We give an explicit construction for the Stone-Cech compactification of a regular a-frame. This enables us to extend the known theory of dimension for regular <7-frames. BackgroundA. frame or locale is a complete lattice L satisfying JC A \fx a = \Jx A x a for x and x a elements of L and a from any index set. Frames are the natural lattice-theoretic generalizations of toplogies, with archetype CIX, the lattice of open sets of a topological space X. A comprehensive reference is the book of Johnstone [10]. A a-frame M is a bounded lattice admitting countable joins and satisfying JC A \Jx n = \/x A x n for x and x n elements of M, and n e co. The less restrictive notion of a a-frame is more than a formal generalization of the concept of a frame. For a-frames which are not frames arise quite naturally; for example coz X, the lattice of all cozero-sets of a topological space X, and Boolean c-algebras. The category Tych are dually adjoint. Here TIL consists of all the a-prime filters of L with topology generated by {T\ a :aeL}, where Il a = {PeTLL:aeP}. A Tychonoff space X is realcompact if and only if X = TIL for some regular

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An estimated 34 million people are living with human immunodeficiency virus (HIV) worldwide. A polymeric drug delivery device for women has been developed to help reduce male to female vaginal transmission of HIV‐1. The device is an intravaginal slow‐release ring containing the antiretroviral drug dapivirine as the active pharmaceutical ingredient (API). In all pharmaceutical delivery systems, API content is a critical control point. Conventional quantification of the API in vaginal rings is a stepwise, destructive and time‐consuming process involving solvent extraction and high pressure liquid chromatography. This study investigates the potential of Raman spectroscopy for fast and non‐destructive quantification of dapivirine in intravaginal rings. Wide‐area illumination measurements were carried out on rotating rings using a custom built ring spinner. Customized reference rings of known concentrations were used to build calibration models, and the models were verified by rings from production using the established method as reference analysis. Bi‐ and multivariate calibration methods were applied. Models based on band ratios and partial least squares (PLS) regression models performed similarly well resulting in low model and prediction errors. Next to an alternative reference analysis for quality assurance, Raman can also be used as a production process performance analysis and optimization tool due to its non‐destructive nature and speed of analysis. Using measurements on connected spots over the entire ring circumference, the within ring variation in API was determined. ANOVA showed that there was a statistical difference in API distribution over the rings, information that can be of use in process optimization, for example. However, Tukey's honestly significant difference (HSD) test proved that no regions of the ring were out of specifications, indicating a stable production system. Copyright © 2014 John Wiley & Sons, Ltd.

We show that there is an adjoint dual equivalence between realcompact Alexandroff spaces and the Alexandroff σ-frames. This gives a corresponding adjoint duality for realcompact Tychonoff spaces. Consequently we characterize lattice theoretically the cozero-sets of a topological space.

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