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EN
Objective: Approximately 50% patients with a deep bite possess anatomically extruded mandibular canines. The objective of this study was to specify the required toe (θ) of the vertical segment of a cantilever from the distal aspect to achieve pure intrusion of a mandibular canine with a segmented arch in lingual orthodontics. Additionally, the optimum magnitude of the required intrusive force by a cantilever was determined assuming non-linear, hyper-elastic behaviour of periodontal ligament (PDL). Methods: The geometrical model of a mandibular canine tooth was developed and the mathematical equation was devised to evaluate θ (positive value: toe-in, negative value: toe-out) based on certain input parameters. To verify this numerical study by finite element analysis (FEA), total eight different positions of point of force application (Pf) on bracket top (occlusal) surface were considered based on different values of input parameters. Results: The results were displayed in terms of nature of tooth movement and Von-Mises (equivalent) stresses generated in the PDL. Additionally, the optimum magnitude of the required intrusive force within the biological limit of a mandibular canine was determined from FEA considering the strength of PDL and factor of safety. Conclusions: The numerical study was developed to compute the value of required toe angle (θ) of the vertical segment of a cantilever for different morphologies of a mandibular canine as well as different positions of Pf. From FEA, the optimum range of an intrusive force within the biological limit of a mandibular canine was found to be 20–30 g.
EN
In this paper, we study the convergence of the sequence of Ishikawa iteration of rank-r to common fixed points of a finite family of asymptotically quasi-nonexpansive mappings in uniformly convex Banach spaces. Our results extend and improve some known recent results.
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