• Journal of the Chungcheong Mathematical Society
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• v.19 no.4
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• pp.409-415
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• 2006

In this paper, we study the condition that a given homeomorphism has a square root and give an example of a wandering homeomorphism without square roots.

• Journal of the Korean Mathematical Society
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• v.52 no.1
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• pp.209-224
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• 2015

Computing square, cube and n-th roots in general, in finite fields, are important computational problems with significant applications to cryptography. One interesting approach to computational problems is by using polynomial representations. Agou, Del$\acute{e}$eglise and Nicolas proved results concerning the lower bounds for the length of polynomials representing square roots modulo a prime p. We generalize the results by considering n-th roots over finite fields for arbitrary n > 2.

• Kyungpook Mathematical Journal
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• v.52 no.3
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• pp.347-357
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• 2012

When Newton's method, or Halley's method is used to approximate the pth root of 1-z, a sequence of rational functions is obtained. In this paper, a beautiful formula for these rational functions is proved in the square root case, using an interesting link to Chebyshev's polynomials. It allows the determination of the sign of the coefficients of the power series expansion of these rational functions. This answers positively the square root case of a proposed conjecture by Guo(2010).

• The Journal of Korean Institute of Communications and Information Sciences
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• v.37A no.12
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• pp.1031-1037
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• 2012

We study an algorithm that can efficiently find square roots and cube roots by modifying Tonelli-Shanks algorithm, which has an application in Number Field Sieve (NFS). The Number Field Sieve, the fastest known factoring algorithm, is a powerful tool for factoring very large integer. NFS first chooses two polynomials having common root modulo N, and it consists of the following four major steps; 1. Polynomial Selection 2. Sieving 3. Matrix 4. Square Root. The last step of NFS needs the process of square root computation in Number Field, which can be computed via square root algorithm over finite field.

• East Asian mathematical journal
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• v.29 no.5
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• pp.511-519
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• 2013

There are various methods for evaluating a matrix square root, which is a solvent of the quadratic matrix equation $X^2-A=0$. We consider new iterative methods for solving matrix square roots of M-matrices. Particulary we show that the relaxed binomial iteration is more efficient than Newton-Schulz iteration in some cases. And we construct a formula to find relaxation coefficients through statistical experiments.

• The Pure and Applied Mathematics
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• v.17 no.1
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• pp.107-113
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• 2010

In this paper, we study the root system of rank 2 symmetric hyperbolic Kac-Moody algebras. We give the sufficient conditions for existence of imaginary roots of square length -2k ($k\;{\in}\;\mathbb{Z}$>0). We also give several relations between the roots on g(A).

• Journal of the Korea Institute of Information Security & Cryptology
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• v.18 no.6A
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• pp.3-15
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• 2008

In this paper we study exponentiation in finite fields $F{_p}{^{k}}$(k is odd) with very special exponents such as they occur in algorithms for computing square roots. Our algorithmic approach improves the corresponding exponentiation independent of the characteristic of $F{_p}{^{k}}$. To the best of our knowledge, it is the first major improvement to the Tonelli-Shanks algorithm, for example, the number of multiplications can be reduced to at least 60% on average when $p{\equiv}1$ (mod 16). Several numerical examples are given that show the speed-up of the proposed methods.

• The Mathematical Education
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• v.57 no.2
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• pp.93-110
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• 2018

One of the biggest changes in the 2015 revised mathematics curriculum is shifting to the second year of middle school in Pythagorean theorem. In this study, the following subjects were studied. First, Pythagoras theorem analyzed the expected problems caused by the shift to the second year middle school. Secondly, we have researched the reconstruction method to solve these problems. The results of this study are as follows. First, there are many different ways to deal with Pythagorean theorem in many countries around the world. In most countries, it was dealt with in 7th grade, but Japan was dealing with 9th grade, and the United States was dealing with 7th, 8th and 9th grade. Second, we derived meaningful implications for the curriculum of Korea from various cases of various countries. The first implication is that the Pythagorean theorem is a content element that can be learned anywhere in the 7th, 8th, and 9th grade. Second, there is one prerequisite before learning Pythagorean theorem, which is learning about the square root. Third, the square roots must be learned before learning Pythagorean theorem. Optimal positions are to be placed in the eighth grade 'rational and cyclic minority' unit. Third, Pythagorean theorem itself is important, but its use is more important. The achievement criteria for the use of Pythagorean theorem should not be erased. In the 9th grade 'Numbers and Calculations' unit, after learning arithmetic calculations including square roots, we propose to reconstruct the square root and the utilization subfields of Pythagorean theorem.

• Imaging Science in Dentistry
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• v.45 no.4
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• pp.221-226
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• 2015

Purpose: This study evaluated the prevalence of distolingual roots in mandibular molars among Koreans, the root canal system associated with distolingual roots, and the concurrent appearance of a distolingual root in the mandibular first molar and a C-shaped canal in the mandibular second molar. Materials and Methods: Cone-beam computed tomographic images of 264 patients were screened and examined. Axial sections of 1056 mandibular molars were evaluated to determine the number of roots. The interorifice distances from the distolingual canal to the distobuccal canal were also estimated. Using an image analysis program, the root canal curvature was calculated. Pearson's chi-square test, the paired t-test, one-way analysis of variance, and post-hoc analysis were performed. Results: Distolingual roots were observed in 26.1% of the subjects. In cases where a distolingual root was observed in the mandibular molar, a significant difference was observed in the root canal curvature between the buccolingual and mesiodistal orientations. The maximum root canal curvature was most commonly observed in the mesiodistal orientation in the coronal portion, but in the apical portion, maximum root canal curvature was most often observed in the buccolingual orientation. Conclusion: The canal curvature of distolingual roots was found to be very complex, with a different direction in each portion. No correlation was found between the presence of a distolingual root in the mandibular first molar and the presence of a C-shaped canal in the mandibular second molar.

• Journal of the Korean Society of Environmental Restoration Technology
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• v.2 no.2
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• pp.43-52
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• 1999

This study was carried out to verify results of the nursery seedbeds. From November of 1997 to September of 1998, the artificial banking slopes in the greenhouse of the College of Agriculture and Life Sciences, Seoul National University were seeded with the mixtures of those species. Most of exotic species showed relatively poor development of root as short as 30cm. Also the green weight of root biomass of the native species was more than two times than that of the exotic species. On the other hand, it was found that the exotic species have relatively well-developed fine roots. Thus, it was concluded that the seed-mixture of the native species with long and thick roots and the exotic species with fine roots be the most effective method for topsoil erosion control on banking-slopes. The artificial rainfall system treatment(30mm/hr, 60mm/hr, 100mm/hr) on $30^{\circ}$ banking-slopes did not cause any significant change in the amount of soil loss by erosion. The root system was best developed in the plot of 1,000 seedlings per square meter and it performed well for soil erosion control. Consequently, in the case of seeding of single herbaceous species without mixing any woody seeds, the expected seedlings were 1,000 to 2,000 per square meter.