• Journal of Communications and Networks
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• v.15 no.2
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• pp.217-221
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• 2013

Quantum low-density parity-check (LDPC) codes based on the Calderbank-Shor-Steane construction have low encoding and decoding complexity. The sum-product algorithm(SPA) can be used to decode quantum LDPC codes; however, the decoding performance may be significantly decreased by the many four-cycles required by this type of quantum codes. All four-cycles can be eliminated using the entanglement-assisted formalism with maximally entangled states (ebits). The proposed entanglement-assisted quantum error-correcting code based on Euclidean geometry outperform differently structured quantum codes. However, the large number of ebits required to construct the entanglement-assisted formalism is a substantial obstacle to practical application. In this paper, we propose a novel class of entanglement-assisted quantum LDPC codes constructed using classical Euclidean geometry LDPC codes. Notably, the new codes require one copy of the ebit. Furthermore, we propose a construction scheme for a corresponding zigzag matrix and show that the algebraic structure of the codes could easily be expanded. A large class of quantum codes with various code lengths and code rates can be constructed. Our methods significantly improve the possibility of practical implementation of quantum error-correcting codes. Simulation results show that the entanglement-assisted quantum LDPC codes described in this study perform very well over a depolarizing channel with iterative decoding based on the SPA and that these codes outperform other quantum codes based on Euclidean geometries.

• Journal of the Institute of Electronics Engineers of Korea CI
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• v.42 no.5
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• pp.69-78
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• 2005

It is crucial to compute the Euclidean distance between two vectors efficiently in high dimensional space for multimedia information retrieval. In this paper, we propose an effective method for approximating the Euclidean distance between two high-dimensional vectors. For this approximation, a previous method, which simply employs norms of two vectors, has been proposed. This method, however, ignores the angle between two vectors in approximation, and thus suffers from large approximation errors. Our method introduces an additional vector called a reference vector for estimating the angle between the two vectors, and approximates the Euclidean distance accurately by using the estimated angle. This makes the approximation errors reduced significantly compared with the previous method. Also, we formally prove that the value approximated by our method is always smaller than the actual Euclidean distance. This implies that our method does not incur any false dismissal in multimedia information retrieval. Finally, we verify the superiority of the proposed method via performance evaluation with extensive experiments.

• Journal of the Korean School Mathematics Society
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• v.19 no.4
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• pp.373-394
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• 2016

The "Figure Equation" chapter of current high school curriculum prevents students from relating the concept with what they studied in middle school Euclidean geometry. Woo(1998) concerns that the curriculum introduces the concept merely in algebraic ways without providing students with opportunities to relate it with their prior understanding of geometry, which is based on Euclidean one. In the present study, a sequence of GeoGebra-embedded-geometry lessons was designed so that students could be introduced to and solve problems of the Analytic Geometry by triggering their prior understanding of the Euclidean Geometry which they had learnt in middle school. The study contributes to the field of mathematics education by suggesting a sequence of geometry lessons where students could introduce to the coordinate geometry meaningfully and conceptually in high school.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.22 no.5
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• pp.1019-1025
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• 2012

Fault attacks manipulate the computation of an algorithm and get information about the private key from the erroneous result. It is the most powerful attack for the cryptographic device. Currently, the research on error detection methods and fault attacks have been studied actively. S. Pontarelli et al. introduced an error detection method in 2009. It can detect an error that occurs during Elliptic Curve Scalar Multiplication (ECSM). In this paper, we present a new fault attack. Our attack can avoid the error detection method introduced by S. Pontarelli et al. We inject a bit flip error in the Euclidean Addition Chain (EAC) on the private key in ECSM and retrieve the private key.

• Honam Mathematical Journal
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• v.38 no.3
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• pp.593-611
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• 2016

This paper is concerned with the classifications of quadric surfaces of first and second kinds in Euclidean 3-space satisfying the Jacobi condition with respect to their curvatures, the Gaussian curvature K, the mean curvature H, second mean curvature $H_{II}$ and second Gaussian curvature $K_{II}$. Also, we study the zero and non-zero constant curvatures of these surfaces. Furthermore, we investigated the (A, B)-Weingarten, (A, B)-linear Weingarten as well as some special ($C^2$, K) and $(C^2,\;K{\sqrt{K}})$-nonlinear Weingarten quadric surfaces in $E^3$, where $A{\neq}B$, A, $B{\in}{K,H,H_{II},K_{II}}$ and $C{\in}{H,H_{II},K_{II}}$. Finally, some important new lemmas are presented.

• Honam Mathematical Journal
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• v.36 no.1
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• pp.67-83
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• 2014

The notion of rectifying curve in the Euclidean space is introduced by Chen as a curve whose position vector always lies in its rectifying plane spanned by the tangent and the binormal vector field t and $n_2$ of the curve, [1]. In this study, we have obtained some characterizations of semi-real spatial quaternionic rectifying curves in $\mathbb{R}^3_1$. Moreover, by the aid of these characterizations, we have investigated semi real quaternionic rectifying curves in semi-quaternionic space $\mathbb{Q}_v$.

• Journal of the Institute of Electronics Engineers of Korea TC
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• v.47 no.11
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• pp.36-42
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• 2010

This paper presents a hybrid approach to the construction of quasi-cyclic (QC) low-density parity-check (LDPC) codes based on parallel bundles in Euclidean geometries and circulant permutation matrices. Codes constructed by this method are shown to be regular with large girth and low density. Simulation results show that these codes perform very well with iterative decoding and achieve reasonably large coding gains over uncoded system.

• Bulletin of the Korean Mathematical Society
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• v.48 no.3
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• pp.601-609
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• 2011

Tensor product immersions of a given Riemannian manifold was initiated by B.-Y. Chen. In the present article we study the tensor product surfaces of two Euclidean plane curves. We show that a tensor product surface M of a plane circle $c_1$ centered at origin with an Euclidean planar curve $c_2$ has harmonic Gauss map if and only if M is a part of a plane. Further, we give necessary and sufficient conditions for a tensor product surface M of a plane circle $c_1$ centered at origin with an Euclidean planar curve $c_2$ to have pointwise 1-type Gauss map.

• Korean Management Science Review
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• v.33 no.4
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• pp.51-64
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• 2016

The maximum weight of single stock in mutual fund is limited by regulations to enforce diversification. Under incomplete information with added constraints on portfolio weights, enhanced performance had been reported in previous researches. We analyze a weight vector to examine the effects of additional constraints on the portfolio's performance by computing the Euclidean distance from the in-sample tangency portfolio, as opposed to previous researches which analyzed ex-post return only. Empirical experiment was performed on Mean-variance and Minimum-variance model with Fama French's 30 industry portfolio and 10 industry portfolio for the last 1,000 months from August 1932 to November 2015. We find that diversification-constrained portfolios have 7% to 26% smaller Euclidean distances with the benchmark portfolio compared to those of unconstrained portfolios and 3% to 11% greater Sharpe Ratio.

• Journal of the Korean Mathematical Society
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• v.49 no.3
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• pp.475-491
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• 2012

In 1950 a class of generalized Petersen graphs was introduced by Coxeter and around 1970 popularized by Frucht, Graver and Watkins. The family of $I$-graphs mentioned in 1988 by Bouwer et al. represents a slight further albeit important generalization of the renowned Petersen graph. We show that each $I$-graph $I(n,j,k)$ admits a unit-distance representation in the Euclidean plane. This implies that each generalized Petersen graph admits a unit-distance representation in the Euclidean plane. In particular, we show that every $I$-graph $I(n,j,k)$ has an isomorphic $I$-graph that admits a unit-distance representation in the Euclidean plane with a $n$-fold rotational symmetry, with the exception of the families $I(n,j,j)$ and $I(12m,m,5m)$, $m{\geq}1$. We also provide unit-distance representations for these graphs.