• 제목/요약/키워드: orientation-preserving

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Equivalence of ℤ4-actions on Handlebodies of Genus g

  • Prince-Lubawy, Jesse
    • Kyungpook Mathematical Journal
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    • 제56권2호
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    • pp.577-582
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    • 2016
  • In this paper we consider all orientation-preserving ${\mathbb{Z}}_4$-actions on 3-dimensional handlebodies $V_g$ of genus g > 0. We study the graph of groups (${\Gamma}(v)$, G(v)), which determines a handlebody orbifold $V({\Gamma}(v),G(v)){\simeq}V_g/{\mathbb{Z}}_4$. This algebraic characterization is used to enumerate the total number of ${\mathbb{Z}}_4$ group actions on such handlebodies, up to equivalence.

Finite, Fiber-preserving Group Actions on Elliptic 3-manifolds

  • Peet, Benjamin
    • Kyungpook Mathematical Journal
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    • 제62권2호
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    • pp.363-388
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    • 2022
  • In two previous papers the author presented a general construction of finite, fiber- and orientation-preserving group actions on orientable Seifert manifolds. In this paper we restrict our attention to elliptic 3-manifolds. For illustration of our methods a constructive proof is given that orientation-reversing and fiber-preserving diffeomorphisms of Seifert manifolds do not exist for nonzero Euler class, in particular elliptic 3-manifolds. Each type of elliptic 3-manifold is then considered and the possible group actions that fit the given construction. This is shown to be all but a few cases that have been considered elsewhere. Finally, a presentation for the quotient space under such an action is constructed and a specific example is generated.

ON THE m-POTENT RANKS OF CERTAIN SEMIGROUPS OF ORIENTATION PRESERVING TRANSFORMATIONS

  • Zhao, Ping;You, Taijie;Hu, Huabi
    • 대한수학회보
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    • 제51권6호
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    • pp.1841-1850
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    • 2014
  • It is known that the ranks of the semigroups $\mathcal{SOP}_n$, $\mathcal{SPOP}_n$ and $\mathcal{SSPOP}_n$ (the semigroups of orientation preserving singular self-maps, partial and strictly partial transformations on $X_n={1,2,{\ldots},n}$, respectively) are n, 2n and n + 1, respectively. The idempotent rank, defined as the smallest number of idempotent generating set, of $\mathcal{SOP}_n$ and $\mathcal{SSPOP}_n$ are the same value as the rank, respectively. Idempotent can be seen as a special case (with m = 1) of m-potent. In this paper, we investigate the m-potent ranks, defined as the smallest number of m-potent generating set, of the semigroups $\mathcal{SOP}_n$, $\mathcal{SPOP}_n$ and $\mathcal{SSPOP}_n$. Firstly, we characterize the structure of the minimal generating sets of $\mathcal{SOP}_n$. As applications, we obtain that the number of distinct minimal generating sets is $(n-1)^nn!$. Secondly, we show that, for $1{\leq}m{\leq}n-1$, the m-potent ranks of the semigroups $\mathcal{SOP}_n$ and $\mathcal{SPOP}_n$ are also n and 2n, respectively. Finally, we find that the 2-potent rank of $\mathcal{SSPOP}_n$ is n + 1.

COREGULARITY OF ORDER-PRESERVING SELF-MAPPING SEMIGROUPS OF FENCES

  • JENDANA, KETSARIN;SRITHUS, RATANA
    • 대한수학회논문집
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    • 제30권4호
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    • pp.349-361
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    • 2015
  • A fence is an ordered set that the order forms a path with alternating orientation. Let F = (F;${\leq}$) be a fence and let OT(F) be the semigroup of all order-preserving self-mappings of F. We prove that OT(F) is coregular if and only if ${\mid}F{\mid}{\leq}2$. We characterize all coregular elements in OT(F) when F is finite. For any subfence S of F, we show that the set COTS(F) of all order-preserving self-mappings in OT(F) having S as their range forms a coregular subsemigroup of OT(F). Under some conditions, we show that a union of COTS(F)'s forms a coregular subsemigroup of OT(F).

Equivalence of Cyclic p-squared Actions on Handlebodies

  • Prince-Lubawy, Jesse
    • Kyungpook Mathematical Journal
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    • 제58권3호
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    • pp.573-581
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    • 2018
  • In this paper we consider all orientation-preserving ${\mathbb{Z}}_{p^2}$-actions on 3-dimensional handlebodies $V_g$ of genus g > 0 for p an odd prime. To do so, we examine particular graphs of groups (${\Gamma}(v)$, G(v)) in canonical form for some 5-tuple v = (r, s, t, m, n) with r + s + t + m > 0. These graphs of groups correspond to the handlebody orbifolds V (${\Gamma}(v)$, G(v)) that are homeomorphic to the quotient spaces $V_g/{\mathbb{Z}}_{p^2}$ of genus less than or equal to g. This algebraic characterization is used to enumerate the total number of ${\mathbb{Z}}_{p^2}$-actions on such handlebodies, up to equivalence.

TYPICALLY REAL HARMONIC FUNCTIONS

  • Jun, Sook Heui
    • Korean Journal of Mathematics
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    • 제8권2호
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    • pp.135-138
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    • 2000
  • In this paper, we study harmonic orientation-preserving univalent mappings defined on ${\Delta}=\{z:{\mid}z{\mid}>1\}$ that are typically real.

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Planar harmonic mappings and curvature estimates

  • Jun, Sook-Heui
    • 대한수학회지
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    • 제32권4호
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    • pp.803-814
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    • 1995
  • Let $\Sigma$ be the class of all complex-valued, harmonic, orientation-preserving, univalent mappings defined on $\Delta = {z : $\mid$z$\mid$ > 1}$ that map $\infty$ to $\infty$.

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HARMONIC MAPPING

  • Jun, Sook Heui
    • Korean Journal of Mathematics
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    • 제10권1호
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    • pp.1-3
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    • 2002
  • In this paper, we obtain some coefficient bounds of harmonic, orientation-preserving, univalent mappings defined on ${\Delta}=\{z:{\mid}z{\mid}>1\}$.

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