Asymptotic Expressions Research Articles

Performance analysis for wireless-powered relaying UWOC-RF systems

In this paper, the system performance for a wireless-powered relaying mixed underwater wireless optical communication-radio frequency (UWOC-RF) network is investigated. In particular, a source sends an optical signal to a power-constrained marine relay, which simultaneously harvests energy from the surrounding environment and processes the received optical signal with an amplify-and-forward protocol and converts it into an electric one before broadcasting it to the destination. Assuming the UWOC and the RF channels undergo a mixture Exponential-Generalized Gamma fading and Nakagami-m fading, the closed-form and asymptotic expressions for outage probability and the diversity order are obtained. Furthermore, the average capacity and average bit error rate are also presented. Monte-Carlo simulations are also used to validate the correctness of those derived results.

Stokes’ Second Flow Problem Revisited for Particle–Fluid Suspensions

Abstract An alternative analytical model is proposed for hydrodynamics of incompressible Newtonian fluids with suspended solid particles. Unlike existing single-phase models that do not distinguish the velocity field of suspended particles from the velocity field of host fluid, the present model accounts for the relative shift between the two velocity fields and assumes that its effect can be largely captured by substituting the inertia term of Navier–Stokes equations with the acceleration field of the mass center of the representative unit cell. The proposed model enjoys a relatively concise mathematical formulation. The oscillating flow of a particle–fluid suspension between two flat plates is studied with the present model, and detailed results are presented for Stokes’ second flow problem on the oscillating flow of a suspension half-space induced by an oscillating plate with specific examples of dusty gases and nanofluids. Remarkably, leading-order asymptotic expressions derived by the present model, for the effect of suspended particles on the decay index and wavenumber of the velocity field, are shown to be identical to known results derived based on the widely adopted Saffman model for dusty gases. It is hoped that the present work could offer a relatively simplified and yet reasonably accurate model for hydrodynamic problems of particle–fluid suspensions.

Performance Analysis of Multiple-RIS-Based NOMA Systems

In this paper, we present a study on a model of multirelay radio network system that utilizes reconfigurable intelligent surfaces (RISs). We investigate the use of nonorthogonal multiple access (NOMA) combined with cooperative RIS systems, using partial RIS selection (PRISs). Specifically, the RISs act as relays to forward data from the base station to the two users. The focus of this paper is to analyze the outage probabilities and throughput for the two users. Based on the results, we examine how PRISs affect the performance of the proposed NOMA scheme. The derived asymptotic expressions show that the proposed model can improve user fairness. Finally, we compare the analysis results with the simulation results and find good agreement.

Dispersion of run-and-tumble microswimmers through disordered media.

Understanding the transport properties of microorganisms and self-propelled particles in porous media has important implications for human health as well as microbial ecology. In free space, most microswimmers perform diffusive random walks as a result of the interplay of self-propulsion and orientation decorrelation mechanisms such as run-and-tumble dynamics or rotational diffusion. In an unstructured porous medium, collisions with the microstructure result in a decrease in the effective spatial diffusivity of the particles from its free-space value. Here, we analyze this problem for a simple model system consisting of noninteracting point particles performing run-and-tumble dynamics through a two-dimensional disordered medium composed of a random distribution of circular obstacles, in the absence of Brownian diffusion or hydrodynamic interactions. The particles are assumed to collide with the obstacles as hard spheres and subsequently slide on the obstacle surface with no frictional resistance while maintaining their orientation, until they either escape or tumble. We show that the variations in the long-time diffusivity can be described by a universal dimensionless hindrance function f(ϕ,Pe) of the obstacle area fraction ϕ and Péclet number Pe, or ratio of the swimmer run length to the obstacle size. We analytically derive an asymptotic expression for the hindrance function valid for dilute media (Peϕ≪1), and its extension to denser media is obtained using stochastic simulations. As we explain, the model is also easily generalized to describe dispersion in three dimensions.

On the block error rate performance of cooperative non‐orthogonal multiple access short‐packet communications with full‐duplex relay and partial relay selection

AbstractIn this paper, we mathematically investigate a downlink non‐orthogonal multiple access (NOMA) system for short‐packet communications (SPC) in which the near users are used as full‐duplex (FD) relays to forward intended signals from the source to a far user. In addition, partial relay selection is employed to enhance the performance of the FD relays under the impact of imperfect interference cancellation. At the far user, selection combining (SC) or maximal ratio combining (MRC) is employed to combine the signals received from the source and the selected FD relay. The analytical expressions for the average block error rate (BLER) of two users over flat Rayleigh fading channels are derived. Furthermore, closed‐form asymptotic expressions of the average BLERs at the near and far users in high signal‐to‐noise ratio (SNR) regimes are obtained. The numerical results show that the analytical BLERs of the near user and far user closely match the simulation results.

Adaptive inference procedures for the concentration parameter of a Fisher-von Mises-Langevin distribution

This paper focuses on the estimation and testing of a shape parameter or concentration parameter κ of a directional distribution named Fisher-von Mises- Langevin distribution. The estimators for κ using standard procedures cannot be obtained in closed or explicit forms. We derive asymptotic expressions up to first order for the bias, mean squared error and variance of the maximum likelihood estimator of κ when its mean direction is unknown. Using these expressions in a selective manner, several nearly-unbiased estimators for κ are proposed. A simulation study is carried out to compare the biases and percentage risk improvement of these new proposed estimators. The proposed estimators are seen to offer substantial improvements in terms of bias and percentage risk improvement over the earlier ones. The simulation study shows that these tests achieve size quite close to their nominal values and have quite good power performances. A nonparametric bootstrap confidence intervals for κ is presented. Finally these estimators and test procedures are applied on a real data set. ‘MATLAB’ functions are developed for practical use.

Reconfigurable Intelligent Surfaces and Capacity Optimization: A Large System Analysis

Reconfigurable Intelligent Surfaces (RISs) have been recently proposed as an enabling technology for programmable wireless environments. In this paper, we present asymptotic closed-form expressions for the mean and variance of the mutual information for a multi-antenna transmitter-receiver pair in the presence of RISs, using statistical physics methods. While nominally valid in the large-system limit, we show that the derived Gaussian approximation for the mutual information can be quite accurate, even for modest-sized antenna arrays and metasurfaces. The above results are particularly useful when fast-fading conditions are present, which renders channel estimation challenging. We find that, when the channel close to an RIS is correlated, for instance due to small angle spread, which is reasonable for wireless systems with increasing carrier frequencies, the communication link benefits significantly from statistical RIS optimization, resulting in gains that are surprisingly higher than the nearly uncorrelated case. Using our novel asymptotic properties of the correlation matrices of the impinging and outgoing signals at the RISs, we can optimize the metasurfaces without brute-force numerical optimization. When the desired reflection from any of the RISs departs significantly from geometrical optics, the metasurfaces can be optimized to provide robust communication links, without significant need for their optimal placement.

IRS-Aided Secure Communications Over an Untrusted AF Relay System

In this paper, an intelligent reflecting surface (IRS) assisted wireless secure communication system is studied. A multi-antenna access point (AP) intends to send confidential information to a single-antenna user in the presence of an amplify-and-forward (AF) untrusted relay, which may eavesdrop the message when helping relay the signal. It is assumed that the direct link between the AP and the user is blocked by the obstacles. The achievable secrecy rate maximization problem is then formulated. To overcome the non-convexity of the formulated problem, an alternating iteration algorithm is proposed to jointly optimize the active and passive beamforming. Specifically, the transmit beamforming vector at the AP, the phase shift matrix at the IRS, and the relay beamforming matrix are jointly optimized to maximize the secrecy rate. Moreover, the asymptotic expressions for the maximum secrecy rates of the proposed scheme in the high and low transmitted signal-to-noise ratio (SNR) regimes are derived as well. Finally, numerical evaluations demonstrate the superiority of the proposed scheme compared with other benchmark schemes, highlight the importance of properly designed phase shifts at the IRS, and validate the theoretical analysis. It is also demonstrated that the number of antennas at the AP should be strictly larger than the number of antennas at the relay to harvest the benefits of increasing the transmit power at the AP in increasing the secrecy rate.

Asymptotic analysis of Jeffreys–Newtonian fluids flowing through a composite vertical porous layered channel: Brinkman–Forchheimer model

This study examines the flow of a Newtonian fluid enclosed between two non-Newtonian Jeffreys fluids with viscosity that varies with temperature within a composite vertical channel. Including a corotational Jeffreys liquid allows for considering stress dependence on the present deformation rate and its history. The proposed study's framework comprises three distinct regions, wherein the intermediate region governs Newtonian fluid flow under temperature-dependent viscosity. However, the outer layers oversee the flow of Jeffreys fluids within the porous medium, demonstrating temperature-dependent viscosity. The Brinkman–Forchheimer equation is employed to establish the governing equations applicable to both low and high permeabilities of the porous medium. This equation is nonlinear, making it challenging to find an analytical solution. Therefore, the regular and singular perturbation methods with matched asymptotic expansions are applied to derive asymptotic expressions for velocity profiles in various regions. The hydrodynamic quantities, such as flow rate, flow resistance, and wall shear stresses, are determined by deriving their expressions using velocities from three distinct regions. The graphical analysis explores the relationships between these hydrodynamic quantities and various parameters, including the Grashof number, Forchheimer number, viscosity parameter, Jeffreys parameter, conductivity ratio, effective viscosity ratio, absorption ratio, and the presence of varying thicknesses of different layers. An interesting finding is that a more pronounced velocity profile is noticed when the permeability is high and the viscosity parameter of the Newtonian region, denoted as α2, is lower than that of the surrounding area. This heightened effect can be linked to a relatively more significant decrease in the viscosity of the Jeffreys fluid, represented by μ1, as compared to the viscosity of the Newtonian fluid, μ2, as the temperature increases. The outcomes of this research hold special significance in situations like the extraction of oil from petroleum reserves, where the oil moves through porous layers with varying viscosities, including sand, rock, shale, and limestone.

NOON state measurement probabilities and outcome fidelities: a Bethe ansatz approach

A recently proposed extended Bose–Hubbard model, one that is a quantum integrable model, provides a framework for a NOON state generation protocol. Here we derive a Bethe ansatz solution for the model. The form of the solution provides the means to obtain exact asymptotic expressions for the energies and eigenstates. These results are used to derive formulae for measurement probabilities and outcome fidelities. We benchmark these results against numerical calculations.

A model for the dissipation rate in linear unsteady flow through porous media

We present a model for the volume-averaged dissipation rate in linear unsteady flow through porous media. The model is derived by blending a new small-time asymptotic expression for the dissipation rate obtained from boundary layer theory with the known large-time asymptotic expression obtained from Darcy's law. The resulting model is a second-order Volterra functional of the volume-averaged acceleration. We validate the model with an analytical solution for transient flow through a porous medium composed of circular tubes and with numerical simulations of transient and oscillatory flow through a cylinder array and through a hexagonal sphere pack.

Study of the Stability Properties for a General Shape of Damped Euler–Bernoulli Beams under Linear Boundary Conditions

We study in this paper a general shape of damped Euler–Bernoulli beams with variable coefficients. Our main goal is to generalize several works already done on damped Euler–Bernoulli beams. We start by studying the spectral properties of a particular case of the system, and then we determine asymptotic expressions that generalize those obtained by other authors. At last, by adopting well-known techniques, we establish the Riesz basis property of the system in the general case, and the exponential stability of the system is obtained under certain conditions relating to the feedback coefficients and the sign of the internal damping on the interval studied of length 1 .

Asymptotic profiles for a nonlinear Kirchhoff equation with combined powers nonlinearity

We study asymptotic behavior of positive ground state solutions of the nonlinear Kirchhoff equation −(a+b∫RN|∇u|2)Δu+λu=uq−1+up−1inRN,(Pλ)as λ→0 and λ→+∞, where N=3 or N=4, 2<q≤p≤2∗, 2∗=2NN−2 is the Sobolev critical exponent, a>0, b≥0 are constants and λ>0 is a parameter. In particular, we prove that in the case 2<q<p=2∗, as λ→0, after a suitable rescaling the ground state solutions of (Pλ) converge to the unique positive solution of the equation −Δu+u=uq−1 and as λ→+∞, after another rescaling the ground state solutions of (Pλ) converge to a particular solution of the critical Emden–Fowler equation −Δu=u2∗−1. We establish a sharp asymptotic characterization of such rescalings, which depends in a non-trivial way on the space dimension N=3 and N=4. We also discuss a connection of our results with a mass constrained problem associated to (Pλ) with normalization constraint ∫RN|u|2=c2. As a consequence of the main results, we obtain the existence, non-existence and asymptotic behavior of positive normalized solutions of such a problem. In particular, we obtain the exact number and their precise asymptotic expressions of normalized solutions if c>0 is sufficiently large or sufficiently small. Our results also show that in the space dimension N=3, there is a striking difference between the cases b=0 and b≠0. More precisely, if b≠0, then both p0≔10/3 and pb≔14/3 play a role in the existence, non-existence, the exact number and asymptotic behavior of the normalized solutions of the mass constrained problem, which is completely different from those for the corresponding nonlinear Schrödinger equation and which reveals the special influence of the nonlocal term.

Variational field theory of macroscopic forces in coulomb fluids.

Based on the variational field theory framework, we extend our previous mean-field formalism [Y. A. Budkov and A. L. Kolesnikov, JStatMech 2022, 053205.2022], taking into account the electrostatic correlations of the ions. We employ a general covariant approach and derive a total stress tensor that considers the electrostatic correlations of ions. This is accomplished through an additional term that depends on the autocorrelation function of the local electric field fluctuations. Utilizing the derived total stress tensor and applying the mechanical equilibrium condition, we establish a general expression for the disjoining pressure of the Coulomb fluids, confined in a pore with a slit-like geometry. Using this equation, we derive an asymptotic expression for the disjoining pressure in a slit-like pore with non-electrified conductive walls. Present theory is the basis for future modeling of the mechanical stresses that occur in electrode pores with conductive charged walls, immersed in liquid phase electrolytes beyond the mean-field theory.

Radiation fluxes of gravitational, electromagnetic, and scalar perturbations in type-D black holes: an exact approach

We present a novel method that solves Teukolsky equations with the source to calculate radiation fluxes at infinity and event horizon for any perturbation fields of type-D black holes. For the first time, we use the confluent Heun function to obtain the exact solutions of ingoing and outgoing waves for the Teukolsky equation. This benefits from our derivation of the asymptotic analytic expression of the confluent Heun function at infinity. It is interesting to note that these exact solutions are not subject to any constraints, such as low-frequency and weak-field. To illustrate the correctness, we apply these exact solutions to calculate the gravitational, electromagnetic, and scalar radiations emitted by a particle in circular orbits around a Schwarzschild black hole. Numerical results show that the proposed exact solution appreciably improves the computational accuracy and efficiency compared with the 23rd post-Newtonian order expansion and the Mano-Suzuki-Takasugi method.

Trajectories of Liquid Particles in a Dark Soliton Field in a Fluid Beneath an Ice Cover

A fluid layer of finite depth described by Euler equations is considered. The ice cover is modeled by a geometrically non-linear elastic Kirchhoff–Love plate. The trajectories of liquid particles under the ice cover are found in the field of a nonlinear surface traveling wave tending to a periodic wave at infinity: the so-called “dark soliton” (which is a nonlinear product of a step-wave and a periodic wave) of small but finite amplitude. The presence of dark solitons in the system is an indicator of the modulation stability of the carrier periodic wave (defocusing). The analysis uses explicit asymptotic expressions for solutions describing wave structures on the water–ice interface such as dark solitons, as well as asymptotic solutions for the velocity field in the liquid column generated by these waves.

Generalized transmit laser selection for vertical underwater wireless optical communications over Gamma-Gamma turbulence channels.

Underwater wireless optical communication (UWOC) is a potential technology for high-speed and reliable underwater data transmission. In underwater environment, ocean turbulence has a strong impact on the performance of UWOC systems. Thus, transmission laser selection (TLS) is proposed as an effective technique for fading mitigation in turbulence channels. In this paper, we present a vertical UWOC system with generalized TLS (GTLS) in multi-layer cascaded Gamma-Gamma ocean turbulence channels. With GTLS, the transmitter is equipped with N laser sources and the nth source is selected for transmission. For the ideal case, the optimal laser source is selected, but in practice, a sub-optimal or worse source may be selected due to channel estimation and/or feedback errors. For the GTLS system, we derive an exact closed-form expression for outage probability. Furthermore, based on the outage probability expression, the diversity order and asymptotical diversity order expression are analyzed. Finally, we give simulation results to validate our analytical results. Numerical results show that the GTLS system performs better than the system without GTLS (i.e. N = 1). The number of cascade channel layers, the number of laser sources and the selection of source index significantly affect the performance of the GTLS system.

A finite element method for modeling diffusion of alpha-emitting particles in tissue.

Diffusing alpha-emitters Radiation Therapy ("DaRT") is a promising new modality for the treatment of solid tumors. Interstitial sources containing 224 Ra are inserted into the tumor, producing alpha particles via the decay of 224 Ra and its daughters. The alpha particles are able to produce a "kill region" of several mm due to the diffusion of the alpha-emitting atoms. The Diffusion-Leakage (D-L) model has been proposed to describe the movement of the alpha-emitters used in DaRT in tumor tissue. To date, estimating the dose delivered under the D-L model has been accomplished with numerical solutions based on finite difference methods, namely DART1D and DART2D, as well as with asymptotic expressions for the long time limit. The aim of this work is to develop a flexible method of finite elements for solving the D-L model and to validate prior solutions of the D-L model. We develop a two-dimensional finite element solution to the D-L model implemented using the FEniCS software library. Our approach solves the variational formulation of the D-L equationson an unstructured mesh of triangular Lagrangian elements. We calculate the local dose in the mid- and axial planes of the source and validate our results against the one- and two-dimensional solutions obtained using the previously proposed numerical scheme, DART1D and DART2D. We use our model to estimate the change in dose in the source midplane as a function of the physical parameters used in the D-Lmodel. The local dose at the end of a 30 day treatment period estimated by our numerical method differs from DART1D and DART2D by less than 1% in the source midplane and less than 3% along the source axis over clinically relevant distances, with the largest discrepancies in high gradient areas where the Finite Element Method (FEM) mesh has a higher element density. We find that within current experimentally estimated ranges for D-L model parameters, the dose in the source midplane at a distance of 2mm can vary by over a factor of 3. The 2D finite element model reproduces the calculated dose obtained with DART1D and DART2D under the assumptions D-L model. The variation in predicted dose within current experimental ranges for model parameters suggests the necessity of further studies to better determine their statistical distributions. Finally, the FEM model can be used to calculate dose from DaRT in a variety of realistic 2D geometries beyond the D-Lmodel.

Asymptotic behavior of the Zhang–Kawazumi's φ-invariants

The φ-invariant was introduced by Shou-Wu Zhang and Nariya Kawazumi. We study the continuity property of the φ-invariants for degenerating graphs, show that this continuity property induces an adelic divisor on the moduli space of Riemann surfaces, and then give an asymptotic expression of the Zhang–Kawazumi's invariants for Riemann surfaces near the boundary of the moduli space. We mainly use Yuan–Zhang's adelic divisors, and follow Yuan's idea of globalization of φ-invariants.

Uncertainty laws of experimental modal analysis with known broadband input

‘Uncertainty law’ aims at closed-form asymptotic formulas for the relationship between the identification uncertainties of modal properties (e.g., natural frequency, damping ratio) and test configuration (e.g., noise level, number and location of sensors, data duration). Existing developments focused on the case of unknown-input (ambient), where it has been found that identification uncertainty does not vanish even for noiseless instruments, essentially because the input is unknown. A natural question is then on how the uncertainty depends on test configuration when the input is known, not to mention how the configuration should be quantified. Motivated by these and related questions, this paper develops the uncertainty laws of modal parameters for well-separated modes with known single broadband input, e.g., vibration test with a single shaker as in experimental modal analysis. Asymptotic expressions for the posterior coefficient of variation of modal parameters are derived via the Fisher Information Matrix for long data and small damping scenarios. Assumptions and theory are validated using synthetic and field test data. Governing factors motivated by the theory are investigated, including the equivalent modal signal-to-noise ratio (for known input), the number of measured degrees of freedom, shaker location, and data duration. By virtue of the Cramér-Rao bound in classical statistics, the developed uncertainty laws represent the lower bound of identification uncertainty with known broadband input that can be achieved by any unbiased estimator. They provide a scientific basis for planning and managing identification uncertainties in vibration tests with known input.