4,391 research outputs found
On the Ambartzumian-Pleijel identity in hyperbolic geometry
We describe a hyperbolic version of the Ambartzumian-Pleijel identity. We use
this identity to prove the hyperbolic Crofton formula and the hyperbolic
isoperimetric inequality. This identity also provides a way to compute the
chord length distribution for an ideal polygon in the hyperbolic plane. The
analogous results for a maximally symmetric, simply connected, -dimensional
Riemannian manifold with constant sectional curvature are given at the end.Comment: 21 pages, 11 figure
Multiple-octave spanning mid-IR supercontinuum generation in bulk quadratic nonlinear crystals
Bright and broadband coherent mid-IR radiation is important for exciting and
probing molecular vibrations. Using cascaded nonlinearities in conventional
quadratic nonlinear crystal like lithium niobate, self-defocusing near-IR
solitons have been demonstrated that led to very broadband supercontinuum
generation in the visible, near-IR and short-wavelength mid-IR. Here we conduct
an experiment where a mid-IR crystal pumped in the mid-IR gives multiple-octave
spanning supercontinua. The crystal is cut for noncritical interaction, so the
three-wave mixing of a single mid-IR femtosecond pump source leads to highly
phase-mismatched second-harmonic generation. This self-acting cascaded process
leads to the formation of a self-defocusing soliton at the mid-IR pump
wavelength and after the self-compression point multiple octave-spanning
supercontinua are observed (covering 1.6-m). The results were recorded
in a commercially available crystal LiInS pumped in the 3-m range,
but other mid-IR crystals can readily be used as well.Comment: submitted to APL Photonic
CLT for linear spectral statistics of normalized sample covariance matrices with the dimension much larger than the sample size
Let
where is a matrix, consisting of independent and
identically distributed (i.i.d.) real random variables with mean zero
and variance one. When , under fourth moment conditions a central
limit theorem (CLT) for linear spectral statistics (LSS) of
defined by the eigenvalues is established. We also explore its applications in
testing whether a population covariance matrix is an identity matrix.Comment: Published at http://dx.doi.org/10.3150/14-BEJ599 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Experimental observation of long-wavelength dispersive wave generation induced by self-defocusing nonlinearity in BBO crystal
We experimentally observe long-wavelength dispersive waves generation in a
BBO crystal. A soliton was formed in normal GVD regime of the crystal by a
self-defocusing and negative nonlinearity through phase-mismatched quatradic
interaction. Strong temporal pulse compression confirmed the formation of
soliton during the pulse propagation inside the crystal. Significant dispersive
wave radiation was measured in the anomalous GVD regime of the BBO crystal.
With the pump wavelengths from 1.24 to 1.4 m, tunable dispersive waves are
generated around 1.9 to 2.2 m. The observed dispersive wave generation is
well understood by simulations.Comment: in preparatio
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