Simple Approximate Formula Research Articles

Cosmological correlators with double massive exchanges: bootstrap equation and phenomenology

Using the recently developed cosmological bootstrap method, we compute the exact analytical solution for the seed integral appearing in cosmological correlators with double massive scalar exchanges. The result is explicit, valid in any kinematic configuration, and free from spurious divergences. It is applicable to any number of fields’ species with any masses. With an appropriate choice of variables, the results contain only single-layer summations. We also propose simple approximate formulas valid in different limits, enabling direct and instantaneous evaluation. Supported by exact numerical results using CosmoFlow, we explore the phenomenology of double massive exchange diagrams. Contrary to single-exchange diagrams with ubiquitous Lorentz-covariant interactions, the size of the cubic coupling constant can be large while respecting perturbativity bounds. Because of this property, the primordial bispectrum from double-exchange diagrams can be as large as, coincidentally, current observational constraints. In addition to being sizable on equilateral configurations, we show that the primordial bispectrum exhibits a large cosmological collider signal in the squeezed limit, making the double massive exchanges interesting channels for the detection of massive primordial fields. We propose to decisively disentangle double-exchange channels from single-exchange ones with cosmological observations by exploiting the phase information of the cosmological collider signal, the inflationary flavor oscillations from multiple fields’ species exchanges and the double soft limit in the primordial trispectrum.

Third harmonic of the dynamic magnetic susceptibility of a concentrated ferrofluid: Numerical calculation and simple approximation formula.

Information about the nonlinear magnetic response of dispersions of magnetic particles is the basis for biomedical applications. In this paper, using analytical and numerical methods, the third harmonic of the dynamic susceptibility of an ensemble of moving magnetic particles in an ac magnetic field with an arbitrary amplitude is studied, taking into account interparticle interactions. A simple approximation formula is proposed to predict the third harmonic as a function of two parameters: the Langevin susceptibility χ_{L}, which is used to estimate the particle dipole-dipole interactions, and the Langevin parameter ξ, which represents the ratio of the energy of the magnetic moment interacting with the magnetic field to the thermal energy. The derived approximation formula corresponds with the known single-particle theories in the limit case of a small particle's concentration and is valid for concentrated dispersions of magnetic particles (with the Langevin susceptibility up to χ_{L}≤3) in high-amplitude ac fields (with the Langevin parameter up to ξ≤10).

Models to predict configurational adiabats of Lennard-Jones fluids and their transport coefficients.

A comparison is made between three simple approximate formulas for the configurational adiabat (i.e., constant excess entropy, sex) lines in a Lennard-Jones (LJ) fluid, one of which is an analytic formula based on a harmonic approximation, which was derived by Heyes et al. [J. Chem. Phys. 159, 224504 (2023)] (analytic isomorph line, AIL). Another is where the density is normalized by the freezing density at that temperature (freezing isomorph line, FIL). It is found that the AIL formula and the average of the freezing density and the melting density ("FMIL") are configurational adiabats at all densities essentially down to the liquid-vapor binodal. The FIL approximation departs from a configurational adiabat in the vicinity of the liquid-vapor binodal close to the freezing line. The self-diffusion coefficient, D, shear viscosity, ηs, and thermal conductivity, λ, in macroscopic reduced units are essentially constant along the AIL and FMIL at all fluid densities and temperatures, but departures from this trend are found along the FIL at high liquid state densities near the liquid-vapor binodal. This supports growing evidence that for simple model systems with no or few internal degrees of freedom, isodynes are lines of constant excess entropy. It is shown that for the LJ fluid, ηs and D can be predicted accurately by an essentially analytic procedure from the high temperature limiting inverse power fluid values (apart from at very low densities), and this is demonstrated quite well also for the experimental argon viscosity.

Unraveling Burst Selection Bias in Single-Molecule FRET of Species with Unequal Brightness and Diffusivity.

Single-molecule free diffusion experiments enable accurate quantification of coexisting species or states. However, unequal brightness and diffusivity introduce a burst selection bias and affect the interpretation of experimental results. We address this issue with a photon-by-photon maximum likelihood method, burstML, which explicitly considers burst selection criteria. BurstML accurately estimates parameters, including photon count rates, diffusion times, Förster resonance energy transfer (FRET) efficiencies, and population, even in cases where species are poorly distinguished in FRET efficiency histograms. We develop a quantitative theory that determines the fraction of photon bursts corresponding to each species and thus obtain accurate species populations from the measured burst fractions. In addition, we provide a simple approximate formula for burst fractions and establish the range of parameters where unequal brightness and diffusivity can significantly affect the results obtained by conventional methods. The performance of the burstML method is compared with that of a maximum likelihood method that assumes equal species brightness and diffusivity, as well as standard Gaussian fitting of FRET efficiency histograms, using both simulated and real single-molecule data for cold-shock protein, protein L, and protein G. The burstML method enhances the accuracy of parameter estimation in single-molecule fluorescence studies.

Symbolic Regression Approaches for the Direct Calculation of Pipe Diameter

This study provides novel and accurate symbolic regression-based solutions for the calculation of pipe diameter when flow rate and pressure drop (head loss) are known, together with the length of the pipe, absolute inner roughness of the pipe, and kinematic viscosity of the fluid. PySR and Eureqa, free and open-source symbolic regression tools, are used for discovering simple and accurate approximate formulas. Three approaches are used: (1) brute force of computing power, which provides results based on raw input data; (2) an improved method where input parameters are transformed through the Lambert W-function; (3) a method where the results are based on inputs and the Colebrook equation transformed through new suitable dimensionless groups. The discovered models were simplified by the WolframAlpha simplify tool and/or the equivalent Matlab Symbolic toolbox. Novel models make iterative calculus redundant; they are simple for computer coding while the relative error remains lower compared with the solution through nomograms. The symbolic-regression solutions discovered by brute force computing power discard the kinematic viscosity of the fluid as an input parameter, implying that it has the least influence.

Continued fractions and the Thomson problem

We introduce new analytical approximations of the minimum electrostatic energy configuration of n electrons, E(n), when they are constrained to be on the surface of a unit sphere. Using 453 putative optimal configurations, we searched for approximations of the form E(n) = (n^2/2) , e^{g(n)} where g(n) was obtained via a memetic algorithm that searched for truncated analytic continued fractions finally obtaining one with Mean Squared Error equal to {5.5549 times 10^{-8}} for the model of the normalized energy (E_n(n) equiv e^{g(n)} equiv 2E(n)/n^2). Using the Online Encyclopedia of Integer Sequences, we searched over 350,000 sequences and, for small values of n, we identified a strong correlation of the highest residual of our best approximations with the sequence of integers n defined by the condition that n^2+12 is a prime. We also observed an interesting correlation with the behavior of the smallest angle alpha (n), measured in radians, subtended by the vectors associated with the nearest pair of electrons in the optimal configuration. When using both sqrt{n} and alpha (n) as variables a very simple approximation formula for E_n(n) was obtained with MSE= 7.9963 times 10^{-8} and MSE= 73.2349 for E(n). When expanded as a power series in infinity, we observe that an unknown constant of an expansion as a function of n^{-1/2} of E(n) first proposed by Glasser and Every in 1992 as -1.1039, and later refined by Morris, Deaven and Ho as -1.104616 in 1996, may actually be very close to −1.10462553440167 when the assumed optima for nle 200 are used.

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.).

Approximation results for the moments of random walk with normally distributed interference of chance

In this study, a random walk process $\left(X\left(t\right)\right)$ with normally distributed interference of chance is considered. In the literature, this process has been shown to be ergodic and the limit form of the ergodic distribution has been found. Here, unlike previous studies, the moments of the $X\left(t\right)$ process are investigated. Although studies investigating the moment problem for various stochastic processes (such as renewal-reward processes) exist in the literature, it has not been considered for random walk processes, as it requires the use of new mathematical tools. Therefore, in this study, firstly, the exact formulas for the first four moments of the ergodic distribution of the $X\left(t\right)$ process, which is a modification of the random walk process, are found. Due to the extremely complex mathematical structure of the exact formulas, in the second part of the study, three-term asymptotic expansions are attained for these moments. Based on the asymptotic expansions, simple and useful approximation formulas, for the moments of the process $X\left(t\right)$ are proposed. In order to show that the approximate formulas are close enough to the exact formulas, a special example is given at the end of the study and the accuracy of the approximate formulas is examined on this example.

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.

The VIX Future in Bergomi Models: Fast Approximation Formulas and Joint Calibration with S&P 500 Skew

Using the exponential generating function of Hermite polynomials, we expand the prices of VIX power payoffs (including VIX futures) in various Bergomi models at any order in the volatility-of-volatility. We introduce the notion of volatility of the VIX squared implied by a VIX power payoff, which we call “implied VIX$^2$ volatility” and also expand at any order. We cover the one-factor and (skewed) two-factor Bergomi models and allow for maturity-dependent and/or time-dependent parameters. When the initial term-structure of variance swaps is flat, we provide the expansions up to order 8 in closed form; otherwise, they simply involve one-dimensional integrals, which are extremely fast to compute. Extensive numerical experiments show that the implied volatility expansion converges much faster than the price expansion and is extremely accurate for a wide range of model parameters, including typical market calibrating parameters with very large volatilities-of-volatility. It leads to new, simple approximation formulas for the price of a VIX power payoff that shed light on how those prices depend on model parameters. We combine the new expansion and the Bergomi--Guyon expansion of the vanilla smile [L. Bergomi and J. Guyon, Risk, May (2012), pp. 60--66] to calibrate the two-factor Bergomi model jointly to the term-structures of S&P 500 at-the-money skew and VIX futures. Very interestingly, the joint fit selects (1) much larger values of volatility-of-volatility and mean reversion than those previously reported in [L. Bergomi, Risk, October (2005), pp. 67--73] and [L. Bergomi, Stochastic Volatility Modeling, CRC Press, 2016], and (2) fully correlated Brownian motions, thus producing a (Markovian) pure path-dependent volatility model with rough-like paths.

To the theory of static magnetizations of ferrogels with heterogeneous clusters. Effect of dimers

Effect of clusters, consisting of two single-domain ferromagnetic nanoparticles (dimers) on magnetostatic properties of ferrogels is studied theoretically. The proposed model of the static magnetization of the motionless dimer is developed by using the mathematically regular saddle point method. A simple approximate formula for the dimer magnetization is derived and compared with the exact formula, following from the Boltzmann law of distribution over physically possible states of the dimer. For large value of parameters of the interparticle dipole–dipole interaction and the particles internal magnetic anisotropy, the approximate formula leads to good agreement with the exact formula and can be recommended for practical calculations to avoid numerical calculation of integrals of high multiplicity. The obtained results can serve as a robust basis for studying of magnetic and other physical properties of ferrogels with more complicated internal structures.

Valuation of volatility derivatives with time-varying volatility: An analytical probabilistic approach using a mixture distribution for pricing nonlinear payoff volatility derivatives in discrete observation case

In this paper, we present an analytical probabilistic approach for pricing nonlinear payoff volatility derivatives with discrete sampling by assuming that the underlying asset price evolves according to the Black–Scholes model with time-varying volatility. A major difficulty to solve the pricing problem analytically is the volatility of the underlying asset price is time-varying, resulting the realized variance is distributed according to a mixture distribution, the probability density function of which is unknown. By utilizing the properties of a linear combination of noncentral chi-square random variables, we can calculate the expectation of square root of the realized variance analytically and provide formulas for pricing volatility swaps, volatility options, and variance options, including put–call parity relationships. Furthermore, we demonstrate an interesting application of our formulas by constructing simple closed-form approximate formulas for pricing the volatility derivatives for the constant elasticity of variance model. Finally, Monte Carlo simulations are conducted to illustrate the performance our approach, and effects of price volatility on the fair strike prices of the volatility derivatives are investigated and analyzed through several numerical experiments.

Low-Frequency Magnetic Shielding of Planar Screens: Effects of Loop Radius and Loop-to-Loop Distance

For the magnetic shielding configuration of both emitting and receiving loops placed coaxially on either side of a planar screen, we investigate the effects of loop radius and loop-to-loop distance on shielding effectiveness by numerically solving the accurate integral expression. The results are in good agreement with finite element simulations and experimental results. Simple and reliable approximate formulas are also proposed for pure conductive screens and pure magnetic screens, respectively. It is shown that the effects are obvious when the loop-to-loop distance is smaller than the loop radius. Also, conductive screens and magnetic screens exhibit opposite and competing shielding effectiveness evolution behaviors. In addition, in the case of conductive screens with a smaller receiving loop compared to the emitting loop, the shielded loop-to-loop mutual inductance does not always decrease with the increase of the loop-to-loop distance but peaks at a distance between the screen thickness and half of the emitting loop radius.

Diabatic Hamiltonian matrix elements made simple

With a view to applying the Generator Coordinate Method to large configuration spaces, we propose a simple approximate formula to compute diabatic many-body matrix elements without having to evaluate two-body interaction matrix elements. The method is illustrated with two analytically solvable Hamiltonians based on the harmonic oscillator.

Error bounds for the spectral approximation of the potential of a homogeneous almost spherical body

Several kinds of approximation of the gravitational potential of a homogeneous body by truncated spherical harmonics series are in use in physical geodesy. However, only one of them is capable of a representation converging to the true potential in the whole layer between the Brillouin sphere and the Bjerhammar sphere of the body. We aim at providing various majorizations, namely upper bounds, of the error with the double purpose of proving explicitly the convergence in the sense of different norms and of giving computable bounds, that might be used in numerical studies. The first aim is reached for all the norms. For the second, however, it turns out that among the bounds, when applied to the example of the terrain correction of the Earth, only those referring to the mean absolute error and the mean squared error at the level of Brillouin sphere of minimum radius give significant and useful results. In order to make the computation an easy exercise, a simple approximate formula has been developed requiring only the use of the distribution function of the heights of the surface of the body with respect to the Bjerhammar sphere.

Analytical expressions for electrodynamic parameters of the shielded microstrip line

On the basis of an electrodynamic model of a screened microstrip line, built on the basis of the projection method using the Chebyshev basis, which explicitly takes into account the edge features of the field, a mathematical model of a microstrip line with a strip conductor was developed. The line width does not exceed the height of the substrate. In this case, the current density on the strip conductor is approximated by only one basis function. Analytical expressions are presented in the form of a sum of slowly and rapidly converging series to determine the main electrodynamic parameters of the line – wave resistance and deceleration coefficient. Due to logarithmic features, slowly converging series are summed up and transformed into rapidly converging power series. In addition, limit expressions in the form of improper integrals are given for the main electrodynamic parameters of an open microstrip line in the quasi-static approximation. Due to the logarithmic features, these integrals are also converted to rapidly converging power series. As a result, simple approximate formulas were obtained. They allow calculating the deceleration coefficient and wave impedance of the line with an error not exceeding 1%, when the width of the strip conductor is less than twice the thickness of the substrate. The results of calculating the electrodynamic parameters obtained on the basis of the developed mathematical model and on the basis of the projection method with an accuracy of up to 5 significant digits are presented. These results make it possible to establish the limits of applicability of the quasi-static approximation and to determine the error in calculating the deceleration coefficient and wave resistance using the obtained analytical expressions. The error does not exceed 0.1%, if the width of the strip conductor is less than twice the thickness of the substrate in a wide range of changes in the substrate dielectric constant and frequency.

Low Ionosphere under Influence of Strong Solar Radiation: Diagnostics and Modeling

Solar flares (SFs) and intense radiation can generate additional ionization in the Earth’s atmosphere and affect its structure. These types of solar radiation and activity create sudden ionospheric disturbances (SIDs), affect electronic equipment on the ground along with signals from space, and potentially induce various natural disasters. Focus of this work is on the study of SIDs induced by X-ray SFs using very low frequency (VLF) radio signals in order to predict the impact of SFs on Earth and analyze ionosphere plasmas and its parameters. All data are recorded by VLF BEL stations and the model computation is used to obtain the daytime atmosphere parameters induced by this extreme radiation. The obtained ionospheric parameters are compared with results of other authors. For the first time we analyzed physics of the D-region—during consecutive huge SFs which continuously perturbed this layer for a few hours—in detail. We have developed an empirical model of the D-region plasma density and gave a simple approximative formula for electron density.

Accurate Gaunt Factors for Nonrelativistic Quadrupole Bremsstrahlung

Abstract The exact result for nonrelativistic quadrupole bremsstrahlung in a Coulomb field was established only recently in Pradler & Semmelrock. It requires the evaluation and integration of hypergeometric functions across a wide range of parameters and arguments, which, in practice, is unfeasible. Here we provide a highly accurate tabulation of the Gaunt factor for quadrupole radiation, its thermal average in a Maxwellian plasma, and the associated cooling function over the entire kinematically relevant range. In addition, we provide a simple approximate formula for the emission cross section that works to within a few percent accuracy for all practical purposes. The results can be applied to the scattering of electrons with themselves, for which quadrupole radiation is the dominant process.

Influence of bit size on stability bottom drill column balance when drilling a horizontal well

The article raises the question that if you change the bit type, the "critical length" of the lower part of the drill string change too. For the first time, the issue of the stability of the lower part of the drill string when drilling a horizontal section of the wellbore was formulated and solved, taking into account the features of the bit designs (roller bit, wing bit, PDC bit) and the mechanism of rock destruction in well drilling. The research uses the generally accepted provisions of the linear theory of elasticity of materials, the theory of stability of rods and shells. It is assumed that the lower part of the drill string is a heavy elastic rod, experiencing the combined action of the forces of the pipe's own weight and the axial load on the bit, and the twisting moment from the rotor table acting on the pipe string is not taken into account. To find the "critical length" of the lower part of the drill string, simple approximate formulas are obtained that take into account the bit size, drill collar and axial load.

Heat transport in shear-driven flow with axial conduction

The classical Graetz problem of fluid entering a long channel possessing different temperature from that of fluid is extended here to the case where the upper wall of the channel is not stationary. Formulation of the current problem incorporates the simple Couette flow driven by the channel movement and taking into account the axial heat conduction, besides the usual convection. Analytical solutions in the thermally developing regime regarding the temperature distribution as well as the heat transfer rates from both the upper moving and the lower fixed channel walls are then obtained by means of the series expansion method making use of the proper orthogonal family of eigenfunctions. Simple approximate formulae for the rates of heat transfer at the channel entrance valid for large Peclet numbers and at the far downstream locations valid for all Peclet numbers are further derived. More simplistic outcomes are also provided when the axial pressure gradient is assumed to be constant. Results are then discussed from low to high Peclet numbers, which are in compliant with the well-known results available in the literature for the channel flow within stationary walls. On the other hand, the rate of heat transfer from the moving wall is shown to be of high importance when Peclet number increases, which requires special attention in technological applications.