5,164 research outputs found
Mutual Dimension
We define the lower and upper mutual dimensions and
between any two points and in Euclidean space. Intuitively these are
the lower and upper densities of the algorithmic information shared by and
. We show that these quantities satisfy the main desiderata for a
satisfactory measure of mutual algorithmic information. Our main theorem, the
data processing inequality for mutual dimension, says that, if is computable and Lipschitz, then the inequalities
and hold for all and . We use this inequality and related
inequalities that we prove in like fashion to establish conditions under which
various classes of computable functions on Euclidean space preserve or
otherwise transform mutual dimensions between points.Comment: This article is 29 pages and has been submitted to ACM Transactions
on Computation Theory. A preliminary version of part of this material was
reported at the 2013 Symposium on Theoretical Aspects of Computer Science in
Kiel, German
Randomness and differentiability in higher dimensions
We present two theorems concerned with algorithmic randomness and
differentiability of functions of several variables. Firstly, we prove an
effective form of the Rademacher's Theorem: we show that computable randomness
implies differentiability of computable Lipschitz functions of several
variables. Secondly, we show that weak 2-randomness is equivalent to
differentiability of computable a.e. differentiable functions of several
variables.Comment: 19 page
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