Terms Of Fox Research Articles

Unified performance analysis of interference-limited and interference-free dual-hop mixed RF/FSO systems with partial relay selection under pointing errors.

This paper investigates the performance of interference-limited and interference-free dual-hop mixed radio frequency (RF)/free space optical (FSO) systems with partial relay selection (PRS) for the variable-gain (VG) amplify-and-forward (AF) relaying scenario. We concentrate on the generalized channel model that not only describes different application scenarios but also allows a more accurate description of the channel characteristics. Specifically, the PRS-aided RF link is modeled by the κ-μ shadowed distribution, and the FSO link is expressed in terms of Fox's H-function, which unifies Fisher-Snedecor F, Gamma-Gamma (GG), and Malága (M) distributions for atmospheric turbulence along with pointing errors and detection modes. The interference signals at the selected relay are modeled by independent identically κ-μ shadowed distributions. Using our analytical framework, new unified closed-form expressions for the cumulative distribution function (CDF), the average bit error rate (BER), and the ergodic capacity are derived. Additionally, we provide asymptotic expressions of the average BER at high SNR. The analysis quantifies the impact of co-channel interference, pointing errors, number of relays, and rank of the selected relay on the considered system's performance. Finally, numerical results and Monte Carlo (MC) simulations are presented to confirm the effectiveness of the derived expressions. Note that our results provide a generalized framework for comprehensive studies of this kind of systems.

CERTAIN RESULTS OF MITTAG-LEFFLER FUNCTION BASED ON HARTLEY TRANSFORM IN TERMS OF FOX'S H-FUNCTION

In this research article, the authors have derived a number of useful results of queen function of fractional calculus known as Mittag- Leffler functions in terms of Fox's H – function.

Analytical framework for the performance of energy detectors over α‐μ fading model with MRC diversity

SummaryThe advancements towards the 5G technology are significantly reliant on the effective utilization of the radio spectrum. The scarcity of the spectrum is a barrier to the employment of emergent wireless multimedia. The cognitive radio (CR) technology has materialized as a reassuring remedy to the present problem of spectral scarcity. This study explores the statistical characterization of energy detector‐based cognitive radio device alongside diversity reception. The analysis is carried out with the postulation that the transmitting signal encounters fading, characterized by α‐μ fading channel. Initially, we present a PDF expression of the α‐μ fading model when subjected to maximal ratio combining (MRC) diversity reception. Utilizing this expression, the closed‐form expression of the average probability of detection (Pd) and the average area under the receiver operating characteristics curve (AUC) are derived in the terms of Fox's H function. The detection threshold has been optimized to improve the detection performance by curtailing the probability of error. Additionally, asymptotic results for Pd with various diversity combining cases are also obtained under high signal‐to‐noise‐ratio (SNR) regime. Furthermore, the result of Pd is applied to explore the execution of cooperative spectrum sensing (CSS) with diversity reception. The accuracy and validation of the presented results are assured by comparison with the Monte Carlo simulation.

Computing the exact distribution of a linear combination of generalized logistic random variables and its applications

We consider methods for numerical evaluation of the probability density function, cumulative distribution function and quantile function of a linear combination of n independent generalized logistic random variables. Here we derive the exact expressions for the probability density function and cumulative distribution function in terms of Fox's H-function. We also provide an algorithm for numerical calculation of Fox H- function and we compare its performance with a method based on the numerical inversion of the characteristic function. The numerical precision, efficiency and computing time of the proposed algorithm were considered. Applicability and comparison of the suggested approaches are illustrated by an application to financial returns.

Molecular Communication With Passive Receivers in Anomalous Diffusion Channels

We consider anomalous diffusion for molecular communication with a passive receiver. We first consider the probability density function of molecules’ location at a given time in a space of arbitrary dimension. The expected number of observed molecules inside a receptor space of the receiver at certain time is derived taking into account the life expectancy of the molecules. In addition, an implicit solution for the time that maximizes the expected number of observed molecules is obtained in terms of Fox’s <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H$ </tex-math></inline-formula> -function. The closed-form expressions for the bit error rate of a single-bit interval transmission and a multi-bit interval transmission are derived. It is shown that lifetime limited molecules can reduce the inter-symbol interference while also enhancing the reliability of MC systems at a suitable observation time.

Performance analysis of ED over air-to-ground and ground-to-ground fading channels: A unified and exact solution

In this work, we study the performance of energy detector (ED) over generalized fading channels expressed in terms of Fox’s H-function over the air-to-ground (A2G) and ground-to-ground (G2G) networks. Firstly, the closed-form expression for the probability of missed-detection (PM) is derived. Then utilizing it, a unified and novel closed-form expression for the average area under the receiver operating characteristic curve (AUC) is proposed for Fox’s H-fading distribution. The proposed formulation is expressed in terms of bivariate Fox’s H-function. Also, we provide the closed-form result for the derivative of PM that helps in obtaining the optimized value of the threshold. The channel model presented here subsumes many fading models of practical interest. To get the simplified expressions of the performance parameters, the asymptotic analysis of the proposed framework has been provided. The diversity gain and coding gain of the performance metrics over the generalized fading channel and its special cases have been evaluated. Lastly, we provide a comparison of the derived expressions with the Monte-Carlo simulations to validate the accuracy of the proposed solutions.

USE OF ATANGANA–BALEANU FRACTIONAL DERIVATIVE IN HELICAL FLOW OF A CIRCULAR PIPE

There is no denying fact that helically moving pipe/cylinder has versatile utilization in industries; as it has multi-purposes, such as foundation helical piers, drilling of rigs, hydraulic simultaneous lift system, foundation helical brackets and many others. This paper incorporates the new analysis based on modern fractional differentiation on infinite helically moving pipe. The mathematical modeling of infinite helically moving pipe results in governing equations involving partial differential equations of integer order. In order to highlight the effects of fractional differentiation, namely, Atangana–Baleanu on the governing partial differential equations, the Laplace and Hankel transforms are invoked for finding the angular and oscillating velocities corresponding to applied shear stresses. Our investigated general solutions involve the gamma functions of linear expressions. For eliminating the gamma functions of linear expressions, the solutions of angular and oscillating velocities corresponding to applied shear stresses are communicated in terms of Fox- H function. At last, various embedded rheological parameters such as friction and viscous factor, curvature diameter of the helical pipe, dynamic analogies of relaxation and retardation time and comparison of viscoelastic fluid models (Burger, Oldroyd-B, Maxwell and Newtonian) have significant discrepancies and semblances based on helically moving pipe.

Molecular Communication With Anomalous Diffusion in Stochastic Nanonetworks

Molecular communication in nature can incorporate a large number of nano-things in nanonetworks as well as demonstrate how nano-things communicate. This paper presents molecular communication where transmit nanomachines deliver information molecules to a receive nanomachine over an anomalous diffusion channel. By considering a random molecule concentration in a space-time fractional diffusion channel, an analytical expression is derived for the first passage time (FPT) of the molecules. Then, the bit error rate of the $\ell $ th nearest molecular communication with timing binary modulation is derived in terms of Fox's $H$ -function. In the presence of interfering molecules, the mean and variance of the number of the arrived interfering molecules in a given time interval are presented. Using these statistics, a simple mitigation scheme for timing modulation is provided. The results in this paper provide the network performance on the error probability by averaging over a set of random distances between the communicating links as well as a set of random FPTs caused by the anomalous diffusion of molecules. This result will help in designing and developing molecular communication systems for various design purposes.

A Versatile Integral in Physics and Astronomy and Fox’s H-Function

This paper deals with a general class of integrals, the particular cases of which are connected to outstanding problems in physics and astronomy. Nuclear reaction rate probability integrals in nuclear physics, Krätzel integrals in applied mathematical analysis, inverse Gaussian distributions, generalized type-1, type-2, and gamma families of distributions in statistical distribution theory, Tsallis statistics and Beck–Cohen superstatistics in statistical mechanics, and Mathai’s pathway model are all shown to be connected to the integral under consideration. Representations of the integral in terms of Fox’s H-function are pointed out.

A fractional diffusion equation with sink term

Approaching dynamical aspects in systems with localised sink term is fundamentally relevant from the physical-experimental point of view. However, theoretical advances have not progressed much in this direction; this is due to the great difficulty of addressing such types of problems from an analytic point of view. In this work, we investigate the systems sink points (like traps) considering a dynamics governed by the like reaction–diffusion equation in the presence of fractional derivatives. In order to do so, making use of the continuous-time random walks theory we constructed a model that contains multiples sink terms, on sequence we analysed the problem for two cases: first, considering only a single sink term and, second, considering multiples sink point associated with a fractal set. In both cases, we present the analytic solution in terms of Fox and Mittag–Leffler functions. Moreover, we perform a calculus of the mean square displacement and survival probability. The proposal and the techniques used in this work are useful to describe anomalous diffusive phenomena and the transport of particles in irregular media.

An Analytical Study of Fractional Klein–Kramers Approximations for Describing Anomalous Diffusion of Energetic Particles

In this article, anomalous transport models of energetic particles in space plasmas are developed by deriving the fractional force-less Fokker–Planck equation and the fractional diffusion-advection equation from the Klein–Kramers equation. Analytical solutions of the space-time fractional equations are obtained by use of the Caputo and the Riesz fractional derivatives with the Laplace–Fourier technique. The solutions are given in terms of Fox’s H-function and the profiles of the particles densities are discussed in each case for different values of fractional orders.

New approximate expressions of average symbol error probability, probability of detection and AUC with MRC over generic and composite fading channels

The present work covers the wide range of distributions and proposes the moment generating function (MGF) based analytical framework to derive the average symbol error probability (SEP), the average probability of detection (PD), and the average area under the receiver operating characteristic curve (AUC) for the wireless system with maximal ratio combining (MRC) diversity. The proposed solutions are expressed in terms of Fox’s H and bivariate Fox’s H-functions which are widely being used in the current literature. Finally, the analytical framework, thus proposed has been used in applications, such as ultra-wideband (UWB) single-input-single-output (SISO) and multiple-input-multiple-output (MIMO) system and the impact of the number of transmitting and receiving antennas for MIMO system over the average SEP has been analysed. The correctness of the proposed solutions has been corroborated by comparing them with Monte-Carlo simulation.

Two discursive frameworks concerning ideology in Australian industrial relations

AbstractThere are two discursive frameworks concerning ideology in Australian industrial relations. In many disciplines concerned with aspects of industrial relations, including political science, law and history, it is the traditional political ideologies of the industrial era which take centre stage: liberalism (classical, social and neoliberalism), socialism (Marxism, social democracy and labourism) and conservatism. By contrast, ideological issues in the discipline of employment relations are chiefly addressed in terms of Fox’s three analytical perspectives: unitarism, pluralism and radicalism. The disjunction between these parallel discourses goes largely unnoted in the literature of the relevant disciplines, which all tend to proceed using their own preferred approach without making reference to the other. This article critically explores the relationship between these two discourses and investigates the broader implications that the existence of the two different discursive traditions has for the analysis of industrial relations phenomena in Australia.

Analytical solutions of the space–time fractional Telegraph and advection–diffusion equations

The aim of this paper is to develop a fractional derivative model of energetic particle transport for both uniform and non-uniform large-scale magnetic field by studying the fractional Telegraph equation and the fractional advection–diffusion equation. Analytical solutions of the space–time fractional Telegraph equation and space–time fractional advection–diffusion equation are obtained by use of the Caputo fractional derivative and the Laplace–Fourier technique. The solutions are given in terms of Fox’s H function. As an illustration they are applied to the case of solar energetic particles.

Solution of fractional bioheat equation in terms of Fox's H-function.

Present paper deals with the solution of time and space fractional Pennes bioheat equation. We consider time fractional derivative and space fractional derivative in the form of Caputo fractional derivative of order alpha in left( 0,1right] and Riesz–Feller fractional derivative of order beta in left( 1,2right] respectively. We obtain solution in terms of Fox’s H-function with some special cases, by using Fourier–Laplace transforms.

Anomalous Diffusion in Molecular Communication

We consider anomalous diffusion to model a molecular communication channel. To account for general and practical molecular propagation, we use the fractional diffusion equation and derive the distribution of first passage time in terms of Fox's $H$ -function. We then analyze the bit error rate for timing and amplitude binary modulation schemes in anomalous diffusion.

Slowing-down of neutrons: a fractional model

The fractional version for the diffusion of neutrons in a material medium is studied. The concept of fractional derivative is presented, in the Caputo and Riesz senses. Using this concept, we discuss a fractional partial differential equation associated with the slowing-down of neutrons, whose analytical solution is presented in terms of Fox's H function. As a convenient limiting case, the classical solution is recovered.

Lévy stable distribution and space-fractional Fokker–Planck type equation

The space-fractional Fokker–Planck type equation ∂p∂t+γ∂p∂x=-D(-Δ)α/2p(0<α⩽2) subject to the initial condition p(x,0)=δ(x) is solved in terms of Fox H functions. The solution as γ=0 expresses the Lévy stable distribution with the index α. From the properties of Fox H functions, the series representation and asymptotic behavior for the solution are also obtained. Lévy stable distribution as 0<α<2 describes anomalous superdiffusion and its diffusion velocity is characterized by xd∝(Dt)1/α.

Solution of Fractional Partial differential Equation Aries in Study of Heat Transfer through Diathermanous Materials

The aim of this paper is to investigate the solutions of fractional partial differential equations with Hilfer composite fractional derivative and the space fractional Laplacian operator, occurring in study of heat transfer through diathermanous materials. The solution of the equation is obtained by applying the Laplace and fourier transforms, in terms of Fox's H-function.

Space-fractional Schrödinger equation for a quadrupolar triple Dirac-δ potential: Central Dirac-δ well and barrier cases

We solve the space-fractional Schrödinger equation for a quadrupolar triple Dirac-δ (QTD-δ) potential for all energies using the momentum-space approach. For the E &lt; 0 solution, we consider two cases, i.e., when the strengths of the potential are V0 &gt; 0 (QTD-δ potential with central Dirac-δ well) and V0 &lt; 0 (QTD-δ potential with central Dirac-δ barrier) and derive expressions satisfied by the bound-state energy. For all fractional orders α considered, we find that there is one eigenenergy when V0 &gt; 0, and there are two eigenenergies when V0 &lt; 0. We also obtain both bound- and scattering-state (E &gt; 0) wave functions and express them in terms of Fox's H-function.