Abstract Swiss Cheese Space and the Classicalisation of Swiss Cheeses

Swiss cheese sets are compact subsets of the complex plane obtained by deleting a sequence of open disks from a closed disk. Such sets have provided numerous counterexamples in the theory of uniform algebras. In this paper, we introduce a topological space whose elements are what we call "abstract Swiss cheeses". Working within this topological space, we show how to prove the existence of "classical" Swiss cheese sets (as discussed in a paper of Feinstein and Heath from 2010) with various desired properties. We first give a new proof of the Feinstein-Heath classicalisation theorem. We then consider when it is possible to "classicalise" a Swiss cheese while leaving disks which lie outside a given region unchanged. We also consider sets obtained by deleting a sequence of open disks from a closed annulus, and we obtain an analogue of the Feinstein-Heath theorem for these sets. We then discuss regularity for certain uniform algebras. We conclude with an application of these techniques to obtain a classical Swiss cheese set which has the same properties as a non-classical example of O'Farrell (1979).Comment: To appear in the Journal of Mathematical Analysis and Application

Using a cognitive prosthesis to assist foodservice managerial decision-making

The artificial intelligence community has been notably unsuccessful in producing intelligent agents that think for themselves. However, there is an obvious need for increased information processing power in real life situations. An example of this can be witnessed in the training of a foodservice manager, who is expected to solve a wide variety of complex problems on a daily basis. This article explores the possibility of creating an intelligence aid, rather than an intelligence agent, to assist novice foodservice managers in making decisions that are congruent with a subject matter expert\u27s decision schema

Regularity points and Jensen measures for R(X)

We discuss two types of `regularity point', points of continuity and R-points for Banach function algebras, which were introduced by the first author and Somerset in [16]. We show that, even for the natural uniform algebras R(X) (for compact plane sets X), these two types of regularity point can be different. We then give a new method for constructing Swiss cheese sets X such that R(X) is not regular, but such that R(X) has no non-trivial Jensen measures. The original construction appears in the first author's previous work. Our new approach to constructing such sets is more general, and allows us to obtain additional properties. In particular, we use our construction to give an example of such a Swiss cheese set X with the property that the set of points of discontinuity for R(X) has positive area

Photometric Observations of the Eta Carinae 2009.0 Spectroscopic Event

We have observed Eta Carinae over 34 nights between 4th January 2009 and 27th March 2009 covering the estimated timeframe for a predicted spectroscopic event related to a suspected binary system concealed within the homunculus nebula. A photometric minimum feature was confirmed to be periodic and comparison to a previous event indicated that the period to within our error at 2022.6 +/-1.0 d. Using the E-region standard star system, the apparent V magnitudes determined for the local comparison stars were HD303308 8.14+/-0.02, HD 93205 7.77 +/-0.03 and HD93162 8.22 +/-0.05. The latter star was found to be dimmer than previously reported.Comment: 5 pages,4 figures, 1 tabl

Swiss cheeses and their applications

Swiss cheese sets have been used in the literature as useful examples in the study of rational approximation and uniform algebras. In this paper, we give a survey of Swiss cheese constructions and related results. We describe some notable examples of Swiss cheese sets in the literature. We explain the various abstract notions of Swiss cheeses, and how they can be manipulated to obtain desirable properties. In particular, we discuss the Feinstein-Heath classicalisation theorem and related results. We conclude with the construction of a new counterexample to a conjecture of S. E. Morris, using a classical Swiss cheese set