On the Normality of Numbers to Different Bases

We prove independence of normality to different bases We show that the set of real numbers that are normal to some base is Sigma^0_4 complete in the Borel hierarchy of subsets of real numbers. This was an open problem, initiated by Alexander Kechris, and conjectured by Ditzen 20 years ago

On zeros of Martin-L\"of random Brownian motion

We investigate the sample path properties of Martin-L\"of random Brownian motion. We show (1) that many classical results which are known to hold almost surely hold for every Martin-L\"of random Brownian path, (2) that the effective dimension of zeroes of a Martin-L\"of random Brownian path must be at least 1/2, and conversely that every real with effective dimension greater than 1/2 must be a zero of some Martin-L\"of random Brownian path, and (3) we will demonstrate a new proof that the solution to the Dirichlet problem in the plane is computable

Martin's conjecture, arithmetic equivalence, and countable Borel equivalence relations

There is a fascinating interplay and overlap between recursion theory and descriptive set theory. A particularly beautiful source of such interaction has been Martin's conjecture on Turing invariant functions. This longstanding open problem in recursion theory has connected to many problems in descriptive set theory, particularly in the theory of countable Borel equivalence relations. In this paper, we shall give an overview of some work that has been done on Martin's conjecture, and applications that it has had in descriptive set theory. We will present a long unpublished result of Slaman and Steel that arithmetic equivalence is a universal countable Borel equivalence relation. This theorem has interesting corollaries for the theory of universal countable Borel equivalence relations in general. We end with some open problems, and directions for future research.Comment: Corrected typo

Pirate Life: measuring success through the balanced score card

The current work project was brought forth in order to understand and measure the performance/success of an entrepreneurial venture, Pirate Life: Children’s Theatre in Toronto, Canada. The Balance Score Card (BSC) was selected as the appropriate tool for this analysis to simply end effectively outline the major streams of the organization and organize them in a manner that would demonstrate success and areas needing improvement – providing the management with a calibration tool for future years to come