By constructing Hopf costructures on closure spaces via homotopy, we give the concepts of cospace (CH-cospace) and cogroup (CH-cogroup). We then prove that retract deformation a CH-cospace are also CH-cospace. construct costructure set with help quotient operator. show space same homotopy type as CH-cogroup is itself CH-cogroup. existence covariant functor between category pointed ($\mathcal{CH...

Bone grafting to repair an alveolar cleft has long been an integral part of the treatment of persons with unilateral and bilateral clefts of the lip and alveolus. The presence of the cleft places a limitation on the orthodontist who would like to move teeth in the area of the cleft. Various grafting materials have been placed in alveolar clefts in an attempt to solve this problem. The case to b...

Let K and X be compact plane sets such that K X. Let P(K)be the uniform closure of polynomials on K. Let R(K) be the closure of rationalfunctions K with poles o K. Dene P(X;K) and R(X;K) to be the uniformalgebras of functions in C(X) whose restriction to K belongs to P(K) and R(K),respectively. Let CZ(X;K) be the Banach algebra of functions f in C(X) suchthat fjK = 0. In this paper, we show th...

Background/Aim: Since both patients and orthodontists want orthodontic treatment to be completed in a short time, various methods have been developed accelerate tooth movement. Vibration applications interrupted force are noninvasive that easily accepted by the patient. Aim of this study is researching modified vibration together how effect movement rate pain perception occurs during canine dis...

Orthodontic forces for sliding tooth movement during space closure are applied at a distance from the centre of resistance of the teeth. For this reason, the teeth will tip until contacts are established between the archwire and diagonally opposite corners of the bracket wings. They will also rotate until the wire contacts opposite corners of the ligature tie or the buccal shield with self-liga...

The first permanent molar as the first permanent tooth in the mouth and with a general insufficient oral hygiene of the children, is commonly subject to significant compromise which may arise due to caries or endodontic complication, or from developmental anomalies such as hypoplasia. Compromised teeth with questionable prognosis may result in shortand long-term clinical dilemmas. The aim of th...

The transitive closure of a reflexive, symmetric, analytic relation is an analytic equivalence relation. Does some smaller class contain the transitive closure of every reflexive, symmetric, closed relation? An essentially negative answer is provided here. Every analytic equivalence relation on an arbitrary Polish space is Borel embeddable in the transitive closure of the union of two smooth Bo...