• Korean Journal of Mathematics
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• v.31 no.2
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• pp.181-188
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• 2023

In 1911, Dubouis determined all positive integers represented by sums of k nonvanishing squares for all k ≥ 4. As a generalization, Y.-S. Ji, M.-H. Kim and B.-K. Oh determined all positive definite binary quadratic forms represented by sums of k nonvanishing squares for all k ≥ 5. In this article, we give a simple proof for Ji-Kim-Oh's theorem for all k ≥ 10.

• Journal of the Korean Mathematical Society
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• v.60 no.6
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• pp.1337-1364
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• 2023

We describe the Gorenstein derived categories of Gorenstein rings via the homotopy categories of Gorenstein injective modules. We also introduce the concept of Gorenstein cosilting complexes and study its basic properties. This concept is generalized by cosilting complexes in relative homological methods. Furthermore, we investigate the existence of the relative version of the Bongartz's theorem and construct a Bongartz's complement for a Gorenstein precosilting complex.

• Journal of the Korean Mathematical Society
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• v.61 no.1
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• pp.149-160
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• 2024

In this paper, we study the unicorn's Landsberg problem from an intrinsic point of view. Precisely, we investigate a coordinate-free proof of Numata's theorem on Landsberg spaces of scalar curvature. In other words, following the pullback approach to Finsler geometry, we prove that all Landsberg spaces of dimension n ≥ 3 of non-zero scalar curvature are Riemannian spaces of constant curvature.

• School Mathematics
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• v.9 no.4
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• pp.523-533
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• 2007

This article focused the meaning of Pythagoras' and Euclid's proof about the Pythagorean theorem in a historical and mathematical perspective. Pythagoras' proof using similarity is based on the arithmetic assumption about commensurability. However, Euclid proved the Pythagorean theorem again only using the concept of dissection-rearrangement that is purely geometric so that it does not need commensurability. Pythagoras' and Euclid's different approaches to geometry have to do with Birkhoff's axiom system and Hilbert's axiom system in the school geometry Birkhoff proposed the new axioms for plane geometry accepting real number that is strictly defined. Thus Birkhoff's metrical approach can be defined as a Pythagorean approach that developed geometry based on number. On the other hand, Hilbert succeeded Euclid who had pursued pure geometry that did not depend on number. The difference between the proof using similarity and dissection-rearrangement is related to the unsolved problem in the geometry curriculum that is conflict of Euclid's conventional synthetical approach and modern mathematical approach to geometry.

• Journal of applied mathematics & informatics
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• v.34 no.3_4
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• pp.269-284
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• 2016

In this paper, we firstly use Krasnosel'skii fixed point theorem to investigate positive solutions for the following three-point boundary value problems for p-Laplacian with a parameter $({\phi}_P(D^{\alpha}_{0}+u(t)))^{\prime}+{\lambda}f(t, u(t))=0$, 0$D^{\alpha}_{0}+u(0)=u(0)=u{\prime}{\prime}(0)=0$, $u^{\prime}(1)={\gamma}u^{\prime}(\eta)$ where ϕ_p(s) = |s|^p−2s, p > 1, $D^{\alpha}_{0^+}$ is the Caputo's derivative, α ∈ (2, 3], η, γ ∈ (0, 1), λ > 0 is a parameter. Then we use Leggett-Williams fixed point theorem to study the existence of three positive solutions for the fractional boundary value problem $({\phi}_P(D^{\alpha}_{0}+u(t)))^{\prime}+f(t, u(t))=0$, 0$D^{\alpha}_{0}+u(0)=u(0)=u{\prime}{\prime}(0)=0$, $u^{\prime}(1)={\gamma}u^{\prime}(\eta)$ where ϕ_p(s) = |s|^p−2s, p > 1, $D^{\alpha}_{0^+}$ is the Caputo's derivative, α ∈ (2, 3], η, γ ∈ (0, 1).

• Journal of applied mathematics & informatics
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• v.7 no.2
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• pp.527-538
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• 2000

We provide sufficient conditions for the monotone convergence of a Chebysheff-Halley-type method or method of tangent hyperbolas in a partially ordered topological space setting. The famous kantorovich theorem on fixed points is used here.

• Korean Journal of Mathematics
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• v.6 no.2
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• pp.231-242
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• 1998

On the basis of Heyer and Zeuner's results we will treat the central limit theorem for probability measures on hypergroup.

• Journal of the Korean Mathematical Society
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• v.36 no.2
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• pp.257-266
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• 1999

We used Carleson and Chang's method to give another proof of the Moser-Trudinger inequality which was known as a limiting case of the Sobolev imbedding theorem and at the same time we get sharper information for the bound.

• Bulletin of the Korean Mathematical Society
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• v.36 no.3
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• pp.533-541
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• 1999

In this paper, we will study controllability of some case s an initial condition $\phi$ included in some approximated phase space. To this prove we used to the Schauder fixed point theorem.

• Bulletin of the Korean Mathematical Society
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• v.37 no.3
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• pp.569-575
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• 2000

In this paper, a unique common fixed point theorem for a sequence of mutually contractive and commuting sequence of self mappings on Menger spaces has been proved.