Quantum randomness and value indefiniteness

As computability implies value definiteness, certain sequences of quantum outcomes cannot be computable.Comment: 13 pages, revise

Randomness extraction and asymptotic Hamming distance

We obtain a non-implication result in the Medvedev degrees by studying sequences that are close to Martin-L\"of random in asymptotic Hamming distance. Our result is that the class of stochastically bi-immune sets is not Medvedev reducible to the class of sets having complex packing dimension 1

How powerful are integer-valued martingales?

In the theory of algorithmic randomness, one of the central notions is that of computable randomness. An infinite binary sequence X is computably random if no recursive martingale (strategy) can win an infinite amount of money by betting on the values of the bits of X. In the classical model, the martingales considered are real-valued, that is, the bets made by the martingale can be arbitrary real numbers. In this paper, we investigate a more restricted model, where only integer-valued martingales are considered, and we study the class of random sequences induced by this model.Comment: Long version of the CiE 2010 paper

Correlations in the T Cell Response to Altered Peptide Ligands

The vertebrate immune system is a wonder of modern evolution. Occasionally, however, correlations within the immune system lead to inappropriate recruitment of preexisting T cells against novel viral diseases. We present a random energy theory for the correlations in the naive and memory T cell immune responses. The non-linear susceptibility of the random energy model to structural changes captures the correlations in the immune response to mutated antigens. We show how the sequence-level diversity of the T cell repertoire drives the dynamics of the immune response against mutated viral antigens.Comment: 21 pages; 6 figures; to appear in Physica

Sub-computable Boundedness Randomness

This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for PSPACE functions. These new notions are robust in that there are equivalent formulations in terms of (1) Martin-L\"of tests, (2) Kolmogorov complexity, and (3) martingales. We show these notions can be equivalently defined with prefix-free Kolmogorov complexity. We prove that one direction of van Lambalgen's theorem holds for relative computability, but the other direction fails. We discuss statistical properties of these notions of randomness