13,391 research outputs found
Combinatorial structures in loops I. Elements of the decomposition theory
Difference sets have been extensively studied in groups, principally in Abelian groups. Here we extend the notion of a difference set to loops. This entails considering the class of k> systems and the special subclasses of k, [lambda]> principal block partial designs (PBPDs) and k, [lambda]> designs. By means of a certain permutation matrix decomposition of the incidence matrices of a system and its complement, we can isomorphically identify an abstract k> system with a corresponding system in a loop. Special properties of this decomposition correspond to special algebraic properties of the loop. Here we investigate the situation when some or all of the elements of the loop are right inversive. We identify certain classes of k, [lambda]> designs, including skew-Hadamard designs and finite projective planes, with designs and difference sets in right inverse property loops and prove a universal existence theorem for k, [lambda]> PBPDs and corresponding difference sets in such loops.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/33925/1/0000192.pd
Group field theories for all loop quantum gravity
Group field theories represent a 2nd quantized reformulation of the loop
quantum gravity state space and a completion of the spin foam formalism. States
of the canonical theory, in the traditional continuum setting, have support on
graphs of arbitrary valence. On the other hand, group field theories have
usually been defined in a simplicial context, thus dealing with a restricted
set of graphs. In this paper, we generalize the combinatorics of group field
theories to cover all the loop quantum gravity state space. As an explicit
example, we describe the GFT formulation of the KKL spin foam model, as well as
a particular modified version. We show that the use of tensor model tools
allows for the most effective construction. In order to clarify the
mathematical basis of our construction and of the formalisms with which we
deal, we also give an exhaustive description of the combinatorial structures
entering spin foam models and group field theories, both at the level of the
boundary states and of the quantum amplitudes.Comment: version published in New Journal of Physic
The complex of pant decompositions of a surface
We exhibit a set of edges (moves) and 2-cells (relations) making the complex
of pant decompositions on a surface a simply connected complex. Our
construction, unlike the previous ones, keeps the arguments concerning the
structural transformations independent from those deriving from the action of
the mapping class group. The moves and the relations turn out to be supported
in subsurfaces with 3g-3+n=1,2 (where g is the genus and n is the number of
boundary components), illustrating in this way the so called Grothendieck
principle.Comment: Minor changes in the introductio
New mathematical structures in renormalizable quantum field theories
Computations in renormalizable perturbative quantum field theories reveal
mathematical structures which go way beyond the formal structure which is
usually taken as underlying quantum field theory. We review these new
structures and the role they can play in future developments.Comment: 26p,4figs., Invited Contribution to Annals of Physics, minor typos
correcte
On the combinatorics of sparsification
Background: We study the sparsification of dynamic programming folding
algorithms of RNA structures. Sparsification applies to the mfe-folding of RNA
structures and can lead to a significant reduction of time complexity. Results:
We analyze the sparsification of a particular decomposition rule, ,
that splits an interval for RNA secondary and pseudoknot structures of fixed
topological genus. Essential for quantifying the sparsification is the size of
its so called candidate set. We present a combinatorial framework which allows
by means of probabilities of irreducible substructures to obtain the expected
size of the set of -candidates. We compute these expectations for
arc-based energy models via energy-filtered generating functions (GF) for RNA
secondary structures as well as RNA pseudoknot structures. For RNA secondary
structures we also consider a simplified loop-energy model. This combinatorial
analysis is then compared to the expected number of -candidates
obtained from folding mfe-structures. In case of the mfe-folding of RNA
secondary structures with a simplified loop energy model our results imply that
sparsification provides a reduction of time complexity by a constant factor of
91% (theory) versus a 96% reduction (experiment). For the "full" loop-energy
model there is a reduction of 98% (experiment).Comment: 27 pages, 12 figure
Dynkin operators and renormalization group actions in pQFT
Renormalization techniques in perturbative quantum field theory were known,
from their inception, to have a strong combinatorial content emphasized, among
others, by Zimmermann's celebrated forest formula. The present article reports
on recent advances on the subject, featuring the role played by the Dynkin
operators (actually their extension to the Hopf algebraic setting) at two
crucial levels of renormalization, namely the Bogolioubov recursion and the
renormalization group (RG) equations. For that purpose, an iterated integrals
toy model is introduced to emphasize how the operators appear naturally in the
setting of renormalization group analysis. The toy model, in spite of its
simplicity, captures many key features of recent approaches to RG equations in
pQFT, including the construction of a universal Galois group for quantum field
theories
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