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Calcium distribution and function during anther development of Torenia fournieri (Linderniaceae)
Potassium antimonite was used to locate calcium in the anthers of Torenia fournieri (Linderniaceae). Abundant calcium precipitates accumulate in the microsporocyte cytoplasm. After meiosis, calcium precipitates are abundant on the microspore wall, as well as the callosic wall of each tetraspore. A large number of calcium precipitates also occur on the outer membranes of the tapetal cells, and in the intercellular spaces of the endothecium and middle layer. The quantity of calcium precipitates in the cytoplasm and nucleus increases at the early microspore stage, then gradually deceases until pollen maturation. Calcium precipitates on the pollen wall gradually increase from the early microspore stage until pollen maturation. Numerous calcium precipitates are observed around the Ubisch bodies. The relation between the distribution of calcium and mitosis, nuclear displacement, the formation of the pollen cell wall, as well as the possible functions of anther walls and Ubisch bodies in the transportation of calcium to the mature pollen are discussed
Energy Distribution in f(R) Gravity
The well-known energy problem is discussed in f(R) theory of gravity. We use
the generalized Landau-Lifshitz energy-momentum complex in the framework of
metric f(R) gravity to evaluate the energy density of plane symmetric solutions
for some general f(R) models. In particular, this quantity is found for some
popular choices of f(R) models. The constant scalar curvature condition and the
stability condition for these models are also discussed. Further, we
investigate the energy distribution of cosmic string spacetime.Comment: 15 pages, accepted for publication in Gen. Relativ. & Gra
Computation of the Generalized F Distribution
Exact expressions are given for the distribution function of the ratio of a
weighted sum of independent chi-squared variables to a single chi-square
variable, scaled appropriately. This distribution is the generalization of the
classical F distribution to mixtures of chi-squared variables. The distribution
is given in terms of the Lauricella functions. The truncation error bounds are
given in terms of hypergeometric functions. Applications to detecting joint
outliers and Hotelling's misspecified T^2 distribution are given.Comment: Latex, 15 page
A Kiefer--Wolfowitz theorem for convex densities
Kiefer and Wolfowitz [Z. Wahrsch. Verw. Gebiete 34 (1976) 73--85] showed that
if is a strictly curved concave distribution function (corresponding to a
strictly monotone density ), then the Maximum Likelihood Estimator
, which is, in fact, the least concave majorant of the empirical
distribution function , differs from the empirical distribution
function in the uniform norm by no more than a constant times almost surely. We review their result and give an updated version of
their proof. We prove a comparable theorem for the class of distribution
functions with convex decreasing densities , but with the maximum
likelihood estimator of replaced by the least squares estimator
: if are sampled from a distribution function
with strictly convex density , then the least squares estimator
of and the empirical distribution function differ in the uniform norm by no more than a constant times almost surely. The proofs rely on bounds on the interpolation error
for complete spline interpolation due to Hall [J. Approximation Theory 1 (1968)
209--218], Hall and Meyer [J. Approximation Theory 16 (1976) 105--122],
building on earlier work by Birkhoff and de Boor [J. Math. Mech. 13 (1964)
827--835]. These results, which are crucial for the developments here, are all
nicely summarized and exposited in de Boor [A Practical Guide to Splines (2001)
Springer, New York].Comment: Published at http://dx.doi.org/10.1214/074921707000000256 in the IMS
Lecture Notes Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
Counting processes with Bern\v{s}tein intertimes and random jumps
We consider here point processes , , with independent increments
and integer-valued jumps whose distribution is expressed in terms of
Bern\v{s}tein functions with L\'evy measure . We obtain the general
expression of the probability generating functions of , the
equations governing the state probabilities of , and their
corresponding explicit forms. We also give the distribution of the
first-passage times of , and the related governing equation. We
study in detail the cases of the fractional Poisson process, the relativistic
Poisson process and the Gamma Poisson process whose state probabilities have
the form of a negative binomial. The distribution of the times
of jumps with height () under the condition for all these special processes is investigated in detail
Statistics of the contact network in frictional and frictionless granular packings
Simulated granular packings with different particle friction coefficient mu
are examined. The distribution of the particle-particle and particle-wall
normal and tangential contact forces P(f) are computed and compared with
existing experimental data. Here f equivalent to F/F-bar is the contact force F
normalized by the average value F-bar. P(f) exhibits exponential-like decay at
large forces, a plateau/peak near f = 1, with additional features at forces
smaller than the average that depend on mu. Computations of the force-force
spatial distribution function and the contact point radial distribution
function indicate that correlations between forces are only weakly dependent on
friction and decay rapidly beyond approximately three particle diameters.
Distributions of the particle-particle contact angles show that the contact
network is not isotropic and only weakly dependent on friction. High
force-bearing structures, or force chains, do not play a dominant role in these
three dimensional, unloaded packings.Comment: 11 pages, 13 figures, submitted to PR
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