• Journal of applied mathematics & informatics
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• v.5 no.3
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• pp.877-888
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• 1998

We show that if dialation operator $\parallel\sigma_r\parallel\; E=\Psi(r)$ of E, then $\Lambda_\subset_{\Psi1}E\subset\;M_\Psi$, We also show that there are lattice-isomorphic copies of $l_\rho \; and \;l_\infty$ in $M \Psi$ under some condition.

• The korean journal of orthodontics
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• v.25 no.3 s.50
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• pp.235-245
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• 1995

The purpose of this paper is to review two recently reported, long-term studies of several chemical methods to control gingivitis and decalcification in adolescent orthodontic patients. The first study(gingivitis study) was designed to determine whether conventional toothbrushing and twice daily use of a brush-on 0.4 per cent $SnF_2$ gel containing more than 90 per cent available $Sn^{2+}$ would be more effective for controlling plaque accumulation and gingivitis in the presence of orthodontic appliances than conventional toothbrushing alone. The second study(decalcification study) was designed to compare the effectiveness of controlling decalcification in orthodontic patients with either a II00 ppm F tooth paste used alone, this same toothpaste and a 0.05 percent NaF rinse or this toothpaste and a 0.4 percent $SnF_2$ gel. In the gingivitis study, sixty-five consecutively treated adolescents who were to receive full-mouth fixed orthodontic appliances were assigned to two groups according to age and sex criteria. In the decalcification study an additional 30 subjects(95 total) were similarly assigned to a third group. The first group(control, n=35) used only toothbrushing with a standard fluoride(1100 ppm F) toothpaste. The second group used toothbrushing with a similar dentifrice supplemented with a 0.4 percent $SnF_2$ gel($SnF_2$ gel group, n=30) used twice daily for the entire 18-month study period. The third group(in the decalcification study only) used a similar toothpaste and 0.05 percent NaF rinse(NgF rinse group, n=30). Clinical assessments of plaque accumulation using the Plaque Index, gingival inflammation using the Gingival Index, and coronal staining were completed single-blinded before appliances were placed and 1, 3, 6, 9, 12 and 18 months after appliances were placed. Decalcification was assessed single blind on all labial surfaces of all erupted teeth before appliances were placed and 3 months after appliances were removed. The results of the gingivitis study indicated that the $SnF_2$ gel gorup had significantly lower scores for the Plaque Index(p<0.01) and Gingival Index(p<0.001) at all examinations during orthodontic treatment than did the control group. In the $SnF_2$ gel group, one subject developed mild coronal staining and two subjects developed moderate staining. In the decalcification study, when pre-treatment levels of decalcification were subtracted from post-treatment values, significantly lower decalcification scores(p<0.05) were found for both whole mouth and first molars in the NaF rinse and gel groups as compared with the control gorup(toothpaste alone). Although the gel group consistently had less decalcification than the rinse group, this difference only approached statistical significance.