969,114 research outputs found
The Causal Boundary of spacetimes revisited
We present a new development of the causal boundary of spacetimes, originally
introduced by Geroch, Kronheimer and Penrose. Given a strongly causal spacetime
(or, more generally, a chronological set), we reconsider the GKP ideas to
construct a family of completions with a chronology and topology extending the
original ones. Many of these completions present undesirable features, like
those appeared in previous approaches by other authors. However, we show that
all these deficiencies are due to the attachment of an ``excessively big''
boundary. In fact, a notion of ``completion with minimal boundary'' is then
introduced in our family such that, when we restrict to these minimal
completions, which always exist, all previous objections disappear. The optimal
character of our construction is illustrated by a number of satisfactory
properties and examples.Comment: 37 pages, 10 figures; Definition 6.1 slightly modified; multiple
minor changes; one figure added and another replace
Strongly minimal pseudofinite structures
We observe that the nonstandard finite cardinality of a definable set in a
strongly minimal pseudofinite structure D is a polynomial over the integers in
the nonstandard finite cardinality of D. We conclude that D is unimodular,
hence also locally modular. We also deduce a regularity lemma for graphs
definable in strongly minimal pseudofinite structures. The paper is elementary,
and the only surprising thing about it is that the results were not explicitly
noted before. In the new version we add a comment on relations to work of
Macpherson and Steinhorn, as well as on the limited nature of the examplesComment: 11 page
Continuous invertibility and stable QML estimation of the EGARCH(1,1) model
We introduce the notion of continuous invertibility on a compact set for
volatility models driven by a Stochastic Recurrence Equation (SRE). We prove
the strong consistency of the Quasi Maximum Likelihood Estimator (QMLE) when
the optimization procedure is done on a continuously invertible domain. This
approach gives for the first time the strong consistency of the QMLE used by
Nelson in \cite{nelson:1991} for the EGARCH(1,1) model under explicit but non
observable conditions. In practice, we propose to stabilize the QMLE by
constraining the optimization procedure to an empirical continuously invertible
domain. The new method, called Stable QMLE (SQMLE), is strongly consistent when
the observations follow an invertible EGARCH(1,1) model. We also give the
asymptotic normality of the SQMLE under additional minimal assumptions
Super- and Anti-Principal Modes in Multi-Mode Waveguides
We introduce a new type of states for light in multimode waveguides featuring
strongly enhanced or reduced spectral correlations. Based on the experimentally
measured multi-spectral transmission matrix of a multimode fiber, we generate a
set of states that outperform the established "principal modes" in terms of the
spectral stability of their output spatial field profiles. Inverting this
concept also allows us to create states with a minimal spectral correlation
width, whose output profiles are considerably more sensitive to a frequency
change than typical input wavefronts. The resulting "super-" and
"anti-principal" modes are made orthogonal to each other even in the presence
of mode-dependent loss. By decomposing them in the principal mode basis, we
show that the super-principal modes are formed via interference of principal
modes with closeby delay times, whereas the anti-principal modes are a
superposition of principal modes with the most different delay times available
in the fiber. Such novel states are expected to have broad applications in
fiber communication, imaging, and spectroscopy.Comment: 8 pages, 5 figures, plus supplementary materia
A slave mode expansion for obtaining ab-initio interatomic potentials
Here we propose a new approach for performing a Taylor series expansion of
the first-principles computed energy of a crystal as a function of the nuclear
displacements. We enlarge the dimensionality of the existing displacement space
and form new variables (ie. slave modes) which transform like irreducible
representations of the space group and satisfy homogeneity of free space.
Standard group theoretical techniques can then be applied to deduce the
non-zero expansion coefficients a priori. At a given order, the translation
group can be used to contract the products and eliminate terms which are not
linearly independent, resulting in a final set of slave mode products. While
the expansion coefficients can be computed in a variety of ways, we demonstrate
that finite difference is effective up to fourth order. We demonstrate the
power of the method in the strongly anharmonic system PbTe. All anharmonic
terms within an octahedron are computed up to fourth order. A proper unitary
transformation demonstrates that the vast majority of the anharmonicity can be
attributed to just two terms, indicating that a minimal model of phonon
interactions is achievable. The ability to straightforwardly generate
polynomial potentials will allow precise simulations at length and time scales
which were previously unrealizable
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