• Title/Summary/Keyword: projective dimension

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The deformation space of real projective structures on the $(^*n_1n_2n_3n_4)$-orbifold

  • Lee, Jungkeun
    • Bulletin of the Korean Mathematical Society
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    • v.34 no.4
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    • pp.549-560
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    • 1997
  • For positive integers $n_i \geq 2, i = 1, 2, 3, 4$, such that $\Sigma \frac{n_i}{1} < 2$, there exists a quadrilateral $P = P_1 P_2 P_3 P_4$ in the hyperbolic plane $H^2$ with the interior angle $\frac{n_i}{\pi}$ at $P_i$. Let $\Gamma \subset Isom(H^2)$ be the (discrete) group generated by reflections in each side of $P$. Then the quotient space $H^2/\gamma$ is a differentiable orbifold of type $(^* n_1 n_2 n_3 n_4)$. It will be shown that the deformation space of $Rp^2$-structures on this orbifold can be mapped continuously and bijectively onto the cell of dimension 4 - \left$\mid$ {i$\mid$n_i = 2} \right$\mid$$.

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ON THE MINIMUM LENGTH OF SOME LINEAR CODES OF DIMENSION 6

  • Cheon, Eun-Ju;Kato, Takao
    • Bulletin of the Korean Mathematical Society
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    • v.45 no.3
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    • pp.419-425
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    • 2008
  • For $q^5-q^3-q^2-q+1{\leq}d{\leq}q^5-q^3-q^2$, we prove the non-existence of a $[g_q(6,d),6,d]_q$ code and we give a $[g_q(6,d)+1,6,d]_q$ code by constructing appropriate 0-cycle in the projective space, where $g_q (k,d)={{\sum}^{k-1}_{i=0}}{\lceil}\frac{d}{q^i}{\rceil}$. Consequently, we have the minimum length $n_q(6,d)=g_q(6,d)+1\;for\;q^5-q^3-q^2-q+1{\leq}d{\leq}q^5-q^3-q^2\;and\;q{\geq}3$.

DERIVED FUNCTOR COHOMOLOGY GROUPS WITH YONEDA PRODUCT

  • Husain, Hafiz Syed;Sultana, Mariam
    • The Pure and Applied Mathematics
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    • v.28 no.2
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    • pp.187-198
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    • 2021
  • This work presents an exposition of both the internal structure of derived category of an abelian category D*(𝓐) and its contribution in solving problems, particularly in algebraic geometry. Calculation of some morphisms will be presented between objects in D*(𝓐) as elements in appropriate cohomology groups along with their compositions with the help of Yoneda construction under the assumption that the homological dimension of D*(𝓐) is greater than or equal to 2. These computational settings will then be considered under sheaf cohomological context with a particular case from projective geometry.

TOTALLY REAL AND COMPLEX SUBSPACES OF A RIGHT QUATERNIONIC VECTOR SPACE WITH A HERMITIAN FORM OF SIGNATURE (n, 1)

  • Sungwoon Kim
    • Journal of the Korean Mathematical Society
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    • v.61 no.3
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    • pp.547-564
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    • 2024
  • We study totally real and complex subsets of a right quarternionic vector space of dimension n + 1 with a Hermitian form of signature (n, 1) and extend these notions to right quaternionic projective space. Then we give a necessary and sufficient condition for a subset of a right quaternionic projective space to be totally real or complex in terms of the quaternionic Hermitian triple product. As an application, we show that the limit set of a non-elementary quaternionic Kleinian group 𝚪 is totally real (resp. commutative) with respect to the quaternionic Hermitian triple product if and only if 𝚪 leaves a real (resp. complex) hyperbolic subspace invariant.

Incremental Linear Discriminant Analysis for Streaming Data Using the Minimum Squared Error Solution (스트리밍 데이터에 대한 최소제곱오차해를 통한 점층적 선형 판별 분석 기법)

  • Lee, Gyeong-Hoon;Park, Cheong Hee
    • Journal of KIISE
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    • v.45 no.1
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    • pp.69-75
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    • 2018
  • In the streaming data where data samples arrive sequentially in time, it is difficult to apply the dimension reduction method based on batch learning. Therefore an incremental dimension reduction method for the application to streaming data has been studied. In this paper, we propose an incremental linear discriminant analysis method using the least squared error solution. Instead of computing scatter matrices directly, the proposed method incrementally updates the projective direction for dimension reduction by using the information of a new incoming sample. The experimental results demonstrate that the proposed method is more efficient compared with previously proposed incremental dimension reduction methods.

$\mathcal{F}_{\mathcal{S}}$-MITTAG-LEFFLER MODULES AND GLOBAL DIMENSION RELATIVE TO $\mathcal{F}_{\mathcal{S}}$-MITTAG-LEFFLER MODULES

  • Chen, Mingzhao;Wang, Fanggui
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.4
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    • pp.961-976
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    • 2019
  • Let R be any commutative ring and S be any multiplicative closed set. We introduce an S-version of $\mathcal{F}$-Mittag-Leffler modules, called $\mathcal{F}_{\mathcal{S}}$-Mittag-Leffler modules, and define the projective dimension with respect to these modules. We give some characterizations of $\mathcal{F}_{\mathcal{S}}$-Mittag-Leffler modules, investigate the relationships between $\mathcal{F}$-Mittag-Leffler modules and $\mathcal{F}_{\mathcal{S}}$-Mittag-Leffler modules, and use these relations to describe noetherian rings and coherent rings, such as R is noetherian if and only if $R_S$ is noetherian and every $\mathcal{F}_{\mathcal{S}}$-Mittag-Leffler module is $\mathcal{F}$-Mittag-Leffler. Besides, we also investigate the $\mathcal{M}^{\mathcal{F}_{\mathcal{S}}$-global dimension of R, and prove that $R_S$ is noetherian if and only if its $\mathcal{M}^{\mathcal{F}_{\mathcal{S}}$-global dimension is zero; $R_S$ is coherent if and only if its $\mathcal{M}^{\mathcal{F}_{\mathcal{S}}$-global dimension is at most one.

BALANCE FOR RELATIVE HOMOLOGY WITH RESPECT TO SEMIDUALIZING MODULES

  • Di, Zhenxing;Zhang, Xiaoxiang;Chen, Jianlong
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.1
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    • pp.137-147
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    • 2015
  • We derive in the paper the tensor product functor -${\otimes}_R$- by using proper $\mathcal{GP}_C$-resolutions, where C is a semidualizing module. After giving several cases in which different relative homologies agree, we use the Pontryagin duals of $\mathcal{G}_C$-projective modules to establish a balance result for such relative homology over a Cohen-Macaulay ring with a dualizing module D.

HOLOMORPHIC MAPS ONTO KÄHLER MANIFOLDS WITH NON-NEGATIVE KODAIRA DIMENSION

  • Hwang, Jun-Muk;Peternell, Thomas
    • Journal of the Korean Mathematical Society
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    • v.44 no.5
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    • pp.1079-1092
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    • 2007
  • This paper studies the deformation theory of a holomorphic surjective map from a normal compact complex space X to a compact $K\"{a}hler$ manifold Y. We will show that when the target has non-negative Kodaira dimension, all deformations of surjective holomorphic maps $X{\rightarrow}Y$ come from automorphisms of an unramified covering of Y and the underlying reduced varieties of associated components of Hol(X, Y) are complex tori. Under the additional assumption that Y is projective algebraic, this was proved in [7]. The proof in [7] uses the algebraicity in an essential way and cannot be generalized directly to the $K\"{a}hler$ setting. A new ingredient here is a careful study of the infinitesimal deformation of orbits of an action of a complex torus. This study, combined with the result for the algebraic case, gives the proof for the $K\"{a}hler$ setting.

MARTENS' DIMENSION THEOREM FOR CURVES OF EVEN GONALITY

  • Kato, Takao
    • Journal of the Korean Mathematical Society
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    • v.39 no.5
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    • pp.665-680
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    • 2002
  • For a smooth projective irreducible algebraic curve C of odd gonality, the maximal possible dimension of the variety of special linear systems ${W^r}_d$(C) is d-3r by a result of M. Coppens et at. [4]. This bound also holds if C does not admit an involution. Furthermore it is known that if dim ${W^r}_d(C)qeq$ d-3r-1 for a curve C of odd gonality, then C is of very special type of curves by a recent progress made by G. Martens [11] and Kato-Keem [9]. The purpose of this paper is to pursue similar results for curves of even gonality which does not admit an involution.

SOME ASPECTS OF ZARISKI TOPOLOGY FOR MULTIPLICATION MODULES AND THEIR ATTACHED FRAMES AND QUANTALES

  • Castro, Jaime;Rios, Jose;Tapia, Gustavo
    • Journal of the Korean Mathematical Society
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    • v.56 no.5
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    • pp.1285-1307
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    • 2019
  • For a multiplication R-module M we consider the Zariski topology in the set Spec (M) of prime submodules of M. We investigate the relationship between the algebraic properties of the submodules of M and the topological properties of some subspaces of Spec (M). We also consider some topological aspects of certain frames. We prove that if R is a commutative ring and M is a multiplication R-module, then the lattice Semp (M/N) of semiprime submodules of M/N is a spatial frame for every submodule N of M. When M is a quasi projective module, we obtain that the interval ${\uparrow}(N)^{Semp}(M)=\{P{\in}Semp(M){\mid}N{\subseteq}P\}$ and the lattice Semp (M/N) are isomorphic as frames. Finally, we obtain results about quantales and the classical Krull dimension of M.