72 research outputs found
Non-splitting Abelian (4 t, 2, 4 t, 2 t) Relative Difference Sets and Hadamard Cocycles
AbstractUsing cohomology we show that in studying the existence of an abelian non-splitting (4 t, 2, 4 t, 2 t) relative difference set, D, we can assume the groups in question have a certain simple form. We obtain an explicit constructive equivalence between generalized perfect binary arrays and cocycles that define Hadamard matrices and thereby show directly that the existence of D corresponds to that of a symmetric Hadamard matrix of a certain form. This extends the well-known equivalence in the case of splitting relative difference sets
Generalized binary arrays from quasi-orthogonal cocycles
Generalized perfect binary arrays (GPBAs) were used by Jedwab to
construct perfect binary arrays. A non-trivial GPBA can exist only if its energy
is 2 or a multiple of 4. This paper introduces generalized optimal binary arrays
(GOBAs) with even energy not divisible by 4, as analogs of GPBAs. We give a
procedure to construct GOBAs based on a characterization of the arrays in terms
of 2-cocycles. As a further application, we determine negaperiodic Golay pairs
arising from generalized optimal binary sequences of small length.Junta de AndalucÃa FQM-01
On quasi-orthogonal cocycles
We introduce the notion of quasi-orthogonal cocycle. This
is motivated in part by the maximal determinant problem for square
{±1}-matrices of size congruent to 2 modulo 4. Quasi-orthogonal cocycles
are analogous to the orthogonal cocycles of algebraic design theory.
Equivalences with new and known combinatorial objects afforded by this
analogy, such as quasi-Hadamard groups, relative quasi-difference sets,
and certain partially balanced incomplete block designs, are proved.Junta de AndalucÃa FQM-01
Almost supplementary difference sets and quaternary sequences with optimal autocorrelation
We introduce almost supplementary difference sets (ASDS). For odd
m, certain ASDS in Zm that have amicable incidence matrices are equivalent to
quaternary sequences of odd length m with optimal autocorrelation. As one consequence,
if 2m − 1 is a prime power, or m 1 mod 4 is prime, then ASDS of
this kind exist. We also explore connections to optimal binary sequences and group
cohomology.Junta de AndalucÃa FQM-01
Quantum charges and spacetime topology: The emergence of new superselection sectors
In which is developed a new form of superselection sectors of topological
origin. By that it is meant a new investigation that includes several
extensions of the traditional framework of Doplicher, Haag and Roberts in local
quantum theories. At first we generalize the notion of representations of nets
of C*-algebras, then we provide a brand new view on selection criteria by
adopting one with a strong topological flavour. We prove that it is coherent
with the older point of view, hence a clue to a genuine extension. In this
light, we extend Roberts' cohomological analysis to the case where 1--cocycles
bear non trivial unitary representations of the fundamental group of the
spacetime, equivalently of its Cauchy surface in case of global hyperbolicity.
A crucial tool is a notion of group von Neumann algebras generated by the
1-cocycles evaluated on loops over fixed regions. One proves that these group
von Neumann algebras are localized at the bounded region where loops start and
end and to be factorial of finite type I. All that amounts to a new invariant,
in a topological sense, which can be defined as the dimension of the factor. We
prove that any 1-cocycle can be factorized into a part that contains only the
charge content and another where only the topological information is stored.
This second part resembles much what in literature are known as geometric
phases. Indeed, by the very geometrical origin of the 1-cocycles that we
discuss in the paper, they are essential tools in the theory of net bundles,
and the topological part is related to their holonomy content. At the end we
prove the existence of net representations
Aharonov-Bohm superselection sectors
We show that the Aharonov-Bohm effect finds a natural description in the
setting of QFT on curved spacetimes in terms of superselection sectors of local
observables. The extension of the analysis of superselection sectors from
Minkowski spacetime to an arbitrary globally hyperbolic spacetime unveils the
presence of a new quantum number labeling charged superselection sectors. In
the present paper we show that this "topological" quantum number amounts to the
presence of a background flat potential which rules the behaviour of charges
when transported along paths as in the Aharonov-Bohm effect. To confirm these
abstract results we quantize the Dirac field in presence of a background flat
potential and show that the Aharonov-Bohm phase gives an irreducible
representation of the fundamental group of the spacetime labeling the charged
sectors of the Dirac field. We also show that non-Abelian generalizations of
this effect are possible only on space-times with a non-Abelian fundamental
group
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