KMS Meeting - Programs and Abstracts
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- Plenary Lecture
- ⋅ 16:50 − 17:40 Chair: YoungJu Choie (POSTECH)
- ⋅ PL 16:50 − 17:40 Quantum invariants of knots and number theory (Don Zagier)
- Don Zagier, Max Planck Institute for Mathematics(Germany), Coll\`ege de France(France), Visiting Professor at POSTECH
Ideas coming originally from quantum field theory, and in particular from the theory of quantum groups, have led to the definition of many new invariants of knots and 3-manifolds. It turns out that these invariants have unexpectedly deep number-theoretical properties related in particular to algebraic $K$-theory and, even more surprisingly, to the theory of modular forms, though most of these relationships are still only conjectural. In the lecture I will describe some of the examples that have led to these conjectures and some of the ideas behind them.
2010 Mathematics Subject Classification:
Key Words and Phrases: quantum invariants
- Invited Lectures
- ⋅ 10:10 − 10:50 [Probability and Statistics] Chair: Hun Hee Lee (Chungbuk National University)
- ⋅ IL 10:10 − 10:50 Distribution of eigenvalues of random normal matrices near the edge of the spectrum (Nam-Gyu Kang, Nikolai Makarov and Yacin Ameur)
- 강남규*(서울대), Nikolai Makarov(Caltech), Yacin Ameur(Lund Univ.)
Nam-Gyu Kang*, Seoul National University, Nikolai Makarov, Caltech, Yacin Ameur, Lund University
Microscopic properties of eigenvalues of random normal matrices change drastically in a narrow belt around the edge of the spectrum. I present an elementary method to prove Borodin and Sinclair's theorem on the scaling limit of correlation kernels for the soft-edge Ginibre ensemble. This method gives new result for the hard-edge Ginibre ensemble. After a discussion of the general properties of this scaling limit, I state a universality conjecture and provide arguments to support it. This is a joint work with Y. Ameur and N. Makarov.
2010 Mathematics Subject Classification: 60B20
Key Words and Phrases: random normal matrices, Ginibre ensemble, universality conjecture
- ⋅ 11:00 − 11:40 [Cryptography] Chair: Hyang-Sook Lee (Ewha Womans University)
- ⋅ IL 11:00 − 11:40 Security of the Even-Mansour scheme and its generalization (Jooyoung Lee)
- 이주영(세종대)
Jooyoung Lee, Sejong University
In this talk, we are going to survey the recent results on the Even-Mansour scheme and its generalization. We give an alternative security proof for the 2-round key alternating construction, and show that the key size can be reduced from 3 keys to 2 keys.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: blockcipher, provable security, Even-Mansour scheme
- ⋅ 13:20 − 14:00 [ Chair: Minkyu Kwak (Chonnam National University)
- ⋅ IL 13:20 − 14:00 Nonlinear degenerate elliptic equation and its application (Myoungjean Bae)
- 배명진(포항공대)
Myoungjean Bae, POSTECH
Motion of inviscid compressible flow is governed by Euler system. In self-similar flow of Euler system, a sonic arc appears when a shock occurs. Mathematically, a sonic arc is a boundary on which ellipticity or hyperbolicity of a nonlinear equation degenerates. This talk will present a regularity result of solutions to a class of nonlinear degenerate elliptic equations, and explain how this result is applied to establish optimal regularity of weak solutions to Euler system for self-similar flow on sonic arcs.
2010 Mathematics Subject Classification: 35J60, 35M10, 76H05
Key Words and Phrases: Euler system, self-similar, sonic arc, nonlinear degenerate elliptic equation, optimal regularity
- ⋅ 14:10 − 14:50 [Geometry] Chair: Hong-Jong Kim (Seoul National University)
- ⋅ IL 14:10 − 14:50 Rigidities, geometric structures, and Lie algebra cohomologies (Jaehyun Hong)
- 홍재현(서울대)
Jaehyun Hong, Seoul National University
I will talk about how to solve a rigidity problem by transforming it into an equivalence problem of geometric structures and by computing the Lie algebra cohomology related to it.
2010 Mathematics Subject Classification: 53
Key Words and Phrases: Euler system, self-similar, sonic arc, nonlinear degenerate elliptic equation, optimal regularity
- ⋅ 14:10 − 14:50 [Mathematics for Information Sciences] Chair: Seunghyun Seo (Kangwon National University)
- ⋅ IL 14:10 − 14:50 Skew Schur P-functions (Soojin Cho)
- 조수진(아주대)
Soojin Cho, Ajou University
There are two combinatorial models for Schur P-functions; marked shifted tableaux and semistandard decomposition tableaux. Skew marked shifted tableaux can be defined in a natural way to define skew Schur P-functions, yet a natural definition of skew semistandard decomposition tableaux is not known. A model for the skew semistandard decomposition tableaux will be given while the falsity of the conjecture by L. Serrano is explained.
2010 Mathematics Subject Classification: 05E05, 05E10
Key Words and Phrases: Schur P-functions, shifted tableaux, semistandard decomposition tableaux
- ⋅ 15:10 − 15:50 [Topology] Chair: Jong Bum Lee (Sogang University)
- ⋅ IL 15:10 − 15:50 A cycle of frustration and rewarding moments (Ki Hyoung Ko)
- 고기형(카이스트)
Ki Hyoung Ko, KAIST
I began my career as a mathematician in 1984 after PhD at Brandeis. It is a great pleasure to retrospect what I have done since then as a researcher in theory of knots and braids -- especially those that are memorable to me. I thank KMS for providing me this opportunity.
2010 Mathematics Subject Classification: 57M25, 20F36
Key Words and Phrases: knot, braid, link
- ⋅ 15:10 − 15:50 [Applied Mathematics] Chair: Do Young Kwak (KAIST)
- ⋅ IL 15:10 − 15:50 Error estimates for the second order semi-discrete stabilized Gauge-Uzawa method for the Navier-Stokes equations (Jae-Hong Pyo)
- 표재홍(강원대)
Jae-Hong Pyo, Kangwon National University
The Gauge-Uzawa method [GUM], which is a projection type algorithm to solve the time depend Navier-Stokes equations, has been constructed in \cite{pyo:gau_uza} and enhanced in \cite{pyo:bou,pyo:density} to apply to more complicated problems. Even though GUM possesses many advantages theoretically and numerically, the studies on GUM have been limited on the first order backward Euler scheme except normal mode error estimate in \cite{pyo:normal}. The goal of this paper is to research the 2nd order GUM. Because the classical 2nd order GUM which is studied in \cite{pyo:normal} needs rather strong stability condition, we modify GUM to be unconditionally stable method using BDF2 time marching. The stabilized GUM is equivalent to the rotational form of pressure correction method and the errors are already estimated in \cite{GS04} for the Stokes equations. In this paper, we will evaluate errors of the stabilized GUM for the Navier-Stokes equations. We also prove that the stabilized GUM is an unconditionally stable method for the Naiver-Stokes equations. So we conclude that the rotational form of pressure correction method in \cite{GS04} is also unconditionally stable scheme and that the accuracy results in \cite{GS04} are valid for the Navier-Stokes equations.
\begin{thebibliography}{99}
\bibitem{GS04} J. L. Guermond and J. Shen {\em On the error estimates of rotational pressure-correction projection methods}, Math. Comp. {\bf 73} (2004), 1719--1737.
\bibitem{pyo:gau_uza} R. H. Nochetto and J.-H. Pyo, {\em A finite element Gauge-Uzawa method. Part I : the Navier-Stokes equations}, SIAM J. Numer. Anal. {\bf 43} (2005), 1043--1068.
\bibitem{pyo:bou} R. H. Nochetto and J.-H. Pyo, {\em A finite element Gauge-Uzawa method. Part II : Boussinesq Equations}, Math. Models Methods Appl. Sci. {\bf 16} (2006), 1599--1626.
\bibitem{pyo:normal} J.-H. Pyo and J. Shen, {\em Normal Mode Analysis of Second-order Projection Methods for Incompressible Flows}, Discrete Contin. Dyn. Syst. Ser. B. {\bf 5} (2005), 817--840.
\bibitem{pyo:density} J.-H. Pyo and J. Shen, {\em Gauge Uzawa methods for incompressible flows with Variable Density}, J. Comput. Phys. {\bf 211} (2007), 181--197.
\end{thebibliography}
2010 Mathematics Subject Classification: 65M60
Key Words and Phrases: Gauge-Uzawa method, projection methods, Navier-Stokes equations
- ⋅ 15:10 − 15:50 [Mathematical Education] Chair: Dal-Won Park (Kongju National University)
- ⋅ IL 15:10 − 15:50 Some overview of Sage and its use in teaching linear algebra, abstract algebra and graph theory with open textbooks (Robert Beezer)
- Robert Beezer, University of Puget Sound
Sage is an open source software system for advanced mathematics, and is a natural choice for use in teaching. I will describe its use in teaching linear algebra, abstract algebra and graph theory, along with open textbooks.
2010 Mathematics Subject Classification: 97Uxx
Key Words and Phrases: Sage, Sage-Math, CAS, algebra
- Recent Trends in Number Theory
- ⋅ 13:20 − 13:50 [Invited Lecture-Algebra] Chair: Jong Hae Keum (KIAS)
- ⋅ 13:20 − 13:50 Application of the strong Artin conjecture to the class number problem (Henry Kim)
- Henry Kim, University of Toronto
As an application of the strong Artin conjecture, we exhibit unconditionally a family of number fields with extreme class numbers whose normal closures have $S_5, S_4, A_4$, and dihedral groups $D_n, n=3,4,5, $ and cyclic groups $C_n, n=4,5,6$, as their Galois groups. This is a joint work with P. J. Cho.
2010 Mathematics Subject Classification: 11R29, 11R42
Key Words and Phrases: class number, strong Artin conjecture, automorphic L-functions
- ⋅ 13:55 − 15:00 Chair: Ja Kyung Koo (KAIST)
- ⋅ 13:55 − 14:25 Traces of singular moduli on congruence subgroups (Chang Heon Kim)
- 김창헌(한양대)
Chang Heon Kim, Hanyang University
Singular moduli are special values of modular functions at imaginary quadratic arguments in the upper half plane which play important roles in number theory. After Zagier proved that the generating function of the traces of singular moduli for the classical modular invariant $j$ is a weakly holomorphic modular form of weight 3/2, various arithmetic properties of the traces of singular moduli have been obtained mostly on the full modular group. In this talk I will explain some results for traces of singular moduli on congruence subgroups.
2010 Mathematics Subject Classification: 11F03, 11F30, 11F37
Key Words and Phrases: modular function, congruence subgroups
- ⋅ 14:30 − 15:00 Larsen's conjecture and some applications of the Hales-Jewett theorem to field arithmetic (Bo-Hae Im, Michael Larsen)
- 임보해*(중앙대), Michael Larsen(Indiana Univ.)
Bo-Hae Im*, Chung-Ang University, Michael Larsen, Indiana University
Let $K$ be a field whose absolute Galois group is finitely generated. If $K$ neither finite nor of characteristic $2$, then every hyperelliptic curve over $K$ with all of its Weierstrass points defined over $K$ has infinitely many $K$-points. If, in addition, $K$ is not locally finite, then every elliptic curve over $K$ with all of its $2$-torsion rational has infinite rank over $K$. These and similar results are deduced from the Hales-Jewett theorem.
2010 Mathematics Subject Classification: 05D10
Key Words and Phrases: the Hales-Jewett theorem, elliptic curves
- ⋅ 15:10 − 15:40 [ Chair: Youn-Seo Choi (KIAS)
- ⋅ 15:10 − 15:40 Special values of Hecke $L$-function of real quadratic fields and class number problem of real quadratic fields (Jungyun Lee, Byungheup Jun)
- 이정연*(고등과학원), 전병흡(고등과학원)
Jungyun Lee*, KIAS, Byungheup Jun, KIAS
It is beyond our knowledge if there exist infinitely many real quadratic fields with class number one. This is mainly due to regulators that are not under control of discriminants. So most of the achievements in class number problem of real quadratic fields are in the restricted form where the regulator is controlled in a way. A best known such family is namely the Richaud-Degert type. In this family, as in the imaginary quadratic fields case, it has been known that there are only finitely many real quadratic fields with given class number. Moreover under the assumption of the generalized Riemann hypothesis, for a given class number, an explicit upper bound of discriminant of Richaud-Degert type can be obtained. Recent progress made by Biro, Byeon and myself suggests a way to obtain explicit upper bound of discriminant for some families of Richaud-Degert type with class number one, not assuming the generalized Riemann hypothesis. For this we used a property that the values of Hecke-L-function at $s=0$ for some families of Richaud-Degert type are expressed as quasi-polynomial of degree 1. Later I discovered that such behavior is closely related to the expression of continued fraction expansion. If we consider the following family of real quadratic fields $K_n=\mathbb{Q}(\sqrt{f(n)})$ for $f(x)\in \mathbb{Z}[x]$ and an ideal $\bf{b}_n$ of $K_n$ such that $$\bf{b}_n^{1}\sim[1,\omega(n)]$$ and $$\omega(n)-1=[a_0(n),a_1(n),\ldots,a_{s-1}(n)]$$ with $a_i(x)\in\mathbb{Z}[X]$. We find that if $a_i(x)$ are all polynomial functions with degree $\leq d$ then the special values $L_{K_n}(0,\chi_n,\bf{b}_n)$ are expressed as quasi polynomial in $n$ of degree $\leq d$. Moreover the family $K_n$ has the regulator in control because of fixed length of continued fraction. This observation suggests families satisfying such a condition generalizes the R-D type. This is a joint work with Byungheup Jun.
2010 Mathematics Subject Classification: 11R42, 11R11, 11R80, 11R29, 11A55
Key Words and Phrases: class number, real quadratic, Hecke-L-function
- ⋅ 15:40 − 16:10 Chair: Youn-Seo Choi (KIAS)
- ⋅ 15:40 − 16:10 A $p$-adic Weil representation of $\operatorname{GL}_2(\mathbb{Z}_p)$ (Jeehoon Park)
- 박지훈(포항공대)
Jeehoon Park, POSTECH
Let $E$ be an unramified quadratic extension of $\mathbb{Q}_p$ and $\mathcal{O}$ be its ring of integers. The goal of this talk is to find a $p$-adic analogue (i.e., an admissible $p$-adic Banach infinite dimensional representation of $\operatorname{GL}_2(\mathbb{Z}_p)$) of the $\mathbb{C}$-linear irreducible finite-dimensional Weil representation $\pi_{m}(\theta)$ of $\operatorname{GL}_2(\mathbb{Z}_p)$, which is constructed from a smooth character $\theta$ of $\mathbb{O}^*$ via the theta correspondence between the similitude orthogonal group $\operatorname{GO}(2)$ and $\operatorname{GL}(2)$.
2010 Mathematics Subject Classification: 11F27, 22E50
Key Words and Phrases: $p$-adic Banach space representation, $p$-adic theta correspondence, $p$-adic Weil representation
- Algebra I
- ⋅ 09:30 − 11:10 Chair: Bo-Hae Im (Chung-Ang University)
- ⋅ AL I 09:30 − 10:00 Coefficient fields and non-vanishing mod $p$ of special modular $L$-values (Hae-Sang Sun)
- 선해상(충북대)
Hae-Sang Sun, Chungbuk National University
By modifying a result of Luo-Ramakrishnan, we show that the coefficient field of a newform can be constructed from modular symbols. Using this, we obtain a result on the non-vanishing modulo prime of special modular $L$-values with cyclotomic twists.
2010 Mathematics Subject Classification: 11Rxx
Key Words and Phrases: modular form, $L$-function, non-vanishing mod $p$, cyclotomic twist
- ⋅ AL I 10:10 − 10:40 Injective modules over Pr\"ufer $v$-multiplication domains (Hwankoo Kim)
- 김환구(호서대)
Hwankoo Kim, Hoseo University
Let $R$ be an integral domain with quotient field $F$. It is shown that $R$ is a strongly discrete Pr\"ufer $v$-multiplication domain if and only if there exists a bijection between the set of the prime $w$-ideals and the set of isomorphism classes of GV-torsionfree indecomposable injective $R$-modules. It is also shown that the $w$-closure of any GV-torsionfree homomorphic image of $F$ is injective if and only if $R$ is a Pr\"ufer $v$-multiplication domain satisfying an almost maximality-type property.
2010 Mathematics Subject Classification: 13A15, 13Bxx, 13Cxx, 13D07, 13Dxx, 13G05
Key Words and Phrases: injective module, indecomposable injective module, GV-torsionfree, semidivisoria, h$_ \mathcal{M}$-local domain, P$v$MD, strongly discrete P$v$MD, almost maximal
- ⋅ AL I 10:40 − 11:10 Star cover number and their preservers (Seok Zun Song)
- 송석준(제주대)
Seok Zun Song, Jeju National University
The star cover number of a matrix $B$ is the minimum number of full stars whose sum dominates $B$. In this paper we investigate linear operators that preserve star cover number of symmetric matrices with zero diagonal. We compare this matrix function with term rank function and we show that a linear operator $T$ on symmetric matrices with zero diagonal preserves star cover number if and only if $T$ preserves star cover number 1 and 2. Other characterizations of star cover number preservers are also given.
2010 Mathematics Subject Classification: 15A86, 15A03, 15A04
Key Words and Phrases: star cover, star cover number, term rank, linear operator
- Algebra II
- ⋅ 09:00 − 11:10 Chair: Yongnam Lee (Sogang University)
- ⋅ AL II 09:00 − 09:20 Quotients of surfaces of general type with $p_{g}=q=0$ by an involution when the bicanonical map is composed with the involution (YongJoo Shin)
- 신용주(서강대)
YongJoo Shin, Sogang University
Let $S$ be a minimal surface of general type with $p_g(S)=q(S)=0$ having an involution $\sigma$. It is well known that if the bicanonical map $\varphi$ of $S$ is composed with $\sigma$ then the quotient $S/\sigma$ is rational or birational to an Enriques surface. In this talk, we review the known examples and discuss the cases when the quotients are rational.
2010 Mathematics Subject Classification: 14J29
Key Words and Phrases: surface of general type, involution
- ⋅ AL II 09:20 − 09:40 Numerical Godeaux surfaces with an involution in positive characteristic (Soon Young Kim)
- 김순영(서강대)
Soon Young Kim, Sogang University
A numerical Godeaux surface is a minimal surface of general type with $\chi(\mathcal{O}_{X})= K_{X}^{2}=1$ . Such a surface is called classical if $p_{g}=h^{0,1}=0$ and non-classical if $p_{g}=h^{0,1}=1$. The classical Godeaux surface with an involution over $C$ is originally studied by JongHae Keum and Yongnam Lee, and completely classified by Calabri, Ciliberto, and Mendes Lopes. In the talk, we work in positive characteristic except char 2. We obtain the same results as the case in characteristic 0 by careful study of the Kawamata-Viehweg vanishing theorem and their numerical calculation in positive characteristic. Also we discuss the examples constructed by Miranda and Liedtke in char 5.
2010 Mathematics Subject Classification: 14J29
Key Words and Phrases: Godeaux surfaces, positive characteristic
- ⋅ AL II 09:40 − 10:00 Some application of the complex multiplication of abelian varieties (Dong Sung Yoon, Ja Kyung Koo)
- 윤동성*(카이스트), 구자경(카이스트)
Dong Sung Yoon*, KAIST, Ja Kyung Koo, KAIST
The fundamental problem of algebraic number theory is to describe the fields of algebraic numbers. The work of Galois made it clear that field extensions are controlled by the Galois groups. The most classical situation is when the group in question is abelian and Hilbert's 12th problem asks what sort of transcendental functions and algebraic numbers are necessary to construct all abelian extensions of given algebraic number fields. The classical theory of complex multiplication of elliptic curves answers the question for the case of any imaginary quadratic field, by using special values of modular functions and elliptic functions. Shimura extended this to CM fields using the abstract theory of abelian varieties. In this talk, we will consider the case of cyclotomic fields. Let $\ell$ and $p$ be odd primes. For a positive integer $\mu$ let $k_\mu$ be the ray class field of $k=\mathbb{Q}(e^\frac{2\pi i}{\ell})$ modulo $2p^\mu$. We present certain class fields $K_\mu$ of $k$ such that $k_\mu\leq K_\mu\leq k_{\mu+1}$, and find the degree of $K_\mu/k_\mu$ explicitly. And by using Shimura's reciprocity law we also construct generators of the field $K_\mu$ over $k_\mu$ in terms of special values of theta constants.
2010 Mathematics Subject Classification: 11G15
Key Words and Phrases: complex multiplication, Siegel modular forms, theta functions
- ⋅ AL II 10:10 − 10:40 Classification of secant defective manifolds near the extremal case (Kangjin Han)
- 한강진(고등과학원)
Kangjin Han, KIAS
Let $X\subset \mathbb{P}^N$ be a nondegenerate irreducible closed subvariety of dimension $n$ over the field of complex numbers and let $SX\subset\mathbb{P}^N$ be its secant variety. $X\subset\mathbb{P}^N$ is called `secant defective' if $\dim(SX)$ is strictly less than the expected dimension $2n+1$. F. L. Zak showed that for a secant defective manifold necessarily $N\le{n+2 \choose n}-1$ and that the Veronese variety $v_2(\mathbb{P}^n)$ is the only boundary case. Recently R. Mu$\tilde{\textrm{n}}$oz, J. C. Sierra, and L. E. Sol\'a Conde classified secant defective varieties next to this extremal case. In this paper, we will consider secant defective manifolds $X\subset\mathbb{P}^N$ of dimension $n$ with $N={n+2 \choose n}-1-\epsilon$ for $\epsilon\ge0$. First, we will prove that $X$ is a $LQEL$-manifold of type $\delta=1$ for $\epsilon\le n-2$ by showing that the tangential behavior of $X$ is good enough to apply Scorza lemma. Then we will completely describe the above manifolds by using the classification of conic-connected manifolds given by P. Ionescu and F. Russo. Our method generalizes above previous results.
2010 Mathematics Subject Classification: 14MXX, 14NXX, 14M22
Key Words and Phrases: secant defective, conic-connected, tangential projection, Scorza lemma, second fundamental form
- ⋅ AL II 10:40 − 11:10 The category of Lie algebroids for generalized complex geometry (Hoil Kim)
- 김호일(경북대)
Hoil Kim, Kyungpook National University
We describe the category of Lie algebroids containing the generalized complex structures and show the cycles, K theory(bundles), and Lie algebroids cohomology and corresponding Chern maps and cycle maps. Finally we compare it with the category of Lie groupoids and explain the behaviour of mirror symmetry in this context.
2010 Mathematics Subject Classification: 14MXX, 14NXX, 14M22
Key Words and Phrases: Lie algebroids, Lie groupoids, cycles, Lie algebroid cohomology, Chern map, K theory
- Analysis I
- ⋅ 09:00 − 10:00 Chair: Sunggeum Hong (Chosun University)
- ⋅ AN I 09:00 − 09:30 Self-reciprocal polynomials with related maximal zeros (Seon-Hong Kim, Jaegug Bae)
- 김선홍*(숙명여대), 배재국(한국해양대)
Seon-Hong Kim*, Sookmyung Women's University, Jaegug Bae, Korea Maritime University
For each real number $n>6$, we prove that there is a sequence $\{ p_k(n,z)\}_{k=1}^{\infty}$ of fourth degree self-reciprocal polynomials such that the zeros of $p_k(n,z)$ are all simple and real, and every $\, p_{k+1}(n,z) $ has the largest (in modulus) zero $\alpha \beta$ where $\alpha$ and $\beta$ are the first and the second largest (in modulus) zeros of $p_{k}(n,z)$, respectively. One such sequence is given by $p_{k}(n,z)$ so that $$ p_{k}(n,z)= z^4 -q_{k-1}(n)\,z^3+(q_k(n)+2)\,z^2-q_{k-1}(n)\,z+1, $$ where $q_0(n)=1$ and other $q_k(n)$'s are polynomials in $n$ defined by the severely nonlinear recurrence $$ 4q_{2m-1}(n)=q_{2m-2}^{2}(n)-(4n+1)\prod_{j=0}^{m-2}q_{2j}^{2}(n), $$ and $$ 4q_{2m}(n)=q_{2m-1}^{2}(n)-(n-2)(n-6)\prod_{j=0}^{m-2}q_{2j+1}^{2}(n) $$ for $m\ge1$, with the usual empty product conventions, i.e., $\prod_{j=0}^{-1}b_j\,=\,1$. Moreover we show that such auxiliary polynomials $q_k$ are some sort of Chebyshev polynomials.
2010 Mathematics Subject Classification: 11B83, 30C15
Key Words and Phrases: self-reciprocal polynomials, polynomials, sequences
- ⋅ AN I 09:30 − 10:00 A time-dependent conditional Fourier-Feynman transform and convolution product on a function 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 $X_{n+1}: C[0,t]\to \mathbb R^{n+2}$ by \begin{eqnarray*} X_{n+1}(x) = (x(t_0), x(t_1), \ldots, x (t_n), x(t_{n+1})), \end{eqnarray*} where $ 0=t_0 < t_1$ $< \cdots < t_n < t_{n+1}$ is a partition of $[0,t]$. In the present talk, using a simple formula for the conditional expectation with the conditioning function $X_{n+1}$, we evaluate the $L_p(1\le p \le \infty)$-analytic conditional Fourier-Feynman transforms and the conditional convolution products of the cylinder functions which have the form, \begin{eqnarray*} f((v_1,x), \ldots, (v_r,x)) \text{ for } x\in C[0,t], \end{eqnarray*} where $\{v_1, \ldots, v_r\}$ is an orthonormal subset of $L_2[0,t]$ and $f\in L_p(\mathbb R^r)$. We then investigate several relationships between the conditional Fourier-Feynman transforms and the conditional convolution products of the cylinder functions.
2010 Mathematics Subject Classification: 28C20
Key Words and Phrases: analogue of Wiener measure, analytic conditional Feynman integral, analytic conditional Fourier-Feynman transform, analytic conditional Wiener integral, conditional convolution product
- ⋅ 10:10 − 11:40 Chair: ChanWoo Yang (Korea University)
- ⋅ AN I 10:10 − 10:40 Standing-waves with a small energy/charge ratio (Daniele Garrisi)
- Daniele Garrisi, Inha University
Standing-waves solutions to the non-linear Klein-Gordon equation \[ u_{tt} - \Delta u + u + f(u) = 0 \] with small energy/charge ratio (hylomorphic charge) exhibits a non-dispersion behaviour. We discuss general hypotheses on the non-linear term for the existence of solutions with small hylomorphic charge to a system of non-linear Klein-Gordon equations and their dispersive behaviour.
2010 Mathematics Subject Classification: 35A15, 35J50, 37K40
Key Words and Phrases: standing-waves, dispersion
- ⋅ AN I 10:40 − 11:10 Some properties of Chebyshev polynomials (Seon-Hong Kim)
- 김선홍(숙명여대)
Seon-Hong Kim, Sookmyung Women's University
In this talk we obtain some new bounds for Chebyshev polynomials and their analogues. They give answers about zero distributions of certain sums of Chebyshev polynomials and their analogues as by-products of such bounds. Finally we get an interesting property about integrals of the sums of Chebyshev polynomials.
2010 Mathematics Subject Classification: 30C15, 26C10
Key Words and Phrases: Chebyshev polynomials, bounds, sums, zeros
- ⋅ AN I 11:10 − 11:40 $h$-stability of nonlinear perturbed differential systems (Yoon Hoe Gu, Dong Man Im and Hyun Muk Lee)
- 구윤회*(한서대), 임동만(청주대), 이현묵(한밭대)
Yoon Hoe Gu*, Hanseo University, Dong Man Im, Cheongju University, Hyun Muk Lee, Hanbat University
In this talk we recall the definitions of some stability. Then we present some results about stability and boundedness of solutions of nonlinear differential equations.
2010 Mathematics Subject Classification: 34D10
Key Words and Phrases: $h$-stable, $t$-infinity similarity, nonlinear nonautonomous system
- ⋅ 14:00 − 15:00 Chair: Moonja Jeong (University of Suwon)
- ⋅ AN I 14:00 − 14:20 Studies on composition operators on the weighted Bergman spaces $A^p_\alpha$ (Inyoung Park, Boo Rim Choe and Hyungwoon Koo)
- 박인영*(서울대), 최부림(고려대), 구형운(고려대)
Inyoung Park*, Seoul National University, Boo Rim Choe, Korea University, Hyungwoon Koo, Korea University
We studied compact difference problem, one of subjects to composition operators. Our results are about the extension of the results on disk by Moorhouse to the polydisks, which are obtained through a different approach. In addition, we calculated the results for the compact difference and related problems on $A^2_\alpha(\bf D)$ to those on $A^p_\alpha(\bf D)$, $0<p<\infty$.
2010 Mathematics Subject Classification: 32Axx
Key Words and Phrases: composition operator, compact difference, Hardy space
- ⋅ AN I 14:20 − 14:40 $L_j$-hyperholomorphic functions in Clifford analysis (Su Jin Lim, Kwang Ho Shon)
- 임수진*(부산대), 손광호(부산대)
Su Jin Lim*, Pusan National University, Kwang Ho Shon, Pusan National University
The Clifford algebra of $\mathcal{A}_n$ is a $2^{n}$-dimensional real vector space with basis $\{e_A : A \subseteq \{1,\ldots,n\}\}$, where $e_\phi =e_0=1$, $e_A=e_{\alpha_1} \cdots e_{\alpha_h}$, $A=\{\alpha_1,\ldots ,\alpha_h\}$, $\alpha_1 < \alpha_2 < \cdots <\alpha_h$; $e_{\alpha_j}e_{\alpha_k}=-e_{\alpha_k}e_{\alpha_j}$ when $j\neq k$ and $e^2_{\alpha_j}=-1$, $j=1,\ldots ,n$. We consider the quaternion valued functions $f(z)=f_1(z)+f_2(z)e_2$ of quaternion variables $z=x_1+e_1x_2+e_2x_3+e_3x_4$ and investigate properties of $L_j$-hyperholomorphic functions for $j=1,2,3,4$ and regular functions on $\mathbb{C}^2$.
2010 Mathematics Subject Classification: 32A99, 30G35, 11E88
Key Words and Phrases: hyperholomorphic, regular function, quaternion, clifford analysis
- ⋅ AN I 14:40 − 15:00 How to compare the absolute values of operator sums and the sums of absolute values (Eun-young Lee)
- 이은영(경북대)
Eun-young Lee, Kyungpook National University
We address the problem of comparing $|A + B|$ and $|A| + |B|$ for $A,B$ in $M_n(C)$. Some results are obtained by using a technique of positive linear maps and several open questions are proposed.
2010 Mathematics Subject Classification: 15A60, 47A30, 47A60
Key Words and Phrases: symmetric norms, operator inequalities, concave
- ⋅ 15:10 − 16:10 Chair: Hyeong-Ohk Bae (Ajou University)
- ⋅ AN I 15:10 − 15:40 Note on $k$-plane transforms in the finite field setting (Doowon Koh)
- 고두원(충북대)
Doowon Koh, Chungbuk National University
In this talk, we study mapping properties of $k$-plane transforms in the finite field setting. We obtain sharp $L^p-L^r$ estimates on the $X$-ray transform and the Radon transform for finite fields. Combinatorics arguments and discrete Fourier analysis play a crucial role in our proofs.
2010 Mathematics Subject Classification: 22E99
Key Words and Phrases: X-ray transform, radon transform, finite field
- ⋅ AN I 15:40 − 16:10 A study on the fuzzy integral with respect to a fuzzy measure on the space of interval-valued functions (Lee-Chae Jang)
- 장이채(건국대)
Lee-Chae Jang, Konkuk University
Since Sugeno introduced the theory of fuzzy measures and fuzzy integrals, many researcher have been studying many new approaches and theories. The concept of interval-valued functions is useful idea to represent uncertain function in the decision making and information theory. In order to represent uncertainty, it has been suggested to use intervals, for examples, fuzzy integral of interval-valued functions Choquet integral of interval-valued functions, Aumann integral of closed set-valued functions, interval-valued probability, and fuzzy random variables. In this paper, we consider interval-valued functions which are motivated by the goal to generalize functions and used for representing uncertain functions. Based on the fuzzy integral of interval-valued functions, we investigate various kinds of convergence theorems for the fuzzy integral with respect to a fuzzy measure of interval-valued functions. In particular, we discuss some representations of nonlinear functionals as the fuzzy integral on the space of interval-valued functions.
2010 Mathematics Subject Classification: 28E10, 28E20, 03E72, 26E50, 11B68
Key Words and Phrases: fuzzy measure, fuzzy integral, interval-valued function, convergence theorem
- Analysis II
- ⋅ 09:00 − 10:00 Chair: Dohan Kim (Seoul National University)
- ⋅ AN II 09:00 − 09:30 Multiplicity result for the parabolic problem with asymptotic linearity (Q-Heung Choi, Tacksun Jung)
- 최규흥*(인하대), 정택선(군산대)
Q-Heung Choi*, Inha University, Tacksun Jung, Kunsan National University
Let $\Omega$ be a bounded open subset of $\mathbb R^{n}$ with smooth boundary $\partial \Omega$. Let $L$ be the self-adjoint strongly elliptic partial differential operator. We consider the multiplicity of the solutions of the following parabolic boundary value problem $$D_{t}u=L u+f(u)-s \phi_{1}-h(x)\qquad\mbox{in}~\mbox{}~\Omega \times \mathbb R,\leqno(1.1)$$ $$u(x,t)=0, \qquad x\in \partial \Omega,\mbox{}~t\in \mathbb R,$$ $$u(x,t)=u(x,t+2\pi), \qquad \mbox{in}~\mbox{}~\Omega \times \mathbb R.$$ We consider the case $f(u)=bu^{+}-au^{-}$, i.e., $$D_{t}u=Lu+bu^{+}-au^{-}-s \phi_{1}-h(x)\qquad\mbox{in}~\mbox{}~\Omega \times \mathbb R,\leqno(1.2)$$ $$u(x,t)=0, \qquad x\in \partial \Omega,\mbox{}~t\in \mathbb R,$$ $$u(x,t)=u(x,t+2\pi), \qquad \mbox{in}~\mbox{}~\Omega \times \mathbb R.$$ \noindent {\bf Theorem 1.1.} {\it Assume that $s>0$. Then $(1.1)$ has at least three periodic solutions.}
2010 Mathematics Subject Classification: 35K20
Key Words and Phrases: parabolic boundary value problem, elliptic boundary problem, inverse compact operator, reduction method, Leray-Schauder degree theory
- ⋅ AN II 09:30 − 10:00 Asymptotically linear Hamiltonian system (Tacksun Jung, Q-Heung Choi)
- 정택선*(군산대), 최규흥(인하대)
Tacksun Jung*, Kunsan National University, Q-Heung Choi, Inha University
We investigate the number of $2\pi$-periodic solutions of the
following Hamiltonian system
$$\dot{z}=J(G_{z}(t,z(t))) \leqno(1.1)$$
with asymptotically linear conditions:
$(G1)$\mbox{}~$G(t,z(t))=0(|z|^{2})$ as $|z| \to 0$, $G(t,\theta)=0$, $G_{z}(t,\theta)=\theta$, where $\theta=(0,\ldots,0)$,
$(G2)$ There exist constants $\alpha$, $\beta$ (without loss of generality, we may assume $\alpha$, $\beta\notin Z$) such that
$$\alpha I\le d^{2}_{z} G(t,z) \le \beta I,\mbox{}~ \forall (t,z)\in \mathbb R^{1}\times \mathbb R^{2n}.$$
$(G3)$ $\mbox{}~$ Let $j_{1}$ be a integer within $[\alpha, \beta]$ such that
$$ j_{1}-1<\alpha<d^{2}_{z}G(t,0)=\lim_{|z|\to 0}\frac{G_{z}(t,z)\cdot z}{|z|^{2}}<j_{1},$$
$(G4)$ $\mbox{}~$ $\lim_{|z|\to \infty}\frac{G_{z}(t,z)\cdot z}{|z|^{2}}$ exists and there exists $j_{2}=j_{1}+1$ which satisfies
$$j_{1}<d^{2}_{z}G(t,\infty)=\lim_{|z|\to \infty}\frac{G_{z}(t,z)\cdot z}{|z|^{2}}<\beta<j_{2},$$
$(G5)$ \mbox{}~ $G$ is $2\pi$-periodic with respect to $t$.
\smallskip
\noindent {\bf Theorem 1.1.} {\it Assume that $G$ satisfies the conditions $(G1)$-$(G5)$.
Then the system $(1.1)$ has at least three $2\pi$-periodic solutions.}
2010 Mathematics Subject Classification: 35Q72
Key Words and Phrases: Hamiltonian system, asymptotically linearity, variational reduction method, critical point theory, variational method
- ⋅ 10:10 − 11:40 Chair: Sun Young Jang (University of Ulsan)
- ⋅ AN II 10:10 − 10:40 On a technique in deformation theory (Hyun Ho Lee)
- 이현호(울산대)
Hyun Ho Lee, University of Ulsan
A fundamental result in the theory of vector bundles over a space $X$ of finite dimension states that when the fiber dimensions of one bundle is sufficiently small than that of another one, the one bundle embeds into the other. We are going to observe that corresponding analytic statements in various situations are useful in dealing with $C_0(X)$-modules.
2010 Mathematics Subject Classification: 46L05, 55R65
Key Words and Phrases: vector bundles, Hilbert modules
- ⋅ AN II 10:40 − 11:10 Some operator equations and subscalarity (Sungeun Jung, Eungil Ko)
- 정성은*(이화여대 수리과학연구소), 고응일(이화여대)
Sungeun Jung*, Institute of Mathematical Sciences, Ewha Womans University, Eungil Ko, Ewha Womans University
In this talk, we consider the operator equations $ABA = A^2$ and $BAB = B^2$. Let $(A,B)$ be a solution to the above system of operator equations. We prove that if $B$ is $p$-hyponormal, then $A$ is subscalar of order $10$. As a corollary, we get that if $B$ is $p$-hyponormal and its spectrum has nonempty interior, then $A$ has a nontrivial invariant subspace. We also show that if 0 does not belong to the essential spectrum of $B$, then $A$ is biquasitriangular if and only if $B$ is biquasitriangular. Finally, we give some connections between $Lat(A)$ and $Lat(B)$.
2010 Mathematics Subject Classification: 47A11
Key Words and Phrases: subscalar, invariant subspace, biquasitriangular
- ⋅ AN II 11:10 − 11:40 Remarks on Schr\"odinger 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
We give a Schr\"odinger uncertainty relation based on the monotone pair skew information. In the special cases the monotone pair skew information is the Wigner-Yanase-Dyson skew information, our result reduces to that of Furuichi and Yanagi.
2010 Mathematics Subject Classification: 81P45, 94A15
Key Words and Phrases: Wigner-Yanase-Dyson skew, monotone pair skew information, Schr\"odinger uncertainty relation
- ⋅ 14:10 − 16:10 Chair: Jinho Lee (Sookmyung Women's University)
- ⋅ AN II 14:10 − 14:40 Elliptic boundary value problem with some blowup data on the boundary (Sungwon Cho)
- 조성원(광주교육대)
Sungwon Cho, Gwangju National University of Education
We consider second order linear elliptic boundary value problem with zero Dirichlet condition on the boundary and allowing some blowup function for Poisson data on the boundary. Without lower order coefficients, a priori estimates is known by the joint work with M. Safonov. Our approach is based on a special growth lemma, and it works for both divergence and non-divergence equations in domains satisfying a general ``exterior measure condition'' (A).
2010 Mathematics Subject Classification: 35Jxx
Key Words and Phrases: elliptic boundary value problem, Dirichlet condition, second order linear elliptic equation
- ⋅ AN II 15:10 − 15:40 Eigenvalue problems for elliptic equations with variable exponents (Yun-Ho Kim)
- 김연호(상명대)
Yun-Ho Kim, Sangmyung University
We are concerned with the eigenvalue problem of the form: \begin{equation*} \begin{cases} -\text{div}(w(x)|\nabla u|^{p(x)-2}\nabla u)=\lambda f(x,u) &\text{in } \Omega \\ \ u=0 \,\ \ \ & \text{on} \ \partial \Omega, \end{cases} \end{equation*} where the variable exponent $p:\bar{\Omega} \to (1,\infty)$ is a continuous function, $w$ is a weighted function in $\mathbb R^{N}$ and $f:\Omega\times\mathbb R\to \mathbb R$ satisfies a Carath\'eodory condition. Under suitable conditions, we consider the existence of a positive eigenvalue for the above problem in the weighted variable exponent Sobolev spaces.
2010 Mathematics Subject Classification: 35J60, 35J70, 47J10
Key Words and Phrases: eigenvalue, weighted variable exponent Lebesgue-Sobolev spaces
- ⋅ AN II 15:40 − 16:10 Micro-macro decomposition and compensating functions in Vlasov- Fokker- Planck coupled systems (Myeongju Chae)
- 채명주(한경대)
Myeongju Chae, Hankyong National University
In this talk we discuss Vlasov-Fokker-Planck coupled systems; Vlasov -Fokker-Planck-Maxwell and compressible Navier-Stokes-Fokker-Planck equations. Micro-macro decomposition is essentially used to have the classical global solutions for those system with suitable smallness on initial data. Kawasima compensating functions is also used to give better stability results on torus or whole space.
2010 Mathematics Subject Classification: 35Lxx
Key Words and Phrases: Vlasov-Fokker-Planck-Maxwell, compressible Navier-Stokes
- Geometry
- ⋅ 09:30 − 10:00 Chair: Jae Hyouk Lee (Ewha Womans University)
- ⋅ 09:30 − 10:00 Half-powered scalar curvature flow (Hyunsuk Kang, Lami Kim and Ki-ahm Lee)
- 강현석*(고등과학원), 김라미(서울대), 이기암(서울대)
Hyunsuk Kang*, KIAS, Lami Kim, Seoul National University, Ki-ahm Lee, Seoul National University
The geometric flows have been useful tools to understand the geometry of manifolds via the immersion of hypersurfaces or metric deformation. As an example of extrinsic curvature flow, we consider the half-powered scalar curvature flow which satisfies properties such as short time existence and long time existence. The flow concerned here, which is a generalization of the scalar curvature flow introduced by B. Chow, starts from a strictly convex smooth hypersurface with a positive curvature term depending on the underlying sphere. Then it converges to a point like other extrinsic flows and when rescaled, it converges to a smooth manifold. The main inputs are the modified pinching estimate and the construction of a monotone quantity in dimension two. This is a joint work with Lami Kim and Ki-ahm Lee.
2010 Mathematics Subject Classification: 58J35
Key Words and Phrases: geometric flow, immersion
- ⋅ 10:10 − 11:40 Chair: Seungsu Hwang (Chung-Ang University)
- ⋅ 10:10 − 10:30 On the simple rotational surfaces of the Euclidean 4-space and their Gauss map (Nurettin Cenk Turgay, Ugur Dursun)
- Nurettin Cenk Turgay*, Istanbul Technical University, Ugur Dursun, Istanbul Technical University
In this work, we study simple rotational surfaces in Euclidean 4-space. We give classification of these type of surfaces with pointwise 1-type Gauss map.
2010 Mathematics Subject Classification: 53C40
Key Words and Phrases: simple rotational surface, Gauss map, finite type mappings
- ⋅ 10:30 − 11:00 Special Lagrangian submanifolds in cotangent bundles of spheres (Jae-Hyouk Lee)
- 이재혁(이화여대)
Jae-Hyouk Lee, Ewha Womans University
Special Lagrangians are calibrated submanifolds appearing in K\"ahler manifolds with Ricci-flat metrics. For each submanifold $X$ in a sphere $S$, we show that the corresponding conormal bundle of $X$ in the cotangent bundle of the sphere is Lagrangian for a K\"ahler form known as the Stenzel form. Furthermore, we correspond an austere submanifold $X$ to a special Lagrangian conormal subbundle in the cotangent bundle of spheres. We also discuss examples of austere submanifolds in the sphere from isoparametric geometry.
2010 Mathematics Subject Classification: 53D12
Key Words and Phrases: special Lagrangian, cotangent bundle
- ⋅ 11:10 − 11:40 Parabolic quasiconformally conjugacy classes in the Heisenberg group (Youngju Kim)
- 김영주(고등과학원)
Youngju Kim, KIAS
In this talk, we study noncompact $3$-dimensional manifolds obtained by quotienting the Heisenberg group by cyclic groups of parabolic automorphisms. In particular, we consider the quasiconformal equivalence classes of such manifolds. This is related to the quasiconformal conjugacy classes of parabolic isometries acting on the complex hyperbolic plane. We here prove that the quotient of the Heisenberg group by the cyclic group generated by a screw parabolic automorphism is quasiconformally distinct from the one generated by a Heisenberg translation.
2010 Mathematics Subject Classification: 57M50
Key Words and Phrases: complex hyperbolic geometry, Heisenberg group, parabolic isometry, quasiconformal mapping
- ⋅ 13:20 − 13:50 Chair: Jinsung Park (KIAS)
- ⋅ 13:20 − 13:50 Spherical versions of geometrical tube design (Hongtaek Hwang)
- 황홍택(금오공대)
Hongtaek Hwang, Kumoh Institute of Technology
튜브디자인 실험을 통하여 얻어진 구 형태의 다선구조들을 소개한다.
2010 Mathematics Subject Classification: 51M20
Key Words and Phrases: polyframe, tube design, spherical version
- ⋅ 15:10 − 16:10 Chair: Young Ho Kim (Kyungpook National University)
- ⋅ 15:10 − 15:40 Characterizations of spheres and elliptic paraboloids (Dong-Soo Kim)
- 김동수(전남대)
Dong-Soo Kim, Chonnam National University
We establish a characterization of spheres in ${\mathbb E}^{3}$ with respect to a surface area property of regions with the aid of a new meaning of Gaussian curvature. Furthermore, with respect to a volume property of regions, we characterize elliptic paraboloids in arbitrary dimensional Euclidean spaces.
2010 Mathematics Subject Classification: 53A05, 53A07
Key Words and Phrases: Gaussian curvature, co-area formula, sphere, paraboloid, Gauss-Kronecker curvature
- ⋅ 15:40 − 16:10 $H$-semi-invariant submersions (Kwang Soon Park)
- 박광순(서울대)
Kwang Soon Park, Seoul National University
In this talk, we introduce the notions of the almost $h$-semi-invariant submersion and the $h$-semi-invariant submersion which may be the extended version of the notion of the semi-invariant submersion. Using them, we obtain some properties. Finally, we give some examples for them.
2010 Mathematics Subject Classification: 53C15, 53C26, 53C43
Key Words and Phrases: Riemannian submersion, semi-invariant submersion, quaternionic manifold
- Topology I
- ⋅ 09:00 − 09:50 Chair: Suyoung Choi (Ajou University)
- ⋅ TO I 09:00 − 09:20 On unstabilized genus three critical Heegaard surfaces (Jungsoo Kim)
- 김정수(중앙대)
Jungsoo Kim, Chung-Ang University
In 2002, D. Bachman introduced the concept ``a critical surface" and he proved several theorems about incompressible surfaces, the number of Heegaard splittings with respect to its genus, and the minimal genus common stabilization. Although a critical Heegaard splitting is a weakly reducible splitting, it shares some common properties with strongly irreducible splittings. For example, if the splitting is strongly irreducible or critical, then the manifold is irreducible and the intersection of an incompressible surface $S$ and the Heegaard surface $F$ can be isotoped essential on both $S$ and $F$. But it is not easy to determine whether a weakly reducible splitting is critical. In this talk, we will define a simplicial complex of weakly reducing pairs $\mathcal{WR}$ and find its fundamental properties. Especially, we will find more properties when the manifold is irreducible and the splitting is unstabilized and of genus three. Using these, we will induce a condition for a weakly reducible splitting to be critical.
2010 Mathematics Subject Classification: 57M50
Key Words and Phrases: Heegaard splitting, critical, weakly reducible, unstabilized
- ⋅ TO I 09:20 − 09:50 A formular for Alexander polynomials of pretzel links (Insook Lee, Yongju Bae)
- 이인숙*(경북대), 배용주(경북대)
Insook Lee*, Kyungpook National University, Yongju Bae, Kyungpook National University
In this talk, we will give a formula for the Alexander polynomial of given pretzel link, and try to find a reccurance relation for Alexander polynomials of locally pretzel type links.
2010 Mathematics Subject Classification: 57M25
Key Words and Phrases: Alexander polynomial, pretzel link
- ⋅ 10:10 − 11:40 Chair: Cheol-Hyun Cho (Seoul National University)
- ⋅ TO I 10:10 − 10:40 The Nielsen type numbers for maps on a 3-dimensional flat Riemannian manifolds (Jong Bum Lee)
- 이종범(서강대)
Jong Bum Lee, Sogang University
Let $f:M\to M$ be a self-map on a 3-dimensional flat Riemannian $M$. We compute the Lefschetz number and the Nielsen number of $f$ by using the infra-nilmanifold structure of M and the averaging formulas for the Lefschetz numbers and the Nielsen numbers of maps on infra-nilmanifolds. For each positive integer $n$, we provide an explicit algorithm for a complete computation of the Nielsen type numbers $NP_n(f)$ and $N\Phi_n(f)$ of $f^n$.
2010 Mathematics Subject Classification: 57S25
Key Words and Phrases: flat manifold, Nielsen type numbers
- ⋅ TO I 10:40 − 11:10 Projective bundles over smooth compact toric surfaces (Seonjeong Park, Suyoung Choi)
- 박선정*(카이스트), 최수영(아주대)
Seonjeong Park*, KAIST, Suyoung Choi, Ajou University
Let $E$ be the Whitney sum of complex line bundles over a smooth compact toric surface. Then the projectivization $P(E)$ is a toric manifold. If the cohomology ring of a quasitoric manifold $M$ is isomorphic to that of a projective bundle $P(E)$, then the orbit space of $M$ can be identified with that of $P(E)$, and, furthermore, $M$ is equivalent or homeomorphic to a projective bundle $P(E')$. Moreover, such quasitoric manifolds are classified by their cohomology rings up to homeomorphism in some cases.
2010 Mathematics Subject Classification: 57S25, 57R19, 57R20
Key Words and Phrases: toric topology, quasitoric manifolds, cohomological rigidity
- ⋅ TO I 11:10 − 11:40 Rational homology of real toric variety over graph associahedra (Hanchul Park, Suyoung Choi)
- 박한철*(아주대), 최수영(아주대)
Hanchul Park*, Ajou University, Suyoung Choi, Ajou University
Given a finite graph, the graph associahedron is a simple, convex polytope whose face poset is based on the connected subgraphs of the graph. For instance, a permutohedron and a Stasheff polytope are examples of graph associahedra corresponding to a complete graph and a path, respectively. It is well-known that they support many interesting real toric varieties. Recently, Henderson computed the rational homology of real Hessenberg variety supported by a permutohedron using geometrical way, and, later, A. Suciu computed it again using new method in toric theory. In this talk, we generalize their result to all graph associahedra.
2010 Mathematics Subject Classification: 57N65
Key Words and Phrases: graph associahedra, real toric variety, rational homology
- ⋅ 13:20 − 14:50 Chair: Seonhee Lim (Seoul National University)
- ⋅ TO I 13:20 − 13:50 Big Cohen-Macaulayness and the integral cohomology of toric orbifolds (Tomoo Matsumura, W. Frank Moore and Shisen Luo)
- Tomoo Matsumura*, KAIST, Frank W. Moore, Wake Forest University, Shisen Luo, Cornell University
The moment angle complex $Z_K$ for a simplicial complex $K$ was introduced by Buchsta\-ber-Panov as a disc-circle decomposition of the Davis-Januszkiewicz universal space which is originally used to introduce a quasi-toric manifold. In this talk, I will explain our recent result that the equivariant cohomology of moment angle complexes with respect to the action of a subgroup of the torus that acts canonically on $Z_K$ can be described in terms of Tor module of the Stanley-Reisner rings, and based on that, I will discuss the condition for the integral cohomology of a toric orbifold to be the quotient of the Stanley-Reisner ring by linear terms
2010 Mathematics Subject Classification: 55N32
Key Words and Phrases: orbifolds, toric orbifolds
- ⋅ TO I 13:50 − 14:20 On the integral cohomology of unipotent groups (Quang M. Hoang)
- Quang M. Hoang, Ulsan University
Let $U_n=U(n,\mathbb Z)$ be the unipotent group consisting of $n\times n$ upper triangular matrices with 1's on diagonals. In their landmark paper, Priddy and Lambe showed that if one omits low torsions, the cohomology of $U_n$ is isomorphic to the cohomology of some associated Lie algebra to the group. This work computes the ring structure of $H^*(U_n;\mathbb Z)$ for small values of $n$. Our result suggests that for large $n$, $H^*(U_n;\mathbb Z)$ is very complicated and contains low torsions.
2010 Mathematics Subject Classification: 55T10
Key Words and Phrases: cohomology, Serre spectral sequences
- ⋅ TO I 14:20 − 14:50 Different moment-angle manifolds arising from two polytopes having the same bigraded Betti numbers (Suyoung Choi)
- 최수영(아주대)
Suyoung Choi, Ajou University
Moment-angle manifolds, which have been studied since the 1960's, are one of the key players in toric topology. In this talk, we discuss about the toric rigidity of polytopes, and we provide two simple polytopes of dimension 3 having the identical bigraded Betti numbers but non-isomorphic Tor-algebras. Furthermore, we shall show that the moment-angle manifolds arising from these two polytopes are homotopically different. Before these examples, as far as I know, in all known examples of combinatorially different polytopes with same bigraded Betti numbers (such as vertex truncations of simplices), the moment-angle manifolds are diffeomorphic.
2010 Mathematics Subject Classification: 55U10
Key Words and Phrases: toric topology, moment-angle manifold, simple polytope, bigraded Betti number, combinatorial rigidity
- Topology II
- ⋅ 13:20 − 14:50 Chair: Yongjin Song (Inha University)
- ⋅ TO II 13:20 − 13:50 A simply connected numerical Campadelli surface with an involution (Heesang Park, Dongsoo Shin and Giancarlo Urzua)
- 박희상*(고등과학원), 신동수(충남대), Giancarlo Urzua(Pointificia Univ. Cat\'olica de Chile, Santiago)
Heesang Park*, KIAS, Dongsoo Shin, Chungnam National University, Giancarlo Urzua, Pointificia Universidad Cat\'olica de Chile, Santiago
We construct a simply connected minimal complex surface of general type with $p_g=0$ and $K^2=2$ which has an involution such that the minimal resolution of the quotient by the involution is a simply connected minimal complex surface of general type with $p_g=0$ and $K^2=1$. In order to construct the example, we combine a double covering and $\mathbb{Q}$-Gorenstein deformation. Especially, we develop a method for proving unobstructedness for deformations of a singular surface by generalizing a result of Burns and Wahl which characterizes the space of first order deformations of a singular surface with only rational double points. We describe the stable model in the sense of Koll\'ar and Shepherd-Barron of the singular surfaces used for constructing the example. We count the dimension of the invariant part of the deformation space of the example under the induced $\mathbb{Z}/2\mathbb{Z}$-action.
2010 Mathematics Subject Classification: 14J29, 14J10, 14J17, 53D05
Key Words and Phrases: surface of general type, involution, Q-Gorenstein smoothing
- ⋅ TO II 13:50 − 14:20 Roots of the Alexander polynomial and Hoste's conjecture (Alexander Stoimenow)
- Alexander Stoimenow, Keimyung University
The Alexander polynomial remains one of the most fundamental invariants of knots and links in 3-space. It topological understanding has led a long time ago to the insight what (Laurent) polynomials occur as Alexander polynomial of an arbitrary knot. Ironically, the question to characterize the Alexander polynomials of alternating knots turns out to be far more difficult, even although in general alternating knots are much better understood. Hoste, based on computer verification, made the following conjecture about 10 years ago: If $z$ is a complex root of the Alexander polynomial of an alternating knot, then Re$z > -1$. We discuss some results toward this conjecture, about 2-bridge (rational) knots or links, 3-braid alternating links, and Montesinos knots.
2010 Mathematics Subject Classification: 57M25
Key Words and Phrases: Alexander polynomial, alternating knot, root of polynomial
- ⋅ TO II 14:20 − 14:50 How to fold a manifold (Scott J. Carter)
- Scott J. Carter, University of South Alabama
This talk is based upon joint work with Seiichi Kamada. Consider a simple irregular branched cover of a $k$-dimensional sphere branched over an embedded $(k-2)$-dimensional manifold. Using a generalization of Kamada's notion of a braid chart, I show how to immerse the covering into a tubular neighborhood of a standard embedding of the $k$ sphere in $(k+2)$-dimensional space. We discuss the problem of lifting this embedding to an immersion.
2010 Mathematics Subject Classification: 57Q45
Key Words and Phrases: branched covering, Kamada chart, higher dimensional knot
- Probability and Statistics
- ⋅ 09:00 − 09:50 Chair: Un-Cig Ji (Chungbuk National University)
- ⋅ 09:00 − 09:20 Valuation of a two-asset option with a barreir in the regime switching model (Hyun Joo Yoo)
- 유현주(고려대)
Hyun Joo Yoo, Korea University
This paper deals with pricing of external-barrier options under the regime switching model. The external-barrier options are path dependent derivatives two correlated assets are involved. We derive the Laplace transform of the price of the external-barrier options under the regime switching model. As a numerical example, we calculate the price of an external-barrier option by numerical inversion of the Laplace transform.
2010 Mathematics Subject Classification: 60H30
Key Words and Phrases: first passage time, option pricing, regime-switching, Laplace transform
- ⋅ 09:20 − 09:50 A note on the solutions for stochastic differential equations with infinite delay (Young-Ho Kim)
- 김영호(창원대)
Young-Ho Kim, Changwon National University
In this note, we consider a $d$-dimensional stochastic functional differential equations: $$ {\rm d} x(t)=f(x_t, t ) {\rm d}t +g (x_t, t ){\rm d}B(t),\quad t_0\leq t\leq T, $$ where $x_{t}=\{x(t+\theta ):-\infty < \theta\leq 0\}$ can be considered as a $BC((-\infty , 0]; R^d )$-value stochastic process, where $$ f : BC((-\infty , 0]; R^d )\times [t_0,T]\rightarrow R^d, g : BC((-\infty , 0]; R^d )\times [t_0,T]\rightarrow R^{d\times m}\notag $$ be Borel measurable. Next, we prove the existence and uniqueness of the solution for the equation with the initial value as followed: $ x_{t_0}=\xi=\{\xi(\theta ):-\infty < \theta\leq 0\}$ is an $\mathcal{F}_{t_0}$-measurable $BC((-\infty , 0];R^d )$-value random variable such that $\xi\in \mathcal{M}^2((-\infty , 0]; R^d ) . $
2010 Mathematics Subject Classification: 60H05
Key Words and Phrases: existence and uniqueness, stochastic differential equations
- ⋅ 11:00 − 11:30 Chair: Hyun Jae Yoo (Hankyong National University)
- ⋅ 11:00 − 11:30 The property of small sample for longitudinal data (Hyun Yung Lee)
- 이현영(신라대)
Hyun Yung Lee, Silla University
Motivated by the need to validly analyse data from small longitudinal designs with count, outcomes, we carried out a simulation study with a two-group longitudinal design with as few as five subjects per group and three measurements per subject. Various correlation structures related to autoregression and/or subject heterogeneity were used to simulate, most of which did not correspond to any of the methods being assessed and so induced some model misspecification. We evaluated validity (Type I error rate) and efficiency (power) for the interaction effect using several methods based on generalized linear mixed models (GLMM) or generalized estimating equations (GEE).
2010 Mathematics Subject Classification: 62-07
Key Words and Phrases: longitudinal data, GEE, GLMM, Type I error rate
- ⋅ 13:20 − 14:10 Chair: Young-Ho Kim (Changwon National University)
- ⋅ 13:20 − 13:50 Distance $k$-graphs of hypercube and $q$-Hermite polynomials (Hun Hee Lee)
- 이훈희(충북대)
Hun Hee Lee, Chungbuk National University
We will extend the quantum probabilistic approach of Hora/Obata to the case of weig\-ht\-ed graph. The main result says that distance $k$-graphs of hypercube can approximate the distribution of $q$-Hermite polynomial of $q$-gaussian variables through the scaling limit. The key ingredients for the proof are Biane's toy Fock model and non-commutative $L_p$ analysis.
2010 Mathematics Subject Classification: 46L53
Key Words and Phrases: $q$-Hermite polynomial, $q$-gaussian, hypercube, distance $k$-graph
- ⋅ 13:50 − 14:10 The rate of convergence 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 rate of convergence 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 rate of convergence for weighted sums of random variables in noncommutative Ciach space and the rate of convergence for weighted additive convolution sums.
2010 Mathematics Subject Classification: 46L53
Key Words and Phrases: noncommutative probability space, weighted sum, rate of convergence
- Applied Mathematics I
- ⋅ 09:00 − 10:00 Chair: Philsu Kim (Kyungpook National University)
- ⋅ AM I 09:00 − 09:20 Higher-order methods for finding multiple roots of nonlinear equations (Young Hee Geum)
- 금영희*(단국대), 김영익(단국대)
Young Hee Geum*, Dankook University, Young Ik Kim, Dankook University
In this paper, we construct a two-parameter family of fourth-order iterative methods using the weighted harmonic mean of two derivative functions to compute approximate multiple roots of nonlinear equations. They are verified to be optimally convergent in the sense of Kung-Traub's optimal order. Numerical experiments well confirm the validity of convergence and asymptotic error constants.
2010 Mathematics Subject Classification: 65H05, 65H99
Key Words and Phrases: asymptotic error constant, multiple root, nonlinear equation, order of convergence, root finding
- ⋅ AM I 09:20 − 09:40 A fast SDIRK4 scheme for solving 1D unsteady convection-diffusion equations (Hyun Jung Choi, Sang Dong Kim, Xiangfan Piao and Philsu Kim)
- 최현정*(경북대), 김상동(경북대), 박상범(경북대), 김필수(경북대)
Hyun Jung Choi*, Kyungpook National University, Sang Dong Kim, Kyungpook National University, Xiangfan Piao, Kyungpook National University, Philsu Kim, Kyungpook National University
In this paper, a fast singularly diagonally implicit Runge-Kutta (FSDIRK) scheme is designed for solving unsteady one-dimensional convection diffusion equations with nonconstant boundary conditions. A compact finite difference approximation for the spatial discretization is used and also a singularly diagonally implicit RK method for the temporal discretization is used. In particular, the boundary values assigned to the internal stages for the implicit RK method are modified with the technique outlined in reference to avoid the order of the reduction. The proposed scheme not only has fourth-order accuracy in both space and time variables but is also computationally efficient requiring only a linear matrix solver for a tridiagonal matrix system. It is also shown that the proposed scheme is unconditionally stable and suitable for stiff problems. Several numerical examples are solved by the new scheme and the numerical efficiency and superiority of the new scheme are compared with the numerical results obtained by other methods in the literature.
2010 Mathematics Subject Classification: 65M12
Key Words and Phrases: fast SDIRK scheme, compact finite difference scheme, singularly diagonally implicit RK method, unconditionally stable, convection-diffusion equation
- ⋅ AM I 09:40 − 10:00 A fast high-order rational scheme for solving 1D unsteady convection-diffusion equations (Hyo Jin Eo, Sang Dong Kim, Jae Min Kweon and Phil Su Kim)
- 어효진*(경북대), 김상동(경북대), 권재민(수리과학연구소), 김필수(경북대)
Hyo Jin Eo*, Kyungpook National University, Sang Dong Kim, Kyungpook National University, Jae Min Kweon, National Fusion Research Institute, Philsu Kim, Kyungpook National University
In this paper, a fast high-order rational (FHOR) scheme is developed for solving unsteady one-dimensional convection diffusion equations. A compact finite difference approximation is used for the spatial discretization and a rational approximation to the exponential function is used for the temporal discretization. The proposed scheme not only has fourth-order accuracy in both space and time variables but is also computationally efficient requiring only a linear matrix solver for a tri-diagonal matrix system. It is also shown that the proposed scheme is unconditionally stable and suitable for stiff problems. Several numerical examples are solved by the new scheme and the numerical efficiency and accuracy of the new scheme are compared with the numerical results obtained by other methods in the literature.
2010 Mathematics Subject Classification: 65M12
Key Words and Phrases: fast high-order rational scheme, compact finite difference scheme, rational approximation, unconditionally stable, convection-diffusion equation
- ⋅ 10:10 − 11:40 Chair: Seki Kim (Sungkyunkwan University)
- ⋅ AM I 10:10 − 10:40 Trinomial tree method for pricing European/American-style Asian options (Gang Joon Yoon, Sung Dae Yang and Min Soo Song)
- 윤강준*(국가수리과학연구소), 양성대(국가수리과학연구소), 송민수(국가수리과학연구소)
Gang Joon Yoon*, National Institute for Mathematical Sciences, Sung Dae Yang, National Institute for Mathematical Sciences, Min Soo Song, National Institute for Mathematical Sciences
We study a trinomial tree method for pricing European/Ameriacn-style Asian options and show its equivalence to an explicit finite difference scheme. Moreover, we prove that the trinomial tree method for European/American-style Asian options converges locally uniformly to a solution of the corresponding Black-Scholes PDE. By numerical experiments we demonstrate the accuracy and efficiency of a trinomial tree method compared to a binomial tree method.
2010 Mathematics Subject Classification: 91B70, 35K65, 60H15
Key Words and Phrases: Asian options, trinomial tree method, viscosity solution
- ⋅ AM I 10:40 − 11:10 Homotopy analysis method for option pricing under stochastic volatility (Sang-Hyeon Park, Jeong-Hoon Kim)
- 박상현*(서울대), 김정훈(연세대)
Sang-Hyeon Park*, Seoul National University, Jeong-Hoon Kim, Yonsei University
Many pricing problems for exotic options do not have analytic solutions under general stochastic volatility environment so that it is hard to compute the solutions or at least it requires much of time to compute the solutions. In this paper, we choose three types of option pricing problem and utilize a semi-analytic approach to connect the well-known Black-Scholes price to the option price under a generalized stochastic volatility model which makes relatively efficient in computing the option prices.
2010 Mathematics Subject Classification: 91G20, 60H10
Key Words and Phrases: option, homotopy method, stochastic volatility
- ⋅ AM I 11:20 − 11:40 A stochastic elasticity correction to perpetual American option prices (Ji Hun Yoon, Jeong Hun Kim and Song Ping Zhu)
- 윤지훈*(연세대), 김정훈(연세대), Song Ping Zhu(Wollongong Univ.)
Ji Hun Yoon*, Yonsei University, Jeong Hun Kim, Yonsei Univerisity, Song Ping Zhu, Wollongong Univerisity
This paper studies pricing the perpetual American options based upon a constant elasticity of variance type of underlying asset price model where the constant elasticity is replaced by a small fast mean-reverting Ornstein-Uhlenbeck process. Through a multiscale asymptotic analysis, we find the impact of the stochastic elasticity of variance on the option prices and the optimal exercise prices given by the renowned Black-Scholes model, which corresponds to zero elasticity, and the constant elasticity of variance model. Our results enhance the flexibility and applicability of the resultant option price structure in terms of the market price of elasticity risk.
2010 Mathematics Subject Classification: 60Gxx
Key Words and Phrases: multiscale analysis, perpetual American option, stochastic elasticity of variance
- ⋅ 13:20 − 14:50 Chair: Sunyoung Kim (Ewha Womans University)
- ⋅ AM I 13:20 − 13:50 Nonconforming linear divergence-free finite element methods for incompressible Sto\-kes equations on square meshes (Chunjae Park)
- 박춘재(건국대)
Chunjae Park, Konkuk University
Recently, a nonconforming linear divergence-free space has been introduced on square meshes. The proposed space has sufficiently many interpolants to approximate a continuous divergence-free function up to the error of $O(h)$ in discrete $H^1$-seminorm. In this talk, an incompressible Stokes problem is solved first for the velocity only using the divergence-free space, and secondly for the pressure. The matrix concerning to the velocity is reduced into the two smaller submatrices of the red squares and black squares.
2010 Mathematics Subject Classification: 65J10
Key Words and Phrases: divergence-free, Stokes, nonconforming
- ⋅ AM I 13:50 − 14:20 Numerical correction methods for solving Stiff problems based on Chebyshev collocation (Xiangfan Piao, Sang Dong Kim and Philsu Kim)
- 박상범*(경북대), 김상동(경북대), 김필수(경북대)
Xiangfan Piao*, Kyungpook National University, Sang Dong Kim, Kyungpook National University, Philsu Kim, Kyungpook National University
In this talk, we present numerical correction methods for solving stiff initial value problems, which not only avoid unnecessary iteration process that may be required in most implicit methods but also have such a good stability as all implicit methods possess. The proposed methods use a Chebyshev collocation technique as well as an asymptotic linear ordinary differential equation of first-order derived from the difference between the exact solution and the Euler's polygon. These methods with or without the Jacobian are analyzed in terms of convergence and stability. In particular, it is proved that the proposed methods have a convergence order up to 4 regardless of the usage of the Jacobian.
2010 Mathematics Subject Classification: 65L04, 65L07, 65L20
Key Words and Phrases: Chebyshev collocation method, stiff initial value problem, A-stability
- ⋅ AM I 14:20 − 14:50 Spectral density analysis for Pohang new harbor (Prashant Kumar, Kim Kwang Ik)
- Prashant Kumar*(포항공대), 김광익(포항공대)
Prashant Kumar*, POSTECH, Kwang Ik Kim, POSTECH
The Pohang New Harbor (PNH), located at the northeast part of Pohang city, has been experienced extreme wave hazards of about 3.0-5.0 meter high in surface elevation due to the wave induced oscillation. To resolve this problem theoretical studies are introduced to investigate the wave induced oscillation in arbitrary shaped harbor, which are based on linearized long wave theory, i.e., derived the Helmholtz equation. Then numerical simulations are conducted to find the wave spectrum (Spectral Density) of the Pohang New Harbor. Firstly, boundary of the PNH, based on the actual topography and bathometry data, is constructed and then the Weber's solution of the Helmholtz equation obtained. The spectral density has been analyzed at the specific record points inside the harbor. The simulation results are compared with real time measurements of wave heights and Tide Gauge (WTG) at the specified eight record points inside the PNH. The model spectral density gave the information about the seiches and harbor oscillation at specific locations inside the harbor with respect to wave period. Further, to visualize the wave spectrum with respect to time and frequency, we proposed some tactics to improve the Pohang New Harbor hazards.
2010 Mathematics Subject Classification: 76B07, 76B15, 35J05
Key Words and Phrases: Helmholtz equation, spectral density, topography and bathometry, track recorder points, Pohang New Harbor (PNH)
- Applied Mathematics II
- ⋅ 09:00 − 09:50 Chair: Jeehyun Lee (Yonsei University)
- ⋅ AM II 09:00 − 09:20 Early effect in a slowly-varying modulated rotating flow (Youngyong Park, Younghae Do and Juan M. Lopez)
- 박영용*(경북대), 도영해(경북대), Juan M. Lopez(경북대, Arizona State Univ.)
Youngyong Park*, Kyungpook National University, Younghae Do, Kyungpook National University, Juan M. Lopez, Kyungpook National University/ Arizona State University
The slow passage through multiple resonances is considered for the flow in an enclosed circular cylinder driven by the harmonic modulation of the rotating end wall whose frequency is slowly ramped linearly in time. The theoretical setup is similar to the familiar slow passage through a Hopf bifurcation problem, where for slow variations of the control parameter, oscillations are delayed until the parameter has exceeded the critical value of the static-parameter problem. In contrast, a single resonance problem modelled from a damped harmonically forced oscillator shows an early onset of resonance. This emerges when the ramped forcing frequency is midway between its initial value and the resonance frequency. Applying the same slow passage protocol to the fluid dynamics problem with multiple resonances, we find an early effect for each resonance frequency that behaves in the same way as in the single resonance model problem.
2010 Mathematics Subject Classification: 37N10
Key Words and Phrases: slow passage, resonance, early effect
- ⋅ AM II 09:20 − 09:50 Nonsmooth semi-infinite multiobjective optimization problems (Thai Doan Chuong, Do Sang Kim)
- Thai Doan Chuong*(부경대), 김도상(부경대)
Thai Doan Chuong*, Pukyong National University, Do Sang Kim, Pukyong National University
We use some advanced tools of variational analysis and differentiation to establish necessary conditions for (weakly) efficient solutions of a nonsmooth semi-infinite multiobjective optimization problem (SIMOP for brevity). Sufficient conditions for (weakly) efficient solutions of a SIMOP are also provided by means of introducing the concepts of $L$-(strictly) invex functions defined in terms of the limiting subdifferential of locally Lipschitz functions. In addition, we propose types of Wolfe and Mond-Weir dual problems for SIMOPs, and explore weak and strong duality relations under assumptions of $L$-(strictly) invexity. Examples are also designed to analyze and illustrate the obtained results.
2010 Mathematics Subject Classification: 49K99, 65K10, 90C29, 90C46
Key Words and Phrases: optimality condition, duality, limiting subdifferential, $L$-invex function, semi-infinite optimization
- ⋅ 13:20 − 15:00 Chair: InKyung Ahn (Korea University)
- ⋅ AM II 13:20 − 13:40 An age-structured model with immune response of HIV infection: modeling and optimal control approaches (Myoung Ho Yoon, Jeehyun Lee and Hee Dae Kwon)
- 윤명호*(연세대), 이지현(연세대), 권희대(인하대)
Myoung Ho Yoon*, Yonsei University, Jeehyun Lee, Yonsei University, Hee Dae Kwon, Inha University
We develop and analyze an age-structured model of HIV infection with compartments including target cells, infected cells, virus and immune effectors to improve our understanding of the interplay between HIV and the immune system. We show that the model has one noninfected steady state and several infected steady states. A local stability analysis for those steady states is investigated using generalization of the standard Jacobian matrix method combined with the Laplace transform. We then consider techniques and ideas from optimal control theory to derive optimal therapy protocols using two types of dynamic treatments representing reverse transcriptase inhibitors(RTIs) and protease inhibitors(PIs). Necessary conditions(optimality system) for optimal control functions are derived by taking variations of a Lagrangian. The optimal therapy protocols were obtained by solving a large optimality system of equations using a difference scheme based on Runge-Kutta method. The numerical results indicated that the optimal therapy protocols can lead to long-term control of HIV by the strong immune response after discontinuation of therapy.
2010 Mathematics Subject Classification: 49Kxx, 92Cxx
Key Words and Phrases: HIV dynamics, age-structured model, optimal control, gradient method
- ⋅ AM II 13:40 − 14:10 Topology of RNA structure and a state sum invariant of links (Myeong-Ju Jeong)
- 정명주(Korea Science Academy)
Myeong-Ju Jeong, Korea Science Academy of KAIST
L. H. Kauffman gave a state sum model of the Jones polynomial of links. We introduce a virtual link invariant using a state model and give a criterion on some type of moves to transform a given link to another one. Using the invariant we analyze behaviors of RNA via secondary structure or pseudoknotted structure.
2010 Mathematics Subject Classification: 57M25
Key Words and Phrases: Links, RNA secondary structures, pseudoknotted structure
- ⋅ AM II 14:20 − 14:40 Optimal control of an influenza model with seasonal forcing and age structure (Jung Eun Kim, Jee Hyun Lee and Hee Dae Kwon)
- 김정은*(연세대), 이지현(연세대), 권희대(인하대)
Jung Eun Kim*, Yonsei University, Jee Hyun Lee, Yonsei University, Hee Dae Kwon, Inha University
This study considers an optimal intervention strategy for influenza outbreaks. Variations in the SEIAR model are considered to include seasonal forcing and age structure, and control strategies include vaccination, antiviral treatment, and social distancing such as school closures. We formulate an optimal control problem by minimizing the incidence of influenza outbreaks while considering intervention costs. We examine the effects of delays in vaccine production, seasonal forcing, and age-dependent transmission rates on the optimal control and suggest some optimal strategies through numerical simulations.
2010 Mathematics Subject Classification: 92D40
Key Words and Phrases: optimal control, deterministic influenza model, seasonal forcing, age-dependent transmission rate
- ⋅ AM II 14:40 − 15:00 Influence of an inhomogeneous magnetic field on the dynamics of the flow of ferrofluid between differentially rotating cylinders (Sebastian Altmeyer, Younghae Do and Juan Lopez)
- Sebastian Altmeyer*(경북대), 도영해(경북대), Juan Lopez(경북대)
Sebastian Altmeyer*, Kyungpook National University, Younghae Do, Kyungpook National University, Juan Lopez, Kyungpook National University/ Arizona State University
The influence of a magnetic field on the dynamics of the flow of a ferrofluid in the gap between two concentric, independently rotating cylinders is investigated numerically. In the absence of an applied magnetic field, the magnetic nano-particles are randomly orientated, the fluid has zero net magnetization. When a sufficiently strong magnetic field is applied, the ferrofluid flows towards regions of the magnetic field, properties of the fluid such as the viscosity are altered, and the hydrodynamics of the system can be significant changed. Based on theoretical estimates to take account of the magnetic boundary conditions, we show modelling the internal field with $r^{-2}$ radial dependence leads to a much reduced difference to the real, locally strong varying internal field.
2010 Mathematics Subject Classification: 76U05
Key Words and Phrases: magneto hydrodynamics, magnetic fluids and ferrofluids, nonlinearity bifurcation, rotating fluids
- Mathematical Education
- ⋅ 13:20 − 14:50 Chair: Byeong Moon Kim (Gangneung-Wonju National University)
- ⋅ 13:20 − 13:50 Special Idea for Auto Part which freshman found in Calculus course (Kim Young Sik)
- 김영식(홍익대)
Young Sik Kim, Hongik University
We introduce the special idea for Auto Part which the freshman in the calculus course found.
2010 Mathematics Subject Classification: 97C40
Key Words and Phrases: 11, calculus. Auto Part
- ⋅ 13:50 − 14:20 기초학력 부진학생에 대한 교양수학 학습지도 방안 (Yong-Soo Pyo, Jong Jin Seo)
- 표용수*(부경대), 서종진(부경대)
Yong-Soo Pyo*, Pukyong National University, Jong Jin Seo, Pukyong National University
이공계열 기피현상과 함께, 특별전형 및 교차지원 허용 등의 다양한 대입전형제도의 도입에 따른 수학교과의 심각한 학력 차이로 대학의 교양수학 교육과정 운영에 상당한 어려움이 있다. 이에 따라, 수학교과에 대한 기초학력 부진학생들의 문제해결력 향상과 학업성취도 고양을 위한 효율적인 학습지도 방안에 대한 사례들을 소개한다.
2010 Mathematics Subject Classification: 97D40
Key Words and Phrases: 11, calculus. Auto Part
- ⋅ 14:20 − 14:50 Geometrical tube designs and math culture activities (Hongtaek Hwang)
- 황홍택(금오공대)
Hongtaek Hwang, Kumoh Institute of Technology
tube design 실험을 통하여 아름답고 실용적이며 창의적인 기하학적 다선구조물 창작 사례를 소개한다.
2010 Mathematics Subject Classification: 97U60
Key Words and Phrases: tube design, polyframe, MathCulture activity
- Mathematics for Information Sciences
- ⋅ 09:00 − 09:50 Chair: Seunghyun Seo (Kangwon National University)
- ⋅ 09:00 − 09:30 Construction of extremal self-dual codes over $F_2 +uF_2$ (Hyun Jin Kim, Yoonjin Lee)
- 김현진*(이화여대), 이윤진(이화여대)
Hyun Jin Kim*, Ewha Womans University, Yoonjin Lee, Ewha Womans University
We complete the classification of the Lee-extremal self-dual codes over the ring $F_2 +uF_2$ of lengths 21 and 22 with a nontrivial automorphism of odd prime order except the case for an automorphism of order 3 with seven cycles, and we partially classify the exceptional case.
2010 Mathematics Subject Classification: 94B05, 11T71
Key Words and Phrases: automorphism, self-dual code, extremal code
- ⋅ 09:30 − 09:50 The number of isotones of tightness 2 and 3 (Do Haeng Lee, Sang-Mok Kim)
- 이도행*(광운대), 김상목(광운대)
Do Haeng Lee*, Kwangwoon University, Sang-Mok Kim, Kwangwoon University
For a given poset $P = (X, \leq_P)$, an injective function $f : X \rightarrow \mathbb{Z}$ is called an isotone if it preserves the order-relation of $P$. The tightness of an isotone $f$ on $P$, written as $T_f(P)$, is the maximum difference between the value of $f$ of incomparable elements of $P$. In this talk, we first introduce the upper triangle matrix representation of a poset with an isotone with certain tightness. Using this representation, we establish the number of isotones of tightness 2 and 3, separately, by solving their recurrence relations, respectively.
2010 Mathematics Subject Classification: 06A07
Key Words and Phrases: poset, tightness, matrix representation, linear discrepancy
- ⋅ 10:10 − 11:40 Chair: Young Soo Kwon (Yeungnam University)
- ⋅ 10:10 − 10:30 Dynamic coloring and list dynamic coloring of planar graphs (Won-Jin Park, Seog-Jin Kim and Sang June Lee)
- 박원진*(서울대), 김석진(건국대), 이상준(Emory Univ.)
Won-Jin Park*, Seoul National University, Seog-Jin Kim, Konkuk University, Sang June Lee, Emory University
A {\em dynamic chromatic number} $\chi_d(G)$ of a graph $G$ is the least number $k$ such that $G$ has a proper $k$-coloring of the vertex set $V(G)$ so that for each vertex of degree at least 2, its neighbors receive at least two distinct colors. We show that $\chi_d(G)\leq 4$ for every planar graph except $C_5$, which was conjectured in [5]. The {\em list dynamic chromatic number} $ch_d(G)$ of $G$ is the least number $k$ such that for any assignment of $k$-element lists to the vertices of $G$, there is a dynamic coloring of $G$ where the color on each vertex is chosen from its list. Based on Thomassen's result [12] that every planar graph is $5$-choosable, an interesting question is whether the list dynamic chromatic number of every planar graph is at most 5 or not. We answer this question by showing that $ch_d(G)\leq 5$ for every planar graph.
2010 Mathematics Subject Classification: 05C15
Key Words and Phrases: dynamic coloring, planar graph
- ⋅ 10:30 − 11:00 Multiscale ensemble clustering for finding modules in complex networks (Eun Youn Kim, Tae Wook Ko)
- 김은연*(국가수리과학연구소), 고태욱(국가수리과학연구소)
Eun Youn Kim*, National Institute for Mathematical Sciences, Tae Wook Ko, National Institute for Mathematical Sciences
The identification of modules in complex networks is important for the understanding of systems. Here, we propose an ensemble clustering method incorporating node groupings in various sizes and the sequential removal of weak ties between nodes which are rarely grouped together. This method successfully detects modules in various networks, such as hierarchical random networks and the American college football network, with known modular structures. Some of the results are compared with those obtained by modularity optimization and $K$-means clustering.
2010 Mathematics Subject Classification: 05C82
Key Words and Phrases: complex network, module, random walk
- ⋅ 11:10 − 11:40 Refined enumeration of ordered trees by the size of maximal decreasing trees (Seunghyun Seo, Heesung Shin)
- 서승현*(강원대), 신희성(인하대)
Seunghyun Seo*, Kangwon National University, Heesung Shin, Inha University
Let ${\mathcal T}_{n}$ be the set of rooted labeled trees with $n+1$ vertices. Given a rooted labeled tree $T$, A maximal decreasing subtree of a rooted labeled tree is defined by the maximal subtree from the root with all edges being decreasing. Recently Seo and Shin have studied a new refinement ${\mathcal T}_{n,k}$ of ${\mathcal T}_n$, which is the set of rooted labeled trees whose maximal decreasing subtree has $k+1$ vertices. In this talk, we introduce the refinement of the set of labeled ordered trees by the size of maximal decreasing trees. This is a joint work with Heesung Shin.
2010 Mathematics Subject Classification: 05A15, 05C30
Key Words and Phrases: ordered tree, decreasing tree, refinement
- ⋅ 13:20 − 13:50 Chair: Sangwook Kim (Chonnam National University)
- ⋅ 13:20 − 13:50 Surreal numbers and Num game (Poo-Sung Park)
- 박부성(경남대)
Poo-Sung Park, Kyungnam University
The surreal number was invented by J. H. Conway to analyze combinatorial games. In this talk, surreal number system will be introduced and analyze the new combinatorial game, Num. This game is a Nim-like game played with numbers. Two players, Left(Lift) and Right(dRop) delete ascending or descending sequence at his/her turn. The last player to move wins the game. The rule is simple but the surreal number structure is so complicated that we know little about it.
2010 Mathematics Subject Classification: 91A46
Key Words and Phrases: surreal numbers, combinatorial games
- ⋅ 15:10 − 16:00 Chair: Kwan Kyu Lee (Chosun University)
- ⋅ 15:10 − 15:40 Computing extensions of linear codes using a greedy algorithm (Sunghyu Han, Markus Grassl)
- 한성휴*(한국기술교육대), Markus Grassl(Centre for Quantum Technologies, National Univ. of Singapore)
Sunghyu Han*, Korea University of Technology and Education, Markus Grassl, Centre for Quantum Technologies, National University of Singapore
This paper deals with the problem of increasing the minimum distance of a linear code by adding one or more columns to the generator matrix. We present a simple greedy algorithm which surprisingly yields many codes improving the previously known lower bounds on the minimum distance. We also discuss variations of the algorithm that succeed when the greedy algorithm fails.
2010 Mathematics Subject Classification: 94B05
Key Words and Phrases: linear code, greedy algorithm, optimal code, extension theorem
- ⋅ 15:40 − 16:00 MacWilliams type equivalence relations (Soohak Choi, Jong Yoon Hyun, Hyun Kwang Kim and Dong Yeol Oh)
- 최수학*(포항공대), 현종윤(이화여대), 김현광(포항공대), 오동렬(한밭대)
Soohak Choi*, POSTECH, Jong Yoon Hyun, Ewha Womans University, Hyun Kwang Kim, POSTECH, Dong Yeol Oh, Hanbat National University
Let $\mathbb{P}$ be a poset of $[n]=\{1,2,\ldots,n\}$ and $E$ an equivalence relation on the set of order ideals of $\mathbb{P}$. First we introduce a new weight distribution, called the $E$-weight distribution, which generalizes the $\mathbb{P}$-weight distribution. Next we introduce the class of equivalence relations so called the MacWilliams-type equivalence relations which enables us to study the MacWilliams duality for $E$-weight distributions. We impose a particular condition on equivalence relation, and finally prove that this condition is an equivalent condition to become a MacWilliams-type equivalence relation. We also derive the MacWilliams duality for the $E$-weight distributions when $E$ satisfies this condition.
2010 Mathematics Subject Classification: 94B05
Key Words and Phrases: MacWilliams identity, poset codes, $E$-weight distribution
- Cryptography
- ⋅ 09:00 − 09:50 Chair: Youngho Park (Sejong Cyber University)
- ⋅ 09:00 − 09:30 MacWilliams duality and a Gleason-type theorem on weakly regular $p$-ary bent functions (Jong Yoon Hyun, Heisook Lee and Yoonjin Lee)
- 현종윤*(이화여대), 이혜숙(이화여대), 이윤진(이화여대)
Jong Yoon Hyun*, Ewha Womans University, Heisook Lee, Ewha Womans University, Yoonjin Lee, Ewha Womans University
In this talk, we first introduce the concept of the sign function of a Boolean function. Using this we derive a MacWilliams duality and a Gleason-type theorem on weakly regular $p$-ary bent functions. They are generalization of the results established by the authors.
2010 Mathematics Subject Classification: 94C10, 94B05
Key Words and Phrases: sign function, weakly regular bent function, MacWilliams duality
- ⋅ 09:30 − 09:50 The improvement of privacy preserving set union protocol (Hyunsook Hong, Jung Hee Cheon and Hyung Tae Lee)
- 홍현숙*(서울대), 천정희(서울대), 이형태(서울대)
Hyunsook Hong*, Seoul National University, Jung Hee Cheon, Seoul National University, Hyung Tae Lee, Seoul National University
In PKC 2012, Seo, Cheon and Katz proposed a protocol for privacy-preserving set union based on polynomial representations of sets and their reversed Laurent series. However, it has one drawback that the number of honest parties should be at least a half of all entities. It comes from the use of secret sharing techniques. In this talk, we provide a new privacy-preserving set union protocol using the homomorphic encryption in order to make up for above drawback. In our protocol, polynomial factorization should be determined uniquely over message space. However, polynomial factorization over message space of the existing homomorphic encryption is not well defined. To solve this problem, we propose a new method for polynomial representation of sets, which makes it possible unique factorization of polynomials. As a result, we obtain construct constant-round privacy-preserving set union protocol using reversed Laurent series without honest majority.
2010 Mathematics Subject Classification: 11T71
Key Words and Phrases: multi-party private set operation, privacy-preserving set operation
- ⋅ 10:10 − 11:00 Chair: Kyung-Ah Shim (National Institute for Mathematical Sciences)
- ⋅ 10:10 − 10:30 Reducing the public key size of Waters signature by using the $\alpha$-CDH problem (Youngjoo An, Hyang-Sook Lee, Seongan Lim and Juhee Lee)
- 안영주*(이화여대), 이향숙(이화여대), 임선간(이화여대), 이주희(이화여대)
Youngjoo An*, Ewha Womans University, Hyang-Sook Lee, Ewha Womans University, Seongan Lim, Ewha Womans University, Juhee Lee, Ewha Womans University
The Waters signature scheme is a building block for many other signature schemes such as group signature, ring signature and verifiably encrypted signature, etc. The scheme is provably secure under the Computational Diffie-Hellman problem(CDHP). The public key of the Waters signature consists of $n+3$ elements of bilinear groups to sign messages of $n$ bits. One practical challenging problem for the Waters signature is to reduce the size of its public key. In this paper, we modify the Waters signature scheme to produce a new weakly secure scheme with short public keys based on the $\alpha$-CDHP. Note that the $\alpha$-CDHP is as hard as the CDHP when $\alpha=1$. The public key size of our scheme, based on the $\alpha$-CDHP, is almost ${1}/({\alpha+1})$ of the public key size of the Waters signature. We also present fully-secure schemes derived from our weakly-secure schemes using a chameleon hash.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: digital signature, computational Diffie-Hellman problems, Waters signature
- ⋅ 10:30 − 11:00 Open-key cryptanalysis against block ciphers (Yongjin Yeom)
- 염용진(국민대)
Yongjin Yeom, Kookmin University
Open-key cryptanalysis is one of the recent techniques for evaluating the strength of block ciphers. In known-key or chosen-key model, under the assumption that the adversary knows the encryption key, it is evaluated whether the target cipher is ideal or not. Open-key model can be of practical use especially when we consider security of hash functions using a block cipher as a building block. In fact, encryption keys of block ciphers correspond to message or chaining variables controlled by attackers. In this talk, we are going to discuss the recent results and the future prospect of open-key attack model.
2010 Mathematics Subject Classification: 96A60
Key Words and Phrases: block cipher, open key attack, cryptanalysis
- ⋅ 13:20 − 14:50 Chair: Hong-Yeop Song (Yonsei University)
- ⋅ 13:20 − 13:50 Quantum Fourier sampling and the McEliece cryptosystem (Soojoon Lee)
- 이수준(경희대)
Soojoon Lee, Kyung Hee University
It was shown [H. Dinh, C. Moore, and A. Russell, ``The McEliece Cryptosystem Resists Quantum Fourier Sampling Attacks", arXiv:1008.2390, 2010] that the McEliece cryptosystem over well-permuted and well-scrambled linear codes resists precisely the quantum computational attacks to which the RSA and El Gamal cryptosystems are vulnerable. In this talk, we introduce this result, and consider the code-based cryptosystems as canditates for post-quantum cryptography.
2010 Mathematics Subject Classification: 81P68
Key Words and Phrases: quantum Fourier sampling, McEliece cryptosystem
- ⋅ 13:50 − 14:20 Recent progress in cryptography (Jung Hee Cheon)
- 천정희(서울대)
Jung Hee Cheon, Seoul National University
In this talk, we introduce a recent progress on cryptology.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: public key cryptography, encryption, homomorphic encryption
- ⋅ 14:20 − 14:50 Identity redactable IBS schemes (Seongan Lim, Juhee Lee and Daegun Ma)
- 임선간*(이화여대 수리과학연구소), 이주희(이화여대 수리과학연구소), 마대건(서울대)
Seongan Lim*, Institute of Mathematical Sciences, Ewha Womans University, Juhee Lee, Institute of Mathematical Sciences, Ewha Womans University, Daegun Ma, Seoul National University
In this talk, we introduce a privacy preserving identity based signature which we call identity redactable IBS, where a redacted identity can be used in the signature generation and verification without any additional interaction with the private key generator(PKG). We formalize its definition and the security requirements. We also construct an identity redactable IBS scheme.
2010 Mathematics Subject Classification: 11T71
Key Words and Phrases: redactable signature, identity redactable IBS
- ⋅ 15:00 − 16:10 Chair: Soonhak Kwon (Sungkyunkwan University)
- ⋅ 15:00 − 15:30 Pairing generating polynomial (Cheolmin Park, Eunjeong Lee and Hyang-Sook Lee)
- 박철민*(국가수리과학연구소), 이은정(이화여대), 이향숙(이화여대)
Cheolmin Park*, National Institute for Mathematical Sciences, Eunjeong Lee, Ewha Womans University, Hyang-Sook Lee, Ewha Womans University
We define a pairing generating polynomial(PGP) and propose the method to construct a family of pairing friendly curves from PGP. We show that pairing over the curve constructed from PGP can be defined directly from PGP and is always non-degenerate under usual cryptographic assumption. Finally, we provide the condition of PGP to obtain optimal pairing.
2010 Mathematics Subject Classification: 11T71
Key Words and Phrases: pairing, pairing-friendly curve
- ⋅ 15:30 − 15:50 Construction of pairing friendly curves with low embedding degree using GLV decomposition (Soo Kyung Eom, Hyang-Sook Lee and Eun Jeong Lee)
- 엄수경*(이화여대), 이향숙(이화여대), 이은정(이화여대)
Soo Kyung Eom*, Ewha Womans University, Hyang-Sook Lee, Ewha Womans University, Eun Jeong Lee, Ewha Womans University
For efficient pairing computation, one can use Vecauteren's method to obtain a sequence of integers to describe a pairing, which we call it as V-ate pairing. When the given embedding degree $k$ is low the V-ate pairing is same with the Tate pairing ($k$=2) or the ate pairing ($k$=4). Since these cases hardly give optimal pairing, it is required to construct a {\em good} pairing friendly curves to have efficient pairing computation. Scott used an efficiently computable endomorphism to have effective representation for Tate pairing and proposed a pairing friendly curves which the endomorphism can be defined. Later Ionica and Joux extended Scott's condition to obtain more pairing friendly curves. However, the restriction of the characteristic polynomial of the endomorphism is too strong to have many pairing friendly curves. To obtain much more curves, we modify their method to construct pairing friendly curves and use GLV decomposition to have effective representation of the ate pairing. We provide examples and the time comparison on the curves of the ate pairing and our modified one.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: elliptic curves
- ⋅ 15:50 − 16:10 Cipolla-Lehmer algorithm and cube roots in finite field (Gookhwa Jo, Namhun Koo, EunHye Ha and Soonhak Kwon)
- 조국화*(성균관대), 구남훈(성균관대), 하은혜(성균관대), 권순학(성균관대)
Gookhwa Jo*, Sungkyunkwan University, Namhun Koo, Sungkyunkwan University, EunHye Ha, Sungkyunkwan University, Soonhak Kwon, Sungkyunkwan University
We study algorithms to find cube roots in finite fields. Tonelli-Shanks and Cipolla-Lehmer algorithms are two standard algorithms and can be used to find the square roots and cube roots in finite fields. We propose new cube root algorithm which is a variant of Cipolla-Lehmer square root algorithm. Our method is based on the linear recurrence relation arising from cubic polynomials and can be efficiently implemented compared with Tonelli-Shanks version of cubic case.
2010 Mathematics Subject Classification: 11Y16
Key Words and Phrases: cube roots, Cipolla-Lehmer algorithm, Tonelli-Shanks algorithm, finite fields, square roots
- Representation Theory
- ⋅ 10:10 − 10:50 Chair: Myungho Kim (KIAS)
- ⋅ 10:10 − 10:30 Littlewood identity and crystal bases (Jae-Hoon Kwon)
- 권재훈(서울시립대)
Jae-Hoon Kwon, University of Seoul
In this talk, we introduce a new combinatorial model for the crystals of integrable highest weight modules over the classical Lie algebras of type $B$ and $C$ in terms of classical Young tableux. We then obtain a new description of its Littlewood-Richardson rule and a maximal Levi branching rule in terms of classical Littlewood-Richardson tableaux, which extends in a bijective way the well-known stable formulas at large ranks. This tableau model admits a natural superization and hence it produces the characters of irreducible highest weight modules over orthosymplectic Lie superalgebras, which correspond to the integrable highest weight modules over the classical Lie algebras of type $B$ and $C$ under the Cheng-Lam-Wang's super duality.
2010 Mathematics Subject Classification: 17B37
Key Words and Phrases: Littlewood identity, quantum group, crystal base
- ⋅ 10:30 − 10:50 Vertex operator algebras associated to type $G$ affine Lie algebras (Jonathan D. Axtell)
- Jonathan D. Axtell, Seoul National University
In this talk, we consider vertex operator algebras (VOA) associated with admissible modules for a type $G$ affine Lie algebra at all admissible one-third integer levels. We present formulas for the singular vectors which are needed to classify irreducible representations from the category $\mathcal{O}$. We will describe in particular our method for computing the singular vectors. This method involves an explicit description of certain invariant subalgebras of a universal enveloping algebra.
2010 Mathematics Subject Classification: 17B69
Key Words and Phrases: vertex operator algebras, singular vectors
- ⋅ 11:00 − 11:40 Chair: Jae-Hoon Kwon (University of Seoul)
- ⋅ 11:00 − 11:20 Khovanov-Lauda-Rouquier algebras and $R$-matrices (Myungho Kim, Seok-Jin Kang and Masaki Kashiwara)
- 김명호*(고등과학원), 강석진(서울대), Masaki Kashiwara(RIMS(Kyoto Univ.))
Myungho Kim*, KIAS, Seok-Jin Kang, Seoul National University, Masaki Kashiwara, RIMS(Kyoto Univ.)
For a finite dimensional representation over a quantum affine algebra, we define an action of a Khovanov-Lauda-Rouquier algebra on a localization of the tensor product of the representation. Using this action, we define a functor from the category of the modules over the KLR algebra to the category of modules over the quantum affine algebra. When the representation is the fundamental representation of $U_q(\widehat sl_n)$ of weight $\varpi_1$, we show that the functor sends an irreducible module to an irreducible module or $0$.
2010 Mathematics Subject Classification: 17B37, 33D80
Key Words and Phrases: Khovanov-Lauda-Rouquier algebras, quantum affine algebras, $R$-matrices, affine Hecke algebras
- ⋅ 11:20 − 11:40 Combinatorial spiders and tableuax for the invariants in the tensor product space of general linear group of rank 2 (Dongho Moon)
- 문동호(세종대)
Dongho Moon, Sejong University
Combinatorial spiders are a model used by Kuperberg to describe the invariants space of the tensor product spaces of representations of rank 2 Lie algebras. On the other hand, using Young tableaux is a common way to describe invariants for linear groups in general. Originally, Khovanov-Kuperberg found a bijection, based on a recursive algorithm, between standard Young tableaux of rectangular shape of three rows and spiders for sl(3) with a certain boundary condition. Recently, an explicit bijection for this has been established by Tymoczko. In this talk, I will describe how we can generalize Tymoczko's bijection to obtain a bijection for general settings of tableaux.
2010 Mathematics Subject Classification: 05E10
Key Words and Phrases: combinatorial spider, invariants, young tableuax
- Ring Theory
- ⋅ 09:10 − 09:55 Chair: Tai Keun Kwak (Daejin University)
- ⋅ 09:10 − 09:25 Skew Hochschild extensions of rings (Nam Kyun Kim)
- 김남균(한밭대)
Nam Kyun Kim, Hanbat National University
Lin and Xi studied the Hochschild extension of reduced and clean rings. Recently, Kosan, Lee and Zhou constructed $H_n(R;\sigma)$ and $B_n(R;\sigma)$ based on a given Hochschild 2-cocycle, extending the results of Lin and Xi. In this note, we introduce a skew Hochschild 2-cocycle $\alpha$, and then study skew Hochschild extension of several rings. Let $\sigma$ be an endomorphism of a ring $R$ and $k\geq 0$. A $\mathbb Z$-bilinear map $\alpha : R\times R\rightarrow M$ is called a $\sigma^k$-skew Hochschild 2-cocycle if $\alpha(ab,\sigma^k(c))-\alpha(a,\sigma^k(b)\sigma^k(c))=a\alpha(b,\sigma^k(c))-\alpha(a,\sigma^k(b))\sigma^k(c)$ for all $a,b,c\in R$.
2010 Mathematics Subject Classification: 16S99
Key Words and Phrases: skew Hochschild 2-cocycle, skew triangular matrix ring, reversible ring, clean ring
- ⋅ 09:25 − 09:40 Pr\"ufer domains and GCD-domains via pullbacks (Jung Wook Lim)
- 임정욱(서강대)
Jung Wook Lim, Sogang University
Let $D \subsetneq E$ be an extension of integral domains, $K$ be the quotient field of $D$, $\Gamma$ be a nonzero torsion-free grading monoid with $\Gamma \cap -\Gamma=\{0\}$, $\Gamma^*=\Gamma \setminus \{0\}$, $E[\Gamma]$ be the semigroup ring of $\Gamma$ over $E$, and $D+E[\Gamma^*]=\{f \in E[\Gamma] \mid f(0) \in D\}$ be the composite semigroup ring. Pullback constructions have for many years been an important tool in the arsenal of commutative algebraists because of their use in producing many examples. In this paper, we devote to study the $D+E[\Gamma^*]$ construction which is a nice example of them. In fact, we show when the domain $D+E[\Gamma^*]$ is a Pr\"ufer domain or a GCD-domain. As a corollary, we characterize a B\'ezout domains of the form $D+E[\Gamma^*]$.
2010 Mathematics Subject Classification: 13A02, 13A15, 13F05, 13G05
Key Words and Phrases: $D+E[\Gamma^*]$, Pr\"ufer domain, GCD-domain
- ⋅ 09:40 − 09:55 On rings over which every module is symmetric (Da Woon Jung)
- 정다운(부산대)
Da Woon Jung, Pusan National University
It is well known that the concept of symmetric module simplifies a kind of torsion submodule relative to a torsion theory. We study in this note the structure of rings over which modules are symmetric. We first show that this concept is placed between commutative and duo-ness. Second, we find the conditions under which the strong duo-ness can be left-right symmetric. Last, we study the minimal noncommutative strongly duo ring.
2010 Mathematics Subject Classification: 16A15
Key Words and Phrases: duo rings, strongly duo rings, regular ring
- ⋅ 10:00 − 10:45 Chair: Nam Kyun Kim (Hanbat National University)
- ⋅ 10:00 − 10:15 On finite local rings (Sang Jo Yun)
- 윤상조(부산대)
Sang Jo Yun, Pusan National University
We study several well-known concepts on finite local rings. We show the following: (1) Let $R$ be a ring and $N$ be a nil ideal of $R$. If $INI=4$, then $N$ is a commutative (hence symmetric and strongly reversible) Armendariz ring without identity such that $N^3=0$. (2) Let $R$ be a local ring with $IJ(R)I=4$. Then $R$ is Armendariz, symmetric, and strongly reversible (hence strongly IFP). and let $R$ be a local ring with $IRI=8$ and $IJ(R)I=4$.Then $R$ is commutative (hence symmetric and strongly reversible) and Armendariz.
2010 Mathematics Subject Classification: 16xx
Key Words and Phrases: local ing, Armendariz ring, insertion-of-factors-property(IFP), symmetric ring, ideal-Armendariz ring
- ⋅ 10:15 − 10:30 Reflexive rings (Tai Keun Kwak)
- 곽태근(대진대)
Tai Keun Kwak, Daejin University
Mason introduced the reflexive property for ideals, and then this concept was generalized by Kim and Baik, defining idempotent reflexive right ideals and rings. In this paper, we characterize aspects of the reflexive and one-sided idempotent reflexive properties, showing that the concept of idempotent reflexive ring is not left-right symmetric. It is proved that a (right idempotent) reflexive ring which is not semiprime (resp., reflexive), can always be constructed from any semiprime (resp., reflexive) ring. It is also proved that the reflexive condition is Morita invariant and that the right quotient ring of a reflexive ring is reflexive. It is shown that both the polynomial ring and the power series ring over a reflexive ring are idempotent reflexive. We obtain additionally that the semiprimeness, reflexive property and one-sided idempotent reflexive property of a ring coincide for right principally quasi-Baer rings.
2010 Mathematics Subject Classification: 16S99, 16U80
Key Words and Phrases: reflexive ring, right (left) idempotent reflexive ring, semiprime ring, matrix ring, polynomial ring, right quotient ring, right principally quasi-Baer ring
- ⋅ 10:30 − 10:45 Regular actions in rings (Juncheol Han, Sangwon Park)
- 한준철*(부산대), 박상원(동아대)
Juncheol Han*, Pusan National University, Sangwon Park, Dong-A University
Let $R$ be a ring with identity, $X(R)$ the set of all nonzero, non-units of $R$ and $G(R)$ the group of all units of $R$. In this paper, by considering left and right regular actions of $G(R)$ on $X(R)$ the following are investigated: (1) For a local ring $R$ such tat $X$ is a union of $n$ distinct orbits under the left (or right) regular action of $G(R)$ on $X(R)$, if $J^{n} \neq (0) = J^{n+1}$, then the set of all the distinct ideals of $R$ is exactly $\{0, J, J^{2}, \dots, J^{n}, R \}$, and each orbit under the left regular action is equal to the one under the right regular action; (2) for a full matrix ring $S$ of $n \times n$ matrices over a commutative ring $R$, the number of orbits under the left regular action of $G(R)$ on $X(R)$ is equal to the number of orbits under the right regular action of $G(R)$ on $X(R)$; the result also holds for the ring of $n \times n$ upper triangular matrices over $R$.
2010 Mathematics Subject Classification: 13H99
Key Words and Phrases: regular action, orbit, local ring, ideal
- ⋅ 10:50 − 11:35 Chair: Chan Yong Hong (Kyung Hee University)
- ⋅ 10:50 − 11:05 On rings in which every lower nilradical coincides with upper nilradical (Chang Ik Lee)
- 이창익(부산대)
Chang Ik Lee, Pusan National University
In this talk we introduce a condition which is some generalization of 2-primal-ness, Armendariz rings and von Neumann regular rings, focusing our attention on nilradicals. We will call a ring $R$ weak 2-primal if the lower nilradical is coinside with the upper nilradical. We study the structure of weak 2-primal rings and give various examples or counter examples of weak 2-primal rings.
2010 Mathematics Subject Classification: 16N40
Key Words and Phrases: 2-primal rings, Armendariz rings, von Neumann regular rings
- ⋅ 11:05 − 11:20 $A[x^{-1}]$-injective modules as $R[x]$-modules (Sangwon Park, Juncheol Han)
- 박상원*(동아대), 한준철(부산대)
Sangwon Park*, Dong-A University, Juncheol Han, Pusan National University
We can define the inverse polynomial $M[x^{-1}]$ as a left $R[x]$-module defined by $$x(m_0+m_1x^{-1}+ \cdots +m_nx^{-n})=m_1+m_2x^{-1}+ \cdots +m_nx^{-n+1}$$ $$r (m_{0} + m_{1}x^{-1} + \cdots + m_{n}x^{-n}) = rm_{0} + rm_{1}x^{-1} + \dots + rm_{n}x^{-n} $$ \noindent where $r \in R$. We call $M[x^{-1}]$ an inverse polynomial module([2]). Define $f : xE[x] \rightarrow E[x]$ by $f(xe) = e$, then $f$ is an $R[x]$-linear map. So consider the following diagram \begin{displaymath} \xymatrix@R=1pc@C=1pc{ 0 \ar[rr] && x E[x] \ar[dd]_{f} \ar[rr] && E[x] \\ \\&& E[x] &&} \end{displaymath} \noindent Then we easily see that the above diagram can not be completed for any $R[x]$-linear map. Thus $E[x]$ is not an injective left $R[x]$-module, if $E \neq 0$. But Northcott showed that if $R$ is a left noetherian ring and $E$ is an injective left $R$-module, then $E[x^{-1}]$ is an injective left $R[x]$-module. Therefore, we see that polynomial module $M[x]$ and inverse polynomial module $M[x^{-1}]$ are not only not isomorphic-they have very different properties as left $R[x]$-modules. Our main object in this paper is to prove that if an $R$-module $M$ is $A$-injective then the inverse polynomial module $M[x^{-1}]$ is $A[x^{-1}]$-injective over the polynomial ring $R[x]$.
2010 Mathematics Subject Classification: 13C11
Key Words and Phrases: inverse polynomial module, $A$-injective module, $A[x^{-1}]$-injective module
- ⋅ 11:20 − 11:35 Strong $\pi$-regularity and $\pi$-regularity of full matrix rings (Yang Lee)
- 이양(부산대)
Yang Lee, Pusan National University
Rowen showed that the 2 by 2 full matrix rings over strongly $\pi$-regular rings need not be strongly $\pi$-regular. In this work we extend the excellent method of Rowen to the $n$ by $n$ full matrix ring case. Applying this argument, we show that the $n$ by $n$ full matrix rings over $\pi$-regular rings need not be $\pi$-regular. In the process we also extend of the concepts ``proper" and ``degenerate" to be defined by Rowen.
2010 Mathematics Subject Classification: 16E50, 16S50
Key Words and Phrases: strongly $\pi$-regular ring, $\pi$-regular ring, full matrix ring, proper, degenerate
- p-Adic Functional Analysis and Their Applications
- ⋅ 09:00 − 09:25 Chair: Seog Hoon Rim (Kyungpook National University)
- ⋅ 09:00 − 09:15 On the twisted $q$-Euler numbers and polynomials with weight $0$ (Joung Hee Jin, Seog Hoon Rim and Joo Hee Jeong)
- 진정희*(경북대), 임석훈(경북대), 정주희(경북대)
Joung Hee Jin*, Kyungpook National University, Seog Hoon Rim, Kyungpook National University, Joo Hee Jeong, Kyungpook National University
The purpose of this paper is to investigate some properties of twisted $q$-Euler numbers and polynomials with weight 0. From those twisted $q$-Euler numbers with weight 0, we derive some identities on the twisted $q$-Euler numbers and polynomials with weight 0.
2010 Mathematics Subject Classification: 11B68, 11S80
Key Words and Phrases: twisted $q$-Euler numbers and polynomials
- ⋅ 09:15 − 09:20 On the twisted $q$-Bernoulli numbers and polynomials (Eun Jung Moon)
- 문은정(경북대)
Eun Jung Moon, Kyungpook National University
Recently, many authors have studied $q$-extension of Bernoulli numbers and polynomials. In this talk, we investigate several results and some properties related with the twisted $q$-Bernoulli numbers and polynomials by using $p$-adic $q$-integral.
2010 Mathematics Subject Classification: 11B68,11S80
Key Words and Phrases: the twisted $q$-Bernoulli numbers, $p$-adic $q$-integral
- ⋅ 09:20 − 09:25 Some identities on the twisted $(h,q)$-Genocchi numbers and polynomials associated with $q$-Bernstein polynomials (Sun Jung Lee, Seog Hoon Rim)
- 이선정*(경북대), 임석훈(경북대)
Sun Jung Lee*, Kyungpook National University, Seog Hoon Rim, Kyungpook National University
In this talk, we give some interesting identities on the twisted $(h,q)$-Genocchi numbers and polynomials associated with $q$-Bernstein polynomials.
2010 Mathematics Subject Classification: 11B68, 11S80
Key Words and Phrases: $q$-Bernstein polynomials, twisted $(h,q)$-Genocchi numbers and polynomials, $p$-adic $q$-integral
- ⋅ 09:25 − 09:40 Chair: Cheon Seoung Ryoo (Hannam University)
- ⋅ 09:25 − 09:30 A note on the generalized higher-order $q$-Bernoulli numbers and polynomials with weight $\alpha$ (Hui Young Lee)
- 이희영(한남대)
Hui Young Lee, Hannam University
In this paper we give some interesting equation of $p$-adic $q$-integrals on $\mathbb Z_p$. From those $p$-adic $q$-integrals, we present a systemic study of some families of extended Carlitz $q$-Bernoulli numbers and polynomials with weight $\alpha$ in $p$-adic number field.
2010 Mathematics Subject Classification: 11B68, 11S40, 11S80
Key Words and Phrases: Bernoulli numbers, Bernoulli polynomials, $p$-adic integrals on $\mathbb Z_p$
- ⋅ 09:30 − 09:35 On the $p$-adic $q$-integral representation associated with the generalized twisted $q$-Euler numbers and polynomials with weight $(\alpha,\beta)$ (Nam Soon Jung, Hui Young Lee, Jung Yoog Kang and Cheon Seoung Ryoo)
- 정남순*(한남대), 이희영(한남대), 강정욱(한남대), 유천성(한남대)
Nam Soon Jung*, Hannam University, Hui Young Lee, Hannam University, Jung Yoog Kang, Hannam University, Cheon Seoung Ryoo, Hannam University
In this paper, by using Dirichlet's character, we construct a new generalized twisted $q$-Euler numbers and polynomials with weight $(\alpha, \beta)$. And we give some interesting relations between the generalized twisted $q$-Euler polynomials with weight $(\alpha, \beta)$ and the generalized twisted $q$-Euler zeta functions with weight $(\alpha, \beta)$
2010 Mathematics Subject Classification: 11B68, 11S40, 11S80
Key Words and Phrases: Euler numbers and polynomials, the twisted q-Euler numbers and polynomials with weight (alpha,beta), the generalized twisted q-Euler numbers and polynomials with weight (alpha,beta)
- ⋅ 09:35 − 09:40 A study on multiple generalized $q$-Genocchi numbers and polynomials with weak weight $\alpha$ (J. Y. Kang, N. S. Jung, H.Y. Lee and C. S. Ryoo)
- 강정욱*(한남대), 정남순(한남대), 이희영(한남대), 유천성(한남대)
J. Y. Kang*, Hannam University, N. S. Jung, Hannam University, H.Y. Lee, Hannam University, C. S. Ryoo, Hannam University
The present papers deal with the various Genocchi numbers and polynomials. We define a new type of multiple generalized $q$-Genocchi numbers and polynomials with weak weight alpha by applying the $p$-adic $q$-integral on $\mathbb Z_p$. We will find a link between their numbers and polynomials with weak weight alpha. Also we will obtain the interesting properties of their numbers and polynomials with weak weight alpha. Moreover, we construct the multiple generalized Hurwitz-type zeta function which interpolates multiple generalized $q$-Genocchi polynomials with weak weight alpha, and find some combinatorial relations.
2010 Mathematics Subject Classification: 11B68, 11S40, 11S80
Key Words and Phrases: generalized $q$-Genocchi numbers and polynomials with weak weight alpha, multiple generalized $q$-Genocchi numbers and polynomials with weak weight alpha, multiple generalized Hurwitz-type zeta function
- ⋅ 09:40 − 10:10 Chair: Kyung Won Hwang (POSTECH)
- ⋅ 09:40 − 09:45 A note on the $p$-adic $q$-Genocchi measures with weight $\alpha$ (Won Joo Kim)
- 김원주(건국대)
Won Joo Kim, Konkuk University
The Genocchi numbers are given by
$$ t\int_{\mathbb Z_p } e^{xt} d\mu_{-1} (x)=\frac{2t}{e^{t}+1}=\sum_{n=0}^{\infty}G_{n}\frac{t^n}{n!}. \leqno(1)
$$
Note that
$$ G_{0}=0, \int_{\mathbb Z_p } x^{n} d\mu_{-1} (x)=\frac{G_{n+1}}{n+1}, n\in \mathbb Z^{+} , \leqno(2)
$$
$$ t\int_{\mathbb Z_p } e^{(x+y)t} d\mu_{-1} (y)=\frac{2t}{e^{t}+1}{e^{xt}}=\sum_{n=0}^{\infty}G_{n}(x)\frac{t^n}{n!}. \leqno(3)
$$
In this paper, we consider to $q$-Genocchi measure with weight $\alpha$ on $\mathbb Z_p$. From these Genocchi measure with weight $\alpha$, we establish some interesting identities and relations.
2010 Mathematics Subject Classification: 11B34, 11B68, 11E95
Key Words and Phrases: Genocchi measure, Genocchi number, Genocchi polynomials
- ⋅ 09:45 − 09:55 A note on the $(w, q)$-Euler numbers and polynomials with weight $\alpha$ (Jongsung Choi)
- 최종성(광운대)
Jongsung Choi, Kwangwoon University
In this talk, we study new extension of Euler numbers and polynomials. Those numbers and polynomials are closely related to the Frobenius-Euler numbers and polynomials.
2010 Mathematics Subject Classification: 11B68, 11S80
Key Words and Phrases: Euler numbers, Bernoulli numbers, fermionic $p$-adic integral
- ⋅ 09:55 − 10:00 A note on the Kim's $q$-Euler numbers and polynomials (Hyun Mee Kim)
- 김현미(국민대)
Hyun Mee Kim, Kookmin University
In this talk, we will consider some arithmetic identities for $q$-Euler numbers and polynomials. We obtain relations between $q$-Euler and $q$-Bernoulli numbers.
2010 Mathematics Subject Classification: 11B68, 11B73
Key Words and Phrases: $q$-Euler numbers, $q$-Bernoulli numbers
- ⋅ 10:00 − 10:10 On the values of Euler zeta function at positive integers (Young-Hee Kim)
- 김영희(광운대)
Young-Hee Kim, Kwangwoon University
Recently, Euler zeta function is introduced by $$ \zeta_E (s)=2 \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s}, \quad (s \in \mathbb C). $$ In this talk, we consider the values of Euler zeta function at positive integers.
2010 Mathematics Subject Classification: 11B68, 42B05
Key Words and Phrases: Euler zeta function
- ⋅ 10:10 − 11:00 Chair: Taekyun Kim (Kwangwoon University)
- ⋅ 10:10 − 10:30 Bernoulli bases and some identities (Dae San Kim, Taekyun Kim and Young-Hee Kim)
- 김대산*(서강대), 김태균(광운대), 김영희(광운대)
Dae San Kim*, Sogang University, Taekyun Kim, Kwangwoon University, Young-Hee Kim, Kwangwoon University
In this talk, we will introduce Bernoulli basis in the space of polynomials of degree less than equal to a fixed degree, express some interesting polynomials in terms of this basis and discuss some of their applications.
2010 Mathematics Subject Classification: 11B68, 11B73, 41A30
Key Words and Phrases: Bernoulli polynomial, Euler polynomial, Bernoulli basis
- ⋅ 10:30 − 10:40 A note on the generalized Euler numbers and polynomials (Cheon Seoung Ryoo)
- 유천성(한남대)
Cheon Seoung Ryoo, Hannam University
In this talk we generalize the Euler numbers and polynomials by the generalized $w$-Euler numbers $E_{n,w}(a)$ and polynomials $E_{n,w}(x:a)$. For the complement theorem, $E_{n,w}(x:a)$ have interesting different properties from the Euler polynomials and we observe an interesting phenomenon of `scattering' of the zeros of the generalized Euler polynomials $ E_{n,w}(x:a)$ in complex plane.
2010 Mathematics Subject Classification: 11B68, 11S40, 11S80
Key Words and Phrases: Euler numbers and polynomials, generalized $w$-Euler numbers and polynomials
- ⋅ 10:40 − 10:50 Divisor functions and Bernoulli numbers (Daeyeoul Kim)
- 김대열(국가수리과학연구소)
Daeyeoul Kim, National Institute for Mathematical Sciences
In this talk, we will explain properties of divisor functions. And then we suggest a problem for divisor functions and Bernouli numbers.
2010 Mathematics Subject Classification: 11A67
Key Words and Phrases: divisor functions, Bernoulli number
- ⋅ 10:50 − 11:00 On the weighted Lebesgue-Radon-Nikodym theorem with respect to $p$-adic $q$-measure on $\mathbb Z_p$ (Seog-Hoon Rim)
- 임석훈(경북대)
Seog-Hoon Rim, Kyungpook National University
In this talk, we will discuss weighted Lebesgue-Radon Nikodym theorem with respect to the weighted $p$-adic integral on $\mathbb Z_p$. In previous my research, I introduced the Lebesgue-Radon-Nikodym theorem with respect to $p$-adic fermionic $p$-adic $q$-measure. The paper to announce in this conference is extension of my previous paper.
2010 Mathematics Subject Classification: 11B68, 11S80
Key Words and Phrases: Bernoulli numbers, Lebesgue-Radon-Nikodym theorem, $p$-adic $q$-integrals, $q$-Volken\-born integral, Euler numbers and polynomials
- ⋅ 11:00 − 11:40 Chair: Daeyeoul Kim (National Institute for Mathematical Sciences)
- ⋅ 11:00 − 11:10 Intersection families and Snevily's conjecture (Kyungwon Hwang, Naeem Sheikh)
- 황경원*(동아대), Naeem Sheikh(동아대)
Kyungwon Hwang*, Donga University, Naeem Sheikh, Donga University
Let $K=\{k_{1},k_{2},\ldots,k_{r}\}$ and $L=\{l_{1},l_{2},\ldots,l_{s}\}$ be sets of nonnegative integers. Let $\mathcal{F} = \{ F_1, F_2, \dots, F_m \}$ be a family of subsets of $[n]$ with $|F_i| \in K$ for each $i$ and $|F_i \cap F_j| \in L$ for any $i \neq j$. Every subset $F_i $ of $[n]$ can be represented by a binary code $\textbf{a}=(a_{1},a_{2},\ldots, a_{n})$ such that $a_{i}=1$ if $i \in F_i$ and $a_{i}=0$ if $i \notin F_i $. Alon-Babai-Suzuki made a conjecture in 1991 in modular version. We prove Alon-Babai-Suzuki's Conjecture in non-modular version: For any $K$ and $L$ with $n\geq s + \operatorname{max}$ $k_{i}$, $|\mathcal{F}|\leq {n \choose s}+{n \choose s-1}+ \cdots +{n \choose s-r+1}$.
2010 Mathematics Subject Classification: 05A19
Key Words and Phrases: Bernoulli numbers, Lebesgue-Radon-Nikodym theorem, $p$-adic $q$-integrals, $q$-Volken\-born integral, Euler numbers and polynomials
- ⋅ 11:10 − 11:20 A study on connection for mathematical learning conditions (Jong Jin Seo)
- 서종진(부경대)
Jong Jin Seo, Pukyong National University
In this paper, a new method of analyzing individual mathematical learning conditions has been searched based on problem solving results of students. First, if $SPq1 \rightarrow SPq2$ and $SPq2 \rightarrow SPq3$, the student forms a pcsp. Second, if $STPq1 \rightarrow STPq2$ and $STPq2 \rightarrow STPq3$, the student is at a state of tpcsp. In general, If $STPqi \rightarrow STPqk (i, k;1, 2, \ldots,n)$, $Q$ forms a tpcsp.
2010 Mathematics Subject Classification: 97A30
Key Words and Phrases: convergence, connection
- ⋅ 11:20 − 11:30 Accelerated parallel projection method for solving the shortest distance problem (Dmitry V. Dolgy)
- Dmitry V. Dolgy, Kwangwoon University
The subject of this work is to develop special final procedure for finding the projection on the simplest form of a convex polyhedron-simplex and to construct of a parallel algorithm for solving this problem. Conducted computational experiments demonstrate high computational efficiency of the algorithm.
2010 Mathematics Subject Classification: 56A35
Key Words and Phrases: simplex, projection, parallel algorithm
- ⋅ 11:30 − 11:40 A note on the $q$-Bernoulli numbers and $q$-Bernstein polynomials (Taekyun Kim)
- 김태균(광운대)
Taekyun Kim, Kwangwoon University
In this talk, we discuss the $q$-extension of Bernoulli numbers and polynomials which are slightly different Carlitz's $q$-Bernoulli numbers and polynomials. From those $q$-Bernoulli numbers and polynomials, we derive some interesting identities between $q$-Bernoulli numbers and $q$-Bernstein polynomials.
2010 Mathematics Subject Classification: 11B68, 11S80
Key Words and Phrases: $p$-adic integrals, $q$-Bernoulli numbers, Witt's formula
- The Existence and Stability of Nonlinear Functional Equations
- ⋅ 09:00 − 10:00 Chair: Choonkil Park (Hanyang University)
- ⋅ 09:00 − 09:20 Functional equations in various matrix spaces (Jung Rye Lee, Dong Yun Shin)
- 이정례*(대진대), 신동윤(서울시립대)
Jung Rye Lee*, Daejin University, Dong Yun Shin, University of Seoul
We define functional equations in various matrix normed (random, fuzzy, non-Arch\-imedean) spaces, and we prove the Hyers-Ulam stability of functional equations in various matrix spaces.
2010 Mathematics Subject Classification: 37B52, 37B72
Key Words and Phrases: Hyers-Ulam stability, functional equation, matrix normed space, matrix random space, matrix fuzzy space, matrix non-Archimedean space
- ⋅ 09:20 − 09:40 Stability of Higher derivations in non-Archimedean Banach algebras (Juri Lee, Hark-Mahn Kim)
- 이주리*(충남대), 김학만(충남대)
Juri Lee*, Chungnam University, Hark-Mahn Kim, Chungnam University
In this paper, we investigate homomorphisms from unital $C^*$-algebras to unital Banach algebras and derivations from unital $C^*$-algebras to Banach $A$-modules related to a Cauchy--Jensen functional inequality.
2010 Mathematics Subject Classification: 39B72, 46H30, 46B06
Key Words and Phrases: Hyers--Ulam stability, unital $C^*$-algebras, unitary group, homomorphisms, derivations
- ⋅ 09:40 − 10:00 Stability of Jordan derivations on $C^*$-algebras (Eun Young Son, Hark-Mahn Kim and Madjid E. Gordji)
- 손은영*(충남대), 김학만(충남대), Madjid E. Gordji(Semnan Univ.)
Eun Young Son*, Chungnam National University, Hark-Mahn Kim, Chungnam National University, Madjid E. Gordji, Semnan University
In this paper, we investigate almost linear Jordan derivations on $C^*$-algebras associated to the following generalized Cauchy--Jensen functional inequality \begin{eqnarray*} \Big\| \sum_{i=1}^{l}f(x_i)+m\sum_{j=1}^{n}f(y_j) \Big\| & \leq& \Big\|m f \Big( \frac{\sum_{i=1}^{l}x_i}{m}+ \sum_{i=1}^{n}y_j \Big) \Big\|. \end{eqnarray*} As results, we apply the almost linear Jordan derivations on $C^*$-algebras to contractive, weakly amenable or amenable Banach modules.
2010 Mathematics Subject Classification: 39B72, 46H25, 46B06
Key Words and Phrases: Cauchy-Jensen functional inequality, $C^*$-algebras, Jordan derivations, weakly amenable Banach modules
- ⋅ 10:10 − 11:40 Chair: Gwang Hui Kim (Kangnam University)
- ⋅ 10:10 − 10:25 Stability of functional inequalities in paranormed spaces (Choonkil Park)
- 박춘길(한양대)
Choonkil Park, Hanyang University
In this paper, we investigate additive functional inequalities in paranormed spaces. Furthermore, we prove the Hyers-Ulam stability of additive functional inequalities in paranormed spaces.
2010 Mathematics Subject Classification: 35A17, 39B52, 39B72
Key Words and Phrases: Hyers-Ulam stability, Jordan-von Neumann functional equation, functional inequality, paranormed space
- ⋅ 10:25 − 10:40 Homomorphisms and derivations related to Cauchy--Jensen functional inequality (Min Young Kim, Madjid E. Gordji)
- 김민영*(충남대), Madjid E. Gordji(Semnan Univ.)
Min Young Kim*, Chungnam National University, Madjid E. Gordji, Semnan University
In this talk, we investigate homomorphisms from unital $C^*$-algebras to unital Banach algebras and derivations from unital $C^*$-algebras to Banach $A$-modules related to a Cauchy--Jensen functional inequality.
2010 Mathematics Subject Classification: 39B72, 46H30, 46B06
Key Words and Phrases: Hyers--Ulam stability, unital $C^*$-algebras, unitary group, homomorphisms, derivations
- ⋅ 10:40 − 10:55 Convergence of an iterative algorithm for variational inequalities and a system of equilibrium problems (Sun Young Cho, Shin Min Kang)
- 조선영*(경상대), 강신민(경상대)
Sun Young Cho*, Gyeongsang National University, Shin Min Kang, Gyeongsang National University
In this paper, we consider the problem of approximating a common element in the solution set of variational inequalities and in the solution set of a system of generalized equilibrium problems of inverse-strongly monotone mappings based on a viscosity iterative algorithm. Strong theorems of common elements are established in real Hilbert spaces.
2010 Mathematics Subject Classification: 47H05, 47H09, 47J25
Key Words and Phrases: variational inequality, equilibrium problem, nonexpansive mapping, inverse-strongly monotone mapping
- ⋅ 10:55 − 11:10 Some iterative algorithms for strict-pseudo contractive mappings and certain variational inequalities (Jong Soo Jung)
- 정종수(동아대)
Jong Soo Jung, Dong-A University
In this talk, we introduce new implicit and explicit iterative algorithms for finding the fixed point of a $k$-strictly pseudo-contractive mapping and for solving variational inequalities related to Lipschitzian and strongly monotone operator in Hilbert spaces. We establish strong convergence of the sequences generated by the proposed algorithms to a fixed point of a $k$-strictly pseudo-contractive mapping. Such a point is also a solution of a certain variational inequality defined on the set of fixed points. As direct consequences, we obtain the unique minimum-norm fixed point of a $k$-strictly pseudo-contractive mapping.
2010 Mathematics Subject Classification: 49J40, 47J20, 47H09, 47H05
Key Words and Phrases: implicit algorithm, explicit algorithm, $k$-strictly pseudo-contractive mapping, variational inequalities, fixed points, nonexpansive mapping, Lipschitzian, $\rho$-Lipschitzian and $\eta$-strongly monotone operator
- ⋅ 11:10 − 11:25 Some problems on generalized differentiation and Aumann integral (Nguyen Huy Chieu, Gue Myung Lee)
- Nguyen Huy Chieu*(Vinh Univ.), 이규명(부경대)
Nguyen Huy Chieu*, Vinh University, Gue Myung Lee, Pukyong National University
Generalized differentiation theories have been developed intensively in the last five decades. The theories create necessary tools for studying many important problems in mathematical programming, the calculus of variations, optimal control and differential inclusions, ect. The Aumann integral of a set-valued mapping has been well recognized to be one the most useful concepts in set-valued analysis. Thus, relationships between generalized differentiation and Aumann integral are definitely remarkable in nonsmooth analysis and related fields. The aim of this talk is to discuss some problems on this topic.
2010 Mathematics Subject Classification: 49J52
Key Words and Phrases: generalized differentiation, Aumann integral, relationships
- ⋅ 11:25 − 11:40 Iterations for variational inequalities and fixed point problems (Yeol Je Cho)
- 조열제(경상대)
Yeol Je Cho, Gyeongsang National University
In this paper, we introduce some iterations for finding a common solution of the equilibrium problem, variational inequality problem and fixed point problem and give some examples to illustrate the main results in this paper.
2010 Mathematics Subject Classification: 47H10, 54H25
Key Words and Phrases: equilibrium problem, variational inequality problem, fixed point problem
- Submanifold and Foliation in Riemannian Geometry
- ⋅ 09:00 − 09:50 Chair: Byung Hak Kim (Kyung Hee University)
- ⋅ 09:00 − 09:25 $\mathfrak D$-parallelism of normal and structure Jacobi operators for real hypersurfaces in complex two-plane Grassmannians (Imsoon Jeong, C. J. G. Machado, J. D. Perez and Young Jin Suh)
- 정임순*(경북대), C. J. G. Machado(Granada Univ.), J. D. Perez(Granada Univ.), 서영진(경북대)
Imsoon Jeong*, Kyungpook National University, C. J. G. Machado, Granada University, J. D. Perez, Granada University, Young Jin Suh, Kyungpook National University
In this talk, we give some non-existence theorems for Hopf hypersurfaces in complex two-plane Grassmannians with $\mathfrak D$-parallel normal Jacobi operator and $\mathfrak D$-parallel structure Jacobi operator if the distribution $\mathfrak D$ or the ${\mathfrak D}^{\bot}$-component of the Reeb vector field are invariant by the shape operator, respectively.
2010 Mathematics Subject Classification: 53C40, 53C15
Key Words and Phrases: complex two-plane Grassmannians, $\mathfrak D$-parallel normal Jacobi operator, $\mathfrak D$-parallel structure Jacobi operator, invariant Reeb vector, Hopf hypersurfaces
- ⋅ 09:25 − 09:50 Transversal vector fields on K\"ahler foliations (Seoung Dal Jung)
- 정승달(제주대)
Seoung Dal Jung, Jeju National University
Let $\mathcal F$ be a K\"ahler foliation on a compact Riemannian manifold $M$. If the transversal scalar curvature of $\mathcal F$ is non-zero constant, then any transversal conformal field is a transversal Killing field; and if the transversal Ricci curvature is non-negative and positive at some point, then there are no transversally holomorphic fields.
2010 Mathematics Subject Classification: 53C
Key Words and Phrases: transversal conformal field, Killing field, holomorphic field, Jacobi field
- ⋅ 10:00 − 10:55 Chair: Seoung Dal Jung (Jeju National University)
- ⋅ 10:00 − 10:15 Real hypersurfaces in complex two-plane Grassmannians with $\eta$-parallel shape operator (Seonhui Kim, Hyunjin Lee and Young Jin Suh)
- 김선희*(경북대), 이현진(경북대), 서영진(경북대)
Seonhui Kim*, Kyungpook National University, Hyunjin Lee, Kyungpook National University, Young Jin Suh, Kyungpook National University
In this talk, we introduce a notion of $\eta$-parallel shape operator in complex two-plane Grassmannian $G_2({\mathbb C}^{m+2})$ and give a non-existence theorem for a Hopf hypersurface $M$ in $G_2({\mathbb C}^{m+2})$ with $\eta$-parallel shape operator.
2010 Mathematics Subject Classification: 53C40, 53C15
Key Words and Phrases: complex two-pane Grassmannians, Hopf hypersurface, $\eta$-parallel shape operator
- ⋅ 10:15 − 10:35 Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster Reeb-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
In this talk, we introduce a notion of Reeb-parallel for a hypersurface in complex two-plane Grassmannians $G_{2}(\mathbb C^{m+2})$, and gave a characterization of a real hypersurface of Type (A) in $G_{2}(\mathbb C^{m+2})$, that is, a Hopf hypersurface $M$ in $G_{2}(\mathbb C^{m+2})$ is locally congruent to 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 hypersurfaces, generalized Tanaka-Webster connection, Reeb-parallel shape operator
- ⋅ 10:35 − 10:55 Results about classification of real hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster invariant shape operator (Eunmi Pak, Young Jin Suh and Imsoon Jeong)
- 박은미*(경북대), 서영진(경북대), 정임순(경북대)
Eunmi Pak*, Kyungpook National University, Young Jin Suh, Kyungpook National University, Imsoon Jeong, Kyungpook National University
In this talk, we introduce new notions of invariant for generalized Tanaka-Webster connection for a hypersurface in complex two-plane Grassmannians, and give classifications of Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster invariant shape operator.
2010 Mathematics Subject Classification: 53C40, 53C15
Key Words and Phrases: real hypersurfaces, complex two-plane Grassmannians, generalized Tanaka-Webster connection, Reeb parallel shape operator, g-T-W invariant shape operator
- ⋅ 10:55 − 11:40 Chair: Young Jin Suh (Kyungpook National University)
- ⋅ 10:55 − 11:15 Characterizations of real hypersurfaces in a nonflat complex space form with composition of two operators (Woon Ha Sohn, In Bae Kim and Dong Ho Lim)
- 손운하*(한국외대), 김인배(한국외대), 임동호(한국외대)
Woon Ha Sohn*, Hankuk University of Foreign Studies, In Bae Kim, Hankuk University of Foreign Studies, Dong Ho Lim, Hankuk University of Foreign Studies
Let $M$ be a real hypersurface in a nonflat complex space form $M_n(c)$. In this paper we prove that if $ L_{\xi} R_{\xi} + R_{\xi} L_{\xi}=0 $ holds on $M$, then $M$ is a Hopf hypersurface. The operator $R_{\xi}$ is the well-known structure Jacobian one and $L_{\xi}$ is the induced operator from the Lie derivative with respect to the structure vector field $\xi$ respectively. We also characterize such Hopf hypersurfaces of $M_n(c)$.
2010 Mathematics Subject Classification: 53C40, 53C15
Key Words and Phrases: real hypersurface, structure Jacobi operator, Hopf hypersurface, model spaces of type A
- ⋅ 11:15 − 11:40 Ricci curvature in the Riemannian manifolds (Byung Hak Kim, Sang Deuk Lee, Jin Hyuk Choi and Young Ok Lee)
- 김병학*(경희대), 이상덕(단국대), 최진혁(경희대), 이영옥(경희대)
Byung Hak Kim*, Kyung Hee University, Sang Deuk Lee, Dankook University, Jin Hyuk Choi, Kyung Hee University, Young Ok Lee, Kyung Hee University
The Ricci curvatures are important and interesting concept in the Riemannian geometry. In this talk, we are to introduce various conditions about Ricci curvatures and geometric characterizations of Riemannian manifolds admitting such Ricci conditions.
2010 Mathematics Subject Classification: 53A55
Key Words and Phrases: Ricci curvature
- Quantum Computation and Quantum Cryptography
- ⋅ 10:10 − 11:40 Chair: Soojoon Lee (Kyung Hee University)
- ⋅ 10:10 − 10:30 Security analysis on arbitrated quantum signature schemes for known quantum messages (Taewan Kim, Jeong Woon Choi and Soojoon Lee)
- 김태완*(이화여대), 최정운(SK텔레콤), 이수준(경희대)
Taewan Kim*, Ewha Womans University, Jeong Woon Choi, SK Telecom, Soojoon Lee, Kyung Hee University
Recently, it has been shown that many arbitrated quantum signature schemes using the optimal quantum one-time encryptions based on Pauli operators are not secure by Choi et al. [Phys. Rev. A. 84, 062330 (2011)]. In order to guarantee the security of the schemes for unknown quantum messages, they have suggested $(U,V)$-type quantum encryptions. Here, we investigate the security analysis on arbitrated quantum signature schemes for known quantum messages.
2010 Mathematics Subject Classification: 81P94
Key Words and Phrases: quantum signature schemes
- ⋅ 10:30 − 10:50 General polygamy inequality of multi-party quantum entanglement (Jeong San Kim)
- 김정산(수원대)
Jeong San Kim, University of Suwon
Using entanglement of assistance, we establish a general polygamy inequality of multi-party entanglement in arbitrary dimensional quantum systems. For multi-party closed quantum systems, we relate our result with the monogamy of entanglement to show that the entropy of entanglement is an universal entanglement measure that bounds both monogamy and polygamy of multi-party quantum entanglement.
2010 Mathematics Subject Classification: 81P45
Key Words and Phrases: multi-party quantum entanglement, entanglement of assistance
- ⋅ 10:50 − 11:10 Randomizing quantum states in Shattem $p$-norms (Kabgyun Jeong)
- 정갑균(고등과학원)
Kabgyun Jeong, KIAS
In this talk, we formularize a method for randomizing quantum states with respect to the Shatten $p$-norms in trace class, including an application such as super-dense coding.
2010 Mathematics Subject Classification: 81P68
Key Words and Phrases: Randomizing quantum states
- ⋅ 11:10 − 11:40 On the relation between quantum and classical computational complexity (Dong Pyo Chi)
- 지동표(서울대)
Dong Pyo Chi, Seoul National University
We will discuss the relation between classical and quantum computational complexity. We hope this relation might suggest another point of view to the problem $P \not= NP$.
2010 Mathematics Subject Classification: 03D15
Key Words and Phrases: computational complexity
- Free Open Source Mathematics Software Sage Day
- ⋅ 09:00 − 10:00 Chair: Heesung Shin (Inha University)
- ⋅ 09:00 − 09:20 How we were able to make a Korean version of Sage and Mobile Sage server (JinYeong Bak, Sang-Gu Lee and Duk-Sun Kim)
- 박진영*(카이스트), 이상구(성균관대), 김덕선(NHN-네이버)
JinYeong Bak*, KAIST, Sang-Gu Lee, Sungkyunkwan University, Duk-Sun Kim, NHN-Naver
Sage is an open-source mathematics software system. It can give us reliable outputs freely through combining the power of many existing open-source packages. We, SKKU team, were able to make a Korean version of Sage server and Mobile Sage server as mobile learning is getting more intesersts.In this short talk, We will share our experiences on building Korean version of Sage and Mobile Sage server.
2010 Mathematics Subject Classification: 97Uxx
Key Words and Phrases: Sage, mathematical educations, Mobile Sage, CAS
- ⋅ 09:20 − 09:40 Use of Sage-Math in visualization of linear algebra concepts (Kyung-Won Kim, Hye-In Jeong and Sang-Gu Lee)
- 김경원*(성균관대), 정혜인(성균관대), 이상구(성균관대)
Kyung-Won Kim*, Sungkyunkwan University, Hye-In Jeong, Sungkyunkwan University, Sang-Gu Lee, Sungkyunkwan University
Linear Algebra is the first abstract mathematics subject for most of new college students. Hence, most of students face some difficulties to deal with various novel mathematical concepts. In this paper, we introduce the visualization of linear algebra concepts which are developed with Sage (a free web site to do mathematical computations). Sage-Math has been developed at the University of Washington in 2005. Sage includes many features and functions are Web-Mathematica enough to surpass. Students don't need any program installed on their computer, since they can access any available Sage server in the world from anywhere at anytime to perform their mathematical computations and simulations. These can be used for students' better understanding in college mathematics education as well as a secondary school mathematics education. You can find these contents for your test or teaching at http://matrix.skku.ac.kr/2012-sage/sage-la.
2010 Mathematics Subject Classification: 15Axx, 97Uxx
Key Words and Phrases: linear algebra, Sage, Sage-Math, mathematical educations, CAS
- ⋅ 09:40 − 10:00 Implementing decoding algorithms in Sage (Kwan Kyu Lee)
- 이관규(조선대)
Kwan Kyu Lee, Chosun University
Sage is a particularly convenient environment for implementing and experimenting with algebraic algorithms. If Sage lacks a feature, then you can add it, which then becomes a part of Sage and every Sage user can benefit. In this short talk, a personal story of how I got into implementing algebraic decoding algorithms in Sage, and contributed to the Sage project along the way.
2010 Mathematics Subject Classification: 94B35
Key Words and Phrases: Sage, decoding algorithm
- ⋅ 10:10 − 11:30 Chair: Sang-il Oum (KAIST)
- ⋅ 10:10 − 10:30 Sage: Ultimate tool for math research and teaching (Robert Beezer)
- Robert Beezer, University of Puget Sound
Sage is a powerful tool for research and teaching, in large measure because of its design as open source software. I will demonstrate several short examples from algebra and discrete mathematics as an introduction to the possibilities.
2010 Mathematics Subject Classification: 97Uxx
Key Words and Phrases: Sage, Sage-Math, CAS, algebra
- ⋅ 10:30 − 10:50 Sage and interactive (Dan Drake)
- Dan Drake, KAIST
It is easy to make interactive applets to demonstrate and explore mathematical ideas -- mathlets -- using Sage's @interact feature. I will demonstrate variety of these mathlets that I have used in my research and teaching and show how you can easily make your own.
2010 Mathematics Subject Classification: 97Pxx
Key Words and Phrases: Sage, CAS, mathlets, Sage-Math
- ⋅ 10:50 − 11:10 Full Sage contents for linear algebra and Sage matrix calculator (Hee-Dong Yoon, Sang-Gu Lee and Jae Hwa Lee)
- 윤희동*(성균관대), 이상구(성균관대), 이재화(성균관대)
Hee-Dong Yoon*, Sungkyunkwan University, Sang-Gu Lee, Sungkyunkwan University, Jae Hwa Lee, Sungkyunkwan University
Over the last 20 years, the issue of using an adequate CAS tool in teaching and learning of linear algebra has been raised constantly. And a variety of CAS tools were introduced in many linear algebra textbooks. In Korea, however, because of some realistic problems, they have not been introduced in the class and the theoretical aspect of linear algebra has been focused on in teaching and learning of it. In this talk, we suggest Sage as an alternative for CAS tools overcoming the problems mentioned above. And, we introduce full contents for linear algebra and matrix calculator that Sage was used to develop. Taking advantage of them, almost all the concepts of linear algebra can be easily covered and the size of matrices can be expanded without difficulty. In addition, we introduce a variety of linear algebra contents and matrix calculator, which are developed in Sage single cell and therefore easy to use without having to log in.
2010 Mathematics Subject Classification: 15-01
Key Words and Phrases: linear algebra, Sage, matrix calculator, web contents
- ⋅ 11:10 − 11:30 Linear algebra and calculus with Sage (Sang-Gu Lee)
- 이상구(성균관대)
Sang-Gu Lee, Sungkyunkwan University
We will show what has been done and can be done on a linear algebra and a calculus with Sage. We will discuss some positive effect of these changes in university mathematics educations. (http://matrix.skku.ac.kr/2012-sage/sage-la, http://matrix.skku.ac.kr/2012-LAwithSage/interact)
2010 Mathematics Subject Classification: 15Axx, 97Uxx
Key Words and Phrases: CAS, SAGE, linear algebra, calculus
- ⋅ 11:30 − 11:40 Q $\&$ A(질의응답) Practice and Discussion with All participant Chair:
- Ergodic Theory and Dynamical Systems
- ⋅ 09:00 − 10:00 Chair: Dong Han Kim (Dongguk University)
- ⋅ 09:00 − 09:20 Decomposition problems in symbolic dynamics (Uijin Jung, Soonjo Hong and In-Je Lee)
- 정의진*(고등과학원), 홍순조(Univ. de Chile), 이인제(고등과학원)
Uijin Jung*, KIAS, Soonjo Hong, Universidad de Chile, In-Je Lee, KIAS
Problems concerning decompositions of one code into a composition of several codes were considered in symbolic dynamics for coding purpose. In this talk, we will present recent developments on decompositions of factor maps and embeddings between symbolic dynamical systems.
2010 Mathematics Subject Classification: 37B10, 37B40
Key Words and Phrases: shift of finite type, sofic shift, decomposition, entropy, Perron number
- ⋅ 09:20 − 09:40 Limit distributions of quantum walks and the interacting Fock spaces (Hyun Jae Yoo, Chul Ki Ko)
- 유현재*(한경대), 고철기(연세대)
Hyun Jae Yoo*, Hankyong National University, Chul Ki Ko, Yonsei University
We investigate the limit distributions of the discrete time quantum walks on the lattices. The moments of the limit distribution will be represented by vacuum expectations of the monomials of a self-adjoint operator in a suitably associated interacting Fock space. As a byproduct, we get some transformation formulas for certain integrals.
2010 Mathematics Subject Classification: 82B41, 60F05, 46L53,
Key Words and Phrases: quantum walks, limit distribution, moments, interacting Fock space
- ⋅ 09:40 − 10:00 Conditions for a sofic measure to be a Gibbs measure (Jisang Yoo)
- 유지상(카이스트)
Jisang Yoo, KAIST
If a factor map between mixing shifts of finite type satisfies the condition we call fiber-mixing, the map projects Markov measures to Gibbs measures. Properties of fiber-mixing maps and continuing maps are also discussed. Let $\pi$ be a factor map from a mixing shift of finite type $X$ to $Y$. The map $\pi$ is fiber-mixing if given $x, \bar{x} \in X$ with $\pi (x) = \pi (\bar{x}) = y \in Y$, there is $z \in \pi^{-1} (y)$ that is left asymptotic to $x$ and right asymptotic to $\bar{x}$. A factor map $\pi$ from a shift space $X$ onto another shift space $Y$ is right continuing if for each $x \in X$ and $y\in Y$ such that $\pi(x)$ is left-asymptotic to $y$, there is at least one $\bar x \in X$ such that $\bar x$ is left asymptotic to $x$ and $\pi(\bar x)=y$.
2010 Mathematics Subject Classification: 37B10
Key Words and Phrases: measures, factor maps
- ⋅ 10:10 − 11:40 Chair: Jungseob Lee (Ajou University)
- ⋅ 10:10 − 10:30 Generic diffeomorphisms with orbital shadowing (Manseob Lee, Keonhee Lee)
- 이만섭*(목원대), 이건희(충남대)
Manseob Lee*, Mokwon University, Keonhee Lee, Chungnam National University
We show that for $C^1$-generic $f$, a locally maximal chain transitive set is hyperbolic if it is orbitally shadowable.
2010 Mathematics Subject Classification: 37C20, 37C50, 37D20, 37B30, 37C29
Key Words and Phrases: chain transitive set, $C^1$-generic, hyperbolic, orbital shadowing, transitive
- ⋅ 10:30 − 10:50 Dynamics of unipotent flow and disjointness of Moebius function (Seonhee Lim, Junehyuk Jung)
- 임선희*(서울대), 정준혁(Princeton Univ.)
Seonhee Lim*, Seoul National University, Junehyuk Jung, Princeton University
We show that the disjointness of Moebius function from unipotent flow (a la Bourgain-Sarnak-Ziegler) holds for some Lie groups more general than $SL(2,R)$, using Ratner theorem on joinings of unipotent flow of semisimple Lie groups. This is a joint work with Junehyuk Jung.
2010 Mathematics Subject Classification: 37Axx
Key Words and Phrases: ergodic theory of Lie groups, unipotent flows
- ⋅ 11:00 − 11:20 Aperiodic mono-tile tiling (Jeong-Yup Lee, R. V. Moody)
- 이정엽*(관동대), R. V. Moody(University of Victoria)
Jeong-Yup Lee*, Kwandong University, R. V. Moody, University of Victoria
In the study of aperiodic tilings, it has long been an open question whether there exists a tiling made by single prototile with matching rules whose translations, rotations, and reflections tile a plane only in aperiodic way. Taylor and Socolar have found such an example (TS-tiling). Recently there have been studies on this tiling from various aspects. In this talk we introduce this aperiodic mono-tile tiling and discuss properties of this tiling from the point of view of mathematical quasicrystals. We consider a parity tiling removing all the coloured information of TS-tiling. It turns out that TS-tiling is mutually locally derivable from the parity tiling.
2010 Mathematics Subject Classification: 52Cxx
Key Words and Phrases: aperiodic tiling, quasicrystal
- ⋅ 11:20 − 11:40 Diophantine type of interval exchange maps (Dong Han Kim)
- 김동한(동국대)
Dong Han Kim, Dongguk University
Diophantine type of irrational numbers have several equivalent arithmetical characterizations as well as several equivalent characterizations in terms of the dynamics of the corresponding circle rotations. In this talk we investigate how to generalize Diophantine conditions to interval exchange maps.
2010 Mathematics Subject Classification: 37E05, 11J70
Key Words and Phrases: interval exchange map, Diophantine type, recurrence time, Rauzy-Veech continued fraction algorithm
- Tools and Techniques in Math Research and Education
- ⋅ 13:20 − 14:50 Chair: Seong-Deog Yang (Korea University)
- ⋅ 13:20 − 14:00 Mathematica와 Symbolic Computing 패키지를 활용한 수학 교육의 새로운 모델 소개 (Youngjoo Chung)
- 정영주(광주과학기술원)
Youngjoo Chung, GIST
기호 계산 (Symbolic Computing)은 수치 계산 (Numerical Computing)에 대비되는 개념으로 변수와 상수 등의 기호를 직접 이용하여 대수학적 규칙에 따라 수식을 다루고 계산을 수행하는 컴퓨팅 기법이며, 이를 통하여 해석적 계산 능력의 획기적인 향상을 기하고 여러 가지 실질적인 문제에도 응용할 수 있다. 이러한 방식의 장점은 알려진 알고리즘과 계산 규칙을 내장하고 있는 컴퓨터 프로그램을 활용함으로써 수식 전개와 계산 과정에서의 오류를 최소화하여 정확성을 기할 수 있다는 점이다. Symbolic Computing 패키지는 기호 계산을 수행할 수 있는 상용 소프트웨어인 Mathematica의 기능을 보완하는 Add-On 패키지로 다양한 수식을 다룰 수 있는 550여 개에 이르는 함수를 포함하고 전통적 표기법에 따른 미분, 적분, 벡터 연산자, 브라켓 등 수식의 입출력을 지원하며 Mathematica에서 제공하는 다른 기능들과 연동하여 사용할 수 있다. 본 발표에서는 이 패키지의 개발 및 사용법과 수학, 과학 및 공학 분야에서의 교육과 연구에 적용된 사례에 대해 설명하고자 한다. 이 패키지를 통해 기호 계산 기법이 널리 알려지고 활용된다면 과학기술 분야의 발전에 크게 기여할 것으로 기대된다.
2010 Mathematics Subject Classification: 37E05, 11J70
Key Words and Phrases: interval exchange map, Diophantine type, recurrence time, Rauzy-Veech continued fraction algorithm
- ⋅ 14:10 − 14:50 Mathematical software for research, education, and exhibition (Tatsuyoshi Hamada)
- Tatsuyoshi Hamada, Fukuoka University/JST CREST/OCAMI
In 1960s, the first generation of computer algebra systems were developed by some research groups. ``REDUCE" was for theoretical physics, and ``Macsyma" was a system as part of a project of artificial intelligence. Both of them are later marketed commercially. In Japan, Professor Hideo Wada published some books of computer algorithms for number theory in 1980s. The community of developing mathematical software was very small, but it became little by little increasing. For example, UBASIC was a very famous software for number theorist in Japan, it was developed by Professor Yuji Kida. From 1990s, several software was released for investigating mathematics, for example, number theory, topology, and algebraic geometry, etc. On the other hand, the idea of Open Source Software changed the environment of mathematical software. Now, many open source mathematical software are freely downloadable, and used for research, education and exhibition. I will introduce the recent circumstances of open source mathematical software in Japan.
2010 Mathematics Subject Classification: 97N80, 97-XX
Key Words and Phrases: mathematical software
- Graduate Student Session I
- ⋅ 09:00 − 10:00 Chair: Seong Sic Yu (Inha University)
- ⋅ GST I 09:00 − 09:30 Level set, phase-field, and immersed boundary methods for two-phase fluid flows (Da Rae Jeong, Jun Seok Kim, Hyun Geun Lee, Jae Min Shin and Yi Bao Li)
- 정다래*(고려대), 김준석(고려대), 이현근(고려대), 신재민(고려대), 이의보(고려대)
Da Rae Jeong*, Korea University, Jun Seok Kim, Korea University, Hyun Geun Lee, Korea University, Jae Min Shin, Korea University, Yi Bao Li, Korea University
Understanding the behavior of fluid flows is often difficult since fluid flows involve inherent nonlinearities, interface deformations, and topological changes. Much effort has been put into studying fluid flows through analysis, asymptotics, and numerical simulation. In this paper, we focus on review on studies of two-phase fluid flows using level set, phase-field, and immersed boundary methods. Level set method is a computational technique for tracking moving interfaces and relies on an implicit formulation of the interface as the zero-level set of an auxiliary function. Phase-field method replaces sharp fluid interfaces by thin but nonzero thickness transition regions where the interfacial forces are smoothly distributed. The basic idea is to introduce a conserved order parameter that varies continuously over thin interfacial layers and is mostly uniform in the bulk phases. Immersed boundary method was originally developed to model blood flow in the heart and has been widely applied to bio fluid dynamics problems with complex geometries and immersed elastic membranes. The main idea of immersed boundary method is to use a regular Eulerian computational grid for the fluid mechanics along with a Lagrangian representation of the immersed boundary. We discuss the essential ideas behind these techniques and various numerical experiments for two-phase fluid flows.
2010 Mathematics Subject Classification: 53C35
Key Words and Phrases: level set, phase-field, immersed boundary method
- ⋅ GST I 09:30 − 10:00 The statistical inferences of fuzzy regression using bootstrap techniques (Woo Joo Lee, Hye Young Jung and Hae Kyung Kim)
- 이우주*(연세대), 정혜영(연세대), 김해경(연세대)
Woo Joo Lee*, Yonsei University, Hye Young Jung, Yonsei University, Hae Kyung Kim, Yonsei University
In this paper, we estimate the parameters of fuzzy regression models and perform statistical inferences with crisp inputs and fuzzy outputs for each alpha-cut. The proposed approach of statistical inference includes fuzzy least squares methods (FLS) and bootstrap techniques. FLS is constructed on the basis of minimizing the square of the total difference between the observed and estimated outputs. One advantage of the bootstrap method is that it does not require specific population shapes such as a normal distribution. A numerical example is illustrated to test hypotheses and to provide the percentile confidence regions of the proposed approach
2010 Mathematics Subject Classification: 62A86
Key Words and Phrases: fuzzy regression, fuzzy least squares method, bootstrap method
- ⋅ 10:10 − 11:10 Chair: Seonja Kim (Chungwoon University)
- ⋅ GST I 10:10 − 10:40 On equivalenced association schemes of valency $4$ (Jeong Rye Park)
- 박정례(부산대)
Jeong Rye Park, 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 equivalenced association scheme of valency $k$ if each nonidentity element of $S$ has valency $k$. In this talk we consider a $4$-equivalenced association scheme $(\Omega,S)$. Every $4$-equivalened association scheme is a symmetric scheme (It was proved by Z. Arad, Y. Erez and M. Muzychuk) and we prove that for any $s\in S$, the complex product $ss$ contains at most three elements.
2010 Mathematics Subject Classification: 05E30
Key Words and Phrases: equivalenced, association scheme
- ⋅ GST I 10:40 − 11:10 Real hypersurfaces in complex two-plane Grassmannians with Bochner maximum principle (Jung Taek Oh, Young Jin Suh and Juan de Dios P\'erez)
- 오정택*(경북대), 서영진(경북대), Juan de Dios P\'erez(University of Granada)
Jung Taek Oh*, Kyungpook National University, Young Jin Suh, Kyungpook National University, Juan de Dios P\'erez, University of Granada
In this paper we give some characterizations of real hypersurfaces of type $A$ in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$ that are tubes over a totally geodesic $G_2({\mathbb C}^{m+1})$ in $G_2({\mathbb C}^{m+2})$ in terms of the squared norm of the covariant derivatives of the shape operator $A$ by estimating the inequality of the Laplacian defined on compact real hypersurfaces in $G_2({\mathbb C}^{m+2})$.
2010 Mathematics Subject Classification: 14M15
Key Words and Phrases: Grassmannians
- ⋅ 11:10 − 11:40 Chair: Jung Taek Oh (Kyungpook National University)
- ⋅ GST I 11:10 − 11:40 How does it look like hopf hypersurfaces with Reeb parallel shape operator in complex two-plane Grassmannians (Chang Hwa Woo, Young Jin Suh and Hyun Jin Lee)
- 우창화*(경북대), 서영진(경북대), 이현진(경북대)
Chang Hwa Woo*, Kyungpook National University, Young Jin Suh, Kyungpook National University, Hyun Jin Lee, Kyungpook National University
We consider a new notion of Reeb parallel shape operator for real hypersurfaces $M$ in complex two plane Grassmannian. When $M$ has Reeb parallel shape operator and non-vanishing geodesic Reeb flow,it becomes a real hypersurfaces of Type(A) with exactly four distinct constant principal curvatures.
2010 Mathematics Subject Classification: 14M15
Key Words and Phrases: Grassmannians, Reeb parallel shape operator
- ⋅ 13:20 − 14:20 Chair: Kyung Pyo Hong (Korea University)
- ⋅ GST I 13:20 − 13:50 Infinitesimal rigidity on space form (An Sook Shin)
- 신안숙*(연세대), 김호범(연세대), 한혜림(연세대)
An Sook Shin*, Yonsei University, Ho Bum Kim, Yonsei University, Hye Lim Han, Yonsei University
In this paper, we show that hyperquadrics are infinitesimally rigid in the semi-Euclidean space. We also show that hypersurfaces of hyperquadrics cut by hyperplanes not passing through the origin are infinitesimally rigid in the hyperquadrics, whereas those cut by hyperplanes through the origin are not infinitesimally rigid in the hyperquadrics. Furthermore, we prove that any hypersurface in semi-Euclidean space, some open subset of which lies in a hyperplane, is not infinitesimally rigid.
2010 Mathematics Subject Classification: 53C40
Key Words and Phrases: infinitesimal rigidity, space form, hyperquadrics, conformal vector field, infinitesimal deformation, isometric deformation, infinitesimal isometric deformation
- ⋅ GST I 13:50 − 14:20 Certain subgroups of the groups of self homotopy equivalences on wedge product of Moore spaces. (Howon Choi, Kee Young Lee)
- 최호원*(고려대), 이기영(고려대)
Howon Choi*, Korea University, Kee Young Lee, Korea University
For a based, 1-connected, finite CW-complex $X$, $\varepsilon (X)$ is the group of classes of self-homotopy equivalences on $X$. In this paper, for wedge products of two Moore spaces $M(Z_q, n+1)$ and $M(Z_p, n)$ for $n\geq 5$ and $r=0, 1$, we calculate subgroups of $\varepsilon (X)$ of homotopy classes of self homotopy equivalences on $X$ which induce the identity homomorphism and the trivial homomorphism on homotopy groups on $X$ in dimensions $\leq \dim X +r$ respectively.
2010 Mathematics Subject Classification: 55P10, 55Q05
Key Words and Phrases: self homotopy equivalence, Moore space
- ⋅ 14:20 − 14:50 Chair: Sung Jong No (Korea University)
- ⋅ GST I 14:20 − 14:50 Exactly 14 intrinsically knotted graphs have 21 edges (Hyoung Jun Kim, Hwa Jeong Lee, Min Jung Lee and Seung Sang Oh)
- 김형준*(고려대), 이화정(중앙대), 이민정(고려대), 오승상(고려대)
Hyoung Jun Kim*, Korea University, Hwa Jeong Lee, Chung-Ang University, Min Jung Lee, Korea University, Seung Sang Oh, Korea University
Johnson, Kidwell and Michael showed that intrinsically knotted graphs have at least 21 edges. Also it is known that K7 and the 13 graphs obtained by triangle-Y moves from K7 are intrinsically knotted graphs with 21 edges. In this paper, we proved that only these 14 graphs are intrinsically knotted graph with 21 edges.
2010 Mathematics Subject Classification: 57M15
Key Words and Phrases: intrinsic, knotted, graphs
- ⋅ 15:10 − 15:40 Chair: Ho Won Choi (Korea University)
- ⋅ GST I 15:10 − 15:40 Lattice sticks number of small knots (Kyung Pyo Hong, Sung Jong No and Seung Sang Oh)
- 홍경표*(고려대), 노성종(고려대), 오승상(고려대)
Kyung Pyo Hong*, Korea University, Sung Jong No, Korea University, Seung Sang Oh, Korea University
Lattice stick number $s_L(K)$ is defined to be the minimal number of sticks required to construct s polygonal representation of the knot $K$ in the cubic lattice. I will prove that lattice stick numbers of $5_1$ and $5_2$ knots are 16.
2010 Mathematics Subject Classification: 57M27
Key Words and Phrases: knot, lattice knot
- ⋅ 15:40 − 16:10 Chair: Hyoung Jun Kim (Korea University)
- ⋅ GST I 15:40 − 16:10 Upper bound on lattice stick numbers of knots (Sung Jong No, Seung Sang Oh and Kyoung Pyo Hong)
- 노성종*(고려대), 오승상(고려대), 홍경표(고려대)
Sung Jong No*, Korea University, Seung Sang Oh, Korea University, Kyoung Pyo Hong, Korea University
The lattice stick number $s_L(K)$ of a knot $K$ is defined to be the minimal number of straight line segments required to construct a stick presentation of $K$ in the cubic lattice. In this paper, we find an upper bound on the lattice stick number of a nontrivial knot $K$, except trefoil knot, in terms of the minimal crossing number $c(K)$ which is $s_L(K) \leq 3 c(K) +2$. Moreover if $K$ is a non-alternating prime knot, then $s_L(K) \leq 3 c(K) - 4$.
2010 Mathematics Subject Classification: 57M27
Key Words and Phrases: knot, lattice stick number, stick number
- Graduate Student Session II
- ⋅ 13:20 − 14:20 Chair: Jae Uk An (Yonsei University)
- ⋅ GST II 13:20 − 13:50 Lyapunov stability for differential equations of fractional order (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 recall the definitions and some properties of two classical Mittag-Leffler functions. Then we present some results about boundedness of solutions of fractional differential equations.
2010 Mathematics Subject Classification: 26A33, 34A08
Key Words and Phrases: fractional differential equations, stability, Lyapunov-type functions, fractional comparison principle
- ⋅ GST II 13:50 − 14:20 Neumann functions for second order elliptic systems with measurable coefficients (Jongkeun Choi, Seick Kim)
- 최종근*(연세대), 김세익(연세대)
Jongkeun Choi*, Yonsei University, Seick Kim, Yonsei University
We study Neumann functions for divergence form, second order elliptic systems with bounded measurable coefficients in a bounded Lipschitz domain or a Lipschitz graph domain. We establish existence, uniqueness, and various estimates for the Neumann functions under the assumption that weak solutions of the system enjoy interior H\"older continuity. Also, we establish global pointwise bounds for the Neumann functions under the assumption that weak solutions of the system satisfy a certain natural local boundedness estimate. Moreover, we prove that such a local boundedness estimate for weak solutions of the system is in fact equivalent to the global pointwise bound for the Neumann function. We present a unified approach valid for both the scalar and the vectorial cases.
2010 Mathematics Subject Classification: 35Jxx
Key Words and Phrases: Neumann function, second-order elliptic system, Green function
- ⋅ 14:20 − 14:50 Chair: Bo Won Kang (Chungnam National University)
- ⋅ GST II 14:20 − 14:50 On regular and singular solutions of a type of Keller-Segel models (Jaewook Ahn, Darae Jeong, Junseok Kim and Kyungkeun Kang)
- 안재욱*(연세대), 정다래(고려대), 김준석(고려대), 강경근(연세대)
Jaewook Ahn*, Yonsei University, Darae Jeong, Korea University, Junseok Kim, Korea University, Kyungkeun Kang, Yonsei University
We study Keller-Segel equations with logarithmic chemotatic sensitivity and non-diffusing chemical. We show that regular solutions exists locally in time, if initial data are sufficiently regular and, in addition, under some restrictions, local regular solutions become global in time. We prove that there exists a critical number of a parameter, beyond which local regular solutions blow up in finite time.
2010 Mathematics Subject Classification: 35Kxx
Key Words and Phrases: Keller-Segel model
- ⋅ 15:10 − 16:10 Chair: Jongkeun Choi (Yonsei University)
- ⋅ GST II 15:10 − 15:40 Robustly transitive sets with orbital shadowing property (Jumi Oh, Manseob Lee and Keonhee Lee)
- 오주미*(충남대), 이만섭(목원대), 이건희(충남대)
Jumi Oh*, Chungnam National University, Manseob Lee, Mokwon University, Keonhee Lee, Chungnam National University
Let $f : M\rightarrow M$ be a diffeomorphism on a closed $C^{\infty}$ manifold. Let $\Lambda\subset M$ be a robustly transitive set. In this talk, we show that, generically robustly transitive sets with orbital shadowing property are hyperbolic.
2010 Mathematics Subject Classification: 37C20, 37C50, 37D20
Key Words and Phrases: robustly transitive set, orbital shadowing property, hyperbolic
- ⋅ GST II 15:40 − 16:10 Semi-cubically hyponormal weighted shift with recursive type (Seunghwan Baek, Mi Ryeong Lee and Chunji Li)
- 백승환*(경북대), 이미령(경북대), 이춘길(동북대)
Seunghwan Baek*, Kyungpook National University, Mi Ryeong Lee, Kyungpook National University, Chunji Li, Northeastern Universty
We discuss the semi-cubic hyponormality of
recursively generated weighted shifts with weight $\alpha \left( x\right) :%
\sqrt{x}, ( \sqrt{a},\sqrt{b},\sqrt{c})^{\wedge }$. In particular,
we consider the semi-cubic hyponormality of weighted shift with
$\alpha ( x,y ) : 1,( 1,\sqrt{x}, \sqrt{y})^{\wedge }$. Using this
weight sequence with first two equal weights, we can show that the
quadratic hyponormality and semi-cubic hyponormality are different
notions one from another.
2010 Mathematics Subject Classification: 47B37, 47B20
Key Words and Phrases: weighted shifts, cubic hyponormality, semi-cubic hyponormality
- Graduate Student Session III
- ⋅ 13:20 − 14:20 Chair: Darae Jeong (Korea University)
- ⋅ GST III 13:20 − 13:50 A phase-field model for tissue regeneration in a cartilage-hydrogel aggregate (Ana Yun, JunSeok Kim)
- 윤아나*(고려대), 김준석(고려대)
Ana Yun*, Korea University, Jun Seok Kim, Korea University
A phase-field model is proposed and an accurate numerical method is developed for regeneration of articular cartilage in a localized defect region. The cylindrical cartilage explant is assumed and it has a removed core region which is replaced with a nutrient-rich hydrogel. We use a phase-field model to capture the hydrogel and cartilage interface where the hydrogel turns into newly generated cartilage. Moreover, we present an accurate numerical method to have the numerical solutions in the phase-field model and reaction diffusion equations for the nutrient and the newly generated cartilage. A variety of numerical simulations are performed to show the effect of parameters of our mathematical model including the temporal evolution of each variables. Moreover, numerical simulations based on biological mechanism are investigated.
2010 Mathematics Subject Classification: 65N06
Key Words and Phrases: phase-field model, articular-cartilage, reaction-diffusion model
- ⋅ GST III 13:50 − 14:20 An optimization based domain decomposition method for PDE with random input data (Yoongu Hwang, Jeehyun Lee and Jang-woon Lee)
- 황윤구*(연세대), 이지현(연세대), 이장운(Univ. of Mary Washington)
Yoongu Hwang*, Yonsei University, Jeehyun Lee, Yonsei University, Jang-woon Lee, University of Mary Washington
In practice, we frequently encounter multi-physics problem, that contains multiple physical models and also we have uncertainties in the input data. In this paper, we introduced an optimization problem based domain decomposition for solving Poisson's equation simplified as the first step for multi-physics problem, with random input data. We need an optimal control and stochastic Galerkin Method to obtain an approximation for solution of Poisson's equation with random input data.
2010 Mathematics Subject Classification: 65N30
Key Words and Phrases: optimal control, domain decomposition, PDE with random input data, stochastic Galerkin, finite elements method
- ⋅ 14:20 − 14:50 Chair: Ana Yun (Korea University)
- ⋅ GST III 14:20 − 14:50 Boundary integral equation formulation for symmetric impinging wall jet in 2D (Sung Sic Yoo, Do Wan Kim)
- 유성식*(인하대), 김도완(인하대)
Sung Sic Yoo*, Inha University, Do Wan Kim, Inha University
We consider a symmetric impinging wall jet in 2-dimensional ideal fluid. The jet issuing from the orifice of a nozzle impinges against the wall and flows away along the wall in two ways. From Bernoulli's principle, on the contact surface between the fluid and the surrounding air, the velocity of fluid is known to be tangent to it and its magnitude as speed is constant. We call this contact surface the free boundary of the jet. Since the free boundary is unknown and free boundary problems are highly non-linear in nature, we propose an iterative algorithm to find the free boundary and stream function by using boundary integral equations and the corresponding perturbed integral equation. Through our numerical algorithm proposed in this talk, we show the properties of free boundaries of impinging wall jets.
2010 Mathematics Subject Classification: 76B10
Key Words and Phrases: boundary element method, impinging wall jet, potential flow, perturbed boundary integral equation
- Poster Session
- ⋅ 11:00 − 16:10 Display Time(게시)\ \ /15:30 − 16:00 Q $\&$ A(질의응답) Chair:
- ⋅ 15:30 − 16:00 Self-intersecting polygons and semi-lattice knots (Hun Kim, Gyo Taek Jin, Seo Yeon Park, Hyun Yui Park, Sung Jin Jeong, Dong Gyu Joo and Hwan Ung Choi)
- 김훈*(한국과학영재학교), 진교택(카이스트), 박서연(한국과학영재학교), 박현이(한국과학영재학교), 정성진(한국과학영재학교), 주동규(한국과학영재학교), 최환웅(한국과학영재학교)
Hun Kim*, Korea Science Academy, Gyo Taek Jin, KAIST, Seo Yeon Park, Korea Science Academy, Hyun Yui Park, Korea Science Academy, Sung Jin Jeong, Korea Science Academy, Dong Gyu Joo, Korea Science Academy, Hwan Ung Choi, Korea Science Academy
Semi lattice knot is a kind of polygon knot in $\mathbf{R}^3$ whose vertices are located at integer coordinates and edges parallel to the $xy$-plane and the $z$-axis. Semi arc index of a knot is the minimum number of edges to represent the knot as semi lattice knots, and it is an invariants of a knot. In this study, we find semi arc index of some kinds of knots and investigate the relation between arc index and semi arc index.
2010 Mathematics Subject Classification: 57M25, 57M27
Key Words and Phrases: knots and links, semi-lattice knots, arc index
- ⋅ 15:30 − 16:00 On the generalized Dunwoody 3-manifold (Soo Hwan Kim)
- 김수환(동의대)
Soo Hwan Kim, Dong-Eui University
We introduce a family of orientable 3-manifolds induced by certain cyclically presented groups and show that this family of 3-manifolds contains all Dunwoody 3-manifolds by using the planar graphs corresponding to the polyhedral description of the 3-manifolds. As applications, we consider two families of cyclically presented groups, and show that these are isomorphic to the fundamental groups of the certain Dunwoody 3-manifolds $D_n$ $(n \ge 2)$ which are the n-fold cyclic coverings of the 3-sphere branched over the certain two-bridge knots, and that $D_n$ is the $(Z_n \oplus Z_2)$-fold covering of the 3-sphere branched over two different $\theta$-curves.
2010 Mathematics Subject Classification: 57M12
Key Words and Phrases: cyclically presented group, cyclically branched covering, Heegaard diagram, 3-manifold, Dunwoody 3-manifold, (1,1)-knot
- ⋅ 15:30 − 16:00 Performance analysis of queueing model for dynamic LB system (Doo Il Choi)
- 최두일(한라대)
Doo Il Choi, Halla University
The LB system to prevent congestion in communication networks has been studied by many researchers. However, almost all works have static token interval and static packet arrival. We propose the dynamic LB system. That is, the LB system has dynamic token interval and dynamic packet arrival. First, the dynamic LB system is modelled by an appropriate queueing model. The queueing model is analyzed by using the embedded Markov chain and the supplementary variable methods of queueing theory. We obtain the performance measures such as loss and waiting. Also. a numerical example is given to show excellence of our proposed system. Finally, this investigation helps the network designer to select an appropriate traffic control, while preventing congestion of network.
2010 Mathematics Subject Classification: 60J27
Key Words and Phrases: queueing theory, traffic control
- ⋅ 15:30 − 16:00 The relationship between approximation and balancing orders of totally interpolating biorthogonal multiwavelet systems (Jaewon Jung, Youngwoo Choi)
- 정재원*(아주대), 최영우(아주대)
Jaewon Jung*, Ajou University, Youngwoo Choi, Ajou University
Totally interpolating biorthogonal multiwavelet systems with FIR two-band multifilter banks are studied. Necessary and sufficient conditions for approximation orders are stated in terms of filter coefficients and a degree of freedom $C$. We apply the definition of balancing orders to compactly supported totally interpolating biorthogonal multiwavelet systems. We show that the concept of approximation orders is completely equivalent to that of balancing orders in our setting. To illustrate our result, suitable examples are provided and their stabilities and regularities are stated.
2010 Mathematics Subject Classification: 42C40
Key Words and Phrases: totally interpolating, biorthogonal multiwavelet, approximation orders, balancing orders
- ⋅ 15:30 − 16:00 Mathematical theory of multiphase flow model equations (Hyeonseong Jin)
- 진현성(제주대)
Hyeonseong Jin, Jeju National University
Multiphase flow equations are generally derived by an averaging process applied to a microphysical description of distinct fluids separated by sharp interfaces. Averaging of nonlinear terms introduces an essential difficulty into multiphase flow, as it does in related subjects such as turbulence modeling. This problem is known as the closure problem, since the unknown averages of nonlinear expressions must be replaced by some expression written in terms of the averaged variables, to yield a closed set of equations. Here we compare various proposed closures. We introduce physical constraints a complete multiphase closure should satisfy to conserve the total energy and phase entropy and to satisfy boundary conditions at the edges of the mixing zones. We find one closure which satisfies all the constraints we have identified. The integral identities for the closures are derived based on a mixing zone homogeneity assumption. A new inequality constraint is presented as a necessary and sufficient condition for the positivity of an entropy of averaging. It relates the motions of the two mixing zone edges.
2010 Mathematics Subject Classification: 76T99, 76N99, 76A99
Key Words and Phrases: multiphase flow, closure
- ⋅ 15:30 − 16:00 Generalized net model of the process of selection and usage of an intelligent e-learning system (Taekyun Kim)
- 김태균(광운대)
Taekyun Kim, Kwangwoon University
A generalized net model of the process of selection and usage of an intelligent e-learning system is constructed. An evaluation of the results of the learning is done. This work is a follow up of previous authors' research in this area
2010 Mathematics Subject Classification: 11A59
Key Words and Phrases: generalized net, e-learning, modelling
- ⋅ 15:30 − 16:00 An array poset and the linear discrepancy (Eol Choi, Sang-Mok Kim)
- 최얼*(광운대), 김상목(광운대)
Eol Choi*, Kwangwoon University, Sang-Mok Kim, Kwangwoon University
The linear discrepancy of a poset $P=(X, \leq_P)$ is defined by $$ld(P)=\min_{f \in \mathcal{F}}{\max_{\substack{x \parallel y \in P}}}| f(x)-f(y)|,$$ where $\mathcal{F}$ is the set of order preserving injections $f:P\rightarrow \mathbb{Z}$. For a positive integer $n$, let $[n]$ denote $\{1, \ldots , n\}$. For any given $m \times n$ array $A_{m \times n}= (a_{ij})$, we define a poset $\mathbf{A}_{m\times n}=(X,\leq_A)$, called the $m\times n$ {\it array poset}, as follows. The ground set $X=\{A_{u \times v}(s,t) \; |\; u \in [m], v \in [n], s \in [m-u+1], t \in [n-v+1]\}$ where $A_{u \times v}(s, t)=(\alpha_{ij})$ is a $u\times v$ subarray of $A$ with $\alpha_{ij}=a_{s-1+i\; \;t-1+j}$ for $s\in [u]$, $t\in [v]$, and $A_{u \times v}(s, t)\leq_{A} A_{u' \times v'}(s', t')$ if and only if $u \in [u']$, $v \in [v']$, $u + s - 1 \in [u' + s' -1]$ and $v + t -1 \in [ v' + t' -1]$. In this talk, we mainly deal with the linear discrepancy of $m \times n$ array posets. We first give some general properties of $\mathbf{A}_{m\times n}$. Especially, we observe that $\mathbf{A}_{1\times n}$ is a subposet of the product poset of two chains by giving its representation by coordinates. Finally, using its coordinate representation, we give an upper bound of the linear discrepancy of $\mathbf{A}_{1\times n}$.
2010 Mathematics Subject Classification: 06A07
Key Words and Phrases: poset, linear discrepancy
- ⋅ 15:30 − 16:00 Spherical harmony between the vertical expansion and the horizontal expansion in tube designs (Hongtaek Hwang)
- 황홍택(금오공대)
Hongtaek Hwang, Kumoh Institute of Technology
수직전개 및 수평전개를 통하여 얻은 입체형 다선구조를 튜브디자인 실험을 통하여 구면상에서 조화를 이루도록 구현한 장면을 소개한다.
2010 Mathematics Subject Classification: 97U60
Key Words and Phrases: tube design, spherical frame, geometrical sculpture