4,191 research outputs found
Basic notions of universal algebra for language theory and graph grammars
AbstractThis paper reviews the basic properties of the equational and recognizable subsets of general algebras; these sets can be seen as generalizations of the context-free and regular languages, respectively. This approach, based on Universal Algebra, facilitates the development of the theory of formal languages so as to include the description of sets of finite trees, finite graphs, finite hypergraphs, tuples of words, partially commutative words (also called traces) and other similar finite objects
Scattering amplitudes in YM and GR as minimal model brackets and their recursive characterization
Attached to both Yang-Mills and General Relativity about Minkowski spacetime
are distinguished gauge independent objects known as the on-shell tree
scattering amplitudes. We reinterpret and rigorously construct them as
minimal model brackets. This is based on formulating YM and GR as
differential graded Lie algebras. Their minimal model brackets are then given
by a sum of trivalent (cubic) Feynman tree graphs. The amplitudes are gauge
independent when all internal lines are off-shell, not merely up to
isomorphism, and we include a homological algebra proof of this fact. Using the
homological perturbation lemma, we construct homotopies (propagators) that are
optimal in bringing out the factorization of the residues of the amplitudes.
Using a variant of Hartogs extension for singular varieties, we give a rigorous
account of a recursive characterization of the amplitudes via their residues
independent of their original definition in terms of Feynman graphs (this does
neither involve so-called BCFW shifts nor conditions at infinity under such
shifts). Roughly, the amplitude with legs is the unique section of a sheaf
on a variety of complex momenta whose residues along a finite list of
irreducible codimension one subvarieties (prime divisors) factor into
amplitudes with less than legs. The sheaf is a direct sum of rank one
sheaves labeled by helicity signs. To emphasize that amplitudes are robust
objects, we give a succinct list of properties that suffice for a dgLa so as to
produce the YM and GR amplitudes respectively.Comment: 51 page
Colored operads, series on colored operads, and combinatorial generating systems
We introduce bud generating systems, which are used for combinatorial
generation. They specify sets of various kinds of combinatorial objects, called
languages. They can emulate context-free grammars, regular tree grammars, and
synchronous grammars, allowing us to work with all these generating systems in
a unified way. The theory of bud generating systems uses colored operads.
Indeed, an object is generated by a bud generating system if it satisfies a
certain equation in a colored operad. To compute the generating series of the
languages of bud generating systems, we introduce formal power series on
colored operads and several operations on these. Series on colored operads are
crucial to express the languages specified by bud generating systems and allow
us to enumerate combinatorial objects with respect to some statistics. Some
examples of bud generating systems are constructed; in particular to specify
some sorts of balanced trees and to obtain recursive formulas enumerating
these.Comment: 48 page
Decidability in the logic of subsequences and supersequences
We consider first-order logics of sequences ordered by the subsequence
ordering, aka sequence embedding. We show that the \Sigma_2 theory is
undecidable, answering a question left open by Kuske. Regarding fragments with
a bounded number of variables, we show that the FO2 theory is decidable while
the FO3 theory is undecidable
Double and Triple Givental's J-functions for Stable Quotients Invariants
We use mirror formulas for the stable quotients analogue of Givental's
J-function for twisted projective invariants obtained in a previous paper to
obtain mirror formulas for the analogues of the double and triple Givental's
J-functions (with descendants at all marked points) in this setting. We then
observe that the genus 0 stable quotients invariants need not satisfy the
divisor, string, or dilaton relations of the Gromov-Witten theory, but they do
possess the integrality properties of the genus 0 three-point Gromov-Witten
invariants of Calabi-Yau manifolds. We also relate the stable quotients
invariants to the BPS counts arising in Gromov-Witten theory and obtain mirror
formulas for certain twisted Hurwitz numbers.Comment: 60 pages, 9 figures, 1 table. arXiv admin note: text overlap with
arXiv:1201.635
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