• Kyungpook Mathematical Journal
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• v.16 no.1
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• pp.27-32
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• 1976

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1133-1138
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• 2015

In 1956, $Je{\acute{s}}manowicz$ conjectured that, for any positive integer n and any primitive Pythagorean triple (a, b, c) with $a^2+b^2=c^2$, the equation $(an)^x+(bn)^y=(cn)^z$ has the unique solution (x, y, z) = (2, 2, 2). In this paper, under some conditions, we prove the conjecture for the primitive Pythagorean triples $(a,\;b,\;c)=(4k^2-1,\;4k,\;4k^2+1)$.

• Journal of the Korean Mathematical Society
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• v.40 no.1
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• pp.109-128
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• 2003

In this paper, we show that the Euler characteristic of an even dimensional closed projectively flat manifold is equal to the total measure which is induced from a probability Borel measure on RP$^{n}$ invariant under the holonomy action, and then discuss its consequences and applications. As an application, we show that the Chen's conjecture is true for a closed affinely flat manifold whose holonomy group action permits an invariant probability Borel measure on RP$^{n}$ ; that is, such a closed affinly flat manifold has a vanishing Euler characteristic.

• Journal of the Korean Mathematical Society
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• v.58 no.5
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• pp.1227-1237
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• 2021

This paper is concerned with a reaction-diffusion logistic model. In [17], Lou observed that a heterogeneous environment with diffusion makes the total biomass greater than the total carrying capacity. Regarding the ratio of biomass to carrying capacity, Ni [10] raised a conjecture that the ratio has a upper bound depending only on the spatial dimension. For the one-dimensional case, Bai, He, and Li [1] proved that the optimal upper bound is 3. Recently, Inoue and Kuto [13] showed that the supremum of the ratio is infinity when the domain is a multi-dimensional ball. In this paper, we generalized the result of [13] to an arbitrary smooth bounded domain in ℝⁿ, n ≥ 2. We use the sub-solution and super-solution method. The idea of the proof is essentially the same as the proof of [13] but we have improved the construction of sub-solutions. This is the complete answer to the conjecture of Ni.

• Communications of the Korean Mathematical Society
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• v.6 no.1
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• pp.159-165
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• 1991

• Bulletin of the Korean Mathematical Society
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• v.55 no.2
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• pp.587-590
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• 2018

We shall prove certain p-indivisibility properties of so-called generalized left-factorials, where p is a prime.

• Communications of the Korean Mathematical Society
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• v.10 no.2
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• pp.317-323
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• 1995

In this note we prove that Apostol's conjecture is true for the operators having totally disconnected spectra and also give a result related on compalence.

• Kyungpook Mathematical Journal
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• v.57 no.3
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• pp.371-383
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• 2017

Let M be a matroid. We study the expansions of M mainly to see how the combinatorial properties of M and its expansions are related to each other. It is shown that M is a graphic, binary or a transversal matroid if and only if an arbitrary expansion of M has the same property. Then we introduce a new functor, called contraction, which acts in contrast to expansion functor. As a main result of paper, we prove that a matroid M satisfies White's conjecture if and only if an arbitrary expansion of M does. It follows that it suffices to focus on the contraction of a given matroid for checking whether the matroid satisfies White's conjecture. Finally, some classes of matroids satisfying White's conjecture are presented.

• Korean Journal of Mathematics
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• v.29 no.4
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• pp.825-831
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• 2021

The derivative of a polynomial p(z) of degree n, with respect to point α is defined by D_αp(z) = np(z) + (α - z)p'(z). Let p(z) be a polynomial having all its zeros in the unit disk |z| ≤ 1. The Sendov conjecture asserts that if all the zeros of a polynomial p(z) lie in the closed unit disk, then there must be a zero of p'(z) within unit distance of each zero. In this paper, we obtain certain results concerning the location of the zeros of D_αp(z) with respect to a specific zero of p(z) and a stronger result than Sendov conjecture is obtained. Further, a result is obtained for zeros of higher derivatives of polynomials having multiple roots.

• Kyungpook Mathematical Journal
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• v.63 no.3
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• pp.373-411
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• 2023

Gompf proposed a conjecture on Cappell-Shaneson matrices whose affirmative answer implies that all Cappell-Shaneson homotopy 4-spheres are diffeomorphic to the standard 4-sphere. We study Gompf conjecture on Cappell-Shaneson matrices using various algebraic number theoretic techniques. We find a hidden symmetry between trace n Cappell-Shaneson matrices and trace 5 - n Cappell-Shaneson matrices which was suggested by Gompf experimentally. Using this symmetry, we prove that Gompf conjecture for the trace n case is equivalent to the trace 5 - n case. We confirm Gompf conjecture for the special cases that -64 ≤ trace ≤ 69 and corresponding Cappell-Shaneson homotopy 4-spheres are diffeomorphic to the standard 4-sphere. We also give a new infinite family of Cappell-Shaneson spheres which are diffeomorphic to the standard 4-sphere.