• Journal of the Korean Mathematical Society
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• v.58 no.1
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• pp.123-132
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• 2021

Let K be a number field and L a finite abelian extension of K. Let E be an elliptic curve defined over K. The restriction of scalars ResKLE decomposes (up to isogeny) into abelian varieties over K $$Res^L_KE{\sim}{\bigoplus_{F{\in}S}}A_F,$$ where S is the set of cyclic extensions of K in L. It is known that if L is a quadratic extension, then AL is the quadratic twist of E. In this paper, we consider the case that K is a number field containing a primitive third root of unity, $L=K({\sqrt[3]{D}})$ is the cyclic cubic extension of K for some D ∈ K×/(K×)3, E = Ea : y2 = x3 + a is an elliptic curve with j-invariant 0 defined over K, and EaD : y2 = x3 + aD2 is the cubic twist of Ea. In this case, we prove AL is isogenous over K to $E_a^D{\times}E_a^{D^2}$ and a property of the Selmer rank of AL, which is a cubic analogue of a theorem of Mazur and Rubin on quadratic twists.

• Journal of Electrochemical Science and Technology
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• v.7 no.4
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• pp.293-297
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• 2016

Here we propose a new equation by which we can estimate the baseline for measuring the peak current of the reverse curve in a cyclic voltammogram. A similar equation already exists, but it is a linear algebraic equation that over-simplifies the voltammetric curve and may cause unpredictable errors when calculating the baseline. In our study, we find a quadratic algebraic equation that acceptably reflects the complexity included in a voltammetric curve. The equation is obtained from a laborious numerical analysis of cyclic voltammetry simulations using the finite element method, and not from the closed form of the mathematical equation. This equation is utilized to provide a virtual baseline current for the reverse peak current. We compare the results obtained using the old linear and new quadratic equations with the theoretical values in terms of errors to ascertain the degree to which accuracy is improved by the new equation. Finally, the equations are applied to practical cyclic voltammograms of ferricyanide in order to confirm the improved accuracy.

• Journal of the Korean School Mathematics Society
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• v.25 no.1
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• pp.61-77
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• 2022

This study looked at the deviation of each textbook, focusing on the detailed learning content related to the quadratic curve properties contained in high school geometry textbooks. Rather than criticizing the diversity of content elements covered in high school geometry textbooks and suggesting alternatives, it focused on analyzing the actual conditions of content element diversity. The curriculum specifies that the practical application of the quadratic curve should be emphasized so that student could recognize the usefulness and value. However, as a result of the analysis, it was confirmed that the purpose of the curriculum and the structure of the textbook did not match somewhat, the deviation of content elements for each textbook was quite large. In terms of acknowledging the diversity of teaching and learning, the diversity of each textbook on the methods of the introduction and the natures related to the quadratic curve can be fully recognized. But in our educational reality, which is aiming for the university entrance examination system through national evaluation such as CSAT, the results are too sensitive in society as a whole, so the diversity of expressions in mathematics textbooks is sometimes interpreted as a disadvantage of evaluation. It is time to reconsider the composition of textbooks that recognizes the diversity of content elements in textbook teaching and learning and at the same time reflects the aspect of equality in evaluation.

• The Transactions of the Korean Institute of Electrical Engineers A
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• v.48 no.1
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• pp.22-30
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• 1999

This paper presents an efficient algorithm to estimate the maximum load level for heavily loaded power systems with the load-generation vector obtained by ELD (Economic Load Dispach) and/or short term load forecasting while utilizing the elliptic pattern of the P-e curve. It is well known the power flow equation in the rectangular corrdinate is jully quadratic. However, the coupling between e and f makes it difficult to take advantage of this quadratic characteristic. In this paper, the elliptic characteristics of P-e curve are illustrated and a simple technique is proposed to reflect the e-f coupling effects on the estimation of maximum loadability with theoretical analysis. An efficient estimation algorithm has been developed with the use of the elliptic properties of the P-e curve. The proposed algorithm is tested on IEEE 14 bus system, New England 39 bus system and IEEE 118 bus system, which shows that the maximum load level can be efficiently estimated with remarkable improvement in accuracy.

• The Transactions of the Korea Information Processing Society
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• v.6 no.12
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• pp.3710-3717
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• 1999

For the development of unmanned autonomous vehicle, it is essential to detect obstacles, especially vehicles, in the forward direction of navigation. In order to reliably exclude regions that do not contain obstacles and save a considerable amount of computational effort, it is often necessary to confine computation only to ROI(region of interest)s. A ROI is usually chosen as the interior region of the lane. We propose a computationally simple and efficient method for the detection of lanes based on Hough transform and quadratic curve fitting. The proposed method first employs Hough transform to get approximate locations of lanes, and then applies quadratic curve fitting to the locations computed by Hough transform. We have experimented the proposed method on real outdoor road scene. Experimental results show that our method gives accurate detection of straight and curve lanes, and is computationally very efficient.

• School Mathematics
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• v.15 no.4
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• pp.995-1013
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• 2013

The purpose of this paper was to investigate mathematics teachers' mathematical content knowledge about quadratic curves. Three components of mathematical knowledge are needed for teaching: (i) knowing school mathematics, (ii) knowing process of school mathematics, (iii) making connections between school mathematics and advanced mathematics. 24 mathematics teachers were asked to perform 10 questions based on mathematics curriculum. The results showed that mathematics teachers had some difficulties in conic section definitions and eccentricity definitions of ellipse and hyperbola. And they also got difficulty in Dandellin sphere proof of the equivalence of conic section definitions and quadratic curve definitions. Especially, no one answered correctly to the question about the definition of eccentricity. The ratio of correct answer for the question about constructing tangent lines of quadratic curves is less than that for the question about the applications of the properties of tangent lines. These findings suggests that it is needed that to provide plenty of opportunities to learn mathematical content knowledge in teacher education programs.

• Journal of Korean Society for Quality Management
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• v.23 no.2
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• pp.53-67
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• 1995

The problem considered here is to find the optimal discriminant analysis method in growth curve model. It has been studied how to find correct prior probability for the effective classification in discriminant analysis. We use the balanced condition to calculate prior probability. From the informative simulation study, new classification rule for the growth curve model is suggested. The suggested classification rule has better classification result than the other previously suggested method in terms of error rate criterion.

• International Journal of Control, Automation, and Systems
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• v.2 no.1
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• pp.107-117
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• 2004

A new algorithm for planning a collision-free path based on an algebraic curve as well as the concept of path space is developed. Robot path planning has so far been concerned with generating a single collision-free path connecting two specified points in a given robot workspace with appropriate constraints. In this paper, a novel concept of path space (PS) is introduced. A PS is a set of points that represent a connection between two points in Euclidean metric space. A geometry mapping (GM) for the systematic construction of path space is also developed. A GM based on the 2$^{nd}$ order base curve, specifically Bezier curve of order two is investigated for the construction of PS and for collision-free path planning. The Bezier curve of order two consists of three vertices that are the start, S, the goal, G, and the middle vertex. The middle vertex is used to control the shape of the curve, and the origin of the local coordinate (p, $\theta$) is set at the centre of S and G. The extreme locus of the base curve should cover the entire area of actual workspace (AWS). The area defined by the extreme locus of the path is defined as quadratic workspace (QWS). The interference of the path with obstacles creates images in the PS. The clear areas of the PS that are not mapped by obstacle images identify collision-free paths. Hence, the PS approach converts path planning in Euclidean space into a point selection problem in path space. This also makes it possible to impose additional constraints such as determining the shortest path or the safest path in the search of the collision-free path. The QWS GM algorithm is implemented on various computer systems. Simulations are carried out to measure performance of the algorithm and show the execution time in the range of 0.0008 ~ 0.0014 sec.

• Journal of the Korean Mathematical Society
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• v.40 no.6
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• pp.977-998
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• 2003

Let k be an imaginary quadratic field, η the complex upper half plane, and let $\tau$ $\in$ η $textsc{k}$, p = $e^{{\pi}i{\tau}}$. In this article, using the infinite product formulas for g2 and g3, we prove that values of certain infinite products are transcendental whenever $\tau$ are imaginary quadratic. And we derive analogous results of Berndt-Chan-Zhang ([4]). Also we find the values of (equation omitted) when we know j($\tau$). And we construct an elliptic curve E : $y^2$ = $x^3$ ＋ 3 $x^2$ ＋ {3-(j/256)}x ＋ 1 with j = j($\tau$) $\neq$ 0 and P = (equation omitted) $\in$ E.

• 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.