• Journal of the Chungcheong Mathematical Society
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• v.23 no.2
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• pp.315-321
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• 2010

Waring's problem deals with representing any nonconstant function in a set of functions as a sum of kth powers of nonconstant functions in the same set. Consider ${\sum}_{i=1}^p\;f_i(z)^k=z$. Suppose that $k{\geq}2$. Let p be the smallest number of functions that give the above identity. We consider Waring's problem for the set of linear fractional transformations and obtain p = k.

• Journal for History of Mathematics
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• v.32 no.2
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• pp.47-59
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• 2019

Hua Loo-Keng(华罗庚, 1910-1985) is one of well-known prominent Chinese mathematicians. While Waring problem is one of his research interests, he made lots of contributions on various mathematical fields including skew fields, geometry of matrices, harmonic analysis, partial differential equations and even numerical analysis and applied mathematics, as well as number theory. He also had devoted his last 20 years to the popularization of mathematics in China. We look at his personal and mathematical life, and consider the meaning of his activity of popularizing mathematics from the cultural perspective to understand the recent rapid developments of China in sciences including mathematics and artificial intelligence.

• Bulletin of the Korean Mathematical Society
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• v.55 no.6
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• pp.1659-1666
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• 2018

Let $P_r$ denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, it is proved that for every sufficiently large even integer N, the equation $$N=x^2+p_1^2+p_2^3+p_3^3+p_4^4+p_5^4$$ is solvable with x being an almost-prime $P_4$ and the other variables primes. This result constitutes an improvement upon that of $L{\ddot{u}}$ [7].

• Journal of the Korean Mathematical Society
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• v.59 no.3
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• pp.439-448
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• 2022

For a natural number n, let R(n) denote the number of representations of n as the sum of one square and five cubes of primes. In this paper, it is proved that the anticipated asymptotic formula for R(n) fails for at most $O(N^{{\frac{4}{9}+{\varepsilon}})$ positive integers not exceeding N.

• Journal of the Korean Mathematical Society
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• v.59 no.2
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• pp.407-420
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• 2022

It is proved that every sufficiently large even integer can be represented as a sum of two squares of primes, two cubes of primes, two fourth powers of primes and 17 powers of 2.

• Journal of Educational Research in Mathematics
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• v.16 no.4
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• pp.327-344
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• 2006

This study examined the proof levels and understanding of constituents of proving by three mathematically gifted 6th grade korean students, who belonged to the highest 1% in elementary school, through observation and interviews on the problem-solving process in relation to constructing a rectangle of which area equals the sum of two other rectangles. We assigned the students with Clairaut's geometric problems and analyzed their proof levels and their difficulties in thinking related to the understanding of constituents of proving. Analysis of data was made based on the proof level suggested by Waring (2000) and the constituents of proving presented by Galbraith(1981), Dreyfus & Hadas(1987), Seo(1999). As a result, we found out that the students recognized the meaning and necessity of proof, and they peformed some geometric proofs if only they had teacher's proper intervention.

• Journal of the Korean Mathematical Society
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• v.49 no.6
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• pp.1139-1161
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• 2012

Let $p$ be an odd prime number and let F be a finite field with $p^m$ elements. We study representations and strict representations of polynomials $M{\in}F$[T] by sums of ($p^r$ + 1)-th powers. A representation $$M=M_1^k+{\cdots}+M_s^k$$ of $M{\in}F$[T] as a sum of $k$-th powers of polynomials is strict if $k$ deg $M_i<k$ + degM.

• Bulletin of the Korean Mathematical Society
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• v.46 no.5
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• pp.941-947
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• 2009

Let q be a power of 16. Every polynomial $P\in\mathbb{F}_q$[t] is a strict sum $P=A^2+A+B^3+C^3+D^3+E^3$. The values of A,B,C,D,E are effectively obtained from the coefficients of P. The proof uses the new result that every polynomial $Q\in\mathbb{F}_q$[t], satisfying the necessary condition that the constant term Q(0) has zero trace, has a strict and effective representation as: $Q=F^2+F+tG^2$. This improves for such q's and such Q's a result of Gallardo, Rahavandrainy, and Vaserstein that requires three polynomials F,G,H for the strict representation $Q=F^2$+F+GH. Observe that the latter representation may be considered as an analogue in characteristic 2 of the strict representation of a polynomial Q by three squares in odd characteristic.

• The Journal of the Korea Contents Association
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• v.15 no.3
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• pp.78-87
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• 2015

Recently, Police integrity has been issued on the media, which cause discredit of police organization. Although high level of morality and integrity are required compared to other occupational groups due to their authority to exert legal force to the citizens and a variety of policies have been enforced by the National Police Agency for the purpose of uplifting the integrity of the officers, in reality, corruption had not yet been eradicated. At this point in time, this study attempted to draw implications for uplifting integrity by utilizing domestic and foreign preceding studies and statistical data related to police corruption and uplifting integrity. The inspection system through whistle-blowing was pointed out as a problem in the institutional framework that hinders uplifting integrity of the police officers and the perception in which police officers are regarded as potential criminals was also pointed out as a problem. Also, vague standards of disciplinary action in examining an offense of a police officer and lack of care for those who were disciplined in the past which affects loyalty to the organization were presented as problems. Based on such suggested concerns, policies for uplifting integrity and restoring citizens' trust in the policies officers were proposed. The proposed agenda were warning the police officers by presenting clear and specific category of corruptive behaviors, expressing the necessity of devising a system that prevents the officers from committing serious crimes by discovering problematic officers earlier through introduction of Early Warning System(EWS) of US and Australian police in order to break away from exposure-oriented inspection system, and reinforcing the testing of integrity in the new employment process.