Objective: To evaluate three-dimensional (3-D) soft tissue facial changes following rapid maxillary expansion (RME) and to compare these changes with an untreated control group. Materials and Methods: Patients who need RME as a part of their orthodontic treatment were randomly divided into two groups of 17 patients each. Eligibility criteria included having maxillary transverse deficiency with crossbite, and to be in the normal range according to body mass index. In the first group (mean age  =  13.4 ± 1.2 years), expansion was performed. The second group received no treatment initially and served as untreated control (mean age  =  12.8 ± 1.3 years). Skeletal and soft tissue changes were evaluated using posteroanterior cephalograms and 3-D facial images. The primary outcome of this study was to assess the soft tissue changes. The secondary outcomes were evaluation hard tissue and soft tissue relations. Randomization was done with preprepared random number tables. Blinding was applicable for outcome assessment only. MANOVA, t-test, and correlation analyses were used (P  =  .05). Results: In both groups, there was a general trend of increase for the transverse skeletal measurements, but these increases were more limited in the control group. Alar base width was greater in the treatment group (P  =  .002). Pogonion soft tissue point (P  =  .022) was located more posteriorly in the expansion group compared with the control group. Conclusions: Soft tissue changes between groups were similar, except for the alar base, which became wider in the treatment group. Weak correlations were found between the skeletal and soft tissue changes.

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Molodtsov [8] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. In this paper we apply the notion of soft sets by Molodtsov to the theory of subtraction algebras. The notion of soft WS-algebras, soft subalgebras and soft deductive systems are introduced, and their basic properties are derived.

More general form of "quasi-coincident with" relation (q) is introduced, and related properties are investigated. The notions of (∈, ∈ ∨ q δ 0 )-fuzzy subgroups, q δ 0 -level sets and ∈ ∨ q δ 0 -level sets are introduced, and related results are investigated. Relations between an (∈, ∈)-fuzzy subgroup and an (∈, ∈ ∨ q δ 0 )-fuzzy subgroup are discussed, and characterizations of (∈, ∈ ∨ q δ 0 )-fuzzy subgroups are displayed by using level sets and ∈ ∨ q δ 0 -level sets. The concepts of ∈ ∨ q δ 0 -admissible fuzzy sets, admissible (∈, ∈ ∨ q δ 0 )-fuzzy subgroups and δ-characteristic fuzzy sets are introduced. Using these notions, characterizations of admissible (∈, ∈ ∨ q δ 0 )-fuzzy subgroups and admissible subgroups are considered.2010 Mathematics Subject Classification. 20N25, 03E72. Keywords. (∈, ∈ ∨ q δ 0 )-fuzzy subgroup, q δ 0 -level set, ∈ ∨ q δ 0 -admissible fuzzy set.