Monotone thematic factorizations of matrix functions

We continue the study of the so-called thematic factorizations of admissible very badly approximable matrix functions. These factorizations were introduced by V.V. Peller and N.J. Young for studying superoptimal approximation by bounded analytic matrix functions. Even though thematic indices associated with a thematic factorization of an admissible very badly approximable matrix function are not uniquely determined by the function itself, R.B. Alexeev and V.V. Peller showed that the thematic indices of any monotone non-increasing thematic factorization of an admissible very badly approximable matrix function are uniquely determined. In this paper, we prove the existence of monotone non-decreasing thematic factorizations for admissible very badly approximable matrix functions. It is also shown that the thematic indices appearing in a monotone non-decreasing thematic factorization are not uniquely determined by the matrix function itself. Furthermore, we show that the monotone non-increasing thematic factorization gives rise to a great number of other thematic factorizations.Comment: To appear in Journal of Approximation Theor

On the sum of superoptimal singular values

We discuss the following extremal problem and its relevance to the sum of the so-called superoptimal singular values of a matrix function: Given an $m\times n$ matrix function $\Phi$ on the unit circle $\mathbb{T}$, when is there a matrix function $\Psi_{*}$ in the set $A_{k}^{n,m}$ such that \int_{\mathbb{T}}{\rm trace}(\Phi(\zeta)\Psi_{*}(\zeta))dm(\zeta)=\sup_{\Psi\in A_{k}^{n,m}}|\int_{\mathbb{T}}{\rm trace}(\Phi(\zeta)\Psi(\zeta))dm(\zeta)|? The set $A_{k}^{n,m}$ is defined by A_{k}^{n,m}={\Psi\in H_{0}^{1}: \|\Psi\|_{L^{1}}\leq 1, {\rm rank}\Psi(\zeta)\leq k{a.e.}\zeta\in T}. We introduce Hankel-type operators on spaces of matrix functions and prove that this problem has a solution if and only if the corresponding Hankel-type operator has a maximizing vector. We also characterize the smallest number $k$ for which \int_{\mathbb{T}}{\rm trace}(\Phi(\zeta)\Psi(\zeta))dm(\zeta) equals the sum of all the superoptimal singular values of an admissible matrix function $\Phi$ for some $\Psi\in A_{k}^{n,m}$. Moreover, we provide a representation of any such function $\Psi$ when $\Phi$ is an admissible very badly approximable unitary-valued $n\times n$ matrix function.Comment: 24 page

Faith in the Algorithm, Part 1: Beyond the Turing Test

Since the Turing test was first proposed by Alan Turing in 1950, the primary goal of artificial intelligence has been predicated on the ability for computers to imitate human behavior. However, the majority of uses for the computer can be said to fall outside the domain of human abilities and it is exactly outside of this domain where computers have demonstrated their greatest contribution to intelligence. Another goal for artificial intelligence is one that is not predicated on human mimicry, but instead, on human amplification. This article surveys various systems that contribute to the advancement of human and social intelligence

Comment on ``Theorem for nonrotating singularity-free universes''

We show that Raychaudhuri's recently proposed theorem on nonrotating universes cannot be used to rule out realistic singularity-free descriptions of the universe, as suggested by him in PRL 80, 654 (1998).Comment: 1 page, to appear in Phys.Rev.Let