• Communications of the Korean Mathematical Society
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• v.33 no.4
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• pp.1141-1158
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• 2018

Hahn difference operator $D_{q,{\omega}}$ which is defined by $$D_{q,{\omega}}g(t)=\{{\frac{g(gt+{\omega})-g(t)}{t(g-1)+{\omega}}},{\hfill{20}}\text{if }t{\neq}{\theta}:={\frac{\omega}{1-q}},\\g^{\prime}({\theta}),{\hfill{83}}\text{if }t={\theta}$$ received a lot of interest from many researchers due to its applications in constructing families of orthogonal polynomials and in some approximation problems. In this paper, we investigate sufficient conditions for stability of the abstract linear Hahn difference equations of the form $$D_{q,{\omega}}x(t)=A(t)x(t)+f(t),\;t{\in}I$$, and $$D^2{q,{\omega}}x(t)+A(t)D_{q,{\omega}}x(t)+R(t)x(t)=f(t),\;t{\in}I$$, where $A,R:I{\rightarrow}{\mathbb{X}}$, and $f:I{\rightarrow}{\mathbb{X}}$. Here ${\mathbb{X}}$ is a Banach algebra with a unit element e and I is an interval of ${\mathbb{R}}$ containing ${\theta}$.

• Journal of the korean academy of Pediatric Dentistry
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• v.28 no.3
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• pp.504-511
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• 2001

Preventive resin restoration, extended concept of occlusal pit and fissure sealing, is procedure composed of as follows. Cavity preparation is limited to areas of initial caries The cavity is then restored with composite resin, while other sound pits and fissures are sealed with pit and fissure sealant. If pit and assure sealant with which microrestoration is possible is used, it may be of great benefit to both patient and operator in case of difficult-to-control children s treatment. However study on preventive resin restoration using this kind of materials have been scarce. The purpose of this study was to compare the microleakage of four different modes of preventive resin restoration. Restoration using only composite resin was compared together Fifty-five bicuspids were prepared with small class I cavity preparation on the occlusal surface, divided into the following groups and restored accordingly. Group 1 : Cavity was restored with Z-100 composite resin Group 2 : Cavity was restored with Z-100 composite resin. Pits and fissures were then sealed with Teethmate F-1 Group 3 : Cavity was restored with Z-100 composite resin Pits and fissures were then sealed with Ultraseal XT-plus Group 4 : Cavity and pits and fissures were restored with Ultraseal XT-plus altogrether Group 5 : Cavity was restored with Ultraseal XT-plus. Pits and fissures were then sealed with the same material. After restoration, the samples were thermocycled 500 times between $5^{\circ}C$ and $55^{\circ}C$ with a dwell time of 30 seconds. After thermocycling, the samples were dipped into 1% methylene blue solution for 24 hours, then rinsed with tap water. The teeth were then embedded in resin and cut buccolingually along the tooth axis and observed with a stereomicroscope to determine the degree of microleakage. The results were as follows : 1. Group 4 showed the greatest microleakage, while group 3, showed the least. The mean microleakage decreased in the following order:4>1>5>2>3. 2. There was no stastically significant difference between group 1 and group 5(p>0.05). However, group 1 showed significantly greater microleakage compared to group 2 and 3(p<0.05) Group 1 showed significantly less microleakage compared to group 4(p<0.05). 3. Group 2 showed no statistically significant difference compared to group 3(p>0.05). However group 2 showed significantly less microleakage compared to group 4 and 5(p<0.05) 4. Group 3 showed significantly less microleakage compared to group 4 and 5(p<0.05). 5. Group 5 showed significantly less microleakage than group 4(p<0.05).