• Kyungpook Mathematical Journal
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• v.27 no.1
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• pp.27-33
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• 1987

Minimal s-Urysohn and minimal s-regular spaces are studied. An s-Urysohn (respectively, s-regular) space (X, $\mathfrak{T}$) is said to be minimal s-Urysohn (respectively, minimal s-regular) if for no topology $\mathfrak{T}^{\prime}$ on X which is strictly weaker than $\mathfrak{T}$, (X, $\mathfrak{T}^{\prime}$) is s-Urysohn (respectively s-regular). Several characterizations and other related properties of these classes of spaces have been obtained. The present paper is a study of minimal P-spaces where P refers to the property of being an s-Urysohn space or an s-regular space. A P-space (X, $\mathfrak{T}$) is said to be minimal P if for no topology $\mathfrak{T}^{\prime}$ on X such that $\mathfrak{T}^{\prime}$ is strictly weaker than $\mathfrak{T}$, (X, $\mathfrak{T}^{\prime}$) has the property P. A space X is said to be s-Urysohn [2] if for any two distinct points x and y of X there exist semi-open set U and V containing x and y respectively such that $clU{\bigcap}clV={\phi}$, where clU denotes the closure of U. A space X is said to be s-regular [6] if for any point x and a closed set F not containing x there exist disjoint semi-open sets U and V such that $x{\in}U$ and $F{\subseteq}V$. Throughout the paper the spaces are assumed to be Hausdorff.

• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.1981-1988
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• 2013

We say a positive integer n satisfies the Lehmer property if ${\phi}(n)$ divides n - 1, where ${\phi}(n)$ is the Euler's totient function. Clearly, every prime satisfies the Lehmer property. No composite integer satisfying the Lehmer property is known. In this article, we show that every composite integer of the form $D_{p,n}=np^n+1$, for a prime p and a positive integer n, or of the form ${\alpha}2^{\beta}+1$ for ${\alpha}{\leq}{\beta}$ does not satisfy the Lehmer property.

• Journal of applied mathematics & informatics
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• v.32 no.1_2
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• pp.285-295
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• 2014

In the setting of semidenite linear complementarity problems on $S^n$, we focus on the Stein Transformation $S_A(X)\;:=X-AXA^T$, and show that $S_A$ is (strictly) monotone if and only if ${\nu}_r(UAU^T{\circ}\;UAU^T)$(<)${\leq}1$, for all orthogonal matrices U where ${\circ}$ is the Hadamard product and ${\nu}_r$ is the real numerical radius. In particular, we show that if ${\rho}(A)$ < 1 and ${\nu}_r(UAU^T{\circ}\;UAU^T){\leq}1$, then SDLCP($S_A$, Q) has a unique solution for all $Q{\in}S^n$. In an attempt to characterize the GUS-property of a nonmonotone $S_A$, we give an instance of a nonnormal $2{\times}2$ matrix A such that SDLCP($S_A$, Q) has a unique solution for Q either a diagonal or a symmetric positive or negative semidenite matrix. We show that this particular $S_A$ has the $P^{\prime}_2$-property.

• Bulletin of the Korean Mathematical Society
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• v.26 no.2
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• pp.109-114
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• 1989

This paper deals with the classifying spaces of finite groups. To any finite group G we associate a space BG with the property that .pi.$_{1}$(BG)=G, .pi.$_{i}$ (BG)=0 for i>1. BG is called the classifying space of G. Consider the problem of finding a stable splitting BG= $X_{1}$$^{V}$ $X_{1}$$^{V}$..$^{V}$ $X_{n}$ localized at pp. Ideally the $X_{i}$ 's are indecomposable, thus displaying the homotopy type of BG in the simplest terms. Such a decomposition naturally splits $H^{*}$(BG). The main purpose of this paper is to give the classification theorem in stable homotopy theory for groups with periodic cohomology i.e. cyclic Sylow p-subgroups for p an odd prime and to calculate some universal stable element. In this paper, all cohomology groups are with Z/p-coefficients and p is an odd prime.prime.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.28 no.9C
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• pp.835-842
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• 2003

For an odd prime p and integer n, m and k such that n=(2m＋1)ㆍk, a new family of p-ary sequences of period p$^{n}$ -1 with optimal correlation property is constructed using the p-ary Helleseth-Gong sequences with ideal autocorrelation, where the size of the sequence family is p$^{n}$ . That is, the maximum nontrivial correlation value R$_{max}$ of all pairs of distinct sequences in the family does not exceed p$^{n}$ 2/ ＋1, which means the optimal correlation property in terms of Welch's lower bound. It is also derived that the linear span of the sequences in the family is (m＋2)ㆍn except for the m-sequence in the family.

• Bulletin of the Korean Mathematical Society
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• v.59 no.2
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• pp.419-429
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• 2022

An integral domain R is an RTP domain (or has the radical trace property) (resp. an LTP domain) if I(R : I) is a radical ideal for each nonzero noninvertible ideal I (resp. I(R : I)R_P = PR_P for each minimal prime P of I(R : I)). Clearly each RTP domain is an LTP domain, but whether the two are equivalent is open except in certain special cases. In this paper, we study the descent of these notions from particular overrings of R to R itself.

• Journal of applied mathematics & informatics
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• v.33 no.1_2
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• pp.229-234
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• 2015

In the setting of semidefinite linear complementarity problems on $S^n$, we focus on the Stein Transformation $S_A(X):=X-AXA^T$ for $A{\in}R^{n{\times}n}$ that is positive semidefinite preserving (i.e., $S_A(S^n_+){\subseteq}S^n_+$) and show that such transformation is strictly monotone if and only if it is nondegenerate. We also show that a positive semidefinite preserving $S_A$ has the Ultra-GUS property if and only if $1{\not\in}{\sigma}(A){\sigma}(A)$.

• Korean Journal of Mathematics
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• v.15 no.1
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• pp.13-26
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• 2007

We prove that every strong null sequence in a Banach space X lies inside the range of a vector measure of bounded variation if and only if the condition $\mathcal{N}_1(X,{\ell}_1)={\Pi}_1(X,{\ell}_1)$ holds. We also prove that for $1{\leq}p&lt;{\infty}$ every strong ${\ell}_p$ sequence in a Banach space X lies inside the range of an X-valued measure of bounded variation if and only if the identity operator of the dual Banach space $X^*$ is ($p^{\prime}$,1)-summing, where $p^{\prime}$ is the conjugate exponent of $p$. Finally we prove that a Banach space X has the property that any sequence lying in the range of an X-valued measure actually lies in the range of a vector measure of bounded variation if and only if the condition ${\Pi}_1(X,{\ell}_1)={\Pi}_2(X,{\ell}_1)$ holds.

• ETRI Journal
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• v.28 no.6
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• pp.745-760
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• 2006

In this paper, we particularly deal with no $F_p$-rational two-torsion elliptic curves, where $F_p$ is the prime field of the characteristic p. First we introduce a shift product-based polynomial transform. Then, we show that the parities of (#E - 1)/2 and (#E' - 1)/2 are reciprocal to each other, where #E and #E' are the orders of the two candidate curves obtained at the last step of complex multiplication (CM)-based algorithm. Based on this property, we propose a method to check the parity by using the shift product-based polynomial transform. For a 160 bits prime number as the characteristic, the proposed method carries out the parity check 25 or more times faster than the conventional checking method when 4 divides the characteristic minus 1. Finally, this paper shows that the proposed method can make CM-based algorithm that looks up a table of precomputed class polynomials more than 10 percent faster.

• Journal of the Korean Mathematical Society
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• v.57 no.2
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• pp.283-312
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• 2020

A sufficient condition for an Artinian complete intersection quotient S = 𝕜[x, y]/(x^m, yⁿ), where 𝕜 is an algebraically closed field of a prime characteristic, to have the strong Lefschetz property (SLP) was proved by S. B. Glasby, C. E. Praezer, and B. Xia in [3]. In contrast, we find a necessary and sufficient condition on m, n satisfying 3 ≤ m ≤ n and p > 2m-3 for S to fail to have the SLP. Moreover we find the Jordan types for S failing to have SLP for m ≤ n and m = 3, 4.