78 research outputs found
Semiclassical Mechanics of the Wigner 6j-Symbol
The semiclassical mechanics of the Wigner 6j-symbol is examined from the
standpoint of WKB theory for multidimensional, integrable systems, to explore
the geometrical issues surrounding the Ponzano-Regge formula. The relations
among the methods of Roberts and others for deriving the Ponzano-Regge formula
are discussed, and a new approach, based on the recoupling of four angular
momenta, is presented. A generalization of the Yutsis-type of spin network is
developed for this purpose. Special attention is devoted to symplectic
reduction, the reduced phase space of the 6j-symbol (the 2-sphere of Kapovich
and Millson), and the reduction of Poisson bracket expressions for
semiclassical amplitudes. General principles for the semiclassical study of
arbitrary spin networks are laid down; some of these were used in our recent
derivation of the asymptotic formula for the Wigner 9j-symbol.Comment: 64 pages, 50 figure
Quaternionic 1-Factorizations and Complete Sets of Rainbow Spanning Trees
A 1-factorization F of a complete graph K2n
is said to be G-regular, or regular under G, if G is an automorphism group of F
acting sharply transitively on the vertex-set. The problem of determining which groups can realize such a situation dates back to a result by Hartman and Rosa (Eur J Comb 6:45–48, 1985) on cyclic groups and it is still open when n is even, although several classes of groups were tested in the recent past. It has been recently proved, see Rinaldi (Australas J Comb 80(2):178–196, 2021) and Mazzuoccolo et al. (Discret Math 342(4):1006–1016, 2019), that a G-regular 1-factorization, together with a complete set of rainbow spanning trees, exists for each group G of order 2n, n odd. The existence for each even n>2
was proved when either G is cyclic and n is not a power of 2, or when G is a dihedral group. Explicit constructions were given in all these cases. In this paper we extend this result and give explicit constructions when n>2
is even and G is either abelian but not cyclic, dicyclic, or a non cyclic 2-group with a cyclic subgroup of index 2
Global integrability of cosmological scalar fields
We investigate the Liouvillian integrability of Hamiltonian systems
describing a universe filled with a scalar field (possibly complex). The tool
used is the differential Galois group approach, as introduced by Morales-Ruiz
and Ramis. The main result is that the generic systems with minimal coupling
are non-integrable, although there still exist some values of parameters for
which integrability remains undecided; the conformally coupled systems are only
integrable in four known cases. We also draw a connection with chaos present in
such cosmological models, and the issues of integrability restricted to the
real domain.Comment: This is a conflated version of arXiv:gr-qc/0612087 and
arXiv:gr-qc/0703031 with a new theory sectio
An effective Chebotarev density theorem for families of number fields, with an application to -torsion in class groups
We prove a new effective Chebotarev density theorem for Galois extensions
that allows one to count small primes (even as small as an
arbitrarily small power of the discriminant of ); this theorem holds for the
Galois closures of "almost all" number fields that lie in an appropriate family
of field extensions. Previously, applying Chebotarev in such small ranges
required assuming the Generalized Riemann Hypothesis. The error term in this
new Chebotarev density theorem also avoids the effect of an exceptional zero of
the Dedekind zeta function of , without assuming GRH. We give many different
"appropriate families," including families of arbitrarily large degree. To do
this, we first prove a new effective Chebotarev density theorem that requires a
zero-free region of the Dedekind zeta function. Then we prove that almost all
number fields in our families yield such a zero-free region. The innovation
that allows us to achieve this is a delicate new method for controlling zeroes
of certain families of non-cuspidal -functions. This builds on, and greatly
generalizes the applicability of, work of Kowalski and Michel on the average
density of zeroes of a family of cuspidal -functions. A surprising feature
of this new method, which we expect will have independent interest, is that we
control the number of zeroes in the family of -functions by bounding the
number of certain associated fields with fixed discriminant. As an application
of the new Chebotarev density theorem, we prove the first nontrivial upper
bounds for -torsion in class groups, for all integers ,
applicable to infinite families of fields of arbitrarily large degree.Comment: 52 pages. This shorter version aligns with the published paper. Note
that portions of Section 8 of the longer v1 have been developed as a separate
paper with identifier arXiv:1902.0200
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