12,403 research outputs found
Counting Maximal-Exponent Factors in Words
This article shows tight upper and lower bounds on the number of occurrences of maximal-exponent factors occurring in a word
Repetitions in infinite palindrome-rich words
Rich words are characterized by containing the maximum possible number of
distinct palindromes. Several characteristic properties of rich words have been
studied; yet the analysis of repetitions in rich words still involves some
interesting open problems. We address lower bounds on the repetition threshold
of infinite rich words over 2 and 3-letter alphabets, and construct a candidate
infinite rich word over the alphabet with a small critical
exponent of . This represents the first progress on an open
problem of Vesti from 2017.Comment: 12 page
Counting faces of randomly-projected polytopes when the projection radically lowers dimension
This paper develops asymptotic methods to count faces of random
high-dimensional polytopes. Beyond its intrinsic interest, our conclusions have
surprising implications - in statistics, probability, information theory, and
signal processing - with potential impacts in practical subjects like medical
imaging and digital communications. Three such implications concern: convex
hulls of Gaussian point clouds, signal recovery from random projections, and
how many gross errors can be efficiently corrected from Gaussian error
correcting codes.Comment: 56 page
On the probabilities of local behaviors in abelian field extensions
For a number field K and a finite abelian group G, we determine the
probabilities of various local completions of a random G-extension of K when
extensions are ordered by conductor. In particular, for a fixed prime p of K,
we determine the probability that p splits into r primes in a random
G-extension of K that is unramified at p. We find that these probabilities are
nicely behaved and mostly independent. This is in analogy to Chebotarev's
density theorem, which gives the probability that in a fixed extension a random
prime of K splits into r primes in the extension. We also give the asymptotics
for the number of G-extensions with bounded conductor. In fact, we give a class
of extension invariants, including conductor, for which we obtain the same
counting and probabilistic results. In contrast, we prove that that neither the
analogy with the Chebotarev probabilities nor the independence of probabilities
holds when extensions are ordered by discriminant.Comment: 28 pages, submitte
Fixed speed competition on the configuration model with infinite variance degrees: unequal speeds
We study competition of two spreading colors starting from single sources on
the configuration model with i.i.d. degrees following a power-law distribution
with exponent tau in (2,3). In this model two colors spread with a fixed but
not necessarily equal speed on the unweighted random graph. We show that if the
speeds are not equal, then the faster color paints almost all vertices, while
the slower color can paint only a random subpolynomial fraction of the
vertices. We investigate the case when the speeds are equal and typical
distances in a follow-up paper.Comment: 44 pages, 9 picture
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