Higher Kurtz randomness
(with André Nies, Frank Stephan, and Liang Yu). Annals of Pure and Applied Logic161 (2010), no. 10, 1280—1290.
I visited Liang Yu at Nanjing University, China in 2008 and 2009. The collaboration for this paper however was later done by email plus a strategy meeting in Marseille in 2009.
This paper grew out of my assignment to teach Math 472, Statistical Inference, in Spring 2009 at UH. By analyzing the proof of the law of large numbers and some other work I showed that Hippocratic and Galenic randomness coincide for Bernoulli measures. [There has been follow-up work to this paper which it would make sense to describe here.]
Preliminary version: The law of the iterated logarithm for algorithmically random paths of Brownian motion.
Logical Foundations of Computer Science, Lecture Notes in Computer Science, vol. 4514, Springer, Berlin, 2007, pp. 310—317.
In this paper we answered a question of Fouché regarding algorithmically random Brownian motion. The key idea was to use Carathéodory’s measure algebra isomorphism theorem.
Mathematics 472 (Statistical Inference), University of Hawaii at Manoa, Spring 2009.
This project concerns the distribution of the nearest walk of length \(n\) to a randomly chosen walk of length \(m>n\),
where the walks have step size \(1/n\) on the \(x\)-axis and \(1/\sqrt n\) on the \(y\)-axis.
It is well known that the distribution of such walks converges to Brownian motion.
Two of the groups in the class found that strings of the form \(1^a0^b\) and \(0^a 1^b\) were the least likely nearest walks, whereas strings of the form \(0101\ldots\) or \(1010\ldots\) were the most likely. Strings of the form \(000\ldots\) or \(111\ldots\) were the second least likely.
In algorithmic randomness we try to determine what it takes to produce randomness, and what you can do with it once you have it.
Logic and computability deal with the borders between the complete and the incomplete, the decidable and the undecidable, the computable and the non-computable. A fundamental result in the area is Gödel’s 1931 incompleteness theorem, which is a precise version of the following sentiment of Ishmail in Moby Dick.
I promise nothing complete; because any human thing supposed to be complete, must for that very reason infallibly be faulty.
Laurent Bienvenu and Paul Shafer discovered an apparent error in the proof of Lemma 4.6. There it is stated that A wtt-reduces to B; however, it seems that the reduction of A to B also requires oracle access to f. Corollary 4.7 also seems false (take f to be a ML random of hyperimmune-free degree, and A=f; then A is complex but not f-complex).
Self-embeddings of computable trees (with Stephen Binns, Manuel Lerman, Jim Schmerl, and Reed Solomon). Notre Dame Journal of Formal Logic49 (2008), no. 1, 1—37.
I was probably the least-involved of the five authors of this paper, which was written in 2005-2006 at UConn.
Professor of Mathematics, University of Hawaii at Manoa