KMS Meeting - Programs and Abstracts
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– Code preceding talk title indicates date-slot-time; for example, "25th-D-09:00 - 10:20" means the talk will be given on 25th, at slot D on time 09:00 -10:20.
- Plenary Lecture
- ⋅ 25th-O-15:30 − 16:20 Chair: Jaigyoung Choe (Korea Institute for Advanced Study)
- ⋅ 25th-O-15:30 − 16:20 Hamilton-Jacobi equation and continuous Hamiltonian dynamics (Yong-Geun Oh)
- 오용근(Univ. of Wisconsin/POSTECH)
Yong-Geun Oh, University of Wisconsin/POSTECH
In this talk, I will explain a Floer theoretic construction of solution of Hamilton-Jacobi equation and its relation to continous Hamiltonian dynamics. I will also indicate how the study is related to problems in area-preserving dynamical systems in 2 dimension.
2010 Mathematics Subject Classification: 53D05, 53D35, 28D10
Key Words and Phrases: Hamilton-Jacobi equation, Topological Hamiltonian flows, Hamiltonian homeomorphisms (or hameomorphisms), Cliffwall Lagrangian surgery
- Public Lecture
- ⋅ 24th-O-16:30 − 17:30 Chair: Hyungju Park (POSTECH)
- ⋅ 24th-O-16:30 − 17:30 세계수학자대회 이야기 (Jun-Muk Hwang)
- 황준묵(고등과학원)
Jun-Muk Hwang, KIAS
2014년 8월 서울에서 세계수학자대회(ICM)가 열립니다. 이 행사를 개최하는 것은 대한민국 수학자 모두의 영광이고 우리 국민 모두가 자랑스러워할 일입니다. 대회에 대한 공식적 안내와 자료는 국제수학연맹(IMU) 홈페이지와 서울 세계수학자대회 홈페이지 및 조직위원회에서 주관하는 각종 홍보와 언론보도를 통해서 얻으실 수 있습니다. 이 강연에서는 강연자의 주관적 견해와 개인적 경험 및 충분히 검증되지 않은 자료를 바탕으로 세계수학자대회에 대해 지극히 비공식적이고 다소 편향된 소개를 해드리고자 합니다.
2010 Mathematics Subject Classification: 00A99
Key Words and Phrases: ICM
- Invited Lectures
- ⋅ 25th-B-11:30 − 12:10 Chair: Jon-Lark Kim (Sogang University)
- ⋅ 25th-B-11:30 − 12:10 {\scriptsize \rm [Mathematics for Information Sciences]} Symmetries of graph embeddings on surfaces (Young Soo Kwon)
- 권영수(영남대)
Young Soo Kwon, Yeungnam University
A map is a cellular embedding of a graph into a surface. Maps are expressed by several ways, for example, drawings, permutations, torsion free subgroups of Dyck's triangle groups, algebraic curves and so on. In this talk, we will consider relations among these expressions. A map is called regular if its map automorphism group acts regularly on flag set which is the set of all incident vertex-edge-face triples. Classification of regular maps are mainly pursued by three different directions: for fixed graphs, fixed surfaces and for fixed groups. In this talk we give some methods to classify regular maps and survey some recent results related to classifications. If time admits, some external symmetries of maps like exponents, self-duality and self-Petrie duality are also dealt with.
2010 Mathematics Subject Classification: 05C10, 05C30
Key Words and Phrases: graph embeddings, regular maps, symmetries
- ⋅ 25th-C-13:30 − 14:10 Chair: Hoil Kim (Kyungpook National University)
- ⋅ 25th-C-13:30 − 14:10 {\scriptsize \rm [Algebra, 2013 대한수학회상 논문상(봄) 수상강연]} A group configuration theorem in model theory (Byunghan Kim)
- 김병한(연세대)
Byunghan Kim, Yonsei University
Group configuration has been an important issue/problem in model theory, a major branch of mathematical logic. In 1970s B. Zilber resolved the group configuration problem when a structure is stable and $\aleph_0$-categorical. In 80s E. Hrushovski removed $\aleph_0$-categoricity which is a notable progress and later the technique is used in his solution of Mordell-Lang conjecture in any characteristic. In mid 90s, due to the author’s work, the scope of stability theory (studying stable structures) is enlarged to simplicity theory dealing more structures such as the random graph and pseudo-finite fields. In this talk the author’s recent work on solving more general group action configuration problem in simplicity theory and its applications will be surveyed.
2010 Mathematics Subject Classification: 03C45
Key Words and Phrases: model theory, stability theory, simplicity theory, group configuration
- ⋅ 25th-C-13:30 − 14:10 Chair: Jongil Park (Seoul National University)
- ⋅ 25th-C-13:30 − 14:10 {\scriptsize\rm [Topology, 2012 상산젊은수학자상 수상강연]} A curve complex for a right-angled Artin group (Sang-hyun Kim, Thomas Koberda)
- 김상현*(카이스트), Thomas Koberda(Yale Univ.)
Sang-hyun Kim*, KAIST, Thomas Koberda, Yale University
For each right-angled Artin group $G$, we associate a quasi-tree $T$. Combinatorially, this quasi-tree is characterized as the ``iterated double'' of the defining graph of $G$. Algebraically, $T$ roughly encodes the isomorphism types of right-angled Artin subgroups of $G$. And geometrically, we observe that $G$ acts on $T$ acylindrically and that $T$ plays an analogous role to a curve complex for a mapping class group. Corollaries include embeddability results between RAAGs, and also from RAAGs to mapping class groups.
2010 Mathematics Subject Classification: 20F36
Key Words and Phrases: right-angled Artin group, mapping class group, curve complex
- ⋅ 25th-C-13:30 − 14:10 Chair: Ja A Jeong (Seoul National University)
- ⋅ 25th-C-13:30 − 14:10 {\scriptsize\rm [Analysis I, 2012 대한수학회상 학술상(가을) 수상강연]} Geometry for separable states (Seung-Hyeok Kye)
- 계승혁(서울대)
Seung-Hyeok Kye, Seoul National University
Distinguishing entanglement from separability is one of the most important question in the theory of quantum entanglement, which is a very useful notion in quantum information thoery as well as quantum physics. In this talk, we compare the local and global geometries for separable states and PPT states for this purpose.
2010 Mathematics Subject Classification: 81P15, 46L05
Key Words and Phrases: separable states, entangled states, positive partial transpose, faces
- ⋅ 25th-C-13:30 − 14:10 Chair: Hyunyong Shin (Korea National University of Education)
- ⋅ 25th-C-13:30 − 14:10 {\scriptsize \rm [Mathematical Education]} On the various situations and its roots in mathematics education (Boo Yoon Kim)
- 김부윤(부산대)
Boo Yoon Kim, Pusan National University
수학교육 현장에 나타나는 여러 상황과 그러한 것이 발생하게 된 뿌리가 무엇인지에 대해 살펴본다. 특히 교육과정, 교과서 제도, 영재교육, 수학 축제, 대학입시, 대학수학교육,각급 학교 학생들의 변화 등에서 보이는 현재 상황으로는 참된 수학교육이 얼마나 어려운가를 던진다.
2010 Mathematics Subject Classification: 97AXX
Key Words and Phrases: situation, mathematics education
- ⋅ 25th-C-13:30 − 14:10 Chair: YunKyong Hyon (NIMS)
- ⋅ 25th-C-13:30 − 14:10 {\scriptsize \rm [Applied Mathematics]} Overlapping Schwarz preconditioners for isogeometric analysis (Durkbin Cho)
- 조덕빈(동국대)
Durkbin Cho, Dongguk University
Isogeometric Analysis (IGA) is a non-standard numerical method for partial differential equations (PDEs), which was introduced by T. J. R. Hughes in [1, 2]. In the isogeometric framework, the ultimate goal is to adopt the geometry description from a Computer Aided Design (CAD) parametrization, and use it for the analysis, that is, within the PDE solver. Non-uniform rational B-splines (NURBS) are a standard in CAD community mainly because they are extremely convenient of the representation of free-form surfaces and there are very efficient algorithms to evaluate them, to refine and derefine them. In IGA, those same basis functions (that represent the CAD geometry) are also used as the basis for the discrete solution space of PDEs, thus following an isoparametric paradigm. IGA methodologies have been studied and applied in fields as diverse as fluid dynamics, structural mechanics and electromagnetics. Domain decomposition methods are a major area of recent research in numerical analysis for PDEs. They provide robust, parallel and scalable preconditioned iterative methods for the large linear systems arising in discretizaton of the continuous problems. In this talk, we give a brief review of B-splines, NURBS and Isogeometric Analysis and then propose overlapping additive Schwarz (OAS) methods for elliptic problems in Isogeometric Analysis [3]. We construct OAS preconditioners both in the parametric space and in the physical space and also prove that our proposed methods in multi-dimensions are scalable. Moreover, we present a set of numerical experiments, including the case with discontinuous coefficients, which is in complete accordance with the theoretical developments.
\begin{thebibliography}{1}
\bibitem{1} T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs, {\it Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement}, Comput. Methods Appl. Mech. Engrg. {\bf 194(39-41)} (2005), 4135--4195.
\bibitem{2} J. A. Cottrell, T. J. R. Hughes, and Y. Bazilevs, {\it Isogeometric Analysis: Toward Integration of CAD and FEA}, John Wiley \& Sons, 2009.
\bibitem{3} L. Beirao da Veiga, D. Cho, L. Pavarino, and S. Scacchi, {\it Overlapping Schwarz methods for isogeometric analysis}, SIAM J. Numer. Anal. {\bf 50} (2012), no. 3, 1394--1416.
\end{thebibliography}
2010 Mathematics Subject Classification: 65N55, 65N30, 65F10
Key Words and Phrases: isogeometric analysis, domain decomposition methods, overlapping additive Schwarz, preconditioners, finite element methods, NURBS
- ⋅ 25th-C-13:30 − 14:10 Chair: Un Cig Ji (Chungbuk National University)
- ⋅ 25th-C-13:30 − 14:10 {\scriptsize \rm [Probability and Statistics]} About the stochastic regularity of distorted Brownian (Gerald Trutnau)
- Gerald Trutnau, Seoul National University
Let $\Omega\subset \mathbb{R}^d$ be an open or closed domain. We consider an energy form $$ {\mathcal E}(f,g) = \frac12\int_{\Omega} \nabla f\cdot \, \nabla \, g \, d\mu, \ \ \ f,g\in D, $$ with $d\mu=\rho\, dx$, $\rho$ locally integrable, $\rho>0$ $dx$-a.e. and test functions $D$ with compact support. If $({\mathcal E}, D)$ is closable on $L^2(\Omega,\mu)$ and $\rho$ is regular enough, then by {\it Fukushima's decomposition} the process $((X_t)_{t\ge 0},(P_x)_{x\in \Omega})$ associated to ${\mathcal E}$ weakly solves \begin{eqnarray*} X_t & = & x + B_t+ \int_0^t \frac{\nabla \rho}{2\rho}(X_s)ds, \ \ \ P_x\mbox{-a.s for}\ x\in \Omega\setminus N, \end{eqnarray*} where $(B_t)_{t\ge 0}$ is a standard Brownian motion. Additional reflection terms may occur if $\rho$ is discontinuous or if we impose boundary conditions on $\partial \Omega$. The set $N$ is in general an abstract capacity zero set and not explicitly known. In this talk we investigate the stochastic regularity of $(X_t)_{t\ge 0}$, i.e., the question whether $N$ can be determined explicitly. One of the difficulties is that $\frac{\nabla \rho}{2\rho}$ can be highly singular on the set $\{\rho =0\}$. We consider different kind of weights and boundary conditions for which we can solve the problem. Our main tools are elliptic regularity results and local heat kernel estimates. This talk is based on joint work with Jiyong Shin (SNU).
2010 Mathematics Subject Classification: 31C25, 60J60, 60J55, 31C15, 35J25
Key Words and Phrases: Dirichlet forms, diffusion processes, distorted Brownian motion, strong Feller property, heat kernel estimates, lomathcal time
- ⋅ 25th-C-14:20 − 15:00 Chair: Ja Kyung Koo (KAIST)
- ⋅ 25th-C-14:20 − 15:00 {\scriptsize\rm [Algebra, 2013 대한수학회상 논문상(봄) 수상강연]} Classification and decomposition of the Witt-Burnside ring and Burnside ring of a profinite group (Young-Tak Oh)
- 오영탁(서강대)
Young-Tak Oh, Sogang University
The Witt-Burnside ring functor attached to a profinite group $G$ was introduced by Dress and Siebeneicher as a group-theoretical generalization of the classical $p$-typical Witt vectors of Teichm\"{u}ller and Witt and the big Witt vectors of Cartier. As its name implies, it has intimate relation with the Burnside ring of $G$. In this talk, I will explain a few fundamental properties of Witt-burnside rings and Burnside rings. Recent developments in this area together with connections to other areas will be dealt with, too.
2010 Mathematics Subject Classification: Primary 13F35; Secondary 19A22, 20E18
Key Words and Phrases: Witt vector, Witt-Burnside ring, Burnside ring, profinite group
- ⋅ 25th-C-14:20 − 15:00 Chair: Gyo Taek Jin (KAIST)
- ⋅ 25th-C-14:20 − 15:00 {\scriptsize\rm [Topology, 특별초청강연]} Groups of self-homeomorphisms of manifolds (Dale Rolfsen, Danny Calegari)
- Dale Rolfsen*, University of British Columbia, Danny Calegari, University of Chicago
In 1920, Kerekjarto proved that the group of homeomorphisms of the 2-dimensional disk which are pointwise fixed on the boundary has no elements of finite order. In joint work with Danny Calegari, I will discuss why the same is true in higher dimensions, at least for the subgroup of piecewise linear homeomorphism. More general results on algebraic properties of groups of PL self-homeomorphisms will be proved.
2010 Mathematics Subject Classification: 37B05, 57Q99
Key Words and Phrases: transformation groups, orderable groups, PL homeomorphism, locally indicable
- ⋅ 26th-E-12:00 − 12:40 Chair: Jeong Hyeong Park (Sungkyunkwan University)
- ⋅ 26th-E-12:00 − 12:40 {\scriptsize \rm [Topology]} Delaunay type domains for an overdetermined elliptic problem (Filippo Morabito, Pieralberto Sicbaldi)
- Filippo Morabito*, KAIST, Pieralberto Sicbaldi, University of Aix-Marseille 3
We construct new domains in $S^n \times R$ and $H^n \times R$ which are critical points for the map $D \to \lambda_1(D),$ where $\lambda_1$ is the first eigenvalue of the Laplace-Beltrami operator and $D$ is a domain, by the technique of the bifurcation. Such domains can be characterized as domains for which an ovedetermined boundary value problem admits a positive solution.
2010 Mathematics Subject Classification: 35N25, 37G10, 53C21
Key Words and Phrases: extremal domain, Laplace-Beltrami operator, first eigenvalue
- ⋅ 26th-E-12:00 − 12:40 Chair: Yongjin Yeom (Kookmin University)
- ⋅ 26th-E-12:00 − 12:40 {\scriptsize\rm[Cryptography, 2012 상산젊은수학자상 수상강연]} (Im)possibility of cryptographic properties in bilinear product groups (Jae Hong Seo)
- 서재홍(명지대)
Jae Hong Seo, Myongji University
Since Boneh and Franklin used bilinear groups to design the first identity-based encryption scheme in 2001, bilinear groups has been used in various areas of cryptography. In 2005, Boneh, Goh, and Nissim found that product groups of bilinear groups have useful cryptographic properties, which are identified and named by Freeman later. In this talk, we introduce recent progress in this field; in particular, we focus on the (im)possibility of these cryptographic properties in prime-order setting.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: bilinear groups, cancelling, projecting
- ⋅ 26th-E-12:00 − 12:40 Chair: Hyeong-Ohk Bae (Ajou University)
- ⋅ 26th-E-12:00 − 12:40 {\scriptsize \rm [Analysis II]} On the regularity of weak solutions to the motion of the degenerate power-law fluids (Bum Ja Jin)
- 진범자(목포대)
Bum Ja Jin, Mokpo National University
Let $\Omega$ be a bounded domain in ${\mathbb R}^n$, $n=2,3$. We consider the steady and unsteady motion of a fluid described by the systems \begin{equation} \label{e1} (u\cdot \nabla )u -\mbox{div}S(Du)+\nabla p =f,\ \mbox{div}u=0 \mbox{ in }\Omega,\ u|_{\partial \Omega}=0\end{equation} and \begin{equation} \label{e2} u_t+(u\cdot \nabla )u -\mbox{div}S(Du)+\nabla p =0,\ \mbox{div}u=0 \mbox{ in }\Omega\times (0,T),\ u|_{\partial \Omega}=0, u|_{t=0}=a, \end{equation} respectively, where \begin{equation} \label{e3} S(Du)=|Du|^{q-2}Du. \end{equation} %Observe that w When $q=2$, the system becomes (incompressible)Navier-Stokes equations, and the well known theories on the linear (partial differential)operator could be applied to study the regularity of weak solutions. When $q\neq 2$, the structure of stress tensor is no more linear, whence we cannot apply any known linear operator theory. To study regularity of weak solutions it is natural to use variational approach such as difference quotient scheme. Since variational approach is based on the integrability of the convection term $(u\cdot \nabla v)\cdot w$ for $u,v,w\in W^{1,q}(\Omega)$, it restricts the range of $q$ to $q>\frac{3n}{n+2}$. The most critical case is that $2q>\frac{9}{5}, n=3$ and $2>q>\frac{3}{2}, n=2$ for the unsteady problem. In this talk, I present two recent works on those cases, which were obtained by myself with other coworkers.
2010 Mathematics Subject Classification: 35Q35, 76A05
Key Words and Phrases: non-Newtonian fluid, weak solution, regularity
- Algebra
- ⋅ 25th-A-09:00 − 10:30 Chair: Insong Choe (Konkuk University)
- ⋅ 25th-A-09:00 − 09:30 Riemann Roch for geometric Lie algebroids (Hoil Kim)
- 김호일(경북대)
Hoil Kim, Kyungpook National University
Riemann Roch theorem on complex manifolds (algebraic varieties) describes holomorphic sections for given holomorphic sheaves on them. This is extended to sheaves on stacks or orbifolds by several people. In this talk we describe them differently (similarly) for geometric Lie algebroids containing generalized complex manifolds which have been developed to combine complex varieties and symplectic manifolds.
2010 Mathematics Subject Classification: 14O00
Key Words and Phrases: Riemann Roch theorem, geometric Lie algebroids, generalized complex manifolds, symplectic manifolds
- ⋅ 25th-A-09:30 − 10:00 Torelli theorem of logarithmic bundles (Edoardo Ballico, Sukmoon Huh, Francesco Malaspina)
- Edoardo Ballico(Univ. di Trento), 허석문*(성균관대), Francesco Malaspina(Politecnico di Torino)
Edoardo Ballico, Universita di Trento, Sukmoon Huh*, Sungkyunkwan University, Francesco Malaspina, Politecnico di Torino
For an arrangement $D$ of smooth hypersurfaces on a nonsingular variety $X$, we can define a logarithmic sheaf to be a sheaf of differential 1 forms with logarithmic poles along $D$. This sheaf is locally free when $D$ has simple normal crossings. When the logarithmic bundle is free, we call the divisor to be free. In particular, we lose many information about the free divisors when we consider their corresponding logarithmic bundles. As an opposite extremal case, we can ask Torelli question, i.e., can we recover the divisor from the logarithmic bundles? This question was answered in the case of projective spaces by Dolgachev-Kapranov and Valles. In this talk, we introduce the recent result on the same question over quadric hypersurfaces.
2010 Mathematics Subject Classification: 14F99, 14J99
Key Words and Phrases: logarithmic bundle, quadric hypersurface, Torelli theorem
- ⋅ 25th-A-10:00 − 10:30 On Schertz's conjecture (Ho Yun Jung, Ja Kyung Koo, Dong Hwa Shin)
- 정호윤(포항수학연구소), 구자경(카이스트), 신동화*(한국외대)
Ho Yun Jung, Pohang Mathematics Institute, Ja Kyung Koo, KAIST, Dong Hwa Shin*, Hankuk University of Foreign Studies
We examine two kinds of Fricke families consisting of Fricke functions and Siegel functions, respectively. We also generates ray class fields of imaginary quadratic fields by means of special values of functions in each Fricke family.
2010 Mathematics Subject Classification: 11G15, 11F03
Key Words and Phrases: complex multiplication, Fricke families, modular functions
- ⋅ 25th-B-10:40 − 12:10 Chair: Yongnam Lee (KAIST)
- ⋅ 25th-B-10:40 − 11:10 Normal complex surface singularities with rational homology disk smoothings (Heesang Park, Dongsoo Shin, Andr\'as I. Stipsicz)
- 박희상(고등과학원), 신동수*(충남대), Andras I. Stipsicz(R\'enyi Institute of Mathematics)
Heesang Park, Korea Institute for Advanced Study, Dongsoo Shin*, Chungnam National University, Andras I. Stipsicz, Renyi Institute of Mathematics
In this talk we show that if the minimal good resolution graph of a normal surface singularity contains at least two nodes (i.e., vertex with valency at least 3) then the singularity does not admit a smoothing with Milnor fiber having rational homology equal to the rational homology of the 4-disk $D^4$ (called a rational homology disk smoothing). This result, combining with earlier ones, then provides a complete classification of resolution graphs of singularities with rational homology disk smoothings, verifying a conjecture of Wahl regarding such singularities.
2010 Mathematics Subject Classification: 14B07, 14J17, 32S30
Key Words and Phrases: surface singularity, Milnor fiber, rational homology disk smoothing
- ⋅ 25th-B-11:10 − 11:40 Orthogonal bundles of odd rank over a curve (Insong Choe, George H. Hitching)
- 최인송*(건국대), George H. Hitching(Hogkolen i Oslo og Akershus)
Insong Choe*, Konkuk University, George H. Hitching, Hogkolen i Oslo og Akershus
The Segre invariant of an orthogonal bundle over an algebraic curve is given by the maximal degree of Lagrangian subbundles. In ths talk, we give a sharp upper bound on the Segre invariant and provide the details of the induced stratification on the moduli of orthogonal bundles. The case of odd rank turns out to be more subtle, because any two Lagrangian subbundles intersect in a subsheaf of positive rank.
2010 Mathematics Subject Classification: 14H60, 14N05
Key Words and Phrases: Segre invarint, orthogonal bundle, algebraic curve, Lagrangian subbundle
- ⋅ 25th-B-11:40 − 12:10 Super duality for Kac-Moody superalgebras and irreducible characters (Jae-Hoon Kwon, Shun-Jen Cheng, Weiqiang Wang)
- 권재훈*(성균관대), Shun-Jen Cheng(Academia Sinica), Weiqiang Wang(Univ. of Virginia)
Jae-Hoon Kwon*, Sungkyunkwan University, Shun-Jen Cheng, Academia Sinica, Weiqiang Wang, University of Virginia
Generalizing the super duality formalism for finite-dimensional Lie superalgebras of type $ABCD$, we establish an equivalence between parabolic BGG categories of a Kac-Moody Lie superalgebra and a Kac-Moody Lie algebra. The characters for a large family of irreducible highest weight modules over a symmetrizable Kac-Moody Lie superalgebra are then given in terms of Kazhdan-Lusztig polynomials for the first time. We formulate a notion of integrable modules over a symmetrizable Kac-Moody Lie superalgebra via super duality, and show that these integrable modules form a semisimple tensor subcategory, whose Littlewood-Richardson tensor product multiplicities coincide with those in the Kac-Moody algebra setting.
2010 Mathematics Subject Classification: 17B56
Key Words and Phrases: Kac-Moody superalgebras, irreducible characters
- ⋅ 26th-D-09:30 − 11:00 Chair: Dong Hwa Shin (Hankuk University of Foreign Studies)
- ⋅ 26th-D-09:30 − 10:00 On the invariant $M(A/K, n)$ of Chen-Kuan for Galois representations (Hyunsuk Moon)
- 문현숙(경북대)
Hyunsuk Moon, Kyungpook National University
Let $A$ be an abelian variety over a number field $K$. For a prime $\mathfrak p$ in $K$, denote the residue field by $\mathbb{F}_{\mathfrak p}$. If $A$ has good reduction at $\mathfrak p$, let $N_{\mathfrak p, n}$ be the number of $n$-torsion $\mathbb{F}_{\mathfrak p}$-rational points of the reduction of $A$ modulo $\mathfrak p$, where $n$ is a positive integer. When dimension of $A$ is one, Chen and Kuan determined the average value $M(A_{/K}, n)$ of $N_{\mathfrak p, n}$ as the prime $\mathfrak p$ varies. In this talk, we generalize their invariant $M(A_{/K}, n)$ for Galois representations. Some of their results follow immediately from our results.
2010 Mathematics Subject Classification: 11F80, 11G05
Key Words and Phrases: Galois representations, torsion points
- ⋅ 26th-D-10:00 − 10:30 Riemann hypothesis and an upper bound of the divisor function for odd integers (Ick Sun Eum, Ja Kyung Koo)
- 엄익선*(서울대 수학연구소), 구자경(카이스트)
Ick Sun Eum*, Research Institute of Mathematics, Seoul National University, Ja Kyung Koo, KAIST
In 1984, Robin showed that the Riemann hypothesis is equivalent to the statement that the Robin's inequality $\sigma(n)<e^{\gamma}n \log\log n $ holds for $n\geq5041$, where $\gamma$ is the Euler-Mascheroni constant. In this paper we provide more sharper bound of $\sigma(n)$ than Robin's one for odd integers by using the ideas of Choie et al.~(2007) and show that Robin's inequality holds for $n\not\equiv0\pmod{3}$ with finite exceptions
2010 Mathematics Subject Classification: 11A25, 11M26
Key Words and Phrases: Riemann hypothesis, Robin's inequality, divisor function, Euler totient function
- ⋅ 26th-D-10:30 − 11:00 On the factorizations of the Pascal matrices via Padovan matrices (GwangYeon Lee, Mustafa Asci)
- 이광연*(한서대), Mustafa Asci(Pamukkale Univ.)
GwangYeon Lee*, Hanseo University, Mustafa Asci, Pamukkale University
Riordan arrays are useful for solving the combinatorial sums by the help of generating functions. Many theorems can be easily proved by Riordan arrays. In this paper, we introduce the first kind of Padovan matrix and the second kind of Padovan matrix and we consider the factorizations of Pascal matrix involving the Padovan matrices.
2010 Mathematics Subject Classification: 05A19, 11B39, 15A23, 15B36
Key Words and Phrases: Riordan group, Pascal matrix, Pell matrix, Fibonacci matrix, Padovan matrix, factorization
- ⋅ 26th-E-11:10 − 12:10 Chair: Euiyong Park (University of Seoul)
- ⋅ 26th-E-11:10 − 11:30 Elementary proofs of the formulas for the representation numbers of integers by certain quadratic forms with parity conditions on the variables (Bumkyu Cho, Ho Park)
- 조범규(동국대), 박호*(동국대)
Bumkyu Cho, Dongguk University, Ho Park*, Dongguk University
In this talk we first introduce some results of Hurwitz and Deutsch about the representation numbers of integers by quadratic forms $x^2+y^2+z^2+w^2$ and $x^2+y^2+2z^2+2w^2$ with certain parity conditions on the variables $x$, $y$, $z$, and $w$. The purpose of this talk is to provide elementary proofs of these results and to give formulas for the representation numbers by the quadratic form $x^2+xy+y^2+z^2+zw+w^2$ with parity conditions on the variables.
2010 Mathematics Subject Classification: 11E25
Key Words and Phrases: number of representations, quadratic forms
- ⋅ 26th-E-11:30 − 11:50 Insertion-of-factors-property and examples of related ring (Sung Hyo Jin, Chan Huh, Hong Kee Kim, Nam Kyun Kim, Chang Ik Lee, Yang Lee)
- 성효진*(부산대), 허찬(부산대), 김홍기(경상대), 김남균(한밭대), 이창익(부산대), 이양(부산대)
Hyo Jin Sung*, Pusan National University, Chan Huh, Pusan National University, Hong Kee Kim, Gyeongsang National University, Nam Kyun Kim, Hanbat National University, Chang Ik Lee, Pusan National University, Yang Lee, Pusan National University
We study the structure of the IFP on powers of zero-dividing elements and introduce $\pi$-IFP as a generalization. We find a simple method by which we can always construct $\pi$-IFP rings but not IFP over any IFP ring. We show that $\pi$-IFP and Armendariz are independent of each other. It is shown that locally finite Abelian rings are $\pi$-IFP.
2010 Mathematics Subject Classification: 16U80
Key Words and Phrases: IFP Ring, Pi-IFP
- ⋅ 26th-E-11:50 − 12:10 Extensions of symmetric rings (Dawoon Jung , Sungju Ryu , Yang Lee )
- 정다운*(부산대), 류성주(부산대), 이양(부산대)
Dawoon Jung *, Pusan National University, Sungju Ryu , Pusan National University, Yang Lee , Pusan National University
In the present note we continue the study of symmetric rings and nearly related conditions for rings, introducing the concepts of an weak symmetric ring and weak reversible. Various kinds of examples are constructed and computed to show that weak symmetric rings are neither weak reversible in containing relations nor left-right symmetric. We next show that polynomial rings over weak reversible rings need not be weak reversible.
2010 Mathematics Subject Classification: 16S70
Key Words and Phrases: symmetric ring, reversible
- Analysis I
- ⋅ 25th-A-09:00 − 10:30 Chair: Seung-Hyeok Kye (Seoul National University)
- ⋅ 25th-A-09:00 − 09:30 An analogue of conditional Wiener integral with drift (Dong Hyun Cho)
- 조동현(경기대)
Dong Hyun Cho, Kyonggi University
Let $C[0,t]$ denote a generalized Wiener space, the space of real-valued continuous functions on the interval $[0,t]$ and define a stochastic process $Y: C[0,t]\times [0,t]\to \mathbb R$ by $Y(x,s)=\int_0^s h(u) dx(u) +a(s)$ for $x\in C[0,t]$ and $s\in[0,t]$, where $h\in L_2[0,t]$ with $h\neq 0$ a.e. and $a$ is continuous on $[0,t]$. Let random vectors $Y_n:C[0,t]\to\mathbb R^n$ and $Y_{n+1}:C[0,t]\to\mathbb R^{n+1}$ be given by $ Y_n(x) = (Y(x,t_1),\ldots, Y(x,t_n)) $ and $ Y_{n+1}(x) = (Y(x,t_1),\ldots, Y(x,t_n), Y(x,t_{n+1}))$. In this talk, we derive a translation theorem for a generalized Wiener integral and then, prove that $Y$ is a generalized Brownian motion process with drift $a$. Furthermore, we derive two simple formulas for generalized conditional Wiener integrals of functions on $C[0,t]$ with the drift and the conditioning functions $Y_n$ and $Y_{n+1}$. As applications of the simple formulas, we evaluate the generalized conditional Wiener integrals of various functions on $C[0,t]$.
2010 Mathematics Subject Classification: 28C20
Key Words and Phrases: conditional Wiener integral, generalized Brownian motion, simple formula for conditional Wiener integral, translation theorem, Wiener integral, Wiener space
- ⋅ 25th-A-09:30 − 09:50 The pre-Schwarzian norm estimate for analytic concave functions (Young Jae Sim, Oh Sang Kwon)
- 심영재*(경성대), 권오상(경성대)
Young Jae Sim*, Kyungsung University, Oh Sang Kwon, Kyungsung University
Let $\mathbb{D}$ denote the open unit disk and let ${\mathcal S}$ denote the class of normalized univalent functions which are analytic in $\mathbb{D}$. Let $Co(\alpha)$ the family of functions $f \in {\mathcal S}$, which have the condition that the opening angle of $f(\mathbb{D})$ at infinity is less than or equal to $\pi\alpha$, $\alpha \in (1,2]$. In this paper, we find a sufficient condition for the Gaussian hypergeometric functions to be in the class $Co(\alpha)$. And we define a class $Co(\alpha,A,B)$, $(-1 \leq B < A \leq 1)$, which is a subclass of $Co(\alpha)$ and we find the set of variabilty for the functional $(1-|z|^{2})(f''(z)/f'(z))$, $f \in Co(\alpha,A,B)$. This gives sharp upper and lower estimates for the pre-Schwarzian norm of functions in $f \in Co(\alpha,A,B)$. We also give a characterization for functions in $f \in Co(\alpha,A,B)$ in terms of Hadamard product.
2010 Mathematics Subject Classification: 30C45
Key Words and Phrases: univalent functions, concave functions, pre-Schwarzian norm, Gaussian hypergeometric functions
- ⋅ 25th-A-09:50 − 10:10 Characterization of split quaternionic functions in Clifford analysis (Su Jin Lim, Kwang Ho Shon)
- 임수진*(부산대), 손광호(부산대)
Su Jin Lim*, Pusan National University, Kwang Ho Shon, Pusan National University
The split complex number has the form $z=a+bj$ where $j^2=1$ and the split quaternions are an expression of the form $z=\sum_{j=0}^3 e_jx_j$ where $x_j~(j=0,\ldots,3)$ are real numbers. The bases $e_j~(j=0,\ldots,3)$ are the split quaternionic units which satisfy the non-commutative multiplication rules $e_1e_2=e_3=-e_2e_1,$ $e_2e_3=-e_1=-e_3e_2,$ $e_3e_1=e_2=-e_1e_3,$ $e_1e_2e_3=1,$ $e_1^2=-1,$ $e_2^2=1,$ $e_3^2=1$. We investigate the properties of regular functions with values in split quaternion.
2010 Mathematics Subject Classification: 32A99, 30G35, 32W50, 11E88
Key Words and Phrases: split quaternion, regular function, Clifford analysis
- ⋅ 25th-A-10:10 − 10:30 Existence of an unbounded branch of the set of solutions for elliptic equations of $p(x)$-Laplace type (Byung-Hoon Hwang, Yun-Ho Kim)
- 황병훈*(상명대), 김연호(상명대)
Byung-Hoon Hwang*, Sangmyung University, Yun-Ho Kim, Sangmyung University
We study the structure of the set of weak solutions of a nonlinear equation of the form \begin{equation}\label{e:JG} -\text{div}(\phi(x,|\nabla u|)\nabla u)=\mu|u|^{p(x)-2}u +f(\lambda,x,u,\nabla u)\quad \textmd{in } \Omega \tag{B} \end{equation} subject to Dirichlet boundary conditions, provided that $\mu$ is not an eigenvalue of the above divergence form. Here the variable exponent $p: \bar \Omega\to (1,\infty)$ is a continuous function and $f:\\
\mathbb R\times\Omega\times\mathbb R\times\mathbb R^{N}\to \mathbb R$ satisfies a Carath\'eodory condition. To do this, we first state and prove some basic results for the variable exponent Lebesgue-Sobolev spaces. Based on these results, we present some properties of the corresponding integral operators. We will prove our main result on global bifurcation for problem \eqref{e:JG}, by using a bifurcation result in abstract setting which gives the existence of a branch of solutions.
2010 Mathematics Subject Classification: 35B32, 47J05, 46E35
Key Words and Phrases: bifurcation, $p(x)$-Laplacian, variable exponent Lebesgue-Sobolev spaces, eigenvalue
- ⋅ 25th-B'-11:00 − 12:30 Chair: Dohan Kim (Seoul National University)
- ⋅ 25th-B'-11:00 − 11:30 Regenerations of regular functions of three variables on a complex Clifford analysis (Ji Eun Kim, Kwang Ho Shon)
- 김지은*(부산대), 손광호(부산대)
Ji Eun Kim*, Pusan National University, Kwang Ho Shon, Pusan National University
We express a new representation $\hat{i}$ of bases of three real variables in a Clifford analysis. The hypercomplex number has the form $z=x_{0}+\hat{i}z_{0}$ where $\hat{i}^{2}=-1$. The bases $e_{j}~~(j=0,1,2)$ satisfy the non-commutative multiplication rules $e_{j}^{2}=-1~~(j=1,2)$, $e_{0}^{2}=1$, and $e_{1}e_{2}=-e_{2}e_{1}$. We investigate the properties of regular functions with values in ternary numbers.
2010 Mathematics Subject Classification: 32A99, 30G35, 32W50, 11E88
Key Words and Phrases: ternary numbers, hypercomplex numbers, regular function, Clifford analysis
- ⋅ 25th-B'-11:30 − 12:00 On elliptic equations of $p(x)$-Laplacian type subject to Dirichlet boundary condition (Yun-Ho Kim)
- 김연호(상명대)
Yun-Ho Kim, Sangmyung University
In this talk, we are concerned with nonlinear elliptic equations of $p(x)$-Laplacian type subject to Dirichlet boundary condition \begin{equation*} \begin{cases} -\text{div}(\phi(x,|\nabla u|)\nabla u)=\lambda f(x,u) &\textmd{in } \Omega \\ u=0 &\textmd{on } \partial\Omega, \end{cases} \end{equation*} where $\Omega$ is a bounded domain in $\mathbb R^{N}$ with Lipschitz boundary $\partial \Omega,$ the function $\phi(x,t)$ is of type $|t|^{p(x)-2}$ with continuous function $p: \overline{\Omega} \to (1,\infty)$ and $f:\Omega\times\mathbb R\to \mathbb R$ satisfies a Carath\'eodory condition. Under suitable conditions on $\phi$ and $f$, employing the variational methods, we show the existence of nontrivial solutions of a class of quasilinear elliptic problems with variable exponents. Also we show the existence of positivity of the infimum of all eigenvalues for the above problem and then give an example to demonstrate our main result.
2010 Mathematics Subject Classification: 35D30, 35J60, 35J70, 35J92, 35P30, 46E35, 47J10
Key Words and Phrases: $p(x)$-Laplacian, variable exponent Lebesgue-Sobolev spaces, weak solution, mountain pass theorem, fountain theorem, eigenvalue
- ⋅ 25th-B'-12:00 − 12:30 Variational results for the semilinear biharmonic problem (Tacksun Jung, Q-Heung Choi)
- 정택선*(군산대), 최규흥(인하대)
Tacksun Jung*, Kunsan National University, Q-Heung Choi, Inha University
We get one theorem that there exists a unique solution of the fourth order semilinear elliptic Dirichlet boundary value problem when the number 0 and the coefficient of the semilinear part belongs to the open interval between two successive eigenvalues of the fourth order eigenvalue problem simultaneously. We prove this result by the contraction mapping principle method. We also get another theorem that there exists at least two solutions when there exist $n$ eigenvalues of the fourth order eigenvalue problem between the coefficient of the semilinear part and the number 0. We prove this result by the critical point theory and the variation linking method.
2010 Mathematics Subject Classification: 35J20, 35J25
Key Words and Phrases: fourth order elliptic boundary value problem, semilinear term, contraction mapping principle, critical point theory, variation linking method
- ⋅ 25th-C-14:20 − 14:50 Chair: Gwang Hui Kim (Kangnam University)
- ⋅ 25th-C-14:20 − 14:50 Change of scale formulas for Wiener integrals related with Fourier-Feynman transform and convolution (Byoung Soo Kim, Bong Jin Kim, Il Yoo)
- 김병수*(서울과학기술대), 김봉진(대진대), 유일(연세대)
Byoung Soo Kim*, Seoul National University of Science and Technology, Bong Jin Kim, Daejin University, Il Yoo, Yonsei University
Cameron and Storvick discovered change of scale formulas for Wiener integrals of functionals in a Banach algebra ${\mathcal S}$ on classical Wiener space. Yoo and Skoug extended these results for functionals in the Fresnel class ${\mathcal F}(B)$ and in a generalized Fresnel class ${\mathcal F}_{A_1,A_2}$ on abstract Wiener space. We express Fourier-Feynman transform and convolution product of functionals in ${\mathcal S}$ as limits of Wiener integrals. Moreover we obtain change of scale formulas for Wiener integrals related with Fourier-Feynman transform and convolution product of these functionals.
2010 Mathematics Subject Classification: 28C20
Key Words and Phrases: Wiener integral, Feynman integral, change of scale formula, Fourier-Feynman transform, convolution
- ⋅ 26th-D-09:30 − 11:00 Chair: Tacksun Jung (Kunsan University)
- ⋅ 26th-D-09:30 − 10:00 Stability of a Pexider type functional equation related to distance measure (Gwang Hui Kim)
- 김광휘(강남대)
Gwang Hui Kim, Kangnam University
The present work continues the study of the stability of the Pexider type functional equations consist with four unknown functions of the type $$f(pr, qs) + f(ps, qr) = f(p,q)\, f(r, s)$$ namely\\
(i) $f(pr, qs) + g(ps, qr) = h(p,q)\, h(r, s)$, and\\
(ii) $f(pr, qs) + g(ps, qr) = h(p,q)\, k(r, s)$ for all $p, q, r, s \in (0,1)$. This work is generalization of our papers. These functional equations arise in the characterization of symmetrically compositive sumform distance measures, products of some multiplicative functions. In reduction, they can be represented an (hyperbolic) cosine(sine, trigonometric) functional equation, exponential, and Jensen functional equation, respectively.
2010 Mathematics Subject Classification: 39B82, 39B52
Key Words and Phrases: distance measure, superstability, multiplicative function, trigonometric equation, stability of functional equation
- ⋅ 26th-D-10:00 − 10:30 Symmetry properties of interpolating refinable function vectors (Jaewon Jung)
- 정재원(아주대)
Jaewon Jung, Ajou University
In this talk, we present two kinds of symmetry of interpolating biorthogonal refinable function vectors. When the dilation factor and multiplicity are same, there is no totally interpolating biorthogonal multiwavelet systems having symmetry properties with respect to filters. Since it is different from a function version, we consider multiscaling functions having symmetry with respect to functions. We are also interested in special symmetry of interpolating refinable functions like a mirror.
2010 Mathematics Subject Classification: 42C40
Key Words and Phrases: interpolating, refinable function vectors, symmetry
- ⋅ 26th-D-10:30 − 11:00 On normality of truncated Toeplitz operators on finite dimensional spaces (Ji Eun Lee, Eungil Ko)
- 이지은*(이화여대 수리과학연구소), 고응일(이화여대)
Ji Eun Lee*, Institute of Mathematical Sciences, Ewha Womans University, Eungil Ko, Ewha Womans University
We say that a truncated Toeplitz operator is the compression of a Toeplitz operator on the Hardy space $H^{2}$ to a model subspace $H^2\ominus BH^{2}$, where $B$ is a finite Blaschke product. In this paper, we study a truncated Toeplitz operator on finite dimensional spaces and give a characterization of normal matrices on such spaces. In particular, we find some conditions for truncated Toeplitz operators on finite dimensional spaces to be normal and provide several examples of such operators. Finally, we consider symbols of normal truncated Toeplitz operators on finite dimensional spaces.
2010 Mathematics Subject Classification: 47B35, 47B15
Key Words and Phrases: truncated Toeplitz operator, normal operator
- ⋅ 26th-E-11:10 − 11:40 Chair: Kun Sik Ryu (Hannam University)
- ⋅ 26th-E-11:10 − 11:40 Variational results for the singular potential Hamiltonian system (Q-Heung Choi, Tacksun Jung)
- 최규흥*(인하대), 정택선(군산대)
Q-Heung Choi*, Inha University, Tacksun Jung, Kunsan National University
We investigate the multiple solutions for the Hamiltonian system with singular potential nonlinearity and periodic condition. We get a theorem which shows the existence of the nontrivial weak periodic solution for the Hamiltonian system with singular potential nonlinearity. We obtain this result by using variational method, critical point theory for indefinite functional.
2010 Mathematics Subject Classification: 35Q72
Key Words and Phrases: Hamiltonian system, singular potential nonlinearity, variational method, critical point theory for indefinite functional, $(P.S.)_{c}$ condition
- Analysis II
- ⋅ 25th-A-09:00 − 10:30 Chair: Dong-Yun Shin (University of Seoul)
- ⋅ 25th-A-09:00 − 09:20 On quasilinear elliptic equations with variable exponents in $\mathbb R^{N}$ (Seung Dae Lee, Yun-Ho Kim)
- 이승대*(상명대), 김연호(상명대)
Seung Dae Lee*, Sangmyung University, Yun-Ho Kim, Sangmyung University
We are concerned with the following elliptic equations with variable exponents \begin{equation*} -\text{div}(\varphi(x,\nabla u))=\lambda f(x,u)\textmd{ in } \mathbb R^{N} \end{equation*} where the functions $\varphi(x,v)$ is of type $|v|^{p(x)-2}v$ with continuous function $p:\mathbb R^{N} \to (1,\infty)$ and $f:\mathbb R^{N}\times\mathbb R \to \mathbb R$ satisfies a Carath\'eodory condition. Here $\varphi$ is not necessarily odd. In this talk, we show the existence of nontrivial solutions for a class of quasilinear elliptic problems with variable exponents in $\mathbb R^{N}$ by using mountain pass theorem and fountain theorem.
2010 Mathematics Subject Classification: 35D30, 35J60, 35J70, 35J92, 35P30, 46E35, 47J10
Key Words and Phrases: $p(x)$-Laplacian, variable exponent Lebesgue-Sobolev spaces, weak solution, mountain pass theorem, fountain theorem
- ⋅ 25th-A-09:20 − 09:40 On a Keller-Segel system with logarithmic sensitivity and non-diffusive chemical (Jaewook Ahn, Kyungkeun Kang)
- 안재욱*(연세대), 강경근(연세대)
Jaewook Ahn*, Yonsei University, Kyungkeun Kang, Yonsei University
We consider a chemotatic system with a logarithmic sensitivity and a non-diffusing chemical. We establish local regular solutions in time and give some characterizations on parameters and initial data for global solutions and blow-up in a finite time. We also prove that there does not exist finite time self-similar solution of the backward type.
2010 Mathematics Subject Classification: 35Kxx
Key Words and Phrases: Keller-Segel system, logarithmic sensitivity, non-diffusive chemical
- ⋅ 25th-A-09:50 − 10:10 Stability of Cauchy--Jensen type functional equation in quasi-$\beta$-normed space (Eunyoung Son, Hark-Mahn Kim)
- 손은영*(충남대), 김학만(충남대)
Eunyoung Son*, Chungnam National University, Hark-Mahn Kim, Chungnam National University
In this talk, we establish general solution of the following Cauchy--Jensen type functional equation $$f(\frac{x+y}{n}+z)+f(\frac{y+z}{n}+x)+f(\frac{z+x}{n}+y)=\frac{n+2}{n}[f(x)+f(y)+f(z)],$$ and then investigate the generalized Hyers--Ulam stability of the equation in quasi-$\beta$-normed spaces for any fixed nonzero integer $n$.
2010 Mathematics Subject Classification: 39B82
Key Words and Phrases: generalized stability, quasi-$\beta$-normed spaces, $(\beta, p)$-Banach spaces
- ⋅ 25th-A-10:10 − 10:30 Blow-up for the discrete reaction-diffusion equation on networks (Jae-Hwang Lee, Soon-Yeong Chung)
- 이재황*(서강대), 정순영(서강대)
Jae-Hwang Lee*, Sogang University, Soon-Yeong Chung, Sogang University
We discuss under what conditions do the blow-up occurs for the solutions of the reaction-diffusion on networks as follows: $$\begin{array}{ll} u_{t} - \Delta_{\omega} = u^{q}, & \hbox{in $S\times (0, \infty)$} \\ u=0, & \hbox{on $\partial S \times (0, \infty)$} \\ u(\cdot, 0)=u_{0}, & \hbox{on $S$,} \end{array}$$ where $S$ is a network with boundary $\partial S$. The main theorem states that (i) $q > 1$ and $y_{0}^{1-q} < \frac{\vert S \vert^{1-q}}{K}$ $\Rightarrow$ the solution blows up, (ii) $q > 1$ and $y_{0}^{1-q} \ge \frac{\vert S \vert^{1-q}}{K}$ $\Rightarrow$ the solution is global, and (iii) $0 < q \le 1$ $\Rightarrow$ the solution is global. In addition, when the solution blows up, we give estimates for the blow-up time and provide blow-up rate. Finally, we show some numerical experiment which illustrate the main results.
2010 Mathematics Subject Classification: 35K57
Key Words and Phrases: reaction-diffusion, discrete Laplacian, comparison principle, blow-up
- ⋅ 25th-B'-11:00 − 12:30 Chair: Minkyu Kwak (Chonnam National University)
- ⋅ 25th-B'-11:00 − 11:30 On Kirchhoff type wave equation with some boundaries (Daewook Kim)
- 김대욱(부산대)
Daewook Kim, Pusan National University
In this talk, firstly, I introduce the mathematical modeling for the system. Secondly, I'm concerned with the existence and uniqueness of global solutions and uniform decay of the total energy for the system associated with Kirchhoff type wave equation with some boundaries. Lastly, I give some numerical results for a special example.
2010 Mathematics Subject Classification: 35L05
Key Words and Phrases: Kirchhoff type wave equation, global solutions, uniform decay, mathematical model
- ⋅ 25th-B'-11:30 − 12:00 Compressible viscous isentropic Navier-Stokes flows through a non-convex polyhedral cylinder in $\mathbb R^3$ (Oh Sung Kwon, Jae Ryong Kweon)
- 권오성*(국가수리과학연구소), 권재룡(포항공대)
Oh Sung Kwon*, NIMS, Jae Ryong Kweon, POSTECH
We investigate edge singularity and regularity for solutions of the compressible viscous Navier-Stokes equations in a non-convex polyhedral cylinder in $\mathbb R^3$. We split the corner singularity occurring at the non-convex edge from the velocity vector and show the $H^{2,q}\times H^{1,q}$-regularity for the velocity remainder and the local sound speed where $3<q<1/(1-\pi/\omega)$ and $\omega\in(\pi,3\pi/2)$ is the opening angle of the edge. The coefficient of the corner singularity is constructed as a function along the non-convex edge and shown to be in the space $H^{2/q'-\pi/\omega,q}(\mathbb R)$. The local sound wave impact by the velocity vector corner singularity occurred at the non-convex edge is propagated into the region along the streamline emanated from the non-convex edge and its derivatives blow up across the streamline.
2010 Mathematics Subject Classification: 35Q30
Key Words and Phrases: compressible viscous flows, edge singularity, regularity
- ⋅ 25th-B'-12:00 − 12:30 Global existence of weak solutions for a Keller-Segel-fluid model with nonlinear diffusion (Yun-Sung Chung, Kyungkeun Kang)
- 정윤성*(연세대), 강경근(연세대)
Yun-Sung Chung*, Yonsei University, Kyungkeun Kang, Yonsei University
In this talk, we consider a mathematical model that is originated
with dynamics of swimming bacteria, so called {\it Bacillus
subtilis}, which live in fluid and consume oxygen. To be more
precise, we study the Cauchy problem on the coupled
Keller-Segel-Navier-Stokes equations in $\mathbb R^d\times (0,T)$
with $0<T\le\infty$ and $d=2, 3$:
\begin{equation*}
\left\{
\begin{array}{l}
\partial_t n+u\cdot\nabla n=\Delta
n^{1+\alpha}-\nabla\cdot\left(\chi(c)n\nabla
c\right),
%&\mathrm{in}~\mathbb R^d\times(0,\infty)
\\
\vspace{-3mm}\\
\partial_t c+u\cdot\nabla c=\Delta c-\kappa(c)n,
%&\mathrm{in}~\mathbb R^d\times(0,\infty)
\\
\vspace{-3mm}\\
\partial_t u+\tau(u\cdot\nabla)u+\nabla p=\Delta
u-n\nabla\phi,
%&\mathrm{in}~\mathbb R^d\times{0,\infty}\\
\qquad {\rm{div}}\, u=0,
%&\mathrm{in}~\mathbb R^d\times(0,\infty)\\
%\left( n,c,u \right)_{|t=0}=\left(n_0,c_0,u_0\right), &\mathrm{on}~\mathbb R^d
\end{array}
\right.
\end{equation*}
where $n$, $c$, $u$ and $p$ are the cell density, oxygen
concentration, velocity field and pressure of the fluid,
respectively. We establish the global-in-time existence of weak
solutions and bounded weak solutions depending on some conditions of
parameters such as chemotatic sensitivity and consumption rate of
oxygen for certain range of diffusive exponents of cell density in
two and three dimensions.
2010 Mathematics Subject Classification: 35Q30, 35Q35
Key Words and Phrases: incompressible fluid, Keller-Segel model, nonlinear-diffusion
- ⋅ 25th-C-14:20 − 15:00 Chair: Eungil Ko (Ewha Womans University)
- ⋅ 25th-C-14:20 − 14:40 Rank-one perturbations of normal operators and hyponormality (Eun Young Lee, Il Bong Jung)
- 이은영*(경북대), 정일봉(경북대)
Eun Young Lee*, Kyungpook National University, Il Bong Jung, Kyungpook National University
Let $T=N+u\otimes v$ be a rank-one perturbation of a normal operator $N$ acting on a separable, infinite dimensional, complex Hilbert space $\mathcal{H}$. It is proved that the hyponormality of $T$ is equivalent to the normality of $T$. Some characterizations of hyponormality[normality] of $T$ are obtained.
2010 Mathematics Subject Classification: 47B20, 47A63, 47A55
Key Words and Phrases: normal operator, hyponormal operator, rank-one perturbation, commutator
- ⋅ 25th-C-14:40 − 15:00 Backward extensions of quadratically hyponormal weighted shifts (George R. Exner, Il Bong Jung, Mi Ryeong Lee , Sun Hyun Park )
- George R. Exner(Bucknell Univ.), 정일봉(경북대), 이미령(대구가톨릭대), 박선현*(경북대)
George R. Exner, Bucknell University, Il Bong Jung, Kyungpook National University, Mi Ryeong Lee, Catholic University of Daegu, Sun Hyun Park *, Kyungpook National University
Given the weight sequence for a subnormal recursively generated weighted shift on Hilbert space, one approach to the study of classes of operators weaker than subnormal has been to form a backward extension of the shift by prefixing weights to the sequence. We discusss backward extensions of quadratically hyponormal weighted shifts in this talk. Especially some backward $3$-step extensions of recursively weighted sequence $\alpha :1,1,\sqrt{x},(\sqrt{u},\sqrt{v},\sqrt{w})^{\wedge }$ of positive real numbers with $1\leq x\leq u\leq v\leq w$ are considered here. And the set of positive real numbers $x$ such that $W_{\alpha }$ is quadratically hyponormal for some $u,v$ and $w$ is described.
2010 Mathematics Subject Classification: 47B37, 47B20
Key Words and Phrases: weighted shifts, quadratic hyponormality, positive quadratic hyponormality, subnormal completion
- ⋅ 26th-E-11:10 − 11:40 Chair: Youngwoo Choi (Ajou University)
- ⋅ 26th-E-11:10 − 11:40 Traveling wave solutions to the half-wave equation (Daniele Garrisi, Vladimir Georgiev)
- Daniele Garrisi*(인하대), Vladimir Georgiev(Universita Degli Studi di Pisa)
Daniele Garrisi*, Inha University, Vladimir Georgiev, Universita Degli Studi di Pisa
We discuss the existence of traveling-wave solutions \begin{equation} \label{eq:3} \psi(t,x) = e^{-i\omega t} \phi(x - tv) \end{equation} to the zero mass evolution equation equation \begin{equation} \label{eq:2} i\partial_t\psi - \sqrt{-\Delta}\,\psi + |\psi|^{p - 1} \psi - |\psi|^{q - 1} \psi = 0,\quad 1 < p < q \end{equation} for velocities $ v $ such that $ |v| < 1 $. Equivalently, we need so solve the equation \begin{equation} \label{eq:1} \sqrt{-\Delta}\,\phi + i\nabla\phi\cdot v - |\phi|^{p - 1} \phi + |\phi|^{q - 1} \phi = \omega\phi,\quad 1 < p < q \end{equation} for some $ \omega $ and $ \phi $. The existence of such solutions can be obtained by showing that minimising sequences of the energy functional \begin{gather*} \mathcal{E}\colon H^{1/2} (\mathbb{R})\rightarrow \mathbb{R}\\ \mathcal{E}(\phi) = H_v (\phi) - \frac{1}{p + 1}\|\phi\|_{p + 1}^{p + 1} + \frac{1}{q + 1}\|\phi\|_{q + 1}^{q + 1}\\ H_v (\phi) = \frac{1}{2}\|\phi\|_{\dot{H}^{1/2} (\mathbb{R})}^ 2 + \frac{i}{2}\int\limits_{-\infty}^{+\infty} \overline{\phi}(x)\nabla \phi(x)\cdot v dx \end{gather*} over the constraint \begin{gather*} S(\lambda) = \{\phi\in H^{1/2} (\mathbb{R})\mid \|\phi\|_{L^ 2}^ 2 = \lambda\} \end{gather*} exhibit a concentration-compactness behaviour. The concentration-compactness, along with other features of positive solutions to \eqref{eq:1}, is an essential premise to the achievement of the orbital stability of the solutions in \eqref{eq:3}.
2010 Mathematics Subject Classification:
Key Words and Phrases: half wave equation, traveling waves, orbital stability
- Geometry
- ⋅ 25th-A-09:30 − 10:30 Chair: Jongsu Kim (Sogang University)
- ⋅ 25th-A-09:30 − 10:00 On the structure of Linearization of the scalar curvature (Seungsu Hwang, Gabjin Yun, Jeongwook Chang)
- 황승수*(중앙대), 윤갑진(명지대), 장정욱(단국대)
Seungsu Hwang*, Chung-Ang University, Gabjin Yun, Myong Ji University, Jeongwook Chang, Dankook University
On a compact $n$-dimensional manifold $M$, it is well-known that a critical point metric $g$ of the total scalar curvature restricted to the space of constant scalar curvature metrics of unit volume satisfies the critical point equation $z_g = D_gdf - (\Delta_g f)g - f r_g$ for a function $f$ with vanishing mean value. The right-hand side in this equation is nothing but the adjoint $s_g'^*$ of the linearization $s_g'$ of the total scalar curvature acting on functions. There are some results on the structure of the kernel space of $s_g'^*$ which plays important role in the geometry of the underlying manifold. In this paper, we study some geometric structure of a given manifold when the kernel space of $s_g'^*$ is nontrivial. As an application, we show that if there are two distinct solutions satisfying the critical point equation mentioned above, then $(M, g)$ should be Einstein. This generalizes our previous result to arbitrary dimension.
2010 Mathematics Subject Classification: 53C25
Key Words and Phrases: total scalar curvature, critical point metric, Einstein metric
- ⋅ 25th-A-10:00 − 10:30 The Erdos distance problem on spheres in the finite field setting (Doowon Koh)
- 고두원(충북대)
Doowon Koh, Chungbuk National University
In this talk, we study the cardinality of the distance set determined by two sets of the $d$-dimensional vector space $\mathbb F_q^d$ over the finite field $\mathbb F_q$ with $q$ elements. We prove that if $E$ and $F$ lie on spheres with non-zero radius and $|E||F|$ is much greater than $q^{d}$, then $E$ and $F$ determine a positive proportion of all possible distances. The key idea is to use a connection between distances with dot-products of two sets on spheres.
2010 Mathematics Subject Classification: 52C10, 11T23
Key Words and Phrases: the Erdos distance problem, Fourier transform, paraboloids
- ⋅ 25th-B-10:40 − 11:40 Chair: Juncheol Pyo (Pusan National University)
- ⋅ 25th-B-10:40 − 11:10 General affine developments (Joon-Sik Park)
- 박준식(부산외대)
Joon-Sik Park, Busan Univesity of Foreign Studies
Let $L(M)$ and $A(M)$ be the linear frame and the affine frame bundles over an $n$-dimensional $C^{\infty}$ manifold $M$ respectively. Let $\tilde {\gamma}:L(M)\hookrightarrow A(M)$ be the homomorphism of $L(M)$ into $A(M)$ with the group homomorphism $\gamma:GL(n;R) \hookrightarrow A(n;R)$, $\tilde {\omega}$ an arbitrarily given general affine connection in $A(M)$, and $\tilde {\gamma}^\star \tilde {\omega}=:\omega + \varphi$ ($\omega$ is the connection (form) in $L(M)$ corresponding to $\tilde {\omega}$, and $\varphi$ is the $R^n$-valued $1$-form on $L(M)$ which is corresponding to $\tilde {\omega}$). Let $\tau=x_t (0 \leqq t \leqq 1)$ be a $C^{\infty}$ curve in $M$, and $\tilde {\tau}^t_0$ the general affine parallel displacement of the affine tangent space $A_{x_t} (M)$ into $A_{x_0} (M)$ with respect to $\tilde {\omega}$ in $A(M)$. Then, the general affine development $\tilde C_t = \tilde {\tau}^t_0 (x_t)(0 \leqq t \leqq 1)$ of the curve $\tau=x_t (0 \leqq t \leqq 1)$ in $M$ into $A_{x_0} (M)$ is given as follows: $$ \tilde C_t = \tilde {\tau}^t_0 (x_t) = \tilde {\tau}^t_0 (\dot {x}_t) - {\tau}^t_0 (\dot {x}_t) \ (0 \leqq t \leqq 1),$$ where $\dot {x}_t:=d x_t/ dt$ and ${\tau}^t_0$ is the linear parallel displacement along $\tau$ from $x_t$ to $x_0$ with respect to $\omega$ in $L(M)$ which is corresponding to $\tilde \omega$ in $A(M)$.
2010 Mathematics Subject Classification: 53C07, 53A15, 53A25
Key Words and Phrases: (affine) development, general affine development, linear (affine, general affine) connection
- ⋅ 25th-B-11:10 − 11:40 Isometric Reeb flow and contact hypersurfaces in Hermitian symmetric space (Young Jin Suh)
- 서영진(경북대)
Young Jin Suh, Kyungpook National University
In this talk, first we introduce the classification of homogeneous hypersurfaces in some Hermitian symmetric spaces of rank 1 or rank 2. In particular, we give a full expression of the geometric structures for hypersurfaces in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$ or in complex hyperbolic two-plane Grassmannians $G_2^{*}({\mathbb C}^{m+2})$. \par Next by using the isometric Reeb flow we give a complete classification for hypersurfaces $M$ in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$, complex hyperbolic two-plane Grassmannians $G_2^{*}({\mathbb C}^{m+2})$ and a complex quadric ${\mathbb Q}^m$. \par Moreover, we introduce the notion of contact in Hermitian symmetric space and give a classification of contact hypersurfaces in Hermitian symmetric space like $G_2({\mathbb C}^{m+2})$, $G_2^{*}({\mathbb C}^{m+2})$ and ${\mathbb Q}^m$.
2010 Mathematics Subject Classification: 53C40
Key Words and Phrases: Hermitian symmetric space, horosphere, isometric Reeb flow, contact hypersurfaces
- ⋅ 25th-C-13:30 − 15:00 Chair: Dae Won Yoon (Gyeongsang National University)
- ⋅ 25th-C-13:30 − 13:50 Log canonical thresholds of complete intersection log del Pezzo surfaces (In-Kyun Kim, Jihun Park)
- 김인균*(포항공대), 박지훈(포항공대)
In-Kyun Kim*, POSTECH, Jihun Park, POSTECH
We compute the global log canonical thresholds of quasi-smooth well-formed complete intersection log del Pezzo surfaces of amplitude one in weighted projective spaces. As a corollary we show the existence of orbifold K\"ahler-Einstein metrics on many of them.
2010 Mathematics Subject Classification: 14J45
Key Words and Phrases: global log canonical threshold, weighted complete intersection, K\"ahler-Einstein metric, exceptional Fano variety
- ⋅ 25th-C-13:50 − 14:20 On standard imbeddings of hyperbolic spaces in the Minkowski space (Dong-Soo Kim)
- 김동수(전남대)
Dong-Soo Kim, Chonnam National University
We establish some characterizations of the standard imbeddings of hyperbolic spaces in the $(n+1)$-dimensional Minkowski space ${\mathbb L}^{n+1}$ with intrinsic and extrinsic properties such as the $n$-dimensional area of the sections cut off by hyperplanes, the $(n+1)$-dimensional volume of regions between parallel hyperplanes, and the $n$-dimensional surface area of regions between parallel hyperplanes. In the same manner, we give a partial affirmative answer to Question A suggested in [1], which is for the characterization of hyperspheres in the $(n+1)$-dimensional Euclidean space ${\mathbb E}^{n+1}$. \begin{thebibliography}{1} \bibitem{1} D. S. Kim and Y. H. Kim, {\it Some characterizations of spheres and elliptic paraboloids II}, Linear Algebra Appl. {\bf 438} (2013), no. 3, 1356--1364.\end{thebibliography}
2010 Mathematics Subject Classification: 53B25, 53B30
Key Words and Phrases: Gauss-Kronecker curvature, surface area, volume, strictly convex hypersurface, Minkowski space, hyperbolic space, parallel hypersurface, spacelike hypersurface
- ⋅ 25th-C-14:30 − 15:00 CMC weighted surfaces in a smooth metric measure space (Juncheol Pyo)
- 표준철(부산대)
Juncheol Pyo, Pusan National University
We explain what is a smooth metric measure space and why do we consider that. Then, we give some rigidity properties of foliations of CMC weighted hypersurfaces in a smooth metric measure space.
2010 Mathematics Subject Classification: 53C42
Key Words and Phrases: constant mean curvature, foliation, smooth metric measure space
- ⋅ 26th-D-09:30 − 10:40 Chair: Byung Hak Kim (Kyung Hee University)
- ⋅ 26th-D-09:30 − 10:00 Basic questions on geometric tube design (Hongtaek Hwang)
- 황홍택(금오공대)
Hongtaek Hwang, Kumoh Institute of Technology
We introduce some basic questions about geometric tube design.
2010 Mathematics Subject Classification: 51P99
Key Words and Phrases: geometric tube design
- ⋅ 26th-D-10:00 − 10:20 Real hypersurfaces with semi-parallel shape operator and structure Jacobi operator in complex two-plane Grassmannians (Young Jin Suh, Changhwa Woo, Hyunjin Lee)
- 서영진(경북대), 우창화*(경북대), 이현진(경북대)
Young Jin Suh, KyungPook National University, Changhwa Woo*, KyungPook National University, Hyunjin Lee, KyungPook National University
In this paper, we introduce notions of semi-parallel shape operator and structure Jacobi operator in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$. We prove that there does not exist a Hopf hypersurface in $G_2({\mathbb C}^{m+2})$, $m \geq 3$, with semi-parallel shape operator. Moreover, we assert another non-existence Hopf hypersurface with semi-parallel structure Jacobi operator.
2010 Mathematics Subject Classification: 53C40
Key Words and Phrases: real hypersurfaces, complextwo-plane Grassmannians, Hopf hypersurface, semi-parallel shape operator, semi-parallel structure Jacobi operator
- ⋅ 26th-D-10:20 − 10:40 Hopf hypersurfaces in complex two-plane Grassmannians with $\mathfrak F$-invariant Ricci tensor (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
Recently, Suh proved that a Hopf hypersurface $M$ in complex two-plane Grassmannians $G_{2}(\mathbb C ^{m+2})$, $m \geq 3$, with Reeb invariant Ricci tensor is locally congruent to an open part of a tube around a totally geodesic $G_{2}(\mathbb C ^{m+1})$ in $G_{2}(\mathbb C ^{m+2})$. Motivated this result, we consider another conditions for the Ricci tensor $S$, namely, $\mathfrak F$-invariant and Lie-vanishing Ricci tensor. Here the distribution $\mathfrak F$ is defined by $\mathfrak F = [\xi] \cup \mathfrak D^{\bot}$ where $[\xi]= \text{Span}\{ \xi \}$ and $\mathfrak D^{\bot}=\text{Span}\{ \xi_{\nu}, \nu=1,2,3\}$, respectively.
2010 Mathematics Subject Classification: 53C40, 53C15
Key Words and Phrases: complex two-plane Grassmannians, Hopf hypersurfaces, Ricci tensor, Lie-vanishing Ricci tensor, Reeb invariant Ricci tensor, $\mathfrak F$-invariant Ricci tensor
- ⋅ 26th-E-11:00 − 12:00 Chair: Keomkyo Seo (Sookmyung Women's University)
- ⋅ 26th-E-11:00 − 11:20 Finiteness of the total curvature of a non-closed curve in $\mathbb{R}^{n}$ (Peter B Gilkey, Chan Yong Kim, Jeong Hyeong Park)
- Peter B Gilkey(Univ. of Oregon), 김찬용*(성균관대), 박정형(성균관대)
Peter B Gilkey, University of Oregon, Chan Yong Kim*, Sungkyunkwan University, Jeong Hyeong Park, Sungkyunkwan University
We use the solution set of a real ordinary differential equation which has order $n\ge2$ to construct a smooth curve $\sigma$ in $\mathbb{R}^n$. We describe when $\sigma$ is a proper embedding of infinite length with finite total first curvature.
2010 Mathematics Subject Classification: 53A04
Key Words and Phrases: finite total curvature, ordinary differential equation, proper embedded curve
- ⋅ 26th-E-11:20 − 11:40 A contact metric manifold and its natural generalization (Jang Hyun Kim, Won Min Shin)
- 김장현*(성균관대), 신원민(성균관대)
Jang Hyun Kim*, Sungkyunkwan University, Won Min Shin, Sungkyunkwan University
We give a characterization of a contact metric manifold as a special almost contact metric manifold and discuss an almost contact metric manifold which is a natural generalization of the contact metric manifolds introduced by Y. Tashiro.
2010 Mathematics Subject Classification: 53B20, 53C20
Key Words and Phrases: almost contact metric manifold
- ⋅ 26th-E-11:40 − 12:00 Orthogonal almost complex structures on products of $S^{2n}$ (Yunhee Euh, Kouei Sekigawa)
- 어윤희*(국가수리과학연구소), Kouei Sekigawa(Niigate Univ.)
Yunhee Euh*, NIMS, Kouei Sekigawa, Niigate University
We discuss the integrability of orthogonal almost complex structures on Riemannian products of even-dimensional spheres and give a partial answer to the question raised by E. Calabi concerning the existence of a complex structure on $S^{2}\times S^{4}$.
2010 Mathematics Subject Classification: 53C15, 53C21, 53C30
Key Words and Phrases: orthogonal complex structure, even-dimensional sphere
- Topology
- ⋅ 25th-A-09:00 − 10:30 Chair: Namjip Koo (Chungnam National University)
- ⋅ 25th-A-09:00 − 09:20 Shadowable chain components and hyperbolicity (Junmi Park)
- 박준미(충남대)
Junmi Park, Chungnam National University
We show that $C^1$-generically, any shadowable chain component of a $C^1$-vector field on a compact smooth manifold containing a hyperbolic periodic orbit is hyperbolic if it is locally maximal.
2010 Mathematics Subject Classification: 37C50, 34D10
Key Words and Phrases: limit shadowing, hyperbolic, chain component
- ⋅ 25th-A-09:20 − 09:40 Measure expansive maps on compact smooth manifolds (Keonhee Lee, Jiweon Ahn)
- 이건희(충남대), 안지원*(충남대)
Keonhee Lee, Chungnam National University, Jiweon Ahn*, Chungnam National University
In this talk, we study the dynamics of measure expansive maps on compact smooth manifolds. In particular, we show that the $C^1$-interior of the set of positively measure expansive maps is equivalent to the set of expanding maps.
2010 Mathematics Subject Classification: 37C50, 37A05
Key Words and Phrases: measure expansive, positively measure expansive map, expanding map
- ⋅ 25th-A-09:50 − 10:10 Positively expansive flows on compact smooth manifolds (Seunghee Lee)
- 이승희(충남대)
Seunghee Lee, Chungnam National University
In this talk, we study the dynamics of the flows which belongs to the $C^1$-interior of the set of positively expansive flows on a compact smooth manifold.
2010 Mathematics Subject Classification: 37C10, 37C50
Key Words and Phrases: positively expansiveness, flows
- ⋅ 25th-A-10:10 − 10:30 A parity and a polynomial invariant for virtual links (Young-Ho Im, Kyoung Il Park)
- 임영호(부산대), 박경일*(한국과학영재학교)
Young-Ho Im, Pusan National University, Kyoung Il Park*, Korea Science Academy
We introduce a parity of classical crossings of virtual link diagrams which extends the Gaussian parity of virtual knot diagrams and the odd writhe of virtual links that extends that of virtual knots introduced by Kauffman. Also, we introduce a multi-variable polynomial invariant for virtual links by using the parity of classical crossings, which refines the index polynomial. As consequences, we give some properties of our invariant, and raise some examples.
2010 Mathematics Subject Classification: 57M25
Key Words and Phrases: virtual link, parity, odd writhe, multi-variable index polynomial, invertibility, amphicheirality
- ⋅ 25th-B-10:40 − 11:40 Chair: Yunhi Cho (University of Seoul)
- ⋅ 25th-B-10:40 − 11:10 Stably expansive homoclinic classes of $C^1$-vector fields (Manseob Lee, Keonhee Lee)
- 이만섭*(목원대), 이건희(충남대)
Manseob Lee*, Mokwon University, Keonhee Lee, Chungnam National University
Let $M$ be a closed $n$-dimensional smooth Riemmanian manifold, and let $X$ be a $C^1$-vector field of $M.$ Let $\gamma$ be a hyperbolic closed orbit of the flow $X_t.$ In this paper, we show that (i) the chain recurrent set $\mathcal{R}(X)$ of $X$ is $C^1$-stably expansive for flows if and only if $X_t$ satisfies both Axiom A and the no-cycle condition. (ii) the homoclinic class $H_X(\gamma)$ of $X$ containing $\gamma$ is $C^1$-stably expansive for flows if and only if $H_X(\gamma)$ is hyperbolic.
2010 Mathematics Subject Classification: 37C20, 37C50, 37C40, 34D05
Key Words and Phrases: expansive, homolinic class, chain recurrent, hyperbolic
- ⋅ 25th-B-11:10 − 11:40 On the cohomology of certain real toric varieties (Suyoung Choi, Hanchul Park, Kyoung Sook Park)
- 최수영(아주대), 박한철*(아주대), 박경숙(아주대)
Suyoung Choi, Ajou University, Hanchul Park*, Ajou University, Kyoung Sook Park, Ajou University
The family of nestohedra contains many famous simple polytopes including permutohedra, associahedra, and cyclohedra. For each of them there corresponds a real toric variety. In this talk, we try to present a formula for the Betti numbers of the real toric variety, generalizing the work of A. Henderson computing the rational cohomology of the real Coxeter toric variety of type A.
2010 Mathematics Subject Classification: 57N65, 55U10, 06B30
Key Words and Phrases: nestohedron, toric topology, real toric variety, hypergraph invariant, poset topology
- ⋅ 26th-D-09:30 − 10:30 Chair: Ki-Heon Yun (SungShin Women's University)
- ⋅ 26th-D-09:30 − 10:00 Smoothings of singularities and symplectic surgery (Heesang Park, Andras I. Stipsicz)
- 박희상*(고등과학원), Andras I. Stipsicz(Renyi Institute of Mathematics)
Heesang Park*, Korea Institute for Advanced Study, Andras I. Stipsicz, Renyi Institute of Mathematics
Suppose that $C=(C_1,\ldots, C_m)$ is a configuration of 2-dimensional symplectic submanifolds in a symplectic 4-manifold $(X,\omega)$ with connected, negative definite intersection graph $\Gamma_C$. We show that by replacing an appropriate neighborhood of $\cup C_i$ with a smoothing $W_S$ of a normal surface singularity $(S, 0)$ with resolution graph $\Gamma_C$, the resulting 4-manifold admits a symplectic structure. This operation generalizes the rational blow-down operation of Fintushel-Stern for other configurations, and therefore extends Symington's result about symplectic rational blow-downs.
2010 Mathematics Subject Classification: 57R17, 14E15, 14J17
Key Words and Phrases: normal surface singularity, symplectic surgery, open book decomposition
- ⋅ 26th-D-10:00 − 10:30 Toric origami manifolds and multi-fans (Mikiya Masuda, Seonjeong Park)
- Mikiya Masuda(Osaka City University), 박선정*(국가수리과학연구소)
Mikiya Masuda, Osaka City University, Seonjeong Park*, NIMS
The notion of a toric origami manifold, which weakens the notion of a symplectic toric manifold, was introduced by Cannas da Silva-Guillemin-Pires and they show that toric origami manifolds bijectively correspond to origami templates via moment maps, where an origami template is a collection of Delzant polytopes with some folding data. Like a fan is assoicated to a Delzant polytope, a multi-fan introduced by Hattori-Masuda can be associated to an oriented origami template. In this talk, we discuss their relationship and show that any simply connected compact smooth 4-manifold with a smooth action of $T^2$ can be a toric origami manifold. We also characterize products of even dimensional spheres which can be toric origami manifolds.
2010 Mathematics Subject Classification: 57S15, 53D20, 14M25
Key Words and Phrases: toric origami manifold, origami template, Delzant polytope, multi-fan, symplectic toric manifold, moment map, torus action
- Probability and Statistics
- ⋅ 25th-A-09:00 − 10:10 Chair: Hyun Jae Yoo (Hankyong University)
- ⋅ 25th-A-09:00 − 09:20 Exponential convergence rates for weighted sums in noncommutative probability space (Byoung Jin Choi, Un Cig Ji)
- 최병진*(충북대), 지운식(충북대)
Byoung Jin Choi*, Chungbuk National University, Un Cig Ji, Chungbuk National University
We study exponential convergence rates for weighted sums of successively independent random variables in a noncommutative probability space of which the weights are in a von Neumann algebra which give upper bounds of large deviation principle for weighted sums in noncommutative probability space. As applications, we study exponential convergence rates for weighted additive convolution sums of probability measures corresponding to the free additive convolution.
2010 Mathematics Subject Classification: 46L53
Key Words and Phrases: noncommutative probability space, weighted sum, weak law of large numbers, exponential convergence rates, large deviations
- ⋅ 25th-A-09:20 − 09:40 Using hidden Markov Model for European option pricing (Youngchul Han, Jungwoo Lee, Jeong-Hoon Kim)
- 한영철*(연세대), 이정우(연세대), 김정훈(연세대)
Youngchul Han*, Yonsei University, Jungwoo Lee, Yonsei University, Jeong-Hoon Kim, Yonsei University
In this paper, we proposed model for European option pricing using predicted cumulative probability distribution of hidden Markov Model. The Black-Scholes model is based on Normal distribution and the model's main drawback is fat-tail problem. Because of fat-tail problem, the implied volatility surface is shaped skew, smirk or other forms. To correct model's drawback, we propose predicted cumulative distribution using hidden Markov Model and simulate the underlying price paths using Monte Carlo method. In the results, our model reflect current implied volatility surface of market satisfactorily.
2010 Mathematics Subject Classification: 46N30
Key Words and Phrases: hidden Markov Model, Monte Carlo simulation, option pricing
- ⋅ 25th-A-09:50 − 10:10 Forecasting time series based on $f$-transform (Woo-Joo Lee, Hae Kyung Kim)
- 이우주*(연세대), 김해경(연세대)
Woo-Joo Lee*, Yonsei University, Hae Kyung Kim, Yonsei University
In this paper, we propose a new methodology for forecasting time series which is based on combination of two techniques: fuzzy transform and fuzzy time series
2010 Mathematics Subject Classification: 62A86
Key Words and Phrases: $f$-transform, fuzzy time series
- ⋅ 25th-B'-11:00 − 12:10 Chair: Jaeseong Heo (Hanyang University)
- ⋅ 25th-B'-11:00 − 11:30 Large deviations for affine diffusion processes on canonical state spaces (Chulmin Kang, Wanmo Kang)
- 강철민*(국가수리과학연구소), 강완모(카이스트)
Chulmin Kang*, NIMS, Wanmo Kang, KAIST
In this talk, I will introduce the large deviation principle for affine diffusion processes with initial values in the interior of the state space. We approach this problem in two different ways. In the first approach, we first prove the large deviation principle for finite dimensional distributions, and then use it to establish the sample path large deviation principle. For this approach, a more careful examination of the affine transform formula is required. The second approach exploits the exponential martingale method of Donati-Martin et al. for the squares of Ornstein-Uhlenbeck processes. We provide an application to importance sampling of affine diffusion models.
2010 Mathematics Subject Classification: 60G99
Key Words and Phrases: affine processes, large deviation principle, affine transform formula
- ⋅ 25th-B'-11:40 − 12:10 Series expansion of super operators in terms of quantum white noise derivatives (Un Cig Ji)
- 지운식(충북대)
Un Cig Ji, Chungbuk National University
In this talk, we study super (continuous linear) operators acting on the space of white noise (generalized) operators. We introduce a notion of quantum white noise derivatives of white noise operators. Then every super operator admits a series expansion in terms of quantum white noise derivatives.
2010 Mathematics Subject Classification: 60H40
Key Words and Phrases: Fock space, generalized operator, Fock expansion, annihilation derivative, creation derivative, super operator
- Applied Mathematics
- ⋅ 25th-A-09:00 − 10:40 Chair: Jun Seok Kim (Korea University)
- ⋅ 25th-A-09:00 − 09:20 Recovery of missing samples for generalized oversampling (Sinuk Kang, Kil Hyun Kwon, Dae Gwan Lee)
- 강신욱(국가수리과학연구소), 권길헌(카이스트), 이대관*(카이스트)
Sinuk Kang, NIMS, Kil Hyun Kwon, KAIST, Dae Gwan Lee*, KAIST
It is well known that for band-limited signals, any finitely many missing samples can be recovered whenever the signal is oversampled at a rate higher than the minimum Nyquist rate. In this work, we consider the problem of recovering missing samples from multi-channel oversampling in a shift-invariant space. We find conditions under which any finite or infinite number of missing samples can be recovered.
2010 Mathematics Subject Classification: 42C15, 94A20
Key Words and Phrases: recovery of missing samples, shift-invariant space, oversampling, multi-channel sampling, frame
- ⋅ 25th-A-09:20 − 09:40 Efficient numerical method for pricing an American put option (Beom Jin Kim, Yong-Ki Ma)
- 김범진*(연세대), 마용기(공주대)
Beom Jin Kim*, Yonsei Univeristy, Yong-Ki Ma, Kongju National University
We present a efficient numerical method to find the optimal exercise boundary in an American put option. We formulate an intermediate function with the fixed free boundary that has Lipschitz character near optimal exercise boundary. Employing it, we can easily determine the optimal exercise boundary by solving a quadratic equation in time-recursive way. We also present several numerical results which illustrate comparison to other methods.
2010 Mathematics Subject Classification: 65C30
Key Words and Phrases: efficient numerical method, optimal exercise boundary, American put option, intermediate function, fixed free boundary
- ⋅ 25th-A-09:40 − 10:00 Valuation of European three-asset options with space and time adaptive finite difference method (Darae Jeong, Junseok Kim)
- 정다래*(고려대), 김준석(고려대)
Darae Jeong*, Korea University, Junseok Kim, Korea University
In this paper, we propose the efficient adaptive technique for pricing European three-asset options. To validate the performance of the proposed new method, we compare the numerical results with those from Monte Carlo simulation in terms of computational cost and accuracy.
2010 Mathematics Subject Classification: 65-XX, 91G60
Key Words and Phrases: option pricing, Black-Scholes model, adaptive technique
- ⋅ 25th-A-10:00 − 10:20 Fast swaption pricing in Gaussian term structure model (Sung Chan Shin, Jaehyuk Choi)
- 신성찬*(카이스트), 최재혁(MIT)
Sung Chan Shin*, KAIST, Jaehyuk Choi, MIT
We propose a fast and accurate numerical method to price European swaptions in a multi-factor Gaussian term structure model that can speed up the calibration to the volatility surface. Pricing an interest rate option in such a multi-factor term structure model involves evaluating a multi-dimensional integration of the underlying claim's payoff on a domain where the payoff is positive. In our method, we approximate the exercise boundary of the state variables as a hyperplane tangent to the maximum probability point on the boundary and simplify the multi-dimensional integration into an analytic form. The maximum probability point can be found using the gradient descent method. We show that the quality of our method is superior to the result of previous studies by directly comparing them to the exact price obtained from the numerical integration.
2010 Mathematics Subject Classification: 91G30
Key Words and Phrases: Gaussian term structure model, volatility surface calibration, fast swaption pricing, swaption analytics
- ⋅ 25th-A-10:20 − 10:40 An enhanced numerical method for pricing Asian options based on binomial tree method (Yunju Jeong, Kyoung-Sook Moon, Hongjoong Kim)
- 정윤주*(고려대), 문경숙(가천대), 김홍중(고려대)
Yunju Jeong*, Korea University, Kyoung-Sook Moon, Gachon University, Hongjoong Kim, Korea University
We propose an enhanced numerical method for pricing path-dependent Asian options based on binomial lattice model. The method provides more accurate option value with rapid convergence. The standard binomial tree method for Asian option evaluates option prices at each node of the tree by setting representative averages. Our algorithm redefines representative averages by considering an interval about each node along the stock price axis and then evaluates option prices by computing the representative averages over each interval. We apply the method to the discrete and continuous monitored Asian options.
2010 Mathematics Subject Classification:
Key Words and Phrases: binomial lattice model, Asian option
- ⋅ 25th-B'-11:00 − 12:30 Chair: Chohong Min (Ewha Womans University)
- ⋅ 25th-B'-11:00 − 11:20 Asymptotical property of a diffusive predator-prey model with stage structure on prey (Seong Lee, Inkyung Ahn)
- 이성*(고려대), 안인경(고려대)
Seong Lee*, Korea University, Inkyung Ahn, Korea University
In this talk, we study the asymptotic property of a diffusive delayed predator-prey model with Beddington-DeAngelis type functional response under homogeneous Neumann boundary conditions, where the discrete time delay covers the period from the birth of immature preys to their maturity. We establish the threshold dynamics of their permanence and the extinction of the predator. Furthermore, we also give sufficient conditions for the global attractiveness and the local stability of the semi-trivial and coexistence equilibria.
2010 Mathematics Subject Classification: 35K40, 35K57, 92D25
Key Words and Phrases: diffusive predator-prey model, Beddington-DeAngelis functional response, time delay, stage structure on prey
- ⋅ 25th-B'-11:20 − 11:40 The phase--field model of dynamical interfaces (Dongsun Lee, Junseok Kim)
- 이동선*(고려대), 김준석(고려대)
Dongsun Lee*, Korea University, Junseok Kim, Korea University
We talk about a phase--field model. The model is an alternative way to approximate the sharp interface models such as mean curvature flow, volume-preserving mean curvature flow, Mullins-Sekerka flow and surface diffusion flow. In this talk, we observe the motions driven by the phase--field model and how they evolve.
2010 Mathematics Subject Classification: 65N06
Key Words and Phrases: phase--field model, geometric evolution
- ⋅ 25th-B'-11:40 − 12:10 Finite difference solver on GPUs: parallel cyclic reduction method (Jaemin Shin, Sungki Kim, Junseok Kim)
- 신재민*(고려대), 김성기(고려대), 김준석(고려대)
Jaemin Shin*, Korea University, Sungki Kim, Korea University, Junseok Kim, Korea University
Graphics Processing Units (GPUs) are specialized for massively parallel computations and those could offer tremendous performance in computing applications. If algorithms are well fit with the GPU's architecture, the computational time can be remarkably reduced. In this work, we study an operator splitting method for solving the heat and Black-Scholes equations. The governing equation is implicitly discretized in time, based on the finite difference method, and the operator splitting method leads to a tridiagonal matrix system. We discuss a parallel algorithm, which is called a parallel cyclic reduction method, to solve this matrix system by using GPUs.
2010 Mathematics Subject Classification: 65Y05
Key Words and Phrases: parallel computing, cyclic reduction method, operator splitting method, Black-Scholes equation
- ⋅ 25th-B'-12:10 − 12:30 Numerical method for solving asymmetric nozzle jet flows impinging on a flat wall using boundary integral equation formulation (Sung Sic Yoo, Do Wan Kim)
- 유성식*(인하대), 김도완(인하대)
Sung Sic Yoo*, Inha University, Do Wan Kim, Inha University
Asymmetric nozzle jet flows impinging on a flat wall are considered. Although the governing equation is harmonic, the fact that the impinging jet has free boundaries makes the problem highly non-linear. In this paper, we propose an efficient algorithm to find the free boundary of the asymmetric jet. Based on the boundary integral equations, our algorithm can find the gradient toward the solution from an assumed free boundary. Particularly, by considering the far-field behavior of solution, our algorithm can deal with the entire computational domain instead of truncated one. In the numerical results, the force exerted on the wall by the pressure and numerical calculations of diverse examples are shown.
2010 Mathematics Subject Classification: 76B07, 76B10
Key Words and Phrases: asymmetric jet, free boundary, boundary integral equations
- ⋅ 25th-C-14:20 − 14:50 Chair: YunKyong Hyon (NIMS)
- ⋅ 25th-C-14:20 − 14:50 A survey on graph wavelets (Sinuk Kang)
- 강신욱(국가수리과학연구소)
Sinuk Kang, NIMS
Thanks to the recent development of computer and communication sciences, an enormous amount of data with a graph-like structure has been produced and stored. Those data are shown to be useful for getting new insights into the structure or into a certain phenomena on the structure. For an efficient way of dealing with the data, there have been several approaches to construct wavelets on a graph. We investigate the pros and cons of the graph wavelets.
2010 Mathematics Subject Classification: 42C40
Key Words and Phrases: graph, wavelets, survey
- Mathematical Education
- ⋅ 25th-A-09:00 − 10:10 Chair: Ik Pyo Kim (Daegu University)
- ⋅ 25th-A-09:00 − 09:20 Development of Sage Grapher and its applications (Shaowei Sun, Victoria Lang, Sanggu Lee)
- Shaowei Sun*(성균관대), Victoria Lang(성균관대), 이상구(성균관대)
Shaowei Sun*, Sungkyunkwan University, Victoria Lang, Sungkyunkwan University, Sanggu Lee, Sungkyunkwan University
As a visual expression of mathematics, graphs make it easy to visualize complex mathematical concepts. The Sage Grapher that we have developed can be used as a supplementary, pedagogical tool for various math courses. Graphs from Sage Grapher can be easily saved on the server so students can review and utilize them at their convenience. On the mobile Sage Grapher, a total of eight different versions have been developed: general function grapher, parametric function grapher, implicit function grapher, etc. We will demonstrate the capabilities of these new and exciting math tools. Manipulating and understanding these graphers can serve as an asset for all math students, regardless of age or level. http://matrix.skku.ac.kr/grapher-html/sage-grapher.html
2010 Mathematics Subject Classification: 97A60, 97U70
Key Words and Phrases: Sage Grapher, mathematical visualization, calculus, linear algebra
- ⋅ 25th-A-09:20 − 09:40 Inquiring for inquiry-based learning in flipped classroom: multivariable calculus (YoungGon Bae, Oh Nam Kwon)
- 배영곤*(서울대), 권오남(서울대)
YoungGon Bae*, Seoul National University, Oh Nam Kwon, Seoul National University
다변수미적분학은 자연,공학,상경계열 대학생들에게 필요한 기초소양 및 상위 전공과목의 선수과목으로서 대다수의 대학에 필수강좌로 개설되어 있다. 또한, 다변수미적분학은 심도있는 이해를 필요로 하는 개념이 집약된 과목으로서 학생들에게 비교적 난이도가 높고 학업부담이 큰 과목으로 여겨지고 있으며, 교수자에 의한 설명식 강의가 주를 이루고 있다. 역전학습(flipped learning, flipped classroom)은 설명식 강의를 디지털 콘텐츠로 대체하고 교실 수업에서는 과제, 토의, 활동 등의 다양한 학습 기회를 제공하는 대안적 수업방법으로서 국내외 대학에서 많은 관심이 집중되고 있다. 본 발표에서는 역전학습의 도입을 통해 온라인 강의와 탐구기반활동 중심의 오프라인 수업으로 구성된 다변수 미적분학 강좌의 운영 사례를 소개하고자 한다.
2010 Mathematics Subject Classification: 97D40
Key Words and Phrases: inquiry-based learning, multivariable calculus, flipped classroom, flipped learning
- ⋅ 25th-A-09:50 − 10:10 Development of Sage Cell for linear algebra (Kyung-Won Kim, Sang-Gu Lee, Jaehwa Lee)
- 김경원*(성균관대), 이상구(성균관대), 이재화(한림대)
Kyung-Won Kim*, Sungkyunkwan University, Sang-Gu Lee, Sungkyunkwan University, Jaehwa Lee, Hallym University
Linear Algebra is the first abstract mathematics subject for most of new college students. Hence, most of students face some difficulties to deal with various novel mathematical concepts. So we have developed various Sage Cells and applied them to our classes. Sage Cell is an open-source, scalable, and easy-to-use web interface for Sage. In this talk, we introduce our Sage Cells for introductory linear algebra course.\\ http://matrix.skku.ac.kr/la-lab
2010 Mathematics Subject Classification: 97H60
Key Words and Phrases: Sage Cell, linear algebra, Sage
- ⋅ 25th-B'-11:00 − 12:30 Chair: Joonkook Shin (Chungnam National University)
- ⋅ 25th-B'-11:00 − 11:30 Introduction of Korean math books in 1890 -- 1945 (Sang-Gu Lee, Jae Hwa Lee, Young-Gu Kim)
- 이상구*(성균관대), 이재화(한림대), 김영구(성균관대)
Sang-Gu Lee*, Sungkyunkwan University, Jae Hwa Lee, Hallym University, Young-Gu Kim, Sungkyunkwan University
본 발표에서는 한국연구재단의 `한국 근대수학사' 연구과제를 수행하는 과정에서 발굴한 `한국 근대수학 사료' 수백 권 중, 일부를 동영상을 만들어 소개한다. 이는 아래 주소와 최근 발간된 책 `한국 근대수학의 개척자들' 부록에서 확인할 수 있다.
\smallskip
\noindent 한국 근대수학의 개척자들 http://matrix.skku.ac.kr/K-Math-History/index.htm \\
한국 근대수학의 개척자들 (사료 1) http://youtu.be/HrlcJ5xK33M\\
사료 모음 (수리-산술신서) http://www.youtube.com/watch?v=sitRMnPvfUY\\
사료 모음 1 http://www.youtube.com/watch?v=vu2mYOH3czs\\
사료 모음 2 http://www.youtube.com/watch?v=2Plf4w5421E\\
사료 모음 3 (고등 산학신편 전편) http://www.youtube.com/watch?v=Z-9xnPGZ72c\\
사료 모음 4 (일제시대) http://www.youtube.com/watch?v=6p4YN-jLAmo\\
사료 모음 5 (해방직후) http://www.youtube.com/watch?v=HrmPnq7f6X0\\
사료 모음 6 (전시 및 책의 이미지) http://www.youtube.com/watch?v=KhXHQkkdyZ4
2010 Mathematics Subject Classification: 01A55, 01A60, 01A13, 97A30
Key Words and Phrases: modern mathematics(근대수학), Korean modern mathematics(한국 근대수학), Sang-Seol Lee(이상설), Rimhak Ree(이임학), Il-Seon Yoo(유일선), Gyu-Dong Choi(최규동), Han-sik Park(박한식), Sung-Hun Lee(이성헌), Kwang Jo Koo(구광조), Jung-K
- ⋅ 25th-B'-11:30 − 11:50 3D-printing with Sage for linear algebra and calculus (Jae Yoon Lee, Yeongjun Lim, Shaowei Sun , Sang-Gu Lee)
- 이재윤*(성균관대), 임영준(성균관대), Shaowei Sun (성균관대), 이상구(성균관대)
Jae Yoon Lee*, Sungkyunkwan University, Yeongjun Lim, Sungkyunkwan University, Shaowei Sun , Sungkyunkwan University, Sang-Gu Lee, Sungkyunkwan University
Recently, 3D-printing technology has been globally emphasized and is now being called `the 3rd Industrial Revolution'. South Korea recently started basic research on 3D-printing technology. We started to look towards possibilities of using 3D-printing for physical representations of mathematics within Korea. This presentation will show how we utilized 3D-printing and generated various 3D objects for Calculus and Linear Algebra courses.
2010 Mathematics Subject Classification: 68N30, 97H60, 97U70
Key Words and Phrases: 3D-printing, Sage, linear algebra, calculus
- ⋅ 25th-B'-12:00 − 12:30 Learning motivation and self-efficacy by feedback in engineering math education (Kim Young Sik)
- 김영식(한양대)
Kim Young Sik, Hanyang University
We investigate the blearning motivation and self-efficacy by feedback in engineering math education in university.
2010 Mathematics Subject Classification: 97C40, 97D10
Key Words and Phrases: ME
- Mathematics for Information Sciences
- ⋅ 25th-A-09:00 − 10:30 Chair: Young Soo Kwon (Yeungnam University)
- ⋅ 25th-A-09:00 − 09:20 Catalan number related compositions (Seoungji Hong, SeungKyung Park)
- 홍성지*(연세대), 박승경(연세대)
Seoungji Hong*, Yonsei University, SeungKyung Park, Yonsei University
We study Catalan number related compositions and its application.
2010 Mathematics Subject Classification: 05A15
Key Words and Phrases: Catalan numbers, compositions, rooted plane trees
- ⋅ 25th-A-09:20 − 09:40 Enumeration of Schr\"{o}der families by type (Suhyung An, Sangwook Kim, Sen-Peng Eu)
- 안수형*(연세대), 김상욱(전남대), Sen-Peng Eu(National Univ. of Kaohsiung)
Suhyung An*, Yonsei University, Sangwook Kim, Chonnam National University, Sen-Peng Eu, National University of Kaohsiung
Schr\"{o}der paths, sparse noncrossing partitions, and partial horizontal strips and three classes of Schr\"{o}der objects which carry a notion of type. We provide type-preserving bijections among these objects and an explicit formula which enumerates these objects according to type and length. We also define a notion of connectivity for these objects and discussion an analogus formula which counts connected objects by type.
2010 Mathematics Subject Classification: 05A15
Key Words and Phrases: Schr\"{o}der paths, sparse noncrossing partitions, partial horizontal strips
- ⋅ 25th-A-09:50 − 10:10 Generalized small Schr\"oder numbers (JiSun Huh, SeungKyung Park)
- 허지선*(연세대), 박승경(연세대)
JiSun Huh*, Yonsei University, SeungKyung Park, Yonsei University
We study a generalization of small Schr\"{o}der paths in the sense of arbitrary sizes of steps. We combinatorially show that the number of generalized small Schr\"{o}der paths is equal to $\sum_{k=1}^{n}N(n,k)5^{n-k}$ for $n \geq 1$.
2010 Mathematics Subject Classification: 05A15
Key Words and Phrases: small Schroder paths, Narayana polynomial, colored Dyck paths
- ⋅ 25th-A-10:10 − 10:30 Avoiding permutations and the Narayana numbers (Youngja Park, Seungkyung Park)
- 박영자*(연세대), 박승경(연세대)
Youngja Park*, Yonsei University, Seungkyung Park, Yonsei University
We study $132$-avoiding permutations that also avoid $(2r+1)(2r+2)\cdot 12$ but contain $(2r-1)(2r)\cdot 12$ pattern. We find an algorithm to get an identity between the number of these permutations and the $\textit{Narayana numbers}$. Also, a generalization of these permutations is obtained.
2010 Mathematics Subject Classification: 05A15
Key Words and Phrases: avoiding permutation, Narayana number
- ⋅ 25th-B-10:40 − 11:10 Chair: Seog-Jin Kim (Konkuk University)
- ⋅ 25th-B-10:40 − 11:10 Some results on Carryless Arithmetic Mod 10 (Joon Yop Lee, Jon-Lark Kim)
- 이준엽(포항공대), 김종락*(서강대)
Joon Yop Lee, POSTECH, Jon-Lark Kim*, Sogang University
We study carryless arithmetic mod 10 introduced by Applegate, LeBrun, and Sloane. We know that this arithmetic corresponds to the algebra in $\mathbb Z_{10}[x]$. In particular, we give an explicit formula for the number of $k$-digit carryless $m$th powers in $\mathbb Z_{10}[x]$ ($m \ge 2$), which was known only for $m=2$. We give a complete characterization of the Pythagorean triples $(f(x), g(x), h(x))$ in $\mathbb Z_{10}[x]$. We also prove the falsity of the analog of Fermat's Last Theorem in carryless arithmetic mod $10$, that is, there are nonzero polynomials $f(x)$, $g(x)$, and $h(x)$ in $\mathbb{Z}_{10}[x]$ such that $f(x)^m + g(x)^m = h(x)^m$ in for any $m\ge 3$. Furthermore, we give a complete solution to $f(x)^{2^n}+g(x)^{2^n}=h(x)^{2^n}$ for any $n\ge 2$.
2010 Mathematics Subject Classification: 05A15
Key Words and Phrases: carryless arithmetic, integers mod 10
- ⋅ 25th-C-13:30 − 15:00 Chair: Gi-Sang Cheon (SungKyunKwan University)
- ⋅ 25th-C-13:30 − 13:50 On the matrix sequence $\{\Gamma(A^m)\}_{m=1}^\infty$ for a Boolean matrix $A$ whose digraph is linearly connected (Jihoon Choi, Suh-Ryung KIM)
- 최지훈*(서울대), 김서령(서울대)
Jihoon Choi*, Seoul National University, Suh-Ryung KIM, Seoul National University
In this talk, we extend the results given by Park~{\em et al.}~\cite{ppk} by studying the convergence of the matrix sequence $\{\Gamma(A^m)\}_{m=1}^\infty$ for a matrix $A \in \mathcal{B}_n$ the digraph of which is linearly connected with an arbitrary number of strong components. In the process for generalization, we concretize ideas behind their arguments. We completely characterize $A$ for which $\{\Gamma(A^m)\}_{m=1}^\infty$ converges. Then we find its limit when all of the irreducible diagonal blocks are of order at least two. We go further to characterize $A$ for which the limit of $\{\Gamma(A^m)\}_{m=1}^\infty$ is a $J$ block diagonal matrix. All of these results are derived by studying the $m$-step competition graph of the digraph of $A$.
\begin{thebibliography}{ppk} \bibitem{ppk} W. Park, B. Park and S. -R. Kim, {\it A matrix sequence $\{\Gamma(A^m)\}_{m=1}^\infty$ might converge even if the matrix A is not primitive}, Lin. Alg. Its Appl. {\bf 438} (2013), 2306--2319. \end{thebibliography}
2010 Mathematics Subject Classification: 05C20, 05C50
Key Words and Phrases: irreducible Boolean $(0,1)$-matrices, powers of Boolean $(0,1)$-matrices, linearly connected digraphs, index of imprimitivity, $m$-step competition graphs, graph sequence, powers of digraphs
- ⋅ 25th-C-13:50 − 14:10 Unavoidable vertex-minors in large prime graphs (O-joung Kwon, Sang-il Oum)
- 권오정*(카이스트), 엄상일(카이스트)
O-joung Kwon*, KAIST, Sang-il Oum, KAIST
A split of a graph $G$ is a partition $(A,B)$ of the vertex set $V(G)$ having subsets $A_0\subseteq A$, $B_0\subseteq B_0$ such that $|A|,|B|\ge 2$ and a vertex $a\in A$ is adjacent to a vertex $b\in B$ if and only if $a\in A_0$ and $b\in B_0$. A graph is prime (with respect to the split decomposition) if it has no split. We prove that for each $n$, there exists $N$ such that every prime graph on at least $N$ vertices contains a vertex-minor isomorphic to either a cycle of length $n$ or a graph consisting of two disjoint cliques of size $n$ joined by a matching. In this talk, we plan to introduce a main tool, which is called a blocking sequence in a prime graph, and we will describe two big steps of the proof. And we will pose some open problems behind this result.
2010 Mathematics Subject Classification: 05C83, 05C55
Key Words and Phrases: vertex-minor, split decomposition, ramsey theory
- ⋅ 25th-C-14:20 − 14:40 Graphical arrangements of compressed graphs (Anh Thi Nguyen, Sangwook Kim)
- Anh Thi Nguyen*(전남대), 김상욱(전남대)
Anh Thi Nguyen*, Chonnam National University, Sangwook Kim, Chonnam National University
We show that if a graph $G$ is compressed, then the proper part of the intersection poset of the corresponding graphical arrangement $A_G$ has the homotopy type of a wedge of spheres. Furthermore, we also indicate the number of spheres in the wedge, based on the number of adjacent edges of vertices in $G$.
2010 Mathematics Subject Classification:
Key Words and Phrases: compressed graph, graphical arrangements
- ⋅ 25th-C-14:40 − 15:00 Minimum rank of a random graph over the binary field (Jisu Jeong, Choongbum Lee, Po-Shen Loh, Sang-il Oum)
- 정지수*(카이스트), 이중범(Massachusetts Institute of Technology),\\ Po-Shen Loh(Carnegie Mellon Univ.), 엄상일(카이스트)
Jisu Jeong*, KAIST, Choongbum Lee, Massachusetts Institute of Technology, Po-Shen Loh, Carnegie Mellon University, Sang-il Oum, KAIST
The minimum rank of a graph $G$ over a field $\mathbb{F}$ is the smallest possible rank of an $n\times n$ symmetric matrix $M$ over $\mathbb{F}$ such that if $i\neq j$, the $(i,j)$-entry of $M$ is nonzero if and only if two vertices $i$ and $j$ are adjacent in $G$. A random graph $G(n,p)$ is a graph on a vertex set $\{1,2,\ldots,n\}$ such that two vertices are adjacent with probability $p$ independently at random.
We investigate the minimum rank of a random graph over the binary field. Friedland and Loewy showed that the minimum rank of $G(n,1/2)$ is at least $n-\sqrt{2n}+1.49$ asymptotically almost surely. We prove that the minimum rank of $G(n,1/2)$ is at least $n-\sqrt{2n}-1.01$ asymptotically almost surely. Furthermore, we prove that if $p(n)$ is a function such that $0<p(n)\le 1/2$ and $np(n)$ tends to infinity, then the minimum rank of $G(n,p(n))$ is at least $n-1.178\sqrt{n/p(n)}$ asymptotically almost surely.
2010 Mathematics Subject Classification: 05C50, 05C80
Key Words and Phrases: minimum rank, random graph
- Cryptography
- ⋅ 26th-D-09:30 − 10:50 Chair: Kyung-Ah Shim (NIMS)
- ⋅ 26th-D-09:30 − 09:50 Affine transformations of Gauss normal bases, and self-dual normal bases over finite fields (Kitae Kim, Jeongil Namgoong, Ikkwon Yie)
- 김기태(인하대), 남궁정일*(인하대), 이익권(인하대)
Kitae Kim, Inha University, Jeongil Namgoong*, Inha University, Ikkwon Yie, Inha University
Normal basis is widely accepted in various applications, especially where fast exponentiations in finite fields are required. Christopolou et al. and Liao investigated the multiplication table and the complexity of a Gauss normal basis which is a normal basis induced from Gauss periods. In this talk, we study the complexity of affine transformations of a Gauss period normal basis, and discuss Liao's characterization of $(n,2)$-Gauss period normal basis. Furthermore, we investigate self-duality of affine transformations of some normal bases.
2010 Mathematics Subject Classification: 11T99, 12E10, 12E20
Key Words and Phrases: finite fields, normal basis, Gauss period, affine transformation, complexity
- ⋅ 26th-D-09:50 − 10:10 A hybrid scheme of public key encryption and fully homomorphic encryption (Jung Hee Cheon, Jinsu Kim)
- 천정희(서울대), 김진수*(서울대)
Jung Hee Cheon, Seoul National University, Jinsu Kim*, Seoul National University
We introduce a hybrid homomorphic encryption by combining public key encryption
(PKE) and somewhat homomorphic encryption (SHE) to reduce storage for most applications of
somewhat or fully homomorphic encryption (FHE). In this model, one encrypts messages with
a PKE and computes on encrypted data using a SHE or a FHE after homomorphic decryption.
To obtain efficient homomorphic decryption, our hybrid schemes is constructed by combining
IND-CPA PKE schemes without complicated message paddings with SHE schemes with large
integer message space. Furthermore, we remark that if the underlying PKE is multiplicative on a
domain closed under addition and multiplication, this scheme has an important advantage that
one can evaluate a polynomial of arbitrary degree without recryption. We propose such a scheme
by concatenating ElGamal and Goldwasser-Micali scheme over a ring $\mathbb Z_N$ for a composite integer
$N$ whose message space is $\mathbb Z_N^\times$.\\
To be used in practical applications, homomorphic decryption of the base PKE is too expensive.
We accelerate the homomorphic evaluation of the decryption by introducing a method to reduce
the degree of exponentiation circuit at the cost of additional public keys. Using same technique,
we give an efficient solution to the open problem in [1] partially.
As an independent interest, we obtain another generic conversion method from private key SHE
to public key SHE. Differently from Rothblum, it is free to choose the message space of SHE.
\begin{thebibliography}{1} \bibitem{1} J. Kim, M. S. Lee, A. Yun and J. H. Cheon, {\it CRT-based fully homomorphic encryption over the integers}, IACR Cryptology ePrint Archive {\bf 57} (2013)\end{thebibliography}
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: ElGamal, Goldwasser-Micali, multiplicative homomorphic encryption, fully homomorphic encryption, bootstrapping, decryption circuit, exponentiation
- ⋅ 26th-D-10:10 − 10:30 An improvement of algorithm for ECDLP over small degree extension fields (Jung Hee Cheon, HeeWon Chung, Hyung Tae Lee)
- 천정희(서울대), 정희원*(서울대), 이형태(서울대)
Jung Hee Cheon, Seoul National University, HeeWon Chung*, Seoul National University, Hyung Tae Lee, Seoul National University
Tag tracing technique is first suggested by Cheon et al. and it enables to solve DLP 10 times faster than before in multiplicative gorups of finite fields. However, for elliptic curves, tag tracing technique they suggested does not effect because of an addition on elliptic curves. To add two points in elliptic curves, we actually need one inversion, two multiplications including a squaring and some additions on the finite field. However, there is no method to compute the partial information of inversion and hence we cannot use tag tracing technique in elliptic curves. In this talk, we give a method to compute the leading coefficient of inversion without full computation and using this method, we can apply tag tracing technique in elliptic curves. Using this algorithm, we can solve ECDLP faster if the degree of extension field is small.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: ECDLP, Pollard rho method, tag tracing
- ⋅ 26th-D-10:30 − 10:50 Fault attacks on the elliptic curve discrete logarithm problem (Taechan Kim, Mehdi Tibouchi)
- 김태찬*(서울대), Mehdi Tibouchi(NTT Secure Platform Laboratories)
Taechan Kim*, Seoul National University, Mehdi Tibouchi, NTT Secure Platform Laboratories
In this talk, we will briefly review on the fault attacks on ECDLP, invalid point attack and modulus fault attack. Further, we shall further analyze the modulus fault attack done by Ciet and Joye. Under the chosen fault model, our analysis reduces the complexity solving ECDLP to subexponential and this extends to the attack on ECDSA.
2010 Mathematics Subject Classification: 94A60
Key Words and Phrases: elliptic curve, discrete logarithm problem
- ⋅ 26th-E-11:10 − 12:00 Chair: JungYeon Hwang (ETRI)
- ⋅ 26th-E-11:10 − 11:40 Polynomial representations for $n$-th roots in finite fields (Seunghwan Chang, Bihtnara Kim, Hyang-Sook Lee)
- 장승환*(이화여대 수리과학연구소), 김빛나라(이화여대), 이향숙(이화여대)
Seunghwan Chang*, Institute of Mathematical Sciences, Ewha Womans University, Bihtnara Kim, Ewha Womans University, Hyang-Sook Lee, Ewha Womans University
Agou, Del\'eglise and Nicolas proved results concerning the lower bounds for the length of polynomials representing square roots modulo a prime $p$. We generalize the results by considering $n$-th roots over finite fields for arbitrary $n$.
2010 Mathematics Subject Classification: 12E20, 68W40
Key Words and Phrases: $n$-th roots, finite fields, polynomial interpolation
- ⋅ 26th-E-11:40 − 12:00 Families of pairing-friendly elliptic curves based on modified DEM method (Hyang Sook Lee, Pa Ra Lee)
- 이향숙(이화여대), 이파라*(이화여대)
Hyang Sook Lee, Ewha Womans University, Pa Ra Lee*, Ewha Womans University
The most general method to construct pairing-friendly elliptic curves is Brezing-Weng construction. Even though the BW method provides complete families of ordinary elliptic curves for almost arbitrary embedding degree $k$, the approach considers the CM discriminant $D$ as the fixed input such as $D=1,2$ or $3$. For the security reason, there are several studies to generate curve parameters for pairing-friendly elliptic curves with various $D$. For example, Scott-Barreto introduced a method to construct the pairing-friendly curves with $D$-variable only for the embedding degree $k=3,4$ and $6$ by extending the MNT curves. In this talk, we provide a new method, so called modified DEM method, to construct families of pairing-friendly elliptic curves with $D$-variables. Our new approach produces the complete families of elliptic curves for arbitrary $k$ and obtains the families of curves with improved $\rho$-values for several $k$.
2010 Mathematics Subject Classification: 11G05
Key Words and Phrases: pairing-friendly, ordinary elliptic curves, complete families
- $p$-Adic Analysis, Umbral Algebra and Their Applications
- ⋅ 25th-A-09:00 − 10:30 Chair: Seog Hoon Rim (Kyungpook National University)
- ⋅ 25th-A-09:00 − 09:20 $p$-adic valued probability of nonspherical majority in a $p$-adic statistical test (Yu Seon Jang, Taekyun Kim, Jong Jin Seo)
- 장유선*(강남대), 김태균(광운대), 서종진(부경대)
Yu Seon Jang*, Kangnam University, Taekyun Kim, Kwangwoon University, Jong Jin Seo, Pukyong National University
Randomness test used to analyze the distribution pattern of a set of data. Even though a universal $p$-adic test for randomness does not exist, it is possible to enumerate effectively all $p$-adic tests for randomness. In this talk, nonspherical majority in a $p$-adic test for randomness is performed by the $Q_p$ valued probability.
2010 Mathematics Subject Classification: 03B48, 03B42, 03B45
Key Words and Phrases: $p$-adic valued probability, $p$-adic statistical test
- ⋅ 25th-A-09:20 − 09:40 A note on the modified $q$-Euler polynomials (Jin-Woo Park, Jongkyum Kwon, Sung-Soo Pyo, Seog-Hoon Rim)
- 박진우*(경북대), 권종겸(경북대), 표성수(경북대), 임석훈(경북대)
Jin-Woo Park*, Kyungpook National University, Jongkyum Kwon, Kyungpook National University, Sung-Soo Pyo, Kyungpook National University, Seog-Hoon Rim, Kyungpook National University
In this paper, we construct new $q$-extension of Euler numbers and polynomials related to fermionic $p$-adic $q$-integral on $\mathbb Z_p$, and give new explicit formulas related to these numbers and polynomials.
2010 Mathematics Subject Classification: 11B68, 11S80
Key Words and Phrases: Bernoulli numbers, $p$-adic $q$-integrals
- ⋅ 25th-A-09:50 − 10:10 Toward the ergodicity of $p$-adic 1-Lipschitz functions represented by the van der Put series (Sang Tae Jeong)
- 정상태(인하대)
Sang Tae Jeong, Inha University
Yurova (2010) and Anashin et al. (2011) characterize the ergodicity of a 1-Lipschitz function on ${\mathbb{Z}}_2$ in terms of the van der Put expansion. Motivated by their recent work, we provide the sufficient conditions for the ergodicity of such a function defined on a more general setting ${\mathbb{Z}}_p$. In addition, we provide alternative proofs of two criteria (because of Anashin et al.,and Yurova) for an ergodic 1-Lipschitz function on ${\mathbb{Z}}_2$, represented by both the Mahler basis and the van der Put basis.
2010 Mathematics Subject Classification: 37P20, 11S82
Key Words and Phrases: ergodic, measure-preserving, 1-Lipschitz, van det Put basis, Mahler basis
- ⋅ 25th-A-10:10 − 10:30 The characteristic polynomial of abelian varieties of dimension three over finite fields (Gyoyong Sohn, Ju-Mok Oh)
- 손교용*(대구교대), 오주목(강릉원주대)
Gyoyong Sohn*, Daegu National University of Education, Ju-Mok Oh, Kanggnung-Wonju National University
In this paper, we provide an explicit bounds for the coefficients of the characteristic polynomial of the Frobenius endomorphism of the abealian varieties of dimension three over finite fields. Based on these bounds, we found an efficient search space that was used in the counting points algorithm on genus 3 curves.
2010 Mathematics Subject Classification: 14G50
Key Words and Phrases: characteristic polynomial
- ⋅ 25th-B-10:40 − 12:10 Chair: Nak Eun Cho (Pukyong National University)
- ⋅ 25th-B-10:40 − 11:00 Higher-order Cauchy of the first kind and poly-Cauchy of the first kind mixed type polynomials (Dae San Kim, Taekyun Kim)
- 김대산*(서강대), 김태균(광운대)
Dae San Kim*, Sogang University, Taekyun Kim, Kwangwoon University
In this paper, we study higher-order Cauchy of the first kind and poly-Cauchy of the first kind mixed type polynomials with viewpoint of umbral calculus and give some interesting identities and formulae of those polynomials which are derived from umbral calculus.
2010 Mathematics Subject Classification: 05A40, 05A19
Key Words and Phrases: poly-Cauchy polynomial of the first kind, higher-order Cauchy number of the first
- ⋅ 25th-B-11:00 − 11:20 On the modified $q$-Euler numbers and polynomials (Jin-Woo Park, Jongkyum Kwon, Sung-Soo Pyo, Seog-Hoon Rim)
- 박진우(경북대), 권종겸*(경북대), 표성수(경북대), 임석훈(경북대)
Jin-Woo Park, Kyungpook National University, Jongkyum Kwon*, Kyungpook National University, Sung-Soo Pyo, Kyungpook National University, Seog-Hoon Rim, Kyungpook National University
In this paper, we construct new $q$-extension of higher-order Euler numbers and polynomialsrelated to fermionic $p$-adic $q$-integral on $\mathbb Z_p$, and give new explicit formulas related to these numbers and polynomials.
2010 Mathematics Subject Classification: 11S80
Key Words and Phrases: Berbouuli numbers, $q$-Bernoulli numbers
- ⋅ 25th-B-11:30 − 11:50 Remarks on Fermat quotient operators over function fields (Sang Tae Jeong, Chunlan Li)
- 정상태(인하대), 이춘란*(인하대)
Sang Tae Jeong, Inha University, Chunlan Li*, Inha University
A study on $Q_n$-quotients and Fermat quotients over function fields was initially undertaken in a previous paper by J. Sauerberg and L. Shu. In this note, we revisit them and further investigate the properties of two closely-related quotients. Then, we provide interesting problems by making conjectures related to the Fermat quotients.
2010 Mathematics Subject Classification: 11R58, 11T55
Key Words and Phrases: quotient operators, Fermat quotient operators, hyperderivatives, function fields
- ⋅ 25th-B-11:50 − 12:10 Some identities of the Bernoulli polynomials with the variable $[x]_q$ (Hui Young Lee, Cheon Seoung Ryoo)
- 이희영*(한남대), 유천성(한남대)
Hui Young Lee*, Hannam University, Cheon Seoung Ryoo, Hannam University
In this paper, we introduce the Bernoulli polynomials with the variable $[x]_q$ and we get polynomials related to the Bernoulli polynomials with the variable $[x]_q$ and the Bernoulli numbers as coefficients. This Bernoulli polynomials is defined by the $p$-adic integral and we get the some interesting properties.
2010 Mathematics Subject Classification: 11B68, 11S40, 11S80
Key Words and Phrases: Bernoulli numbers, Bernoulli polynomials, Bernoulli numbers and polynomials with weight $\alpha$
- ⋅ 26th-D-09:30 − 11:00 Chair: Dae San Kim (Sogang University)
- ⋅ 26th-D-09:30 − 09:45 A study on symmetric property for $q$-Genocchi polynomials and zeta function (Jung Yoog Kang, Cheon Seoung Ryoo )
- 강정욱*(한남대), 유천성(한남대)
Jung Yoog Kang*, Hannam University, Cheon Seoung Ryoo , Hannam University
The present papers deal with the various $q$-Genocchi numbers and polynomials. We find the symmetric identity of $q$-Genocchi zeta function. By using the symmetric identity of $q$-Genocchi zeta function, we study a few interesting symmetric properties of $q$-Genocchi polynomials.
2010 Mathematics Subject Classification: 11B68, 11S40, 11S80
Key Words and Phrases: symmetric identities of $q$-Genocchi polynomials, symmetric property of $q$-Genocchi zeta function
- ⋅ 26th-D-09:45 − 10:00 A study of the new approach of $q$-Bernoulli and $q$-Euler polynomials (WonJoo Kim)
- 김원주(광운대)
WonJoo Kim, Kwangwoon University
Recently, Kim introduced new $q$-Euler numbers and polynomials \begin{equation} \label{eq:10} \left(\frac{2}{e^t+1}\right)e^{xt}=\sum_{n=0}^{\infty}E_{n}(x)\frac{t^n}{n!}\end{equation} and Hegazi and Mansour introduced new $q$-extension of Bernoulli numbers and polynomials: \begin{eqnarray} \label{eq:20} \sum_{n=0}^{\infty}B_{n,q}(x)\frac{z^n}{(q:q)_n}=\frac{z}{(1-q)(e^{\frac{z}{1-q}}-1)}e_q\left(\frac{zx}{1-q} \right). \end{eqnarray} In this paper, we investigate generalized $q$-Bernoulli and $q$-Euler numbers and we give the Witt's type formula for the $q$-Bernoulli and $q$-Euler polynomials.
2010 Mathematics Subject Classification: 12H25, 11D88
Key Words and Phrases: $q$-Bernoulli polynomials, $q$-Euler polynomials
- ⋅ 26th-D-10:00 − 10:15 A note on the modified $q$-Bernoulli polynomials (Seog-Hoon Rim, Jin-Woo Park, Sung-Soo Pyo, Jongkyum Kwon)
- 임석훈*(경북대), 박진우(경북대), 표성수(경북대), 권종겸(경북대)
Seog-Hoon Rim*, Kyungpook National University, Jin-Woo Park, Kyungpook National University, Sung-Soo Pyo, Kyungpook National University, Jongkyum Kwon, Kyungpook National University
In this paper, we construct new $q$-extension of Bernoulli polynomials. These $q$-Bernoulli polynomials are useful to study various identities of Carlitz's $q$-Bernoulli numbers.
2010 Mathematics Subject Classification: 11B68
Key Words and Phrases: Bernoulli numbers, $p$-adic $q$-integrals, $q$-Bernoulli numbers
- ⋅ 26th-D-10:30 − 10:45 The twisted $q$-Euler numbers and polynomials with weight $0$ (Seog-Hoon Rim, Joohee Jeong, Joung Hee Jin)
- 임석훈(경북대), 정주희(경북대), 진정희*(경북대)
Seog-Hoon Rim, Kyungpook National University, Joohee Jeong, Kyungpook National University, Joung Hee Jin*, Kyungpook National University
The purpose of this paper is to give some identities related the twisted weak $q$-Euler numbers and polynomials with weight $0$.
2010 Mathematics Subject Classification:
Key Words and Phrases: $q$-Euler numbers, $q$-Euler polynomials
- ⋅ 26th-D-10:45 − 11:00 On the twisted weak weight $q$-Bernoulli polynomials and numbers (Sun-Jung Lee, Dongjin Kang, Jin-Woo Park, Seog-Hoon Rim)
- 이선정*(경북대), 강동진(경북대), 박진우(경북대), 임석훈(경북대)
Sun-Jung Lee*, Kyungpook National University, Dongjin Kang, Kyungpook National University, Jin-Woo Park, Kyungpook National University, Seog-Hoon Rim, Kyungpook National University
Recently, many authors have studied twisted $q$-Bernoulli polynomials by using the $p$-adic invariant $q$-integral on $\mathbb Z_p$. In this paper, we define and study the twisted weak weight $q$-Bernoulli polynomials and numbers. And, we derive some various identities related to the twisted weak weight $q$-Bernoulli polynomials.
2010 Mathematics Subject Classification:
Key Words and Phrases: twisted weak weight $q$-Bernoulli polynomials
- ⋅ 26th-E-11:10 − 12:40 Chair: Sang Tae Jeong (Inha University)
- ⋅ 26th-E-11:10 − 11:25 Some identities of $q$-Bernoulli numbers associated $p$-adic convolutions (Jong-Jin Seo)
- 서종진(부경대)
Jong-Jin Seo, Pukyong National University
In this paper, we give some interesting and new identities of $q$-Bernoulli numbers which are derived from convolutions on the ring of $p$-adic integers.
2010 Mathematics Subject Classification: 11B68
Key Words and Phrases: $p$-adic integral, Bernoulli numbers, $q$-Bernoulli numbers
- ⋅ 26th-E-11:25 − 11:40 A note on the twisted $q$-Euler polynomials (Eun-Jung Moon)
- 문은정(경북대)
Eun-Jung Moon, Kyungpook National University
In this paper, we give an identity for the twisted $q$-Euler polynomials of higher order. By using the $p$-adic $q$-integral on $\mathbb Z_p$, we will derive some relationship between the power sums and the twisted $q$-Euler polynomials.
2010 Mathematics Subject Classification: 11B68
Key Words and Phrases: twisted $q$-Euler polynomials , $p$-adic $q$-integral on $\mathbb Z_p$
- ⋅ 26th-E-11:55 − 12:10 Conditions for Caratheodory functions with some applications (Hyo Jeong Lee, Nak Eun Cho)
- 이효정*(부경대), 조낙은(부경대)
Hyo Jeong Lee*, Pukyong National University, Nak Eun Cho, Pukyong National University
The purpose of the present talk is to consider some sufficient conditions for Caratheo\-dory functions. Moreover, we introduce interesting special cases of main results associated with univalent (close-to-convex) and (strongly) starlike functions.
2010 Mathematics Subject Classification: 30C45, 30C80
Key Words and Phrases: Caratheodory function, univalent function, (strongly) starlike function, (strongly) convex function, integral operator
- ⋅ 26th-E-12:10 − 12:25 Control of the linear systems under uncertainty (Dolgy Dmitriy)
- Dolgy Dmitriy(광운대 한림원)
Dolgy Dmitriy, Hanrimwon, Kwangwoon University
The method of control of the linear systems subjected to perturbations is offered. The method is based on approximation of terminal set by cube.
2010 Mathematics Subject Classification: 11A50
Key Words and Phrases: controlled system, terminal set, approximation
- ⋅ 26th-E-12:25 − 12:40 A note on Dahee numbers and polynomials (Taekyun Kim)
- 김태균(광운대)
Taekyun Kim, Kwangwoon University
In this paper, we introduce Daehee polynomials and numbers and we give Witt's formula for those numbers and polynomials.
2010 Mathematics Subject Classification: 11B68, 11S80
Key Words and Phrases: Bernoulli numbers, polynomials, Daehee numbers and polynomials
- Several complex variables
- ⋅ 26th-D-09:30 − 10:30 Chair: Heungju Ahn (Deagu Gyeongbuk Institute of Science and Technology )
- ⋅ 26th-D-09:30 − 09:50 Method of characteristics for systems of quasilinear PDE of first order (Chong-Kyu Han)
- 한종규(서울대)
Chong-Kyu Han, Seoul National University
We consider a quasi-linear PDE of first order $\sum_{k=1}^n a^k(x,u) \frac{\partial u}{\partial x^k} = b(x,u)$ for one unknown function $u(x)$ of $n$ variables $x=(x^1,\ldots, x^n).$ We disucss in this talk
1) degree of freedom of solutions with Cauchy data on lower dimensions
2) existence of solutions for systems
3) generalizations to fully non-linear cases.
These are subsequent results to the author's joint papers with Jong-Do Park on systems of quasi-linear PDE.
2010 Mathematics Subject Classification: 35A07, 35A30, 58A17
Key Words and Phrases: quasi-linear, first order, underdetermined Cauchy data, characteristic vector field
- ⋅ 26th-D-09:50 − 10:10 On a theorem of Paul Yang on negatively pinched bisectional curvature (Aeryeong Seo)
- 서애령(고등과학원)
Aeryeong Seo, KIAS
In 1976, P. Yang proved the following theorem \cite{yang}:\medskip
\noindent{\bf Theorem {\rm [P. Yang 1976]}.}
The polydisc $D^n$ and bounded symmetric domains with rank $\geq 2$ do not admit any complete K\"ahler metrics with its bisectional
curvature bounded between two negative constants.
\medskip
In this talk, I will present examples of domains that do not admit any complete K\"ahler
metric with bisectional curvature bounded between prescribed two negative
constants by a modification of a method of P. Yang.
\begin{thebibliography}{1} \bibitem{yang} P. Yang, {\it On Kahler manifolds with negative holomorphic bisectional curvature}, Duke Math. J. {\bf 43} (1976), 871 -- 874. \end{thebibliography}
2010 Mathematics Subject Classification: 32Q05, 32Q15
Key Words and Phrases: bisectional curvature
- ⋅ 26th-D-10:10 − 10:30 Variation of Kahler-Einstein metrics (Yoing-Jun Choi)
- 최영준(고등과학원)
Yoing-Jun Choi, KIAS
Let $(z,s)\in\mathbb{C}^n\times\mathbb{C}^m$ and $\pi:\mathbb{C}^n\times\mathbb{C}^m\rightarrow\mathbb{C}^m$ be the projection on the second factor. Let $D$ be a smooth domain in $\mathbb{C}^{n+m}$ which satisfies that for each $s\in\pi(D)$, the fiber $D_s:=\pi^{-1}(s)$ is a bounded strongly pseudoconvex domain with smooth boundary. The celebrated theorem of Cheng and Yau implies that on each fiber $D_s$ there exists a unique complete K\"{a}hler metric $h_{\alpha\bar\beta}(z,s):=h^s_{\alpha\bar\beta}(z)$ which satisfies:
\begin{align*}
-(n+1)h_{\alpha\bar\beta}(z,s)
=-\frac{\partial{^2}}{{\partial {z^\alpha{z}^{\bar\beta}}}}\log\det
{(h_{\gamma\bar\delta}(z,s))}_{1\le\gamma,\delta\le{n}},
\end{align*}
namely, the Ricci curvature is negative constant $-(n+1)$. This unique complete K\"{a}hler metric is called \emph{K\"{a}hler-Einstein metric}. Hence on each fiber $D_s$,
\begin{equation*}
\frac{1}{n+1}\log\det{(h_{\gamma\bar\delta}(z,s))}_{1\le\gamma,\delta\le{n}}
\end{equation*}
is a potential function of the K\"{a}hler-Einstein metric $h_{\alpha\bar\beta}$. Denote it by $h(z,s)$. Consider the function $h(z,s)$ on $D$. It is an immediate consequence of the K\"{a}hler-Einstein condition is that the restriction of $H$ to each fiber $D_s$ is strictly plurisubharmonic. But it is not obvious that it is also plurisubharmonic or strictly plurisubharmonic in base direction ($s$-direction).\\
\medskip
\noindent{\bf Theorem.} {\it
If $n\ge3$, then $h(z,s)$ is a plurisubharmonic function on $D$ provided $D$ is pseudoconvex. Moreover if $D$ is strongly pseudoconvex, then $h(z,s)$ is a strictly plurisubharmonic function.}
2010 Mathematics Subject Classification: 32Q20
Key Words and Phrases: variation, Kahler-Einstein metric, strongly pseudoconvex domains, plurisubharmonic
- ⋅ 26th-E-10:40 − 11:40 Chair: Jisoo Byun (Kyungnam University)
- ⋅ 26th-E-10:40 − 11:00 Hermite-Sobolev and Fock-Sobolev spaces of fractional order (Hong Rae Cho)
- 조홍래(부산대)
Hong Rae Cho, Pusan National University
Let $s\geq 0$. The Fock-Sobolev spaces $F^{s,2}_R$ of fractional order $s$ are introduced through the fractional radial derivatives $R^s$. We show that the Bargmann transform is a unitary isomorphism between the Hermite-Sobolev saces $W^{s,2}(\mathbb R^n)$ with the Fock-Sobolev spaces $F^{s,2}_R$. Moreover, we prove that the Fock-Sobolev spaces $F^{s,2}_R$ are identified with the weight\-ed Fock spaces $F^2_s$ that do not involve derivatives. So, the study on the Fock-Sobolev spaces is reduced to that on the weighted Fock spaces.
2010 Mathematics Subject Classification: 32A37
Key Words and Phrases: Hermite-Sobolev space, fractional Hermite operator, Fock-Sobolev space
- ⋅ 26th-E-11:00 − 11:20 The automorphism group of a certain unbounded non-hyperbolic domain (Hyeseon Kim, Van Thu Ninh, Atsushi Yamamori)
- 김혜선*(포항공대 기하학연구센터), Van Thu Ninh(포항공대 기하학연구센터), Atsushi Yamamori(포항공대 기하학연구센터)
Hyeseon Kim*, The Center for Geometry and its Applications, POSTECH, Van Thu Ninh, The Center for Geometry and its Applications, POSTECH, Atsushi Yamamori, The Center for Geometry and its Applications, POSTECH
The purpose of this talk is to determine the automorphism group of the Fock-Bargmann-Hartogs domain $D_{n,m}$ which is defined by \begin{equation*} D_{n,m}:=\{(z,\zeta)\in\mathbb{C}^n\times\mathbb{C}^m:\;{\|\zeta\|}^2<\exp(-\mu{\|z\|}^2) \},\quad\mu>0. \end{equation*}For a complex manifold, the automorphism group is the set of all biholomorphic self maps that forms a group under the law of composition. For a domain in $\mathbb{C}^k~(k\geq2)$, the automorphism group is not easy to describe explicitly. The automorphism groups for the cases of bounded or hyperbolic in the sense of Kobayashi, have been already studied extensively. In contrast to these cases, the Fock-Bargmann-Hartogs domain is concerned with an unbounded non-hyperbolic case. In describing the associated automorphism group, the main ingredients are the celebrated Cartan's theorem by using the Bergman representative mapping and an explicit form of the Bergman kernel function of $D_{n,m}$. This talk is based on the collaboration with Van Thu Ninh and Atsushi Yamamori.
2010 Mathematics Subject Classification: 32M17
Key Words and Phrases: automorphism group, Bergman kernel, Cartan theorem
- ⋅ 26th-E-11:20 − 11:40 Cesaro operators in the Bergman spaces with exponential weights on the unit ball (Hong Rae Cho, Su Kyung Han, Inyoung Park)
- 조홍래(부산대), 한수경(부산대), 박인영*(부산대)
Hong Rae Cho, Pusan National University, Su Kyung Han, Pusan National University, Inyoung Park*, Pusan National University
We study operators of the form $T_gf(z)=\int^z_0 f(\xi)Rg(\xi)dv(\xi)$ with holomorphic symbol $g$ in $\mathbf{B_n}$ on weighted Bergman spaces with exponential weights $A^p_w(\mathbf{B_n})$.
2010 Mathematics Subject Classification: 32A37
Key Words and Phrases: Bergman space, exponential weights, unit ball, Cesaro operators
- ⋅ 26th-E-11:50 − 12:30 Chair: Jong-Do Park (Kyung Hee University)
- ⋅ 26th-E-11:50 − 12:10 Exponentionally weighted $L^p$-estimates for $\overline{\partial}$ on the unit disc (Hong Rae Cho, Su Kyung Han)
- 조홍래(부산대), 한수경*(부산대)
Hong Rae Cho, Pusan National University, Su Kyung Han*, Pusan National University
Let $\mathbb D$ be the unit disc. Let $\varphi\in C^2(\mathbb D)$ be a radial function such that $\Delta\varphi(z)\geq C_\varphi>0$ for some positive constant $C_\varphi$ depending only on the function $\varphi$. We prove the $ L^p $-estimates for the $ \overline{\partial} $-equation in the exponentionally weighted $L^p$ spaces on the unit disc. We use the holomorphic peak functions with precise growth conditions in the construction of the explicit solutions to the equation.
2010 Mathematics Subject Classification: 97I80, 32T40, 47A13
Key Words and Phrases: complex analysis, operator theory
- ⋅ 26th-E-12:10 − 12:30 Equidistribution in higher codimension for holomorphic endomorphisms of projective spaces (Taeyong Ahn)
- 안태용(SRC-GaiA, 포항공대)
Taeyong Ahn, SRC-GaiA, POSTECH
In this talk, we discuss equidistribution phenomena. As a tool, we study super-potentials introduced by T.-C. Dinh and N. Sibony and present some difficulties in higher codimensional cases. Then, using Lojasiewicz inequality, we briefly show a new result on equidistribution in higher codimensioal cases.
2010 Mathematics Subject Classification: 37F10, 32H50, 32U40
Key Words and Phrases: Green current, equidistribution, exceptional set, super-potential
- Recent Results of Young PDE Researchers
- ⋅ 26th-D-09:30 − 11:00 Chair: Hyeong-Ohk Bae (Ajou University)
- ⋅ 26th-D-09:30 − 09:45 Hydrodynamic Cucker-Smale flocking models (Bongsuk Kwon)
- 권봉석(울산과학기술대)
Bongsuk Kwon, UNIST
We briefly discuss the Cucker-Smale flocking models at the particle level and kinetic level, and the asymptotic behaviors of the solutions. We then derive the hydrodynamic model of the pressureless Euler equations with a nonlocal flocking dissipation term, which describes the dynamics of a large number of the Cucker-Smale particles in a collision free regime. For the proposed hydrodynamic model, we discuss the global well-posedness of classical solutions, and show that the classical solutions exhibit the asymptotic flocking as time goes on. If time permits, we also discuss about some recent progress on free boundary problem arising in the flocking model.
2010 Mathematics Subject Classification:
Key Words and Phrases: Cucker-Smale model, flocking, global existence, asymptotic behavior
- ⋅ 26th-D-09:45 − 10:00 Prandtl-Meyer reflection for supersonic flow past a solid ramp (Myoungjean Bae, Gui-Qiang Chen, Mikhail Feldman)
- 배명진*(포항공대), Gui-Qiang Chen(Oxford Univ.), Mikhail Feldman(Univ. of Wisconsin-Madison)
Myoungjean Bae*, POSTECH, Gui-Qiang Chen, Oxford University, Mikhail Feldman, University of Wisconsin-Madison
When a steady supersonic flow passes a solid ramp, there are two possible configurations: the weak shock solution and the strong shock solution. Elling-Liu’s theorem (2008) indicates that the steady supersonic weak shock solution can be regarded as a long-time asymptotic state of an unsteady flow for a class of physical parameters determined by certain assumptions for potential flow. In this talk, I present recent progress in removing these assumptions and establishing the stability theorem for steady supersonic weak shock solutions as the long-time asymptotics of unsteady flows for all the physical parameters for potential flow. This talk is based on joint work with Gui-Qiang Chen(Oxford) and Mikhail Feldman(UW-Madison).
2010 Mathematics Subject Classification: Primary 35M10, 35M12, 35B65, 35L65, 35L70, 35J70, 76H05, 35L67, 35R35; Secondary 35L15, 35L20, 35J67, 76N10, 76L05
Key Words and Phrases: Prandtl-Meyer reflection, supersonic flow, unsteady flow, steady flow,weak shock solution, strong shock solution, stability, self-similar, transonic shock, sonic boundary, free boundary
- ⋅ 26th-D-10:00 − 10:15 Elliptic obstacle problems with measurable coefficients (Sun-Sig Byun, Dian Palagachev, Seungjin Ryu)
- 변순식(서울대), Dian Palagachev(Politecnico di Bari), 유승진*(서울시립대)
Sun-Sig Byun, Seoul National University, Dian Palagachev, Politecnico di Bari, Seungjin Ryu*, University of Seoul
We discuss the $W^{1,p}$ regularity of solutions to variational inequalities and obstacle problems for divergence form elliptic equations with measurable coefficients. We are dealing here with differential operators having only measurable coefficients and irregular obstacles.
2010 Mathematics Subject Classification: Primary 35J88 ; Secondary 35R05
Key Words and Phrases: ellipitic system, obstacle problem, measurable coefficients, Reifenberg flat domain
- ⋅ 26th-D-10:15 − 10:30 Results on standing waves for Chern-Simons-Schr\"{o}dinger equations (Jinmyoung Seok, Jaeyoung Byeon, Hyungjin Huh)
- 석진명*(고등과학원), 변재형(카이스트), 허형진(중앙대)
Jinmyoung Seok*, Korea Institute for Advanced Study, Jaeyoung Byeon, KAIST, Hyungjin Huh, Chung-Ang University
The Chern-Simons theory is proposed in the 1980's to explain electromagnetic phenomena of anyon physics such as the high temperature super-conductivity or the fractional quantum Hall effect. In this talk, I will introduce the nonlinear Schr\"{o}dinger equation coupled with the Chern-Simons gauge fields, proposed by Jackiw and Pi in 1990 and present recent results about the existence and nonexistence of the standing wave solutions. We will see that there is no nontrivial standing wave solution if the Chern-Simons coupling constant $\lambda$ is less than 1 and there is a standing wave solution with a vortex point of arbitrary order $N$ if $\lambda > 1$. If $\lambda = 1$, it turns out that every standing wave solution is gauge equivalent to a solution of the first order self-dual system.
2010 Mathematics Subject Classification: 35J20
Key Words and Phrases: Chern-Simons, Schr\"{o}dinger, standing wave
- ⋅ 26th-D-10:30 − 10:45 Nonlinear gradient estimates for parabolic problems with irregular obstacles (Sun-Sig Byun, Yumi Cho)
- 변순식(서울대), 조유미*(서울대)
Sun-Sig Byun, Seoul National University, Yumi Cho*, Seoul National University
We establish the natural Calder\'{o}n-Zygmund theory for solutions to parabolic variational inequalities satisfying an irregular obstacle constraint and involving degenerate/sin\-gular operators in divergence form of general type, and proving that the (spatial) gradient of solutions is as integrable as both the (spatial) gradient of the obstacles and the inhomogeneous terms, under the assumption that the involved nonlinearities have small a BMO semi-norm in the spatial variables while they are allowed to be merely measurable in the time variable.
2010 Mathematics Subject Classification: Primary 35J60; Secondary 35R05
Key Words and Phrases: irregular obstacle, Calder\'{o}n-Zygmund estimate, measurable nonlinearity, BMO space
- ⋅ 26th-D-10:45 − 11:00 On weighted $W^{2,p}$ estimates for elliptic equations with BMO coefficients (Sun-Sig Byun, Mikyoung Lee)
- 변순식 (서울대), 이미경*(서울대)
Sun-Sig Byun, Seoul National University, Mikyoung Lee*, Seoul National University
In this talk, we discuss weighted $W^{2,p}$ estimates for the solution to the Dirichlet problem for an elliptic equation in nondivergence form under the assumption that the matrix of the coefficients has a small BMO semi-norm.
2010 Mathematics Subject Classification: 35J25
Key Words and Phrases: weighted Sobolev space, elliptic equation, BMO space, strong solution
- ⋅ 26th-E-11:00 − 12:00 Chair: Bongsuk Kwon (UNIST)
- ⋅ 26th-E-11:00 − 11:15 Gradient estimates for nonlinear elliptic equations with a measurable coefficient (You-Chan Kim)
- 김유찬(서울대)
You-Chan Kim, Seoul National University
We consider divergence form type elliptic equations. The coefficients are assumed to be merely measurable on one variable. We talk about the recent results on the regularity theory of these equations and discuss about the nonlinear case.
2010 Mathematics Subject Classification: 35J60, 35Q74
Key Words and Phrases: elliptic equations, laminate
- ⋅ 26th-E-11:15 − 11:30 Gradient estimates for solutions of elliptic systems with measurable coefficients from composite material (Sun-Sig Byun, Yunsoo Jang, Seungjin Ryu)
- 변순식(서울대), 장윤수*(서울대), 유승진(서울시립대)
Sun-Sig Byun, Seoul National University, Yunsoo Jang*, Seoul National University, Seungjin Ryu, University of Seoul
In this talk, we obtain a global gradient estimate for the solution of an elliptic system in divergence form with measurable coefficients from composite material in a non-smooth bounded domain. The principal 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. As a consequence, an optimal regularity estimate is essentially established in this literature.
2010 Mathematics Subject Classification: 35J57
Key Words and Phrases: elliptic system, global estimate, measurable coefficient, BMO space, Reifenberg domain
- ⋅ 26th-E-11:30 − 11:45 $W^{1,p(\cdot)}$-regularity for elliptic equations with measurable coefficients in nonsmooth domains (Sun-Sig Byun, Jihoon Ok, Lihe Wang)
- 변순식(서울대), 옥지훈*(서울대), Lihe Wang(Shanghai Jiaotong Univ.)
Sun-Sig Byun, Seoul National University, Jihoon Ok*, Seoul National University, Lihe Wang, Shanghai Jiaotong University
We establish global $W^{1,p(\cdot)}$-estimates for second order elliptic equations in divergence form under the natural assumption that $p(\cdot)$ is \textit{log-H\"older continuous}. To this end, we assume that the coefficients are \textit{measurable in one variable and have small BMO semi-norms in the other variables} and the boundary of the domain is \textit{Reifenberg flat}. Our work is an optimal and natural extension of $W^{1,p}$-regularity for such equations with merely measurable coefficients beyond Lipschitz domains.
2010 Mathematics Subject Classification: Primary 35J60; Secondary 46E30
Key Words and Phrases: $W^{1,p(\cdot)}$-estimate, elliptic equations, generalized Lebesgue space, laminate, fractal
- ⋅ 26th-E-11:45 − 12:00 $L^\infty$ estimates of solution for $m$-Laplacian type elliptic equation with Morrey data (Sun-Sig Byun, Dian Palagachev, PilSoo Shin)
- 변순식(서울대), Dian Palagachev(Politecnico di Bari), 신필수*(서울대)
Sun-Sig Byun, Seoul National University, Dian Palagachev, Politecnico di Bari, PilSoo Shin*, Seoul National University
We prove global essential boundness of weak solutions for $m$-Laplacian type elliptic equation in divergence form with data belonging to Morrey space. The nonlinear terms are given in terms of Carath\'eodory functions and controlled growth assumptions.
2010 Mathematics Subject Classification: 35J60
Key Words and Phrases: $m$-Laplacian type, elliptic equation, Morrey space, essential boundness
- Theory of Ring and Module
- ⋅ 25th-A-09:00 − 10:30 Chair: Jung Wook Lim (Kyungpook National University)
- ⋅ 25th-A-09:00 − 09:20 Dimension theory of mixed polynomial and power series rings (Mi Hee Park)
- 박미희(중앙대)
Mi Hee Park, Chung-Ang University
We consider the mixed polynomial and power series ring extensions, or, for short, mixed extensions. A mixed extension over a ring $R$ in variables $x_1, \ldots, x_n$ is denoted by $R[x_1]\!]\ldots [x_n]\!]$, where each $[x_i]\!]$ is fixed as either $[x_i]$ or $[\![x_i]\!]$. One extreme is the polynomial extension $R[x_1, \ldots, x_n]$ and the other extreme is the power series extension $R[\![x_1, \ldots, x_n]\!]$. The motivation for our work comes from the following question raised by Robert Gilmer and Jim Coykendall: Let $R$ be a commutative ring with identity. When $\text{dim} (R[\![x]\!])<\infty$, is $\text{dim} (R[\![x]\!])\leq 2(\text{dim}\, R) +1$? In a joint paper with B. G. Kang, we gave an answer to the question in the negative by computing the Krull dimension of mixed extensions over a Pr\"{u}fer domain. In this talk, we discuss similarities and differences among several mixed extensions over fields. Especially, given a field extension $K\subset L$, we recall the algebraicity of the extension $K[x_1]\!]\cdots [x_n]\!]\hookrightarrow L[x_1]\!]\cdots [x_n]\!]$ and then compute the dimension of its generic fiber ring. We also characterize the field extensions $K\subset L$ such that the prime spectra $\text{Spec}(L[x_1]\!]\cdots [x_n]\!])$ and $\text{Spec}(K[x_1]\!]\cdots [x_n]\!])$ are homeomorphic. Finally, as an application, we compute the Krull dimension of mixed extensions over a globalized pseudo-valuation domain.
2010 Mathematics Subject Classification: 13F25
Key Words and Phrases: dimension, mixed extension, geberic fibre, SFT, pullback
- ⋅ 25th-A-09:20 − 09:40 Irreducible elements in commutative ring (Sangmin Chun, Dan Anderson)
- 천상민*(서울대), Dan Anderson(Univ. of Iowa)
Sangmin Chun*, Seoul National University, Dan Anderson, University of Iowa
Let $R$ be a commutative ring with identity and let $a$ be a nonunit. Then $a$ is irreducible if $a=bc$ implies $(a)=(b)$ or $(a)=(c)$. We study the various forms of atomicity. In this talk we introduce the various characterizations of the different types of irreducible elements.
2010 Mathematics Subject Classification: Primary 13A05; Secondary 13A15, 13F99
Key Words and Phrases: irreducible element, atomic ring
- ⋅ 25th-A-09:50 − 10:10 The diameter and girth of line graphs of zero-divisor graphs (Jonguk Baik)
- 백종욱(경희대)
Jonguk Baik, KyungHee University
In this talk, we study the line graphs of zero-divisor graphs and investigate their some properties. Also, we find some relationship between zero-divisor graphs and their line graphs and then we characterize the diameter and the girth of line graphs of zero-divisor graphs.
2010 Mathematics Subject Classification: 13P25
Key Words and Phrases: zero-divisor, graph, line graph, diameter, girth
- ⋅ 25th-A-10:10 − 10:30 On a module theoretic setting of the left Rickart property of rings (Gangyong Lee, S. Tariq Rizvi, Cosmin Roman)
- 이강용*(The Ohio State Univ.), S. Tariq Rizvi(The Ohio State Univ. at Lima), Cosmin Roman(The Ohio State Univ. at Lima)
Gangyong Lee*, The Ohio State University, S. Tariq Rizvi, The Ohio State University at Lima, Cosmin Roman, The Ohio State University at Lima
Kaplansky, Maeda, and Hattori introduced the Rickart property of rings, independently. A ring $R$ is said to be \emph{right} \emph{Rickart} if the right annihilator of any single element of $R$ is a direct summand of $R$ as a right ideal. A Rickart ring is not left-right symmetric, shown by Chase. Rizvi and Roman, in 2007, defined the notion of the \emph{right} Rickart property of rings to a general module theoretic setting, called a Rickart module. Now, we introduce the notion of a $\mathfrak{L}$-Rickart module as a general module theoretic setting of the \emph{left} Rickart ring. A right $R$-module $M_R$ an $\mathfrak{L}$-\emph{Rickart} module if the left annihilator in $S=\text{End}_R(M)$ of any single element of $M$ is a direct summand of $S$ as a left ideal of $S$. We provide results on this new concept for a right $R$-module $M_R$ where $R$ is any ring. For example, while it is known that the endomorphism ring of a Rickart module is a right Rickart ring, we show that the endomorphism ring of an $\mathfrak{L}$-Rickart module is not a left Rickart ring, in general. If $M_R$ is a finitely generated $\mathfrak{L}$-Rickart module, we prove that $\text{End}_R(M)$ is a left Rickart ring. Also, an $\mathfrak{L}$-Rickart module with no set of infinitely many nonzero orthogonal idempotents in its endomorphism ring is a Baer module.
2010 Mathematics Subject Classification: 16D10, 16S50
Key Words and Phrases: (left) Rickart rings, Rickart modules, Baer modules, endomorphism rings
- ⋅ 25th-B'-10:40 − 12:30 Chair: Mi Hee Park (Chung-Ang University)
- ⋅ 25th-B'-10:40 − 11:00 Reflexive-idempotents-property on rings (Tai Keun Kwak)
- 곽태근(대진대)
Tai Keun Kwak, Daejin University
The reflexive property for ideals was introduced by Mason and has important roles in noncommutative ring theory. In this note we study the structure of idempotents satisfying the reflexive property and introduce reflexive-idempotents-property (simply, RIP) as a generalization. It is proved that the RIP can go up to polynomial rings, power series rings, and Dorroh extentions. The structure of non-Abelian RIP rings of minimal order (with or without identity) is completely investigated.
2010 Mathematics Subject Classification: 16S99, 16U80
Key Words and Phrases: reflexive property, reflexive-idempotents-property (RIP), polynomial ring, Dorroh extension, minimal RIP ring
- ⋅ 25th-B'-11:00 − 11:20 Valuation overrings of a Noetherian domain (Gyu Whan Chang, Dong Yeol Oh)
- 장규환(인천대), 오동렬*(포항공대)
Gyu Whan Chang, University of Incheon, Dong Yeol Oh*, POSTECH
Let $R$ be a Noetherian domain and $0 = P_0 \subsetneq P_1 \subsetneq \cdots \subsetneq P_n$ be a saturated chain of prime ideals of $R$. Let $V$ be a valuation overring of $R$ that has a chain of prime ideals $\{Q_{\alpha}\}_{\alpha \in \Lambda}$ such that $\{Q_{\alpha} \cap R\}_{\alpha \in \Lambda} = \{P_i\}_{i=0}^n$. In this talk, we prove that $\{Q_{\alpha}\}_{\alpha \in \Lambda} = \{0 = Q_0 \subsetneq Q_1 \subsetneq \cdots \subsetneq Q_n\}$ and $V_{Q_n}$ is discrete, i.e., $Q_iV_{Q_i}$ is principal for all $i = 1, \dots , n$. Let $D$ be an integral domain with property that for any prime ideal $P$ of $D$, ht$(P) < \infty$ if and only if $P$ is finitely generated. As a corollary, we show that if $\{P_k\}$ is a chain of prime ideals of $D$ such that ht$P_k < \infty$ for each $k$, then there exists a discrete valuation overring of $D$ which has a chain of prime ideals lying over $\{P_k\}$.
2010 Mathematics Subject Classification: 13A15, 13A18, 13E05
Key Words and Phrases: Noetherian domain, saturated chain of prime ideals, discrete valuation domain
- ⋅ 25th-B'-11:30 − 11:50 The minimal prime spectrum of rings with annihilator conditions (Chan Yong Hong, Nam Kyun Kim, Yang Lee, Pace P. Nielsen)
- 홍찬용(경희대), 김남균*(한밭대), 이양(부산대), Pace P. Nielsen(Brigham Young Univ.)
Chan Yong Hong, Kyung Hee University, Nam Kyun Kim*, Hanbat National University, Yang Lee, Pusan National University, Pace P. Nielsen, Brigham Young University
In this paper we study rings with the annihilator condition (a.c.)\ and rings whose space of minimal prime ideals, $\text{Min}(R)$, is compact. We begin by extending the definition of (a.c.) to noncommutative rings. We then show that several extensions over semiprime rings have (a.c.). Moreover, we investigate the annihilator condition under the formation of matrix rings and classical quotient rings. Finally, we prove that if $R$ is a reduced ring then: the classical right quotient ring $Q(R)$ is strongly regular if and only if $R$ has a Property (A) and $\text{Min}(R)$ is compact, if and only if $R$ has (a.c.) and $\text{Min}(R)$ is compact. This extends several results about commutative rings with (a.c.) to the noncommutative setting.
2010 Mathematics Subject Classification: 16D25
Key Words and Phrases: compact space of minimal prime ideals, annihilator condition (a.c.), Property (A)
- ⋅ 25th-B'-11:50 − 12:10 On archimedean rings (Jung Wook Lim)
- 임정욱(경북대)
Jung Wook Lim, Kyungpook National University
A commutative ring $R$ is said to be archimedean if $\bigcap_{n \geq 1} r^nR=(0)$ for all nonzero nonunits $r\in R$. In this talk, we investigate archimedean rings in special pullbacks.
2010 Mathematics Subject Classification: 13E99
Key Words and Phrases: Archimedean ring, strongly presimplifiable
- ⋅ 25th-B'-12:10 − 12:30 Some studies on a generalization of Artinian rings (Chang Ik Lee)
- 이창익(부산대)
Chang Ik Lee, Pusan National University
In this talk, we investigate a ring whose finitely generated subring is right Artinian. Such a ring is said to be locally right Artinian. First we show that there is a ring which is locally right Artinian but not right Artinian. We also show properties and structures of locally right Artinian rings. Moreover we give some concept which satisfies two conditions are equivalent.
2010 Mathematics Subject Classification: 16P20
Key Words and Phrases: Artinian rings, locally finite rings
- ⋅ 25th-C-13:30 − 14:30 Chair: Tai Keun Kwak (Daejin University)
- ⋅ 25th-C-13:30 − 13:50 Ideal-symmetric and semiprime rings (Yang Lee)
- 이양(부산대)
Yang Lee, Pusan National University
Lambek extended the usual commutative ideal theory to ideals in noncommutative rings, calling an ideal $A$ of a ring $R$ symmetric if $rst \in A$ implies $rts\in A$ for $r, s, t\in R$. $R$ is usually called symmetric if $0$ is a symmetric ideal. This naturally gives rise to extending the study of symmetric ring property to the lattice of ideals. In the process we introduce the concept of an {\it ideal-symmetric} ring. We first characterize the class of ideal-symmetric rings and show that this ideal-symmetric property is Morita invariant. We provide a method of constructing an ideal-symmetric ring (but not semiprime) from any given semiprime ring, noting that semiprime rings are ideal-symmetric. We investigate the structure of minimal ideal-symmetric rings completely, finding two kinds of basic forms of finite ideal-symmetric rings. It is also shown that the ideal-symmetric property can go up to right quotient rings in relation with regular elements. The polynomial ring $R[x]$ over an ideal-symmetric ring $R$ need not be ideal-symmetric, but it is shown that the factor ring $R[x]/x^nR[x]$ is ideal-symmetric over a semiprime ring $R$.
2010 Mathematics Subject Classification: 16U80, 16S70
Key Words and Phrases: ideal-symmetric ring, semiprime ring, symmetric ring, matrix ring, polynomial ring, right quotient ring
- ⋅ 25th-C-13:50 − 14:10 On right primelike ring (Hong Kee Kim)
- 김홍기(경상대)
Hong Kee Kim, Gyeongsang National University
We consider a ring property that is stronger than the behavior of right half part of primeness, introducing right primelike. It is shown that right primelike and prime are independent of each other. The primelike property is shown to be left-right symmetric for left or right Artinian rings. It is proved that the right primelike property can go up to classical right quotient rings. The class of primelike rings contains semiprime right Goldie rings and von Neumann regular rings.
2010 Mathematics Subject Classification: 16N40, 16P99
Key Words and Phrases: right primelike ring, matrix ring, polynomial ring, classical right quotient ring, von Neumann regular ring
- ⋅ 25th-C-14:10 − 14:30 Graph of equivalence classes of zero divisors of a graph-designable ring (Juncheol Han, Yang Lee, Sangwon Park)
- 한준철*(부산대), 이양(부산대), 박상원 (동아대)
Juncheol Han*, Pusan National University, Yang Lee, Pusan National University, Sangwon Park, Dong-A University
Let $R$ be a ring with identity, $X$ be the set of all nonzero, nonunits of $R$ and $G$ be the group of all units of $R$. A ring $R$ is called $graph$-$designable$ $ring$ if $[x]_{\ell} = [x]_{r}$ for all $x \in R$ where $[x]_{\ell} = \{ux \ | \ u \in G\}$ (resp. $[x]_{r} = \{xu \ | \ u \in G\}$) which are equivalence classes on $X$. We study the zero divisor graph (denoted $\widetilde{\Gamma} (R)$) determined by equivalence classes of zero divisors of a graph-designable ring $R$. We find some class of graph-designable rings and investigate some properties of $\widetilde{\Gamma}(R)$ including $\widetilde{\Gamma} (\mathbb Z_{n})$ over the ring of integers modulo $n$.
2010 Mathematics Subject Classification: 05C20, 16P20
Key Words and Phrases: graph-designable ring, zero divisor graph
- Geometry, Dynamics, and Operator algebras
- ⋅ 25th-A-09:00 − 11:00 Chair: Hyun Ho Lee (University of Ulsan)
- ⋅ 25th-A-09:00 − 09:30 Finite group actions on higher dimensional noncommutative tori (Ja A Jeong, Jae Hyup Lee)
- 정자아 (서울대), 이재협*(서울대)
Ja A Jeong, Seoul National University, Jae Hyup Lee*, Seoul National University
The group $SL_2(\mathbb Z)$ acts canonically as automorphisms on the 2-dimensional rotation algebras $A_\theta$, $\theta\in \mathbb R$. If we denote the action restricted on its finite subgroups that are necessarily isomorphic to $\mathbb Z_n$($n=2,3,4,6$) by $\alpha$, the crossed products $A_\theta\times_\alpha \mathbb Z_n$ is recently known to be classifiable via Huaxin Lin's classification theorem. Moreover the classification theorem together with K-theoretic data of the crossed products shows that every $A_\theta\times_\alpha \mathbb Z_n$ is an AF algebra. It would be interesting to explore the same question for higher dimensional noncommutative tori. For this we first need to have actions of finite groups on higher dimensional tori. In this paper concrete finite group actions on 4-dimensional tori that are isomorphic to a 2-dimensional rotation algebra tensored by itself are provided, and then it is shown that their crossed products are classifiable via Lin's theorem. We also show that simple 3-dimensional tori do not admit any canonical finite group actions but the flip action.
2010 Mathematics Subject Classification: 46L35, 46L55
Key Words and Phrases: noncommutative torus, crossed product
- ⋅ 25th-A-09:30 − 10:10 Quasidiagonal unitary representations of groups (Caleb Eckhardt)
- Caleb Eckhardt(Univ. of Miami (Ohio))
Caleb Eckhardt, University of Miami (Ohio)
Quasidiagonality is a natural representation theoretic property of $C^*$-algebras, but has not been heavily studied with regards to unitary group representations. In this talk we discuss some recent progress in this direction concerning polycyclic groups.
2010 Mathematics Subject Classification: 46L35, 43A07
Key Words and Phrases: quasidiagonality, unitary representation
- ⋅ 25th-A-10:20 − 11:00 Uniform Roe algebra and coarse amenability for discrete metric spaces (Hiroki Sako)
- Hiroki Sako(Tokai Univ.)
Hiroki Sako, Tokai University
We study property A for coarse (metric) spaces introduced by Guoliang Yu. Property A is an amenability-type condition, which is less restrictive than amenability for groups. The property is often called coarse amenability. Skandalis, Tu, and Yu clarified a connection with amenability in the theory of operator algebras. They proved that a coarse metric space $X$ is coarsely amenable if and only if $C^*_u(X)$ is nuclear. We prove that exactness and local reflexivity of $C^*_u(X)$ also characterize property A.
2010 Mathematics Subject Classification: 22D25, 46L05
Key Words and Phrases: uniform Roe algebra, coarse metric space, coarse amenability
- ⋅ 25th-B'-11:10 − 12:40 Chair: Ja A Jeong (Seoul National University)
- ⋅ 25th-B'-11:10 − 11:50 $C^*$-algebras arising from classification problems in symbolic dynamics (Uijin Jung)
- 정의진(아주대)
Uijin Jung, Ajou University
Cuntz-Krieger algebras are $C^*$-algebras associated with subshifts of finite type. In 90s, Matsumoto associated to each subshift a $C^*$-algebra which becomes a Cuntz-Krieger if the subshift is of finite type. These $C^*$-algebras are used to characterize several equivalences between subshifts. In this talk, we introduce several links between the classification problems in symbolic dynamics and isomorphisms of $C^*$-algebras.
2010 Mathematics Subject Classification: 37B10, 46L35
Key Words and Phrases: subshift of finite type, shift space, Cuntz-Krieger algebra
- ⋅ 25th-B'-12:00 − 12:40 Weighted Fourier algebras and complexification of Lie groups: non-compact cases (Hun Hee Lee)
- 이훈희(서울대)
Hun Hee Lee, Seoul National University
In this talk we will demonstrate that the spectrum of weighted Fourier algebras are closely related to the complexification of Lie groups focusing on non-compact cases. Our examples include the full and reduced Heisenberg groups and the ax+b group.
2010 Mathematics Subject Classification: 47L25, 46B07
Key Words and Phrases: Fourier algebra, complexfication of Lie groups, Heisenberg group
- Poster Session
- ⋅ 25th-C-10:00 − 15:00 Display Time(게시)\ \ /14:30 − 15:00 Q $\&$ A(질의응답) Chair:
- ⋅ 25th-C-10:00 − 15:00 A numerical scheme for pricing barrier options by the domain decomposition (Hyeong-Ohk Bae, Gwang-Eun Choi, Yongsik Kim)
- 배형옥(아주대), 최광은*(아주대), 김용식(아주대)
Hyeong-Ohk Bae, Ajou University, Gwang-Eun Choi*, Ajou University, Yongsik Kim, Ajou University
We apply the finite difference scheme for pricing barrier options, which has the second order accuracy in time. The barrier structure of the option is one of the most common features in financial derivatives, and it is appeared as knock-in, knock-out or early redemption conditions. It is well known that the conventional Crank-Nicholson scheme could produce worse result than the implicit scheme for pricing the barrier option due to the numerical jump on the barrier level. We consider the barrier level as an interface of decomposed domains, so that we could impose the Poincare-Steklov operator on the barrier level. The advantage of our scheme is that we do not solve the partial differential equations on the barrier level and the Crank-Nicholson type time discretization could be blended. We tested our algorithm to several benchmark problems including real world problems such as the Equity-Linked-Security(ELS).
2010 Mathematics Subject Classification: 65N06
Key Words and Phrases: barrier option, FDM, Crank-Nicholson
- ⋅ 25th-C-10:00 − 15:00 A finite difference scheme for pricing of the participating life insurance (Hyeong-Ohk Bae, Tae-Hee Hong, Yongsik Kim)
- 배형옥(아주대), 홍태희*(아주대), 김용식(아주대)
Hyeong-Ohk Bae, Ajou University, Tae-Hee Hong*, Ajou University, Yongsik Kim, Ajou University
We apply a finite difference scheme to evaluate the premium of an insurance which have complicated participating payoff structure. The participating policy is an insurance contract such that the policy holder is to be paid dividends. Since the rebate structure of the objective insurance is quite similar to an exotic option (Cliquet option) in the financial derivatives market, we apply theories and numerical schemes from option pricing problems to the premium pricing. For that, we solve systems of one dimensional Black-Scholes type partial differential equations where the size of the system shrinks annually. In this article, we suggest parallel computation techniques such as MPI(Message Passing Interface) for improving computational efficiency in our algorithm. We numerically validated our algorithm through several benchmark problems including complicated contract such as the participating policy with mortality
2010 Mathematics Subject Classification: 97M30
Key Words and Phrases: participating life insurance, finite difference, Black-Scholes equation, MPI, option pricing
- ⋅ 25th-C-10:00 − 15:00 Numerical method for pricing discretely monitored Barrier option (Yicheng Hong, Sungchul Lee)
- 홍의성*(연세대), 이승철(연세대)
Yicheng Hong*, Yonsei University, Sungchul Lee, Yonsei University
A Barrier option is a path-dependent exotic option that become activated (or extinguished) only if the underlying reaches a predetermined level (the barrier). Most models assume that the barrier is continuously monitored. However in practice most, the terms of a contract state that the asset price, S, is observed periodically. Unlike their continuous counterparts, there is essentially no closed form solution available. Some authors provide a way of adjusting the formulas for the continuous situation where the price of the underlying is observed discretely. However, this change continuous model to approximate method of discrete model not only convergence rate is slow, but also have great differences with real option price. In this paper, we develops a numerical pricing method for discretely monitored Barrier option which monitoring of the barrier condition is done at discrete time intervals. Sullivan (2000) used the Gauss-Legendre quadrature and the Chebyshev approximation to price the American put option. Starting from his work we construct a systematic way of pricing the discretely monitored Barrier option with long-term maturity using numerical integrations. At each monitored time we approximate the option values multiplied by the discounted transition density function as a whole by the Chebyshev approximation. Then, at one step monitored time we evaluate the option values by the exact integration of the Chebyshev approximation, i.e., by the summation of all the Chebyshev extremal nodes, which is known as the Clenshaw-Curtis weights at Chebyshev extremal nodes, which is known as the Chenshaw-Curtis quadrature. In this way we construct a recombining tree and using this recombining tree we price the discretely monitored Barrier option.So we call this pricing method the recombining Clenshaw-Curtis quadrature. We search several different quadratures, construct recombining trees, and devolop pricing methods. We compare the performance of various recombining quadratures with the benchmark trapezoidal and Simpson method. Among all these pricing methods including the trapezoidal and Simpson method we find our recombining Clenshaw-Curtis and Gauss-Legendre-Lobatto quadrature are by far the most efficient and accurate in pricing the discretely monitored Barrier option.
2010 Mathematics Subject Classification: 62P05
Key Words and Phrases: Barrier option, quadrature, Chebyshev polynomial