• Journal of the Korean Mathematical Society
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• v.47 no.3
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• pp.547-572
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• 2010

We study existence of positive solutions of the classical nonlinear Schr$\ddot{o}$dinger equation $-{\Delta}u\;+\;V(x)u\;-\;f(x,\;u)\;-\;H(x)u^{2*-1}\;=\;0$, u > 0 in $\mathbb{R}^n$ $u\;{\rightarrow}\;0\;as\;|x|\;{\rightarrow}\;{\infty}$. In fact, we consider the following more general quasi-linear Schr$\ddot{o}$odinger equation $-div(|{\nabla}u|^{m-2}{\nabla}u)\;+\;V(x)u^{m-1}$ $-f(x,\;u)\;-\;H(x)u^{m^*-1}\;=\;0$, u > 0 in $\mathbb{R}^n$ $u\;{\rightarrow}\;0\;as\;|x|\;{\rightarrow}\;{\infty}$, where m $\in$ (1, n) is a positive number and $m^*\;:=\;\frac{mn}{n-m}\;>\;0$, is the corresponding critical Sobolev embedding number in $\mathbb{R}^n$. Under appropriate conditions on the functions V(x), f(x, u) and H(x), existence and non-existence results of positive solutions have been established.

• Journal of Applied and Pure Mathematics
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• v.4 no.3_4
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• pp.85-97
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• 2022

Let a graph G = (V, E) be a (p, q) graph. Define $${\rho}\;=\;\{\array{{\frac{p}{2}}&p\text{ is even}\\{\frac{p-1}{2}}\;&p\text{ is odd,}}$$ and M = {±1, ±2, ⋯ ± 𝜌} called the set of labels. Consider a mapping λ : V → M by assigning different labels in M to the different elements of V when p is even and different labels in M to p - 1 elements of V and repeating a label for the remaining one vertex when p is odd. The labeling as defined above is said to be a pair mean cordial labeling if for each edge uv of G, there exists a labeling $\frac{{\lambda}(u)+{\lambda}(v)}{2}$ if λ(u) + λ(v) is even and $\frac{{\lambda}(u)+{\lambda}(v)+1}{2}$ if λ(u) + λ(v) is odd such that ${\mid}\bar{\mathbb{S}}_{{\lambda}_1}-\bar{\mathbb{S}}_{{\lambda}^c_1}{\mid}{\leq}1$ where $\bar{\mathbb{S}}_{{\lambda}_1}$ and $\bar{\mathbb{S}}_{{\lambda}^c_1}$ respectively denote the number of edges labeled with 1 and the number of edges not labeled with 1. A graph G for which there exists a pair mean cordial labeling is called a pair mean cordial graph. In this paper, we investigate the pair mean cordial labeling of graphs which are obtained from path and cycle.

• Journal of Applied and Pure Mathematics
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• v.5 no.3_4
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• pp.237-253
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• 2023

Let a graph G = (V, E) be a (p, q) graph. Define $${\rho}=\{\array{{\frac{p}{2}} & \;\;p\text{ is even} \\ {\frac{p-1}{2}} & \;\;p\text{ is odd,}$$ and M = {±1, ±2, … ± ρ} called the set of labels. Consider a mapping λ : V → M by assigning different labels in M to the different elements of V when p is even and different labels in M to p - 1 elements of V and repeating a label for the remaining one vertex when p is odd. The labeling as defined above is said to be a pair mean cordial labeling if for each edge uv of G, there exists a labeling ${\frac{{\lambda}(u)+{\lambda}(v)}{2}}$ if λ(u) + λ(v) is even and ${\frac{{\lambda}(u)+{\lambda}(v)+1}{2}}$ if λ(u) + λ(v) is odd such that ${\mid}{\bar{{\mathbb{S}}}}_{\lambda}{_1}-{\bar{{\mathbb{S}}}}_{{\lambda}^c_1}{\mid}{\leq}1$ where ${\bar{{\mathbb{S}}}}_{\lambda}{_1}$ and ${\bar{{\mathbb{S}}}}_{{\lambda}^c_1}$ respectively denote the number of edges labeled with 1 and the number of edges not labeled with 1. A graph G for which there exists a pair mean cordial labeling is called a pair mean cordial graph. In this paper, we investigate the pair mean cordial labeling behavior of few graphs including the closed helm graph, web graph, jewel graph, sunflower graph, flower graph, tadpole graph, dumbbell graph, umbrella graph, butterfly graph, jelly fish, triangular book graph, quadrilateral book graph.

• Journal of Applied and Pure Mathematics
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• v.6 no.1_2
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• pp.55-69
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• 2024

Let a graph G = (V, E) be a (p, q) graph. Define $${\rho}=\{\array{{\frac{p}{2}} && p\;\text{is even} \\ {\frac{p-1}{2}} && p\;\text{is odd,}}$$ and M = {±1, ±2, … ± 𝜌} called the set of labels. Consider a mapping λ : V → M by assigning different labels in M to the different elements of V when p is even and different labels in M to p - 1 elements of V and repeating a label for the remaining one vertex when p is odd. The labeling as defined above is said to be a pair mean cordial labeling if for each edge uv of G, there exists a labeling $\frac{{\lambda}(u)+{\lambda}(v)}{2}$ if λ(u) + λ(v) is even and $\frac{{\lambda}(u)+{\lambda}(v)+1}{2}$ if λ(u) + λ(v) is odd such that ${\mid}\bar{\mathbb{s}}_{{\lambda}_1}-\bar{\mathbb{s}}_{{\lambda}^c_1}{\mid}\,{\leq}\,1$ where $\bar{\mathbb{s}}_{{\lambda}_1}$ and $\bar{\mathbb{s}}_{{\lambda}^c_1}$ respectively denote the number of edges labeled with 1 and the number of edges not labeled with 1. A graph G with a pair mean cordial labeling is called a pair mean cordial graph. In this paper, we investigate the pair mean cordial labeling behavior of some union of graphs.

• Koreanishche Zeitschrift fur Deutsche Sprachwissenschaft
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• v.1
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• pp.69-90
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• 1999

Die vorliegende Arbeit setzte sich zum Ziel, deutsche verbale Phraseolexeme(vPL) auf der Grundlage einer Mehrebenenbeschreibung syntaktisch sowie semantisch zu untersuchen. Dabei $st\"{u}tzten$ wir uns vor allem auf die Valenz- und Kasustheorie sowie das 6-Stufen-Modell von G. Helbig(1983) und das Mehrebenenmodell $f\"{u}r$ vPL von B. Wotjak(1992), das seinerseits vom methodischen Ansatz her in engem Zusammenhang zu dem Beschreibungsansatz steht. Aber Wir modifizierten teilweise das Modell yon B. Wotjak(1992) und stellten einige vPL exemplarisch dar. Dieser Untersuchung lag eine verbozentrische Auffassung des Satzes zugrunde: Das Verb bildet das organisierende Zentrum des Satzes und ist fur die Grundstruktur des Satzes verantwortlich. Das Verb als $Valenztr\"{a}ger\;er\"{o}ffnet$ um sich herum bestimmte Leerstellen, deren Zahl und Art letztlich von der Verbbedeutung her determiniert sind. Auch vPL kann als $Valenztr\"{a}ger$ - wie das Verb - das strukturelle Zentrum des Satzes darstellen, d.h. vPL bildet als Ganzes das $Pr\"{a}dikat$, das wendungsextern eine bestimmte Zahl von Leerstellen $er\"{o}ffnet$ und determiniert im Satz mit seinen $inh\"{a}renten$ semantischen Merkmalen seine semantische und syntaktische Umgebung. Bei der Analyse von vPL gingen wir von der Auffassung von B. Wotjak aus, $da\ss$ Modelle zur Beschreibung von Verben prinzipiell auch auf die Beschreibung von verbalen PL anwendbar sind. In dieser Untersuchung verwendeten wir - in Anlehnung an B. Wotjak(1996:4f.) - den Terminus 'Phraseologismus' als Oberbegriff, der a) Kollokationen, $b)\;Funktionsverbgef\"{u}ge\;(FVG)$, c) Wortidiome (wortwertige idiomatische Redewendungen oder Phraseolexeme), d) Satzidiome(satzwertige idiomatische Redewendungen bzw. kommunikative Formeln/ Routineformeln), e) $Sprichw\"{o}rter\;umfa{\ss}t$. Wir $beschr\"{a}nkten$ uns den Gegenstand unserer Untersuchung auf verbale Phraseolexeme(vPL), d.h. wie oben genannte Wortidiome. VPL lassen sich $prim\"{a}r$ in bezug auf das wendungsexterne ($\"{a}u{\ss}ere$) Aktantenpotential(PLe) und $sekund\"{a}r$ in bezug auf die interne Kom­ponentenstruktur des PL(PLi) klassifizieren. Unsere Studie soil nicht nur Hintergrundwissen $f\"{u}r$ Lexikonein­tragungen vertiefen helfen, sondern auch $f\"{u}r$ den Fremdsprachen­unterricht von Nutzen sein.

• Communications of the Korean Mathematical Society
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• v.27 no.3
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• pp.441-448
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• 2012

Let R be a prime ring with center Z and characteristic different from two, U a nonzero Lie ideal of R such that $u^2{\in}U$ for all $u{\in}U$ and $d$ be a nonzero (${\sigma}$, ${\tau}$)-derivation of R. We prove the following results: (i) If $[d(u),u]_{{\sigma},{\tau}}$ = 0 or $[d(u),u]_{{\sigma},{\tau}}{\in}C_{{\sigma},{\tau}}$ for all $u{\in}U$, then $U{\subseteq}Z$. (ii) If $a{\in}R$ and $[d(u),a]_{{\sigma},{\tau}}$ = 0 for all $u{\in}U$, then $U{\subseteq}Z$ or $a{\in}Z$. (iii) If $d([u,v])={\pm}[u,v]_{{\sigma},{\tau}}$ for all $u{\in}U$, then $U{\subseteq}Z$.

• Korean Journal of Materials Research
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• v.23 no.2
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• pp.116-122
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• 2013

For fabricating silicon solar cells with high conversion efficiency, texturing is one of the most effective techniques to increase short circuit current by enhancing light trapping. In this study, four different types of textures, large V-groove, large U-groove, small V-groove, and small U-groove, were prepared by a wet etching process. Silicon substrates with V-grooves were fabricated by an anisotropic etching process using a KOH solution mixed with isopropyl alcohol (IPA), and the size of the V-grooves was controlled by varying the concentration of IPA. The isotropic etching process following anisotropic etching resulted in U-grooves and the isotropic etching time was determined to obtain U-grooves with an opening angle of approximately $60^{\circ}$. The results indicated that U-grooves had a larger diffuse reflectance than V-grooves and the reflectances of small grooves was slightly higher than those of large grooves depending on the size of the grooves. Then amorphous Si:H thin film solar cells were fabricated on textured substrates to investigate the light trapping effect of textures with different shapes and sizes. Among the textures fabricated in this work, the solar cells on the substrate with small U-grooves had the largest short circuit current, 19.20 mA/$cm^2$. External quantum efficiency data also demonstrated that the small, U-shape textures are more effective for light trapping than large, V-shape textures.

• Korean Journal of Mathematics
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• v.27 no.4
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• pp.831-841
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• 2019

A Radiocoloring (RC) of a graph G is a function f from the vertex set V (G) to the set of all non-negative integers (labels) such that |f(u) - f(v)| ≥ 2 if d(u, v) = 1 and |f(u) - f(v)| ≥ 1 if d(u, v) = 2. The number of discrete labels and the range of labels used are called order and span, respectively. In this paper, we concentrate on the minimum order span Radiocoloring Problem (RCP) of trees. The optimization version of the minimum order span RCP that tries to find, from all minimum order assignments, one that uses the minimum span. We provide attainable lower and upper bounds for trees. Moreover, a complete characterization of caterpillars (as a subclass of trees) with the minimum order span is given.

• Journal of applied mathematics & informatics
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• v.41 no.3
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• pp.469-485
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• 2023

In this paper, we investigate the Hyers-Ulam-Rassias stability for a Drygas functional equation g(u + v) + g(u - v) = 2g(u) + g(v) + g(-v) in the setting of quasi-Banach space using fixed point approach. Also, we give general results on hyperstability of a Drygas functional equation. The results obtain in this paper extend various previously known results in the setting of quasi-Banach space. Some examples are also illustrated.

• Journal of the Chungcheong Mathematical Society
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• v.9 no.1
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• pp.77-87
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• 1996

For any map $v:X{\rightarrow}Y$, the generalized Gottlieb set $G({\Sigma}A;X,v,Y)$ with respect to v is a subgroup of $[{\Sigma}A,Y]$. If $v:X{\rightarrow}Y$ has a left homotopy inverse $u:X{\rightarrow}Y$, then for any $f{\in}G({\Sigma}A;X,v,Y)$, $g{\in}G({\Sigma}A;X,u,Y)$, the function spaces $L({\Sigma}A,X;uf)$ and $L({\Sigma}A,X;g)$ have the same homotopy type.