• East Asian mathematical journal
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• v.40 no.1
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• pp.75-83
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• 2024

For a reduced, irreducible and nondegenerate curve C ⊂ ℙ^r of degree d, it was shown that the arithmetic genus g of C has an upper bound π₀(d, r) by G. Castelnuovo. And he also classified the curves that attain the extremal value. These curves are arithmetically Cohen-Macaulay and contained in a surface of minimal degree. In this paper, we investigate the arithmetic genus of curves lie on a surface of minimal degree - the Veronese surface, smooth rational normal surface scrolls and singular rational normal surface scrolls. We also provide a construction of curves on singular rational normal surface scroll S(0, 2) ⊂ ℙ³ which attain the maximal arithmetic genus.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.2
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• pp.345-349
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• 2011

In this work, we find the genera of arithmetic subgroups $\langle{\Gamma}_{\Delta}(N),{\Phi}\rangle$ of $GL_2^+(\mathbb{R})$ generated by congruence subgroup ${\Gamma}_{\Delta}(N)$ and the Fricke involution $\Phi$.

• Journal of the Korean Mathematical Society
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• v.61 no.1
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• pp.161-181
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• 2024

We describe the moduli space of Higgs pairs on an irreducible nodal curve of arithmetic genus one and its geometric structures in terms of the Hitchin map and a flat degeneration of the moduli space of Higgs bundles on an elliptic curve.

• Journal of the Chungcheong Mathematical Society
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• v.27 no.4
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• pp.563-569
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• 2014

We investigate a minimal set of generators of a homogeneous ideal of a curve of degree d on a linearly normal smooth surface scroll W in $P^r$ whose arithmetic genus is maximal among curves of degree d on W.

• Journal of the Korean Mathematical Society
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• v.56 no.5
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• pp.1355-1370
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• 2019

A curve $X{\subset}\mathbb{P}^r$ has maximal rank if for each $t{\in}\mathbb{N}$ the restriction map $H^0(\mathcal{O}_{\mathbb{P}r}(t)){\rightarrow}H^0(\mathcal{O}_X(t))$ is either injective or surjective. We show that for all integers $d{\geq}r+1$ there are maximal rank, but not arithmetically Cohen-Macaulay, smooth curves $X{\subset}\mathbb{P}^r$ with degree d and genus roughly $d^2/2r$, contrary to the case r = 3, where it was proved that their genus growths at most like $d^{3/2}$ (A. Dolcetti). Nevertheless there is a sector of large genera g, roughly between $d^2/(2r+2)$ and $d^2/2r$, where we prove the existence of smooth curves (even aCM ones) with degree d and genus g, but the only integral and non-degenerate maximal rank curves with degree d and arithmetic genus g are the aCM ones. For some (d, g, r) with high g we prove the existence of reducible non-degenerate maximal rank and non aCM curves $X{\subset}\mathbb{P}^r$ with degree d and arithmetic genus g, while (d, g, r) is not realized by non-degenerate maximal rank and non aCM integral curves.

• East Asian mathematical journal
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• v.40 no.3
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• pp.287-293
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• 2024

For a nondegenerate irreducible projective variety, it is a classical problem to describe its defining equations and the syzygies among them. In this paper, we precisely determine a minimal generating set and the minimal free resolution of defining ideals of some rational curves of maximal genus in ℙ³.

• Journal of applied mathematics & informatics
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• v.32 no.1_2
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• pp.17-26
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• 2014

In this paper, we present an algorithm for computing the number of points on the Jacobian varieties of genus 3 hyperelliptic curves of type $y^2=x^7+ax$ over finite prime fields. The problem of determining the group order of the Jacobian varieties of algebraic curves defined over finite fields is important not only arithmetic geometry but also curve-based cryptosystems in order to find a secure curve. Based on this, we provide the explicit formula of the characteristic polynomial of the Frobenius endomorphism of the Jacobian variety of hyperelliptic curve $y^2=x^7+ax$ over a finite field $\mathbb{F}_p$ with $p{\equiv}1$ modulo 12. Moreover, we also introduce some implementation results by using our algorithm.

• ETRI Journal
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• v.37 no.1
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• pp.107-117
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• 2015

This paper presents an energy-efficient (low power) prime-field hyperelliptic curve cryptography (HECC) processor with uniform power draw. The HECC processor performs divisor scalar multiplication on the Jacobian of genus 2 hyperelliptic curves defined over prime fields for arbitrary field and curve parameters. It supports the most frequent case of divisor doubling and addition. The optimized implementation, which is synthesized in a $0.13{\mu}m$ standard CMOS technology, performs an 81-bit divisor multiplication in 503 ms consuming only $6.55{\mu}J$ of energy (average power consumption is $12.76{\mu}W$). In addition, we present a technique to make the power consumption of the HECC processor more uniform and lower the peaks of its power consumption.

• Korean Journal of Medicinal Crop Science
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• v.21 no.6
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• pp.455-460
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• 2013

Collected germplasms of five representative species belonging to Curcuma genus (C. longa, C. aromatica, C. zedoaria, C. phaeocaulis and C. kwangsiensis) were 52 samples from different farmhouse in Korea and China. To elucidate the genetic diversity among the species, 52 samples were analyzed by genomic fingerprinting method using amplified fragment length polymorphism (AFLP). AFLP results of 6 primer combinations were revealed 643 total DNA fragments and 349 polymorphic bands with the 54.3% ratio of polymorphism. In the analysis of coefficient similarity using unweight pair group method with arithmetic averages (UPGMA), 52 Curcuma germplasm lines were ranged from 0.60 to 0.99 and clustered distinct five groups according to the species and collected geographical levels. However, the result of principal coordinate analysis (PCA) by multi-variate analysis was shown significantly greater differences among species than geographical origins based on AFLP profiling data of these samples.

• Korean Journal of Plant Taxonomy
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• v.49 no.1
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• pp.13-27
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• 2019

Numerical taxonomy is employed to determine the phenetic proximity of the Egyptian taxa belonging to the genus Ononis L. A classical clustering analysis and a principal component analysis (PCA) were used to separate 57 macro- and micromorphological characters in order to circumscribe 11 taxa of Ononis. A clustering analysis using the unweighted pair-group method with the arithmetic means (UPGMA) method gives the highest co-phenetic correlation. Results from clustering and PCA revealed the segregation of five groups. Our results are in line, to some certain degree, with the traditional sub-sectional concept, as can be seen in the grouping of the representative members of the subsections Diffusae and Mittisimae together and the representative members of the subsections Viscosae and Natrix. The phenetic uniqueness of Ononis variegata and O. reclinata subsp. mollis was formally established. However, our findings contradict the classic sectional concept; this opinion was suggested earlier in previous phylogenetic circumscriptions of the genus. The most useful characters that provide taxonomic clarity were discussed.