7,751 research outputs found
A priori bounds for co-dimension one isometric embeddings
We prove a priori bounds for the trace of the second fundamental form of a
isometric embedding into of a metric of non-negative
sectional curvature on , in terms of the scalar curvature, and the
diameter of . These estimates give a bound on the extrinsic geometry in
terms of intrinsic quantities. They generalize estimates originally obtained by
Weyl for the case and positive curvature, and then by P. Guan and the
first author for non-negative curvature and . Using
interior estimates of Evans and Krylov for concave fully nonlinear elliptic
partial differential equations, these bounds allow us to obtain the following
convergence theorem: For any , the set of metrics of non-negative
sectional curvature and scalar curvature bounded below by which are
isometrically embedable in Euclidean space is closed in the H\"older
space , . These results are obtained in an effort to
understand the following higher dimensional version of the Weyl embedding
problem which we propose: \emph{Suppose that is a smooth metric of
non-negative sectional curvature and positive scalar curvature on \S^nR^{n+1}(S^n,g)R^{n+1}$?
Selective Categories and Linear Canonical Relations
A construction of Wehrheim and Woodward circumvents the problem that
compositions of smooth canonical relations are not always smooth, building a
category suitable for functorial quantization. To apply their construction to
more examples, we introduce a notion of highly selective category, in which
only certain morphisms and certain pairs of these morphisms are "good". We then
apply this notion to the category of linear canonical
relations and the result of our version of the WW
construction, identifying the morphisms in the latter with pairs
consisting of a linear canonical relation and a nonnegative integer. We put a
topology on this category of indexed linear canonical relations for which
composition is continuous, unlike the composition in itself.
Subsequent papers will consider this category from the viewpoint of derived
geometry and will concern quantum counterparts
Equilibrium Configuration of Black Holes and the Inverse Scattering Method
The inverse scattering method is applied to the investigation of the
equilibrium configuration of black holes. A study of the boundary problem
corresponding to this configuration shows that any axially symmetric,
stationary solution of the Einstein equations with disconnected event horizon
must belong to the class of Belinskii-Zakharov solutions. Relationships between
the angular momenta and angular velocities of black holes are derived.Comment: LaTeX, 14 pages, no figure
Mass and angular-momentum inequalities for axi-symmetric initial data sets. II. Angular-momentum
We extend the validity of Dain's angular-momentum inequality to maximal,
asymptotically flat, initial data sets on a simply connected manifold with
several asymptotically flat ends which are invariant under a U(1) action and
which admit a twist potential.Comment: added Appendix C on uniquenes
Template-based Gravitational-Wave Echoes Search Using Bayesian Model Selection
The ringdown of the gravitational-wave signal from a merger of two black
holes has been suggested as a probe of the structure of the remnant compact
object, which may be more exotic than a black hole. It has been pointed out
that there will be a train of echoes in the late-time ringdown stage for
different types of exotic compact objects. In this paper, we present a
template-based search methodology using Bayesian statistics to search for
echoes of gravitational waves. Evidence for the presence or absence of echoes
in gravitational-wave events can be established by performing Bayesian model
selection. The Occam factor in Bayesian model selection will automatically
penalize the more complicated model that echoes are present in
gravitational-wave strain data because of its higher degree of freedom to fit
the data. We find that the search methodology was able to identify
gravitational-wave echoes with Abedi et al.'s echoes waveform model about 82.3%
of the time in simulated Gaussian noise in the Advanced LIGO and Virgo network
and about 61.1% of the time in real noise in the first observing run of
Advanced LIGO with significance. Analyses using this method are
performed on the data of Advanced LIGO's first observing run, and we find no
statistical significant evidence for the detection of gravitational-wave
echoes. In particular, we find combined evidence of the three events
in Advanced LIGO's first observing run. The analysis technique developed in
this paper is independent of the waveform model used, and can be used with
different parametrized echoes waveform models to provide more realistic
evidence of the existence of echoes from exotic compact objects.Comment: 16 pages, 6 figure
Uniqueness of radial solutions for the fractional Laplacian
We prove general uniqueness results for radial solutions of linear and
nonlinear equations involving the fractional Laplacian with for any space dimensions . By extending a monotonicity
formula found by Cabre and Sire \cite{CaSi-10}, we show that the linear
equation in has at most one radial and
bounded solution vanishing at infinity, provided that the potential is a
radial and non-decreasing. In particular, this result implies that all radial
eigenvalues of the corresponding fractional Schr\"odinger operator
are simple. Furthermore, by combining these findings on
linear equations with topological bounds for a related problem on the upper
half-space , we show uniqueness and nondegeneracy of ground
state solutions for the nonlinear equation in for arbitrary space dimensions and all
admissible exponents . This generalizes the nondegeneracy and
uniqueness result for dimension N=1 recently obtained by the first two authors
in \cite{FrLe-10} and, in particular, the uniqueness result for solitary waves
of the Benjamin--Ono equation found by Amick and Toland \cite{AmTo-91}.Comment: 38 pages; revised version; various typos corrected; proof of Lemma
8.1 corrected; discussion of case \kappa_* =1 in the proof of Theorem 2
corrected with new Lemma A.2; accepted for publication in Comm. Pure. Appl.
Mat
Multilingual Speech Recognition With A Single End-To-End Model
Training a conventional automatic speech recognition (ASR) system to support
multiple languages is challenging because the sub-word unit, lexicon and word
inventories are typically language specific. In contrast, sequence-to-sequence
models are well suited for multilingual ASR because they encapsulate an
acoustic, pronunciation and language model jointly in a single network. In this
work we present a single sequence-to-sequence ASR model trained on 9 different
Indian languages, which have very little overlap in their scripts.
Specifically, we take a union of language-specific grapheme sets and train a
grapheme-based sequence-to-sequence model jointly on data from all languages.
We find that this model, which is not explicitly given any information about
language identity, improves recognition performance by 21% relative compared to
analogous sequence-to-sequence models trained on each language individually. By
modifying the model to accept a language identifier as an additional input
feature, we further improve performance by an additional 7% relative and
eliminate confusion between different languages.Comment: Accepted in ICASSP 201
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