• Title/Summary/Keyword: projective dimension

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THE DIMENSION OF THE SPACE OF STABLE MAPS TO THE RELATIVE LAGRANGIAN GRASSMANNIAN OVER A CURVE

  • Daewoong Cheong
    • Journal of the Chungcheong Mathematical Society
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    • v.36 no.1
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    • pp.1-8
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    • 2023
  • Let C be a smooth projective curve and W a symplectic bundle over C of degree w. Let π : 𝕃𝔾(W) → C be the relative Lagrangian Grassmannian over C and Sd(W) be the space of Lagrangian subbundles of degree w -d. Then Kontsevich's space ${\bar{\mathcal{M}}}_g$(𝕃𝔾(W), βd) of stable maps to 𝕃𝔾(W) is a compactification of Sd(W). In this article, we give an upper bound on the dimension of ${\bar{\mathcal{M}}}_g$(𝕃𝔾(W), βd), which is an analogue of a result in [8] for the relative Lagrangian Grassmannian.

ON THE CONSTRUCTION OF OPTIMAL LINEAR CODES OF DIMENSION FOUR

  • Atsuya Kato;Tatsuya Maruta;Keita Nomura
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.5
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    • pp.1237-1252
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    • 2023
  • A fundamental problem in coding theory is to find nq(k, d), the minimum length n for which an [n, k, d]q code exists. We show that some q-divisible optimal linear codes of dimension 4 over 𝔽q, which are not of Belov type, can be constructed geometrically using hyperbolic quadrics in PG(3, q). We also construct some new linear codes over 𝔽q with q = 7, 8, which determine n7(4, d) for 31 values of d and n8(4, d) for 40 values of d.

A Semi-supervised Dimension Reduction Method Using Ensemble Approach (앙상블 접근법을 이용한 반감독 차원 감소 방법)

  • Park, Cheong-Hee
    • The KIPS Transactions:PartD
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    • v.19D no.2
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    • pp.147-150
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    • 2012
  • While LDA is a supervised dimension reduction method which finds projective directions to maximize separability between classes, the performance of LDA is severely degraded when the number of labeled data is small. Recently semi-supervised dimension reduction methods have been proposed which utilize abundant unlabeled data and overcome the shortage of labeled data. However, matrix computation usually used in statistical dimension reduction methods becomes hindrance to make the utilization of a large number of unlabeled data difficult, and moreover too much information from unlabeled data may not so helpful compared to the increase of its processing time. In order to solve these problems, we propose an ensemble approach for semi-supervised dimension reduction. Extensive experimental results in text classification demonstrates the effectiveness of the proposed method.

STALE REDUCTIONS OF SINGULAR PLANE QUARTICS

  • Kang, Pyung-Lyun
    • Communications of the Korean Mathematical Society
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    • v.9 no.4
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    • pp.905-915
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    • 1994
  • Let $M_g$ be the moduli space of isomorphism classes of genus g smooth curves. It is a quasi-projective variety of dimension 3g - 3, when $g > 2$. It is known that a complete subvariety of $M_g$ has dimension $< g-1 [D]$. In general it is not known whether this bound is rigid. For example, it is not known whether $M_4$ has a complete surface in it. But one knows that there is a complete curve through any given finite points [H]. Recently, an explicit example of a complete curve in moduli space is given in [G-H]. In [G-H] they constructed a complete curve of $M_3$ as an intersection of five hypersurfaces of the Satake compactification of $M_3$. One way to get a complete curve of $M_3$ is to find a complete one dimensional family $p : X \to B$ of plane quartics which gives a nontrivial morphism from the base space B to the moduli space $M_3$. This is because every non-hyperelliptic smooth curve of genus three can be realized as a nonsingular plane quartic and vice versa. This paper has come out from the effort to find such a complete family of plane quartics. Since nonsingular quartics form an affine space some fibers of p must be singular ones. In this paper, due to the semistable reduction theorem [M], we search singular plane quartics which can occur as singular fibers of the family above. We first list all distinct plane quartics in terms of singularities.

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REPRESENTATIONS OVER GREEN ALGEBRAS OF WEAK HOPF ALGEBRAS BASED ON TAFT ALGEBRAS

  • Liufeng Cao
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.6
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    • pp.1687-1695
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    • 2023
  • In this paper, we study the Green ring r(𝔴0n) of the weak Hopf algebra 𝔴0n based on Taft Hopf algebra Hn(q). Let R(𝔴0n) := r(𝔴0n) ⊗ ℂ be the Green algebra corresponding to the Green ring r(𝔴0n). We first determine all finite dimensional simple modules of the Green algebra R(𝔴0n), which is based on the observations of the roots of the generating relations associated with the Green ring r(𝔴0n). Then we show that the nilpotent elements in r(𝔴0n) can be written as a sum of finite dimensional indecomposable projective 𝔴0n-modules. The Jacobson radical J(r(𝔴0n)) of r(𝔴0n) is a principal ideal, and its rank equals n - 1. Furthermore, we classify all finite dimensional non-simple indecomposable R(𝔴0n)-modules. It turns out that R(𝔴0n) has n2 - n + 2 simple modules of dimension 1, and n non-simple indecomposable modules of dimension 2.

ON THE ORDER OF SPECIALITY OF A SIMPLE, SPECIAL, AND COMPLETE LINEAR SYSTEM ON A CURVE

  • Ballico, Edoardo;Homma, Masaaki;Ohbuchi, Akira
    • Journal of the Korean Mathematical Society
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    • v.39 no.4
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    • pp.593-609
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    • 2002
  • The order of speciality of an ample invertible Sheaf L on a curve is the least integer m so that $L^{ m}$ is nonspecial. There is a reasonable upper bound of the order of speciality for a simple invertible sheaf in terms of its degree and projective dimension. We study the case where it reaches the upper bound. Moreover we for mulate Castelnuovo's genus bound involving the order of speciality.ality.

CHARACTERIZING ALMOST PERFECT RINGS BY COVERS AND ENVELOPES

  • Fuchs, Laszlo
    • Journal of the Korean Mathematical Society
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    • v.57 no.1
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    • pp.131-144
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    • 2020
  • Characterizations of almost perfect domains by certain covers and envelopes, due to Bazzoni-Salce [7] and Bazzoni [4], are generalized to almost perfect commutative rings (with zero-divisors). These rings were introduced recently by Fuchs-Salce [14], showing that the new rings share numerous properties of the domain case. In this note, it is proved that admitting strongly flat covers characterizes the almost perfect rings within the class of commutative rings (Theorem 3.7). Also, the existence of projective dimension 1 covers characterizes the same class of rings within the class of commutative rings admitting the cotorsion pair (𝒫1, 𝒟) (Theorem 4.1). Similar characterization is proved concerning the existence of divisible envelopes for h-local rings in the same class (Theorem 5.3). In addition, Bazzoni's characterization via direct sums of weak-injective modules [4] is extended to all commutative rings (Theorem 6.4). Several ideas of the proofs known for integral domains are adapted to rings with zero-divisors.

(Σ, ∆)-Compatible Skew PBW Extension Ring

  • Hashemi, Ebrahim;Khalilnezhad, Khadijeh;Alhevaz, Abdollah
    • Kyungpook Mathematical Journal
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    • v.57 no.3
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    • pp.401-417
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    • 2017
  • Ever since their introduction, skew PBW ($Poincar{\acute{e}}$-Birkhoff-Witt) extensions of rings have kept growing in importance, as researchers characterized their properties (such as primeness, Krull and Goldie dimension, homological properties, etc.) in terms of intrinsic properties of the base ring, and studied their relations with other fields of mathematics, as for example quantum mechanics theory. Many rings and algebras arising in quantum mechanics can be interpreted as skew PBW extensions. Our aim in this paper is to study skew PBW extensions of Baer, quasi-Baer, principally projective and principally quasi-Baer rings, in the case when the base ring R is not assumed to be reduced. We just impose some mild compatibleness over the base ring R, and prove that these properties are stable over this kind of extensions.

ON SEMI-REGULAR INJECTIVE MODULES AND STRONG DEDEKIND RINGS

  • Renchun Qu
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.4
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    • pp.1071-1083
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    • 2023
  • The main motivation of this paper is to introduce and study the notions of strong Dedekind rings and semi-regular injective modules. Specifically, a ring R is called strong Dedekind if every semi-regular ideal is Q0-invertible, and an R-module E is called a semi-regular injective module provided Ext1R(T, E) = 0 for every 𝓠-torsion module T. In this paper, we first characterize rings over which all semi-regular injective modules are injective, and then study the semi-regular injective envelopes of R-modules. Moreover, we introduce and study the semi-regular global dimensions sr-gl.dim(R) of commutative rings R. Finally, we obtain that a ring R is a DQ-ring if and only if sr-gl.dim(R) = 0, and a ring R is a strong Dedekind ring if and only if sr-gl.dim(R) ≤ 1, if and only if any semi-regular ideal is projective. Besides, we show that the semi-regular dimensions of strong Dedekind rings are at most one.

North Korean Defectors' depression through the CES-D and the Rorschach test (CES-D와 로샤검사를 통해 본 북한이탈주민의 우울감)

  • Jong-Nam Kim;Yun-Kyung Choi;Jung-Min Chae
    • Korean Journal of Culture and Social Issue
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    • v.14 no.2
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    • pp.41-61
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    • 2008
  • This study was performed to investigate the North Korean defectors's depression through the CES-D and a projective test simultaneously. So 40 North Korean defectors and 40 South Koreans participated in this study. The results are the following. First, the North Korean defectors demonstrated more higher CES-D scores than the South Koreans, while there were not significant differences in the Rorschach depression-related indexes, DEPI, CDI, S-CON between two groups. Second, at the analysis of the sex difference effect, women among the North Korean defectors showed the most high CDI index scores. In addition, we analyzed the mean differences between two groups for morbid content, shading response, achromatic response, human movement response, form dimension response, Lamda, and the number of total response. However, there are not significant differences between two groups for those variables, except for form dimension(FD) response. It suggested the probability that the North Korean women defectors could experience more psychological distress underlyingly than the North Korean men defectors. Third, most importantly, the correlation between CES-D and DEPI in the Rorschach test was significantly different in the two groups(the North Korean defectors and the South Koreans). In conclusion, the self-report of the North Korean defectors on depression could be exaggerated. So we should be cautious to interpret the self report results of the North Korean defectors.

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