• Journal of the Korean Data and Information Science Society
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• v.26 no.6
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• pp.1295-1303
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• 2015

In a mixture experiment, one may be interested in estimating not only main effects but also some interactions. Main effects and interactions may be estimated through appropriate designs such as simplex-centroid designs. However, the estimability problems, implied by the sum to one functional relationship among the factors, have strong consequences on the confounding and identifiability of models for such designs. To handle these problems, we address homogeneous polynomial model based on the computational commutative algebra (CCA) instead of using $Scheff{\acute{e}}s$ canonical model which is typically used. The problem posed here is to give how to choose estimable main effects and also some low-degree interactions. The theory is tested using a fraction of simplex-centroid designs aided by a modern computational algebra package CoCoA.

• Bulletin of the Korean Mathematical Society
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• v.57 no.1
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• pp.207-218
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• 2020

Some modular representations of reflection groups related to Weyl groups are considered. The rational cohomology of the classifying space of a compact connected Lie group G with a maximal torus T is expressed as the ring of invariants, H*(BG; ℚ) ≅ H*(BT; ℚ)^W(G), which is a polynomial ring. If such Lie groups are locally isomorphic, the rational representations of their Weyl groups are equivalent. However, the integral representations need not be equivalent. Under the mod p reductions, we consider the structure of the rings, particularly for the Weyl group of symplectic groups Sp(n) and for the alternating groups A_n as the subgroup of W(SU(n)). We will ask if such rings of invariants are polynomial rings, and if each of them can be realized as the mod p cohomology of a space. For n = 3, 4, the rings under a conjugate of W(Sp(n)) are shown to be polynomial, and for n = 6, 8, they are non-polynomial. The structures of H*(BT^n-1; 𝔽_p)^A_n will be also discussed for n = 3, 4.

• Journal of the Korean Mathematical Society
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• v.58 no.3
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• pp.643-702
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• 2021

Fix ℤ/2 is the prime field of two elements and write 𝒜₂ for the mod 2 Steenrod algebra. Denote by GL_d := GL(d, ℤ/2) the general linear group of rank d over ℤ/2 and by ${\mathfrak{P}}_d$ the polynomial algebra ℤ/2[x₁, x₂, …, x_d] as a connected unstable 𝒜₂-module on d generators of degree one. We study the Peterson "hit problem" of finding the minimal set of 𝒜₂-generators for ${\mathfrak{P}}_d$. Equivalently, we need to determine a basis for the ℤ/2-vector space $$Q{\mathfrak{P}}_d:={\mathbb{Z}}/2{\otimes}_{\mathcal{A}_2}\;{\mathfrak{P}}_d{\sim_=}{\mathfrak{P}}_d/{\mathcal{A}}^+_2{\mathfrak{P}}_d$$ in each degree n ≥ 1. Note that this space is a representation of GL_d over ℤ/2. The problem for d = 5 is not yet completely solved, and unknown in general. In this work, we give an explicit solution to the hit problem of five variables in the generic degree n = r(2^t - 1) + 2^ts with r = d = 5, s = 8 and t an arbitrary non-negative integer. An application of this study to the cases t = 0 and t = 1 shows that the Singer algebraic transfer of rank 5 is an isomorphism in the bidegrees (5, 5 + (13.2⁰ - 5)) and (5, 5 + (13.2¹ - 5)). Moreover, the result when t ≥ 2 was also discussed. Here, the Singer transfer of rank d is a ℤ/2-algebra homomorphism from GL_d-coinvariants of certain subspaces of $Q{\mathfrak{P}}_d$ to the cohomology groups of the Steenrod algebra, $Ext^{d,d+*}_{\mathcal{A}_2}$ (ℤ/2, ℤ/2). It is one of the useful tools for studying these mysterious Ext groups.

• Journal of the Korean Mathematical Society
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• v.42 no.6
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• pp.1287-1309
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• 2005

A key exchange protocol using commutative subalge-bras of a full matrix algebra is considered. The security of the protocol depends on the difficulty of solving matrix equations XRY = T, with given matrices R and T. We give a polynomial time algorithm to solve XRY = T for the choice of certain types of subalgebras. We also compare the efficiency of the protocol with the Diffie-Hellman key exchange protocol on the key computation time and the key size.

• Communications of the Korean Mathematical Society
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• v.10 no.2
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• pp.293-300
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• 1995

Let $R = k[y_1,\cdots,y_n] \otimes E[x_1, \cdots, x_n]$ with characteristic $k = p > 2$ (odd prime), where $$\mid$y_i$\mid$ = 2, $\mid$x_i$\mid$ = 1$ and $y_i = \betax_i, \beta$ is the Bockstein homomorphism. Topologically, $R = H^*(B(Z/p)^n,k)$. For a symmetric group $\sum_n, R^{\sum_n} = k[\sigma_1,\cdots,\sigma_n] \otimes E[d\sigma_1, \cdots, d\sigma_n]$ where d is the derivation satisfying $d(y_i) = x_i$ and $d(x_iy_i) = x_iy_i + x_jy_i, 1 \leq i, j \leq n$. We give a direct proof of this theorem by using induction.

• Journal of the Chungcheong Mathematical Society
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• v.31 no.1
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• pp.47-52
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• 2018

Let ${\delta}$ be a derivation in a noetherian integral domain A. It is shown that a natural map induces a homeomorphism between the spectrum of $A[z;{\delta}]$ and the Poisson spectrum of $A[z;{\delta}]_p$ such that its restriction to the primitive spectrum of $A[z;{\delta}]$ is also a homeomorphism onto the Poisson primitive spectrum of $A[z;{\delta}]_p$.

• Communications of the Korean Mathematical Society
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• v.37 no.4
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• pp.957-967
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• 2022

Recently, Brešar's Jordan {g, h}-derivations have been investigated on triangular algebras. As a first aim of this paper, we extend this study to an interesting general context. Namely, we introduce the notion of Jordan 𝒢_n-derivations, with n ≥ 2, which is a natural generalization of Jordan {g, h}-derivations. Then, we study this notion on path algebras. We prove that, when n > 2, every Jordan 𝒢_n-derivation on a path algebra is a {g, h}-derivation. However, when n = 2, we give an example showing that this implication does not hold true in general. So, we characterize when it holds. As a second aim, we give a positive answer to a variant of Lvov-Kaplansky conjecture on path algebras. Namely, we show that the set of values of a multi-linear polynomial on a path algebra KE is either {0}, KE or the space spanned by paths of a length greater than or equal to 1.

• Bulletin of the Korean Mathematical Society
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• v.42 no.4
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• pp.679-690
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• 2005

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, d a non-zero derivation of R, I a non-zero right ideal of R, f($x_1,{\cdots},\;x_n$) a multilinear polynomial in n non-commuting variables over K, a $\in$ R. Supppose that, for any $x_1,{\cdots},\;x_n\;\in\;I,\;a[d(f(x_1,{\cdots},\;x_n)),\;f(x_1,{\cdots},\;x_n)]$ = 0. If $[f(x_1,{\cdots},\;x_n),\;x_{n+1}]x_{n+2}$ is not an identity for I and $$S_4(I,\;I,\;I,\;I)\;I\;\neq\;0$$, then aI = ad(I) = 0.

• Bulletin of the Korean Mathematical Society
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• v.29 no.1
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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.

• Kyungpook Mathematical Journal
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• v.57 no.1
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• pp.99-108
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• 2017

In this paper we construct quadratic ${\beta}-algebras$ on a field, and we discuss both linear-quadratic ${\beta}-algebras$ and quadratic-linear ${\beta}-algebras$ in a field. Moreover, we discuss some relations of binary operations in ${\beta}-algebras$.