Closed-form Analytical Expressions Research Articles

Effect of aberrations on the beam quality factor of Hermite-Gauss beams.

The effect of aberrations on the beam quality factor (M2) of Hermite-Gauss (HG) beams is examined. Using the method of moments, we derive closed-form analytical expressions of M2 due to astigmatism and spherical aberration. Our analysis reveals that the radius of the HG beams plays a significant role in determining the effect of the aberrations on M2. For each aberration, we establish a critical width that separates the region where M2 changes infinitesimally from the region where it changes sharply. The analytical results are validated through numerical simulations.

Geometric-algebraic approach to aqueous solutions of diprotic acids and its buffer mixtures

A closed-form analytical expression for [H3O+]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${[\\hbox {H}_{3}\\hbox {O}^{+}]}$$\\end{document} has been obtained for aqueous solutions of diprotic acids and its soluble salts. This formula allows to calculate the pH of aqueous solutions of diprotic acids, their buffer solutions, and the titrations of these two by a strong base, from the values of pK1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_1$$\\end{document}, pK2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_2$$\\end{document}, and the effective concentrations of the acid and the base, C¯a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{C}_\ extrm{a}$$\\end{document} and C¯b\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{C}_\ extrm{b}$$\\end{document} respectively. It is shown that a strong base titration of an acid, or its buffer solutions, is always a linear path in the C¯a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{C}_\ extrm{a}$$\\end{document}–C¯b\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{C}_\ extrm{b}$$\\end{document} plane, which allows a simple analysis of the pH stability of buffer solutions. The mathematical analysis of the equilibrium equations of the dissolution of a diprotic acid in water and the physical constraints allowed to obtain two approximate equations for the diprotic acids. One of the approximations is useful for acids with pK2-pK1≤log104\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{p}K_2-\ extrm{p}K_1\\le \\log _{10}4$$\\end{document}, the other for acids with pK2-pK1≤-log104\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{p}K_2-\ extrm{p}K_1\\le -\\log _{10}4$$\\end{document}.

On the capacity of a SIM-based cooperative NLOS UVC system with best relay selection

Ultraviolet communication (UVC) is a promising technology due to its ability to operate in non-line-of-sight (NLOS) mode thereby eliminating the pointing acquisition and tracking (PAT) requirement as needed by infrared and visible light communications. However, NLOS UVC suffers from very high attenuation and turbulence-induced fading when operated over a long distance. Due to these limitations, the existing literature on the NLOS UVC is mostly restricted to short-distance communications only. Therefore, this paper addresses these challenges by proposing an outdoor subcarrier-intensity-modulation (SIM) based multi-relay cooperative communication system employing the best relay selection and decode-and-forward (DF) relaying protocol. The turbulence-induced fading is modelled using lognormal distribution under weak atmospheric turbulence conditions. We derive novel closed-form analytical expressions of outage probability and ergodic capacity. Correctness of the derived analytical expressions is validated through numerical simulations.

The directional flow generated by peristalsis in perivascular networks-Theoretical and numerical reduced-order descriptions.

Directional fluid flow in perivascular spaces surrounding cerebral arteries is hypothesized to play a key role in brain solute transport and clearance. While various drivers for a pulsatile flow, such as cardiac or respiratory pulsations, are well quantified, the question remains as to which mechanisms could induce a directional flow within physiological regimes. To address this question, we develop theoretical and numerical reduced-order models to quantify the directional (net) flow induceable by peristaltic pumping in periarterial networks. Each periarterial element is modeled as a slender annular space bounded internally by a circular tube supporting a periodic traveling (peristaltic) wave. Under reasonable assumptions of a small Reynolds number flow, small radii, and small-amplitude peristaltic waves, we use lubrication theory and regular perturbation methods to derive theoretical expressions for the directional net flow and pressure distribution in the perivascular network. The reduced model is used to derive closed-form analytical expressions for the net flow for simple network configurations of interest, including single elements, two elements in tandem, and a three element bifurcation, with results compared with numerical predictions. In particular, we provide a computable theoretical estimate of the net flow induced by peristaltic motion in perivascular networks as a function of physiological parameters, notably, wave length, frequency, amplitude, and perivascular dimensions. Quantifying the maximal net flow for specific physiological regimes, we find that vasomotion may induce net pial periarterial flow velocities on the order of a few to tens of m/s and that sleep-related changes in vasomotion pulsatility may drive a threefold flow increase.

New analytical results in solid state physics using the Lambert W function

Analytical solutions in physics are always preferred for the sake of mathematical completeness. For this, using the Lambert W function, I derive closed-form analytical expressions for the equilibrium interionic distance in an ionic crystal, the formation energy of a vacancy in a crystal, the zero-temperature energy gap of a clean-limit superconductor and the critical Kosterlitz–Thouless temperature for the phase transition in 2D-XY model. Besides their theoretical interest, some of the present results suggest alternative experimental determinations of the relevant physical quantities. In addition, I similarly derive an explicit analytical formula based on the Lambert W function for a determination of the Boltzmann constant. This formula is proposed here as a theoretical basis for an experimental method to measure this constant. This method is suitable for undergraduate physics students.

Coverage Analysis of LoRa and NB-IoT Technologies on LPWAN-Based Agricultural Vehicle Tracking Application

This study focuses on the recently emerged Internet of Vehicles (IoV) concept to provide an integrated agricultural vehicle/machinery tracking system through two leading low power wide area network (LPWAN) technologies, namely LoRa and NB-IoT. The main aim is to investigate the theoretical coverage limits by considering the urban, suburban, and rural environments. Two vehicle tracking units (VTUs) have been designed for LoRa and NB-IoT connectivity technologies that can be used as reference hardware in coverage analysis. On this basis, the closed-form explicit analytical expressions of the maximum transmission range have been derived using the Hata path loss model. Besides, the computer simulation results have been validated via the maps from XIRIO online radio planning tool. In light of the obtained findings, several evaluations have been made to enhance the LPWAN-based agricultural vehicle tracking feasibility in smart farms.

Symplectic stiffness method for the buckling analysis of hierarchical and chiral cellular honeycomb structures

Structure-function-material integration design based on the buckling instability of flexible cellular structures has attracted much attention. According to the reversible and repeatable characteristics of buckling deformation, phonon bandgap switches, the autonomous drive of soft robots, programmable logic controllers, and reusable spacecraft landing buffer devices can be designed. Based on the Hamiltonian variational principle and symplectic eigen expansion, a theoretical analysis model of the symplectic stiffness method is established, and analytical expressions for the stiffness of several typical buckling modes of elastically supported beams are obtained. The analytical closed-form expressions for the macroscopic buckling strength of honeycombs are obtained according to the bending moment equilibrium equation. The buckling modes of regular hexagonal, hierarchical hexagonal, and tri-chiral honeycombs are obtained by numerical simulation and experimental verification, and some novel higher-order modes are observed. Uniaxial compression experiments reveal that these higher-order buckling modes can be triggered by adjusting the geometric parameters of the unit cells. The high agreement between experimental, numerical, and existing literature results for the failure surfaces of cellular structures verifies the effectiveness of the proposed symplectic stiffness method. In a word, the buckling modes can be designed to meet different functional requirements by changing the initial configuration of flexible cellular structures. This provides a new idea and theoretical guidance for the development of metamaterials and structures with tailored properties.

Liquid−vapor order parameter and coexistence-curve diameter of nitrogen, ethylene, and sulfur hexafluoride: From the triple point to the critical scaling regime

Analytic closed-form expressions are obtained for the liquid and vapor saturation densities defining the coexistence curve. The densities are modeled with broken power laws and Weibull distributions. Specifically, the coexistence curves of nitrogen, ethene and sulfur hexafluoride are derived, without the use of perturbative expansions, based on high-precision data extending from the triple point into the critical regime. The analytic continuation of the vapor branch below the triple point decays exponentially at low temperature. The order parameter and coexistence-curve diameter of the fluids are assembled from the regressed liquid and vapor branches of the coexistence curve, and the critical power-law scaling of these quantities is examined. The scaling exponents of the order parameter and the reduced diameter are regressed and compared with the calculated critical exponents of the 3D Ising universality class. Index functions representing the Log−Log slopes (temperature-dependent effective exponents) of the liquid and vapor densities, order parameter and diameter are used to determine the onset of the ideal power-law scaling regime and to illustrate the slope evolution of these quantities in the subcritical regime.

Strip-saturation model for arbitrary polarized electro-elastic material weakened by an eccentrically situated anti-plane semi-permeable crack

A strip-saturation model is studied to observe the effect of changing polarization axis on an electro-elastic strip, weakened by an eccentrically situated anti-plane semipermeable crack. The model is modified by considering that the saturated limit of electric displacement over rims of the saturation zone is quadratic interpolating polynomial times. The proposed model is formulated and then solved by using the Fourier integral transform technique and integral equations method. Closed-form analytic expressions are derived for developed saturation zone and fracture parameters viz. electric crack condition parameter, crack-sliding displacement, crack-opening potential drop, field intensity factors, energy release rate, etc. The influence of poling angle, prescribed electro-mechanical loadings, crack length, and material constants on fracture parameters are presented and analyzed graphically.

Quantum Learning on Structured Code With Computing Traps for Secure URLLC in Industrial IoT Scenarios

Resilient and secure ultra-reliable low-latency communications (URLLC) over radio interface is expected to play a crucial role in next generation Industrial IoT scenarios. However, attacking wireless pilot signals has been a potential easy way to interrupt URLLC services. In this work, we propose a random structured code to encode and decode pilot signals on multidimensional physical resources, and also design a quantum learning framework to make this code secure and reliable. Specifically, the code suggests using random encoding with little structures to disperse the effect of attacks. We find that the decoding process can be modeled as a computing trap if the group spatial channel features are employed. The security problem is therefore transformed as a random computing with redundancy. We employ a quantum algorithm to learn the computing trap model such that the computing redundancy can be removed quickly while the dispersed attack can be eliminated. In this respect, we can prove the existence of the quantum black-box model corresponding to the computing trap, and derive a precise expression of computing performance. Based on the result, we can formulate novel analytical closed-form expressions of system failure probability to characterize the reliability of the URLLC. Numerical results show that the proposed system can maintain ultra-high reliability and low latency against attacks on wireless pilots.

An Insight to the Outage Performance of Multi-Hop Mixed RF/FSO/UWOC System

In this paper, we investigate the outage performance of the three-hop mixed system integrating radio frequency (RF), free space optics (FSO), and under water optical communication (UWOC) system. The closed-form analytical expressions for the outage probability of the system are derived. In the considered system, the RF channel follows the Nakagami-m distribution, the FSO channel observes the Gamma-Gamma fading statistics, and the UWOC link experiences a mixture Exponential Generalized Gamma (EGG) fading distribution. To verify the derived analytical expressions, numerical simulations are also carried out, and we present the influence of the various link parameters such as path loss, atmospheric turbulence, pointing errors, angle-of-arrival fluctuations, water salinity, and scintillation on the performance of the decode and forward (DF) relayed multi-hop communication system.

A model for the initial stiffness of the face plate component

The component method for joint analysis relies on the formulation of stiffness and strength of individual parts to derive the global properties of the joint. One of these components is the web of an open I-section, or the face of a rectangular hollow section, hereby referred to as face plate. It is currently not codified despite its frequent occurrence in the engineering practice. Hereby, a new mechanical model is proposed to estimate the initial stiffness of the face plate component under out-of-plane loading, leading to closed-form analytical expressions. The model is validated against experimental test results and an extensive numerical parametric study, showing excellent agreement.

Closed-form analytic expressions for shadow estimation with brickwork circuits

Properties of quantum systems can be estimated using classical shadows, which implement measurements based on random ensembles of unitaries. Originally derived for global Clifford unitaries and products of single-qubit Clifford gates, practical implementations are limited to the latter scheme for moderate numbers of qubits. Beyond local gates, the accurate implementation of very short random circuits with two-local gates is still experimentally feasible and, therefore, interesting for implementing measurements in near-term applications. In this work, we derive closed-form analytical expressions for shadow estimation using brickwork circuits with two layers of parallel two-local Haar-random (or Clifford) unitaries. Besides the construction of the classical shadow, our results give rise to sample-complexity guarantees for estimating Pauli observables.We then compare the performance of shadow estimation with brickwork circuits to the established approach using local Clifford unitaries and find improved sample complexity in the estimation of observables supported on sufficiently many qubits.

Probability Distribution of Tree Age for the Simple Birth-Death Process, with Applications to Distributions of Number of Ancestral Lineages and Divergence Times for Pairs of Taxa in a Yule Tree.

In this contribution, a general expression is derived for the probability density of the time to the most recent common ancestor (TMRCA) of a simple birth-death tree, a widely used stochastic null-model of biological speciation and extinction, conditioned on the constant birth and death rates and number of extant lineages. This density is contrasted with a previous result which was obtained using a uniform prior for the time of origin. The new distribution is applied to two problems of phylogenetic interest. First, that of the probability of the number of taxa existing at any time in the past in a tree of a known number of extant species, and given birth and death rates, and second, that of determining the TMRCA of two randomly selected taxa in an unobserved tree that is produced by a simple birth-only, or Yule, process. In the latter case, it is assumed that only the rate of bifurcation (speciation) and the size, or number of tips, are known. This is shown to lead to a closed-form analytical expression for the probability distribution of this parameter, which is arrived at based on the known mathematical form of the age distribution of Yule trees of a given size and branching rate, which is derived here de novo, and a similar distribution which additionally is conditioned on tree age. The new distribution is the exact Yule prior for divergence times of pairs of taxa under the stated conditions and is potentially useful in statistical (Bayesian) inference studies of phylogenies.

Analytical Prediction of Static Flux-Linkage Characteristics of Switched Reluctance Machine

This paper deals with the analytical prediction of static flux-linkage characteristics of switched reluctance (SR) machines, which have emerged as competitive candidates in various applications, including automotive, aerospace, machine tools, and renewable energy. Specifically, this paper proposes a novel flux-tube-based approach to derive closed-form analytical expressions for the effective air-region permeance at the aligned and unaligned rotor positions in terms of machine dimensions. The proposed method does not require preliminary finite element analysis (FEA) and includes the effect of rotor pole height on the effective air-region permeance. The predictions by the proposed method are compared with the results of FEA and three existing analytical methods, considering seven different SRM designs and different sets of machine dimensions. The results of the proposed method are found to be accurate and comparable to those of FEA in all cases. Further, based on magnetic material characteristics and machine dimensions, the static flux-linkage characteristics at the aligned and unaligned rotor positions are predicted. The aligned and unaligned flux-linkage characteristics of a 10000-rpm, 5-kW SR machine, thus predicted, are verified through FEA and experiments. As an application of the proposed analytical method, electromagnetic design of a 40-kW, 80000-rpm, 600-V SRM is carried out and is validated through static FEA and simulations.

Blind reconfigurable intelligent surface-aided fixed non-orthogonal multiple access for intelligent vehicular networks

In intelligent vehicular networks, vehicles should be able to communicate with their surroundings while traveling. This results in more efficient, safer, and comfortable driving experiences, as well as new commercial prospects in a variety of industries. Connected vehicles and autonomous vehicles expect 100% reliable connectivity without any compromise in quality. However, due to challenges such as difficult channel terrains in urban scenarios and dead zones, the reliability of current vehicle-to-infrastructure (V2I) and vehicle-to-vehicle communication systems cannot be guaranteed. The performance of vehicular networks can be considerably enhanced with reconfigurable intelligent surfaces (RIS). Non-orthogonal multiple access (NOMA) allows for massive connectivity with the surroundings. In vehicular networks, the RIS-assisted NOMA can ensure regulated channel gains, better coverage, throughput, and energy efficiency. In this work, a blind RIS-assisted fixed NOMA (FNOMA) system is proposed for a downlink V2I scenario. The closed-form analytical outage probability and throughput expressions are derived by considering RIS as an intelligent reflector and as a roadside unit. It is observed that the analytical and Monte Carlo simulation results are closely related. In simulations, it has been discovered that RIS-assisted FNOMA outperforms the traditional NOMA variants in terms of outage and throughput. Even without precise channel knowledge, blind RIS transmission outperforms traditional NOMA variants due to huge array gain. The increase in the number of reflective elements also results in a significant improvement in signal-to-noise ratio gains.

Passive Cooling Method Analysis & Optimization of PV Solar Panel Heat Sink

The development of solar photovoltaic (PV) systems in arid climates is hindered primarily by high temperatures, which adversely affect system performance and lifespan. This manuscript presents a comprehensive approach to address this challenge by developing an analytical model for predicting the temperature of PV panels under a passive cooling system specifically designed for arid environments. The analytical model takes into account the crucial relationship between solar panel temperature and its conversion efficiency. By applying Kirchhoff's and Ohm's laws for a complex circuit, the model accurately calculates the heat flux within the solar panel system, allowing for the determination of temperatures for each layer in the system. The study deduces closed-form analytical expressions that describe the temperature, output power, and conversion efficiency of the solar panel as functions of various factors, including solar irradiance, ambient temperature, emissivity, wind velocity, tilt angle, and dimensions of fins. These expressions provide valuable insights into the thermal behaviour of the system and enable the evaluation of its performance under different environmental conditions.

Estimation of the dissipation due to radiation damping for the pile embedded in soil: A closed-form solution

In the present work, the overall energy dissipation of a pile embedded in the soil, modeled as visco-elastic springs, is thoroughly investigated. The term dissipation is used to measure the rate of decay of amplitude of displacement response in the time domain. The estimation of dissipation in piles is necessary to understand how quickly a pile reaches its steady state or equilibrium under dynamic loads or in cases of free vibration, respectively. A higher level of dissipation indicates a shorter duration of transient vibration, which is a deserving trait for vibration mitigation in extreme events, such as earthquakes, storms, blasts, shocks, etc. The dispersion relation which is a function of wavenumber and free wave frequency is obtained for the pile embedded in the soil medium by implementing the generalized form of Bloch’s theorem. The dispersion relationship is solved in the free-wave approach, in which a real wave number yields a set of complex frequencies. For predicting the overall dissipation of the system from the complex frequency values, an analytical closed-form expression is proposed in this paper. Further, the predicted dissipation is validated with a finite element-based numerical model of the soil–pile system.

An Eigenanalysis of Angle-Based Deformation Energies

Angle-based energies appear in numerous physics-based simulation models, including thin-shell bending and isotropic elastic strands. We present a generic analysis of these energies that allows us to analytically filter the negative eigenvalues of the second derivative (Hessian), which is critical for stable, implicit time integration. While these energies are usually formulated in terms of angles and positions, we propose an abstract edge stencil that succinctly parameterizes the edge deformation, and allows us to derive generic, closed-form analytical expressions for the energy eigensystems. The resultant eigenvectors have straightforward geometric interpretations. We demonstrate that our method is readily applicable to a variety of 2D and 3D angle-based elastic energies, including both cloth and strands, and is up to 7× faster than numerical eigendecomposition.

Performance Analysis of IRS-Assisted THz Communication Systems over α-μ Fading Channels with Pointing Errors.

In this paper, we analyze the performance of an intelligent reflecting surface (IRS)-aided terahertz (THz) wireless communication system with pointing errors. Specifically, we derive closed-form analytical expressions for the upper bounded ergodic capacity and approximate expression of the outage probability. We adopt an α-μ fading channel model for our analysis that is experimentally demonstrated to be a good fit for THz small-scale fading statistics, especially in indoor communication scenarios. In the proposed analysis, the statistical distribution of the α-μ fading channel is used to derive analytical expressions for the ergodic capacity and outage probability. Our proposed analysis considers not only the IRS reflected channels, but also the direct channel between the communication nodes. The results of the derived analytical expressions are validated through Monte Carlo simulations. Through simulations, it has been noticed that pointing errors degrade the performance of the IRS-assisted THz wireless communication system which can be compensated by deploying an IRS having a large number of reflecting elements.