90 research outputs found
Generalized Measure of Entropy, Mathai's Distributional Pathway Model, and Tsallis Statistics
The pathway model of Mathai (2005) mainly deals with the rectangular
matrix-variate case. In this paper the scalar version is shown to be associated
with a large number of probability models used in physics. Different families
of densities are listed here, which are all connected through the pathway
parameter 'alpha', generating a distributional pathway. The idea is to switch
from one functional form to another through this parameter and it is shown that
basically one can proceed from the generalized type-1 beta family to
generalized type-2 beta family to generalized gamma family when the real
variable is positive and a wider set of families when the variable can take
negative values also. For simplicity, only the real scalar case is discussed
here but corresponding families are available when the variable is in the
complex domain. A large number of densities used in physics are shown to be
special cases of or associated with the pathway model. It is also shown that
the pathway model is available by maximizing a generalized measure of entropy,
leading to an entropic pathway. Particular cases of the pathway model are shown
to cover Tsallis statistics (Tsallis, 1988) and the superstatistics introduced
by Beck and Cohen (2003).Comment: LaTeX, 13 pages, title changed, introduction, conclusions, and
references update
A certain class of Laplace transforms with applications to reaction and reaction-diffusion equations
A class of Laplace transforms is examined to show that particular cases of
this class are associated with production-destruction and reaction-diffusion
problems in physics, study of differences of independently distributed random
variables and the concept of Laplacianness in statistics, alpha-Laplace and
Mittag-Leffler stochastic processes, the concepts of infinite divisibility and
geometric infinite divisibility problems in probability theory and certain
fractional integrals and fractional derivatives. A number of applications are
pointed out with special reference to solutions of fractional reaction and
reaction-diffusion equations and their generalizations.Comment: LaTeX, 12 pages, corrected typo
Pathway Model, Superstatistics, Tsallis Statistics, and a Generalized Measure of Entropy
The pathway model of Mathai (2005) is shown to be inferable from the
maximization of a certain generalized entropy measure. This entropy is a
variant of the generalized entropy of order 'alpha', considered in Mathai and
Rathie (1975), and it is also associated with Shannon, Boltzmann-Gibbs, Renyi,
Tsallis, and Havrda-Charvat entropies. The generalized entropy measure
introduced here is also shown to haveinteresting statistical properties and it
can be given probabilistic interpretations in terms of inaccuracy measure,
expected value, and information content in a scheme. Particular cases of the
pathway model are shown to be Tsallis statistics (Tsallis, 1988) and
superstatistics introduced by Beck and Cohen (2003). The pathway model's
connection to fractional calculus is illustrated by considering a fractional
reaction equation.Comment: LaTeX, 22 page
Random volumes under a general matrix-variate model
AbstractThe convex hull generated by p linearly independent points in Euclidean n-space, n⩾p will almost surely determine a p-simplex and the corresponding p-parallelotope. The volume of this p-parallelotope is v=|XX′|12 where the rows of the p×n,n⩾p matrix of rank p represent the p linearly independent points. If the points are random points in some sense then v becomes a random volume. The distribution of this random volume v when the matrix X has a very general real rectangular matrix-variate density is the topic of this paper. The complicated classical procedures based on integral geometry techniques for dealing with such problems are replaced by a simpler procedure based on Jacobians of matrix transformations and functions of matrix argument. Apart from the distribution of v under this general model, arbitrary moments of v, connection to the likelihood ratio statistic or λ-criterion for testing hypotheses on the parameters of multivariate normal distributions, connections to Mellin–Barnes integrals and Meijer’s G-function, connection to the concept of generalized variance, various structural decompositions of v and special cases are also examined here
Enumeration of almost cubic maps
AbstractThis paper deals with the enumeration of rooted planar maps in which the root vertex is of arbitrary valence and all other vertices are trivalent. A formula, in explicit form, is given and closed form expressions are given for several cases of interest. Some interesting summation formulae are also obtained towards the end of the paper
On Generalized Fractional Kinetic Equations
In a recent paper, Saxena et al. [1] developed the solutions of three
generalized fractional kinetic equations in terms of Mittag-Leffler functions.
The object of the present paper is to further derive the solution of further
generalized fractional kinetic equations. The results are obtained in a compact
form in terms of generalized Mittag-Leffler functions. Their relation to
fundamental laws of physics is briefly discussed.Comment: 10 pages, LaTe
Computational aspects of the gravitational instability problem for a multicomponent cosmological medium
The paper presents results for deriving closed-form analytic solutions of the
non-relativistic linear perturbation equations, which govern the evolution of
inhomogeneities in a homogeneous spatially flat multicomponent cosmological
model. Mathematical methods to derive computable forms of the perturbations are
outlined.Comment: 20 pages in LaTeX, McGill University preprin
Solar Structure in terms of Gauss' Hypergeometric Function
Hydrostatic equilibrium and energy conservation determine the conditions in
the gravitationally stabilized solar fusion reactor. We assume a matter density
distribution varying non- linearly through the central region of the Sun. The
analytic solutions of the differential equations of mass conservation,
hydrostatic equilibrium, and energy conservation, together with the equation of
state of the perfect gas and a nuclear energy generation rate
, are given in terms of Gauss' hypergeometric
function. This model for the structure of the Sun gives the run of density,
mass pressure, temperature, and nuclear energy generation through the central
region of the Sun. Because of the assumption of a matter density distribution,
the conditions of hydrostatic equilibrium and energy conservation are separated
from the mode of energy transport in the Sun.Comment: Invited Paper (A.M.Mathai) at the Fourth UN/ESA Workshop on Basic
Space Science, Cairo, Egypt, July 1994, 10 pages LaTeX,4 figures available on
reques
Astrophysical thermonuclear functions
As theoretical knowledge and experimental verification of nuclear cross
sections increases it becomes possible to refine analytic representations for
nuclear reaction rates. In this paper mathematical/statistical techniques for
deriving closed-form representations of thermonuclear functions are summarized
and numerical results for them are given.The purpose of the paper is also to
compare numerical results for approximate and closed-form representations of
thermonuclear functions.Comment: 17 pages in LaTeX, 8 figures available on request from
[email protected]
The United Nations Basic Space Science Initiative: The TRIPOD concept
Since 1990, the United Nations is annually holding a workshop on basic space
science for the benefit of the worldwide development of astronomy. Additional
to the scientific benefits of the workshops and the strengthening of
international cooperation, the workshops lead to the establishment of
astronomical telescope facilities through the Official Development Assistance
(ODA) of Japan. Teaching material, hands-on astrophysics material, and variable
star observing programmes had been developed for the operation of such
astronomical telescope facilities in an university environment. This approach
to astronomical telescope facility, observing programme, and teaching astronomy
has become known as the basic space science TRIPOD concept. Currently, a
similar TRIPOD concept is being developed for the International Heliophysical
Year 2007, consisting of an instrument array, data taking and analysis, and
teaching space science.Comment: 8 pages, LaTe
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