• 한국수학교육학회지시리즈D:수학교육연구
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• 제7권1호
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• pp.1-10
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• 2003

This research reviewed literatures on proof in mathematics education. Several views of proof can be classified (and identified) such as psychological approach (Platonism, empiricism), structural approach (logicism, formalism, intuitionism) and social approach (ontology, axiomatic systems). All these views of proof are valuable in mathematics education society. The concept of proof can be found in the form of analytic knowledge not of constructive knowledge. Human beings developed their knowledge in the sequence of constructive knowledge to analytic knowledge. Therefore, in mathematics education, the curriculum of mathematics should involve the process of cognitive knowledge development.

• 대한수학회논문집
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• 제3권2호
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• pp.225-229
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• 1988

• Korean Journal of Mathematics
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• 제25권3호
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• pp.349-358
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• 2017

The cyclic group $C_n={\langle}(12{\cdots}n){\rangle}$ acts on the set $(^{[n]}_k)$ of all k-subsets of [n]. In this action of $C_n$ the number of orbits of size d, for d | n, is $$O^{n,k}_d={\frac{1}{d}}{\sum\limits_{{\frac{n}{d}}{\mid}s{\mid}n}}{\mu}({\frac{ds}{n}})(^{n/s}_{k/s})$$. Stanton and White [6] generalized the above identity to construct the orbit polynomials $$O^{n,k}_d(q)={\frac{1}{[d]_{q^{n/d}}}}{\sum\limits_{{\frac{n}{d}}{\mid}s{\mid}n}}{\mu}({\frac{ds}{n}})[^{n/s}_{k/s}]_{q^s}$$ and conjectured that $O^{n,k}_d(q)$ have non-negative coefficients. In this paper we give a constructive proof for the positivity of coefficients of the orbit polynomial $O^{n,2}_d(q)$.

• 대한수학회보
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• 제60권6호
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• pp.1439-1451
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• 2023

This paper provides a constructive proof of the weak factorizations of the classical Hardy space H¹(ℝⁿ) in terms of multilinear fractional integral operator on the variable Lebesgue spaces, which the result is new even in the linear case. As a direct application, we obtain a new proof of the characterization of BMO(ℝⁿ) via the boundedness of commutators of the multilinear fractional integral operator on the variable Lebesgue spaces.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제13권2호
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• pp.121-136
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• 2006

We formulate and analyze a hybrid system model that involves Volterra integral operators with multiple integrals and two types of impulsive terms. We give a constructive proof, via an iteration method, of existence and uniqueness of solutions.

• 대한수학회지
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• 제46권3호
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• pp.463-473
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• 2009

Two $C^1$-mappings, whose domain is a connected compact $C^1$-Banach manifold modelled over a Banach space X over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ and whose range is a Banach space Y over $\mathbb{K}$, are introduced. Sufficient conditions are given to assert they share only a value. The proof of the result, which is based upon continuation methods, is constructive.

• Journal of applied mathematics & informatics
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• 제5권3호
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• pp.867-875
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• 1998

In this paper we prove that any continuous function on a bounded closed interval of can be approximated by the superposition of a bounded sigmoidal function with a fixed weight. In addition we show that any continuous function over $\mathbb{R}$ which vanishes at infinity can be approximated by the superposition f a bounded sigmoidal function with a weighted norm. Our proof is constructive.

• Journal of applied mathematics & informatics
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• 제29권1_2호
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• pp.505-510
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• 2011

We give a constructive proof that a (partial) evaluation of a multivariate algebraic function with algebraic numbers is again an algebraic function. Especially, we obtain a bound on the degree of an evaluation with the degrees of the original algebraic function and the algebraic numbers evaluated. Furthermore, we show that our bound is sharp with an example.

• 대한수학회지
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• 제45권2호
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• pp.527-535
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• 2008

Sufficient conditions are given to assert that between any two Banach spaces over $\mathbb{K}$ Fredholm mappings share exactly N values in a specific open ball. The proof of the result is constructive and is based upon continuation methods.

• 대한수학회보
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• 제59권3호
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• pp.547-566
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• 2022

Let T be a bilinear Calderón-Zygmund operator, $b{\in}U_q>_1L^q_{loc}(G)$. We firstly obtain a constructive proof of the weak factorisation of Hardy spaces. Then we establish the characterization of BMO spaces by the boundedness of the commutator [b, T]_j in variable Lebesgue spaces.