46,382 research outputs found
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 matrix function on the unit circle , when is there a
matrix function in the set 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 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
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 for some . Moreover, we provide a
representation of any such function when is an admissible very
badly approximable unitary-valued 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
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