Analytical Equations Research Articles

Description of Excitonic Absorption Using the Sommerfeld Enhancement Factor and Band-Fluctuations

Abstract One of the challenges of excitonic materials is the accurate determination of the exciton binding energy and bandgap from optical measurements. The difficulty arises from the overlap of the discrete and continuous excitonic absorption at the band edge. Many researches have modeled the shape of the absorption edge of such materials on the seminal formulation proposed by Elliott in 1957 and its several modifications such as non-parabolic bands, magnetic potentials and electron-hole-polaron interactions. However, exciton binding energies obtained from optical absorption often vary strongly depending on the chosen ``Elliott formula". Here, we propose an alternative and rather simple approach, which has previously been successful in the determination of the optical bandgap of amorphous, direct and indirect semiconductors, based on the band-fluctuations model. In this model, the fluctuations due to disorder, temperature or lattice vibrations give rise to the well known exponential shape of band tail states. The formulation results in an analytic equation for the fundamental absorption with 6 parameters only. To test it, the binding energy and optical bandgap of GaAs and the family of tri-halide perovskites (MAPbX3), X=Br,I,Cl, over a wide range of temperatures, are obtained by fitting the modified Elliott model. The results for the bandgap, linewidth and exciton binding energy are in good agreement with reports based on non-optical measurements. Moreover, due to the polar nature of perovskites, the retrieved binding energies can be compared with those computed with a model proposed by Kane et al. (1978). In the latter model, the exciton is surrounded by a cloud of virtual phonons interacting via the Frölich interaction. As a consequence, the upper bound for the binding energy of the exciton-polaron system can be estimated. These results are in good agreement with the optical parameters obtained with the proposed Elliott equation including band-fluctuations.

Design of optimum spacecraft reorientation control with a combined criteria of quality based on the quaternions

Specifical original problem of attitude controlling for spacecraft was proposed in this paper. Problem of optimal rotation from a known initial state in a prescribed spatial orientation was studied in detail (turnaround time is not fixed). Design of optimal program of reorientation is based on new indicator of quality that combines energy costs including the contribution of controlling torques and integral of rotary energy (in a known proportion) and reorientation time; presence of duration factor bounds time of rotation finish. To construct an optimal control of angular momentum changing, quaternionic method and the maximum principle were applied. Differential equation that relates spacecraft angular momentum and quaternion of spacecraft orientation is a base to obtain analytic solution to a problem. We reveal the properties of optimum control program analytically, and study key features of optimum motion in details. Also, we write the formalized equations, mathematical formulas to design optimal law for change of spacecraft’s angular momentum. Analytic relations and equations are given for finding the optimal solution. Control law (in as explicit dependence between phase variables and con-trolling variables has been formulated. Main relations determining optimum values of parameters for rotation control algorithm were given. The closed-form law for rotation was obtained for dynamically symmetric solids. Numerical example as well as results of mathematical modeling of spacecraft motion that formed using optimum control are presented. This data as addition to the made theoretic descriptions shows reorientation process (in virtual form) and demonstrates practical feasibility of the developed control method. A designed algorithm for optimal control of rotation improves an efficiency of attitude system, and originates more economical performing of space vehicle during its flight along orbit.

A consistent treatment of mixed‐phase saturation for atmospheric thermodynamics

AbstractIce processes are integral to the formation and maintenance of high‐level clouds and can contribute substantially to the buoyancy of deep moist convection. These processes are represented in modern microphysics parameterisations but are commonly ignored in theoretical treatments of moist thermodynamics. While many thermodynamic variables can be defined uniquely for saturation with respect to liquid water and ice, it is desirable to account for the presence of mixed‐phase condensate, which is common in real clouds. Here, a simple yet novel representation of mixed‐phase saturation is introduced. Using this approach, analytical equations are derived for a variety of thermodynamic variables under mixed‐phase conditions. These include mixed‐phase versions of the saturation vapour pressure, saturated lapse rates, isobaric wet‐bulb temperature, and equivalent potential temperature. A new formulation of the ice–liquid water potential temperature is also presented, together with a method for calculating the mixed‐phase pseudo wet‐bulb temperature and wet‐bulb potential temperature. In addition, two new thermodynamic quantities are introduced: the isobaric saturation‐point temperature, a mixed‐phase analogue of the dew‐point and frost‐point temperatures, and the lifting saturation level, a mixed‐phase analogue of the lifting condensation and lifting deposition levels. Differences between each mixed‐phase variable and its liquid‐ and ice‐only counterparts are quantified across a wide range of atmospheric conditions. A Python library for evaluating these variables, developed during the course of this work, is also briefly introduced.

Horizontal walls effect on the subsonic flow around airfoil by the image method

The aim of this work consists in developing a new numerical method and a calculation program making it possible to determine the effect of one and two horizontal walls on the distribution of the pressure ratio, and the aerodynamics coefficients of a subsonic flow around airfoils. The model is considered for the perfect gas. The calculation is presented by adding the effect of one and two walls on the discretization of the airfoils boundary to obtain a closed surface. The method is called by the image method. So for the case of the existence of one wall, we will see a single image of the main airfoil, and in the case of two walls, we will have an infinity of images. The equation system is obtained by considering the effect of all images using alternating digital series, whose convergence is ensured numerically. We therefore obtain a large system of equations, the resolution of which is done by the Khaletski method to obtain the velocities and pressure. The application is made for some practical interest’s airfoils. Since the airfoils boundary is given by tabulated values, the cubic spline interpolation is used to obtain an analytical equation of the upper and the lower surfaces. The comparison is made with the case of open air flow by extending the distance between the walls and the airfoil chord towards infinity.

Short Communication: Numerically simulated time to steady state is not a reliable measure of landscape response time

Abstract. Quantifying the timescales over which landscapes evolve is critical for understanding past and future environmental change. Computational landscape evolution models are one tool among many that have been used in this pursuit. We compare numerically modeled times to reach steady state for a landscape adjusting to an increase in rock uplift rate. We use three different numerical modeling libraries and explore the impact of time step, grid type, numerical method for solving the erosion equation, and metric for quantifying the time to steady state. We find that modeled time to steady state is impacted by all of these variables. Time to steady state varies inconsistently with time step length, both within a single model and among different models. In some cases, drainage rearrangement extends the time to reach steady state, but this is not consistent in all models or grid types. The two sets of experiments operating on Voronoi grids have the most consistent times to steady state when comparing across time step and metrics. On a raster grid, if we force the drainage network to remain stable, time to steady state varies much less with computational time step. In all cases we find that many measures of modeled time to steady state are longer than that predicted by an analytical equation for bedrock river response time. Our results show that the predicted time to steady state from a numerical model is, in many cases, more reflective of drainage rearrangement and numerical artifacts than the time for an uplift wave to propagate through a fixed drainage network.

Numerical analysis of stress distribution due to upper protective layer mining and its impact on gas extraction

Pressure-relief gas extraction through the floor directional long boreholes is an advanced and effective gas control technology for the upper protective layer of contiguous coal seams. This study systematically analyzed the effects of mining activities on stress distribution in the underlying strata and developed an analytical equation to calculate the permeability distribution of the protected layer under mining-induced conditions. The results indicate that the effective extraction radius of directional long boreholes increased by 186% as the mining distance of the upper protective layer extended from 50 to 150 m. Furthermore, increasing the borehole diameter from 89 to 153 mm led to a 14.8% improvement in gas extraction efficiency, while raising the negative pressure from 15 to 35 kPa resulted in a 19.6% increase in the effective extraction radius. These findings provide valuable guidance for optimizing borehole design parameters and gas extraction efficiency, ensuring safer and more effective gas control in coal mining operations.

Quantification of Salt Transports Due To Exchange Flow and Tidal Flow in Estuaries

AbstractTo understand mechanisms of salt intrusion in estuaries, we develop a semi‐analytical model, that explicitly accounts for salt transport by both exchange flow and tidal flow. This model, after calibration, successfully hindcasts hydrodynamics and salinity dynamics in three estuaries that have strongly different characteristics. We find, from analyzing the model results for these three estuaries, that salt transport processes by exchange flow and tidal flow interact through the subtidal stratification. Transport by exchange flow creates stratification, thereby generating a phase shift of tidal salinity with respect to the tidal flow, which is important for the magnitude of the tidal salt transport. Conversely, the strength of tidal currents determines the vertical mixing that breaks down stratification. A new analytical formulation is presented for the component of the salt transport driven by the depth‐averaged tidal flow. This salt transport is larger than the component associated with the vertical shear of the tidal current. Finally, a method that yields analytical equations that quantify the importance of different contributions to the salt transport using only primary information is developed using approximate solutions for the subtidal stratification. This method performs well for the estuaries considered.

The results of a magnetic field on flow past an inclined isothermal vertical plate with heat and varying mass diffusion

This work describes the influence of a magnetic environment on the circulation of matter and thermal energy over an accelerating advanced inclined isothermal sectional surface. The temperature is elevated to . The proximity intensity is increased to . The Laplace-transform approach is applied to solve the dimensionless analytical equation. In this research work different physical variables including thermal grashof number (Tg), mass grashof number (Tm), Schmidt number (Sc), Prandtl number(Pr), time(t), velocity profile ( ), temperature ( ), and intensity ( ) are investigated. The output of the graph produced by the Matlab software commands an energy equation, momentum equation, and concentration equation. The values are shown in an aligned pattern for a variety of flow variables. Diagrammatic representations of the fluid velocity profiles are provided. The velocity rises corresponding to different Tg and the velocity rises corresponding to different Tm. As the angle is decreased, the velocity increases differently. , As the different magnetic environment values lower, the velocity rises. Elevated values of angle ( ), Sc, Pr, and contribute to an enhancement in local skin friction, and an upsurge in Gr, Gc, and t leads to a reduction. A tabulated presentation of Nusselt quantities reveals a positive correlation with increasing Pr. Similarly, Sherwood quantities, as tabulated for various parameters, demonstrate a proportional increase with rising Sc.

Fatigue Life Prediction of 2024-T3 Al Alloy by Integrating Particle Swarm Optimization-Extreme Gradient Boosting and Physical Model.

The multi-parameter characteristics of the physical model pose a challenge to the fatigue life prediction of 2024-T3 aluminum (Al) alloy. In response to this issue, a parameter-solving method that integrates particle swarm optimization (PSO) with extreme gradient boosting (XGBoost) is proposed in this study. The fatigue performance and failure mechanism of the 2024-T3 Al alloy are analyzed. Furthermore, the fatigue life prediction physical model of the 2024-T3 Al alloy is established by using the energy method of fracture mechanics. The physical model incorporates critical physical parameters. Meanwhile, the PSO algorithm optimizes the hyperparameters of the XGBoost model based on fatigue data of the 2024-T3 Al alloy. Eventually, the optimized XGBoost model is used to solve the parameters of the physical model. Furthermore, the analytical equation of the fatigue life prediction model is obtained. This paper provides a new method for solving the parameters of the fatigue life prediction model, which reduces the error and cost of obtaining the model parameters in the experiment and shortens the time required.

Complementary use of analytical equations and numerical models for composite liner designs

The finite element (FE) method is used to identify limitations and extend the applicability of an analytical solution for leakage through holed wrinkles in composite liners. The limiting assumption in the analytical model is relaxed based on FE results, providing a means of obtaining good agreement through the two methods over a wide spectrum of cases. Despite the flexibility of the FE, the analytical model has the advantage of readily accommodating a wide range of dimensions and material properties that, even with a very large mesh, may be difficult to analyze using FE. In addition, the analytical equation is well-suited for Monte Carlo simulation. This is illustrated by the use of the modified analytical equation and the Monte Carlo simulations to compare the performance of two types of composite liners currently approved for use in Chinese landfills with a generic design in Canada and other parts of the world. It is shown that even with a relatively small number of holed wrinkles leakage through the geomembrane composite liner can differ by up to two orders of magnitude between composite liners considered to be equivalent. The paper also notes the need to revise regulations to reflect the evolution of knowledge.

Design discharge estimation for small urban districts – an analytical approach

ABSTRACT This paper presents a straightforward design discharge estimation procedure for small urban districts. In this approach, the configuration of the small urban district was simplified into either a rectangular porous overland plane or a V-shaped porous overland model. The Horton equation was used to estimate the infiltration on the overland planes, and the kinematic-wave approximation was applied for runoff simulations. In deriving the runoff equation, the Darcy–Weisbach coefficient was adopted to account for the influences of spatially various flow depths on overland flow resistance. The theoretical derivations were applied to a series of cases and verified by numerical models to demonstrate the applicability of the proposed method. The results showed that the design discharges obtained from the analytical equations are close to those simulated by the numerical model. The analysis procedure developed herein can provide a convenient way for engineers to conduct urban sewer designs.

Spin angular momentum and optical chirality of Poincaré vector vortex beams

Abstract The optical chirality and spin angular momentum of structured scalar vortex beams has been intensively studied in recent years. The pseudoscalar topological charge $\ell$ of these beams is responsible for their unique properties. Constructed from a superposition of scalar vortex beams with topological charges $\ell_\text{A}$ and $\ell_\text{B}$, cylindrical vector vortex beams are higher-order Poincar'e modes which possess a spatially inhomogeneous polarization distribution. Here we highlight the highly tailorable and exotic spatial distributions of the optical spin and chirality densities of these higher-order structured beams under both paraxial (weak focusing) and non-paraxial (tight focusing) conditions. Our analytical theory can yield the spin angular momentum and optical chirality of each point on any higher-order or hybrid-order Poincar'e sphere. It is shown that the tunable Pancharatnam topological charge $\ell_{\text{P}} = (\ell_\text{A} + \ell_\text{B})/2$ and polarization index $m = (\ell_\text{B} -\ell_\text{A})/2$ of the vector vortex beam plays a decisive role in customizing their spin and chirality spatial distributions. We also provide the correct analytical equations to describe a focused, non-paraxial scalar Bessel beam.

MANTA-Ray: Supercharging speeds for calculating the optical properties of fractal aggregates in the long-wavelength limit

Abstract Correctly modelling the absorptive properties of dust and haze particles is of great importance for determining the abundance of solid matter within protoplanetary disks and planetary atmospheres. Rigorous analyses such as the discrete dipole approximation (DDA) can be used to obtain accurate absorption cross-sections, but these require significant computing time and are often impractical to use in models. A simple analytical equation exists for spherical particles in the long-wavelength limit (where the wavelength is much larger than the size of the dust particle), but we demonstrate that this can significantly underestimate the absorption. This effect is found to depend strongly on refractive index, with values of m = 1 + 11i corresponding to an underestimate in absorption by a factor of 1,000. Here we present MANTA-Ray (Modified Absorption of Non-spherical Tiny Aggregates in the RAYleigh regime): a simple model that can calculate absorption efficiencies within 10-20% of the values predicted by DDA, but 1013 times faster. MANTA-Ray is very versatile and works for any wavelength and particle size in the long wavelength regime. It is also very flexible with regards to particle shape, and can correctly model structures ranging from long linear chains to tight compact clusters, composed of any material with refractive index 1+0.01i ≤m ≤ 11+11i. The packaged model is provided as publicly-available code for use by the astrophysical community.

Analytical Second-Order Extended Kalman Filter for Satellite Relative Orbit Estimation

This study considers a relative orbit estimation problem wherein an observing spacecraft navigates with respect to a target space object at a large separation distance (several kilometers) using only the bearing angles obtained by a single onboard camera. Generally, the extended Kalman filter (EKF), which is based on linear relative motion equations such as the Clohessy–Wiltshire equation, is used for the relative navigation of satellites. The EKF linearizes the estimation error around the current estimate and applies the Kalman filter equations to this linearized system. However, it has been shown that nonlinearities of the orbit determination problem can make the linearization assumption insufficient to represent the actual uncertainty. Therefore, an analytical second-order extended Kalman filter (ASEKF) for relative orbit estimation is proposed in this study. The ASEKF, to sequentially estimate the relative states of satellites and their associated uncertainties, is formulated based on a second-order analytic relative-motion equation under J2-perturbtation, which can overcome the deficiencies of existing approaches that mainly focus on applications in two-body, near-circular, and linearized orbit dynamics. Numerical results show that the proposed method provides superior robustness and mean-square error performance compared to linear estimators under the conditions considered.

Charge Carrier Collection Losses in Lead‐Halide Perovskite Solar Cells

AbstractThe collection of photogenerated charges in halide perovskite solar cells depends on the thickness of the absorber layer, with larger thicknesses leading to a reduced collection efficiency. This observation has traditionally been associated with insufficiently high electron and hole diffusion lengths in the absorber layers. However, it is shown that in the presence of low‐mobility contact layers, charge collection can be thickness‐dependent, even if the absorber layer has infinite mobility. Here, analytical equations are derived for the thickness dependence of charge collection losses in situations where recombination is bulk or interface‐limited and show how to relate these equations to voltage‐dependent photoluminescence data. The analytical equations are compared to experimental data and numerical simulations and it is observed that experimental data on triple‐cation perovskite devices with different thicknesses approximately follows the case, where bulk recombination dominates.

Two-loop five-point two-mass planar integrals and double Lagrangian insertions in a Wilson loop

We consider the complete set of planar two-loop five-point Feynman integrals with two off-shell external legs. These integrals are relevant, for instance, for the calculation of the second-order QCD corrections to the production of two heavy vector bosons in association with a jet or a photon at a hadron collider. We construct pure bases for these integrals and reconstruct their analytic differential equations in canonical form through numerical sampling over finite fields. The newly identified symbol alphabet, one of the most complex to date, provides valuable data for bootstrap methods. We then apply our results to initiate the study of double Lagrangian insertions in a four-cusp Wilson loop in planar maximally supersymmetric Yang-Mills theory, computing it through two loops. We observe that it is finite, conformally invariant in four dimensions, and of uniform transcendentality. Furthermore, we provide numerical evidence for its positivity within the amplituhedron region through two loops.

Flexural response of metallurgically bonded steel-aluminium foam-steel panels and composite beams

This paper presents the results of an experimental and numerical campaign aimed at investigating the flexural behaviour of both metallurgically bonded steel-aluminium foam-steel sandwich (MBSAFS) panels and steel-MBSAFS composite beams. Three-point bending tests were carried out and the obtained results were used to validate the accuracy of analytical models to predict the stiffness and resistance of both MBSAFS panels and the composite beams. The predictive capacity of the analytical equations was further verified against the results of parametric finite element simulations, which were also carried out to investigate the influence of the thickness of the steel sheet, the foam thickness, the strength of the materials, the span length, and the type of beam profile. The comparative analysis demonstrated that the predictive formulae accurately provide both stiffness and resistance. The stiffness was predicted with a mean ratio (analytical/FEM) equal to 1, accompanied by a coefficient of variation (CoV) equal to 0.02. The plastic resistance of the composite beam was predicted with a mean accuracy of 0.96 and a coefficient of variation of 0.01.

Influence of VIVO Mobile Brand Sponsorship of Sports Events on Consumer Purchase Intention

Effective sports sponsorship necessitates consideration of multiple factors. Previous scholars have primarily focused on the brand assets of sponsors and enhancement of the reputation that sponsoring brings to businesses. There has been limited exploration from the perspective of sports events themselves in terms of brand assets and the direct impact of these assets on sponsor audience purchase intention. This research aims to analyze the influences of VIVO mobile brand sponsorship including brand assets, brand fit, and audience engagement of sports events on consumer purchase intention. The research framework was laid on the stimulus-organism-response model. The sample size was 400 audience who attended the 22nd FIFA World Cup held in Guangzhou, China. Statistics used to analyze data were descriptive statistics including frequency, percentage, mean, and standard deviation; and inferential statistics including independent samples t-test, One-way ANOVA, LSD, simple linear regression, and multiple linear regression at the statistical significance level of 0.05. The results found that most of the respondents were male, married, aged 25-35 years old, educational level high school or vocational school, had monthly income of 5,000-10,000 yuan, and company employee. The hypotheses results found that age and occupation affect consumer purchase intention. Event brand assets including event brand image and event brand awareness has positive relation with consumer purchase intention with multiple correlation (R) = .566. The ability to predict the analytical equation is 31.70%. Brand fit and audience engagement influenced consumer purchase intention with correlation (R) = 0.485 and 0.469 accordingly. Keywords: brand assets, brand fit, audience engagement, purchase intention, sports sponsorship

Energies of an Electron in a One-Dimensional Lattice Using the Dirac Equation: The Coulomb Potential

The energies of an electron in a one-dimensional crystal are studied with both the Schrödinger and Dirac equations using the plane wave expansion method. The crystalline potential sensed by the electron in a cell was calculated by accounting for the Coulombic (electrostatic) interaction between the electron and the surrounding cores (immobile positive ions at the center of the crystal cells). The energies and wave functions of the electron were calculated as a function of four parameters: the period ap of the lattice, the dimension ndim of the matrix in the momentum space, the partition number lpa in which the unit cell is divided to calculate the potential and the number of cores nco that affect the electron. It was found that 8000 cores (surrounding the electron) were needed to reach our convergence criterion. An analytical equation that accurately describes the behavior of the energies in function of the cores that affect the electron was also found. As case studies, the energies for pseudo-lithium and pseudo-graphene were obtained as a first approximation for one-dimensional lattices. Subsequently, the energies of an isolated dimer nanoparticle were also calculated using the supercell method.

Energy levels and thermodynamic models of the general molecular oscillator under a 2D electromagnetic potential field

This study employs the parametric Nikiforov-Uvarov approach (PNUA) to resolve the radial Schrödinger equation (RSE) for the general molecular oscillator with a 2D electromagnetic potential coupling. Analytical approximations are developed for the energy levels, molar enthalpy, and constant-pressure molar heat capacity, with a focus on their applicability to diatomic molecules. The generated equations are employed to investigate the physical properties of real substances like BeCl (X 2Σ+), CsF (X 1Σ+), CuCl (X 1Σ+), CO+ (X 2Σ+), 7Li2 (1 3Δg), and P2 (X 1Σg +) molecules. The percentage average absolute deviations (PAAD) deduced with the analytical model equations are found to agree with the findings on diatomic molecules. Analysis of PAAD values also reveals that the predicted molar enthalpy and heat capacity of the diatomic molecules are better if the magnetic and Aharonov-Bohm components of the EM potential fields are finite.