KMS Meeting - Program and Abstracts

Algebra

⋅ 29th-A-09:00 − 10:30   Chair: Sunah Kim (Chosun University)

⋅ 29th-A-09:00 − 09:20 Complex surfaces of general type with geometric genus zero from an Enriques surface (Dongsoo Shin, Yoonjeong Yang)

신동수(충남대), 양윤정*(충남대)
Dongsoo Shin, Chungnam National University, Yoonjeong Yang*, Chungnam National University

I construct complex surfaces of general type with $K^{2}=3,4$ and $p_{g}=q=0$ as double coverings of an Enriques surface which are known as Keum-Naie surfaces. These surfaces are constructed by bidegree (1,1) divisors on $\mathbb{P}^{1}\times\mathbb{P}^{1}$. In particular, the surface of $K^2=3$ has $(-4)$-curve. Our goal is getting the surface of $K^2=4$ from the surface of $K^2=3$ by deformation. This is a joint work in progress with Dongsoo Shin.

2010 Mathematics Subject Classification: 14J29
Key Words and Phrases: complex surface, general type, $p_g=q=0$, double covering, Enriches surface, Keum-Naie, $(-4)$-curve

⋅ 29th-A-09:20 − 09:40 Invariants of deformations of quotient surface singularities (Byoungcheon Han, Jaekwan Jeon, Dongsoo Shin)

한병천(충남대), 전재관*(충남대), 신동수(충남대)
Byoungcheon Han, Chungnam National University, Jaekwan Jeon*, Chungnam National University, Dongsoo Shin, Chungnam National University

For a quotient surface singularity $X$, there is an one-to-one correspondence between the irreducible components of the reduced base space of the versal deformation $\mathrm{Def}(X)$ and certain partial resolutions, called \textit{$P$-resolutions} of $X$. We first find all $P$-resolutions of quotient surface singularities. Then we compute the dimensions and Milnor numbers of the corresponding irreducible components. Furthermore, we realize Milnor fibers corresponding to $P$-resolutions as complements of certain special divisors (called \textit{compactifying divisors}) in certain rational surfaces by using the minimal model program of 3-folds. This is a part of the master's thesis of Byoungcheon Han and Jaekwan Jeon at Chungnam National University.

2010 Mathematics Subject Classification: 14J17
Key Words and Phrases: quotient surface singularity, Milnor fiber, minimal model program

⋅ 29th-A-09:50 − 10:10 Identities of ordered Bell numbers arising from differential equations (Hyuck In Kwon)

권혁인(광운대)
Hyuck In Kwon, Kwangwoon University

In this talk, we consider the Bell numbers arising from differential equations and give some new identities of these numbers and polynomials.

2010 Mathematics Subject Classification: 11B68, 11S80
Key Words and Phrases: differential equations, ordered Bell numbers

⋅ 29th-A-10:10 − 10:30 Identities for Daehee numbers arising from nonlinear differential equations (GwanWoo Jang)

장관우(광운대)
GwanWoo Jang, Kwangwoon University

Kwon-Kim-Seo introduced some interesting identities of Daehee numbers arising from differential equation. In this talk, we consider the inverse problem for the some identities of Daehee numbers which are derived from differential equations.

2010 Mathematics Subject Classification: 11B68, 11S80
Key Words and Phrases: Daehee numbers, differential equations

⋅ 29th-B-10:40 − 12:10   Chair: Nam Kyun Kim (Hanbat National University)

⋅ 29th-B-10:40 − 11:00 A study on Fourier series of sums of products of ordered Bell functions (Lee-Chae Jang, Taekyun Kim)

장이채*(건국대, 교육대학원), 김태균(광운대)
Lee-Chae Jang*, Konkuk University, The Graduate School of Education, Taekyun Kim, Kwangwoon University

The Genocchi polynomials $G_m(x)$ are given by the generating function \begin{equation}\begin{split}\label{101} \frac{2t}{e^t+1}e^{xt} =\sum_{m=0}^\infty G_m(x) \frac{t^m}{m!}. \end{split}\end{equation} The ordered Bell polynomials $b_m(x)$ are a natural companion to ordered Bell numbers and defined by the generating function \begin{equation}\begin{split}\label{112} \frac{1}{2-e^t} e^{xt} = \sum_{m=0}^\infty b_m(x) \frac{t^m}{m!} . \end{split}\end{equation} The ordered Bell numbers $b_m=b_m(0)$ have been studied in many counting problems in enumerative combinatorics and number theory, the first appearance of which goes back to as early as 1859. In this paper, we will study three types of sums of products of ordered Bell and Genocchi functions, and derive their Fourier expansions. Further, we will express those functions in terms of Bernoulli functions as follows: \vspace{.1cm} (1) $\alpha_m(<x>)=\sum_{k=0}^{m-1} b_k(<x>) G_{m-k}(<x>), \;\;(m\geq 2) $; (2) $\beta_m(<x>)= \sum_{k=0}^{m-1}\frac{1}{k!(m-k)!} b_k(<x>) G_{m-k}(<x>), \;\;(m\geq 2) $; (3) $\gamma_m(<x>)=\sum_{k=1}^{m-1}\frac{1}{k(m-k)} b_k(<x>) G_{m-k}(<x>), \;\;(m\geq 2) $.

2010 Mathematics Subject Classification: 11B83, 42A16
Key Words and Phrases: Fourier series, ordered Bell polynomials, Genocchi polynomials

⋅ 29th-B-11:00 − 11:20 Rational period functions and indefinite binary quadratic forms in higher level cases (SoYoung Choi, Chang Heon Kim)

최소영*(경상대), 김창헌(성균관대)
SoYoung Choi*, Gyeongsang National University, Chang Heon Kim, Sungkyunkwan University

Let $q(z)$ be a rational period function. As for the location of poles of $q(z)$, it was shown by Knopp and the authors that if $\tau$ is a finite pole of $q(z)$, then $\tau \in \mathbb{Q}(\sqrt{n})$ for some square-free positive integer $n$, and if $\tau$ is a finite rational pole of $q(z),$ then $\tau = 0$. Moreover, in the case of $p = 1$, Gethner proved that rational period functions with irrational poles are not Hecke eigenfunctions. We gave the evidence of such phenomenon in higher level cases and suggested that we can construct rational period polynomials by using binary quadratic forms. In this talk by extending Gethner's and Choie and Parson's results we solve these problems in higher level cases. This is a joint work with Chang Heon Kim

2010 Mathematics Subject Classification: 11F11, 11F67
Key Words and Phrases: rational period functions, indefinite binary quadratic forms

⋅ 29th-B-11:30 − 11:50 Ulrich bundles on rational normal scrolls (Marian Aprodu, Sukmoon Huh, Francesco Malaspina, Joan Pons-Llopis)

Marian Aprodu(Univ. of Bucharest), 허석문*(성균관대), Francesco Malaspina(Politecnico di Torino), Joan Pons-llopis(Kyoto Univ.)
Marian Aprodu, University of Bucharest, Sukmoon Huh*, Sungkyunkwan University, Francesco Malaspina, Politecnico di Torino, Joan Pons-Llopis, Kyoto University

We classify the Ulrich vector bundles of arbitrary rank on smooth rational normal scrolls. Main ingredients are the construction of new full exceptional collection of derived category by mutations and the Beilinson spectral sequence.

2010 Mathematics Subject Classification: 14J60, 14D20
Key Words and Phrases: Ulrich bundle, rational normal scroll, Beilinson spectral theorem

⋅ 29th-B-11:50 − 12:10 Brill-Noether theory for rank 2 vector bundles (Youngook Choi, Flaminio Flamini, Seonja Kim)

최영욱*(영남대), Flaminio Flamini(Universita' di Roma ``Tor Vergata"), 김선자(청운대)
Youngook Choi*, Yeungnam University, Flaminio Flamini, Universita' di Roma ``Tor Vergata", Seonja Kim, Chungwoon University

In this talk, we discuss on the Brill Noether components of rank 2, (semi) stable vector bundles with speciality two on a general $k$-gonal curve. Also, we describe a general element of each component in terms of extensions of line bundles with suitable minimality properties.

2010 Mathematics Subject Classification: 14H60, 14D20
Key Words and Phrases: stable rank-two vector bundles, Brill-Noether loci, gonality

⋅ 29th-C-14:30 − 15:10   Chair: Bokhee Im (Chonnam National University)

⋅ 29th-C-14:30 − 14:50 Zero-term rank preservers of symmetric matrices over semirings (Seok-Zun Song)

송석준(제주대)
Seok-Zun Song, Jeju National University

Let $B$ be a matrix. A line of $B$ is a row or column of $B$. The term rank of $B$ is the fewest number of lines of $B$ that contain all the nonzero entries of $B$. The zero-term rank of $B$ is the fewest number of lines that contain all the zeros of $B$. We study linear operators that preserve zero-term rank of $n$ by $n$ symmetric matrices. We determine the form of linear operators $T$ on $n$ by $n$ symmetric matrices that preserves zero-term rank. We also show that a linear operator $T$ on symmetric matrices preserves zero-term rank if and only if $T$ preserves zero-term ranks 1 and 2. Other characterizations of zero-term rank preservers are also given.

2010 Mathematics Subject Classification: 15A03, 15A04, 15A86
Key Words and Phrases: semiring, semimodule, upper ideal, linear preserver, zero-term rank

⋅ 29th-C-14:50 − 15:10 Basis for the minus space of weakly holomorphic modular forms in certain level cases (SoYoung Choi, Chang Heon Kim, Kyung Seung Lee)

최소영(경상대), 김창헌(성균관대), 이경승*(성균관대)
SoYoung Choi, Gyeongsang National University, Chang Heon Kim, Sungkyunkwan University, Kyung Seung Lee*, Sungkyunkwan University

Let $M_{k}^{!}(p)$ be the space of weakly holomorphic modular forms of weight $k$ on $\Gamma_{0}(p)$, and let $M_{k}^{!-}(p)$ be the minus space which is the subspace of $M_{k}^{!}(p)$ consisting of every eigenforms $f$ of Fricke involution $W_{p}$ with eigenvalue $-1$. Our goal is to find a canonical basis for the minus space $M_{k}^{!-}(p)$ in certain level cases. Using this result, along with previous works of Choi and Kim (``Basis for the space of weakly holomorphic modular forms in higher level cases", J. Number Theory, 133, (2013), 1300-1311), we obtain the basis for the space of $M_{k}^{!}(p)$.

2010 Mathematics Subject Classification: 11F11, 11F03, 11F37
Key Words and Phrases: weakly holomorphic modular forms
Analysis I

⋅ 29th-A-09:00 − 10:30   Chair: Lee-Chae Jang (Konkuk University)

⋅ 29th-A-09:00 − 09:20 Quadratic type functional inclusions on square-symmetric groupoids and Hyers-Ulam stability (Gwanghui Kim, Hwan Yong Shin)

김광휘(강남대), 신환용*(충남대)
Gwanghui Kim, Kangnam Universaty, Hwan Yong Shin*, Chungnam National University

We consider that a set-valued $F:X \rightarrow \mathcal{P}_0(Y)$ satisfying the functional inclusion $ F(x \ast y)\Diamond F(x \ast y^{-1}) \subseteq \sigma_{\Diamond}(F(x)\Diamond F(y))$(or $ \sigma_{\Diamond}(F(x)\Diamond F(y)) \subseteq \sigma_{\Diamond} (F(x\ast y) \Diamond F(x \ast y^{-1})))$ admits, in appropriate conditions, a unique selection $f : X \rightarrow Y$ satisfying the functional equation $ f(x \ast y)\diamond f(x \ast y^{-1})= \sigma_{\diamond} (f(x)\diamond f(y))$, where $ (X, \ast) $, $(Y, \diamond)$ are square-symmetric groupoids and $\Diamond$ is the extension of $ \diamond$ to the collection $\mathcal{P}_0(Y)$ of all nonempty parts of $Y$.

2010 Mathematics Subject Classification: 39B72, 54C60
Key Words and Phrases: Hyers-Ulam stability, square-symmetric groupoid, functional inclusion

⋅ 29th-A-09:20 − 09:40 Hyponormality of singular integral operators with Cauchy kernel on $L^2$ (Jongrak Lee, Takahiko Nakazi, Eungil Ko, Yoenha Kim)

이종락*(이화여대), Takahiko Nakazi(Hokkaido Univ.), 고응일(이화여대), 김연하(이화여대)
Jongrak Lee*, Ewha Womans University, Takahiko Nakazi, Hokkaido University, Eungil Ko, Ewha Womans University, Yoenha Kim, Ewha Womans University

In this paper, we study a class of hyponormal singular Cauchy integral operators with matrix-valued symbols. We characterize hyponormal singular Cauchy integral operators $S_{\Phi,\Psi}$ by properties of the symbols $\Phi$ and $\Psi$.

2010 Mathematics Subject Classification: 45E10, 47B35, 47B20
Key Words and Phrases: singular integral operator, Toeplitz operator, hyponormal operator

⋅ 29th-A-09:50 − 10:10 Approximate quadratic mappings in modular spaces (Hark-Mahn Kim, Young Soon Hong)

김학만(충남대), 홍영순*(충남대)
Hark-Mahn Kim, Chungnam National University, Young Soon Hong*, Chungnam National University

In this article, we investigate an alternative generalized Hyers--Ulam stability theorem of a modified quadratic functional equation in a modular space $X_\rho$ using $\Delta_3$-condition without the Fatou property on the modular function $\rho.$

2010 Mathematics Subject Classification: 39B82, 39B72, 16W25
Key Words and Phrases: generalized Hyers--Ulam stability, modular spaces, Fatou property, $\Delta_3$-condition

⋅ 29th-A-10:10 − 10:30 The existence and uniqueness of weighted pseudo $S$-asymptotically $\omega$- periodic mild solution in the Stepanov sense with finite delay for some partial differential equation (Hyun Mork Lee)

이현목(충남대)
Hyun Mork Lee, Chungnam National University

The theory of almost periodic functions was introduced in the literature around 1924-1926 the Denish mathematician Harald Bohr. However real systems usually exhibit internal variations or external perturbations which are only approximately periodic. Many authors have furthermore generalized in different directions. In this talk, we investigate the existence and uniqueness of weighted pseudo $S$-asymptotically $\omega$-periodic mild solution in the Stepanov sense with finite delay. Our analysis is based on the theory of a strongly continuous semigroup of operators and the Banach contraction mapping principle.

2010 Mathematics Subject Classification: 34K06, 34K20, 34K30, 47D06
Key Words and Phrases: almost periodicity, $s$-asymptotically $\omega$-periodic, $S^P$-weighted pseudo $S$-asymptotic periodicity, weighted pseudo $S$-asymptotic periodicity, mild solutions

⋅ 29th-B-10:40 − 12:10   Chair: Han Ju Lee (Dongguk University)

⋅ 29th-B-10:40 − 11:00 Lie-Trotter means of positive definite operators (Jinmi Hwang, Sejong Kim)

황진미(충북대), 김세정*(충북대)
Jinmi Hwang, Chungbuk National University, Sejong Kim*, Chungbuk National University

As extension of the Lie-Trotter product formula, we define the two-variable and multivariable Lie-Trotter means with several examples including the Sagae-Tanabe and Hansen inductive means. We show that the weighted means satisfying the arithmetic-geometric-harmonic mean inequalities are the multivariable Lie-Trotter means. We finally give a few interesting versions of the extended Lie-Trotter formula derived from multivariable Lie-Trotter means.

2010 Mathematics Subject Classification: 47B65
Key Words and Phrases: Lie-Trotter formula, geometric mean, spectral geometric mean, inductive mean, arithmetic-geometric-harmonic mean inequalities

⋅ 29th-B-11:00 − 11:20 Convergences of alternating projections in ${\rm CAT}(\kappa)$ spaces (Byoung Jin Choi)

최병진(충북대)
Byoung Jin Choi, Chungbuk National University

We establish the asymptotic regularity and the $\Delta$-convergence of the sequence constructed by the alternating projections to closed convex sets in a ${\rm CAT}(\kappa)$ space with $\kappa > 0$. Furthermore, the strong convergence of the alternating von Neumann sequence is presented under certain regularity or compactness. This is a joint work with Un Cig Ji and Yongdo Lim.

2010 Mathematics Subject Classification: 47J25
Key Words and Phrases: ${\rm CAT}(\kappa)$ space, alternating projection method, alternating von Neumann sequence, asymptotic regularity, $\Delta$-convergence

⋅ 29th-B-11:30 − 11:50 $n$-tuple Jordan product commuting maps with $\lambda$-Aluthge transform (Younjin Kim, Eungil Ko)

김연진*(이화여대), 고응일 (이화여대)
Younjin Kim*, Ewha Womans University, Eungil Ko, Ewha Womans University

Let $\mathcal{B}(H, K)$ be the Banach space of all bounded linear operators from $H$ into $K$, where $H$ and $K$ are complex Hilbert spaces. In this talk, we prove that a bijective map $\Phi$ between the Banach spaces commutes with a $\lambda$-Aluthge transform under the $n$-tuple Jordan product, if and only if there exists an unitary operator $U: H \rightarrow K$ such that $ \Phi(T) = U T U^*$ for every $T \in \mathcal{B}(H, H) := \mathcal{B}(H).$

2010 Mathematics Subject Classification: 47A63
Key Words and Phrases: $\lambda$-Aluthge transform, Jordan product

⋅ 29th-B-11:50 − 12:10 Properties of $(\infty,C)$-isometric operators (Ji Eun Lee, Eungil Ko, Muneo Ch\=o)

이지은*(세종대), 고응일(이화여대), Muneo Ch\=o(Kanagawa University)
Ji Eun Lee*, Sejong University, Eungil Ko, Ewha Womans University, Muneo Ch\=o, Kanagawa University

In this paper we focus on properties of $(\infty,C)$-isometric operators. In particular, we show that if $T$ is an $(\infty,C)$-isometry and $Q$ is a quasinilpotent operator, then $T+Q$ is an $(\infty,C)$-isometry under some conditions. Moreover, we show that the class of $(\infty,C)$-isometric operators is norm closed. Finally, we study properties of products and tensor products of $(\infty,C)$-isometric operators.

2010 Mathematics Subject Classification: Primary 47A11; Secondary 47B25
Key Words and Phrases: $(\infty,C)$-isometric operator, $m$-isometric operator, quasinilpotent operator

⋅ 29th-C-14:30 − 16:00   Chair: Sejong Kim (Chungbuk National University)

⋅ 29th-C-14:30 − 14:50 On square roots of self-adjoint weighted composition operators on $H^2$ (Sungeun Jung, Yoenha Kim, Eungil Ko)

정성은(한국외대), 김연하*(이화여대), 고응일(이화여대)
Sungeun Jung, Hankuk University of Foreign Studies, Yoenha Kim*, Ewha Womans University, Eungil Ko, Ewha Womans University

Let ${\mathbb D}$ denote the open unit disk in the complex plane $\mathbb C$. The Hardy space $H^{2}(\mathbb D)=H^2$ consists of all the analytic functions on ${\mathbb D}$ having power series representation with square summable complex coefficients. For an analytic function $f$ on ${\mathbb D}$ and an analytic selfmap $\varphi$ of $\mathbb D$, the operator $W_{f, \varphi}:H^{2} \rightarrow H^{2}$ given by $$W_{f, \varphi}h=f\cdot(h\circ \varphi), \ h\in H^2$$ is called a weighted composition operator. In this talk we study square roots of self-adjoint weighted composition operators on the Hardy space $H^2$.

2010 Mathematics Subject Classification: 47B20, 47B38, 47B33
Key Words and Phrases: weighted composition operator, Hermitian, self adjoint

⋅ 29th-C-14:50 − 15:10 Existence of mild solutions in the $\alpha$-norm for some partial functional integrodifferential equations with nonlocal conditions (Yoon Hoe Goo, Dong Man Im)

구윤회*(한서대), 임동만(청주대)
Yoon Hoe Goo*, Hanseo University, Dong Man Im, Cheongju University

In this work, we discuss the existence of mild solutions in the $\alpha$-norm for some partial functional integrodifferential equations with infinite delay. We assume that the linear part generates an analytic semigroup on a Banach space $X$ and the nonlinear part is a Lipschitz continuous function with respect to the fractional power norm of the linear part.

2010 Mathematics Subject Classification: 34K06, 34K20, 34K30, 47D06
Key Words and Phrases: an analytic semigroup, $\alpha$-norm, nonlocal conditions, partial functional integrodifferential equations

⋅ 29th-C-15:20 − 15:40 Remarks on nearly equivalent operators (Mee-Jung Lee, Eungil Ko)

이미정*(이화여대), 고응일(이화여대)
Mee-Jung Lee*, Ewha Womans University, Eungil Ko, Ewha Womans University

An operator $S\in\mathcal{L(H)}$ is said to be nearly equivalent to $T$ if there exists an invertible operator $V\in\mathcal{L(H)}$ such that $S^{\ast}S=V^{-1}T^{\ast}TV$. In this paper, we study several properties of nearly equivalent operators, and investigate their local spectral properties and invariant subspaces.

2010 Mathematics Subject Classification: 47B20, 47A10
Key Words and Phrases: nearly equivalent operators, local spectral property, invariant subspace

⋅ 29th-C-15:40 − 16:00 Behavior of a scale factor for the Wiener integral (Young Sik Kim)

김영식(한양대)
Young Sik Kim, Hanyang University

We wish to investigate the behavior of the scale factor for the Wiener integral of bounded and unbounded functions on the Wiener space.

2010 Mathematics Subject Classification: 28C20
Key Words and Phrases: Wiener integral
Analysis II

⋅ 29th-A-09:00 − 10:30   Chair: Jongmin Han (Kyung Hee University)

⋅ 29th-A-09:00 − 09:20 Estimates for the Bergman kernel of generalized Fock spaces (Hong Rae Cho, Soohyun Park)

조홍래(부산대), 박수현*(부산대)
Hong Rae Cho, Pusan National University, Soohyun Park*, Pusan National University

We will present some estimates for the Bergman kernel of generalized Fock spaces. We use the canonical solution to the $\bar\partial$-equation. Precisely, a kind of sub-mean-value property and improved $L^2$-estimate for the solution are used.

2010 Mathematics Subject Classification: 30H20, 32A25
Key Words and Phrases: Bergman kernel, generalized Fock space, exponential type weight

⋅ 29th-A-09:20 − 09:40 Classification and existence of solutions for Einstein-Maxwell-Higgs model (Jongmin Han, Ju Hee Son)

한종민(경희대), 손주희*(경희대)
Jongmin Han, Kyung Hee University, Ju Hee Son*, Kyung Hee University

In this talk, we are concerned with an elliptic system arising from the Einstein-Maxwell-Higgs model which describes electromagnetic dynamics coupled with gravitationalfields in space-time. Reducing this system to a single equation and setting up the radial ansatz, we classify solutions into three cases: topological solutions, nontopological solutions of type I, and nontopological solutions of type II. Also, we prove the existence of topological multi-string solutions to the elliptic equations arising from self-dual Abelian gauge field theories coupled with the Einstein equations. By applying the super/subsolution method, we prove that topological multi-string solutions exist for any coupling constant.

2010 Mathematics Subject Classification: 35J61
Key Words and Phrases: Einstein-Maxwell-Higgs model, topological solutions, nontopological solutions

⋅ 29th-A-09:50 − 10:10 Characterizations for the Fock type spaces (Jeong Min Ha, Hong Rae Cho)

하정민*(부산대), 조홍래(부산대)
Jeong Min Ha*, Pusan National University, Hong Rae Cho, Pusan National University

In this talk, we consider the Fock type spaces with an exponential type weight and find the equivalent norm of an entire function in these $n$ dimensional complex spaces. In particular, we prove that Fock type spaces with the weight $exp(-{\frac{|z|^m}{2}})$ can be characterized by Lipschitz type and double integrability.

2010 Mathematics Subject Classification: 32A37, 30H20
Key Words and Phrases: exponential type weighted integrals, double integral characterization

⋅ 29th-A-10:10 − 10:30 Classification of solutions of elliptic equations arising from the gravitational Maxwell gauged $O(3)$ sigma model (Nari Choi, Jongmin Han)

최나리*(경희대), 한종민(경희대)
Nari Choi*, Kyung Hee University, Jongmin Han, Kyung Hee University

In this paper, we study an elliptic equation arising from the self-dual equations for the Maxwell gauged $O(3)$ sigma model coupled with gravitation. We completely classify all radial solutions for single vortex case according to values of $aN$ where $a$ is a scaled gravitational constant and $N$ is the total vortex number.

2010 Mathematics Subject Classification: 81T13, 35B40
Key Words and Phrases: Maxwell O(3) sigma model, gravitation, radially symmetric solutions, nontopological solutions

⋅ 29th-B-10:40 − 12:10   Chair: Soyeun Jung (Kongju National University)

⋅ 29th-B-10:40 − 11:00 Fujita type blow-up for discrete reaction-diffusion (Soon Yeong Chung, Min Jun Choi, Jea Hyun Park )

정순영(서강대), 최민준*(서강대), 박재현(군산대)
Soon Yeong Chung, Sogang University, Min Jun Choi*, Sogang University, Jea Hyun Park, Kunsan National University

This talk is concerned with long time behaviors of solutions to the reaction-diffusion equations $u_{t} = \Delta_{\omega}u + \psi(t)|u|^{q-1}u$ with the non-trivial and nonnegaive initial data. The propose of this paper to introduce a critical set $\mathcal{C}(\psi)$ in the following sense: (i) solutions blow up in finite time, for $q\in\mathcal{C}(\psi)$. (ii) solutions with small initial data are exponentially decreasing, for $q\not\in\mathcal{C}(\psi)$. In order to prove the main theorems, we first derive the comparison principles for the solutions of the equation above, which play an important role throughout this paper. In addition, we finally give some numerical illustrations which exploit the main results.

2010 Mathematics Subject Classification: 39A12, 39A13, 39A70, 35K57
Key Words and Phrases: reaction-diffusion equations, discrete Laplacian, comparison principle, Fujita blow-up

⋅ 29th-B-11:00 − 11:20 Relations between quaternionic differential and the corresponding Cauchy Riemann system in Clifford analysis (Hang Ul Kang)

강한울(부산대)
Hang Ul Kang, Pusan National University

We provide some properties of several left regular functions in Clifford analysis. We find the corresponding Cauchy-Riemann system in the sense of several complex variables. And we investigate several properties of quaternionic functions. We research some differentials of quaternionic functions, and relations between the differentials and the corresponding Cauchy Riemann system in Clifford analysis.

2010 Mathematics Subject Classification: 32A99, 30G35, 11E88
Key Words and Phrases: Clifford analysis, corresponding Cauchy Riemann system, differential of quaternionic function

⋅ 29th-B-11:30 − 11:50 Uniform decay for a Kirchhoff-Carrier system with an internal time-varying delay term (Daewook Kim)

김대욱(서원대)
Daewook Kim, Seowon University

In this talk we prove the uniform decay for a Kirchhoff-Carrier system with an internal time-varying delay term. First, we introduce the main theorem and some lemmas. We also use the smallness condition and taking into account the appropriate Lyapunov functional. Finally, we get the uniform decay rate of the main energy.

2010 Mathematics Subject Classification: 35L05
Key Words and Phrases: Kirchhoff-Carrier system, internal time-varying delay, Lyapunov functional, uniform decay

⋅ 29th-B-11:50 − 12:10 Some applications of the interval-valued Choquet expected utility related with the level of animal product exports between Korea and trading partners (Lee-Chae Jang, Jacob Wood)

장이채*(건국대, 교육대학원), Jacob Wood(충남대, 무역학과)
Lee-Chae Jang*, Konkuk University, Graduate School of Education, Jacob Wood, Chungnam National University, Department of International Trade

In this presentation, we consider the discrete Choquet integral with respect to a fuzzy measure and define the Choquet expected utility as representing an act that utilizes for HS product codes to demonstrate the level of animal product exports between Korea and selected trading partners for years 2010-2013. We also consider the discrete interval-valued Choquet integral with respect to a fuzzy measure and define the interval-valued Choquet expected utility as representing an act that assesses for animal product exports between Korea and trading partners for HS Product Codes $i=1,2,3,4,5$. In particular, we investigate the following applications: (1) the ranking and the level of contribution, from an economic value perspective, for animal exports with HS product code $i=1,2,3,4,5$ between Korea and selected trade partners for years 2010-2013 and (2) the ranking and the level of contribution from an economic value perspective for total animal product exports between Korea and its trade partners for both years 2010-2013 and the respective HS product codes $i=1,2,3,4,5$.

2010 Mathematics Subject Classification: 28E10, 28E20, 03E72, 26E50 11B68
Key Words and Phrases: Choquet integral, Choquet expected utility, fuzzy set

⋅ 29th-C-14:30 − 15:40   Chair: Daewook Kim (Seowon University)

⋅ 29th-C-14:30 − 14:50 Bifurcating spatially periodic solutions and their stability of the Brusselator model (Soyeun Jung)

정소연(공주대)
Soyeun Jung, Kongju National University

Applying the Lyapunov-Schmidt reduction approach introduced by Mielke and Schneider in their analysis of the fourth-order scalar Swift-Hohenberg equation, we find (small-amplitude) spatially periodic solutions bifurcating from the uniform solution by Turing instability for the Bruseelator Model. Moreover, we carry out a rigorous small-amplitude stability analysis of the bifurcating periodic solutions.

2010 Mathematics Subject Classification: 35B10, 35B35
Key Words and Phrases: periodic solutions, turing instability, Lyapunov-Schmidt reduction, stability

⋅ 29th-C-14:50 − 15:10 On a coefficient problem for starlike functions (Young jae Sim, Oh sang Kwon, Nak eun Cho)

심영재*(경성대), 권오상(경성대), 조낙은(부경대)
Young Jae Sim*, Kyungsung University, Oh Sang Kwon, Kyungsung Univeristy, Nak Eun Cho, Pukyong National University

In this talk, the estimate of the third Hankel determinant \begin{equation*} \begin{split} H_{3,1}(f) &= \begin{vmatrix} a_{1} &a_{2} &a_{3} \\ a_{2} &a_{3} &a_{4} \\ a_{3} &a_{4} &a_{5} \end{vmatrix} \end{split} \end{equation*} for the class of starlike functions, i.e., for the class of analytic functions $f$ standardly normalized such that $Re(zf'(z)/f(z)) > 0,\ z\in{\mathbb D}:=\{z \in \mathbb{C} : |z|<1\},$ is shown.

2010 Mathematics Subject Classification: 30C45
Key Words and Phrases: univalent functions, starlike functions, Carath\'eodory functions, Hankel determinant

⋅ 29th-C-15:20 − 15:40 Averaged decay estimates for a compact hyperboloid in $\mathbb R^3$ (Chu Hee Cho)

조주희(서울대)
Chu Hee Cho, Seoul National University

We consider a compact hyperboloid in $\mathbb R^3$. We use polynomial partitioning to get the related restriction estimates for the hyperboloid with respect to a general measure. We also obtain averaged decay estimates for the hyperboloid.

2010 Mathematics Subject Classification: 42B20
Key Words and Phrases: restriction estimate, averaged decay estimate
Geometry

⋅ 29th-B-10:40 − 11:50 Chair: Dae Won Yoon (Gyeongsang National University)

⋅ 29th-B-10:40 − 11:00 Various centroids of catenaries and catenary rotation hypersurfaces (Dong-Soo Kim)

김동수(전남대)
Dong-Soo Kim, Chonnam National University

We study positive $C^1$ functions $z=f(x), x=(x_1,\ldots, x_n)$ defined on the $n$-dimensional Euclidean space $ \mathbb R^{n}$. For $x=(x_1,\ldots, x_n)$ with nonzero numbers $x_1, \ldots, x_n$, we consider the rectangular domain $I(x)=I(x_1)\times \cdots \times I(x_n)\subset \mathbb R^{n}$, where $I(x_i)= [0, x_i]$ if $x_i>0$ and $I(x_i)= [x_i,0]$ if $x_i<0$. We denote by $V$, $S$, $(\bar{x}_{V},\bar{z}_{V})$ and $(\bar{x}_{S},\bar{z}_{S})$ the volume of the domain under the graph of $z=f(x)$, the surface area $S$ of the graph of $z=f(x)$, the geometric centroid of the domain under the graph of $z=f(x)$ and the surface centroid of the graph itself over the rectangular domain $I(x)$, respectively. In this paper, first we show that among $C^2$ functions with isolated singularities, $V=kS$, $k\in \mathbb R$ characterizes the family of catenary rotation hypersurfaces $f(x)=k\cosh(r/k)$, $r=|x|$. Next, we show that the equality of $n$ coordinates of $(\bar{x}_{S},\bar{z}_{S})$ and $(\bar{x}_{V},2\bar{z}_{V})$ for every rectangular domain $I(x)$ characterizes the family of catenary rotation hypersurfaces among $C^2$ functions with isolated singularities.

2010 Mathematics Subject Classification: 53A07, 26A30
Key Words and Phrases: centroid, surface centroid, volume, surface area, catenary rotation hypersurface

⋅ 29th-B-11:00 − 11:20 Bounded cohomology and the Cheeger isoperimetric constant (Sungwoon Kim, Inkang Kim)

김성운*(제주대), 김인강(고등과학원)
Sungwoon Kim*, Jeju National University, Inkang Kim, KIAS

We study equivalent conditions for the Cheeger isoperimetric constant of Riemannian manifolds to be positive. We first give a proof of Gromov's assertion for locally symmetric spaces with infinite volume, which states that the existence of a bounded primitive of the Riemannian volume form is equivalent to the positivity of the Cheeger isoperimetric constant. Furthermore, under the assumption of pinched negative sectional curvature, we obtain another equivalent condition in terms of bounded cohomology classes. This generalizes Soma's result (Duke Math J 88(2):357–370, 1997) for hyperbolic 3-manifolds to RR-rank one locally symmetric spaces.

2010 Mathematics Subject Classification: 53C23, 53C35, 20F67
Key Words and Phrases: bounded cohomology, Cheeger isoperimetric constant, bounded primitive

⋅ 29th-B-11:30 − 11:50 On compact convex domains in complex hyperbolic space (Joonhyung Kim)

김준형(한남대)
Joonhyung Kim, Hannam University

J. Abardia and E. Gallego proved the following theorem in 2013. Theorem Let $\{\Omega(t)\}_{t \in \mathbb{R}^{+}}$ be a piecewise $C^2$ family of $\lambda$-convex compact domains, $0 \le \lambda \le k$, expanding over the whole space $\mathbb{C}H^n(-4k^2)$, $n \ge 2$. Then, $$ \frac{\lambda}{4nk^2} \le \liminf_{t\rightarrow \infty}\frac{vol(\Omega(t))}{vol(\partial\Omega(t))} \le \limsup_{t\rightarrow \infty}\frac{vol(\Omega(t))}{vol(\partial\Omega(t))} \le \frac{1}{2nk}. $$ Moreover, the upper bound is sharp. In this talk, I will explain this inequality and prove the sharpness of the lower bound by finding an example.

2010 Mathematics Subject Classification: 52A20
Key Words and Phrases: complex hyperbolic space, convex domain

⋅ 29th-C-14:30 − 15:40 Chair: Joonhyung Kim (Hannam University)

⋅ 29th-C-14:30 − 14:50 Super generalized complex geometry (Hoil Kim)

김호일(경북대)
Hoil Kim, Kyungbook National University

Generalized complex geometry has been developed to combine complex geometry and symplectic geometry by several people including Hitchin, Gualtieri and many others. It is very useful to understand mirror symmetry and is also used to interpret physical model of M theory and double field theory by Kapustin and Zwiebach. We extend these results to super version.

2010 Mathematics Subject Classification: 53D18, 53D37, 53D45
Key Words and Phrases: generalized complex geometry, symplectic geometry, superspace, double field theory, mirror symmetry

⋅ 29th-C-14:50 − 15:10 Harmonic Finsler torus without conjugate points (Chang-Wan Kim)

김창완(목포해양대)
Chang-Wan Kim, Mokpo National Maritime University

In 1994 Burago and Ivanov proved Hopf's conjecture, that is, Riemannian tori without conjugate points are flat. However, this conjecture does not hold in Finsler cases. Burago and Ivanov may conjecture that the geodesic flow on a Finsler torus without conjugate points is smoothly conjugate to that of a flat one, and all known examples confirm this guess. In this talk we show that complete noncompact harmonic Finsler manifolds with minimal horospheres are flat. In particular, harmonic Finsler tori without conjugate points are flat. And we offer some future open problems to work on.

2010 Mathematics Subject Classification: 53C23, 53C60
Key Words and Phrases: Finsler geometry, harmonic manifolds, minimal horospheres, without conjugate points

⋅ 29th-C-15:20 − 15:40 Classification of flat surfaces in $L^3$ (Jae Won Lee)

이재원(경상대)
Jae Won Lee, Gyeongsang National University

We study surfaces in the $3$-dimensional Minkowski space ${\mathbb L}^{3}$ with $\Delta G=fG+gC$, called {\it the general $1$-type Gauss map}, where $G, C$ and $f,~ g$ are respectively the Gauss map, a constant vector and some functions. As a result, first of all, we classify both conical and cylindrical surfaces with generalized 1-type Gauss map in the Minkowski 3-space $\mathbb L^3$. Second, we prove that nonplanar tangent developable surfaces in the Minkowski $3$-space $\mathbb L^3$ cannot have generalized 1-type Gauss map. Finally, we classify null scrolls as non-flat surfaces in$\mathbb L^3$.

2010 Mathematics Subject Classification: 53A05, 53B25
Key Words and Phrases: conical surface, developable surface, generalized 1-type Gauss map, null scrolls
Topology I

⋅ 29th-A-09:00 − 10:30   Chair: Sang Youl Lee (Pusan National University)

⋅ 29th-A-09:00 − 09:20 3-manifolds and circles (Hyungryul Baik)

백형렬(한국과학기술원)
Hyungryul Baik, KAIST

Thurston's geometrization program revolutionized 3-manifold theory, and had guided researches in low-dimensional topology until Perelman completed the program by proving Poincare conjecture. Just before Perelman announced his result, Thurston developed so-called universal circle program as an attempt to prove geometrization theorem via foliation theory. This beautiful piece of work is still mysterious and connected to many interesting features of 3-manifolds. I will give a short introduction to this theory, and discuss some recent progresses.

2010 Mathematics Subject Classification: 37C85, 37E30, 57M60, 20H10
Key Words and Phrases: 3-manifolds, geometrization, universal circle, orderable group, foliations

⋅ 29th-A-09:20 − 09:40 Irreducible 3-manifolds that are not obtained by 0-surgery along a knot (Matt Hedden, Min Hoon Kim, Kyungbae Park)

Matt Hedden(Michigan State Univ.), 김민훈(고등과학원), 박경배*(고등과학원)
Matt Hedden, Michigan State University, Min Hoon Kim, KIAS, Kyungbae Park*, KIAS

It is well known, due to Lickorish and Wallace, that any closed orientable 3-manifold can be obtained by Dehn-surgery along a link. The surgery number is a natural invariant of 3-manifold $M$, defined as the smallest number of components in a link in a Dehn surgery description of $M$. In their recent textbook, Aschenbrenner, Friedl and Wilton asked the following question regarding this invariant. Let $M$ be a closed, orientable, irreducible 3-manifold such that $b_1(M)=1$ and $\pi_1(M)$ has weight 1. Is $M$ the result of Dehn surgery along a knot in $S^3$? (i.e., Is the surgery number of $M$ one?) We answer the question negatively by presenting two infinite families of such 3-manifolds that cannot be obtained by surgery of a single knot. We use an obstruction from Heegaard Floer correction term, introduced by Ozsv\'ath and Szab\'o.

2010 Mathematics Subject Classification: 57M27, 57R58, 57R65
Key Words and Phrases: 3-manifolds, Dehn surgery, Heegaard Floer homology

⋅ 29th-A-09:50 − 10:10 On intersection forms of definite 4-manifolds bounded by a rational homology 3-sphere (Dong Heon Choe, Kyungbae Park)

최동헌*(서울대), 박경배(고등과학원)
Dong Heon Choe*, Seoul National University, Kyungbae Park, KIAS

We show that, if a rational homology 3-sphere $Y$ bounds a positive definite 4-manifold, then there are finitely many negative definite lattices, up to the stable-equivalence, which can be realized as the intersection form of a smooth 4-manifold bounded by $Y$. To this end, we make use of Donaldson's diagonalization theorem, and Ozsv\'ath and Szab\'o's Heegaard Floer correction term. We also present some families of Seifert fibered 3-manifolds that bound both positive and negative definite smooth 4-manifolds.

2010 Mathematics Subject Classification: 11H06 (57R57 57R58)
Key Words and Phrases: intersection form, definite 4-manifolds, Heegaard Floer correction term

⋅ 29th-A-10:10 − 10:30 On dissolving knot surgery 4-manifolds under $\mathbb{CP}^2$-connected sum (Hakho Choi, Jongil ParkK, Ki-Heon Yun)

최학호*(서울대), 박종일(서울대), 윤기헌(성신여대)
Hakho Choi*, Seoul National University, Jongil ParkK, Seoul National University, Ki-Heon Yun, Sungshin Women's University

In this talk we show that all knot surgery 4-manifolds $E(n)_K$ are mutually diffeomorphic after a connected sum with $\mathbb{CP}^2$. Hence, by combining a known fact that every simply connected elliptic surface is almost completely decomposable, we conclude that every knot surgery $4$-manifold $E(n)_K$ is also almost completely decomposable.

2010 Mathematics Subject Classification: 57N13, 57R17, 53D35
Key Words and Phrases: knot surgery 4-manifold, almost completely decomposable

⋅ 29th-B-10:40 − 12:10   Chair: Se-Goo Kim (Kyung Hee University)

⋅ 29th-B-10:40 − 11:00 On the volume and the Chern-Simons invariant for the hyperbolic alternating knot orbifolds (Ji-Young Ham, Hyuk Kim, Joongul Lee, SeokBeom Yoon)

함지영*(서울대), 김혁(서울대), 이준걸(홍익대), 윤석범(서울대)
Ji-Young Ham*, Seoul National University, Hyuk Kim, Seoul National University, Joongul Lee, Hongik University, SeokBeom Yoon, Seoul National University

We extend the Neumann's methods and give the explicit formulae for the volume and the Chern-Simons invariant for hyperbolic alternating knot orbifolds. Dedicated to Professor Joan Birman on the occasion of her ninetieth birthday.

2010 Mathematics Subject Classification: 57M27, 57M25
Key Words and Phrases: volume, Chern-Simons invariant, orbifold, explicit formula, alternating knot, Riley-Mednykh polynomial

⋅ 29th-B-11:00 − 11:20 Two-pointed Heegaard diagrams of $(1,1)$-knots up to 10 crossings (Hyeran Cho, Geunyoung Kim, Sang Youl Lee, Hyun-Jong Song)

조혜란(부산대), 김근영*(부산대), 이상율(부산대), 송현종(부경대)
Hyeran Cho, Pusan National University, Geunyoung Kim*, Pusan National University, Sang Youl Lee, Pusan National University, Hyun-Jong Song, Pukyong National University

A knot $K$ in the 3-sphere $S^3$ is a called $(1,1)$-knot if $K$ is split up into a trivial arc in each handlebody determined by a genus one Heegaard splitting of $S^3$. A $(1,1)$-knot in $S^3$ is represented by a two-pointed Heegaard diagram $D(a,b,c,r)$, where $a,b,c,$ and $r$ are nonnegative integers. In this talk, we introduce an algorithm for finding a two-pointed Heegaard diagram by using an unknotting tunnel of a given $(1,1)$-knot and present a list of all two-pointed Heegaard diagrams of $(1,1)$-knots up to 10 crossings in Rolfsen's knot table.

2010 Mathematics Subject Classification: 57M25, 57M27
Key Words and Phrases: knot theory, (1,1)-knot, Heegaard diagram

⋅ 29th-B-11:30 − 11:50 On $(1,1)$-diagrams of $2$-bridge knots and their duals (Hyeran Cho, Geunyoung Kim, Sang Youl Lee, Hyun-Jong Song)

조혜란*(부산대), 김근영(부산대), 이상율(부산대), 송현종(부경대)
Hyeran Cho*, Pusan National University, Geunyoung Kim, Pusan National University, Sang Youl Lee, Pusan National University, Hyun-Jong Song, Pukyong National University

A genus one $1$-bridge knot in $S^3$(simply called a $(1,1)$-knot) is a knot that can be decomposed into two trivial arcs embedded in two solid tori in a genus one Heegaard splitting of $S^3$. A $(1,1)$-knot can be described by a $(1,1)$-diagram $D(a,b,c,r)$ determined by four integers $a,b,c$ and $r$. It is well-known that all $2$-bridge knots are $(1,1)$-knots and have $(1,1)$-diagrams of the form $D(a,0,1,r)$. In this talk, we prove that the dual $(1,1)$-diagram of $D(a,0,1,r)$ is $D(\frac{r^{-1}}{2},0,2a+1-r^{-1},r^{-1})$ if $r^{-1}$ is even where $r^{-1}$ is the multiplicative inverse of $r$ (mod $2a+1$) and $D(\frac{2a+1-r^{-1}}{2},0,r^{-1},r^{-1})$, otherwise.

2010 Mathematics Subject Classification: 57M25, 57M27
Key Words and Phrases: $(1,1)$-knot, 2-bridge knot, $(1,1)$-diagram, dual $(1,1)$-diagram

⋅ 29th-B-11:50 − 12:10 On a rational type of surface-tangles (Yongju Bae, Seonmi Choi)

배용주(경북대), 최선미*(경북대)
Yongju Bae, Kyungpook National University, Seonmi Choi*, Kyungpook National University

In 1990, Viro introduced 2-dimensional briads, called surface braids. Satoh introduced surface-tangles as a generalization of classical tangles and showed that any surface-link can be represented as the closure of a surface-tangle. In this talk, I will introduce a rational type of surface-tangles defined as similar to surface braids via classical rational tangles and we will study some properties related to the closure of a rational type of surface-tangles.

2010 Mathematics Subject Classification: 57M25, 57M27
Key Words and Phrases: surface-tangle, rational tangle, surface braid

⋅ 29th-C-14:30 − 16:00   Chair: Jung Hoon Lee (Chonbuk National University)

⋅ 29th-C-14:30 − 14:50 Rope length of the standard double helix (Hyoungjun Kim, Youngsik Huh, Kyungpyo Hong, Sungjong No, Seungsang Oh)

김형준*(이화여대), 허영식(한양대), 홍경표(국가수리과학연구소), 노성종(이화여대), 오승상(고려대)
Hyoungjun Kim*, Ewha Womans University, Youngsik Huh, Hanyang University, Kyungpyo Hong, NIMS, Sungjong No, Ewha Womans University, Seungsang Oh, Korea University

In this paper we prove the unique existence of a ropelength-minimizing conformation of the double helix in a mathematically rigorous way, and find the minimal ropelength $\mbox{Rop}_{*}(\pi) = - \frac{8 \pi}{t}$ where $t$ is the unique solution in $[-\pi, 0]$ of the equation $2 - 2 \cos (t + \pi) = t^2$. Using this result, we also find the ropelength of the standard $N$-helix. The pitch angles of the standard double, triple and quadruple helices are around $39.3771^{\circ}$, $42.8354^{\circ}$ and $43.8351^{\circ}$, respectively, which are almost identical with the approximated pitch angles of the zero-twist structures previously known by Olsen and Bohr.

2010 Mathematics Subject Classification: 57M25
Key Words and Phrases: rope length, double helix

⋅ 29th-C-14:50 − 14:55 On the tricolorability of alternating $(1,1)$-tangles (Yongju Bae, Hun Lee)

배용주(경북대), 이헌*(경북대)
Yongju Bae, Kyungpook National University, Hun Lee*, Kyungpook National University

The tricolorability of an $(n,n)$-tangle diagram can be defined as similar to that of a knot diagram. In this talk, we will prove that an alternating $(1,1)$-tangle diagram T is tricolorable if and only if the closure of $T$ is tricolorable, provided that an alternating $(1,1)$-tangle has no circle components.

2010 Mathematics Subject Classification: 57M25 57M27
Key Words and Phrases: tangle, tricolorability

⋅ 29th-C-14:55 − 15:00 On Jones polynomials of Kanenobu braid type of links (Yongju Bae, Yongjae Park)

배용주(경북대), 박용재*(경북대)
Yongju Bae, Kyungpook National University, Yongjae Park*, Kyungpook National University

In 1986, Taizo Kanenobu defined a family of knots $K_{p,q}$, called Kanenobu knots, where $p$ and $q$ are the number of full twists. In 2005, Liam Watson introduced a generalization of Kanenobu knots. In this talk, we can deform Watson's generalization via $(2,2)$-tangles $T$, $U$ and a $3$-braid $b$, called Kanenobu braid type of links $L_{b}(T,U)$. A family of links is obtained from $L_{b}(T,U)$ by $\sigma_{2}$-action and it is denoted by $L_{b^{\sigma_{2}}}(T,U)$. We observe a relation of the Jones polynomials between $L_{b}(T,U)$ and $L_{b^{\sigma_{2}}}(T,U)$.

2010 Mathematics Subject Classification: 57M25, 57M27
Key Words and Phrases: tangle, Jones polynomial, braid, Kanenobu knots

⋅ 29th-C-15:20 − 15:40 On decomposition of finite quandle operation tables (Yongju Bae, Byeorhi Kim)

배용주(경북대), 김벼리*(경북대)
Yongju Bae, Kyungpook National University, Byeorhi Kim*, Kyungpook National University

S. Nelson, C.-Y. Wong and G. Ehrman, A. Gurpinar, D. N. Yetter studied that every finite quandle can be decomposed as subquandles in 2006 and 2008, respectively. In this talk, we observe an operation table $Q$ which consists of four sub-operation tables where diagonal tables are quandle operation tables. We will study precise conditions for off-diagonal tables which $Q$ can be a quandle operation table.

2010 Mathematics Subject Classification: 57M25, 57M27
Key Words and Phrases: quandle, inner automorphism group

⋅ 29th-C-15:40 − 16:00 Lattice stick number of spatial graphs (SeungSang Oh, HyungKee Yoo, ChaeRyn Lee)

오승상(고려대), 유형기*(고려대), 이채린(고려대)
SeungSang Oh, Korea University, HyungKee Yoo*, Korea University, ChaeRyn Lee, Korea University

The lattice stick number of a knot is defined to be the minimal number of straight line segments required to construct a stick presentation of the knot in the cubic lattice. For a nontrivial knot $K$, except the trefoil knot, Hong, No and Oh found an upper bound on the lattice stick number $s_{L}(K)$ in terms of the crossing number $c(K)$ which is $s_{L}(K) \leq 3c(K)+2$. As a sequel to this research program, we similarly define the stick number $s_{L}(G)$ of a spatial graph $G$ with vertices of degree at most six (necessary for embedding into the cubic lattice), and present an upper bound in terms of the crossing number $c(G)$ $$ s_{L}(G) \leq 3c(G)+6e-4v-2s+3b+k, $$ where $G$ has $e$ edges, $v$ vertices, $s$ cut-components, $b$ bouquet cut-components, and $k$ knot components. We mainly use the recent result of Lee, No and Oh who found arc presentations of spatial graphs with a fairly small arc index. (Joint work with Seungsang Oh, Chaeryn Lee)

2010 Mathematics Subject Classification: 57M25
Key Words and Phrases: lattice stick number, spatial graph
Topology II

⋅ 29th-A-09:00 − 10:30   Chair: Ensil Kang (Chosun University)

⋅ 29th-A-09:00 − 09:20 General expansive flows (Ju Mi Oh, Manseob Lee)

오주미*(성균관대), 이만섭(목원대)
Ju Mi Oh*, Sungkyunkwan University, Manseob Lee, Mokwon University

In light of the rich results of expansiveness in the dynamics of homeomorphisms, it is natural to consider another notions of expansiveness such as countably-expansive, measure expansive, $N$-expansive and so on. In this talk, we introduce the notion of $N$-expansiveness for flows on a $C^{\infty}$ compact connected manifold $M$ which is extension of the $N$-expansive homeomorphisms. And we prove that a vector field $X$ on $M$ is $C^1$ robustly $N$-expansive then $X$ satisfies quasi-Anosov. This work is partially supported by the Research Center for Nonlinear Ergodic Theory of Sungkyunkwan University.

2010 Mathematics Subject Classification: 37D05
Key Words and Phrases: flows, expansive, Anosov

⋅ 29th-A-09:20 − 09:40 Entropy expansiveness and measure expansiveness for continuous flows (Jaehyun Jeong, Namjip Koo)

정재현*(충남대), 구남집(충남대)
Jaehyun Jeong*, Chungnam National University, Namjip Koo, Chungnam National University

In this talk, we recall the notions of entropy expansiveness and measure expansiveness for homeomorphisms and continuous flows, and discuss the relationship between them. Moreover we study the dynamics of entropy expansive flows and measure expansive flows on compact smooth Riemannian manifolds.

2010 Mathematics Subject Classification: 37B40, 54H20
Key Words and Phrases: topological entropy, measure expansive

⋅ 29th-A-09:50 − 10:10 Expansive flows from measure theoretical viewpoint (Nguyen Ngoc Thach, Keonhee Lee)

Nguyen Ngoc Thach*(충남대), 이건희(충남대)
Nguyen Ngoc Thach*, Chungnam National University, Keonhee Lee, Chungnam National University

In this talk, we introduce a concept of weak expansive measure for flows which is really weaker than that of expansive measure, and show that there is a weak measure-expansive flow on a closed surface. Moreover we prove that if any nonempty open subset of $X$ intersects with uncountably many orbits of a flow $\phi$, then the set of $\phi$-orbit-vanishing measures on $X$ is dense in the set of all Borel probability measures on $X$ with weak*-topology. The result is applied to characterize the set of weak measure-expansive flows using the notion of countably expansive flows. Finally we show that the integrated flow of a $C^1$ vector field without singularities on a compact smooth manifold is $C^1$ stably expansive if and only if it is $C^1$ stably weak measure-expansive. {\it This is a joint work with K. Lee.}

2010 Mathematics Subject Classification: 37C05
Key Words and Phrases: expansive, flows, weak measure-expansive, C1 stably weak measure-expansive

⋅ 29th-A-10:10 − 10:30 Stability of homeomorphisms on noncompact metric spaces (Yang Yinong, Keonhee Lee)

Yinong Yang*(충남대), 이건희(충남대)
Yinong Yang*, Chungnam National University, Keonhee Lee, Chungnam National University

The dynamics of homeomorphisms on compact metric spaces are well understood. In this talk, we extend the notions of expansiveness, POTP and topological stability of homeomorphisms on compact metric spaces to homeomorphisms (or continuous flows) on noncompact metric spaces. Furthermore, we prove that any weak expansive homeomorphism (or flow) with extended POTP is extended topologically stable.

2010 Mathematics Subject Classification: 37B20
Key Words and Phrases: weak expansive, extended POTP, extended topological stable

⋅ 29th-B-10:40 − 12:10   Chair: Keonhee Lee (Chungnam National University)

⋅ 29th-B-10:40 − 11:20 Topological stability from Gromov-Hausdorff viewpoint (A. Arbieto, C. A. Morales)

A. Arbieto(Universidade Federal do Rio de Janeiro), C. A. Morales*(Universidade Federal do Rio de Janeiro)
A. Arbieto, Universidade Federal do Rio de Janeiro, C. A. Morales*, Universidade Federal do Rio de Janeiro

We combine Gromov-Hausdorff's metric with the $C^0$ distance to obtain the $C^0$-{\it Gromov-Hausdorff distance} between metric space's maps. The latter is then combined with Walters's topological stability to obtain the notion of {\it topologically GH-stable homeomorphism}. We prove that there are topologically stable homeomorphism which are not topologically GH-stable. Also that every topological GH-stable circle homeomorphism is topologically stable. Afterwards, we prove that every expansive homeomorphism with the pseudo-robit tracing property of a compact metric space is topologically GH-stable. This is related to Walters's stability theorem. Finally, we extend the topological GH-stability to continuous maps and prove that the constant maps on compact homogeneous manifolds are topologically GH-stable.

2010 Mathematics Subject Classification: 54H20, 53C23
Key Words and Phrases: topological stability, topological GH-stability

⋅ 29th-B-11:30 − 11:50 On the quasi-weakly almost periodic points of continuous flows (Jiandong Yin)

Jiandong Yin(Nanchang Univ.)
Jiandong Yin, Nanchang University

Let $X$ be a compact metric space, $F: X\times \mathbb{R}\rightarrow X$ a continuous flow and $x\in X$ a proper quasi-weakly almost periodic point; that is, $x$ is quasi-weakly almost periodic but not weakly almost periodic. In this talk, we will investigate whether there exists an invariant measure $\mu$ on $X$ generated by the orbit of $x$ such that the support of this measure $\mu$ coincides with the minimal center of attraction of $x$.

2010 Mathematics Subject Classification: 37B20
Key Words and Phrases: quasi-weakly almost point, minimal center of attraction, invariant measure

⋅ 29th-B-11:50 − 12:10 Topological transitivity and various shadowing properties for shift maps (Namjip Koo, Nyamdavaa Tsegmid)

구남집(충남대), Nyamdavaa Tsegmid*(충남대)
Namjip Koo, Chungnam National University, Nyamdavaa Tsegmid*, Chungnam National University

We briefly recall the shift map having expansiveness, topological transitivity and the pseudo-orbit-tracing property on compact metric spaces. Then, various shadowing properties are considered for shift maps

2010 Mathematics Subject Classification: 37B99, 54D20
Key Words and Phrases: expansive, transitive, POTP, shift map, Markov subshift, Lipschitz shadowing property, average shadowing property

⋅ 29th-C-14:30 − 16:00   Chair: Yongjin Song (Inha University)

⋅ 29th-C-14:30 − 14:50 Topologically stable measures for finitely generated group actions (Meihua Dong, Sang Jin Kim)

동미화*(충남대), 김상진(충남대)
Meihua Dong*, Chungnam National University, Sang Jin Kim, Chungnam National University

Recently, Chung and Lee [to appear in Proc AMS] introduced a concept of topologically stable group action and proved that any expansive group action with POTP is topologically stable. In this talk, we consider a measurable version of the above results. More precisely, we introduce a notion of topologically stable measure for finitely generated group action on a compact metric space, and prove that any expansive measure with the pseudo orbit tracing property is topologically stable. Moreover we show that any Borel probability measure is topologically stable for a topologically stable group action. \textit{This is a joint work with Sang Jin Kim.}

2010 Mathematics Subject Classification: 37B05
Key Words and Phrases: topologically stable measure, group action, POTP

⋅ 29th-C-14:50 − 15:10 Configuration spaces of graphs (Byung Hee An, Gabriel C. Drummond-Cole, Ben Knudsen)

안병희*(기초과학연구원), Gabriel C. Drummond-Cole(Institute for Basic Science), Ben Knudsen(Harvard Univ.)
Byung Hee An*, Institute for Basic Science, Center for Geometry and Physics, Gabriel C. Drummond-Cole, Institute for Basic Science, Center for Geometry and Physics, Ben Knudsen, Harvard University

We consider total configuration space $B\Gamma$ of a given graph $\Gamma$ equipped with an action of edges $E(\Gamma)$ and introduce a minimal model $S_*(\Gamma)$ for its singular chain complexes $C_*(B\Gamma)$, which is a DG-module over $\mathbb{Z}[E]$. By using this model, we compute homology groups for several interesting examples, and also discuss about the formality. This is a joint work with Gabriel C. Drummond-Cole (IBS-CGP) and Ben Knudson (Harvard University).

2010 Mathematics Subject Classification: 55R80, 16E45
Key Words and Phrases: configuration spaces, graphs, formality

⋅ 29th-C-15:20 − 15:40 Vanishing odd degree cohomology of toric orbifolds (Anthony Bahri, Soumen Sarkar, Jongbaek Song)

Anthony Bahri(Rider Univ.), Soumen Sarkar(IIT-Madras), 송종백*(한국과학기술원)
Anthony Bahri, Rider University, Soumen Sarkar, IIT-Madras, Jongbaek Song*, KAIST

There are several advantages to studying topological spaces having vanishing odd degree cohomology groups. For example, Atiyah—Bredon sequence derives its equivariant cohomology ring and Atiyah—Hirzebruch spectral sequence allows to compute complex K-theory or complex cobordism ring. In toric topology, it is well-known that every smooth toric manifold, a $2n$-dimensional manifold equipped with well-behaved $n$-dimensional real torus action, has vanishing odd degree cohomology. However, there are many examples of toric orbifolds with non-trivial odd degree cohomology groups. In this talk, we introduce a sufficient condition for vanishing odd degree cohomology, which can be described by a combinatorial way. This is a joint work with Anthony Bahri and Soumen Sarkar.

2010 Mathematics Subject Classification: 55N91
Key Words and Phrases: toric orbifolds

⋅ 29th-C-15:40 − 16:00 A flagged Bott tower and the torus orbit closure (Shintaro Kuroki, Eunjeong Lee, Jongbaek Song, Dong Youp Suh)

Shintaro Kuroki(Okayama University of Science), 이은정*(한국과학기술원), 송종백(한국과학기술원), 서동엽(한국과학기술원)
Shintaro Kuroki, Okayama University of Science, Eunjeong Lee*, KAIST, Jongbaek Song, KAIST, Dong Youp Suh, KAIST

The flag manifold $\mathcal{F}lag(n+1)$ is the set of full-flags in $\mathbb{C}^{n+1}$. More precisely, a point in $\mathcal{F}lag(n+1)$ is an increasing sequence of vector spaces. It has been studied that the flag manifold has an effective $(\mathbb{C}^{\ast})^n$-action and the closure of the torus orbit of a generic point is the permutohedral variety, which is a nonsingular projective toric variety whose orbit space can be identified with the permutohedron $\Xi_n$. Recently, an extended notion of the flag manifold has been introduced, called flagged Bott tower, which is a sequence of $\mathcal{F}lag(n_j+1)$-fiber bundles. A flagged Bott tower is not a toric variety but it has a torus action. We study the torus orbit closure of a generic point in a flagged Bott tower. This talk is based on an on-going project with Jongbaek Song, Shintaro Kuroki, and Dong Youp Suh.

2010 Mathematics Subject Classification: 14M17, 57S25, 14M25
Key Words and Phrases: flag manifold, flagged Bott tower, toric variety
Probability and Statistics

⋅ 29th-B-10:40 − 12:10 Chair: Panki Kim (Seoul National University)

⋅ 29th-B-10:40 − 11:00 Optimal information disclosure policies in a strategic queueing model (Bara Kim, Jeongsim Kim)

김바라*(고려대), 김정심(충북대)
Bara Kim*, Korea University, Jeongsim Kim, Chungbuk National University

We find the optimal policy for the information disclosure problem of an M/M/1 queue studied by Simhon et al. (2016). The optimal disclosure policy is as follows: the service provider informs all customers about the queue length when the queue length is above the specified threshold and does not inform them when the queue length is below the threshold.

2010 Mathematics Subject Classification: 60K25
Key Words and Phrases: queueing models, equilibrium strategies, optimal policies

⋅ 29th-B-11:00 − 11:20 Girsanov theorem for quantum stochastic processes (Un Cig Ji)

지운식(충북대)
Un Cig Ji, Chungbuk National University

A quantum extension of the Girsanov theorem is formulated as an implementation problem for Boson fields in terms of quantum white noise calculus. For our purpose, we first prove that the space of all admissible white noise operators is a commutative $*$-algebra with the Wick product. Secondly, we develop the concept of quantum white noise derivatives of admissible white noise operators and as applications, we study general forms of implementation problem of admissible white noise operators. Finally, implementation problems with solutions as quantum martingales are applied to Girsanov theorem for quantum stochastic processes. This talk is based on a series of joint works with Nobuaki Obata.

2010 Mathematics Subject Classification: 81S25
Key Words and Phrases: admissible white noise operators, quantum white noise calculus, quantum white noise derivatives, implementation problem, quantum stochastic process, Girsanov theorem

⋅ 29th-B-11:30 − 11:50 Mathematical modeling and analysis of interference coordination based on stochastic geometry (Jonghun Yoon, Ganguk Hwang)

윤종훈(한국과학기술원), 황강욱*(한국과학기술원)
Jonghun Yoon, KAIST, Ganguk Hwang*, KAIST

We use stochastic geometry theory to model a small cell network and analyze the performance of interference coordination. While most of the previous works do not consider the spatial distribution of so-called edge users in performance modeling, we carefully consider the spatial distribution of edge users and derive the optimal cooperation radius in interference coordination.

2010 Mathematics Subject Classification: 60D05
Key Words and Phrases: stochastic geometry, small cell network, interference coordination

⋅ 29th-B-11:50 − 12:10 Non-symmetric jump process with exponential decaying jumping kernel (Jaehoon Lee )

이재훈(서울대)
Jaehoon Lee, Seoul National University

Consider a non-symmetric and non-local operator of the following form: $$ \mathcal{L}^\kappa f(x)= \lim_{\epsilon \rightarrow 0} \int_{|z|>\epsilon} (f(x+z)-f(x))\kappa(x,z)J(|z|)dz, \quad \kappa(x,z)=\kappa(x,-z). $$ Panki Kim, Renming Song and Zoran Vondracek obtained some heat kernel estimates of $\mathcal{L}^\kappa$ when $J$ be the L\'evy density of a symmetric L\'evy process in $\mathbb{R}^d$ with its L\'evy exponent satisfying a weak lower scaling condition at infinity. In this talk, we discuss the heat kernel estimates of $\mathcal{L}^\kappa$ when $J$ has additional exponentially decaying condition, i.e., $J(r) \le c_1 exp(-c_2 |r|^\beta)$, $r>1$ for some $c_1, c_2>0$ and $0 \le \beta \le 1$. The talk is based on a joint work with Panki Kim.

2010 Mathematics Subject Classification: 60J35
Key Words and Phrases: heat kernel estimates, subordinate Brownian motion, symmetric Levy process, non-symmetric operator, non-symmetric Markov process

⋅ 29th-C-14:30 − 15:40 Chair: Un Cig Ji (Chungbuk National University)

⋅ 29th-C-14:30 − 14:50 Intrinsic ultracontractivity of symmetric jump processes on unbounded domains (Panki Kim)

김판기(서울대)
Panki Kim, Seoul National University

In this talk, we consider a symmetric pure-jump Markov process on Euclidean space generated by a non-local Dirichlet form with jumping kernel $J(x,y)$. We first discuss sufficient conditions for the compactness and the intrinsic ultracontractivity of the Dirichlet Markov semigroup on $D$ when $D$ is an unbounded open set. When $D$ is the horn-shaped domain, we will discuss sharp criterion for the intrinsical ultracontractivity and the sharp estimates of the ground state. This is a joint work with Xin Chen (Shanghai Jiao Tong University) and Jian Wang (Fujian Normal University).

2010 Mathematics Subject Classification: 60G51, 60G52, 60J25, 60J7
Key Words and Phrases: symmetric jump process, non-local Dirichlet form, intrinsic ultracontractivity, ground state, intrinsic super Poincar\'{e} inequality

⋅ 29th-C-14:50 − 15:10 Sufficient conditions for ergodicity of stochastic reaction networks and mixing times (Jinsu Kim, David Anderson)

김진수*(Univ. of Wisconsin-Madison), David Anderson(Univ. of Wisconsin-Madison)
Jinsu Kim*, University of Wisconsin-Madison, David Anderson, University of Wisconsin-Madison

Reaction networks are graphical configurations that can be used to describe biological interaction networks. If the abundances of the constituent species of the system are low, we can model the dynamics of species counts in a jump by jump fashion as a continuous time Markov chain. In this talk, we will mainly focus on which conditions of the graph imply ergodicity (existence of a stationary distribution) for the associated continuous time Markov chain. I will also present results related to their mixing times, which give the time required for the distribution of the continuous time Markov chain to get close to the stationary distribution.

2010 Mathematics Subject Classification: 47N30
Key Words and Phrases: reaction networks, Markov chains, stationary distributions

⋅ 29th-C-15:20 − 15:40 Dirichlet heat kernel estimates for subordinate Brownian motions with Gaussian component (Joohak Bae)

배주학(서울대)
Joohak Bae, Seoul National University

The transition density (if it exists) of Markov process is the heat kernel of the generator of the process. The transition density of a general Markov process rarely admits an explicit expression. Thus obtaining sharp estimates on transition density is a fundamental problem both in probability theory and in analysis. In this talk, we discuss the behavior of transition density (Dirichlet heat kernel) for subordinate Brownian motions in $C^{1,1}$-open subsets whose laplace exponent $\psi(\lambda)=\lambda+\phi(\lambda)$ and scaling orders of $\phi$ are not necessarily strictly less than 1. This is a joint work with Panki Kim.

2010 Mathematics Subject Classification: no number
Key Words and Phrases: Browinan motion
Applied Mathematics I

⋅ 29th-A-09:20 − 10:30   Chair: Shinuk Kim (Sangmyung University)

⋅ 29th-A-09:20 − 09:40 Optimal harvest strategies of sailfin sandfish (Giphil Cho, Il Hyo Jung)

조기필*(부산대), 정일효(부산대)
Giphil Cho*, Pusan National University, Il Hyo Jung, Pusan National University

We discuss the optimal control strategies for an objective function in the impulsive system. There are two objectives in fisheries management. The biological purpose is to maximize the total amount of cumulative fishing resources obtained by fishing, and the goal of economic profit is to maximize the cost and profit of fishing. We demonstrates the existence and necessary condition for the optimal control problem. We simulate the optimal control problem from biological and economic perspectives, and compare the result of the control strategies. Sensitivity analysis is conducted based on the cost. Next, we estimate the optimal harvest strategies for maximizing catch and profit as an example of sandfish.

2010 Mathematics Subject Classification: 92B00
Key Words and Phrases: optimal harvest, Sailfin sandfish, impulsive system

⋅ 29th-A-09:50 − 10:10 A predator-prey model with a general starvation driven diffusion (Wonhyung Choi, Inkyung Ahn)

최원형*(고려대), 안인경(고려대)
Wonhyung Choi*, Korea University, Inkyung Ahn, Korea University

In this talk, we present a predator-prey model with starvation driven diffusion(SDD) under Neumann boundary condition. The proposed model describes the predator-prey interaction with a Lotka-Volterra type and a general mobility in common heterogeneous environment. We investigate the stability of semi-trivial solutions to the model with SDD and compare with the condition for the model with constant diffusion. Furthermore, the biological interpretation will be presented.

2010 Mathematics Subject Classification: 92B05
Key Words and Phrases: SDD, predator-prey, starvation driven diffusion

⋅ 29th-A-10:10 − 10:30 Dynamical behavior of an oncolytic virus dynamics model with time delay (kwangsu kim, Il Hyo Jung, Sangil Kim)

김광수*(부산대), 정일효(부산대), 김상일(한국외대 차세대 도시$\cdot$농림융합기상사업단)
Kwangsu Kim*, Pusan National University, Il Hyo Jung, Pusan National University, Sangil Kim, WISE Institute, HanKuk University of Foreign Studies

In this talk, we consider a delayed mathematical model for the interactions of Cancer-viral therapy. We first investigate the existence of therapy failure equilibrium and partial success equilibrium. Sufficient criteria for local asymptotic stability for two equilibrium point and existence of Hopf bifurcation by using delay as a bifurcation parameter are given. We show that the time delay can unstabilize the partial success equilibrium, leading to Hopf bifurcations and stable periodic oscillations. We will focus on the Hopf bifurcation point that is calculated depending on viral cytotoxicity and infection rate. Secondly, the optimal control problem is suggested to investigate the impact of delay for Oncolytic immunotherapy, by reformulating the model with two control; cancer-viral therapy and immunotherapy. Numerical simulations are presented to support the analytical results.

2010 Mathematics Subject Classification: 92D25, 49J15, 93D20
Key Words and Phrases: cancer-viral therapy, Hopf bifurcation, optimal treatment strategy, oncolyticimmunotherapy, time-delay

⋅ 29th-B-10:40 − 12:10   Chair: Eun-Hee Park (Kangwon National University)

⋅ 29th-B-10:40 − 11:00 A dual iterative substructuring method with a small penalty parameter (Chang-Ock Lee, Eun-Hee Park)

이창옥(한국과학기술원), 박은희*(강원대)
Chang-Ock Lee, KAIST, Eun-Hee Park*, Kangwon National University

In this talk we will discuss a non-overlapping domain decomposition method which is a variant of the dual-primal finite element tearing and interconnecting (FETI-DP) method. Differently from the FETI-DP method which enforces the continuity across the interface by introducing Lagrange multipliers, the proposed method introduces a penalty term in addition to Lagrange multipliers in order to strengthen the continuity across the interface. Due to the penalty term, the proposed method has a better convergence property than the standard FETI-DP method. Theoretical results on the condition number will be presented along with numerical results. The condition number estimate shows the relationship between the condition number and a penalty parameter as well as a close spectral connection of the proposed method with the FETI-DP method.

2010 Mathematics Subject Classification: 65F10, 65N30, 65N55
Key Words and Phrases: augmented Lagrangian, domain decomposition, dual substructuring

⋅ 29th-B-11:00 − 11:20 Multigrid algorithms for elliptic equations with interface problems based on immersed finite element discretization (Gwanghyun Jo, Do Y. Kwak)

조광현*(한국과학기술원), 곽도영(한국과학기술원)
Gwanghyun Jo*, KAIST, Do Y. Kwak, KAIST

Multigrid algorithm is one of the most efficient algorithms for solving sparse linear systems arising from the discretization of partial differential equations (PDEs). However, when the coefficients of the PDE become discontinuous along the interface, it is hard to apply the multigrid algorithms especially when the geometry of interface is complex. It is because conventional methods are based on the fitted grids, which makes the discretized system to have complex data structure. Recently, there has been some new developments in finite element methods. The immersed finite element method (IFEM) uses uniform grids for the interface problems allowing the interfaces to cut through the element. Instead, the basis functions are modified so that the flux becomes continuous along the interface. The applicability of the multigrid solvers for the IFEM was one of the features of the IFEM. However, the performance of multigrid solvers of IFEM was not reported until the work [1] appears. In this talk, we report the performance of multigrid solvers for IFEM. We observe the optimal scalability for the various interface problems even when the discontinuity is up to 1000 times (1 versus 1000) along the interfaces. [1] Gwanghyun Jo, Do Y. Kwak, An IMPES scheme for a two-phase flow in heterogeneous porous media using a structured grid, Computer Methods in Applied Mechanics and Engineering, 2017, published online

2010 Mathematics Subject Classification: 65N55
Key Words and Phrases: immersed finite element method, multigrid

⋅ 29th-B-11:30 − 11:50 Numerical implementation of the shallow water equation on a curved surface (Seunggyu Lee, Yong-Jung Kim)

이승규*(국가수리과학연구소), 김용정(한국과학기술원)
Seunggyu Lee*, NIMS, Yong-Jung Kim, KAIST

We present a finite difference method for solving the shallow water equation numerically on a curved surface. Although most of the flow dynamics on the earth are modeled by the shallow water equation, it is too complicated to solve the shallow water equation analytically in the majority of cases even on flat surfaces. Using the closest point method, embedding a problem within a structured Cartesian domain of an original equation, the conventional numerical solver and grid mesh and can be applied instead of complex ones. To examine the method, we perform the several numerical simulations.

2010 Mathematics Subject Classification: 65M06, 65M22, 76D27
Key Words and Phrases: finite difference, shallow water equation, closest point method, curved surface

⋅ 29th-B-11:50 − 12:10 A class of Hermite subdivision schemes reproducing polynomials (Byeongseon Jeong, Jungho Yoon)

정병선*(이화여대 수리과학연구소), 윤정호(이화여대)
Byeongseon Jeong*, Institute of Mathematical Sciences, Ewha Womans University, Jungho Yoon, Ewha Womans University

We propose a new class of Hermite subdivision schemes with tension parameters. This class unifies well-known interpolatory and noninterpolatory Hermite subdivision schemes. Our Hermite schemes preserve polynomials up to a certain degree, guaranteeing high accuracy. The approximation order of the proposed schemes is discussed. The smoothness analysis has been performed by using the factorization framework of subdivision operators. Some numerical examples illustrate the performance of the proposed Hermite schemes.

2010 Mathematics Subject Classification: 65D18
Key Words and Phrases: Hermite subdivision scheme, polynomial reproduction

⋅ 29th-C-14:30 − 16:00   Chair: Chi-Ok Hwang (Gwangju Institute of Science and Technology)

⋅ 29th-C-14:30 − 14:50 Diffusion Monte Carlo methods for charge density on a conducting surface: Review (Chi-Ok Hwang)

황치옥(광주과학기술원)
Chi-Ok Hwang, Gwangju Institute of Science and Technology

Diffusion Monte Carlo methods, first-passage and last-passage methods, have been used for charge density on a conducting surface. Diffusion Monte Carlo methods are based on probability potential theory. To expedite the diffusion simulations, the isomorphism between electrostatic Poisson problem and homogeneous and isotropic diffusion has been used. I will review all the expediting algorithms for the homogeneous and isotropic diffusion.

2010 Mathematics Subject Classification: 60J65
Key Words and Phrases: Monte Carlo, diffusion, first-passage, last-passage

⋅ 29th-C-14:50 − 15:10 Mean-field limit of the thermodynamic Cucker-Smale model (Jeongho Kim)

김정호(서울대)
Jeongho Kim, Seoul National University

The Cucker-Smale (CS) model is one of the most well-known model to describe the collective dynamics. But in CS model, there is no variables for each particle's internal property. Therefore, we generalize the CS model in which each particle has its own internal property such as temperature, which is called a Thermodynamic Cucker-Smale (TCS) model. We will derive a uniform mean-field limit from the TCS model to the Vlasov type kinetic model in the phase space for some class of initial data whose Dirac comb approximations guarantees exponential flocking in the TCS model.

2010 Mathematics Subject Classification: 70F10
Key Words and Phrases: thermodynamic Cucker-Smale model, mean-field limits, uniform stability

⋅ 29th-C-15:20 − 15:40 A semi-analytic formula for European option price approximation under a multi-scale Heston stochastic volatility model (Veng Sotheara, Jai Heui Kim)

Veng Sotheara*(부산대), 김재희(부산대)
Veng Sotheara*, Pusan National University, Jai Heui Kim, Pusan National University

In this talk, we consider an option pricing problem under a model extension of Heston stochastic volatility model in which a slow varying factor of volatility is incorporated to the multi-scale stochastic volatility model of [Fouque and Lorig, SIAM J. Finan. Math., 2011]. An asymptotic analysis consisting of singular and regular perturbation expansions is used to obtain an approximation for European option prices. Consequently, the pricing formulas are still semi-analytical which can be expressed as integral forms in the same level of computational complexity as those derived in [Fouque and Lorig, SIAM J. Finan. Math., 2011]. A proof of accuracy of the asymptotic approximation is also provided.

2010 Mathematics Subject Classification: 91G20, 41A60
Key Words and Phrases: asymtotic analysis, Heston model, stochastic volatility, mean reverting

⋅ 29th-C-15:40 − 16:00 Recurrence relations-based convergence analysis of a two-step Newton-like method in Banach spaces under milder condition with applications (Dohee Jung, Changbum Chun)

정도희*(성균관대), 천창범(성균관대)
Dohee Jung*, Sungkyunkwan University, Changbum Chun, Sungkyunkwan University

In this paper we study the semilocal convergence of a well-known multipoint third order Newton-like method for solving $F(x)=0$ in Banach spaces. The convergence of this method is established based on recurrence relations. We derive the recurrence relations for the method under the assumption that the Fr$\rm \acute{e}$chet derivative of $F$ satisfies the $\omega$-continuity condition which includes the Lipchitz and H$\rm\ddot{o}$lder conditions as particular cases. Numerical examples are worked out to demonstrate the applicability of our result.

2010 Mathematics Subject Classification: 41A25, 65H10
Key Words and Phrases: Newton-like method, semilocal convergence, recurrence relations, Fr\'{e}chet derivative, Banach space
Applied Mathematics II

⋅ 29th-B-10:40 − 12:10 Chair: Younhee Lee (Chungnam National University)

⋅ 29th-B-10:40 − 11:00 Error estimates for derivative pricing problems under a regime-switching L\'{e}vy process with local volatility (Sunju Lee, Younhee Lee)

이순주(충남대), 이윤희*(충남대)
Sunju Lee, Chungnam National University, Younhee Lee*, Chungnam National University

In this talk we consider a regime-switching jump-diffusion process with local volatility to evaluate the prices of European and American options which are traded in the financial market. The stability of implicit methods with three time levels to evaluate the prices of the European option is proved and then these numerical methods have the second-order convergence rate in the time and spatial variables. Several experiments are carried out to illustrate the stability analysis of the proposed methods.

2010 Mathematics Subject Classification: 91G60
Key Words and Phrases: partial integro-differential equation, regime-switching L\'{e}vy process, numerical method

⋅ 29th-B-11:00 − 11:20 Mobility scaling of extreme events in spatiotemporal dynamical systems (Beomseok Kim, Sebastian Altmeyer, Bongsoo Jang, Younghae Do, Ying-Cheng Lai)

김범석*(경북대), Sebastian Altmeyer(Institute of Science and Technology Austria, Austria), 장봉수(울산과학기술원), 도영해(경북대), Ying-Cheng Lai(School of Electrical, Computer, and Energy Engineering)
Beomseok Kim*, KNU-Center for Nonlinear Dynamics, Kyungpook National University, Sebastian Altmeyer, Institute of Science and Technology Austria, Austria, Bongsoo Jang, Ulsan National Institute of Science and Technology (UNIST), Younghae Do, KNU-Center for Nonlinear Dynamics, Kyungpook National University, Ying-Cheng Lai, School of Electrical, Computer, and Energy Engineering

Extreme or rare intense events are ubiquitous in spatiotemporal dynamical systems. Given a closed spatial domain, such events occur at different spatial locations with time. Does a scaling law governing the movement or the ``hopping" of the extreme events exist? Utilizing the complex Ginzburg-Landau equation (CGLE) as a paradigmatic model, we investigate the dynamical behaviors of the extreme events by defining their mobility or ``velocity" as the ratio of the distance between the locations of two consecutive events to the time interval of their occurrences. We uncover an algebraic scaling law for the mobility. The scaling is universal as it holds not only for extreme events in the CGLE model system, but also for actual earthquakes. In particular, we analyze the Southern California and Japan earthquake data sets in a 36-year span (from 1980 to 2015), and obtain the same scaling law for the earthquake mobility. The algebraic scaling indicates the existence of ``rapidly moving" earthquakes with relatively large mobility, implying a fundamental difficulty in predicting these events.

2010 Mathematics Subject Classification: 37F99
Key Words and Phrases: extreme events, spatiotemporal dynamical systems, complex Ginzburg-Landau equation, mobility, scaling law, earthquake

⋅ 29th-B-11:30 − 11:50 The Hermitian positive definite solution to a nonlinear matrix equation (Jie Meng, Hyun-Min Kim)

Jie Meng*(부산대), 김현민(부산대)
Jie Meng*, Pusan National University, Hyun-Min Kim, Pusan National University

In this talk, we consider a nonlinear matrix equation $X^s-A^*X^{-t}A=Q$, where $s$ and $t$ are positive integers and $s\geq t$, $A$ is an arbitrary $n\times n$ matrix and $Q$ is an $n\times n$ Hermitian positive definite matrix. The uniqueness of the Hermitian positive definite solution without any extra condition is obtained. One basic fixed-point iteration with Stepsize Parameter for computing the Hermitian positive definite solution is proposed. One condition number of the unique positive definite solution is derived. Some numerical examples to show the efficiency of the proposed iterative methods are provided.

2010 Mathematics Subject Classification: 15A24, 65F10, 65H10
Key Words and Phrases: matrix equation, Banach space, Hermitian positive definite, fixed-point iteration, condition number

⋅ 29th-B-11:50 − 12:10 Spreading of rumors with different preferences (Byeong-kyu Park, Younghae Do)

박병규*(경북대), 도영해(경북대)
Byeong-kyu Park*, Kyungpook National University, younghae Do, Kyungpook National University

Recently, many spreading models of rumor have been studied. Due to the similarity of the spreading, these models are related to the epidemic models. So epidemic threshold and final state give more information about the rumors. In [1], they consider a situation that two different types of rumors are propagating on the same contact network. More precisely, one rumor, say rumor 1, is considered first when an individual decides to which rumor to adapt. This model shows that the rumor 1 is dominant and the other is expelled when the degree of nodes is high or the network has large clustered groups of nodes. However, in real situations, the priority of the rumors is different personally. So we would introduce a modified the spreading model of rumors and we show the numerical results which is supported to our analytic analysis.

2010 Mathematics Subject Classification: 37N99
Key Words and Phrases: network theory, spreading dynamics, rumor spreding

⋅ 29th-C-14:30 − 15:10 Chair: Youngyong Park (Kyungpook National University)

⋅ 29th-C-14:30 − 14:50 Early effect in nonsmooth dynamical systems excited under frequency-dependent amplitude (Youngyong Park, Younghae Do, Ying-Cheng Lai)

박영용*(경북대), 도영해(경북대), Ying-Cheng Lai(Arizona State University, USA)
Youngyong Park*, Kyungpook National University, Younghae Do, KNU-Center for Nonlinear Dynamics, Kyungpook National University, Ying-Cheng Lai, Arizona State University, USA

We investigate bifurcations in nonsmooth dynamical systems under adiabatic parameter variations. Against expectation that it is effective to detect the possible bifurcations by using a single parameter sweep in experimental studies of nonlinear dynamical systems, nonstationary dynamical system with varying external forcing frequency as a time dependent bifurcation parameter can not detect the true bifurcations, when the forcing amplitude depends on the forcing frequency in the nonsmooth dynamical system. To resolve the problem, we consider the approximate solution of the corresponding smooth nonimpact system which is confirmable analytically and supports the necessity for amplitude adjustment to detect the true bifurcations. The external amplitude adjustment suggests that, in general, any true bifurcation of a nonlinear dynamical system can be clearly uncovered through slowly varying parameter in time, which has a useful value to experimental studies of nonlinear dynamical systems.

2010 Mathematics Subject Classification: 34C23
Key Words and Phrases: bifurcation, early effect, nonsmooth, slow passage, recovering

⋅ 29th-C-14:50 − 15:10 Global existence and blow-up phenomena for a quasilinear parabolic equation with time-dependent coefficient source (Rui Yang)

Rui Yang(부산대)
Rui Yang, Pusan National University

In this paper, we deal with a class of quasilinear parabolic equations with time-dependent coefficient source under Dirichlet boundary condition. By the technique of differential inequalities, we discuss the behaviour of the solution and we establish the conditions on the nonlinearities respectively to guarantee that $u(x, t)$ exists globally or blows up at some finite time. We also derive upper and lower bounds of the blow-up time if blow-up does occur.

2010 Mathematics Subject Classification: 34B16, 34B18, 35B40
Key Words and Phrases: quasilinear parabolic, time-dependent, global, blow-up, lower bound, upper bound
Mathematical Education

⋅ 29th-C-14:30 − 15:40 Chair: Young Hee Kye (Kosin University)

⋅ 29th-C-14:30 − 14:50 Analysis of teaching of van Hiele geometry in schools in Korea (Young Rock Kim)

김영록(한국외대)
Young Rock Kim, Hankuk University of Foreign Studies

In this work we tested van Hiele Geometry tests to 2,413 students from elementary schools, middle schools, and high schools in Korea for year 2015 and 2016. The methodology of van Hiele model was used in questionnaires, and in analysis of data and results. In this work we present the results of the study applied to students in schools in Korea about van Hiele Geometry tests and their analysis. Therefore we determined the distribution of the 5 van Hiele levels of thinking through 12 grades in schools in Korea. Applying this tests to every students through 12 grades in Korean schools, we analyzed 5 van Hiele levels of thinking (Visualization, Analysis, Informal Deduction, Deduction, and Rigor). Later, EXCEL (spreadsheet program) was used to analyze the results.

2010 Mathematics Subject Classification: 97C40, 97G99
Key Words and Phrases: Van Hiele, levels of thinking, phases of learning, teaching mathematics, Korea

⋅ 29th-C-14:50 − 15:10 Pacioli as an educator of mathematics (Sung Sook Kim)

김성숙(배재대)
Sung Sook Kim, Paichai University

파치올리(Luca Pacioli)는 프란체스코회 수도사였으며 많은 책을 저술한 헌신된 선생님이었으며 수학자이고 레오나르도 다 빈치의 절친한 친구였다. 그의 대표적인 저서는 산술집성과 신성비례이다. 그는 수학자들에게 많은 영향을 주어 유럽의 수학 발전에 기여하였다. 이 발표에서는 수학 교육자로서의 그의 생애를 살펴보고 그의 저술서 중 산술집성과 신성비례를 중심으로 그가 후세에 끼친 영향에 대하여 살펴보려고 한다.

2010 Mathematics Subject Classification: 97
Key Words and Phrases: Pacioli, 산술집성, 신성비례

⋅ 29th-C-15:20 − 15:40 Interactive digital linear algebra textbook (Sang-Gu Lee, Jae Hwa Lee, Kyung-Eun Park)

이상구*(성균관대), 이재화(성균관대), 박경은(성균관대)
Sang-Gu Lee*, Sungkyunkwan University, Jae Hwa Lee, Sungkyunkwan University, Kyung-Eun Park, Sungkyunkwan University

In this talk, we introduce recently developed interactive digital linear algebra textbook. Our digital textbook includes all available materials such as PDF e-book, web contents, video clips of lectures, interactive labs. Using this linear algebra digital textbook, students can freely use any mobile devices to access diverse learning materials, lessons, and hands-on exercises without any limitations. Also, times saved in the computation, coding, and typing process can be used to have more discussions for deeper understanding of mathematical concepts. This type of linear algebra digital textbook can be considered as a new model for the next generation of math digital textbook. Refer : http://matrix.skku.ac.kr/LA/ or http://matrix.skku.ac.kr/LA-K/

2010 Mathematics Subject Classification: 97U20, 97U70
Key Words and Phrases: digital math textbook, linear algebra, video lectures, interactive laboratory, SageMath, free e-book
Discrete Mathematics

⋅ 29th-A-09:20 − 10:30   Chair: Mitsugu Hirasaka (Pusan National University)

⋅ 29th-A-09:20 − 09:40 On Jacobian group and complexity of the generalized Petersen graph $GP(n, k)$ through Chebyshev polynomials (Young Soo Kwon, Alexander Mednykh, Ilya Mednykh)

권영수*(영남대), Alexander Mednykh(Novosibirsk State Univ.), Ilya Mednykh(Novo\-sibirsk State Univ.)
Young Soo Kwon*, Yeungnam University, Alexander Mednykh, Novosibirsk State University, Ilya Mednykh, Novosibirsk State University

In the present paper we give a new method for counting Jacobian group $Jac(GP(n,k))$ of the generalized Petersen graph $GP(n, k).$ We show that the minimum number of generators of $Jac(GP(n,k))$ is at least two and at most $2k+1.$ Both estimates are sharp. Also, we obtain a closed formula for the number of spanning trees of this graph in terms of Chebyshev polynomials and investigate some arithmetical properties of this number. This is a joint work with Alexander Mednykh and Ilya Mednykh.

2010 Mathematics Subject Classification: 05C30
Key Words and Phrases: spanning tree, Jacobian group, Petersen graph, Laplacian matrix, Chebyshev polynomial

⋅ 29th-A-09:50 − 10:10 On the Jordan product of weighted shifts on directed trees (Younjin Kim, Eungil Ko)

김연진*(이화여대), 고응일(이화여대)
Younjin Kim*, Ewha Womans University, Eungil Ko, Ewha Womans University

Let $\mathcal{T} (V,E)$ be a directed tree, where $V$ and $E$ are the set of vertices and edges, respectively. For any vertex $u \in V$, we set $\mathbf{Chi(u)} = \{ v \in V : (u,v) \in E \}$. Given a system $\lambda =\{ \lambda_v \}$, which is a family of complex numbers, the operator $S_{\lambda}$ defined by $S_{\lambda}e_u=\sum_{v \in \mathbf{Chi(u)}} \lambda_v e_v$ for $u\in V$, where $\{e_v\}_{v \in V}$ is the standard orthonormal basis of $ \ell^2(V)$, is the weighted shift on the directed tree $\mathcal{T} (V,E)$. Let $\mathcal{B}(H, K)$ be the Banach space of all bounded linear operators from $H$ into $K$, where $H$ and $K$ are complex Hilbert spaces. The Jordan product of $A$ and $B$ $(A \circ B)$ is defined as $A \circ B = \frac{1}{2} (AB+BA)$ for $A, B \in \mathcal{B}(H, H)$. In this talk, we establish a formula for the Jordan product of a weighted shift on the directed tree.

2010 Mathematics Subject Classification: 47B37
Key Words and Phrases: weighted shift, Jordan product, directed tree

⋅ 29th-A-10:10 − 10:30 On the weighted information graph game (Doyoung Gil, Jaehyun Kim, Boram Park, Seung Cheol Shin)

길도영(아주대), 김재현*(아주대), 박보람(아주대), 신승철(아주대)
Doyoung Gil, Ajou University, Jaehyun Kim*, Ajou University, Boram Park, Ajou University, Seung Cheol Shin, Ajou University

This paper introduces a weighted version of an information graph game by Kuipers (1993), called a weighted information graph game, then generalizes the results. More precisely, we give a necessary and sufficient condition for a game being concave, and it is proved that the core of a weighted information graph game can be described by a set of at most $2n-1$ linear constraints, where $n$ is the number of players. Furthermore, we prove that the core of a concave game is stable and the converse is not true.

2010 Mathematics Subject Classification: 05C57
Key Words and Phrases: weighted information graph game, concave game, core

⋅ 29th-B-10:40 − 12:10   Chair: Sang-il Oum (Korea Advanced Institute of Science and Technology)

⋅ 29th-B-10:40 − 11:00 Characteristic polynomial of certain hyperplane arrangements through finite field method (Joungmin Song)

송정민(광주과학기술원)
Joungmin Song, Gwangju Institute of Science and Technology

We use the finite field method developed by C. Athanasiadis based on Crapo-Rota's theorem to give a complete formula of the characteristic polynomial of hyperplane arrangements $\mathcal J_n$ consisting of the walls $x_i+x_j = 1, x_k=0, x_l = 1$ for $1\leq i,j,k,l \leq n.$ This provides an alternate to the graph theoretical method we obtained in 2015 in which enumeration of bipartite graphs plays a central role.

2010 Mathematics Subject Classification: 32S22, 05C30
Key Words and Phrases: finite field method, hyperplane arrangements

⋅ 29th-B-11:00 − 11:20 A refinement of the non-symmetric Cauchy identity (Seung Il Choi)

최승일(서울대)
Seung Il Choi, Seoul National University

This talk gives a crystal-theoretic proof for the Lascoux's non-symmetric Cauchy identity. More precisely, Lascoux showed that the non-symmetric Cahuchy kernel $\prod_{i+j \leq n}(1-x_iy_j)^{-1}$ has an expansion of products of Demzaure characters for type $A$. Here it provides a refinement of this expansion mostly using the properties of semistandard Young tableaux and the Robinson-Schensted-Knuth correspondence.

2010 Mathematics Subject Classification: 05E05, 05E10
Key Words and Phrases: Demazure crystal, right key, Bruhat order, Cauchy identity

⋅ 29th-B-11:30 − 11:50 Reinforcement learning and its application to a math game (Jon-Lark Kim, Byung-Sun Won)

김종락(서강대), 원병선*(서강대)
Jon-Lark Kim, Sogang University, Byung-Sun Won*, Sogang University

Artificial intelligence has become so popular in Korea due to an event between Lee Sedol and AlphaGo. In fact, AlphaGo is a program based on two main algorithms. One of them is the reinforcement learning (RL simply). RL allows the machine or software agent to learn its behaviour based on feedback from the environment. This behaviour can be learned once and for all, or keep on adapting as time goes by. Several games such as poker, back-gammom, chess, and Go have been tackled somewhat successfully. Is there any other interesting mathematical game that can be tackled by the reinforcement learning? In this talk, we apply Q-learning, one of the reinforcement learning algorithms, to the factorization game, a new game based on the factorization of a natural number invented by Jon-Lark Kim. We show that learning is properly done for the factorization game by observing a change in the winning rate.

2010 Mathematics Subject Classification: 91A05, 91A46, 97A20
Key Words and Phrases: reinforcement learning, factorization game

⋅ 29th-B-11:50 − 12:10 Association schemes all of whose symmetric fusion schemes are integral (Semin Oh, Mitsugu Hirasaka, Kijung Kim)

오세민*(부산대), Mitsugu Hirasaka(부산대), 김기정(부산대)
Semin Oh*, Pusan National University, Mitsugu Hirasaka, Pusan National University, Kijung Kim, Pusan National University

In this talk we aim to characterize association schemes all of whose symmetric fusion schemes have only integral eigenvalues, and classify those obtained from a regular action of a finite group by taking its orbitals.

2010 Mathematics Subject Classification: 05E15
Key Words and Phrases: association scheme, group

⋅ 29th-C-14:30 − 16:00   Chair: Sangwook Kim (Chonnam National University)

⋅ 29th-C-14:30 − 14:50 Unavoidable induced subgraphs in large graphs with no homogeneous sets (Maria Chudnovsky, Ringi Kim, Sang-il Oum, Paul Seymour)

Maria Chudnovsky(Princeton Univ.), 김린기(Univ. of Waterloo), 엄상일*(한국과학기술원), Paul Seymour(Princeton Univ.)
Maria Chudnovsky, Princeton University, Ringi Kim, University of Waterloo, Sang-il Oum*, KAIST, Paul Seymour, Princeton University

A \emph{homogeneous} set of an $n$-vertex graph is a set $X$ of vertices ($2\le \lvert X\rvert\le n-1$) such that every vertex not in $X$ is either complete or anticomplete to $X$. A graph is called \emph{prime} if it has no homogeneous set. A \emph{chain} of length $t$ is a sequence of $t+1$ vertices such that for every vertex in the sequence except the first one, its immediate predecessor is its unique neighbor or its unique non-neighbor among all of its predecessors. We prove that for all $n$, there exists $N$ such that every prime graph with at least $N$ vertices contains one of the following graphs or their complements as an induced subgraph: (1) the graph obtained from $K_{1,n}$ by subdividing every edge once, (2) the line graph of $K_{2,n}$, (3) the line graph of the graph in (1), (4) the half-graph of height $n$, (5) a prime graph induced by a chain of length $n$, (6) two particular graphs obtained from the half-graph of height $n$ by making one side a clique and adding one vertex.

2010 Mathematics Subject Classification: 05C55
Key Words and Phrases: modular decomposition, induced subgraph, prime graph, Ramsey

⋅ 29th-C-14:50 − 15:10 A new complexity function of repetition and irrationality exponents (Yann Bugeaud, Dong Han Kim)

Yann Bugeaud(Univ. of Strasbourg), 김동한*(동국대)
Yann Bugeaud, University of Strasbourg, Dong Han Kim*, Dongguk University

We introduce and study a new complexity function in combinatorics on words, which takes into account the smallest return time of a factor of an infinite word. We characterize the eventually periodic words and the Sturmian words by means of this function. Then, we establish a new result on repetitions in Sturmian words and show that it is best possible. We deduce a lower bound for the irrationality exponent of real numbers whose sequence of $b$-ary digits is a Sturmian sequence over $\{0,1,\ldots,b-1\}$ and we prove that this lower bound is best possible. If the irrationality exponent of $\xi$ is equal to 2 or slightly greater than 2, then the $b$-ary expansion of $\xi$ cannot be `too simple', in a suitable sense. Our result applies, among other classical numbers, to badly approximable numbers, non-zero rational powers of $e$, and $\log(1+1/a)$, provided that the integer $a$ is sufficiently large. It establishes an unexpected connection between the irrationality exponent of a real number and its $b$-ary expansion.

2010 Mathematics Subject Classification: 11A63, 68R15
Key Words and Phrases: combinatorics on word, $b$-ary expansion, complexity, irrationality exponent, Sturmian word

⋅ 29th-C-15:20 − 15:40 Horizontal and vertical formulas for exponential Riordan matrices and their applications (Ji-Hwan Jung, Gi-Sang Cheon)

정지환*(성균관대), 천기상(성균관대)
Ji-Hwan Jung*, Sungkyunkwan University, Gi-Sang Cheon, Sungkyunkwan University

In this talk, we show that an infinite lower triangular matrix $A=[a_{ij}]_{i,j\in\mathbb{N}_0}$ is an exponential Riordan matrix $A={\mathcal E}(g,f)$ given by $\sum_{i\ge j}a_{ij}{z^i/i!}=gf^j/j!$ if and only if there exist both horizontal pair $\{h_n;\tilde{h}_{n}\}_{n\ge0}$ and vertical pair $\{v_n;\tilde{v}_{n}\}_{n\ge0}$ of the sequences that represent all the elements in the matrix. As a consequence, we obtain that if the horizontal and vertical pairs of an exponential Riordan matrix are identical then the matrix is involution. In addition, this concept can be applied to obtain the determinants of the production matrix and some conditions for $d$-orthogonality of the Sheffer polynomial sequences.

2010 Mathematics Subject Classification: 05A15
Key Words and Phrases: exponential Riordan matrix, production matrix, Sheffer sequence, $d$-orthogonal polynomials

⋅ 29th-C-15:40 − 16:00 Arithmetic properties of Pascal matrices (Jaesung Jeong, Soonhak Kwon)

정재성*(성균관대), 권순학(성균관대)
Jaesung Jeong*, Sungkyunkwan University, Soonhak Kwon, Sungkyunkwan Univesity

Pascal matrices related to Pascal graphs have many interesting arithmetical properties, and some of the known properties are not proven yet. We discuss determinant, rank, and short basis vectors of Pascal matrices. Recurrence sequences and generating functions related to Pascal matrices will also be discussed.

2010 Mathematics Subject Classification: 05A10
Key Words and Phrases: Pascal matrix, rank, basis
Cryptography

⋅ 29th-C-14:30 − 16:00 Chair: Younhwan Cheon (Korea Army Academy at Young-Cheon)

⋅ 29th-C-14:30 − 14:50 A necessary and sufficient condition for shortest vectors in lattices of dimension 2, 3 (Hyang-Sook Lee, Jeongeun Park)

이향숙(이화여대), 박정은*(이화여대)
Hyang-Sook Lee, Ewha Womans University, Jeongeun Park*, Ewha Womans University

A lattice in the Euclidean space is an important issue for cryptography. Finding a nonzero shortest vector in a lattice of high dimension is in general very difficult to compute. There are several results on a sufficient condition for a basis to contain a shortest vector of a given lattice. However, not many results are known for necessary and sufficient conditions, even in low dimensional cases. In this talk, we present a necessary and sufficient condition for a basis to contain a shortest vector of a given three dimensional lattice.

2010 Mathematics Subject Classification: 11P21
Key Words and Phrases: lattices, SVP, Gaussian, pairwise Gaussian, lattice basis reduction

⋅ 29th-C-14:50 − 15:10 Secure GLS recomposition for sum-of-square cofactors (Eunkyung Kim, Mehdi Tibouchi)

김은경*(이화여대), Mehdi Tibouchi(NTT Secure Platform Laboratories)
Eunkyung Kim*, Ewha Womans University, Mehdi Tibouchi, NTT Secure Platform Laboratories

The GLV/GLS decomposition method let us speed up scalar multiplications [k]P on certain class of elliptic curves, but applying this decomposition method should be done carefully since it tends to leak side-channel information during implementation. To avoid this weakness, Aranha et al. proposed the recomposition method, and they showed that with this technique, one can securely compute [k]P where k is a random scalar and P is a point on an GLS curve of prime order. In this talk, we extend their result from GLS curves of prime order to GLS curves with any cofactor which is a sum of two squares of integers.

2010 Mathematics Subject Classification: 11R04
Key Words and Phrases: elliptic curve scalar multiplication, Gaussian integers

⋅ 29th-C-15:20 − 15:40 About lattice signature (Jaeyoon Kim, Junghee Cheon)

김재윤*(서울대), 천정희(서울대)
Jaeyoon Kim*, Seoul National University, Junghee Cheon, Seoul National University

Post Quantum era is very close. Lattice is one of the most plausible base theory for post quantum cryptography, with the most various possible applications such as homomophic encryption, homomorphic signature. But Lattice Signature is studied less compared to Encryption.In this talk, we will introduce various lattice signatures and propose new technik/scheme for lattice signature.

2010 Mathematics Subject Classification: 11T71
Key Words and Phrases: lattice, signature, post quantum

⋅ 29th-C-15:40 − 16:00 A refinement Frobenius representation algorithm for solving discrete logarithm problem over a finite field (Jung Hee Cheon, Minki Han, Jinhyuck Jeong, Jiseung Kim)

천정희(서울대), 한민기(서울대), 정진혁(서울대), 김지승*(서울대)
Jung Hee Cheon, Seoul National University, Minki Han, Seoul National University, Jinhyuck Jeong, Seoul National University, Jiseung Kim*, Seoul National University

In Eurocrypt 2014, Barbulescu $\textit{et al.}$ suggested a heuristic quasi-polynomial time algorithm, called Frobenius Representation Algorithm, for solving the discrete logarithm problem (DLP) on a finite field ${\mathbb F}_{q^{2 n}}$ with a small characteristic. In this paper, we generalize the identity $X^q - X = \prod_{\alpha \in {\mathbb F}_q} (X- \alpha)$ used in FRA into $q^{2k-1}-q$ identities in a special case. These multiple identities enable us to collect more equations among the linear terms in the target and express the target with polynomials of lower degree in each descent step without increasing the complexity of each step much. That reduces the depth of the descent tree and improves the asymptotic complexity of the algorithm. Our algorithm has the (heuristic) asymptotic complexity $(q^k n)^{\frac{\log n}{\log u -\log 3}}$ where $u^u = \frac{q^{2k} – q^2}{q^{2k}} \times (q^{2}+1)$ while the Frobenius Representation Algorithm has a time complexity $q^{O(log n)}$.

2010 Mathematics Subject Classification: 11T71
Key Words and Phrases: discrete logarithm problem, function field sieve

Ring Theory and Related Topics

⋅ 29th-A-09:05 − 09:35   Chair: Chan Yong Hong (Kyung Hee University)

⋅ 29th-A-09:05 − 09:20 t-almost Dedekind graded domains (Gyu Whan Chang, Dong Yeol Oh)

장규환(인천대), 오동렬*(조선대)
Gyu Whan Chang, Incheon National University, Dong Yeol Oh*, Chosun University

Let $\Gamma$ be a torsion-free cancellative monoid with quotient group $\langle \Gamma \rangle$, $R = \bigoplus_{\alpha \in \Gamma}R_{\alpha}$ be a graded integral domain graded by $\Gamma$, $H$ be the set of nonzero homogeneous elements of $R$, and $N(H) = \{f \in R \mid C(f)_v = R\}$. We introduce the notion of graded $t$-almost Dedekind domains. We then show that $R$ is a $t$-almost Dedekind domain if and only if $R$ is a graded $t$-almost Dedekind domain and $R_H$ is a $t$-almost Dedekind domains. We also show that if $R = D[\Gamma]$ is the monoid domain of $\Gamma$ over an integral domain $D$, then $R$ is a graded $t$-almost Dedekind domain if and only if $D$ and $\Gamma$ are $t$-almost Dedekind, if and only if $R_{N(H)}$ is an almost Dedekind domain. In particular, if $\langle \Gamma \rangle$ satisfies the ascending chain condition on its cyclic subgroups, then $R = D[\Gamma]$ is a $t$-almost Dedekind domain if and only if $R$ is a graded $t$-almost Dedekind domain.

2010 Mathematics Subject Classification: 13A02, 13A15, 13F05, 20M25
Key Words and Phrases: graded integral domain, t-almoste Dedekind domain, graded t-almost Dedekind domain

⋅ 29th-A-09:20 − 09:35 Amitsur's property for differential operator rings (Nam Kyun Kim)

김남균(한밭대 기초과학부)
Nam Kyun Kim, School of Basic Sciences, Hanbat National University

We investigate when radicals $\mathfrak{F}$ satisfy Amitsur's property on skew polynomials of derivation type, namely $\mathfrak{F}(R[x;\delta]) = (\mathfrak{F}(R[x;\delta]) \cap R)[x;\delta]$. In particular, we give a new argument that the Brown-McCoy radical has this property. We also give a new characterization of the prime radical of $R[x;\delta]$.

2010 Mathematics Subject Classification: 16S36, 16N60
Key Words and Phrases: Amitsur's property, Brown-McCoy radical, derivation, prime radical, skew polynomial

⋅ 29th-A-09:45 − 10:30   Chair: Dong Yeol Oh (Chosun University)

⋅ 29th-A-09:45 − 10:00 Locally almost GCD domains (Gyu Whan Chang)

장규환(인천대)
Gyu Whan Chang, Incheon National University

Let $D$ be an integral domain. We say that $D$ is an almost GCD domain (AGCD domain) if for all $0 \neq a,b \in D$, there is an integer $n =n(a,b) \geq 1$ such that $a^nD \cap b^nD$ is principal. Hence, a GCD domain is an AGCD domain, while $\mathbb{Z}_2[X^2, X^3]$ is an AGCD domain but not a GCD domain. In this talk, we study integral domains $D$ such that $D_M$ is a GCD domain or an AGCD domain for all maximal ideals $M$ of $D$.

2010 Mathematics Subject Classification: 13F05
Key Words and Phrases: almost GCD domain

⋅ 29th-A-10:00 − 10:15 On nil-reflexive rings (Tai Keun Kwak)

곽태근(대진대)
Tai Keun Kwak, Daejin University

We in this note consider the reflexive ring property on nil ideals, introducing the concept of a nil-reflexive ring as a generalization of the reflexive ring property. We will call a ring $R$ nil-reflexive if $IJ = 0$ implies $JI = 0$ for nil ideals $I$, $J$ of $R$. The polynomial and the power series rings over a right Noetherian ring (or an NI ring) R are shown to be nil-reflexive if $(aRb)^2 = 0$ implies $aRb = 0$ for all $a, b \in N(R)$. We further investigate the structure of nil-reflexive rings, related to various sorts of ring extensions which have roles in ring theory.

2010 Mathematics Subject Classification: 16N40, 16S70
Key Words and Phrases: Nil-reflexive ring, nil ideal, matrix ring, polynomial ring, power series ring, right quotient ring

⋅ 29th-A-10:15 − 10:30 On rings whose simple singular R-modules are flat (Sang Bok Nam)

남상복(경동대)
Sang Bok Nam, Kyungdong University

The right SSF-ring, a generalization of right SF-rings, has been introduced, and checked about its relation and properties. It is shown that R is a strongly regular ring if and only if R is a left or right nil-IFP right (left) SF-ring if and only if R is a weakly nil-IFP right (left) SSF-ring.

2010 Mathematics Subject Classification: 16Y30
Key Words and Phrases: reduced ring, left or right nil-IFP ring, weakly nil-IFP, IFP, strongly regular ring

⋅ 29th-B-10:40 − 11:10   Chair: Gyu Whan Chang (Incheon National University)

⋅ 29th-B-10:40 − 10:55 Insertion of units at zero products (Hong Kee Kim)

김홍기(경상대)
Hong Kee Kim, Gyeongsang National University

The purpose of this note is to provide useful connections between units and zero divisors, by investigating the structure of a class of rings in which K\"{o}the's conjecture (i.e., the sum of two nil left ideals is nil) holds. We introduce the concept of unit-IFP for the purpose, in relation with the inserting property of units at zero products. We first study the relation between unit-IFP rings and related ring properties in a kind of matrix rings which has roles in noncommutative ring theory.. The Jacobson radical of the polynomial ring over a unit-IFP ring is shown to be nil. We also provide equivalent conditions to the commutativity via the unit-IFP of such matrix rings. We construct examples and counterexamples which are necessary to the naturally raised questions.

2010 Mathematics Subject Classification: 16U60, 16U80
Key Words and Phrases: unit-IFP ring, unit, IFP ring, nilpotent element, Armendariz ring, matrix ring, polynomial ring

⋅ 29th-B-10:55 − 11:10 On a ring structure related to annihilators (Chan Yong Hong, Chan Huh, Hong Kee Kim, Nam Kyun Kim, Yang Lee, Jeong Sook Park, Sung Ju Ryu, Yun Sang Jo)

홍찬용(경희대), 허찬(부산대), 김홍기(경상대), 김남균(한밭대 기초과학부), 이양(대진대 기초과학연구소), 박정숙(경상대), 류성주(부산대), 윤상조*(동아대)
Chan Yong Hong, Kyung Hee University, Chan Huh, Pusan National University, Hong Kee Kim, Gyeongsang National University, Nam Kyun Kim, School of Basic Sciences, Hanbat National University, Yang Lee, Institute of Basic Science, Daejin University, Jeong Sook Park, Gyeongsang National University, Sung Ju Ryu, Pusan National University, Sang Jo Yun*, Dong-A University

In this note we focus our attention on a new ring structure related to annihilators, and consider a ring property that contains many kinds of ring classes, introducing right ZAFS. This property is shown to be not left-right symmetric but left-right symmetric for left or right Artinian rings. The left (right) ZAFS property is shown to pass to Ore extensions with automorphisms. The left (resp., right) ZAFS property is shown to pass also to classical left (resp., right) quotient rings, yielding that semiprime right Goldie rings are ZAFS.

2010 Mathematics Subject Classification: 16N40, 16P99, 16S50
Key Words and Phrases: right ZAFS ring, matrix ring, polynomial ring, classical right quotient ring

⋅ 29th-B-11:20 − 11:50   Chair: Hong Kee Kim (Gyeongsang National University)

⋅ 29th-B-11:20 − 11:35 Symmetric ring property on nil ideals (Yang Lee, Zhelin Piao)

이양(대진대 기초과학연구소), 박철림*(부산대)
Yang Lee, Institute of Basic Science, Daejin University, Zhelin Piao*, Pusan National University

The usual commutative ideal theory was extended to ideals in noncommutative rings by Lambek, introducing the concept of symmetric. Camillo et al. naturally extended the study of symmetric ring property to the lattice of ideals, defining the new concept of an ideal-symmetric ring. This note focuses on the symmetric ring property on nil ideals, as a generalization of an ideal-symmetric ring. A ring $R$ will be said to be right(resp.,left) nil-ideal-symmetric if $IJK=0$ implies $IKJ=0$ (resp., $JIK=0$) for nil ideals $I, J, K$ of $R$. This concept generalizes both ideal-symmetric rings and weak nil-symmetric rings in which the symmetric ring property have been observed in some restricted situations. The structure of nil-ideal-symmetric rings is studied in relation to the near concepts and ring extensions which have roles in ring theory.

2010 Mathematics Subject Classification: 16N40, 16S70
Key Words and Phrases: nil-ideal-symmetric ring, nil-ideal, matrix ring, polynomial ring, right quotient ring, Dorroh extension

⋅ 29th-B-11:35 − 11:50 Commutativity of a Schur ring over some groups (Eun-Kyung Cho)

조은경(부산대)
Eun-Kyung Cho, Pusan National University

Consider a group ring $\mathds{C}[G]$ and a partition $\mathcal{P} = \{D_0=\{e\}, D_1, \ldots, D_d\}$ of $G$. Let $D_i^{*} = \{g^{-1} \mid g \in D_i\}$ and $\bar{D_i}$ is the sum of all the elements of $D_i$ for all $i \in \{0, 1, \ldots, d\}$. If $\mathcal{P}$ satisfies the conditions (1) $\mathcal{P} = \{D_0^*, D_1^*, \ldots, D_d^*\}$ and (2) $\bar{D_i} \bar{D_j} = \sum_{k=0}^d p_{ij}^k \bar{D_k}$ for some $p_{ij}^k \in \mathds{Z}$ for all $D_i, D_j \in \mathcal{P}$, then a subalgebra of $\mathds{C}[G]$ generated by $\bar{D_0}, \bar{D_1}, \ldots, \bar{D_d}$ is called a Schur ring over $G$ and is denoted by $\mathfrak{G} =( G ; \mathcal{P})$. In this talk, we classify finite groups which induces only commutative proper Schur rings.

2010 Mathematics Subject Classification: 97H40
Key Words and Phrases: Schur ring, commutative
Abstract Harmonic Analysis, Dynamical System, and Operator Algebras

⋅ 29th-A-09:00 − 10:30 Chair: Hyun Ho Lee (University of Ulsan)

⋅ 29th-A-09:00 − 09:20 Dixmier's similarity problem on locally compact quantum groups (Michael Brannan, SangGyun Youn)

Michael Brannan(Texas A\&M Univ.), 윤상균*(서울대)
Michael Brannan, Texas A\&M University, SangGyun Youn*, Seoul National University

In this talk, we will first introduce the Dixmier's similarlity problem and how it is described in the cases of locally compact quantum groups. Then we will talk about one of the ongoing research with the results so far known.

2010 Mathematics Subject Classification: 20G42, 46L07, 46L89
Key Words and Phrases: Dixmier's similarlity problem and Locally compact quantum groups

⋅ 29th-A-09:20 − 09:40 Twisted group algebra and Fourier spaces, amenability and similarity (Xiong Xiao, Hun Hee Lee, Nico Spronk)

Xiong Xiao*(서울대), Hun Hee Lee(서울대), Nico Spronk(Univ. of Waterloo)
Xiong Xiao*, Seoul National University, Hun Hee Lee, Seoul National University, Nico Spronk, University of Waterloo

For a discrete group $G$ and a $2$-cocycle $\sigma$ on it, we consider the twisted version of group algebras and Fourier spaces. Denote by $C_r^*(G,\sigma)$ and $VN(G,\sigma)$ the reduced twisted group algebra and twisted von Neumann algebra respectively, and $B(G,\sigma)$ the dual of $C_r^*(G,\sigma)$ and $A(G,\sigma)$ the predual of $VN(G,\sigma)$. For nontrivial $\sigma$, $B(G,\sigma)$ is not a Banach algebra any more. But we still have $B(G,\sigma)\subset M_{cb}\big(A(G),A(G,\sigma)\big)\subset M\big(A(G),A(G,\sigma)\big)$, where $M$ (resp. $M_{cb}$) denotes the space of (resp. completely) bounded Fourier multipliers. Moreover, the equivalence of the three characterizes the amenability of $G$. Applying this assertion, we prove the twisted analogue of similarity degree conclusion for group algebras obtained by Pisier.

2010 Mathematics Subject Classification: 43A20
Key Words and Phrases: twisted group algebra, amenability, similarity

⋅ 29th-A-09:50 − 10:10 Rigidity for group actions (Nhan-Phu Chung, Yongle Jiang, Keonhee Lee)

Nhan-Phu Chung*(성균관대), Yongle Jiang(SUNY at Buffalo), 이건희(충남대)
Nhan-Phu Chung*, Sungkyunkwan University, Yongle Jiang, SUNY at Buffalo, Keonhee Lee, Chungnam National University

In this talk, we will present certain rigidity results for group actions on compact spaces. In the first part, we will provide a new characterization of one end groups via cocycle superrigidity of their full shifts. As a consequence, we have an application in continuous orbit equivalence rigidity. In the second part, we prove that if an action of a finitely generated group is expansive and has the pseudo-orbit tracing property then it is $C^0$ local rigid. A new characterization of subshifts of finite type over finitely generated groups in term of pseudo-orbit tracing property is also mentioned. The first part is joint with Yongle Jiang and the second part is joint work with Keonhee Lee.

2010 Mathematics Subject Classification: 37A20, 37B05
Key Words and Phrases: rigidity, cocycle, shift of finite type

⋅ 29th-A-10:10 − 10:30 Truncated moment problems and the Hadamard product (Seonguk Yoo)

유성욱(성균관대)
Seonguk Yoo, Sungkyunkwan University

Up to the present day, the best solution to the truncated moment problem (TMP) is probably the Flat Extension Theorem. It says if the corresponding moment matrix of a moment sequence admits a rank-preserving positive extension, then the sequence has a representing measure. However, constructing a flat extension for most higher-order moment sequence cannot be executed easily because it requires to allow many parameters. Recently, the author has considered various decompositions of a moment matrix to find a solution to the TMP instead of an extension; in doing so, we can find an economical solution to the TMP. Using a new approach with the Hadamard product, the author would like to introduce more techniques related to moment matrix decompositions.

2010 Mathematics Subject Classification: 44A60, 47B35, 15A83, 15A60, 47A30, 15-04
Key Words and Phrases: truncated moment problems, moment matrix extensions, rank-one decomposition, Hadamard product

⋅ 29th-B-10:40 − 12:10 Chair: Seung-Hyeok Kye (Seoul National University)

⋅ 29th-B-10:40 − 11:00 Loops in labeled spaces and infiniteness of labeled graph $C^*$-algebras (Ja A Jeong, Eun Ji Kang, Sun Ho Kim, Gi Hyun Park)

정자아*(서울대), 강은지(BK21+ 창의인재양성 수리과학사업단), 김선호(BK21+ 서강 이산및기하구조 연구팀), 박기현 (한신대)
Ja A Jeong*, Seoul National University, Eun Ji Kang, BK21+ Mathematical Science Division, Sun Ho Kim, BK21+ Sogang Mathematics, Gi Hyun Park, Hanshin University

A $C^*$-algebra is called infinite if it contains an infinite projection. If every nonzero hereditary subalgebra of every quotient of $A$ is infinite, then $A$ is known to be purely infinite in the sense of Kirchberg and R\o rdam. While the converse is not true in general, it is well known to hold true for graph $C^*$-algebras $C^*(E)$. Moreover, if a graph $C^*$-algebra $C^*(E)$ is infinite, the graph $E$ must contain a loop with an exit. This implies that if $C^*(E)$ is purely infinite, then $E$ has a loop and every vertex on a loop must admit another loop. We discuss if these sort of results would be true for the class of labeled graph $C^*$-algebras. For this, we provide an appropriate definition of loop in a labeled space. With this definition we obtain results similar to the case of graph $C^*$-algebras, and also show by examples that more intricate $C^*$-algebras than the graph algebras can occur as labeled graph $C^*$-algebras. This talk is based on the following joint papers: \noindent 1. {\it AF labeled graph $C^*$-algerbas} (with E. J. Kang and Sun Ho Kim) \noindent 2. {\it Purely infinite labeled graph $C^*$-algebras} (with E. J Kang and G. H. Park)

2010 Mathematics Subject Classification: 37A55, 46L05, 46L55
Key Words and Phrases: infinite $C^*$-algebras, labeled graphs, loops

⋅ 29th-B-11:00 − 11:20 Integration over the quantum diagonal subgroup and associated Fourier-like algebras (Hun Hee Lee, Uwe Franz, Adam Skalski)

이훈희*(서울대), Uwe Franz(Univ. of Besancon), Adam Skalski(Institute of Mathematics, Polish Academy of Sciences)
Hun Hee Lee*, Seoul National University, Uwe Franz, University of Besancon, Adam Skalski, Institute of Mathematics, Polish Academy of Sciences

By analogy with the classical construction due to Forrest, Samei and Spronk, we associate to every compact quantum group $G$, a completely contractive Banach algebra $A_\triangle(G)$, which can be viewed as a deformed Fourier algebra of $G$. To motivate the construction, we first analyze in detail the quantum version of the integration over the diagonal subgroup, showing that although the quantum diagonal subgroups in fact never exist, as noted earlier by Kasprzak and Soltan, the corresponding integration represented by a certain idempotent state on $C(G)$ makes sense as long as $G$ is of Kac type. Finally, we analyze as an explicit example the algebras $A_\triangle(O+_N), N \geq 2$, associated to Wang's free orthogonal groups, and show that they are not operator weakly amenable.

2010 Mathematics Subject Classification: 46L65
Key Words and Phrases: compact quantum group, Fourier algebra, diagonal subgroup

⋅ 29th-B-11:30 − 12:10 Rohklin property for an inclusion of $C^*$-algebras (Hiroyuli Osaka)

Hiroyuli Osaka(Ritsumeikan Univ.)
Hiroyuli Osaka, Ritsumeikan University

We introduce the Rokhlin property (respectively, the tracial Rokhlin property) for a conditional expectation for an inclusion of (unital) $C^*$-algebras $P \subset A$ with index finite, and show that an action $\alpha$ from a finite group $G$ on a simple unital $C^*$-algebra A has the Rokhlin property in the sense of M. Izumi (respectively, the tracial Rokhlin property in the sense of N. C. Phillips) if and only if the canonical conditional expectation $E : A \rightarrow AG$ has the Rokhlin property (respectively, the tracial Rokhlin property). Let $\mathcal{C}$ be a class of infinite dimensional stably finite separable unital $*$-algebras which is closed under the following conditions: (1) If $A\in \mathcal{C}$ and $B\sim A$, then $B\in \mathcal{C}$. (2) If $A \in \mathcal{C}$, the matrix algebra $M_n(A) \in \mathcal{C}$ for any $n$. (3) If $A\in \mathcal{C}$ and $p\in A$ a nonzero projection, $pAp \in \mathcal{C}$. Suppose that any $C^*$-algebra in $C$ is weakly semiprojective. We present that if $A$ is a local $C$-algebra (respectively, a local tracial $C$-algebra) and a conditional expectation $E : A \rightarrow P$ is of index-finite type with the Rokhlin property (respectively, the tracial Rokhlin property), then $P$ is a unital local $C$-algebra (respectively, a local tracial $C$-algebra). The main result is that several properties such that stable rank one, real rank zero, rank of nuclear dimension, strict comparison property in the Cuntz semigroup, $D$-absorbing for $P$ are preserved when $A$ satisfies one of them and $E : A \rightarrow P$ has the Rokhlin property. Moreover, if $A$ is simple, separable, unital, nuclear, Jiang-Su absorbing and $E : A \rightarrow P$ has the tracial Rokhlin property, then $P$ is Jiang-Su absorbing. Moreover, when an action $\alpha$ from a finite group $G$ on a unital $C^*$-algebra $A$ has the Rokhlin poperty, then for any subgroup $H$ of $G$ the restricted $E$ to $A^H$, which is a conditional expectation from $A^H$ onto $A^G$, has the Rokhlin property. As an application. if $A$ is $D$-absorbing. When an action $\alpha$ has the tracial Rokhlin property on a simple unital $C^*$-algebra $A$, then the fixed point algebra $A^H$ and the crossed product algebra $A \rtimes {}_{\alpha |H} H$ is Jiang-Su absorbing. We also show that the strict comparison property for a Cuntz semigroup $W(A)$ is hereditary to $W(P)$ if $A$ is simple, separable, exact, unital, and $E : A \rightarrow P$ has the tracial Rokhlin property.

2010 Mathematics Subject Classification: 46L35, 46L55
Key Words and Phrases: Rohklin property, an inclusion of $C^*$-algebras, index of $C^*$-subalgebras, Toms-Winter conjecture
On Hermitian Symmetric Spaces and Twisted Manifolds

⋅ 29th-A-09:00 − 09:20   Chair: Sun Hyang Chun (Chosun University)

⋅ 29th-A-09:00 − 09:20 Hopf hypersurfaces in complex two-plane Grassmannians with Lie derivative Ricci tensors (Chang Hwa Woo, Imsoon Jeong, Young Jin Suh)

우창화*(우석대), 정임순(배제대), 서영진(경북대)
Chang Hwa Woo*, Woosuk University, Imsoon Jeong, Paichai University, Young Jin Suh, Kyngpook National University

We consider a complex two-plane Grassmannian which consists of all complex two-dimensional linear subspaces in $(m+2)$-dimensional complex space. The ambient space has a remarkable geometric structure. It was known that the complex two-plane Grassmannian is the unique compact irreducible Riemannian symmetric space equipped with both a Kaehler structure and a quaternionic Kaehler structure. In a paper due to Lee, Perez, Suh and Woo, we have introduced a certain condition of the Ricci tensor for real hypersurfaces in complex two-plane Grassmannians. Motivated by this fact, in this talk, we consider Lie derivation instead of Levi-Civita connection and we give a partial classification of real hypersurfaces in complex two-plane Grassmannians with the given condition.

2010 Mathematics Subject Classification: Primary 53C40; Secondary 53C15
Key Words and Phrases: real hypersurfaces, complex two-plane Grassmannians, Hopf hypersurface, shape operator, Ricci tensor, Lie derivation

⋅ 29th-A-09:20 − 09:40   Chair: Im Soon Jeong (Paichai University)

⋅ 29th-A-09:20 − 09:40 Levi-Civita and generalized Tanaka–Webster covariant derivatives for structure Jacobi operator of a real hypersurface in a complex two-plane Grassmannian (Hyunjin Lee, Gyu Jong Kim, Young Jin Suh)

이현진*(경북대), 김규종(경북대), 서영진(경북대)
Hyunjin Lee*, Kyungpook National University, Gyu Jong Kim, Kyungpook National University, Young Jin Suh, Kyungpook National University

It is known that submanifolds in Kaehler manifolds have many kinds of connection. Among them, we consider two kinds of connections named Levi-Civita $\nabla$ and Tanaka-Webster ${\hat \nabla}^{(k)}$ connecions for real hypersurfaces in complex two-plane Grassmannians. Related to these conncetion, when the structure Jacobi operator $R_{\xi}=R( \,\cdot\, , \xi)\xi$ of a real hypersurface in a complex two-plane Grassmannian satisfies the condition $\nabla_{X}R_{\xi} = {\hat \nabla}_{X}^{(k)}R_{\xi}$, we give some characterizations for Hopf hypersurfaces in complex two-plane Grassmannians.

2010 Mathematics Subject Classification: 53C40, 53C15
Key Words and Phrases: Hopf hypersurface, complex two-plane Grassmannian, structure Jacobi operator, Levi-Civita connection, generalized Tanaka-Webster connection

⋅ 29th-A-09:50 − 10:10   Chair: Chang Hwa Woo (Woosuk National University)

⋅ 29th-A-09:50 − 10:10 Reeb flow invariant unit tangent sphere bundles (Sun Hyang Chun, Jong Taek Cho)

전선향*(조선대), 조종택(전남대)
Sun Hyang Chun*, Chosun University, Jong Taek Cho, Chonnam National University

In a contact manifold $(\bar{M},\eta)$, we have a fundamental property that the Reeb vector field $\xi$ generates a contact diffeomorphism, that is, $L_\xi \eta=0$. For an associated Riemannian metric $\bar{g}$, if $\xi$ generates an isometric flow, that is, $\bar M$ satisfies $L_\xi \bar{g}=0$,then $\bar{M}$ is said to be K-contact. It is very intriguing to study the interplay between Riemannian manifolds $(M,g)$ and their unit tangent sphere bundles $T_1 M$ with the standard contact metric structure $(\eta, \bar g,\phi,\xi)$. In particular, the geodesic flow generated by the Reeb vector field $\xi$ has a crucial role on the geometry of Riemannian manifold $(M,g)$. As a classical result, Y. Tashiro proved that $(T_1 M, \eta, \bar g)$ is a K-contact manifold if and only if $(M,g)$ has constant sectional curvature $1$. In this paper, we study unit tangent sphere bundles $T_1 M$ whose Ricci operator $\bar{S}$ is Reeb flow invariant, that is, $L_\xi \bar{S}=0$.

2010 Mathematics Subject Classification: 53C25, 53D10
Key Words and Phrases: unit tangent sphere bundle, Reeb vector field

⋅ 29th-A-10:10 − 10:30   Chair: Hyunjin Lee (Kyungpook National University)

⋅ 29th-A-10:10 − 10:30 On curvature related conditions of the twisted product space (Byung Hak Kim, Sang Deok Lee, Jin Hyuk Choi)

김병학*(경희대), 이상덕(단국대), 최진혁(경희대)
Byung Hak Kim*, Kyung Hee University, Sang Deok Lee, Dankook University, Jin Hyuk Choi, Kyung Hee University

The twisted product space is a Riemannian product manifold with twisted metric. In this talk, we will report of the recent results of the twisted product space with various curvature conditions.

2010 Mathematics Subject Classification: 53C25
Key Words and Phrases: twisted product space, curvature related conditions

⋅ 29th-B-10:40 − 11:20   Chair: Byung Hak Kim (Kyung Hee University)

⋅ 29th-B-10:40 − 11:20 Isometric Reeb flow on Hermitian symmetric spaces (Young Jin Suh)

서영진(경북대)
Young Jin Suh, Kyngpook National University

In this talk, we give introduce some notions of isometric Reeb flow, contact real hypersurfaces in the complex quadric $Q^{m}$ or in the complex hyperbolic quadric $Q^{m^{*}}$ and give some classifications of such hypersurfaces. As generalizations of such notions, we introduce a new notion of isometric Reeb flow for real hypersurfaces in Hermitian symmetric spaces and give a new classification of real hypersurfaces in generalized complex Grassmannians $G_{k}(\mathbb C^{r+1})$ with isometric Reeb flow.

2010 Mathematics Subject Classification: 53C40, 53C55
Key Words and Phrases: isometric Reeb flow, contact, Hermitian symmetric spaces, complex quadric, complex hyperbolic quadric, complex $k$-Grassmannians

⋅ 29th-B-11:30 − 12:10   Chair: Young Jin Suh (Kyungpook National University)

⋅ 29th-B-11:30 − 12:10 The intersection of two real flag manifolds in a complex flag manifold (Takashi Sakai)

Takashi Sakai(Tokyo Metropolitan Univ.)
Takashi Sakai, Tokyo Metropolitan University, Japan

An orbit of the adjoint representation of a compact connected Lie group $G$ admits a $G$-invariant Kaehler structure, and is called a complex flag manifold. In this talk, we give a definition of an antipodal set of a complex flag manifold using a torus action, which is a generalization of that of a compact Hermitian symmetric space introduced by Chen-Nagano. We show that a maximal antipodal set of a complex flag manifold is characterized as an orbit of the Weyl group of $G$. An orbit of the linear isotropy representation of a compact symmetric space $G/K$ is embedded in a complex flag manifold as a real form, and is called a real flag manifold. We give a necessary and sufficient condition for two real flag manifolds, which are not necessarily congruent with each other, in a complex flag manifold to intersect transversally in therms of symmetric triads. If the intersection is discrete, then it is given as an orbit of a certain Weyl group, in particular it is an antipodal set of the complex flag manifold. This talk is based on a joint work with Osamu Ikawa, Hiroshi Iriyeh, Takayuki Okuda, and Hiroyuki Tasaki.

2010 Mathematics Subject Classification: 53C30
Key Words and Phrases: complex flag manifold, real flag manifold, symmetric triad, antipodal set
Applied Analysis and Computations

⋅ 29th-A-09:00 − 10:30 Chair: TongKeun Chang (Yonsei University)

⋅ 29th-A-09:00 − 09:20 Optimal prediction problem with loss cut (Hi Jun Choe, Jongeun Kim)

최희준(연세대), 김종은*(연세대)
Hi Jun Choe, Yonsei University, Jongeun Kim*, Yonsei University

In order to maximize an expected payoff or to minimize an expected cost, traders has to determine the right time to trade some financial instruments. In the problem of determining the time to sell the stocks, the selling before the prices take a nosedive is also important problem in order to cut the loss of assets. We can formulate the financial problem to the optimal prediction problem with loss cut. We show that it can be stated as a free-boundary problem.

2010 Mathematics Subject Classification: 60G40
Key Words and Phrases: optimal prediction problem, loss cut, first hitting probability, Brownian motion, drawdown, maximal process

⋅ 29th-A-09:20 − 09:40 Machine learning on sequential datas (Eunho Koo)

구은호(연세대)
Eunho Koo, Yonsei University

We introduce various methods to analyze given sequential datas such as stock price or precipitation. We choose certain types or arrangements of inputs which make the result better. Neural network and recurrent neural network would be essential technics. We also use Bayesian classifier to deal with sequential datas

2010 Mathematics Subject Classification: 68T50
Key Words and Phrases: recurrent neural network, long short term memory, neural network, backpropagation through time

⋅ 29th-A-09:50 − 10:10 Stochastic volatility effects on Gaussian copula (Yong-Ki Ma)

마용기(공주대)
Yong-Ki Ma, Kongju National University

Gaussian copula is one of representative copulas used in the financial industry. The main assumption of the Gaussian copula is to consider that the random variables are normally distributed and their joint distribution is multivariate normal. However, it has a major drawback: it does not exhibit tail dependence. To overcome this problem, we suggest a multiscale stochastic volatility model. More precisely, we consider volatility processes which are driven by two diffusions; one fluctuating on a fast time scale and the other fluctuating on a slowly varying diffusion process. Based on this model, an approximate formula is obtained for the joint transition density and then it is applied to the Gaussian copula.

2010 Mathematics Subject Classification: 91G30
Key Words and Phrases: Gaussian copula, multiscale stochastic volatility model, perturbation theory

⋅ 29th-A-10:10 − 10:30 H\"{o}lder continuity of Keller-Segel equations of porous medium type coupled to fluid equations (YunSung Chung, Sukjung Hwang, Kyungkeun Kang, Jaewoo Kim)

정윤성(연세대), 황숙정*(연세대), 강경근(연세대), 김재우(연세대)
YunSung Chung, Yonsei University, Sukjung Hwang*, Yonsei University, Kyungkeun Kang, Yonsei University, Jaewoo Kim, Yonsei University

We consider a coupled system consisting of a degenerate porous medium type of Keller-Segel system and Stokes system modeling the motion of swimming bacteria living in fluid and consuming oxygen. We establish the global existence of weak solutions and H\"{o}lder continuous solutions in dimension three, under the assumption that the power of degeneracy is above a certain number depending on given parameter values. To show H\"{o}lder continuity of weak solutions, we consider a single degenerate porous medium equation with lower order terms, and via a unified method of proof, we obtain H\"{o}lder regularity, which is of independent interest.

2010 Mathematics Subject Classification: 35B45, 35K65
Key Words and Phrases: Keller-Segel system, porous medium equations, boundedness, Holder continuity

⋅ 29th-B-10:40 − 12:10 Chair: Sukjung Hwang (Yonsei University)

⋅ 29th-B-10:40 − 11:00 Local kinetic energy and singularities of the incompressible Navier--Stokes equations (Hi Jun Choe, Minsuk Yang)

최희준(연세대), 양민석*(고등과학원)
Hi Jun Choe, Yonsei University, Minsuk Yang*, KIAS

We study the partial regularity problem of the incompressible Navier–Stokes equations. In this paper, we show that a reverse H\"older inequality of velocity gradient with increasing support holds under the condition that a scaled functional corresponding the local kinetic energy is uniformly bounded. As an application, we give a new bound for the Hausdorff dimension and the Minkowski dimension of singular set when weak solutions $v$ belong to $L^\infty(0,T;L^{3,\infty}(\mathbb R^3))$ where $L^{3,\infty}(\mathbb R^3)$ denotes the standard weak Lebesgue space.

2010 Mathematics Subject Classification: 35K20
Key Words and Phrases: Navier–Stokes equations, partial regularity, singular set

⋅ 29th-B-11:00 − 11:20 Gradient estimates for transmission problems with nonsmooth internal boundaries (Yunsoo Jang)

장윤수(연세대 응용해석 및 계산센터)
Yunsoo Jang, Center for Mathematical Analysis and Computation(CMAC), Yonsei University

In this talk, we obtain an interior gradient estimate for a weak solution of a transmission problem with nonsmooth internal boundaries. The coefficients are assumed to be merely measurable in one variable and have small BMO semi-norms in the other variables on each subdomain whose boundary satisfies the so-called $\delta$-Reifenberg flat condition. Under these assumptions, we prove a Calder\'{o}n-Zygmund type estimate.

2010 Mathematics Subject Classification: 35B65, 35D30, 35J47
Key Words and Phrases: elliptic system, transmission problem, measurable coefficient, Reifenberg domain

⋅ 29th-B-11:30 − 11:50 Weighted $L_{p,q}$-estimates for higher order elliptic and parabolic systems with BMO coefficients (Jongkeun Choi, Doyoon Kim)

최종근*(고려대), 김도윤(고려대)
Jongkeun Choi*, Korea University, Doyoon Kim, Korea University

We prove weighted $L_{p,q}$-estimates for divergence type higher order elliptic and parabolic systems with irregular coefficients on Reifenberg flat domains. In particular, in the parabolic case the coefficients do not have any regularity assumptions in the time variable. As functions of the spatial variables, the leading coefficients are permitted to have small mean oscillations. The weights are in the class of Muckenhoupt weights $A_p$.

2010 Mathematics Subject Classification: 35B45, 35J48, 35K41, 35R05
Key Words and Phrases: weighted $L_{p,q}$-estimates

⋅ 29th-B-11:50 − 12:10 $L^q$-estimates for the stationary rotating Oseen equations on exterior domains (Dugyu Kim)

김두규(연세대 응용해석 및 계산센터)
Dugyu Kim, Center for Mathematical Analysis and Computation(CMAC), Yonsei University

In this talk, we study the Dirichlet problem for the stationary rotating Oseen equations on exterior smooth domains $\Omega$ in $\mathbb{R}^3$. Our main results are the existence and uniqueness of weak and very weak solutions satisfying appropriate $L^q$-estimates. The existence of very weak solutions is deduced by a duality argument from our existence result and estimates for strong solutions. Using this and interior regularity, we prove the existence of a weak solution $v$ satisfying $\nabla v \in L^{r}(\Omega)$ for $4/3 < r < 4$.

2010 Mathematics Subject Classification: 35J25
Key Words and Phrases: Oseen equations, exterior domain, very weak solution, weak solution
Numerical Modeling and Computation

⋅ 29th-B-10:40 − 12:10 Chair: Eun-Jae Park (Yonsei University)

⋅ 29th-B-10:40 − 11:00 Space-time adaptivity in mixed FEM with Crank-Nicolson and backward Euler schemes (Dongho Kim, Eun-Jae Park, Bo Yoon Seo)

김동호(연세대), 박은재(연세대), 서보윤*(연세대)
Dongho Kim, Yonsei University, Eun-Jae Park, Yonsei University, Bo Yoon Seo*, Yonsei University

In this talk we present space-time a posteriori error estimators for fully discrete mixed methods for linear parabolic problems. The lowest order Raviart-Thomas space is considered for mixed variational formulation. For time discretization, a variant Crank-Nicolson scheme is suggested, which comes from discontinuous Galerkin method in time. We analyze a posteriori error estimations and we prove second order convergence in time. These estimations suggest several indicators for space-time adaptivity. This adaptive algorithm is performed with the change of meshes at each time-interval for refining or coarsening. Numerical results are given to support the theory developed in this work. Compared with Backward Euler scheme, advantages of our method are discussed.

2010 Mathematics Subject Classification: 65M12, 65M50
Key Words and Phrases: mixed method, a posteriori error estimation, adaptive algorithm, Crank-Nicolson scheme

⋅ 29th-B-11:00 − 11:20 A multiscale mixed method for nonlinear elliptic problems (Muhammad Arshad, Eun-Jae Park, Dong-wook Shin)

Muhammad Arshad*(연세대), 박은재(연세대), 신동욱(연세대)
Muhammad Arshad*, Yonsei University, Eun-Jae Park, Yonsei University, Dong-wook Shin, Yonsei University

We consider the approximation of nonlinear second order elliptic partial differential equation by multiscale mortar mixed finite element methods. The full partial differential equation is restricted to small local subproblems by decomposing the domain into nonoverlapping subdomains. A coarse mortar finite element space is introduced on interface grid to impose the weak continuity of flux across the subdomain interface. The coarse mortar grid is compensated by using the higher degree of polynomial approximation. We derived a priori error estimates for pressure and flux. We prove the optimal order convergence on fine scale by an appropriate choice of mortar space and polynomial degree of approximation. We also prove the second order convergence of Newton's method for the nonlinear algebraic system arising from mixed formulation of model problem. The numerical results are presented to confirm the theoretical results and to illustrate the behavior of method.

2010 Mathematics Subject Classification: 65N30, 65N55
Key Words and Phrases: multiscale, mixed method, domain decomposition, Newton's method, error estimates

⋅ 29th-B-11:30 − 11:50 A $C^0$-discontinuous Galerkin method for a wind-driven ocean circulation model (Dong-wook Shin, Eun-Jae Park, Younghun Kang)

신동욱*(연세대), 박은재(연세대), 강영훈(연세대)
Dong-wook Shin*, Yonsei University, Eun-Jae Park, Yonsei University, Younghun Kang, Yonsei University

In this work, we present a nonconforming finite element method for the stationary quasi-geostrophic equations, which describe the large scale wind-driven ocean circulation. The nonconforming method is developed based on the streamfunction formulation which is a fourth-order nonlinear PDE. Since the method is based on $C^0$ basis functions, the method enforces weak continuity of the normal flux across interelement boundaries. Existence and uniqueness of the approximation are proved and error estimates are demonstrated under a small data assumption. Numerical results from several benchmark problems on rectangular and curved domains are provided to demonstrate the accuracy and robustness of the method. We also provide the Mediterranean sea example to demonstrate the capability of the approach to the wind-driven ocean circulation simulation. Moreover, two-grid algorithms are presented to efficiently resolve nonlinearity and computational results are given to demonstrate the efficiency of the algorithms.

2010 Mathematics Subject Classification: 65N12, 65N30
Key Words and Phrases: nonconforming finite-element method, fourth-order partial-differential equation, two-grid algorithm

⋅ 29th-B-11:50 − 12:10 Discontinuous Galerkin methods for Hodgkin-Huxley model (Do-Hyun Kim)

김도현(연세대)
Do-Hyun Kim, Yonsei University

In this talk we present discontinuous Galerkin methods for the Hodgkin-Huxley model. Discontinuous Galerkin methods provide arbitrary high order convergence and mass conservation for the potential equation. This work is the first DG finite element method for the model equation and demonstrates effectivity of the method. Several boundary conditions are considered including Dirichlet, Neumann, and two other special types. The first special is a soma boundary condition which accounts for the soma presence and the second is a junction boundary condition which accounts connection of multiple domains at a point. For dendrite network problem, mortar method is considered to connect multiple domains. For time discretization, we use staggered Crank-Nicolson method which does not require nonlinear solver in time evolution. Optimal convergences are tested by using manufactured solutions. Several numerical simulations will be given from a single cable case to dendrite network cases. This is joint work with Namyong Lee (Minnesota State University), Mi-young Kim (Inha University), and Eun-jae Park (Yonsei University).

2010 Mathematics Subject Classification: 65
Key Words and Phrases: discontinuous Galerkin methods, Mortar methods, staggered Crank-Nicolson, Hodgkin-Huxley, neuro-science, potential propagation