Decay chain differential equations: Solution through matrix algebra

A matricial method to solve the decay chain differential equations system is presented. The quantity of each nuclide in the chain at a time t may be evaluated by analytical expressions obtained in a simple way using recurrence relations. This method may be applied to problems of radioactive buildup and decay and can be easily implemented computationally. (C) 2009 Elsevier B.V. All rights reserved.

Renormalizing the kinetic energy operator in elementary quantum mechanics

In this paper, we consider solutions to the three-dimensional Schrodinger equation of the form psi(r) = u(r)/r, where u(0) not equal 0. The expectation value of the kinetic energy operator for such wavefunctions diverges. We show that it is possible to introduce a potential energy with an expectation value that also diverges, exactly cancelling the kinetic energy divergence. This renormalization procedure produces a self-adjoint Hamiltonian. We solve some problems with this new Hamiltonian to illustrate its usefulness.

Class I malocclusion treatment: Influence of a missing mandibular incisor on anterior guidance

This case report describes the orthodontic treatment of a patient with a deep-overbite Angle Class I malocclusion, agenesis of a mandibular central incisor, and 2 supernumerary teeth, which caused impaction of the mandibular first premolars. The 15-year-old patient also had a convex profile, maxillary dentoalveolar protrusion, and deficiency of space for the correct alignment of teeth. Therefore, treatment consisted of fixed appliance therapy, cervical headgear, extraction of the supernumeraries and the mandibular and maxillary first premolars, and mesiodistal reduction of the maxillary incisors to solve the arch perimeter discrepancy as much as possible with interproximal stripping. This method of treatment significantly improved the patient`s facial and dental esthetics and provided a good functional occlusion, despite the absence of a mandibular incisor, which generally impairs achieving adequate incisal guidance. (Am J Orthod Dentofacial Orthop 2010;138:109-17)