• Honam Mathematical Journal
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• v.33 no.4
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• pp.467-485
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• 2011

We consider the Hyers-Ulam-Rassias stability problem $${\parallel}u{\circ}A-{\upsilon}{\circ}P_1-w{\circ}P_2{\parallel}{\leq}{\varepsilon}({\mid}x{\mid}^p+{\mid}y{\mid}^p)$$ for the Schwartz distributions u, ${\upsilon}$, w, which is a distributional version of the Pexider generalization of the Hyers-Ulam-Rassias stability problem ${\mid}(x+y)-g(x)-h(y){\mid}{\leq}{\varepsilon}({\mid}x{\mid}^p+{\mid}y{\mid}^p)$, x, $y{\in}\mathbb{R}^n$, for the functions f, g, h : $\mathbb{R}^n{\rightarrow}\mathbb{C}$.

• Bulletin of the Korean Mathematical Society
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• v.32 no.2
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• pp.211-214
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• 1995

Let M and N be compact Riemannian manifolds and $f : M \to N$ be a smooth map. Following J. Eells, f is exponentially harmonic if it represents a critical point of the exponential energy integral $$ E(f) = \int_{M} exp(\left\$\mid$ df \right\$\mid$^2) dM $$ where $(\left\ df $\mid$\right\$\mid$^2$ is the energy density defined as $\sum_{i=1}^{m} \left\$\mid$ df(e_i) \right\$\mid$^2$, m = dimM, for orthonormal frame $e_i$ of M. The Euler- Lagrange equation of the exponential energy functional E can be written $$ exp(\left\$\mid$ df \right\$\mid$^2)(\tau(f) + df(\nabla\left\$\mid$ df \right\$\mid$^2)) = 0 $$ where $\tau(f)$ is the tension field along f. Hence, if the energy density is constant, every harmonic map is exponentially harmonic and vice versa.

• Bulletin of the Korean Mathematical Society
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• v.50 no.4
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• pp.1235-1242
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• 2013

We give area and height estimates for cmc-graphs over a bounded planar $C^{2,{\alpha}}$ domain ${\Omega}{\subset}\mathbb{R}^3$. For a constant H satisfying $H^2{\mid}{\Omega}{\mid}{\leq}9{\pi}/16$, we show that the height $h$ of H-graphs over ${\Omega}$ with vanishing boundary satisfies ${\mid}h{\mid}$ < $(\tilde{r}/2{\pi})H{\mid}{\Omega}{\mid}$, where $\tilde{r}$ is the middle zero of $(x-1)(H^2{\mid}{\Omega}{\mid}(x+2)^2-9{\pi}(x-1))$. We use this height estimate to prove the following existence result for cmc H-graphs: for a constant H satisfying $H^2{\mid}{\Omega}{\mid}$ < $(\sqrt{297}-13){\pi}/8$, there exists an H-graph with vanishing boundary.

• Honam Mathematical Journal
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• v.31 no.4
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• pp.505-513
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• 2009

Let {$X_k;k{\in}N^d$} be centered and identically distributed random field which is asymptotically negative dependent in a certain case. In this note we prove that for $p{\alpha}$ > 1 and ${\alpha}$ > ${\frac{1}{2}}$ $E{\mid}X_1{\mid}^p(log^+{\mid}X_1{\mid}^{d-1})$ < ${\infty}$ if and only if ${\sum}_n{\mid}n{\mid}^{p{\alpha}-2}P$($max_{1{\leq}k{\leq}n{\mid}S_k{\mid}}$ > ${\epsilon}{\mid}n{\mid}$) < ${\infty}$ for all ${\epsilon}$ > 0, where log$^+$x = max{1,log x}.

• Honam Mathematical Journal
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• v.24 no.1
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• pp.121-130
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• 2002

In this paper we prove a functional central limit theorem for a field $\{X_{\underline{j}}:{\underline{j}}{\in}Z_+^d\}$ of nonstationary associated random variables with $EX{\underline{j}}=0,\;E{\mid}X_{\underline{j}}{\mid}^{r+{\delta}}<{\infty}$ for some $r>2,\;{\delta}>0$and $u(n)=O(n^{-{\nu}})$ for some ${\nu}>0$, where $u(n):=sup_{{\underline{i}}{\in}Z_+^d{\underline{j}}:{\mid}{\underline{j}}-{\underline{i}}{\mid}{\geq}n}{\sum}cov(X_{\underline{i}},\;X_{\underline{j}}),\;{\mid}{\underline{x}}{\mid}=max({\mid}x_1{\mid},{\cdots},{\mid}x_d{\mid})\;for\;{\underline{x}}{\in}{\mathbb{R}}^d$. Our investigation implies and analogous result in the case associated random measure.

• Communications of the Korean Mathematical Society
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• v.26 no.2
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• pp.261-271
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• 2011

We consider the Hyers-Ulam-Rassias stability problem ${\parallel}2u{\circ}\frac{A}{2}-u{\circ}P_1-u{\circ}P_2{\parallel}{\leq}{\varepsilon}({\mid}x{\mid}^p+{\mid}y{\mid}^p)$, $x,y{\in}{\mathbb{R}}^n$ for the Schwartz distributions u, which is a distributional version of the Hyers-Ulam-Rassias stability problem of the Jensen functional equation ${\mid}2f(\frac{x+y}{2})-f(x)-F(y){\mid}{\leq}{\varepsilon}({\mid}x{\mid}^p+{\mid}y{\mid}^p)$, $x,y{\in}{\mathbb{R}}^n$ for the function f : ${\mathbb{R}}^n{\rightarrow}{\mathbb{C}}$.

• Bulletin of the Korean Mathematical Society
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• v.34 no.3
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• pp.385-394
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• 1997

In this paper, we prove in p-uniforlmy convex space a fixed point theorem for a class of mappings T satsfying: for each x, y in the domain and for n = 1, 2, 3, $\cdots$, $$ \left\$\mid$ T^n x - T^n y \right\$\mid$ \leq a \cdot \left\$\mid$ x - y \right\$\mid$ + b(\left\$\mid$ x - T^n x \right\$\mid$ + \left\$\mid$ y - T^n y \right\$\mid$) + c(\left\$\mid$ c - T^n y \right\$\mid$ + \left\$\mid$ y - T^n x \right\$\mid$, $$ where a, b, c are nonnegative constants satisfying certain conditions. Further we establish some fixed point theorems for these mappings in a Hilbert space, in $L^p$ spaces, in Hardy spaces $H^p$ and in Sobolev spaces $H^{p,k}$ for 1 < p < $\infty$ and k $\leq$ 0. As a consequence of our main result, we also the results of Goebel and Kirk [7], Lim [8], Lifshitz [12], Xu [20] and others.

• Bulletin of the Korean Mathematical Society
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• v.58 no.5
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• pp.1279-1300
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• 2021

Let P be a finite point set in ℝ² with the set of distance n-chains defined as ∆_n(P) = {(|p₁ - p₂|, |p₂ - p₃|, …, |p_n - p_n+1|) : p_i ∈ P}. We show that for 2 ⩽ n = O_|P|(1) we have ${\mid}{\Delta}_n(P){\mid}{\gtrsim}{\frac{{\mid}P{\mid}^n}{{\log}^{\frac{13}{2}(n-1)}{\mid}P{\mid}}}$. Our argument uses the energy construction of Elekes and a general version of Rudnev's rich-line bound implicit in [28], which allows one to iterate efficiently on intersecting nested subsets of Guth-Katz lines. Let G is a simple connected graph on m = O(1) vertices with m ⩾ 2. Define the graph-distance set ∆_G(P) as ∆_G(P) = {(|p_i - p_j|)_{i,j}∈E(G) : p_i, p_j ∈ P}. Combining with results of Guth and Katz [17] and Rudnev [28] with the above, if G has a Hamiltonian path we have ${\mid}{\Delta}_G(P){\mid}{\gtrsim}{\frac{{\mid}P{\mid}^{m-1}}{\text{polylog}{\mid}P{\mid}}}$.

• Journal of KIISE:Computer Systems and Theory
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• v.32 no.5
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• pp.255-259
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• 2005

To search a pattern P in a text, such index data structures as suffix trees and suffix arrays are widely used in diverse applications of string processing and computational biology. It is well known that searching in suffix trees is faster than suffix ways in the aspect of time complexity, i.e., it takes O(${\mid}P{\mid}$) time to search P on a constant-size alphabet in a suffix tree while it takes O(${\mid}P{\mid}+logn$) time in a suffix way where n is the length of the text. In this paper we present a linear-tim8 search algorithm in suffix arrays for constant-size alphabets. For a gene.al alphabet $\Sigma$, it takes O(${\mid}P{\mid}log{\mid}{\Sigma}{\mid}$) time.

• Journal of Korean Academy of Nursing
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• v.38 no.6
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• pp.874-880
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• 2008

Purpose: The purpose of this study was to identify muscle atrophy in critically ill patients on ventilators. A comparison was made between limb circumferences and muscle areas on the day of admission to an intensive care unit (ICU) and on the 8th day after admission. Methods: The data were collected from 30 critically ill patients using ventilators at S hospital ICU in Seoul during the period from October 2005 to April 2006. Limb circumferences and skinfold thickness were measured on the day of admission and on the 8th day after admission to the ICU. Limb circumferences and skinfold thickness were measured on the right mid-arm, right mid-thigh, and right mid-calf using a measuring tape and a skinfold calipers. Limb muscle areas were calculated by an equation after measuring limb circumferences and skinfold thickness. The data were analyzed by paired t-test and independent t-test. Results: 1) Mid-arm circumference, mid-thigh circumference and mid-calf circumference on the 8th day after admission to the ICU were significantly less than those on the day of admission to the ICU, 2) Mid-arm muscle area, mid-thigh muscle area and mid-calf muscle area on the 8th day after admission to the ICU were significantly less than those on the day of admission to the ICU, 3) Steroid and vecuronium medication didn't affect changes in limb circumferences (mid-arm, mid-thigh, mid-calf) and muscle areas (mid-arm, mid-thigh, mid-calf on the 8th day after admission to the ICU compared to the day of admission. Conclusion: Limb muscle atrophy may occur on the 8th day after admission to the ICU in critically ill patients using ventilator.