14,494 research outputs found
Coincidence site modules in 3-space
The coincidence site lattice (CSL) problem and its generalization to
Z-modules in Euclidean 3-space is revisited, and various results and
conjectures are proved in a unified way, by using maximal orders in quaternion
algebras of class number 1 over real algebraic number fields.Comment: 25 page
Multiple planar coincidences with N-fold symmetry
Planar coincidence site lattices and modules with N-fold symmetry are well
understood in a formulation based on cyclotomic fields, in particular for the
class number one case, where they appear as certain principal ideals in the
corresponding ring of integers. We extend this approach to multiple
coincidences, which apply to triple or multiple junctions. In particular, we
give explicit results for spectral, combinatorial and asymptotic properties in
terms of Dirichlet series generating functions.Comment: 13 pages, two figures. For previous related work see math.MG/0511147
and math.CO/0301021. Minor changes and references update
Pauli graphs, Riemann hypothesis, Goldbach pairs
Let consider the Pauli group with unitary quantum
generators (shift) and (clock) acting on the vectors of the
-dimensional Hilbert space via and , with
. It has been found that the number of maximal mutually
commuting sets within is controlled by the Dedekind psi
function (with a prime)
\cite{Planat2011} and that there exists a specific inequality , involving the Euler constant , that is only satisfied at specific low dimensions . The set is closely related to
the set of integers that are totally Goldbach, i.e.
that consist of all primes ) is equivalent to Riemann hypothesis.
Introducing the Hardy-Littlewood function (with the twin prime constant),
that is used for estimating the number of
Goldbach pairs, one shows that the new inequality is also equivalent to Riemann hypothesis. In this paper,
these number theoretical properties are discusssed in the context of the qudit
commutation structure.Comment: 11 page
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