• Bulletin of the Korean Mathematical Society
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• v.54 no.6
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• pp.1991-1999
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• 2017

Let A and B be two Banach algebras and $T:B{\rightarrow}A$ be a bounded homomorphism, with ${\parallel}T{\parallel}{\leq}1$. Recently, Dabhi, Jabbari and Haghnejad Azar (Acta Math. Sin. (Engl. Ser.) 31 (2015), no. 9, 1461-1474) obtained some results about the n-weak amenability of $A{\times}_TB$. In the present paper, we address a gap in the proof of these results and extend and improve them by discussing general necessary and sufficient conditions for $A{\times}_TB$ to be n-weakly amenable, for an integer $n{\geq}0$.

• Bulletin of the Korean Mathematical Society
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• v.54 no.6
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• pp.2043-2051
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• 2017

Consider a sequence of compactly supported Hamiltonian diffeomorphisms ${\phi}_k$ of an exact symplectic manifold, all of which are "graphical" in the sense that their graphs are identified by a Darboux-Weinstein chart with the image of a one-form. We show by an elementary argument that if the ${\phi}_k$ $C^0$-converge to the identity, then their Calabi invariants converge to zero. This generalizes a result of Oh, in which the ambient manifold was the two-disk and an additional assumption was made on the Hamiltonians generating the ${\phi}_k$. We discuss connections to the open problem of whether the Calabi homomorphism extends to the Hamiltonian homeomorphism group. The proof is based on a relationship between the Calabi invariant of a $C^0$-small Hamiltonian diffeomorphism and the generalized phase function of its graph.

• Communications of the Korean Mathematical Society
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• v.17 no.1
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• pp.165-173
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• 2002

We first review Ramaswami's find Apostol's identities involving the Zeta function in a rather detailed manner. We then present corrected, or generalized formulas, or a different method of proof for some of them. We also give closed-form evaluation of some series involving the Riemann Zeta function by an integral representation of ζ(s) and Apostol's identities given here.

• Bulletin of the Korean Mathematical Society
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• v.31 no.1
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• pp.93-97
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• 1994

Previously T.Porter [3] has given a relative Chinese Remainder Theorem under the hypothesis that given ring R has at least one .tau.-closed maximal ideal (by his notation Ma $x_{\tau}$(R).neq..phi.). In this short paper we drop his overall hypothesis that Ma $x_{\tau}$(R).neq..phi. and give the proof and some related results with this Theorem. In this paper R will always denote a commutative ring with identity element and all modules will be unitary left R-modules unless otherwise specified. Let .tau. be a given hereditarty torsion theory for left R-module category R-Mod. The class of all .tau.-torsion left R-modules, dented by J is closed under homomorphic images, submodules, direct sums and extensions. And the class of all .tau.-torsionfree left R-modules, denoted by F, is closed under taking submodules, injective hulls, direct products, and isomorphic copies ([2], Proposition 1.7 and 1.10).

• Communications of the Korean Mathematical Society
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• v.10 no.4
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• pp.925-929
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• 1995

Consider the problem $-div($\mid$\bigtriangledown_u$\mid$^{p-2}\bigtriangledown_u) = $\mid$u$\mid$^{p^*-2}u + \lambda$\mid$u$\mid$^{q-2}u$ in B, u = 0 on $\partial B$; where $B \subset R^n$ is a ball, $\lambda < 0, 1 < p < n$ and $p^* = \frac{np}{n-p}$ is the critical Sobolev exponent. For given $\lambda > 0$, we show that there exists $k = k(\lambda) \in N$ such that any radial solutions to this problem have at most k noda curves when $p \leq q \leq p^* - 1$.

• Communications of the Korean Mathematical Society
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• v.23 no.3
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• pp.401-411
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• 2008

In this paper, we characterize Sobolev spaces $H^s(\mathbb{R}^n),\;s{\in}\mathbb{R}$ by the initial value of solutions of heat equation with a growth condition. By using an idea in its proof, we also discuss a stability problem for Cauchy functional equation in the Sobolev spaces.

• Journal of the Korean Mathematical Society
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• v.57 no.6
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• pp.1435-1449
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• 2020

In this article we make a local classification of n-dimensional Riemannian manifolds (M, g) with harmonic curvature and less than four Ricci eigenvalues which admit a smooth non constant solution f to the following equation $$(1)\hspace{20}{\nabla}df=f(r-{\frac{R}{n-1}}g)+x{\cdot} r+y(R)g,$$ where ∇ is the Levi-Civita connection of g, r is the Ricci tensor of g, x is a constant and y(R) a function of the scalar curvature R. Indeed, we showed that, in a neighborhood V of each point in some open dense subset of M, either (i) or (ii) below holds; (i) (V, g, f + x) is a static space and isometric to a domain in the Riemannian product of an Einstein manifold N and a static space (W, gW, f + x), where gW is a warped product metric of an interval and an Einstein manifold. (ii) (V, g) is isometric to a domain in the warped product of an interval and an Einstein manifold. For the proof we use eigenvalue analysis based on the Codazzi tensor properties of the Ricci tensor.

• Journal of the Korean Mathematical Society
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• v.41 no.4
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• pp.629-646
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• 2004

Let G be a compact semialgebraic group and M a semi-algebraic G-set. We prove that there exists a semialgebraic slice at every point of M. Moreover M can be covered by finitely many semialgebraic G-tubes. As an application we give a different proof that every semialgebraic G-set admits a semi algebraic G-embedding into some semialgebraic orthogonal representation space of G, which has been proved in [15].

• Journal of the Korean Mathematical Society
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• v.37 no.1
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• pp.31-44
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• 2000

We study a free boundary problem satisfying Bernoulli type boundary condition along which the gradient of a piecewise harmonic solution jumps zero to a given constant value. In such problem, the free boundary splits the domain into two regions, the zero set and the harmonic region. Our main interest is to identify the global shape and the location of the zero set. In this paper, we find the lower and the upper bound of the zero set. In a convex domain, easier estimation of the upper bound and faster disk test technique are given to find a rough shape of the zero set. Also a simple proof on the convexity of zero set is given for a connected zero set in a convex domain.

• Communications of the Korean Mathematical Society
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• v.20 no.4
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• pp.695-702
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• 2005

We study the transformed measures with respect to the real parameters of a self-similar measure on a self-similar Can­tor set to give a simple proof for some result of its multifractal spectrum. A transformed measure with respect to a real parameter of a self-similar measure on a self-similar Cantor set is also a self­similar measure on the self-similar Cantor set and it gives a better information for multifractals than the original self-similar measure. A transformed measure with respect to an optimal parameter deter­mines Hausdorff and packing dimensions of a set of the points which has same local dimension for a self-similar measure. We compute the values of the transformed measures with respect to the real parameters for a set of the points which has same local dimension for a self-similar measure. Finally we investigate the magnitude of the local dimensions of a self-similar measure and give some correlation between the local dimensions.