• Bulletin of the Korean Mathematical Society
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• v.54 no.6
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• pp.1981-1990
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• 2017

Let R be a root system in $\mathbb{R}^d$ with Coxeter-Weyl group W and let k be a nonnegative multiplicity function on R. The generalized volume mean of a function $f{\in}L^1_{loc}(\mathbb{R}^d,m_k)$, with $m_k$ the measure given by $dmk(x):={\omega}_k(x)dx:=\prod_{{\alpha}{\in}R}{\mid}{\langle}{\alpha},x{\rangle}{\mid}^{k({\alpha})}dx$, is defined by: ${\forall}x{\in}\mathbb{R}^d$, ${\forall}r$ > 0, $M^r_B(f)(x):=\frac{1}{m_k[B(0,r)]}\int_{\mathbb{R}^d}f(y)h_k(r,x,y){\omega}_k(y)dy$, where $h_k(r,x,{\cdot})$ is a compactly supported nonnegative explicit measurable function depending on R and k. In this paper, we prove that for almost every $x{\in}\mathbb{R}^d$, $lim_{r{\rightarrow}0}M^r_B(f)(x)= f(x)$.

• Bulletin of the Korean Mathematical Society
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• v.33 no.3
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• pp.455-463
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• 1996

In [3] and [4], Kitahara, Pak and the author obtained the conformally invariant tensor $B_0$, which is an algebraic Hermitian analogue of the Weyl conformal curvature tensor W in the Riemannian geometry, by the decomposition of the curvature tensor H of the Hermitian connection and the notion of semi-curvature-like tensors of Tanno (see[7]). In [5], the author defined a conformally invariant tensor $B_0$ on a Hermitian manifold as a modification of $B_0$. Moreover he introduced the notion of local conformal Hermitian-flatness of Hermitian manifolds and proved that the vanishing of this tensor $B_0$ together with some condition for the scalar curvatures is a necessary and sufficient condition for a Hermitian manifold to be locally conformally Hermitian-flat.

• Communications of the Korean Mathematical Society
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• v.35 no.2
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• pp.401-415
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• 2020

It is well-known that if char𝕜 = 0, then an Artinian monomial complete intersection quotient 𝕜[x₁, …, x_n]/(x₁^a₁, …, x_n^a_n) has the strong Lefschetz property in the narrow sense, and it is decomposed by the direct sum of irreducible 𝖘𝖑₂-modules. For an Artinian ring A = 𝕜[x₁, x₂, x₃]/(x₁⁶, x₂⁶, x₃⁶), by the Schur-Weyl duality theorem, there exist 56 trivial representations, 70 standard representations, and 20 sign representations inside A. In this paper we find an explicit basis for A, which is compatible with the S₃-module structure.

• Communications of the Korean Mathematical Society
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• v.21 no.1
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• pp.45-52
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• 2006

Every non-associative algebra L corresponds to its symmetric semi-Lie algebra $L_{[,]}$ with respect to its commutator. It is an interesting problem whether the equality $Aut{non}(L)=Aut_{semi-Lie}(L)$ holds or not [2], [13]. We find the non-associative algebra automorphism groups $Aut_{non}\; \frac\;{(WN_{0,0,1}_{[0,1,r_1...,r_p])}$ and $Aut_{non-Lie}\; \frac\;{(WN_{0,0,1}_{[0,1,r_1...,r_p])}$ where every automorphism of the automorphism groups is the composition of elementary maps [3], [4], [7], [8], [9], [10], [11]. The results of the paper show that the F-algebra automorphism groups of a polynomial ring and its Laurent extension make easy to find the automorphism groups of the algebras in the paper.

• Communications of the Korean Mathematical Society
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• v.17 no.2
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• pp.349-361
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• 2002

The ill-posed elliptic wave propagation problems can be transformed into well-posed initial value problems of the reflection and transmission operators characterizing the material structure of the given model by the combination of wave field splitting and invariant imbedding methods. In general, the derived scattering operator equations are of first-order in range, nonlinear, nonlocal, and stiff and oscillatory with a subtle fixed and movable singularity structure. The phase space and path integral analysis reveals that construction and reconstruction algorithms depend crucially on a detailed symbol analysis of the scattering operators. Some information about the singularity structure of the scattering operator symbols is presented and analyzed in the transversely homogeneous limit.

• The Pure and Applied Mathematics
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• v.11 no.4
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• pp.287-292
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• 2004

The operator $A \; {\in} \; L(H_{i})$, the Banach algebra of bounded linear operators on the complex infinite dimensional Hilbert space $\cal H_{i}$, is said to be p-hyponormal if $(A^\ast A)^P \geq (AA^\ast)^p$ for $p\; \in \; (0,1]$. Let (equation omitted) denote the completion of (equation omitted) with respect to some crossnorm. Let $I_{i}$ be the identity operator on $H_{i}$. Letting (equation omitted), where each $A_{i}$ is p-hyponormal, it is proved that the commuting n-tuple T = ($T_1$,..., $T_{n}$) satisfies Bishop's condition ($\beta$) and that if T is Weyl then there exists a non-singular commuting n-tuple S such that T = S + F for some n-tuple F of compact operators.

• Journal of the Korean Mathematical Society
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• v.40 no.1
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• pp.1-28
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• 2003

The Iwahori-Hecke algebra $H_{k}$ ( $q^2$) of type A acts on the k-fold tensor product space of the natural representation of the quantum superalgebra (equation omitted)$_{q}$(gl(m, n)). We show the Hecke algebra $H_{k}$ ( $q^2$) and the quantum superalgebra (equation omitted)$_{q}$(gl(m n)) have commuting actions on the tensor product space, and determine the centralizer of each other. Using this result together with Gyoja's q-analogue of the Young symmetrizers, we construct highest weight vectors of irreducible summands of the tensor product space.

• Bulletin of the Korean Mathematical Society
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• v.54 no.5
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• pp.1779-1801
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• 2017

Let X be a minimal del Pezzo surface of degree 2 over a finite field ${\mathbb{F}}_q$. The image ${\Gamma}$ of the Galois group Gal(${\bar{\mathbb{F}}}_q/{\mathbb{F}}_q$) in the group Aut($Pic({\bar{X}})$) is a cyclic subgroup of the Weyl group W($E_7$). There are 60 conjugacy classes of cyclic subgroups in W($E_7$) and 18 of them correspond to minimal del Pezzo surfaces. In this paper we study which possibilities of these subgroups for minimal del Pezzo surfaces of degree 2 can be achieved for given q.

• Journal of the Korean Mathematical Society
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• v.57 no.4
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• pp.845-871
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• 2020

We consider generalized Dedekind sums in dimension n, defined as sum of products of values of periodic Bernoulli functions. For the generalized Dedekind sums, we associate a Laurent polynomial. Using this, we associate an exponential sum of a Laurent polynomial to the generalized Dedekind sums and show that this exponential sum has a nontrivial bound that is sufficient to fulfill the equidistribution criterion of Weyl and thus the fractional part of the generalized Dedekind sums are equidistributed in ℝ/ℤ.

• Bulletin of the Korean Mathematical Society
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• v.34 no.1
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• pp.1-8
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• 1997

Let G be a finiteg oroup. We denote BG a classifying space of G, which a contractible universal principal G bundle EG. The stable type of BG does not determine G up to isomorphism. A simple example [due to N. Minami]is given by $Q_{4p} \times Z/2$ and $D_{2p} \times Z/4$ where ps is an odd prime, $Q_{4p} is the generalized quarternion group of order 4p and $D_{2p}$ is the dihedral group of order 2p. However the paper [6] gives us a necessary and sufficient condition for $BG_1$ and $BG_2$ to be stably equivalent localized et pp. The local stable type of BG depends on the conjegacy classes of homomorphisms from the p-groups Q into G. This classification theorem simplifies if G has a normal sylow p-subgroup. Then the stable homotopy type depends on the Weyl group of the sylow p-subgroup.