Real Scalar Case Research Articles

Statistical Distribution Theory and Fractional Calculus

This is an overview paper. This paper is an attempt to show that fractional calculus can be reached through statistical distribution theory. This paper brings together results on fractional integrals and fractional derivatives of the first and second kinds in the real and complex domains in the scalar, vector, and matrix-variate cases, and shows that all these results can be reached through statistical distribution theory. It is shown that the whole area of fractional integrals can be reached through distributions of products and ratios in the scalar variable case and distributions of symmetric products and symmetric ratios in the matrix-variate cases. While summarizing the materials, the real domain results are also listed side by side with the complex domain results so that a comparative study is possible. Fractional integrals and derivatives in the real domain mean that the parameters involved could be real or complex with appropriate conditions, the arbitrary function is real-valued, and the variables involved are all real. These in the complex domain mean that the parameters could be real or complex and the arbitrary function is still real-valued but the variables involved are in the complex domain. Fully complex domain means the variables as well as the arbitrary function are in the complex domain. Most of the materials on fractional integrals and fractional derivatives involving a single matrix or a number of matrices in the real or complex domain are of this author. Slight modifications of the results, compared with the published works in various papers, are there in various sections. In the paragraph on notations, the lemmas that are taken from this author’s own book on Jacobians are common with published works and hence the similarity index with this author’s works will be high. Section Matrix-Variate Joint Distributions and Fractional Integrals in Many Matrix-Variate Cases material on a statistical approach to Kiryakova’s multi-index fractional integral and its extension to the real scalar case of second kind integrals as well as extensions of first and second kind integrals to real and complex matrix-variate cases are believed to be new. Matrix differential operators are introduced in Section Fractional Derivatives and, with the help of these operators, fractional derivatives are constructed from the corresponding fractional integrals. These operators are applicable in a large variety of functions. Applicability is shown through identities created from scale transformed gamma random variables. Some concluding remarks are given and some open problems are pointed out in Section Concluding Remarks.

Multivariate and Matrix-Variate Logistic Models in the Real and Complex Domains

Several extensions of the basic scalar variable logistic density to the multivariate and matrix-variate cases, in the real and complex domains, are given where the extended forms end up in extended zeta functions. Several cases of multivariate and matrix-variate Bayesian procedures, in the real and complex domains, are also given. It is pointed out that there are a range of applications of Gaussian and Wishart-based matrix-variate distributions in the complex domain in multi-look data from radar and sonar. It is hoped that the distributions derived in this paper will be highly useful in such applications in physics, engineering, statistics and communication problems, because, in the real scalar case, a logistic model is seen to be more appropriate compared to a Gaussian model in many industrial applications. Hence, logistic-based multivariate and matrix-variate distributions, especially in the complex domain, are expected to perform better where Gaussian and Wishart-based distributions are currently used.

Effective field theory of dark matter: a global analysis

We present global fits of an effective field theory description of real, and complex scalar dark matter candidates. We simultaneously take into account all possible dimension 6 operators consisting of dark matter bilinears and gauge invariant combinations of quark and gluon fields. We derive constraints on the free model parameters for both the real (five parameters) and complex (seven) scalar dark matter models obtained by combining Planck data on the cosmic microwave background, direct detection limits from LUX, and indirect detection limits from the Fermi Large Area Telescope. We find that for real scalars indirect dark matter searches disfavour a dark matter particle mass below 100 GeV. For the complex scalar dark matter particle current data have a limited impact due to the presence of operators that lead to p-wave annihilation, and also do not contribute to the spin-independent scattering cross- section. Although current data are not informative enough to strongly constrain the theory parameter space, we demonstrate the power of our formalism to reconstruct the theoretical parameters compatible with an actual dark matter detection, by assuming that the excess of gamma rays observed by the Fermi Large Area Telescope towards the Galactic centre is entirely due to dark matter annihilations. Please note that the excess can very well be due to astrophysical sources such as millisecond pulsars. We find that scalar dark matter interacting via effective field theory operators can in principle explain the Galactic centre excess, but that such interpretation is in strong tension with the non-detection of gamma rays from dwarf galaxies in the real scalar case. In the complex scalar case there is enough freedom to relieve the tension.

Quantum corrections in Higgs inflation: the real scalar case

We present a critical discussion of quantum corrections,renormalisation, and the computation of the beta functions and theeffective potential in Higgs inflation. In contrast with claims inthe literature, we find no evidence for a disagreement between theJordan and Einstein frames, even at the quantum level. For clarity ofdiscussion we concentrate on the case of a real scalar Higgs. Wefirst review the classical calculation and then discuss the backreaction of gravity. We compute the beta functions for the Higgsquartic coupling and non-minimal coupling constant. Here, themid-field regime is non-renormalisable, but we are able to give anupper bound on the 1-loop corrections to the effective potential. Weshow that, in computing the effective potential, the Jordan andEinstein frames are compatible if all mass scales are transformedbetween the two frames. As such, it is consistent to take a constantcutoff in either the Jordan or Einstein frame, and both prescriptionsyield the same result for the effective potential. Our results areextended to the case of a complex scalar Higgs.

A Pathway Idea for Model Building.

Models, mathematical or stochastic, which move from one functional form to another through pathway parameters, so that in between stages can be captured, are examined in this article. Models which move from generalized type-1 beta family to type-2 beta family, to generalized gamma family to generalized Mittag-Leffler family to Lévy distributions are examined here. It is known that one can likely find an approximate model for the data at hand whether the data are coming from biological, physical, engineering, social sciences or other areas. Different families of functions are connected through the pathway parameters and hence one will find a suitable member from within one of the families or in between stages of two families. Graphs are provided to show the movement of the different models showing thicker tails, thinner tails, right tail cut off etc.

On generalized distributions and pathways

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).

Invariant manifolds of hypercyclic vectors for the real scalar case

We show that every hypercyclic operator on a real locally convex vector space admits a dense invariant linear manifold of hypercyclic vectors.

Non-Euclidean motions in projective matrix spaces

Among the projectivities S of the ( m−1)-dimensional left-projective matrix space P m−1 ( M n ( C)), we characterize those which map the unit ball Δ onto itself. If the matrices S corresponding to S are of the form S=s T , where T is a J -unitary matrix and s ≠ 0 is an arbitrary complex number, then S maps Δ onto itself and keeps the hyperbolic distance invariant. Conversely, if the projectivity S maps Δ onto itself, then the corresponding matrices are of the above form. For the real scalar case P m−1 ( R) these results reduce to known facts of ordinary ( m−1)-dimensional projective geometry.