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Convex Sobolev inequalities and spectral gap
This note is devoted to the proof of convex Sobolev (or generalized
Poincar\'{e}) inequalities which interpolate between spectral gap (or
Poincar\'{e}) inequalities and logarithmic Sobolev inequalities. We extend to
the whole family of convex Sobolev inequalities results which have recently
been obtained by Cattiaux and Carlen and Loss for logarithmic Sobolev
inequalities. Under local conditions on the density of the measure with respect
to a reference measure, we prove that spectral gap inequalities imply all
convex Sobolev inequalities with constants which are uniformly bounded in the
limit approaching the logarithmic Sobolev inequalities. We recover the case of
the logarithmic Sobolev inequalities as a special case
Stark quenching of rovibrational states of H2+ due to motion in a magnetic field
The motional electric field experienced by an H2+ ion moving in a magnetic
field induces an electric dipole, so that one-photon dipole transitions between
rovibrational states become allowed. Field induced spontaneous decay rates are
calculated for a wide range of states. For an ion stored in a high-field (B ~
10 T) Penning trap, it is shown that the lifetimes of excited rovibrational
states can be shortened by typically 1-3 orders of magnitude by placing the ion
in a large cyclotron orbit. This can greatly facilitate recently proposed [E.
G. Myers, Phys. Rev. A 98, 010101 (2018)] high-precision spectroscopic
measurements on H2+ and its antimatter counterpart for tests of CPT symmetry
Analysis of Scarp Profiles: Evaluation of Errors in Morphologic Dating
Morphologic analysis of scarp degradation can be used quantitatively to determine relative ages of different scarps formed in cohesionless materials, under the same climatic conditions. Scarps of tectonic origin as well as wavecut or rivercut terraces can be treated as topographic impulses that are attenuated by surface erosional processes. This morphological evolution can be modelled as the convolution of the initial shape with erosion (or degradation) function whose width increases with time. Such modeling applies well to scarps less than 10m high, formed in unconsolidated fanglomerates. To a good approximation, the degradation function is Gaussian with a variance measuring the degree of rounding of the initial shape. This geometric parameter can be called the degradation coefficient. A synthetic experiment shows that the degradation coefficient can be obtained by least squares fitting of profiles levelled perpendicular to the scarp. Gravitational collapse of the free face is accounted for by assuming initial scarp slopes at the angle of repose of the cohesionless materials (30°–35°). Uncertainties in the measured profiles result in an uncertainty in degradation coefficient that can be evaluated graphically. Because the degradation coefficient is sensitive to the regional slope and to three-dimensional processes (gullying, loess accumulation, stream incision, etc.), a reliable and accurate determination of degradation coefficient requires several long profiles across the same scarp. The linear diffusion model of scarp degradation is a Gaussian model in which the degradation coefficient is proportional to numerical age. In that case, absolute dating requires only determination of the propotionality constant, called the mass diffusivity constant. For Holocene scarps a few meters high, in loose alluvium under arid climatic conditions, mass diffusivity constants generally range between 1 and 6 m^2/kyr. Morphologic analysis is a reliable method to compare ages of different scarps in a given area, and it can provide approximate absolute ages of Holocene scarplike landforms
Variation of the constants in the late and early universe
Recent key observational results on the variation of fine structure constant,
the proton to electron mass ratio and the gravitational constant are reviewed.
The necessity to substantiate the dark sector of cosmology and to test gravity
on astrophysical scales is also emphasized.Comment: 19 pages, 6 figures, Invited talk at the conference "The Quest for
Cosmological Scalar Fields", Porto, 8-10 July 200
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