Terms Of Meijer-G Function Research Articles

New results for Shannon capacity over generalized multipath fading channels with MRC diversity

In this article, we investigate the Shannon capacity for L-branch maximal combining ratio (MRC) over generalized multipath fading channel. We derive closed-form expressions of the maximal spectral efficiency over Rayleigh, Rician, Nakagami-m, and Weibull multipath fading channel under flat fading conditions. The results are expressed in terms of Meijer G-functions, which can be evaluated numerically using mathematical tools such as Mathematica and Maple. We show, in particular, that the more the number L increases, the larger the Shannon capacity is. We deduce that four branches are sufficient in several cases to mitigate the fading effect and the channel model will approaches the one of AWGN.

Exact expression of the density of the sample generalized variance and applications

We give the exact expression of the density of the determinant of a sample covariance matrix obtained from a multivariate normal distribution, in terms of Meijer G-function. Applications include the full computation of this density, the exact confidence interval of the determinant of the population covariance matrix, as well as the expressions of the densities of the product and ratio of independent sample generalized variances.

Photon-added Barut$ndash$Girardello coherent states of the pseudoharmonic oscillator

In a previous paper (Popov D 2001 J. Phys. A: Math. Gen. 34 5283) we have constructed the coherent states (CSs) of the pseudoharmonic oscillator (PHO). These states are of the Barut–Girardello type. In the present paper, we have constructed and investigated some properties of the photon-added Barut–Girardello coherent states (PA-BGCSs) of the PHO. These states are not orthogonal but are normalized and satisfy a unity resolution relation (the completeness relation). We have found the analytical form for the positive weight function in the resolution of unity. By using these states, we have calculated some expectation values (the first two powers of the number operator N), which have evinced some non-classical properties. In order to examine the statistical properties of a PHO gas, which obeys the quantum canonical distribution, the diagonal P-representation of the density operator ρ(m) is constructed and the thermal expectation values are calculated. All these quantities are expressed in terms of Meijer G-functions, and so the PA-BGCSs are a new field of application for these functions. Also, the time dependence of the PA-BGCSs is examined. If in the obtained results we put m = 0, we recover all the results from our previous paper.

Note on an Exact Solution for Magnetoatmospheric Waves

Solutions for magnetoatmospheric waves in an isothermal plane stratified atmosphere with uniform vertical magnetic field have long been known in terms of Meijer G-functions. It is pointed out that they may alternatively be expressed using the more familiar hypergeometric 2F3 functions, with significant advantages for ease of use and physical interpretation. The nature of these solutions in different regions of the frequency-wavenumber plane is fully discussed, with particular reference to reflection, transmission, and mode conversion. Reflection, transmission, and mode conversion coefficients for slow and fast waves incident from below, including the effects of tunnelling, are calculated exactly. The exact solutions are useful in interpreting observational results and numerical simulations of more complex magnetoatmospheric waves.

Moments and distributions of certain multivariate test criteria in the canonical correlation case under violation

The joint probability density function (pdf) and the moments of order h of the general multivariate statistic defined by Y (p) (a,b) = П p i = 1 ℓ a i(1 - ℓ i) b , where a and b are positive real numbers, are obtained in this paper in the canonical correlation case and expressed in terms of the H-function. Alternative forms of some test criteria are also furnished in terms of Meijer G-function. Explicit forms of the pdf and the cdf are also supplemented in the case p = 2. Here ℓ 1, ℓ 2, …, ℓ p are the latent roots of the matrix L = λR(1 + λR) −1, R = S 1 S −1 2, and λ > 0. The covariance matrices S 1 and S 2 are independently distributed and have, respectively, the noncentral Wishart distribution W( p, n 1, Σ 1, Щ) , Щ = 1 2 MM′Σ −1 1 and the central Wishart distribution W( p, n 2, Σ 2, 0). The derivation of this pdf is based on Pillai's density function of S 1 S −1 2 under violation. It is necessary to point out that violation means the assumption of normality is violated and the common covariance matrix is disturbed. Furthermore, the moments of order h and the pdf of some test criteria such as, Wilks, Wilks-Lawely and the modified likelihood ratio may be deduced from the results associated with the general statistic as special cases. The results of this paper may be used further to investigate the exact robustness against nonnormality of the test of independence.

Electrical double-layer interaction energy for two cylinders

Equations are derived for the free energy of interaction between electrical double layers about two parallel cylinders of different radii. Integration of the Bessel functions which describe the potential distribution about the cylinders is facilitated by writing them in terms of Meijer G-functions. Finally the equations for free energy are obtained as truncated expansions of the integrated G-functions. A comparison is made with the results of Brenner and McQuarrie ( J. Colloid Interface Sci. 44, 298 (1973)).

Glauber theory of atomic helium ionization by electron impact

Glauber scattering amplitude is expressed in the form of a one-dimensional integral in terms of Meijer G-functions. This may further be expressed in a closed form by using the series representation for Meijer function.

E2 ¯ − Parametrization of SL(2, C)

We consider the E2¯−parametrization of unimodular 2 × 2 matrices A ∈ SL(2, C), which is of the form A=E1e−12aσ3VE2, with V=e12iπσ2 and E=(e−12iφ0ze12iφ)∈E2¯. E2¯ is a covering of the group of Euclidean motions in the plane. We compute the correspondingly factorized matrix elements of the unitary representations of SL(2, C) in an E2¯−basis the result is given in Eq. (6). As a fringe benefit we obtain an integral transform which amounts to expansion in terms of Meijer G-functions and which generalizes the familiar Hankel transform. The results of this paper are useful, e.g., for computing vertex functions in the theory of massless particles with continuous spin.