Analytical Formulas Research Articles

Propagation properties of a partially coherent electromagnetic hyperbolic-sine-Gaussian vortex beam through anisotropic atmospheric turbulence.

Using the extended Huygens-Fresnel principle and the Rytov approximation, the analytical formula for the propagation of a partially coherent electromagnetic hyperbolic-sine-Gaussian vortex beam (PCEShVB) in anisotropic atmospheric turbulence has been theoretically derived. Detailed studies have been conducted on the evolution characteristics of the average intensity, the degree of coherence (DOC), and the degree of polarization (DOP) of the beam in turbulence. The results show that during propagation, the intensity distribution of the beam will exhibit a spiral structure, and the overall distribution of the light spots will rotate in a direction related to the sign of the topological charge. The DOC distribution of PCEShVB will display a pattern reminiscent of beam interference fringes with an increase in propagation distance, with the number of "interference fringes" greatly impacted by the hyperbolic sine parameter. Furthermore, PCEShVB with a large initial coherent length and hyperbolic sine parameter will increase the degree of separation of the spots and yield a large DOP. Finally, for the validation of the theoretical findings, the random phase screen method was employed to simulate the propagation of PCEShVB through anisotropic atmospheric turbulence. The studies revealed a consistent alignment between the simulation results and the theoretical predictions.

The effect of loss/gain and Hamiltonian perturbations of the Ablowitz—Ladik lattice on the recurrence of periodic anomalous waves

Using the finite gap method, in this paper we extend the recently developed perturbation theory for anomalous waves (AWs) of the periodic nonlinear Schrödinger (NLS) type equations to lattice equations, using as basic model the Ablowitz–Ladik (AL) lattices, integrable discretizations of the focusing and defocusing NLS equations. We study the effect of physically relevant perturbations of the AL equations, like linear loss, gain, and/or Hamiltonian corrections, on the AW recurrence, in the simplest case of one unstable mode. We show that these small perturbations induce O(1) effects on the periodic AW dynamics, generating three distinguished asymptotic patterns. Since dissipation and higher order Hamiltonian corrections can hardly be avoided in natural phenomena involving AWs, we expect that the asymptotic states described analytically in this paper will play a basic role in the theory of periodic AWs in natural phenomena described by discrete systems. The quantitative agreement between the analytic formulas of this paper and numerical experiments is excellent.

Sampling rate requirement for accurate calculation of heart rate and its variability based on the electrocardiogram.

Objective. To develop analytical formulas which can serve as quantitative guidelines for the selection of the sampling rate for the electrocardiogram (ECG) required to calculate heart rate (HR) and heart rate variability (HRV) with a desired level of accuracy.Approach. We developed analytical formulas which relate the ECG sampling rate to conservative bounds on HR and HRV errors: (i) one relating HR and sampling rate to a HR error bound and (ii) the others relating sampling rate to HRV error bounds (in terms of root-mean-square of successive differences (RMSSD) and standard deviation of normal sinus beats (SDNN)). We validated the formulas using experimental data collected from 58 young healthy volunteers which encompass a wide HR and HRV ranges through strenuous exercise.Main results. The results strongly supported the validity of the analytical formulas as well as their tightness. The formulas can be used to (i) predict an upper bound of inaccuracy in HR and HRV for a given sampling rate in conjunction with HR and HRV as well as to (ii) determine a sampling rate to achieve a desired accuracy requirement at a given HR or HRV (or its range).Significance. HR and its variability (HRV) derived from the ECG have been widely utilized in a wide range of research in physiology and psychophysiology. However, there is no established guideline for the selection of the sampling rate for the ECG required to calculate HR and HRV with a desired level of accuracy. Hence, the analytical formulas may guide in selecting sampling rates for the ECG tailored to various applications of HR and HRV.

A reduced-order-model-based equivalent circuit for piezoelectric micro-electro-mechanical-system loudspeakers modeling.

Piezoelectric micro-electro-mechanical-system (MEMS) speakers are emerging as promising implementations of loudspeakers at the microscale, as they are able to meet the ever-increasing requirements for modern audio devices to become smaller, lighter, and integrable into digital systems. In this work, we propose a finite element model (FEM)-assisted lumped-parameters equivalent circuit for a fast and accurate modeling of these types of devices. The electro-mechanical parameters are derived from a pre-stressed FEM eigenfrequency analysis, to account for arbitrarily complex geometries and for the shift of the speaker resonance frequency due to an initial non-null pre-deflected configuration. The parameters of the acoustical circuit are instead computed through analytical formulas. The acoustic short-circuit between the speaker front and rear sides is taken into account through a proper air-gaps modeling. The very good matching in terms of radiated sound pressure level among the equivalent circuit predictions, FEM simulations, and experimental data proves the ability of the proposed method to accurately simulate the speaker performance. Moreover, due to its generality, it represents a versatile tool for designing piezoelectric MEMS speakers.

A Theory for the Balance between Warm Rain and Ice Crystal Processes of Precipitation in Mixed-Phase Clouds

Abstract Mixed-phase clouds contain both supercooled cloud liquid and ice crystals. In principle, precipitation may be initiated either by the liquid phase or by the ice phase. Ice crystals may grow by vapor diffusion to become snow (“ice crystal process”), forming “cold” precipitation. Equally, cloud droplets, when large enough, coalesce to form “warm” precipitation by the “warm rain process.” Warm rain could be supercooled and freeze as “warm” graupel. In the present paper, a new simplified theoretical analysis is provided to examine the microphysical system consisting of three species of hydrometeor, namely, cloud liquid, “cold ice” (crystals, snow), and “warm rain” (frozen or supercooled). This is obtained by nondimensionalizing and simplifying the evolution equations for the mass of each species. Analytical formulas are given for equilibria. Feedback analysis shows that the sign of the feedback is linked to the abundance of precipitation, with a neutral surface in the 3D phase space. The system’s precipitation amount explodes while in the initial unstable regime, crossing the neutral surface and approaching the equilibrium point that is a stable attractor. Positive and negative feedbacks are elucidated. In a standard case, the cold ice mass is about 1000 times larger than the warm rain mass. To illustrate the physical behavior of the theory, sensitivity tests are performed with respect to environmental conditions (e.g., aerosol, updraft speed) and microphysical parameters (e.g., riming and sedimentation rates for cold ice). Cold ice prevails, especially in fast ascent, due to its low bulk density, favoring slow sedimentation and a wide cross-sectional area for riming. Significance Statement The theory elucidates how the ice phase can prevail in the precipitation from any mixed-phase clouds with supercooled cloud liquid and crystals. The ice phase radically suppresses cloud liquid by riming when active and “wins” the competition against coalescence. This prevalence of ice is shown to arise from the low bulk density of snow. The cloud is viewed as a system of negative and positive feedbacks that prevail in realms of stability and instability in a 3D phase space.

Passive damping of vibrations of prestressed thin-walled structures using piezoelectric elements connected to electric circuits

In this paper we estimate the influence of static pressure on the efficiency of passive damping of vibrations of a circular cylindrical shell using an open piezoelectric ring connected to a passive electric circuit of different configurations. The mathematical formulation of the problem of the dynamic behavior of an electroelastic structure is based on the variational principle of virtual displacements, which is written in matrix form taking into account the prestressed non-deformed state caused by applied loads, and as well as the work done by external surface forces, inertial forces and external electric field to transfer the free charge. The resulting equation is supplemented by the equation for the electric circuit, which relates the voltage at the piezoelectric ring with the corresponding electric charge. The problem is solved in the framework of 3D formulation using the finite element method. As examples, we consider the serial RL- and resistive R-circuits. Their parameters providing the most effective damping of free vibrations in terms of their decay rate are determined by solving the optimization problem. The results obtained by numerical simulations are compared with the results calculated by the analytical formulas at different values of static pressure.

Analytical and numerical modelling of the electrostatic behaviour of highly insulating materials in the time domain

The usual response of an insulator subjected to a voltage step involves time power laws. We present here mathematical tools allowing to calculate this time domain response in open circuit after an initial charge deposit, within the framework of linear systems theory, using linear fractional transfer functions. In the time domain, the inverse Laplace transform of the data taken from the frequency domain leads to Mittag-Leffler functions, generalizing the Debye exponential response to an extended fractional α-order response. The open-circuit boundary conditions are different from the closed-circuit ones. We nevertheless demonstrate that using a transfer function deduced from the Cole-Cole response in closed-circuit, a precise analytical formula of the potential decay after an initial charge deposit may be established, and a numerical computation of this decay may be performed using easily available software. Applying the superposition principle, the voltage return following a brief short circuit may also be deduced. Experimental results are presented and the limits of the superposition principle applied to real materials are discussed.

Transient slow motion of a porous sphere

The start-up creeping motion of a porous spherical particle, which models a permeable polymer coil or floc of nanoparticles, in an incompressible Newtonian fluid generated by the sudden application of a body force is investigated for the first time. The transient Stokes and Brinkman equations governing the fluid velocities outside and inside the porous sphere, respectively, are solved by using the Laplace transform. An analytical formula for the transient velocity of the particle as a function of relevant parameters is obtained. As expected, the particle velocity increases over time, and a particle with greater mass density lags behind a corresponding less dense particle in the growth of the particle velocity. In general, the transient velocity is an increasing function of the porosity of the particle. On the other hand, a porous particle with a higher fluid permeability will have a greater transient velocity than the same particle with a lower permeability, but may trail behind the less permeable particle in the percentage growth of the velocity. The acceleration of the porous particle is a monotonic decreasing function of the elapsed time and a monotonic increasing function of its fluid permeability. In particular, the transient behavior of creeping motions of porous particles may be much more important than that of impermeable particles.

Polarization of disk electrodes in high-conductivity electrolyte solutions.

We investigate the polarization of disk electrodes immersed in an electrolyte solution and subjected to a small external AC voltage over a wide range of frequencies. A mathematical model is developed based on the Debye-Falkenhagen approximation to the coupled Poisson-Nernst-Planck equations. Analytical techniques are used for predicting the spatial distribution of the electric potential and the complex impedance of the system. Scales for impedance and frequency are identified, which lead to a self-similar behavior for a range of frequencies. Experiments are conducted with gold electrodes of sizes in the range 100-350 μm immersed in a high-conductivity KCl solution over five orders of magnitude in frequency. A collapse of data on impedance magnitude and phase angle onto universal curves is observed with scalings motivated by the mathematical model. A direct comparison with the approximate analytical formula for impedance is made without any fitting parameters, and a good agreement is found for the range of frequencies where the analytical model is valid.

The Study on the Critical Temperature and Gap-to-Tc Ratio of Yttrium Hydride Superconductors

This study investigates the gap-to-Tc ratio (R) of yttrium hydride superconductors within the weak coupling limit. We derived an analytical formula for the gap-to-Tc ratio. The ratio of the gap-to-Tc is dependent on the pressure applied to each superconductor. The maximum ratio, approximately 3.85, is observed in one superconductor, while the lowest ratio, roughly 3.21, is found in another superconductor. Based on the findings of our study, it can be deduced that yttrium hydride superconductors exhibit attributes commonly associated with weak-coupling superconductors. The influence of the Coulomb potential is more pronounced at a critical temperature compared to the ratio of the gap to the critical temperature.

Delay-Aware resource allocation to increase efficiency over LTE network for M2M communications in a noisy channel

Currently, the predominant source of Internet traffic stems from interpersonal user interactions, commonly referred to as user-to-user or H.2.H communications, facilitated through computer systems or mobile devices. The Internet of Things (IoT) is a transformative concept that has gained prominence over the past few years. It represents a paradigm shift from isolated systems to a vast interconnected network of devices that are capable of collecting environmental data, processing it, and making informed decisions based on the processed information. Therefore, it is imperative to have an effective resource scheduler in place for the management of an Long-Term Evolution (L.T.E) network system that encompasses both Machine-to-Machine (M.2.M) devices and Human-to-Human (H.2.H) users. In order to develop analytical formulas for assessing the performance of the suggested programs in respect to average waiting time, average system delay, and average number in the system for M.2.M and H.2.H users, this study uses the M / G / 1 priority queue model. It is possible to provide M.2.M consumers with good Quality of Service (QoS) while maintaining the targeted QoS levels for H.2.H users, according to the analysis of the collected data. The relaxing method is employed to cater to the needs of users in the M.2.M queue. To provide QoS for delay-sensitive traffic generated by H.2.H users, a classification scheme is employed to categorize H.2.H traffic into two primary classes. The findings show that the proposed systems have provided 87.34% improvement in machine-to-machine (M.2.M) performance while managing 92.26% QoS for H.2.H services.

A Rapid Single-view Radar Imaging Method with Window Functions

Monostatic rapid single-view radar imaging technology is a technique that employs single incidence angle and single frequency point information to implement rapid monostatic radar imaging within a small angular field. Owing to its analytical expression, this technique can substitute the traditional frequency-angle-scanning imaging in a small angular range, facilitating the rapid generation of highly realistic radar imaging data slices for complex targets and environments. This technology has been significantly applied in scatter hotspot diagnostics and target recognition. In order to achieve the windowing effect equivalent to that of frequency-angle-scanning imaging, and to enhance the scattering feature of monostatic imaging while controlling sidelobes, this paper derives analytic windowed imaging formulas for monostatic radar. It then obtains analytical expressions for various typical monostatic windowing rapid radar imaging scenarios. This enables the monostatic rapid imaging technology to maintain high efficiency in its analytical expressions while achieving the windowing effect equivalent to traditional imaging. The validity and correctness of the analytical formula and software implementation have been confirmed through 1D, 2D, and 3D imaging verifications. This technology can provide a vast amount of training data for modern radars.

A New Analytical Solution for Calculating Rock Block Volume

The study of rock slope stability and evolution suffers from many uncertainty factors related to block size and shape, and slope morphology. While nothing can be done to remove the aleatory component of these uncertainties, efforts in reducing the epistemic ones are desirable. This research aims to propose a new analytical solution for calculating rock block volume in the case of three discontinuity sets whose orientation and true spacing are known. Researchers and practitioners can take advantage of such a correct analytical formula thanks to its easiness of use: guidelines based on stereogram are provided in order to explain how to obtain the required input data. The correctness of the equation is demonstrated by comparing the results of the new solution applied to 12 theoretical blocks with those obtained with 3DEC (Itasca Consulting Group). Also, the differences with respect to results obtained with the well-known Palmstrøm’s formula are reported. The new methodology is applied to the case study of Elva valley road (Northern Italy), which is overhung by steep rocky cliffs and is subject to the consequences of frequent rockfall phenomena. The results are used to discuss the proposed method’s applicability: while it is evident that such a formula is not able to compete with the great potentiality of DFNs, this user-friendly tool can quickly and at no cost assess rock block volume in rockfall or rock slope stability studies.

Analytical and experimental study on shaking effects for improved stone column foundations

Stone column technology is a method used to reduce liquefaction in saturated sediments. By using materials with greater permeability than the surrounding soil, stone columns can help drain excess pore water pressure and mitigate liquefaction potential during seismic loading. In this study, analytical solutions are proposed to address the liquefaction problem in saturated soils during earthquakes, taking into account the influence of stone columns and the surrounding soil. These solutions consider the excess pore water pressure at the boundary of the zones and analyze the vertical and horizontal drainages. The study also includes centrifuge tests on sandy soils improved with stone columns, calculating the total seepage flow and converting it into settlement based on the relationship between drainage flow rate and volumetric changes in the soil. The calculated settlement aligns well with the measured value in the centrifuge test, confirming the accuracy of the analytical formula for calculating settlement in stone column foundations. The study also evaluates the combined effects of radial and vertical drainages in stone column foundations using the derived analytical solutions.

New orthogonality relationships of the Gaunt coefficients

A new analytical formula for the Gaunt coefficients is derived using expressions of Clebsch Gordan coefficients in terms of the generalized hypergeometric functions of unit argument and binomial coefficients. In addition, new sum expressions and orthogonality relations containing the Gaunt coefficients are written. These expressions are used to test the accuracy of the numerical calculation for the Gaunt coefficients. These formulae are coded in the Mathematica programming language, and the Gaunt coefficients are calculated numerically and compared with the values from the literature.

Analytic description of the gas flow around planets embedded in protoplanetary disks

Context. A growing planet embedded in a protoplanetary disk induces three-dimensional gas flow, which exhibits a midplane outflow that can suppress dust accretion onto the planet and form global dust substructures (rings and gaps). Aims. Because analytic formulae for the planet-induced outflow are useful for modeling its influences on local and global dust surface densities and planet accretion, we derived analytic formulae that describe the morphology and velocity of the planet-induced outflow. Methods. We first performed three-dimensional, nonisothermal hydrodynamical simulations of the gas flow past a planet, which enabled us to introduce a fitting formula that describes the morphology of the outflow. We then derived an analytic formula for the outflow speed using Bernoulli’s theorem. Results. We successfully derived a fitting formula for the midplane outflow morphology (the shape of the streamline), which is valid when the dimensionless thermal mass falls below m ≲ 0.6. The obtained analytic formulae for the outflow, such as the maximum outflow speed and the velocity distributions of the outflow in the radial and vertical directions to the disk, show good agreement with the numerical results. We find the following trends: (1) the maximum outflow speed increases with the planetary mass and has a peak of ~30–40% of the sound speed when the dimensionless thermal mass is m ~ 0.3, corresponding to a super-Earth mass planet at 1 au for the typical steady accretion disk model, and (2) the presence of the headwind (namely, the global pressure force acting in the positive radial direction of the disk) enhances (reduces) the outflow toward the outside (inside) of the planetary orbit. Conclusions. The planet-induced outflow of the gas affects the dust motion when the dimensionless stopping time of dust falls below St ≲ min(10 m2, 0.1), which can be used to model the dust velocity influenced by the outflow.

Linked shrinkage to improve estimation of interaction effects in regression models.

The addition of two-way interactions is a classic problem in statistics, and comes with the challenge of quadratically increasing dimension. We aim to a) devise an estimation method that can handle this challenge and b) to aid interpretation of the resulting model by developing computational tools for quantifying variable importance. Existing strategies typically overcome the dimensionality problem by only allowing interactions between relevant main effects. Building on this philosophy, and aiming for settings with moderate n to p ratio, we develop a local shrinkage model that links the shrinkage of interaction effects to the shrinkage of their corresponding main effects. In addition, we derive a new analytical formula for the Shapley value, which allows rapid assessment of individual-specific variable importance scores and their uncertainties. We empirically demonstrate that our approach provides accurate estimates of the model parameters and very competitive predictive accuracy. In our Bayesian framework, estimation inherently comes with inference, which facilitates variable selection. Comparisons with key competitors are provided. Large-scale cohort data are used to provide realistic illustrations and evaluations. The implementation of our method in RStan is relatively straightforward and flexible, allowing for adaptation to specific needs. Our method is an attractive alternative for existing strategies to handle interactions in epidemiological and/or clinical studies, as its linked local shrinkage can improve parameter accuracy, prediction and variable selection. Moreover, it provides appropriate inference and interpretation, and may compete well with less interpretable machine learners in terms of prediction.

Synthesis of Optimal Control of Spacecraft Angular Momentum for Spatial Turn Taking into Account Energy Costs Using Quaternions

Background: In this paper, we propose solving the specific original problem of control synthesis of spacecraft attitude. Optimization of the control program is made with the use of a new criterion of quality that combines energy costs and duration of reorientation under restrictions on control (the presence of a time factor limits the duration of slew maneuver). Methods: The construction of optimal control for angular momentum change is based on the quaternion method and L.S. Pontryagin maximum principle. An analytical solution to the problem was obtained on the base of a differential equation relating the orientation quaternion and angular momentum of a spacecraft. Results: Key properties of the optimal solution are formulated in analytical form; the features of optimal motion are studied in detail. The control law is formulated in the form of explicit dependence between control and phase variables. In a case when the controlling torque is limited by the given restriction (at the beginning and end of a turn), analytical formulas have been written for the duration of braking and acceleration. Main relations which determine optimal values of parameters of the algorithm for control of angular momentum are given. Examples and results of mathematical modeling of spacecraft motion formed by optimal control were given. This data in addition to the theoretical descriptions illustrates the process of reorientation in evident form and demonstrates the practical feasibility of a designed method for control of angular momentum during spatial turn. Conclusion: The designed optimal algorithm of control of spacecraft motion improves the efficiency of spacecraft attitude system, and originates more economical performance of spacecraft during flight on orbit.

α-particle preformation factors in heavy and superheavy nuclei* *Supported in part by the National Natural Science Foundation of China (12175100, 11975132), the Construct Program of the Key Discipline in Hunan Province, the Research Foundation of Education Bureau of Hunan Province, China (21B0402, 18A237, 22A0305), the Natural Science Foundation of Hunan Province, China (2018JJ2321), the Innovation Group of Nuclear and Particle

In this study, α-particle preformation factors in heavy and superheavy nuclei from 220Th to 294Og are investigated. By combing experimental α decay energies and half-lives, the α-particle preformation factors are extracted from the ratios between theoretical α decay half-lives calculated using the Two-Potential Approach (TPA) and experimental data. We find that the α-particle preformation factors exhibit a noticeable odd-even staggering behavior, and unpaired nucleons inhibit α-particle preformation. Moreover, we find that both the α decay energy and mass number of parent nucleus exhibit considerable regularity with the extracted experimental α-particle preformation factors. After considering the major physical factors, we propose a local phenomenological formula with only five valid parameters for α-particle preformation factors . This analytic expression has a clear physical meaning as well as good precision. As an application, this analytic formula is extended to estimate the α-particle preformation factors and further predict the α decay half-lives for unknown even-even nuclei with Z = 118 and 120.

Diborides of Multielement Transition Metals: Methods for Calculating Physical and Mechanical Characteristics

From the first principles simulation (using the method of “a priori pseudopotential” and the “quasi-harmonic approximation” method- author’s developments), the basic characteristics of diborides and diborides of multielement transition metals (DMTMs) with an AlB2 type structure were calculated. For both diborides and DMTMs, the linear coefficients of thermal expansion (LCTE) along the axial axes differ little from each other, i.e., transition metal diborides and hexagonal lattice DMTMs are quasi-isotropic. Quasi-isotropy makes it possible to estimate the LCTE using an analytical formula that depends on the melting temperature. In the absence of experimental data on the melting point of DMTMs, a method for calculating it from first principles is presented. The theoretical hardness values of transition metal diborides and DMTMs with averaged parameters were calculated from the first principles. The hardness of both bulk and nano-sized DMTMs was assessed using a hybrid method. There is agreement between the calculated and available experimental data.