• Bulletin of the Korean Mathematical Society
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• v.50 no.4
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• pp.1109-1126
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• 2013

Let $\mathbb{M}^3_q(c)$ denote the 3-dimensional space form of index $q=0,1$, and constant curvature $c{\neq}0$. A curve ${\alpha}$ immersed in $\mathbb{M}^3_q(c)$ is said to be a Bertrand curve if there exists another curve ${\beta}$ and a one-to-one correspondence between ${\alpha}$ and ${\beta}$ such that both curves have common principal normal geodesics at corresponding points. We obtain characterizations for both the cases of non-null curves and null curves. For non-null curves our theorem formally agrees with the classical one: non-null Bertrand curves in $\mathbb{M}^3_q(c)$ correspond with curves for which there exist two constants ${\lambda}{\neq}0$ and ${\mu}$ such that ${\lambda}{\kappa}+{\mu}{\tau}=1$, where ${\kappa}$ and ${\tau}$ stand for the curvature and torsion of the curve. As a consequence, non-null helices in $\mathbb{M}^3_q(c)$ are the only twisted curves in $\mathbb{M}^3_q(c)$ having infinite non-null Bertrand conjugate curves. In the case of null curves in the 3-dimensional Lorentzian space forms, we show that a null curve is a Bertrand curve if and only if it has non-zero constant second Frenet curvature. In the particular case where null curves are parametrized by the pseudo-arc length parameter, null helices are the only null Bertrand curves.

• Honam Mathematical Journal
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• v.38 no.3
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• pp.467-477
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• 2016

In this paper, we study the timelike Bertrand curves in Minkowski 3-space. Since the principal normal vector of a timelike curve is spacelike, the Bertrand mate curve of this curve can be a timelike curve, a spacelike curve with spacelike principal normal or a Cartan null curve, respectively. Thus, by considering these three cases, we get the necessary and sufficient conditions for a timelike curve to be a Bertrand curve. Also we give the related examples.

• Kyungpook Mathematical Journal
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• v.61 no.4
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• pp.805-812
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• 2021

In this study, we obtain various conditions for the non-null curve flows to be inextensible in the 6-dimensional Lorentzian space 𝕃⁶. Then, we find partial differential equations which characterize the family of inextensible non-null curves.

• Bulletin of the Korean Mathematical Society
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• v.52 no.6
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• pp.2071-2093
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• 2015

In this paper, we investigate the similarity transformations in the Minkowski n-space. We study the geometric invariants of non-null curves under the similarity transformations. Besides, we extend the fundamental theorem for a non-null curve according to a similarity motion of ${\mathbb{E}}_1^n$. We determine the parametrizations of non-null self-similar curves in ${\mathbb{E}}_1^n$.

• Communications of the Korean Mathematical Society
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• v.15 no.3
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• pp.533-540
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• 2000

In the present paper, we study a non-degenrate ruled surface along a null curve in a 3-dimensional Minkowski space E31, which is called a null scroll, an investigate some characterizations of null scrolls satisfying the condition H=AH, A Mat(3, ), where denotes the Laplacian of the surface with respect to the induced metric, H the mean curvature vector and Mat(3, ) the set of 3$\times$3-real matrices.

• Bulletin of the Korean Mathematical Society
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• v.60 no.5
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• pp.1299-1320
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• 2023

In the present paper, firstly we obtain the general expression of the canal hypersurfaces that are formed as the envelope of a family of pseudo hyperspheres, pseudo hyperbolic hyperspheres and null hyper-cones whose centers lie on a non-null curve with non-null Frenet vector fields in E⁴₁ and give their some geometric invariants such as unit normal vector fields, Gaussian curvatures, mean curvatures and principal curvatures. Also, we give some results about their flatness and minimality conditions and Weingarten canal hypersurfaces. Also, we obtain these characterizations for tubular hypersurfaces in E⁴₁ by taking constant radius function and finally, we construct some examples and visualize them with the aid of Mathematica.

• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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• pp.1035-1059
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• 2023

Let p be an odd prime. Let E be an elliptic curve defined over a quadratic imaginary field, where p splits completely. Suppose E has supersingular reduction at primes above p. Under appropriate hypotheses, we extend the results of [17] to ℤ²_p-extensions. We define and study the fine double-signed residual Selmer groups in these settings. We prove that for two residually isomorphic elliptic curves, the vanishing of the signed 𝜇-invariants of one elliptic curve implies the vanishing of the signed 𝜇-invariants of the other. Finally, we show that the Pontryagin dual of the Selmer group and the double-signed Selmer groups have no non-trivial pseudo-null submodules for these extensions.

• Journal of the Korean Mathematical Society
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• v.43 no.6
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• pp.1339-1355
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• 2006

This paper develops in detail the differential geometry of ruled surfaces from two perspectives, and presents the underlying relations which unite them. Both scalar and dual curvature functions which define the shape of a ruled surface are derived. Explicit formulas are presented for the computation of these functions in both formulations of the differential geometry of ruled surfaces. Also presented is a detailed analysis of the ruled surface which characterizes the shape of a general ruled surface in the same way that osculating circle characterizes locally the shape of a non-null Lorentzian curve.

• Archives of Pharmacal Research
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• v.28 no.6
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• pp.709-715
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• 2005

The purpose of this study was to investigate the effect of atropine on peripheral vasodilation and the mechanisms involved. The isometric tension of rat mesenteric artery rings was recorded in vitro on a myograph. The results showed that atropine, at concentrations greater than 1$\mu$M, relaxed the noradrenalin (NA)-precontracted rat mesenteric artery in a concentration-dependent manner. Atropine-induced vasodilatation was mediated, in part, by an endothelium-dependent mechanism, to which endothelium-derived hyperpolarizing factor may contribute. Atropine was able to shift the NA-induced concentration-response curve to the right, in a non-parallel manner, suggesting the mechanism of atropine was not mediated via the ${\alpha}_1$-adrenoreceptor. The $\beta$-adrenoreceptor and ATP sensitive potassium channel, a voltage dependent calcium channel, were not involved in the vasodilatation. However, atropine inhibited the contraction derived from NA and $CaCl_2$ in $Ca^{2+}$-free medium, in a concentration dependent manner, indicating the vasodilatation was related to the inhibition of extracellular $Ca^{2+}$ influx through the receptor-operated calcium channels and intracellular $Ca^{2+}$ release from the $Ca^{2+}$ store. Atropine had no effect on the caffeine-induced contraction in the artery segments, indicating the inhibition of intracellular $Ca^{2+}$ release as a result of atropine most likely occurs via the IP3 pathway rather than the ryanodine receptors. Our results suggest that atropine-induced vasodilatation is mainly from artery smooth muscle cells due to inhibition of the receptor-mediated $Ca^{2+}$-influx and $Ca^{2+}$-release, and partly from the endothelium mediated by EDHF.