 13:40 − 14:00 Speeding up final exponentiation in pairing computation (Taechan Kim, Jung Hee Cheon)
- 김태찬*(서울대), 천정희(서울대)
Taechan Kim*, Seoul National University, Jung Hee Cheon, Seoul National University
Pairing computation is composed of Miller's step and final exponentiation step. In many cases, shortening the loop length in Miller's algorithm is the main focus to obtain efficient pairing. Here, we exploit the general approach to obtain efficiently computable final exponent.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: pairing, Miller's algorithm, final exponentiation
- ⋅ 14:00 − 14:20 5-term GP of $O(N^{2/3})$ in special case (Namhun Koo, Gooc Hwa Jo and Soonhak Kwon)
- 구남훈*(성균관대), 조국화(성균관대), 권순학(성균관대)
Namhun Koo*, Sungkyunkwan University, Gooc Hwa Jo, Sungkyunkwan University, Soonhak Kwon, Sungkyunkwan University
The number field sieve(NFS) algorithm is the (asymptotically) fastest known algorithm to factor large integer $N$. The most important part of NFS is how to select good polynomial pair. Montgomery proposed a method to select two quadratic sieving polynomial using LLL algorithm and a geometric progression(GP) of 3-term with size $O(N^{1/2})$. To generate 2 cubic sieving polynomials of coefficient size $O(N^{1/6})$, we need a 5-term GP with size $O(N^{2/3})$. But it is still an open problem to find a GP generally, when $N$ is large. In this presentation, we will give several special form of 5-term GP of $O(N^{2/3})$ and some examples.
2010 Mathematics Subject Classification: 11A51
Key Words and Phrases: number field sieve(NFS), factorization, nonlinear polynomial selection
- ⋅ 14:20 − 14:40 Privacy-preserving top-$k$ aggregation (Myungsun Kim, Jung Hee Cheon, Abedelaziz Mohaisen and Yongdae Kim)
- 김명선*(서울대), 천정희(서울대), Abedelaziz Mohaisen(Univ. of Minnesota), 김용대(Univ. of Minnesota)
Myungsun Kim*, Seoul National University, Jung Hee Cheon, Seoul National University, Abedelaziz Mohaisen, University of Minnesota, Yongdae Kim, University of Minnesota
In this paper we revise the privacy-preserving top-$k$ data aggregation problem and introduce the new notion of owner privacy, which is necessitated by both theoretical and practical needs in many distributed computing settings. We show that a recently proposed work (by Applebaum et al., PETS 2010) does not satisfy this notion, other than being insecure against a simple coalition attack. To realize the notion of owner privacy, and to strike a balance between efficiency and functionality, we devise a cryptographic construction that comes into two forms: fully decentralized simple construction and its practical variant that is semi-decentralized. We analyze the complexity of our construction, and show it is comparable---and sometimes superior---to other work in literature albeit outperforming them in the sense that it meets the new defined notion of owner privacy.
2010 Mathematics Subject Classification: 11T71
Key Words and Phrases: privacy-preserving, top-k
- ⋅ 14:50 − 15:20 The square root Diffie-Hellman problem (Dongyoung Roh, Sang Geun Hahn)
- 노동영*(국가수리과학연구소), 한상근(카이스트)
Dongyoung Roh*, National Institute for Mathematical Sciences, Sang Geun Hahn, KAIST
Many cryptographic schemes are based on the computationally hard problems. The computational Diffie-Hellman problem is the most well-known hard problem. And there are many variants of it. Two of them are the square Diffie-Hellman problem and the square root Diffie-Hellman problem. There were no known reductions from one problem to the other in either way. In this paper we show that these two problems are polynomial time equivalent under a certain condition. But this condition is weak and almost all of the parameters of cryptographic schemes satisfy this condition. Therefore our reductions are valid for almost all of the cryptographic schemes.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: discrete logarithm problem, computational Diffie-Hellman problem, square Diffie-Hellman problem, square root Diffie-Hellman problem
- Poster Session
- ⋅ 14:50 − 15:20 On wreath products of cyclic schemes (Kijung Kim)
- 김기정(포항공대)
Kijung Kim, POSTECH
The Terwilliger algebra was first introduced by terwilliger as new tool for commutative association schemes. In general, this algebra is non-commutative, finite dimensional and semi-simple $\mathbb{C}$-algebra. G. Bhattacharyya, S.Y. Song and R. Tanaka began to study the wreath product of one-class association schemes. It was shown that all irreducible modules except for the primary module of the Terwilliger algebra are one-dimensional. Moreover, R. Tanaka proved that the only class of association schemes coming as the wreath product of one-class association schemes and group schemes of finite abelian groups satisfies this property. Recently, S.Y. Song and B. Xu gave a complete structural description of the Bose-Mesner algebra and Terwilliger algebra for wreath products of one-class association schemes. In this paper, we describe explicitly the algebraic structure of the Terwilliger algebra of wreath products of cyclic schemes.
2010 Mathematics Subject Classification: 05E30
Key Words and Phrases: cyclic scheme, wreath product
- ⋅ 14:50 − 15:20 Generalized nonlinear variational-like inequalities with generalized monotone multivalued mappings in Banach spaces (Jae Ug jeong, Soo Hwan Kim)
- 정재욱*(동의대), 김수환(동의대)
Jae Ug jeong*, Dongeui University, Soo Hwan Kim, Dongeui University
In this paper, we study a new clsee of generalized nonlinear variational-like inequalities in reflexive Banach spaces. By using the KKM technique and the concept of the Hausdorff metric, we obatin some existence results for generalized nonlinear variational-like inequalities with generalized monotone multi-valued mappings in Banach spaces. These results improve and generalized many known results in recent literature.
2010 Mathematics Subject Classification: 49A29, 49J40
Key Words and Phrases: Variational-like inequality, generalized monotone multi-valued mapping, KKM mapping, Hausdorff metric, reflexive Banach space
- ⋅ 14:50 − 15:20 A study on the comparison of classes conducted by modified Moore method (Seong-A Kim)
- 김성아(동국대)
Seong-A Kim, Dongguk University
We have examined the modified Moore methods that were applied to college mathematics courses in several researches. We introduce, compare and analyze the concrete teaching methods that the researchers conducted in various modified Moore methods, and propose the appropriate form of modified Moore method most suitable for the present situations of the mathematics and related departments in Korea.
2010 Mathematics Subject Classification: 97C70
Key Words and Phrases: Modified Moore method, Self-directed learning