• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.83-96
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• 2013

Let K be a number field and fix a prime number $p$. For any set S of primes of K, we here say that an elliptic curve E over K has S-reduction if E has bad reduction only at the primes of S. There exists the set $B_{K,p}$ of primes of K satisfying that any elliptic curve over K with $B_{K,p}$-reduction has no $p$-torsion points under certain conditions. The first aim of this paper is to construct elliptic curves over K with $B_{K,p}$-reduction and a $p$-torsion point. The action of the absolute Galois group on the $p$-torsion subgroup of E gives its associated Galois representation $\bar{\rho}_{E,p}$ modulo $p$. We also study the irreducibility and surjectivity of $\bar{\rho}_{E,p}$ for semistable elliptic curves with $B_{K,p}$-reduction.

• Communications of the Korean Mathematical Society
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• v.27 no.3
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• pp.497-503
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• 2012

In this paper, we study elliptic curves E over $\mathbb{Q}$ such that the 3-torsion subgroup E[3] is split as ${\mu}_3{\oplus}\mathbb{Z}/3{\mathbb{Z}}$. For a non-zero intege $m$, let $C_m$ denote the curve $x^3+y^3=m$. We consider the relation between the set of integral points of $C_m$ and the elliptic curves E with $E[3]{\simeq}{\mu}_3{\oplus}\mathbb{Z}/3{\mathbb{Z}}$.

• Kyungpook Mathematical Journal
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• v.56 no.3
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• pp.755-761
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• 2016

Let A be an abelian variety over a global field K. We show that, in "many" cases, Chen-Kuan's invariant M(A[n]), that is the average number of n-torsion points of A over various residue fields of K, has the minimal possible value.

• Journal of Biomedical Engineering Research
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• v.26 no.3
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• pp.133-138
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• 2005

This paper presents a new method for measuring ocular torsion using the optical flow. Images of the iris were cropped and transformed into rectangular images that were orientation invariant. Feature points of the iris region were selected from a reference and a target image, and the shift of each feature was calculated using the iterative Lucas-Kanade method. The feature points were selected according to the strength of the corners on the iris image. The accuracy of the algorithm was tested using printed eye images. In these images, torsion was measured with $0.15^{\circ}$ precision. The proposed method shows robustness even with the gaze directional changes and pupillary reflex environment of real-time processing.

• Journal of the Korean Mathematical Society
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• v.55 no.4
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• pp.1005-1018
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• 2018

Let ${\Sigma}_{g,n,b}$ denote the orientable surface obtained from the closed orientable surface ${\Sigma}_g$ of genus $g{\geq}2$ by deleting the interior of $n{\geq}1$ distinct topological disks and $b{\geq}1$ points. Using the notion of symplectic chain complex, the present paper establishes a formula for computing Reidemeister torsion of the surface ${\Sigma}_{g,n,b}$ in terms of Reidemeister torsion of the closed surface ${\Sigma}_g$, Reidemeister torsion of disk, and Reidemeister torsion of punctured disk.

• Honam Mathematical Journal
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• v.46 no.2
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• pp.279-290
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• 2024

In this study, tubular surfaces were introduced according to the modified orthogonal frame defined at the points where the torsion is different from zero in the 3-dimensional Euclidean space. First, the relations between the Frenet frame and the modified orthogonal frame with torsion are given. Then, the singularity, Gaussian curvature, mean curvature and basic forms of the tubular surface given according to the modified orthogonal frame with torsion were calculated. In addition, the conditions for the parameter curves of the tubular surface to be geodesic, asymptotic and line of curvature were examined. Finally, tubular surface examples based on both the Frenet frame and the modified orthogonal frame with torsion were given to support the study.

• Journal of the Chungcheong Mathematical Society
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• v.33 no.2
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• pp.227-236
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• 2020

A set {a₁, a₂,_…, a_m} of positive integers is called Diophantine m-tuple if a_ia_j+1 is a perfect square for all 1 ≤ i < j ≤ m. In this paper, we find the structure of torsion group of elliptic curve E_k constructed by Diophantine triple, and find all integer points on E_k under assumption that rank(E_k(ℚ)) = 1.

• Honam Mathematical Journal
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• v.20 no.1
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• pp.1-14
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• 1998

In this short note we study the torsion theories over a commutative ring R and discuss a relative dimension related to such theories for R-modules. Let ${\sigma}$ be a torsion functor and (T, F) be its corresponding partition of Spec(R). The concept of ${\sigma}$-Cohen Macaulay (abbr. ${\sigma}$-CM) module is defined and some of the main points concerning the usual Cohen-Macaulay modules are extended. In particular it is shown that if M is a non-zero ${\sigma}$-CM module over R and S is a multiplicatively closed subset of R such that, for all minimal element of T, $S{\cap}p={\emptyset}$, then $S^{-1}M$ is a $S^{-1}{\sigma}$-CM module over $S^{-1}$R, where $S^{-1}{\sigma}$ is the direct image of ${\sigma}$ under the natural ring homomorphism $R{\longrightarrow}S^{-1}R$.

• Kyungpook Mathematical Journal
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• v.56 no.4
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• pp.1135-1140
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• 2016

Let A be an abelian variety over a global field K. We know [6, 7] that, in many cases, the average number of n-torsion points of A over various residue fields of K, takes the minimal possible value. In this article, we study several defect cases by calculating the number of Galois orbits.

• Journal of the Korean Orthopaedic Association
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• v.55 no.1
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• pp.62-70
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• 2020

Purpose: External tibia torsion and proximal tibial vara have been reported in severe varus deformed osteoarthritis, which is a tibio-femoral angle of more than 20°. The radiology measurements were compared with those of control group and the preoperative and follow-up radiology and clinical results were examined. Materials and Methods: From January 2007 to March 2016, 43 knees from 37 persons, who underwent total knee arthroplasty for a severe varus deformity of more than 20° on the tibio-femoral angle on the standing radiographs and had a follow-up period more than two years, were examined. The mean follow-up period was 45.7 months. The control group, who underwent conservative treatments, had Kellgren-Lawrence grade three osteoarthritis and a tibio-femoral angle of less than 3° varus. The external tibial torsion of enrolled patients and control group were estimated using the proximal tibio-fibular overlap length and the tibial torsion values on computed tomography. The proximal tibia vara was measured using the proximal tibial tilt angle. The preoperative and postoperative proximal tibio-fibular overlap length, tibial torsion value, proximal tibial tilt angle, and hospital for special surgery (HSS) score were evaluated. Results: The mean proximal tibio-fibular overlap length was 18.6 mm preoperatively and 11.2 mm (p=0.031) at the follow-up. The control group had a mean proximal tibio-fibular overlap length of 8.7 mm (p=0.024). The mean tibial torsion value was 13.8° preoperatively and 14.0° (p=0.489) at the follow-up. The control group had a mean tibial torsion value of 21.9° (p=0.012). The mean proximal tibial tilt angle was 12.2° preoperatively and 0° (p<0.01) at the follow-up. The control group had a mean proximal tilt angle of 1.2° (p<0.01). The preoperative tibiofemoral angle and mechanical axis deviation were corrected from preoperative 28.3° and medial 68.4 mm to postoperative 0.7° and medial 3.5 mm (p<0.01, p<0.01), respectively. The HSS scores increased from 34 points of preoperatively to 87 points at the last follow-up (p=0.028). Conclusion: Patients with advanced osteoarthritis with a severe varus deformity of more than 20° had significant increases in the external tibial torsion and varus of the proximal tibia. The tibial torsion value before and after surgery in the enrolled patients was not changed statistically, but good clinical results without complications were obtained.