• Journal of the Korean Mathematical Society
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• v.55 no.6
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• pp.1359-1380
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• 2018

In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Hardy type inequality with the same exponent n ($n{\geq}3$), then it has exactly the n-dimensional volume growth. Besides, three interesting applications of this fact have also been given. The first one is that we prove that complete noncompact smooth metric measure space with non-negative weighted Ricci curvature on which the Hardy type inequality holds with the best constant are isometric to the Euclidean space with the same dimension. The second one is that we show that if a complete n-dimensional Finsler manifold of nonnegative n-Ricci curvature satisfies the Hardy type inequality with the best constant, then its flag curvature is identically zero. The last one is an interesting rigidity result, that is, we prove that if a complete n-dimensional Berwald space of non-negative n-Ricci curvature satisfies the Hardy type inequality with the best constant, then it is isometric to the Minkowski space of dimension n.

• Bulletin of the Korean Mathematical Society
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• v.60 no.3
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• pp.623-636
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• 2023

In this paper, we prove some vanishing theorems under the assumptions of weighted BiRic curvature or m-Bakry-Émery-Ricci curvature bounded from below.

• Bulletin of the Korean Mathematical Society
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• v.41 no.1
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• pp.153-166
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• 2004

We prove the Radon-Nikodym theorem for a nonabsolute integral on measure spaces endowed with metric topologies and hence provide a descriptive definition of the integral.

• Journal of the Chungcheong Mathematical Society
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• v.27 no.4
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• pp.581-590
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• 2014

In approach theory, we can provide arbitrary products of ${\infty}p$-metric spaces with a natural structure, whereas, classically only if we rely on a countable product and the question arises, then, whether properties which are derived from countability properties in metric spaces, such as sequential and countable compactness, can also do away with countability. The classical results which simplify the study of compactness in pseudometric spaces, which proves that all three of the main kinds of compactness are identical, suggest a further study of the category $pMET^{\infty}$.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.3
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• pp.457-467
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• 2011

In this paper, we establish some comparison theorems about volumes of tubes in metric spaces with nonpositive curvature. First we compare the Hausdorff measure of tube about a metric ball contained in an (n-1)-dimensional totally geodesic subspace of an n-dimensional locally compact, geodesically complete Hadamard space with Lebesgue measure of its corresponding tube in Euclidean space ${\mathbb{R}}^n$, and then develop the result to the case of an m-dimensional totally geodesic subspace for 1 < m < n with an additional condition. Also, we estimate the Hausdorff measure of the tube about a shortest curve in a metric space of curvature bounded above and below.

• Journal of the Chungcheong Mathematical Society
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• v.35 no.1
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• pp.85-90
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• 2022

In this article, we consider the notion of expansiveness on compact metric spaces for symbolically point of view. And we show that a homeomorphism is symbolically countably expansive if and only if it is symbolically measure expansive. Moreover, we prove that a homeomorphism is symbolically N-expansive if and only if it is symbolically measure N-expanding.

• Journal of the Chungcheong Mathematical Society
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• v.25 no.3
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• pp.491-499
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• 2012

The hyperbolic metric is invariant under the action of M$\ddot{o}$bius maps and unbounded. For 0 < $r$ < 1, there is an r-lattice in the Bergman metric. Using this r-lattice, we get the measure ${\mu}_r$ and the Toeplitz operator $T^{\alpha}_{\mu}_r$ and we prove that $T^{\alpha}_{\mu}_r$ is bounded and $T^{\alpha}_{\mu}_r$ is compact under some condition.

• Journal of the Korean Mathematical Society
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• v.57 no.4
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• pp.935-955
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• 2020

In this paper we present a measurable version of the Smale's spectral decomposition theorem for homeomorphisms on compact metric spaces. More precisely, we prove that if a homeomorphism f on a compact metric space X is invariantly measure expanding on its chain recurrent set CR(f) and has the eventually shadowing property on CR(f), then f has the spectral decomposition. Moreover we show that f is invariantly measure expanding on X if and only if its restriction on CR(f) is invariantly measure expanding. Using this, we characterize the measure expanding diffeomorphisms on compact smooth manifolds via the notion of Ω-stability.

• Journal of the Korean Mathematical Society
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• v.39 no.5
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• pp.783-800
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• 2002

Let B denote the unit ball in $C^n$, and ν the normalized Lebesgue measure on B. For $\alpha$ > -1, define $dv_\alpha$(z) ＝ $c_\alpha$$(1-\midz\mid^2)^{\alpha}$dν(z), z $\in$ B. Here $c_\alpha$ is a positive constant such that $v_\alpha$(B) ＝ 1. Let H(B) denote the space of all holomorphic functions in B. For $p\geq1$, define the Bergman-Privalov space $(AN)^{p}(v_\alpha)$ by $(AN)^{p}(v_\alpha)$ = ${f\inH(B)$ : $\int_B{log(1+\midf\mid)}^pdv_\alpha\;<\;\infty}$ In this paper we prove that a function $f\inH(B)$ is in $(AN)^{p}$$(v_\alpha)$ if and only if $(1+\midf\mid)^{-2}{log(1+\midf\mid)}^{p-2}\mid\nablaf\mid^2\;\epsilon\;L^1(v_\alpha)$ in the case 1＜p＜$\infty$, or $(1+\midf\mid)^{-2}\midf\mid^{-1}\mid{\nabla}f\mid^2\;\epsilon\;L^1(v_\alpha)$ in the case p ＝ 1, where $nabla$f is the gradient of f with respect to the Bergman metric on B. This is an analogous result to the characterization of the Hardy spaces by M. Stoll [18] and that of the Bergman spaces by C. Ouyang-W. Yang-R. Zhao [13].

• Genomics & Informatics
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• v.17 no.1
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• pp.4.1-4.7
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• 2019

In this paper, we propose a window-based mechanism visualization approach as an alternative way to measure the seriousness of the difference among data-insights extracted from a composite biodata point. The approach is based on two components: undirected graph and Mosaab-metric space. The significant application of this approach is to visualize the segmented genome of a virus. We use Influenza and Ebola viruses as examples to demonstrate the robustness of this approach and to conduct comparisons. This approach can provide researchers with deep insights about information structures extracted from a segmented genome as a composite biodata point, and consequently, to capture the segmented genetic variations and diversity (variants) in composite data points.