774 research outputs found
Finite-State Dimension and Real Arithmetic
We use entropy rates and Schur concavity to prove that, for every integer k
>= 2, every nonzero rational number q, and every real number alpha, the base-k
expansions of alpha, q+alpha, and q*alpha all have the same finite-state
dimension and the same finite-state strong dimension. This extends, and gives a
new proof of, Wall's 1949 theorem stating that the sum or product of a nonzero
rational number and a Borel normal number is always Borel normal.Comment: 15 page
Climbing down Gaussian peaks
How likely is the high level of a continuous Gaussian random field on an
Euclidean space to have a "hole" of a certain dimension and depth? Questions of
this type are difficult, but in this paper we make progress on questions
shedding new light in existence of holes. How likely is the field to be above a
high level on one compact set (e.g. a sphere) and to be below a fraction of
that level on some other compact set, e.g. at the center of the corresponding
ball? How likely is the field to be below that fraction of the level {\it
anywhere} inside the ball? We work on the level of large deviations
LRM-Trees: Compressed Indices, Adaptive Sorting, and Compressed Permutations
LRM-Trees are an elegant way to partition a sequence of values into sorted
consecutive blocks, and to express the relative position of the first element
of each block within a previous block. They were used to encode ordinal trees
and to index integer arrays in order to support range minimum queries on them.
We describe how they yield many other convenient results in a variety of areas,
from data structures to algorithms: some compressed succinct indices for range
minimum queries; a new adaptive sorting algorithm; and a compressed succinct
data structure for permutations supporting direct and indirect application in
time all the shortest as the permutation is compressible.Comment: 13 pages, 1 figur
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