32,701 research outputs found
An Atypical Survey of Typical-Case Heuristic Algorithms
Heuristic approaches often do so well that they seem to pretty much always
give the right answer. How close can heuristic algorithms get to always giving
the right answer, without inducing seismic complexity-theoretic consequences?
This article first discusses how a series of results by Berman, Buhrman,
Hartmanis, Homer, Longpr\'{e}, Ogiwara, Sch\"{o}ening, and Watanabe, from the
early 1970s through the early 1990s, explicitly or implicitly limited how well
heuristic algorithms can do on NP-hard problems. In particular, many desirable
levels of heuristic success cannot be obtained unless severe, highly unlikely
complexity class collapses occur. Second, we survey work initiated by Goldreich
and Wigderson, who showed how under plausible assumptions deterministic
heuristics for randomized computation can achieve a very high frequency of
correctness. Finally, we consider formal ways in which theory can help explain
the effectiveness of heuristics that solve NP-hard problems in practice.Comment: This article is currently scheduled to appear in the December 2012
issue of SIGACT New
Spectra of D-branes with Higgs vevs
In this paper we continue previous work on counting open string states
between D-branes by considering open strings between D-branes with nonzero
Higgs vevs, and in particular, nilpotent Higgs vevs, as arise, for example,
when studying D-branes in orbifolds. Ordinarily Higgs vevs can be interpreted
as moving the D-brane, but nilpotent Higgs vevs have zero eigenvalues, and so
their interpretation is more interesting -- for example, they often correspond
to nonreduced schemes, which furnishes an important link in understanding old
results relating classical D-brane moduli spaces in orbifolds to Hilbert
schemes, resolutions of quotient spaces, and the McKay correspondence. We give
a sheaf-theoretic description of D-branes with Higgs vevs, including nilpotent
Higgs vevs, and check that description by noting that Ext groups between the
sheaves modelling the D-branes, do in fact correctly count open string states.
In particular, our analysis expands the types of sheaves which admit on-shell
physical interpretations, which is an important step for making derived
categories useful for physics.Comment: 46 pages, LaTeX; v2: typos fixed; v3: more typos fixe
Brane Inflation, Solitons and Cosmological Solutions: I
In this paper we study various cosmological solutions for a D3/D7 system
directly from M-theory with fluxes and M2-branes. In M-theory, these solutions
exist only if we incorporate higher derivative corrections from the curvatures
as well as G-fluxes. We take these corrections into account and study a number
of toy cosmologies, including one with a novel background for the D3/D7 system
whose supergravity solution can be completely determined. This new background
preserves all the good properties of the original model and opens up avenues to
investigate cosmological effects from wrapped branes and brane-antibrane
annihilation, to name a few. We also discuss in some detail semilocal defects
with higher global symmetries, for example exceptional ones, that could occur
in a slightly different regime of our D3/D7 model. We show that the D3/D7
system does have the required ingredients to realise these configurations as
non-topological solitons of the theory. These constructions also allow us to
give a physical meaning to the existence of certain underlying homogeneous
quaternionic Kahler manifolds.Comment: Harvmac, 115 pages, 9 .eps figures; v2: typos corrected, references
added and the last section expanded; v3: Few minor typos corrected and
references added. Final version to appear in JHE
Optimal coding and the origins of Zipfian laws
The problem of compression in standard information theory consists of
assigning codes as short as possible to numbers. Here we consider the problem
of optimal coding -- under an arbitrary coding scheme -- and show that it
predicts Zipf's law of abbreviation, namely a tendency in natural languages for
more frequent words to be shorter. We apply this result to investigate optimal
coding also under so-called non-singular coding, a scheme where unique
segmentation is not warranted but codes stand for a distinct number. Optimal
non-singular coding predicts that the length of a word should grow
approximately as the logarithm of its frequency rank, which is again consistent
with Zipf's law of abbreviation. Optimal non-singular coding in combination
with the maximum entropy principle also predicts Zipf's rank-frequency
distribution. Furthermore, our findings on optimal non-singular coding
challenge common beliefs about random typing. It turns out that random typing
is in fact an optimal coding process, in stark contrast with the common
assumption that it is detached from cost cutting considerations. Finally, we
discuss the implications of optimal coding for the construction of a compact
theory of Zipfian laws and other linguistic laws.Comment: in press in the Journal of Quantitative Linguistics; definition of
concordant pair corrected, proofs polished, references update
String theory and the Kauffman polynomial
We propose a new, precise integrality conjecture for the colored Kauffman
polynomial of knots and links inspired by large N dualities and the structure
of topological string theory on orientifolds. According to this conjecture, the
natural knot invariant in an unoriented theory involves both the colored
Kauffman polynomial and the colored HOMFLY polynomial for composite
representations, i.e. it involves the full HOMFLY skein of the annulus. The
conjecture sheds new light on the relationship between the Kauffman and the
HOMFLY polynomials, and it implies for example Rudolph's theorem. We provide
various non-trivial tests of the conjecture and we sketch the string theory
arguments that lead to it.Comment: 36 pages, many figures; references and examples added, typos
corrected, final version to appear in CM
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