8,406 research outputs found
Counting Abelian Squares for a Problem in Quantum Computing
In a recent work I developed a formula for efficiently calculating the number
of abelian squares of length over an alphabet of size , where may
be very large. Here I show how the expressiveness of a certain class of
parameterized quantum circuits can be reduced to the problem of counting
abelian squares over a large alphabet, and use the recently developed formula
to efficiently calculate this quantity
Abelian-Square-Rich Words
An abelian square is the concatenation of two words that are anagrams of one
another. A word of length can contain at most distinct
factors, and there exist words of length containing distinct
abelian-square factors, that is, distinct factors that are abelian squares.
This motivates us to study infinite words such that the number of distinct
abelian-square factors of length grows quadratically with . More
precisely, we say that an infinite word is {\it abelian-square-rich} if,
for every , every factor of of length contains, on average, a number
of distinct abelian-square factors that is quadratic in ; and {\it uniformly
abelian-square-rich} if every factor of contains a number of distinct
abelian-square factors that is proportional to the square of its length. Of
course, if a word is uniformly abelian-square-rich, then it is
abelian-square-rich, but we show that the converse is not true in general. We
prove that the Thue-Morse word is uniformly abelian-square-rich and that the
function counting the number of distinct abelian-square factors of length
of the Thue-Morse word is -regular. As for Sturmian words, we prove that a
Sturmian word of angle is uniformly abelian-square-rich
if and only if the irrational has bounded partial quotients, that is,
if and only if has bounded exponent.Comment: To appear in Theoretical Computer Science. Corrected a flaw in the
proof of Proposition
Sandpiles and Dominos
We consider the subgroup of the abelian sandpile group of the grid graph
consisting of configurations of sand that are symmetric with respect to central
vertical and horizontal axes. We show that the size of this group is (i) the
number of domino tilings of a corresponding weighted rectangular checkerboard;
(ii) a product of special values of Chebyshev polynomials; and (iii) a
double-product whose factors are sums of squares of values of trigonometric
functions. We provide a new derivation of the formula due to Kasteleyn and to
Temperley and Fisher for counting the number of domino tilings of a 2m x 2n
rectangular checkerboard and a new way of counting the number of domino tilings
of a 2m x 2n checkerboard on a M\"obius strip.Comment: 35 pages, 24 figure
Counting generalized Jenkins-Strebel differentials
We study the combinatorial geometry of "lattice" Jenkins--Strebel
differentials with simple zeroes and simple poles on and of the
corresponding counting functions. Developing the results of M. Kontsevich we
evaluate the leading term of the symmetric polynomial counting the number of
such "lattice" Jenkins-Strebel differentials having all zeroes on a single
singular layer. This allows us to express the number of general "lattice"
Jenkins-Strebel differentials as an appropriate weighted sum over decorated
trees.
The problem of counting Jenkins-Strebel differentials is equivalent to the
problem of counting pillowcase covers, which serve as integer points in
appropriate local coordinates on strata of moduli spaces of meromorphic
quadratic differentials. This allows us to relate our counting problem to
calculations of volumes of these strata . A very explicit expression for the
volume of any stratum of meromorphic quadratic differentials recently obtained
by the authors leads to an interesting combinatorial identity for our sums over
trees.Comment: to appear in Geometriae Dedicata. arXiv admin note: text overlap with
arXiv:1212.166
Edge Mode Combinations in the Entanglement Spectra of Non-Abelian Fractional Quantum Hall States on the Torus
We present a detailed analysis of bi-partite entanglement in the non-Abelian
Moore-Read fractional quantum Hall state of bosons and fermions on the torus.
In particular, we show that the entanglement spectra can be decomposed into
intricate combinations of different sectors of the conformal field theory
describing the edge physics, and that the edge level counting and tower
structure can be microscopically understood by considering the vicinity of the
thin-torus limit. We also find that the boundary entropy density of the
Moore-Read state is markedly higher than in the Laughlin states investigated so
far. Despite the torus geometry being somewhat more involved than in the sphere
geometry, our analysis and insights may prove useful when adopting entanglement
probes to other systems that are more easily studied with periodic boundary
conditions, such as fractional Chern insulators and lattice problems in
general.Comment: 13 pages, 8 figures, published version on PR
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
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