Closed-form Analytical Expressions Research Articles

Three-Dimensional Green Tensor of One-Dimensional Hexagonal Quasicrystals

In this work, the elastic 4×4 Green tensor of one-dimensional quasicrystals is given and has phonon, phason and phonon–phason coupling components. Using the residue method, a closed-form expression of the elastic 4×4 Green tensor for one-dimensional hexagonal quasicrystals of Laue class 10, which possess 10 independent material constants, is derived. The 10 independent components of the obtained 4×4 Green tensor are numerically presented in contour plots, revealing features of anisotropy as well as the interesting result that the phason component of the Green tensor has the strongest contribution in comparison with all the other components. In the case of vanishing phonon–phason coupling, the phonon part of the derived Green tensor reproduces Kröner’s well-known elastic 3×3 Green tensor for hexagonal crystals. The analytical closed-form expression of the derived Green tensor provides an advantage for efficient computational calculations in various applications.

UAV-assisted dual-hop RF communication: Performance analysis of outage probability and altitude optimization for multi-user systems

Unmanned Aerial Vehicles (UAV) are growingly used as communication relays in this modern era due to their unique capabilities and advantages which includes enhanced connectivity, cost effectiveness, flexibility, risk reduction and real time data surveillance. UAVs offer a versatile and efficient solution for enhancing communication networks, especially in situations where traditional infrastructure is inadequate or unavailable. In this manuscript, we present the dual hop UAV enabled wireless communication system where the data is broadcasted from Source (S) to Destination (D) through a UAV working as decode and forward (DF) relay (R). The data is transmitted from S to R via Radio frequency (RF) link and then the decoded data from R is forward to D with multiple users. Closed form analytical expressions of outage probability is analyzed which is further utilized to do the optimization analysis to find the optimum altitude of the UAV in order to exaggerate its system performance. Numerical results shows that there is need to find out the optimal altitude of the UAV so as to keep down the overall outage probability of the proposed system and to enhance the system quality. We also find the outage probability at higher Signal-to-Noise ratio (SNR) to study the system behavior deeply. It is clearly observable from the numerical analysis that the exact outage probability and asymptotic outage probability are exactly matching at higher SNR values concluding the validity of the proposed system. The derived expressions are authenticated by simulated results via Monte Carlo simulations.

The influence of bi-directional material gradation on a mode-III crack in functionally graded material via strain gradient elasticity theory

In contrast to classical mechanics, which primarily relies on continuum assumptions and neglects micro-structural effects, the strain gradient elasticity (SGE) theory represents a paradigm shift in understanding the mechanical behavior of materials at small length scales. In this article, the influence of the bi-directional material gradation on a mode-III crack in functionally graded material via SGE theory is studied. The SGE theory uses two material characteristic lengths, ℓ and ℓ′, to account for volumetric and surface strain-gradient factors, respectively. Our investigation is centered on a material gradation model assumed to vary exponentially, with the shear modulus represented as G(x,y)=G0eβx+γy, where β and γ are material gradation constants. To address the crack boundary value problem under consideration, we employ a methodology combining Fourier transforms and an innovative hyper-singular integro-differential equation approach. Using this approach, we systematically formulate a system of equations, which can be solved by selecting suitable collocation points. The closed-form analytical expressions are derived for the standard fracture parameters such as crack surface displacement (CSD), stress intensity factor (SIF), and energy release rate (ERR). Numerical studies are illustrated for the derived standard fractures, and the influence of these parameters β, γ, ℓ, ℓ′, and applied shear load is graphically presented. Through comprehensive analysis, our aim is to provide insights into the complex interplay between material parameters, loading conditions, and crack behavior in functionally graded materials.

Statistical theory of neutron-induced nuclear fission and of heavy-ion fusion

For both reactions I use an approach similar to that of compound-nucleus reaction theory. For neutron-induced fission, the compound system generated by absorption of the neutron and the nuclear system near the scission point are described as two statistically independent systems governed by random-matrix theory. The systems are connected either by a barrier penetration factor or by a set of transition states above the barrier. Each system is coupled to a different set of channels. An analogous model is used for heavy-ion fusion. Under the assumption that (seen from the entrance channel) the system on the other side of the barrier is in the regime of strongly overlapping resonances, closed-form analytical expressions for the total probability for fission and for fusion are obtained for each value of spin and parity. Parts of these expressions can be calculated reliably within existing compound-nucleus reaction theory. The remaining parts are the probabilities for passage through or over the barrier. These may be determined theoretically from the liquid-drop model or experimentally from total fission or fusion cross sections. Published by the American Physical Society 2024

Secrecy performance of D2D assisted cooperative uplink NOMA system with optimal power allocation strategy

This paper investigates the physical-layer security (PLS) aspects of the uplink cooperative non-orthogonal multiple access (NOMA) system in the context of two user scenario. More specifically, this work employs device-to-device (D2D) pair users (i.e., D1, and D2) to further improve the spectral efficiency (SE) of the proposed cooperative uplink NOMA (CU-NOMA) system. One D2D user acts as a decode-and-forward relay, improving the performance of both the cell-edge user (CU) and another D2D user i.e., (D2). We analyze the secrecy performance of CU and D2 under perfect and imperfect successive interference cancellation (SIC), and optimizes power allocation (PA) to boost the performance. The proposed CU-NOMA system is evaluated in terms of its performance metrics like ergodic secrecy capacity (ESC), ergodic secrecy sum capacity (ESSC), non-zero secrecy capacity (NSC), effective secrecy throughput (EST), and secrecy outage probability (SOP) under the presence of an external eavesdropper (Eav). In addition, this work derives the closed-form analytical expressions of ESC, NSC, EST, and SOP metrics for both CU and D2 under perfect SIC (pSIC) and imperfect (ipSIC) cases in order to characterize the secrecy performance of the proposed secure CU-NOMA network. Further, the outcomes of the simulations are shown as evidence for both the validation of the mathematical analysis and the performance of the method being suggested. The simulation result analysis of this work infers that the secrecy performance of CU-NOMA system with optimal PA exhibits superior performance than that of the fixed PA scheme.

Closed-form BER analysis of FBMC-OQAM system over long-distance horizontal underwater acoustic transmission in shallow water

The filter bank multi-carrier with offset quadrature amplitude modulation (FBMC-OQAM) waveform demonstrates strong anti-interference capabilities due to its sub-carrier-level filtering. As a result, it is increasingly being adopted for underwater acoustic communication (UWAC) in challenging transmission environments. This paper presents a closed-form analytical expression for the bit error rate (BER) performance of FBMC-OQAM in shallow water, long-distance horizontal communication scenarios. The expression considers the residual Doppler frequency offset, multipath propagation, and ambient noise characteristic of underwater acoustic channels, accounting for the limitations of traditional non-uniform Doppler coarse compensation methods. Theoretical numerical results and Monte Carlo simulation results show that under any number of symbols, the derived closed-form expression consistently exhibits high computational accuracy, when facing the impact of Doppler spread or compression at any scale.

Evolution of modal properties in the non-proportionally damped coupled vehicle–bridge system

The modal properties of bridges are crucial parameters in engineering applications, and their variation caused by moving vehicles has been increasingly recognized in recent years. However, existing closed-form analytical expressions for the varying modal properties during vehicle passage exist only for simple structural configurations. This paper proposes an innovative method for analyzing the instantaneous modal properties of the system, considering general boundary conditions and the damping effect. First, only general boundary conditions are considered, and the results show that the location of the maximum system frequency shift depends on the mode shape. Then the damping effect is considered, observing that the system is non-proportionally damped in most cases, making it difficult to obtain a closed-form solution using existing methods. To solve this, we transform the system of second-order ordinary differential equations into a set of fourth-order ordinary differential equations and derive the closed-form solution for system’s modal properties. Based on this solution, we introduce the concept of Critical Coupling Damping (CCD), which defines the transition between the coupled description and the moving mass case. This work deepens understanding of changing modal properties during the traverse of sprung masses, and so has numerous potential engineering applications.

Mechanical Coupling Effect on Tide‐Induced Fluctuations in a Multilayer Aquifer System

ABSTRACTWe present a new analytical solution to study the mechanical coupling on tidal‐induced head fluctuations in a coastal aquifer system. The conceptual model consists of two overlying confined units that extend infinitely under the sea. The proposed model assumes that the tidal fluctuations in the overlying aquifer are described by the classical equation derived by van der Kamp. For the underlying aquifer, a closed form analytical expression is derived by solving a boundary value problem that considers the loading effect produced by the overlying unit. The amplitude and phase lag of the underlying aquifer depend on both the specific storage and the thickness of the overlying unit. A parametric study shows that the mechanical coupling effect can be important for both large values of specific storage and thickness of the overlying aquifer.

Improved PHY-layer authentication utilizing multi-modal features for mmWave MIMO UAV-enabled systems

This paper exploits multi-modal Physical (PHY)-layer features in terms of artificial fingerprint, In-phase/Quadrature (IQ) imbalance and Angle of Arrival (AoA) to propose a novel PHY-layer authentication framework for a Millimeter Wave (mmWave) Multiple-Input Multiple-Output (MIMO) Unmanned Aerial Vehicle (UAV)-enabled communication system. First, we resort to the AoA-based spatial fingerprint to effectively address the challenge of channel fingerprint instability induced by high-speed UAV mobility. To further enhance the low discriminability of hardware fingerprints caused by refined manufacturing techniques, artificial Gaussian noise is injected into the transmission signals to assist the receiver in better distinguishing between legitimate and illegitimate UAVs. Then, we jointly combine with inherent IQ imbalance and AoA features to design a hybrid authentication scheme and thus construct a multi-dimensional fingerprint space for a comprehensive characterization of UAV identities. To theoretically evaluate the effectiveness of the proposed authentication framework, the analytical closed-form expressions of performance metrics like false alarm and detection probabilities are also exactly derived based on the statistical signal processing technology and composite hypothesis testing. Finally, we provide large simulation results to validate the correctness and feasibility of the proposed theoretical models, and also discuss the relation between system security and communication service quality under different artificial fingerprint level.

Additional applications of the Lambert W function to solid state physics

Analytical solutions are always desirable in physics for the sake of mathematical exactness and physical insight. In this study, I obtain closed-form analytical expressions for various physical quantities taken from solid state physics. More precisely, I derive analytical expressions in terms of the Lambert W function for the work function and the field-enhancement factor of a field emitting material from the Fowler–Nordheim equation. Additionally, I derive similar analytical expressions for the localization length and the density of states in amorphous semiconductors from the Mott hopping conductivity. Similarly, I also derive analytical formulae based on the Lambert W function to compute the extrinsic-intrinsic transition temperature in a partially compensated semiconductor and the Kondo exchange coupling constant. All the obtained results are exact and explicit. Moreover, some of them allow a direct determination of some physical quantities of interest compared to their indirect determination from semi-logarithmic experimental plots. The findings of this paper are accessible and suitable for students enrolled in graduate solid state physics courses.

Fault tolerant motion planning for a quadrotor subject to complete rotor failure

Assuring safety of a quadrotor subject to rotor failure has been heavily investigated at the control level in view of fault tolerant control (FTC) approach. Yet, the existing FTCs are often concerned with tracking the reference motion even when that reference may not be safely trackable due to the physical constraints of the quadrotor. This paper tackles the faulty quadrotor safety at the planner level, proposing a fault tolerant motion planner. Starting from the formal backward reachability problem formulation, the proposed motion planner generates the time trajectory of the coupled rotational and translational motions that safely guide the faulty quadrotor. The generated trajectory is theoretically guaranteed to be tracked by the embedded FTC without violating the physical constraints. Further, the trajectory is prescribed as an analytical closed-form expression and thus suitable for real-time emergency maneuvers. The effectiveness of the proposed motion planner is numerically validated in conjunction with the different FTC techniques and compared to the existing planning method. The simulation results clearly signify that the proposed planner can successfully complement the fault tolerance of quadrotor. The supplements including code implementations are available on GitHub repository: https://github.com/HMCL-UNIST/Fault-tolerant-motion-planner.

Effective secrecy throughput optimization for GTLS-based UWOC systems with eavesdropper outage constraint.

In this work, the effective secrecy throughput (EST) which explicitly captures both the reliability and secrecy features of wiretap channels for an underwater wireless optical communication (UWOC) system has been optimized by considering the generalized transmit laser selection (GTLS) scheme. Specifically, the instantaneous channel state information (CSI) of the main channel and eavesdropper's channel is assumed to be unavailable at the transmitter, which is more practical for the underwater application. The fading of salinity induced oceanic turbulence channels is described by the Weibull distribution considering angular pointing error. The GTLS scheme is first introduced into the UWOC eavesdropping system to improve the quality of the main channel. Based on these models, the analytical closed-form expression of the EST with a constraint on the maximum allowed secrecy outage probability (SOP) is mathematically derived in terms of Meijer's G function and Gauss hypergeometric function. Then, an optimization problem with the objective of maximizing the constrained EST of the proposed UWOC system while ensuring wiretap code rate constraints are formulated. To solve this problem, a whale optimization algorithm (WOA) based wiretap code rates optimization method is effectively developed to acquire the maximum constrained EST, which is validated by the exhaustive search (ES) method. Simulation results suggest that the maximum constrained EST of this UWOC system can be significantly improved by the GTLS scheme. Besides, the maximum allowed SOP has a pronounced impact on the maximum constrained EST. The presented results could offer valuable insights into the physical layer security (PLS) performance enhancement of UWOC systems.

Performance analysis of CSI independent cache-aided NOMA ordering

This paper introduces a Non-Orthogonal Multiple Access (NOMA) ordering technique, called the region-based ordering method, which operates independently of channel state information (CSI) for vehicular-to-infrastructure (V2I) networks. Caching has been Integrated with NOMA to significantly enhance outage probability, ergodic sum rate, and latency performance compared to traditional NOMA systems. The performance of the proposed method is evaluated under various cache content availability scenarios and compared to dynamic ordered NOMA, which relies on instantaneous CSI for user ordering. A closed-form analytical expression for outage probability is derived for a Rayleigh fading channel and validated through Monte Carlo simulations. The analysis demonstrates that the proposed method is as effective as the dynamic ordering method. Additionally, cache-aided NOMA significantly improves outage performance and reduces delivery delay compared to conventional NOMA.

Physical layer security in SWIPT-based cooperative vehicular relaying networks

The future of autonomous transportation systems depends on energy sustainability and secure information exchange from low-power vehicular sensors, hence the increased interest in vehicular sensor charging using simultaneous wireless information and power transfer (SWIPT). This study investigates the physical layer security of a SWIPT-based radio frequency energy harvesting cooperative vehicular relaying network subjected to cascade Nakagami-m and double Nakagami-m (DN) fading channels. In the considered system model, a stationary source communicates with a mobile destination through a power-splitting-based decode-and-forward relay in the presence of a mobile passive eavesdropper. Based on the Gamma-distributed first term of the Laguerre series, new statistical probability density function (PDF) and cumulative distribution function (CDF) expressions for the DN are derived to accurately model the complex cascaded fading scenario. The secrecy performance metrics analyzed are the secrecy outage probability (SOP), the probability of non-zero secrecy capacity (PNZSC), and the intercept probability (IP). In addition, the asymptotic SOP (ASOP) is investigated in the high signal-to-noise ratio (SNR) to enhance the comprehension of the secrecy performance. Based on the derived ASOP, the secrecy diversity order (SDO) of the proposed system is determined and examined. Particularly, we present analytical closed-form expressions for the secrecy performance metrics and provide a detailed understanding of the impact of the system parameters under the cascade fading scenario. Then, a power splitting (PS) optimization problem is formulated to minimize the SOP. The results demonstrate a reduction in the SOP with the proposed PS scheme compared to the equal PS scheme. The obtained analytical findings are validated using Monte Carlo simulations.

Performance Evaluation of UOWC Systems from an Empirical Channel Model Approach for Air Bubble-Induced Scattering.

Underwater optical wireless communication (UOWC) systems provide the potential to establish secure high-data-rate communication links in underwater environments. The uniqueness of oceanic impairments, such as absorption, scattering, oceanic turbulence, and air bubbles demands accurate statistical channel models based on empirical measurements for the development of UOWC systems adapted to different types of water and link conditions. Recently, generalized Gamma and a mixture of two generalized Gamma probability density functions (PDF) were proposed to describe the statistical behavior of small and large air bubbles, respectively, when considering several levels of particle-induced scattering. In this paper, we derive novel closed-form analytic expressions to compute the bit error rate (BER) and outage performance using both proposed PDFs for various scattering conditions. Furthermore, simple asymptotic expressions are obtained to determine the diversity order of each scenario. Monte Carlo simulation results verify the obtained theoretical expressions. Our results also reveal that UOWC systems present lower BER and outage performance under more turbid water cases with respect to the tap water case due to the higher diversity order and despite the significant increases in pathloss at short link distances. Particle-induced scattering provides an inherent mechanism of turbid waters to mitigate air bubble-induced fluctuations and light blockages.

On the electromechanical bandgap of microscale vibration isolation metamaterial beams with flexoelectric effect

In this paper, a type of metamaterial isolation beam consisting a cantilever beam with flexoelectric film attached is provided. Electrodes interconnect on the surface to create parallel resonant shunt circuits, enabling band-stop filtering. The electromechanical control equations are derived using the Hamiltonian principle. A closed-form analytical expression for the bandgap range is obtained via the root locus method. Theoretical results are validated through finite element methods. Findings reveal that the operational bandgap of millimeter-scale flexoelectric metamaterial isolation beams is 5.2 times greater than that of conventional piezoelectric metamaterial beams. Furthermore, the vibration excitation amplitude is reduced by 99.9%. These results highlight the significant advantages of flexoelectric materials for microscale high-frequency passive isolation, providing superior stability in complex vibration environments. The research investigates the effects of end mass, electrode plate count, and external resistance on the vibration isolation performance of flexoelectric metamaterial beams. The study also examines the variations in relative bandgap width between flexoelectric and piezoelectric metamaterial beams concerning material, size, and structural parameters, identifying optimal values. Notable differences in bandgap characteristics are observed between flexoelectric and piezoelectric beams; for instance, while piezoelectric beams have an optimal substrate elastic modulus, flexoelectric beams do not, and the optimal film thickness for flexoelectric beams is 2.2 times that for piezoelectric beams.

Machine learning for the identification of phase transitions in interacting agent-based systems: A Desai-Zwanzig example.

Deriving closed-form analytical expressions for reduced-order models, and judiciously choosing the closures leading to them, has long been the strategy of choice for studying phase- and noise-induced transitions for agent-based models (ABMs). In this paper, we propose a data-driven framework that pinpoints phase transitions for an ABM-the Desai-Zwanzig model-in its mean-field limit, using a smaller number of variables than traditional closed-form models. To this end, we use the manifold learning algorithm Diffusion Maps to identify a parsimonious set of data-driven latent variables, and we show that they are in one-to-one correspondence with the expected theoretical order parameter of the ABM. We then utilize a deep learning framework to obtain a conformal reparametrization of the data-driven coordinates that facilitates, in our example, the identification of a single parameter-dependent ordinary differential equation(ODE) in these coordinates. We identify this ODE through a residual neural network inspired by a numerical integration scheme (forward Euler). We then use the identified ODE-enabled through an odd symmetry transformation-to construct the bifurcation diagram exhibiting the phase transition.

Mathematical analysis of batch reactor performance for the enzymatic synthesis of cephalexin: Laplace Homotopy perturbation method and Adomian decomposition method

In this article, a mathematical model of diffusion reaction in kinetically controlled cephalexin synthesis in the batch reactor with penicillin acylase immobilized in glyoxyl-agarose is analyzed. The kinetic model is a non-linear non-stead-state reaction–diffusion equation with non-linear terms related to the Fick’s law. We have presented the approximate analytical expression for the non-steady-state non-linear reaction–diffusion equation by utilizing the Laplace homotopy perturbation method (LHPM) and for steady-state by using LHPM and Adomian decomposition method (ADM). The approximate analytical solution obtained from these methods proved that they are fit for every values of the reaction–diffusion and kinetic parameters. We also present the numerical solution of the considered reaction–diffusion equation by using pdepe tool in MATLAB software. When comparing the semi-analytical solution with the numerical solution, a satisfactory result is noted for all the possible values of the parameters. The closed-form analytical expressions for the effectiveness factor in non-steady and steady-state conditions are obtained and analyzed using LHPM and ADM.

The resurgence of the plateau in supersymmetric N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 Jackiw-Teitelboim gravity

Significant progresses have been made recently in understanding the spectral form factor of Jackiw-Teitelboim gravity, particularly at late times where non-perturbative effects are expected to play a dominant role. By focusing on a peculiar regime of large time and fixed temperature, called τ-scaling limit, it was found that it is possible to analytically investigate the late-time plateau directly through the gravitational genus expansion. We extend this analysis to the supersymmetric N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 generalization of the bosonic theory, revealing an interesting structure. First, we notice that in the τ-scaling limit the perturbative sum over genera truncates after a single term, which solely accounts for the ramp behaviour. Instead a non-perturbative completion, responsible for the plateau, is encoded into an exact formula coming from the properties of the chiral gaussian ensemble, governing the spectral properties of the supersymmetric theory. We are able to recover the non-perturbative contributions by slightly deforming the genus of the involved surfaces and using resurgence theory. We derive a closed-form analytical expression for the late-time plateau and a trans-series expansion that captures the ramp-plateau transition.

On the transmission performance of DS-CDMA-based drone swarm

In drone swarm communication, multi-user interference can occur in the transmission within the drone swarm due to the randomness of the drone positions. This article mainly investigates the transmission performance of drone swarm communication, including outage probability and ergodic capacity, using direct sequence code division multiple access (DS-CDMA). In this work, the randomness of the drone position distribution is modeled using random geometry theory. Specifically, the author uses wireless communication modeling and stochastic geometry theories to establish a signal model using DS-CDMA technology. Next, closed-form analytical expressions for outage probability and asymptotic analysis are derived. Based on this, the influence of system factors such as DS-CDMA parameters and transmission distance on the transmission performance is summarized, and numerical results are provided to investigate the effect of the randomness of drone position distribution on the transmission performance between target communication nodes. In addition, the comparison between theoretical numerical results and Monte Carlo simulation results also verifies the correctness of the transmission performance analysis model proposed in this paper.