Approximate Analytical Formula Research Articles

Analytical formula for signal optimization in stimulated photon-photon scattering setup with three laser pulses

We consider a setup to detect stimulated photon-photon scattering using high-power lasers. Signal photons are emitted from an overlap of the incoming intense laser pulses focused in vacuum from three sides. We derive and justify a general approximate analytical formula for the angular distribution and total yield of such signal photons in terms of the parameters of the incoming pulses, including their intensity, carrier frequencies, durations, focusing, polarizations, mutual orientation and overlap. Using the obtained formula a parametric study of the signal is carried out and optimization is performed. Published by the American Physical Society 2024

Impact of the LZ experiment on the DM phenomenology and naturalness in the MSSM

This paper uses approximate analytical formulas and numerical results with the bino-dominated dark matter (DM) as an example to analyze the impact of the LUX-ZEPLIN (LZ) experiment on the DM phenomenology and naturalness in the Minimal Supersymmetric Standard Model (MSSM). We conclude that the limitation of the latest LZ experiment worsens the naturalness of the MSSM, as the predictions of the Z-boson mass and DM relic density demonstrate, particularly in the regions where the correct DM relic density is obtained by the Z- or h-mediated resonant annihilations.

Investigating the impact of applied magnetic field and current on anode power deposition in AF-MPDT

The high temperature superconductors enhanced magnetoplasmadynamic thruster is a promising device for various space mission scenarios. However, a critical issue hindering the enhancement of the thruster’s performance is the problem of anode power deposition. Previous studies have identified the anode fall voltage as a major contributor to this power deposition. This paper introduces an approximate analytical formula for calculating anode power deposition resulting from anode fall voltage. The paper further analyzes and calculates the variation of anode fall voltage with an applied magnetic field and discharge current. The findings reveal a direct correlation between applied magnetic field and both anode fall voltage and power, indicating an increase with higher magnetic field values. In contrast, while the anode fall voltage experiences a slight decline with increased discharge current, the anode power deposition remains elevated due to the transition from field ionization to thermal ionization. To gain a comprehensive understanding of anode power deposition, an experiment was conducted to obtain measurements under different operating conditions, involving varying levels of applied magnetic field and discharge current. The experimental results align with the theoretical predictions. These insights have the potential to serve as a valuable reference for enhancing the efficiency of magnetoplasmadynamic thrusters, bringing us closer to unlocking the full potential of these remarkable propulsion systems in the realm of space exploration.

IMPACT OF SOLID BOUNDARIES AND VISCOSITY OF THE MEDIUM ON THE CONTRIBUTION OF THE INERTIAL AND VORTEX COMPONENTS OF THE NORMAL FORCE OF A ROTATING PLATE. PART 2

Within the framework of the improved method of discrete vortices, generalized for viscous media, a method has been developed for determining the contribution of forces of inertial, vortex and circulation nature to the normal force of a plate moving in a stationary viscous boundless medium accord- ing to an arbitrary law, in the presence of a wall and in a channel. The developed method was tested for the case of instantaneous angular start of the plate and subsequent constant angular speed of rotation (Wagner's problem) in a viscous boundless medium, in the presence of a wall and in a channel, in laminar and turbulent modes. The application of approximate analytical formulas for the components of the induced velocity from a discrete vortex in a viscous medium near the wall and in the channel made it possible to obtain velocity fields that visualize two vortices separated from the edges of the plate and the dynamics of their dispersion in laminar and turbulent regimes of rotation of the plate near the wall and in the channel. The inertial- vortical nature of the normal force of the plate (with the dominance of inertial forces) is confirmed, which rotates after an instantaneous start with the separation of the flow from both its edges, regardless of the presence of solid boundaries and laminar or turbulent flow regimes. It was found that in the case of the laminar regime, the effect of the presence of the wall on the reduced inertial component of the normal force of the plate is insignificant, but the influence of the channel leads to a faster departure of the laminar vortex from the moving edge of the plate, which leads to a gradual increase in the contribution of the inertial component of the normal force of the plate up to 100 % and more when turning the plate perpendicular to the channel walls. Approximately the same happens in the case of the absence of solid boundaries, when a turbulent vortex of a larger size and intensity than the corresponding laminar one moves away from the moving edge of the plate.

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.

Effective reduced models from delay differential equations: Bifurcations, tipping solution paths, and ENSO variability

Conceptual delay models have played a key role in the analysis and understanding of El Niño-Southern Oscillation (ENSO) variability. Based on such delay models, we propose in this work a novel scenario for the fabric of ENSO variability resulting from the subtle interplay between stochastic disturbances and nonlinear invariant sets emerging from bifurcations of the unperturbed dynamics.To identify these invariant sets we adopt an approach combining Galerkin–Koornwinder (GK) approximations of delay differential equations and center-unstable manifold reduction techniques. In that respect, GK approximation formulas are reviewed and synthesized, as well as analytic approximation formulas of center-unstable manifolds. The reduced systems derived thereof enable us to conduct a thorough analysis of the bifurcations arising in a standard delay model of ENSO. We identify thereby a saddle–node bifurcation of periodic orbits co-existing with a subcritical Hopf bifurcation, and a homoclinic bifurcation for this model. We show furthermore that the computation of unstable periodic orbits (UPOs) unfolding through these bifurcations is considerably simplified from the reduced systems.These dynamical insights enable us in turn to design a stochastic model whose solutions – as the delay parameter drifts slowly through its critical values – produce a wealth of temporal patterns resembling ENSO events and exhibiting also decadal variability. Our analysis dissects the origin of this variability and shows how it is tied to certain transition paths between invariant sets of the unperturbed dynamics (for ENSO’s interannual variability) or simply due to the presence of UPOs close to the homoclinic orbit (for decadal variability). In short, this study points out the role of solution paths evolving through tipping “points” beyond equilibria, as possible mechanisms organizing the variability of certain climate phenomena.

Common mode far field radiation mechanisms and prediction of a buck converter with cables based on common mode voltage mutation method

AbstractIn the field of electromagnetic compatibility (EMC) research, the knowledge of the formation mechanism of the far field electromagnetic radiation and its prediction is essential for the electromagnetic noise suppression design of a switched mode power supply. This paper investigates the equivalent driving sources of the common mode (CM) current of a Buck converter with cables. Based on the concept of CM voltage mutation, this paper proposes two mechanism models of the same‐direction CM voltage driving mode and the opposite‐direction CM voltage driving mode and derives the approximate analytical formulas for the far field intensity estimation of the horizontal cables. To evaluate the predictive effectiveness of the CM far field radiation model, a 30 MHz very high frequency (VHF) Buck prototype with shielded cables is built and tested, and a shielding shell and low impedance ground conductor suppression designs are then verified. The CM electromagnetic radiation mechanisms of a non‐isolated power converter are demonstrated here. The proposed electromagnetic radiation models, approximate analytical formulas, and improvement design can help to advance the CM radiation analysis and optimization design.

Accuracy of analytical approximation formula for bond prices in a three-factor convergence model of interest rates

We consider a convergence model of interest rates, in which the behaviour of the domestic instantaneous interest rate (so called short rate) depends on the short rate in a monetary union that the country is going to join. The short rate in the monetary union is modelled by a two-factor model, which leads to a three-factor model for the domestic rate. In this setting, term structures of interest rates are computed from bond prices, which are obtained as solutions to a parabolic partial differential equation. A closed-form solution is known only in special cases. An analytical approximation formula for the domestic bond prices has been proposed, with the error estimate only for certain parameter values, when the solution has a separable form. In this paper, we derive the order of accuracy in the general case. We also study a special case, which makes it possible to model the phenomenon of negative interest rates that were observed in the previous years. It turns out that it leads to a higher accuracy than the one achieved in the general case without restriction on parameters.

Dynamic and Pareto-improving swapping of vehicles to enhance integrated and modular mobility services

Efficiency of many mobility services depends on effective matching of service vehicles and customers over time and space. New technologies, such as those related to service integrators and modularized autonomous vehicle platforms (i.e., chassis with built-in power and control systems), further allow pooling of vehicles from traditionally separate service operators to flexibly serve multiple types of customers (e.g., passengers and freight). Dynamic matching systems, if not carefully managed, suffer from the so-called wild goose chase (WGC) phenomenon, which describes an inefficient system equilibrium with a large number of vehicles trapped in unproductive deadheading. This paper proposes a new dynamic vehicle swap strategy that can be used to enhance system efficiency by reducing the expected waiting/deadheading time. This strategy can not only mitigate the WGC and reduce the needed fleet size, but also ensure that the vehicle swap achieves a Pareto improvement for each involved participant. Approximate analytic formulas are derived from a series of differential equations and spatial probability models to estimate the expected system performance in the steady state. This system of nonlinear equations can be solved numerically to yield the equilibrium points and the associated system performance metrics. Agent-based simulations are used to verify the accuracy of the derived formulas, and to demonstrate the effectiveness of the proposed dynamic and Pareto-improving swap strategy in a variety of application contexts (e.g., taxi, service integrator, or pooled modular chassis).

Polyrate 2023: A computer program for the calculation of chemical reaction rates for polyatomics. New version announcement

Polyrate 2023: A computer program for the calculation of chemical reaction rates for polyatomics. New version announcement

Mathematical Model of the Flow in a Nanofiber/Microfiber Mixed Aerosol Filter

A new mathematical model of an aerosol fibrous filter, composed of a variety of nano- and microfibers, is developed. The combination of nano- and microfibers in a mixed-type filter provides a higher overall quality factor compared with filters with monodisperse fibers. In this paper, we propose a mathematical model of the flow of an incompressible fluid in a porous region consisting of a set of cylinders of various diameters in the range of nano- and micrometers to describe a mixed-type aerosol filter. The flow domain is a rectangular periodic cell with one microfiber and many nanofibers. The motion of the carrier medium is described by the boundary value problem in Stokes flow approximation with the no-slip boundary condition for microfibers and the slip condition for nanofibers. The boundary element method taking into account the slip and non-slip conditions is developed. The calculated velocity field, streamlines, vorticity distribution, and drag of separate fibers and the entire periodic cell are presented. Numerical results for the drag force of the porous medium of a mixed-type filter for the various ratios of mass proportion of nano- and microfibers, porosity, and filtration velocity are presented. The obtained results are compared with the analytical formulas based on the approximate theory of filtration of bimodal filters and with known experimental data. It is shown that with an increase in the mass fraction of nanofibers, the total drag force of the cell increases, while the relative contribution of nanofibers to the total drag force tends toward the value that is less than unity. An approximate analytical formula for the drag coefficient of a mixed aerosol filter is derived. The developed flow model and analytical formulas allow for estimating the aerodynamic drag of a mixed filter composed by nano- and microfibers.

Formulas for Beam Shift and Beam Narrowing in 1-D Leaky-Wave Antennas Due to Element Pattern

The radiation pattern of one-dimensional (1-D) unidirectional leaky-wave antennas (LWAs) is typically derived from the analysis of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">space factor</i> (SF) only. However, when the antenna aperture is characterized by a longitudinal radiating current, a cosine-type <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">element pattern</i> term has to be included: this factor may considerably affect the beam properties. Specifically, the beam peak is shifted and the beamwidth is narrowed when the element pattern is included. These effects are more prominent as the antenna size decreases and the beam angle increases. Approximate but accurate analytical formulas are here provided and numerically validated to predict the beam angle and the half-power beamwidth for these kinds of 1-D unidirectional LWAs when the element pattern is included.

Early universe dynamics of PQ field with very small self-coupling and its implications for axion dark matter

Axion-like particles (ALPs) are often considered as good candidates for dark matter. Several mechanisms generating relic abundance of ALP dark matter have been proposed. They may involve processes which take place before, during or after cosmic inflation. In all cases an important role is played by the potential of the corresponding Peccei-Quinn (PQ) field. Quite often this potential is assumed to be dominated by a quartic term with a very small coupling. We show that in such situation it is crucial to take into account different kinds of corrections especially in models in which the PQ field evolves during and after inflation. We investigate how such evolution changes due to radiative, thermal and geometric corrections. In many cases those changes are very important and result in strong modifications of the predictions of a model. They may strongly influence the amount of ALP contributions to cold and warm components of dark matter as well as the power spectrum of associated isocurvature perturbations. Models with a quasi-supersymmetric spectrum of particles to which the PQ field couples seem to be especially interesting.Qualitative features of such models are discussed with the help of approximate analytical formulae. However, the dynamics of the PQ field with the considered corrections taken into account is more complicated than in the case without corrections so dedicated numerical calculations are necessary to obtain precise predictions.We present such results for some characteristic benchmark points in the parameter space.

ELF-EM fields in the multi-layer spherical ‘Earth-ionosphere’ model based on WKB

Abstract With a high signal-to-noise ratio and a great depth of exploration, the wireless electromagnetic method (WEM) has wide applications in the exploration of deep mineral resources and oil and gas reservoirs. Extremely low-frequency electromagnetic (ELF) waves emitted from a horizontal antenna are used to achieve synchronous acquisition for different receivers of multi-coverage information in a global region. However, previous research based on a planar model ignored the curvature of the Earth. This work focuses on the electromagnetic fields (EM fields) in the model of a spherical ‘Earth ionosphere’ to extend the coverage of WEM. By transferring the EM fields from a vertical electric dipole (VED) as well as a vertical magnetic dipole (VMD) in the multi-layered medium of the Earth, we obtain the formulae for the EM fields emitted by a horizontal electric dipole (HED) by using a reciprocity theorem. The correctness of the proposed method is verified by comparing it with the approximate analytical formula and previous work. Based on the above results, we have studied the propagation and frequency characteristics of electromagnetic fields in a spherical waveguide consisting of the ionosphere and earth. The results show that the electromagnetic fields under the spherical model produce interference effects that are different from those of the planar model. The electromagnetic response of the layered Earth was then discussed, and its potential as an electromagnetic technique for exploring the deep Earth was demonstrated.

A simulation method on target strength and circular SAS imaging of X-rudder UUV including multiple acoustic scattering

Target strength (TS) and circular synthetic aperture sonar (CSAS) images provide essential information for active acoustic detection and recognition of non-cooperative unmanned undersea vehicles (UUVs), which pose a significant threat to underwater preset facilities. To access them, we propose an iterative physical acoustics (IPA)-based method to simulate the multiple acoustic scattered fields on rigid surfaces in high-frequency cases. It uses the Helmholtz integral equation with an appropriate Green's function in terms of the Neumann series, and then incorporates the ideas of triangulation and iteration into a numerical implementation. Then two approximate analytic formulae with precise physical meanings are derived to predict the TS and CSAS images of concave targets, respectively. There are no restrictions on the surface's curvature and the order of multiple scattering. The method is validated against the finite element method (FEM) for acoustic scattering from a sphere segment and against an experiment involving an X-rudder UUV's stern. On this basis, we simulate and analyze the TS and CSAS images of an X-rudder UUV. In addition, the influence of the angle of adjacent rudders on the multiple scattering characteristics is discussed. Results show that this method can potentially predict accurate UUV features, especially the multiple scattered features.

Steady-state equation of thermal explosion in a distributed activation energy medium: numerical solution and approximations

This work presents a mathematical model of thermal explosion in a medium described by a Gaussian distribution of reactivity, along with the determination of critical values for model parameters and their dependence on the distribution dispersion. The numerical solution of boundary value problems for steady-state temperature distribution in a reaction medium (a sweep method along with the iterative refinement of a source function, a half-interval method to find the critical value of the Frank-Kamenetskii parameter) was used. The grid convergence was investigated for the used difference scheme; the first order of accuracy was observed as a result of numerical evaluation of the critical value of the Frank-Kamenetskii parameter. Calculations were carried out with accuracy to three decimal places. Numerical methods were implemented as programs in the MATLAB environment. Numerical approximations were obtained for solutions of the thermal explosion equation characterised by distributed activation energy in the quasi-steady-state approximation. It was shown that the critical value of the Frank-Kamenetskii parameter is associated with the dispersion of the distribution and the Arrhenius parameter by a simple approximate analytical formula, confirmed by comparing with numerical estimates. Since the dependence of the critical value of the Frank-Kamenetskii parameter on the dispersion is described by a Gaussian function, the reaction medium becomes thermally unstable even at small values of the distribution dispersion. Calculations showed that a significant dispersion of reactivity (on the order of tenths of the average) can be observed only for chemical reactions characterised by low sensitivity to temperature (i.e. a small heat effect or low activation energy). Approximate formulas for critical conditions were also obtained for asymmetrical distribution functions. The analysis allows the proposed mathematical model to be used for assessing the thermal stability of reactive media having distributed reactivity (for example, natural materials, polymers, heterogeneous catalytic systems, etc.).

Analytical approximation of European option prices under a new two-factor non-affine stochastic volatility model

&lt;abstract&gt;&lt;p&gt;In this paper, the pricing of European options under a new two-factor non-affine stochastic volatility model is studied. In order to reduce the computational complexity, we use the Taylor expansion and Fourier-cosine method to derive an analytical approximation formula for European option prices. Numerical experiments prove that the proposed formula is fast and efficient for pricing European options compared with Monte Carlo simulations. The sensitivity of the parameters is analyzed to explain the rationality of the model. Finally, we present some preliminary empirical analysis revealing that the pricing performance of our proposed model is superior to that of the single-factor model.&lt;/p&gt;&lt;/abstract&gt;

Projectile Motion in Midair Using Simple Analytical Approximations

The study of the motion of a projectile, thrown at an angle to the horizon, is a wonderful classical problem. This issue has been the subject of great interest to investigators for centuries. Currently, the study of parabolic motion, in the absence of any drag force, is a common example in introductory physics courses. The theory of parabolic motion allows you to analytically determine the trajectory and all important characteristics of the movement of the projectile. Introduction of air resistance forces into the study of the motion, however, complicates the problem and makes it difficult to obtain analytical solutions. This especially applies to the movement of the projectile, subjected to quadratic air drag force. Numerical studies of projectile motion with quadratic dependence on projectile speed have been performed in many works. From an educational point of view, many students are not confident with numerical methods and gain little insight from using them. So the description of the projectile motion by means of simple approximate analytical formulas under air resistance has great methodological and educational importance. The main goal of this work is to give analytical approximations for the projectile trajectory and movement characteristics as simple as possible from a technical point of view, in order to be grasped even by first-year undergraduates.

Theoretical and Experimental Study of Joint Osmotic and Electroosmotic Water Transfer through a Cation-Exchange Membrane.

Using the previously developed cell model of a charged membrane and the principles of linear thermodynamics of irreversible processes (the Onsager approach), exact and approximate (in the case of an ideally selective membrane) analytical formulae for calculating the osmotic and electroosmotic permeability of the membrane in aqueous solutions of 1:1 electrolyte at constant electric current density and concentration gradient were suggested. The formulae have been successfully verified by our own experimental data for the extrusion cation-exchange membrane MF-4SC p.29 in NaCl solution up to concentrations of 3 M. The contribution of electroosmotic and osmotic water fluxes to the total water transport through the mentioned individual perfluorinated ion-exchange membrane under conditions close to the process of electrodialysis concentrating was experimentally estimated. The cases of co- and counter-directed osmotic and electroosmotic water fluxes are studied. A good correspondence between theoretical and experimental results was obtained, which made it possible to determine the physicochemical parameters of the electromembrane system (the diffusion coefficients of individual ions and the coefficient of equilibrium distribution of electrolyte molecules in the membrane matrix, the characteristic exchange capacity of the cell model). The achieved results make it possible to fully characterize existing and promising types of ion-exchange membranes based on the developed cell model of a charged membrane.

A General Neural Network Model for Complex Refractive Index Extraction of Low-Loss Materials in the Transmission-Mode THz-TDS

The complex refractive index for low-loss materials is conventionally extracted by either approximate analytical formula or numerical iterative algorithm (such as Nelder-Mead and Newton-Raphson) based on the transmission-mode terahertz time domain spectroscopy (THz-TDS). A novel 4-layer neural network model is proposed to obtain optical parameters of low-loss materials with high accuracy in a wide range of parameters (frequency and thickness). Three materials (TPX, z-cut crystal quartz and 6H SiC) with different dispersions and thicknesses are used to validate the robustness of the general model. Without problems of proper initial values and non-convergence, the neural network method shows even smaller errors than the iterative algorithm. Once trained and tested, the proposed method owns both high accuracy and wide generality, which will find application in the multi-class object detection and high-precision characterization of THz materials.