Last quarter, we introduced the closed diagonal condition for a prevariety to be a prevariety, and the “universally closed” condition for a variety to be complete. Earlier this quarter, we verified that in the case of complex prevarieties, these conditions are equivalent to the analytic topology being Hausdorff and compact, respectively. As we’ve discussed, in the scheme context a prevariety co...

We discuss conditions under which certain compactifications of topological spaces can be obtained by composing the ultrafilter space monad with suitable reflectors. In particular, we show that these inherit their categorical properties from monad. further observe various constructions such as prime open filter defined H. Simmons, closed compactification studied Bentley and Herrlich, well separa...

In 1998 Burago and Kleiner (independently) McMullen gave examples of separated nets in Euclidean space which are bilipschitz non-equivalent to the integer lattice. We study weaker notions than equivalence demonstrate that such also distinguish between nets. Put differently, we find occurrences particularly strong divergence from Our approach generalises takes place largely a continuous setting....

In our earlier paper [BuK] (see also McMullen’s paper [M]), we constructed examples of separated nets in the plane R2 which are not biLipschitz equivalent to the integer lattice Z2. These examples gave a negative answer to a question raised by H. Furstenberg and M. Gromov. Furstenberg asked this question in connection with Kakutani equivalence for R2-actions, [F]. Return times for a section of ...

Background The use of ultrasonic devices has been shown to be very successful for the removal of various canal obstructions.3-5 The use of ultrasonics is advantageous over conventional methods for removal of separated instruments as it is able to set separated instruments free from canals while making the sacrifice of dentin significantly less.6 However, there are several negative factors when ...