For a centerless group G, we can define its automorphism tower. We define G: G = G, G = Aut (G) and for limit ordinals G = ⋃ α<δ G. Let τG be the ordinal when the sequence stabilizes. Thomas’ celebrated theorem says τG < ( 2 + and more. If we consider Thomas’ proof too set theoretical, we have here a shorter proof with little set theory. However, set theoretically we get a parallel theorem with...

BACKGROUND Accurate impression making is an essential prerequisite for achieving a passive fit between the implant and the superstructure. The aim of this in vitro study was to compare the three-dimensional accuracy of open-tray and three closed-tray impression techniques. MATERIALS AND METHODS Three acrylic resin mandibular master models with four parallel implants were used: Biohorizons (BI...

Using sie®e-tray air stripping as a means of remo®ing contaminants from surfactant solutions was studied to determine if tray air stripping can be applied successfully to treating surfactant-containing wastewaters and, if so, to de®elop a quantitati®e approach for designing tray air-stripping systems for treating surfactant-containing wastewaters. High-concentration surfactant solutions were pr...

The Hanoï Tower problem is a classic exercise in recursive programming: the solution has a simple recursive definition, and its complexity and the matching lower bound correspond to the solution of a simple recursive function (the solution is so simple that most students memorize it and regurgitate it at exams without truly understanding it). We describe how some minor change in the rules of th...

The problem of the Hanoï Tower is a classic exercise in recursive programming: the solution has a simple recursive definition, and its complexity and the matching lower bound are the solution of a simple recursive function (the solution is so easy that most students memorize it and regurgitate it at exams without truly understanding it). We describe how some very minor changes in the rules of t...

STATEMENT OF PROBLEM Impressions are made using high-viscosity putty vinyl polysiloxane impression materials in conjunction with disposable plastic stock trays. The impression materials have been shown to be dimensionally stable. However, it remains unclear if the disposable plastic stock trays are rigid enough to resist deformation thus yielding potentially unreliable results. PURPOSE This s...

I knocked on Denise’s door, and a weak voice told me to come in. When I entered, Denise was lying in bed, barely able to lift her head from the pillow. Her body was skeletal, and her arms were the bruised color of cancer and chemotherapy. Intravenous lines dangled, as saline and morphine flowed into her veins. A breakfast tray sat untouched on the bedside table. The room smelled of sweat, sickn...

The 3-way tray technique is popular as it provides master and opposing arch impressions and an interocclusal record at the same time. Excellent clinical results can be achieved with appropriate case selection, use of rigid tray and impression materials, attention to tray positioning and other details throughout the process, and clear understanding of the limitations of the technique by the oper...