124,777 research outputs found
Disjointness properties for Cartesian products of weakly mixing systems
For we consider the class JP() of dynamical systems whose every
ergodic joining with a Cartesian product of weakly mixing automorphisms
() can be represented as the independent extension of a joining of the
system with only coordinate factors. For we show that, whenever
the maximal spectral type of a weakly mixing automorphism is singular with
respect to the convolution of any continuous measures, i.e. has the
so-called convolution singularity property of order , then belongs to
JP(). To provide examples of such automorphisms, we exploit spectral
simplicity on symmetric Fock spaces. This also allows us to show that for any
the class JP() is essentially larger than JP(). Moreover, we
show that all members of JP() are disjoint from ergodic automorphisms
generated by infinitely divisible stationary processes.Comment: 24 pages, corrected versio
Dynamical versus diffraction spectrum for structures with finite local complexity
It is well-known that the dynamical spectrum of an ergodic measure dynamical
system is related to the diffraction measure of a typical element of the
system. This situation includes ergodic subshifts from symbolic dynamics as
well as ergodic Delone dynamical systems, both via suitable embeddings. The
connection is rather well understood when the spectrum is pure point, where the
two spectral notions are essentially equivalent. In general, however, the
dynamical spectrum is richer. Here, we consider (uniquely) ergodic systems of
finite local complexity and establish the equivalence of the dynamical spectrum
with a collection of diffraction spectra of the system and certain factors.
This equivalence gives access to the dynamical spectrum via these diffraction
spectra. It is particularly useful as the diffraction spectra are often simpler
to determine and, in many cases, only very few of them need to be calculated.Comment: 27 pages; some minor revisions and improvement
Pauli graphs when the Hilbert space dimension contains a square: why the Dedekind psi function ?
We study the commutation relations within the Pauli groups built on all
decompositions of a given Hilbert space dimension , containing a square,
into its factors. Illustrative low dimensional examples are the quartit ()
and two-qubit () systems, the octit (), qubit/quartit () and three-qubit () systems, and so on. In the single qudit case,
e.g. , one defines a bijection between the maximal
commuting sets [with the sum of divisors of ] of Pauli
observables and the maximal submodules of the modular ring ,
that arrange into the projective line and a independent set
of size [with the Dedekind psi function]. In the
multiple qudit case, e.g. , the Pauli graphs rely on
symplectic polar spaces such as the generalized quadrangles GQ(2,2) (if
) and GQ(3,3) (if ). More precisely, in dimension ( a
prime) of the Hilbert space, the observables of the Pauli group (modulo the
center) are seen as the elements of the -dimensional vector space over the
field . In this space, one makes use of the commutator to define
a symplectic polar space of cardinality , that
encodes the maximal commuting sets of the Pauli group by its totally isotropic
subspaces. Building blocks of are punctured polar spaces (i.e. a
observable and all maximum cliques passing to it are removed) of size given by
the Dedekind psi function . For multiple qudit mixtures (e.g.
qubit/quartit, qubit/octit and so on), one finds multiple copies of polar
spaces, ponctured polar spaces, hypercube geometries and other intricate
structures. Such structures play a role in the science of quantum information.Comment: 18 pages, version submiited to J. Phys. A: Math. Theo
Boundary quotients and ideals of Toeplitz C*-algebras of Artin groups
We study the quotients of the Toeplitz C*-algebra of a quasi-lattice ordered
group (G,P), which we view as crossed products by a partial actions of G on
closed invariant subsets of a totally disconnected compact Hausdorff space, the
Nica spectrum of (G,P). Our original motivation and our main examples are drawn
from right-angled Artin groups, but many of our results are valid for more
general quasi-lattice ordered groups. We show that the Nica spectrum has a
unique minimal closed invariant subset, which we call the boundary spectrum,
and we define the boundary quotient to be the crossed product of the
corresponding restricted partial action. The main technical tools used are the
results of Exel, Laca, and Quigg on simplicity and ideal structure of partial
crossed products, which depend on amenability and topological freeness of the
partial action and its restriction to closed invariant subsets. When there
exists a generalised length function, or controlled map, defined on G and
taking values in an amenable group, we prove that the partial action is
amenable on arbitrary closed invariant subsets. Our main results are obtained
for right-angled Artin groups with trivial centre, that is, those with no
cyclic direct factor; they include a presentation of the boundary quotient in
terms of generators and relations that generalises Cuntz's presentation of O_n,
a proof that the boundary quotient is purely infinite and simple, and a
parametrisation of the ideals of the Toeplitz C*-algebra in terms of subsets of
the standard generators of the Artin group.Comment: 26 page
Effects of Residue Background Events in Direct Dark Matter Detection Experiments on the Determination of the WIMP Mass
In the earlier work on the development of a model-independent data analysis
method for determining the mass of Weakly Interacting Massive Particles (WIMPs)
by using measured recoil energies from direct Dark Matter detection experiments
directly, it was assumed that the analyzed data sets are background-free, i.e.,
all events are WIMP signals. In this article, as a more realistic study, we
take into account a fraction of possible residue background events, which pass
all discrimination criteria and then mix with other real WIMP-induced events in
our data sets. Our simulations show that, for the determination of the WIMP
mass, the maximal acceptable fraction of residue background events in the
analyzed data sets of O(50) total events is ~20%, for background windows of the
entire experimental possible energy ranges, or in low energy ranges; while, for
background windows in relatively higher energy ranges, this maximal acceptable
fraction of residue background events can not be larger than ~10%. For a WIMP
mass of 100 GeV with 20% background events in the windows of the entire
experimental possible energy ranges, the reconstructed WIMP mass and the
1-sigma statistical uncertainty are ~97 GeV^{+61%}_{-35%} (~94
GeV^{+55%}_{-33%} for background-free data sets).Comment: 27 pages, 22 eps figures; v2: revised version for publication,
references added and update
Equicontinuous factors, proximality and Ellis semigroup for Delone sets
We discuss the application of various concepts from the theory of topological
dynamical systems to Delone sets and tilings. We consider in particular, the
maximal equicontinuous factor of a Delone dynamical system, the proximality
relation and the enveloping semigroup of such systems.Comment: 65 page
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