• 대한수학회보
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• 제29권1호
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• pp.137-143
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• 1992

The purpose of this paper is to provide the affirmative solution of the following conjecture due to Davis and Geramita. Conjecture; Let A=R[T] be a polynomial ring in one variable, where R is a regular local ring of dimension d. Then maximal ideals in A are complete intersection. Geramita has proved that the conjecture is true when R is a regular local ring of dimension 2. Whatwadekar has rpoved that conjecture is true when R is a formal power series ring over a field and also when R is a localization of an affine algebra over an infinite perfect field. Nashier also proved that conjecture is true when R is a local ring of D[ $X_{1}$,.., $X_{d-1}$] at the maximal ideal (.pi., $X_{1}$,.., $X_{d-1}$) where (D,(.pi.)) is a discrete valuation ring with infinite residue field. The methods to establish our results are following from Nashier's method. We divide this paper into three sections. In section 1 we state Theorems without proofs which are used in section 2 and 3. In section 2 we prove some lemmas and propositions which are used in proving our results. In section 3 we prove our main theorem.eorem.rem.

• 대한수학회보
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• 제56권5호
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• pp.1273-1283
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• 2019

In matrix analysis, the Wielandt-Mirsky conjecture states that $$dist({\sigma}(A),{\sigma}(B)){\leq}{\parallel}A-B{\parallel}$$ for any normal matrices $A,B{\in}{\mathbb{C}}^{n{\times}n}$ and any operator norm ${\parallel}{\cdot}{\parallel}$ on $C^{n{\times}n}$. Here dist(${\sigma}(A),{\sigma}(B)$) denotes the optimal matching distance between the spectra of the matrices A and B. It was proved by A. J. Holbrook (1992) that this conjecture is false in general. However it is true for the Frobenius distance and the Frobenius norm (the Hoffman-Wielandt inequality). The main aim of this paper is to study the Hoffman-Wielandt inequality and some weaker versions of the Wielandt-Mirsky conjecture for matrix polynomials.

• Kyungpook Mathematical Journal
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• 제57권3호
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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.

• Kyungpook Mathematical Journal
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• 제46권1호
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• pp.1-9
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• 2006

M. Reni has shown that there are at most nine mutually inequivalent knots in the 3-sphere whose 2-fold branched covering spaces are mutually homeomorphic, hyperbolic 3-manifolds. By observing that the Z-homology sphere version of M. Reni's result still holds, M. Mecchia and B. Zimmermann showed that there are exactly nine mutually inequivalent, knots in Z-homology 3-spheres whose 2-fold branched covering spaces are mutually homeomorphic, hyperbolic 3-manifolds, and conjectured that there exist exactly nine mutually inequivalent, knots in the true 3-sphere whose 2-fold branched covering spaces are mutually homeomorphic, hyperbolic 3-manifolds. Their proof used an argument of AID imitations published in 1992. The main result of this paper is to solve their conjecture affirmatively by combining their argument with a theory of strongly AID imitations published in 1997.

• Kyungpook Mathematical Journal
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• 제56권2호
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• pp.639-645
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• 2016

In [3] the main theorem was erroneously stated. We needed to assume that the irrationality measure of 1/r is finite to prove the theorem.

• Korean Journal of Mathematics
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• 제6권2호
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• pp.221-229
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• 1998

Let $k$ be a real abelian field and $k_{\infty}$ be its $\mathbb{Z}_p$-extension for an odd prime $p$. Let $A_n$ be the Sylow $p$-subgroup of the ideal class group of $k_n$, the $nth$ layer of the $\mathbb{Z}_p$-extension. By using the main conjecture of Iwasawa theory, we have the following: If $p$ does not divide $\prod_{{{\chi}{\in}\hat{\Delta}_k},{\chi}{\neq}1}B_{1,{\chi}{\omega}^{-1}$, then $A_n$ = {0} for all $n{\geq}0$, where ${\Delta}_k=Gal(k/\mathbb{Q})$ and ${\omega}$ is the Teichm$\ddot{u}$ller character for $p$. The converse of this statement does not hold in general. However, we have the following when $k$ is of prime conductor $q$: Let $q$ be an odd prime different from $p$. and let $k$ be a real subfield of $\mathbb{Q}({\zeta}_q)$. If $p{\mid}{\prod}_{{\chi}{\in}\hat{\Delta}_{k,p},{\chi}{\neq}1}B_{1,{\chi}{\omega}}-1$, then $A_n{\neq}\{0\}$ for all $n{\geq}1$, where ${\Delta}_{k,p}$ is the $Gal(k_{(p)}/\mathbb{Q})$ and $k_{(p)}$ is the decomposition field of $k$ for $p$.

• 대한수학회논문집
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• 제21권2호
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• pp.337-345
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• 2006

John Harer conjectured that the canonical map from braid group to mapping class group induces zero homology homomorphism. To prove the conjecture it suffices to show that this map preserves the first Araki-Kudo-Dyer-Lashof operation. To get information on this homology operation we need to investigate the image of braiding under the Harer map. The main result of this paper is to give both algebraic and geometric interpretations of the image of braiding under the Harer map. For this we need to calculate long chains of consecutive actions of Dehn twists on the fundamental group of surface.

• Kyungpook Mathematical Journal
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• 제59권1호
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• pp.47-63
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• 2019

For $y{\in}{\mathbb{R}}$, a sequence $x=(x_n){\in}{\ell}^{\infty}$, and a non-negative regular matrix A, Bartoszewicz et. al., in 2015, defined the notion of the A-density ${\delta}_A(y)$ of the indices of those $x_n$ that are close to y. Their main result states that if the set of limit points of ($x_n$) is countable and density ${\delta}_A(y)$ exists for any $y{\in}\mathbb{R}$ where A is a non-negative regular matrix, then ${\lim}_{n{\rightarrow}{\infty}}(Ax)_n={\sum}_{y{\in}{\mathbb{R}}}{\delta}_A(y){\cdot}y$. In this note we first show that the result can be extended to a more general class of matrices and then consider a conjecture which naturally arises from our investigations.

• Kyungpook Mathematical Journal
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• 제60권2호
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• pp.387-399
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• 2020

In this paper, we study a problem of a non-constant entire function f that shares a set S = {a, b, c} with its k-th derivative f^(k), where a, b and c are any three distinct complex numbers. We have found a gap in the statement of the main result of Chang-Fang-Zalcman [10], and with the help of their method, we have generalize their result in a more compact form. As an application, we generalize the famous Brück conjecture [9] with the idea of set sharing.

• 대한수학회지
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• 제47권2호
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• pp.311-329
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• 2010

In this paper we will prove that the groups $D_{p+1}$(2) and $D_{p+1}$(3), where p is an odd prime number, are uniquely determined by their sets of order components. A main consequence of our result is the validity of Thompson's conjecture for the groups $D_{p+1}$(2) and $D_{p+1}$(3).