A theoretical study of the modified equal scalar and vector Manning-Rosen potential within the deformed Klein-Gordon and Schrödinger in RNCQM and NRNCQM symmetries

In this research work, within the framework of relativistic and nonrelativistic noncommutative quantum mechanics, the deformed Klein–Gordon and Schrödinger equations were solved with the modified equal vector scalar Manning-Rosen potential that has been of significance interest in recent years using Bopp's shift method and standard perturbation theory in the first-order in the noncommutativity parameters in 3-dimensions noncommutative quantum mechanics. By employing the improved approximation of the centrifugal term, the relativistic and nonrelativistic bound state energies were obtained for some diatomic molecules such as (HCl, CH, LiH, CO, NO, O2, I2, N2, H2, and Ar2). The obtained energy eigenvalues appear as a function of the generalized Gamma function, the parameters of noncommutativity, and the parameters of studied potential, in addition to the atomic quantum numbers . In both relativistic and nonrelativistic problems, we show that the corrections on the spectrum energy are smaller than the main energy in the ordinary cases of RQM and NRQM. A straightforward limit of our results to ordinary quantum mechanics shows that the present result is consistent with what is obtained in the literature. We have seen that the improved approximation of the centrifugal term is better than the other approximations in finding the approximate analytical solutions of the Klein-Gordon and Schrödinger equations for the modified Manning–Rosen potential in RNCQM and NRNCQM.

The use of optical coherence tomography envelope statistics to quantify local temperature changes in tissue-mimicking phantoms

There exists a multitude of extreme thermally induced therapies for treating both benign and malignant tumors. Advancement in optics has shown prospect in clinical applications, especially monitoring oncological clinical treatments like thermal ablation. Optical coherence tomography (OCT) backscattered spectrum has demonstrated sensitivity to structural changes on the microscopic level. Envelope statistics analysis on OCT images of tissue-mimicking phantoms that are thermally modulated can provide structural information that correlates to changes in temperature. Several probability distribution functions were analyzed by looking at suitable theoretical matches to the empirical OCT data. Results indicate that the generalized gamma function was the best fit and has potential in relating the scale parameter to the size of the scatterers in the phantom. Moreover, the parameters revealed sensitivity to temperature changes, which can be further studied to understand the biological response of tissue that are exposed to extreme thermal conditions in order to improve patient care.

The use of optical coherence tomography envelope statistics to quantify local temperature changes in tissue-mimicking phantoms

There exists a multitude of extreme thermally induced therapies for treating both benign and malignant tumors. Advancement in optics has shown prospect in clinical applications, especially monitoring oncological clinical treatments like thermal ablation. Optical coherence tomography (OCT) backscattered spectrum has demonstrated sensitivity to structural changes on the microscopic level. Envelope statistics analysis on OCT images of tissue-mimicking phantoms that are thermally modulated can provide structural information that correlates to changes in temperature. Several probability distribution functions were analyzed by looking at suitable theoretical matches to the empirical OCT data. Results indicate that the generalized gamma function was the best fit and has potential in relating the scale parameter to the size of the scatterers in the phantom. Moreover, the parameters revealed sensitivity to temperature changes, which can be further studied to understand the biological response of tissue that are exposed to extreme thermal conditions in order to improve patient care.

Some inequalities involving two generalized beta functions in n variables

The beta and gamma functions have recently seen several developments and various extensions because of their nice properties and interesting applications. The contribution of this paper falls within this framework. After introducing a generalized gamma function and two generalized beta functions in several variables, we investigate some inequalities involving these generalized functions.

Characterizations of Some Probability Distributions with Completely Monotonic Density Functions

For a non-negative continuous random variable , Chaudhry and Zubair (2002, p. 19) introduced a probability distribution with a completely monotonic probability density function based on the generalized gamma function, and called it the Macdonald probability function. In this paper, we establish various basic distributional properties of Chaudhry and Zubair’s Macdonald probability distribution. Since the percentage points of a given distribution are important for any statistical applications, we have also computed the percentage points for different values of the parameter involved. Based on these properties, we establish some new characterization results of Chaudhry and Zubair’s Macdonald probability distribution by the left and right truncated moments, order statistics and record values. Characterizations of certain other continuous probability distributions with completely monotonic probability density functions such as Mckay, Pareto and exponential distributions are also discussed by the proposed characterization techniques.

The Generalized Fourier Transform: A Unified Framework for the Fourier, Laplace, Mellin and Z Transforms

<div>This paper introduces Generalized Fourier transform (GFT) that is an extension or the generalization of the Fourier transform (FT). The Unilateral Laplace transform (LT) is observed to be the special case of GFT. GFT, as proposed in this work, contributes significantly to the scholarly literature. There are many salient contribution of this work. Firstly, GFT is applicable to a much larger class of signals, some of which cannot be analyzed with FT and LT. For example, we have shown the applicability of GFT on the polynomially decaying functions and super exponentials. Secondly, we demonstrate the efficacy of GFT in solving the initial value problems (IVPs). Thirdly, the generalization presented for FT is extended for other integral transforms with examples shown for wavelet transform and cosine transform. Likewise, generalized Gamma function is also presented. One interesting application of GFT is the computation of generalized moments, for the otherwise non-finite moments, of any random variable such as the Cauchy random variable. Fourthly, we introduce Fourier scale transform (FST) that utilizes GFT with the topological isomorphism of an exponential map. Lastly, we propose Generalized Discrete-Time Fourier transform (GDTFT). The DTFT and unilateral $z$-transform are shown to be the special cases of the proposed GDTFT. The properties of GFT and GDTFT have also been discussed. </div><div><br></div>

The Generalized Fourier Transform: A Unified Framework for the Fourier, Laplace, Mellin and Z Transforms

<div>This paper introduces Generalized Fourier transform (GFT) that is an extension or the generalization of the Fourier transform (FT). The Unilateral Laplace transform (LT) is observed to be the special case of GFT. GFT, as proposed in this work, contributes significantly to the scholarly literature. There are many salient contribution of this work. Firstly, GFT is applicable to a much larger class of signals, some of which cannot be analyzed with FT and LT. For example, we have shown the applicability of GFT on the polynomially decaying functions and super exponentials. Secondly, we demonstrate the efficacy of GFT in solving the initial value problems (IVPs). Thirdly, the generalization presented for FT is extended for other integral transforms with examples shown for wavelet transform and cosine transform. Likewise, generalized Gamma function is also presented. One interesting application of GFT is the computation of generalized moments, for the otherwise non-finite moments, of any random variable such as the Cauchy random variable. Fourthly, we introduce Fourier scale transform (FST) that utilizes GFT with the topological isomorphism of an exponential map. Lastly, we propose Generalized Discrete-Time Fourier transform (GDTFT). The DTFT and unilateral $z$-transform are shown to be the special cases of the proposed GDTFT. The properties of GFT and GDTFT have also been discussed. </div><div><br></div>

A New Approach to the Approximate Analytic Solution of the Three-Dimensional Schrӧdinger Equation for Hydrogenic and Neutral Atoms in the Generalized Hellmann Potential Model

Within the framework of nonrelativistic noncommutative quantum mechanics using the improved approximation scheme to the centrifugal term for any l-states via the generalized Bopp’s shift method and standard perturbation theory, we have obtained the energy eigenvalues of a newly proposed generalized Hellmann potential model (the GHP model) for the hydrogenic atoms and neutral atoms. The potential is a superposition of the attractive Coulomb potential plus Yukawa one, and new central terms appear as a result of the effects of noncommutativity properties of space and phase in the Hellmann potential model. The obtained energy eigen-values appear as a function of the generalized gamma function, the discrete atomic quantum numbers (j, n, l, s and m), infinitesimal parameters (a, b, б) which are induced by the position-position and phase-phase noncommutativity, and, the dimensional parameters (Θ, 0) of the GHP model, in the nonrelativistic noncommutative three-dimensional real space phase (NC: 3D-RSP). Furthermore, we have shown that the corresponding Hamiltonian operator with (NC: 3D-RSP) symmetries is the sum of the Hamiltonian operator of the Hellmann potential model and two operators, the first one is the modified spin-orbit interaction, while the second is the modified Zeeman operator for the hydrogenic and neutral atoms.

Nonrelativistic treatment of hydrogen-like and neutral atoms subjected to the generalized perturbed Yukawa potential with centrifugal barrier in the symmetries of noncommutative quantum mechanics

We have obtained the approximate analytical solutions of the nonrelativistic Hydrogen-like atoms such as [Formula: see text] and [Formula: see text] and neutral atoms such as ([Formula: see text] and [Formula: see text]) atoms with a newly proposed generalized perturbed Yukawa potential with centrifugal barrier (GPYPCB) model using the generalized Bopp’s shift method and standard perturbation theory in the symmetries of noncommutative three-dimensional real space phase (NC: 3D-RSP). By approximating the centrifugal term through the Greene–Aldrich approximation scheme, we have obtained the energy eigenvalues and generalized Hamiltonian operator for all orbital quantum numbers [Formula: see text] in the symmetries of NC: 3D-RSP. The potential is a superposition of the perturbed Yukawa potential and new terms proportional with [Formula: see text]) appear as a result of the effects of noncommutativity properties of space and phase on the perturbed Yukawa potential model. The obtained energy eigenvalues appear as functions of the generalized Gamma function, the discreet atomic quantum numbers [Formula: see text], two infinitesimal parameters [Formula: see text], which are induced by (position–position and phase–phase). In addition, the dimensional parameters [Formula: see text] of perturbed Yukawa potential with centrifugal barrier model in NC: 3D-RSP. Furthermore, we have shown that the corresponding Hamiltonian operator in (NC: 3D-RSP) symmetries is the sum of the Hamiltonian operator of perturbed Yukawa potential model and the two operators are modified spin–orbit interaction and the modified Zeeman operator for the previous Hydrogenic and neutral atoms.