Equation For Function Research Articles

Нерелятивистское приближение в 39-компонентной теории для частицы со спином 2

The goal of the present paper is investigation of the nonrelativistic approximation in the 39-component theory for a spin 2 particle. We apply explicit expressions for four main matrices Γa with dimension 39×39 in the relevant first-order system of equations, written in Cartesian coordinates and in presence of external electromagnetic fields. For distinguishing the large and small constituents in the complete wave function, we apply three projective operators constructed on the base of the minimal polynomial of the 7-th order for the matrix Γ0. The relevant large and small components are found in explicit form. In each group, we have found independent variables; in particular, among the large components there exist only five independent ones. We have derived the nonrelativistic equation for 5-component wave function; it has the term describing interaction of the magnetic moment of the spin 2 particle with the external magnetic field we distinguished. This additional term is constructed by the projections of spin operator Si and the components of the magnetic field Bi.

Leading order track functions in a hot and dense QGP

We study the modifications to the fragmentation pattern of partons into charged particles in the presence of a hot and dense quark gluon plasma. To this end, we analyze the perturbative renormalization group equations of the track functions, which describe the energy fraction carried by charged hadrons. Focusing on pure Yang-Mills theory, we compute the lowest-order moments of the medium-modified track functions, which are found to be sensitive to the reduced phase space for emissions in the medium and to energy loss. We use the extracted moments to calculate the energy energy correlator (EEC) on tracks in the collinear limit. The EEC on medium-evolved tracks does not differ qualitatively from the EEC on vacuum tracks despite being sensitive to the color decoherence transition and suppressing the distribution due to quenching, as seen in other jet observables. Published by the American Physical Society 2024

On the dependence of the thermal module of elasticity of a two-component magnetic fluid on frequency, concentration and magnetic field

Purpose. Study of the dependence of the thermal modulus of elasticity of a two-component magnetic fluid on the magnitude of the magnetic field strength, the frequency of external disturbance and the volumetric concentration of magnetic particles.Method. The research method is based on the kinetic theory of liquid systems. Based on previously constructed kinetic equations for one-particle and two-particle distribution functions and a microscopic expression for the heat flux vector, an explicit dynamic expression for the thermal modulus of elasticity of magnetic fluids is obtained. In fast processes in liquids, heat transfer occurs in waves and their propagation is similar to the propagation of second sound in helium II. The thermal modulus of elasticity in liquids appears at high frequencies and ensures the propagation of second sound. The expression for the thermal modulus of elasticity consists of potential and kinetic parts, taking into account structural and translational relaxation processes, respectively. To study the thermoelastic properties of magnetic fluids, appropriate expressions of potential interaction energies were selected for each subsystem, allowing for numerical calculations.Results. Numerical calculations of the frequency and concentration dependence of the dynamic thermal modulus of elasticity in the presence of an external magnetic field in a kerosene-based magnetic fluid were carried out. The calculation results show that an increase in the influence of external disturbances leads to a nonlinear increase in the thermal modulus of elasticity in the magnetic fluid. An increase in the volume concentration of magnetic particles and an increase in the magnetic field strength also led to a nonlinear increase in the thermal modulus of elasticity in the magnetic fluid.Conclusion. It has been established that, due to taking into account translational and structural relaxation processes, the region of frequency dispersion of the thermal elastic modulus is wide. Numerical calculations carried out at different values of the external magnetic field and the volume concentration of magnetic particles showed that although an increase in the magnetic field and concentration of magnetic particles leads to an increase in the thermal modulus of elasticity, their increase does not affect the change in the frequency dispersion region.

Eliashberg Theory for Dynamical Screening in Bilayer Exciton Condensation.

We study the effect of dynamical screening of interactions on the transition temperatures (T_{c}) of exciton condensation in a symmetric bilayer of quadratically dispersing electrons and holes by solving the linearized Eliashberg equations for the anomalous interlayer Green's functions. We find that T_{c} is finite for the range of density and layer separations studied, decaying exponentially with interlayer separation. T_{c} is suppressed well below that predicted by a Hartree Fock mean field theory with unscreened Coulomb interaction, but is above the estimates from the statically screened Coulomb interaction. Furthermore, using a diagrammatic framework, we show that the system is always an exciton condensate at zero temperature but T_{c} is exponentially small for large interlayer separation.

An analytical model of tornado generation

A new analytical model for the generation of axisymmetric tornado-type vortices has been developed. A solution to the nonlinear equation for the stream function in an unstable stratified atmosphere is obtained and analyzed within the framework of ideal hydrodynamics. The solution is sought by smooth connecting continuous solutions for the internal region (eye), the central region (“wall” with maximum velocities), and the external region of the tornado. Expressions describing radial dependences for the radial and vertical velocity components include combinations of Bessel functions. The vortex is spatially localized by radius and height. Convective instability of a stratified atmosphere leads to an increase in the radial and vertical components of velocities according to the hyperbolic sine law. A downward flow is observed near the tornado axis. The maximum speed of the upward flow is achieved at a certain radial distance at a certain height. Below this height, radial flows converge toward the central part of the tornado, and above this height, there is an outflow from the wall to the axis and to the periphery. The radial structure of the azimuthal velocity is determined by the structure of the initial disturbance and can change with height. Maximum rotation is achieved in the tornado wall at a certain height. The increase in azimuthal velocity can occur according to a superexponential law. Possible structures of movements, scenarios for the development of a tornado, and its dynamics are discussed.

Deret Fourier dengan Pendekatan Numerik Menggunakan Metode Trapezium

Fourier series has the ability to decompose a periodic function into a collection of sine and cosine wave sums. Another way to express a function into an infinite series is with the Maclaurin series or Taylor series but both of these series require that the function must have the first derivative, the second derivative and so on of the function is defined at its expansion point and there must be no discontinuity value at that point. However, the advantage of the Fourier series is that we can decompose the function into an infinite series containing the sum of sine and cosine terms even though the function cannot be differentiated and has discontinuity at some points. The Fourier series equation of an analytical function is obtained by first finding the values ​​of the Fourier coefficients using analytical integrals. In the case of functions formed from numerical data that are assumed to have a periodic pattern, the Fourier series formula that uses analytical integrals to find its Fourier coefficients can no longer be used so the numerical approach using the Trapezium method is an alternative that allows us to find the Fourier series equation for a periodic function expressed numerically. We use 12 points as the basis for the calculation in the Trapezium method to approximate the Fourier series numerically. This paper presents a numerical approach to solving the problem.

Wigner Function Method for Describing Electromagnetic Field in Plasma-Like Media with Spatial Dispersion and Resonant Dissipation

The paper gives a systematic presentation of the Wigner function method (Weyl formalism) for modeling the propagation and absorption of electromagnetic waves in anisotropic and gyrotropic dissipative media with spatial dispersion. A general kinetic equation for the Wigner function (tensor) is formulated, and its asymptotic expansion up to the second order for smoothly inhomogeneous and weakly dissipative media is constructed. As a result, a modification of the method of the kinetic equation for rays is proposed, based on the stochastic description of rays, making it possible to increase the accuracy of numerical modeling of wave problems with strong transverse inhomogeneity of the absorption coefficient without increasing the amount of calculations. The technique developed can be used to describe the propagation, absorption, and scattering of electron-cyclotron waves in high-temperature plasma of magnetic traps for controlled fusion in cases where standard modeling methods do not provide the necessary accuracy.

A KP-mKP hierarchy via pseudo-differential operators with two derivations

By using pseudo-differential operators containing two derivations, we extend the Kadomtsev–Petviashvili (KP) hierarchy to a certain KP-mKP hierarchy. For the KP-mKP hierarchy, we derive its Bäcklund transformations, bilinear equations of Baker–Akhiezer functions and Hirota equations of tau functions. Moreover, we show that this hierarchy is equivalent to a subhierarchy of the dispersive Whitham hierarchy associated to the Riemann sphere with its infinity point and one movable point marked.

Exploring self-gravitating cylindrical structures in modified gravity: Insights from scalar-vector-tensor theory

We investigate static cylindrical solutions within an extended theory of modified gravity. By incorporating various coupling functions through a straightforward boost symmetry approach, we establish the equations of motion in a self-consistent manner and subsequently determine the linear scalar field profile. Utilizing analytical methods, we solve the system of equations for the metric functions and the U(1) gauge field, revealing their dependence on Bessel's functions. To comprehend gravito-objects exhibiting cylindrical symmetry, we develop a perturbative framework aimed at identifying all nontrivial solutions for the scalar profiles. Introducing first-order truncated perturbation equations for the gauge field, synchronized with metric gauges and electromagnetic field considerations, we demonstrate their integrability and obtain solutions through quadrature. Our findings suggest the feasibility of obtaining self-gravitating cylindrical structures within the scalar-vector-tensor theory. These cylindrical structures could provide insights into the behavior of gravitational and gauge fields in modified gravity, potentially offering new perspectives on astrophysical phenomena such as cosmic strings and cylindrical gravitational waves.

Dynamics of nucleation in binary alloys

ABSTRACT Nucleation in binary alloys is studied in the capillary approximation of classical theory. By allowing both the size of the cluster and its composition to vary, the phase transition is investigated in a two-dimensional space. This parametrisation allows to derive diffusivities of the Fokker-Planck equation for the distribution function of clusters from the Cahn–Hilliard equation. It is shown how kinetics, together with thermodynamics, determine the direction of the nucleation current as well as the magnitude of the nucleation rate. The properties of the critical clusters and the direction of the nucleation current at the saddle point are derived for a generic model, from the binodal line to the spinodal limit. The critical clusters are found to exhibit properties that are very similar to that of non-classical theories. Once moderate supersaturation is reached, the interplay between kinetics and thermodynamics is found to invalidate the classical picture by modifying the direction of the nucleation current. The consequences on the magnitude of the rate of nucleation are discussed. The model is then applied to decomposition of FeCr solid solutions and is shown to constitute a reasonable sharp-interface approximation of the diffuse-interface theory of nucleation for the determination of both the cluster properties and the nucleation rate.

Transient nonlocal heat flux in simple metals at low excitations

In this paper we discuss the ultrafast transfer of charge and energy in a metal in the kinetic approximation. An exact solution of the Boltzmann transport equation for the electron distribution function in the relaxation time approximation is obtained, which is used to calculate transport coefficients for charge and heat fluxes. Different regimes of energy transfer under conditions of significant spatial dispersion and non-equilibrity of the electron distribution function are discussed.

Precision modelling of leaf area index for enhanced surface temperature partitioning and improved evapotranspiration estimation

Remote Sensing-based two-source model is widely used to estimate crop evapotranspiration (ET), involving one key step of partitioning land surface temperature (LST) into canopy and soil temperatures (Tc and Ts). Leaf area index (LAI) plays a significant part in available energy allocation during this process. However, the asymptotic saturation problem makes the mismatch between vegetation index and LAI. In this study, two-stage LAI models were developed through the red-edge chlorophyll index (CIred-edge) considering the hysteresis between them. Considering the distinct characteristics, modeling LAI by one-degree linear equations for sunflower (C3), linear and exponential functions for maize (C4) were presented in the distinguished grow-up and senescence periods. The two-source energy balance (TSEB) and hybrid dual-source scheme and trapezoid framework-based evapotranspiration (HTEM) models were selected to estimate Tc, Ts, ET, and its components contrastively. The established LAI models and other modified parameters were then integrated into the two models to improve the estimation of Tc, Ts, and ET (named the R-TSEB and R-HTEM models, respectively). Results demonstrated that the partitioned Tc & Ts became closer to the measurements after utilizing the presented LAI models. For daily ET, the R-TSEB and R-HTEM models alleviated the overestimation and underestimation existing in the original two models, respectively. At monthly and seasonal scales compared to the water balance results (ETwb), the ET of R-TSEB model had significant promotion, including the determination coefficient (R2), mean relative error (RE), root mean square error (RMSE), and model agreement index (d) with values of 0.87, 6.54%, 11.65 mm, and 0.95, from the according values of 0.80, 12.85%, 17.60 mm, and 0.90 for the TSEB model, respectively. The estimated ET by the R-HTEM model was more consistent with ETwb than the HTEM model. These results indicate that the established LAI models can enhance ET estimation and further advance water cycle understanding.

Study of salt free-diffusion by 1D transport numerical simulations and shadowgraph experiments

Applying the Nernst-Planck formalism of 1D diffusive flux equations for ionic species of salts dissolved in water, Pitzer formalism for activity coefficients and PHREEQC software for species diffusion coefficients, with ionic strength-dependence coefficients optimized, we obtained expressions for apparent diffusion coefficients of sodium chloride, calcium chloride, and sodium sulphate, prevalent in deep aquifers. PHREEQC 1D transport simulations of free-diffusion experiments yielded temporal evolution of the concentration and gradient vertical profiles. The dynamics of concentrations non-equilibrium fluctuations during the free-diffusion experiments have been recorded by shadowgraphy. Thanks to the calculated gradient profiles the thickness of the sample has been integrated in the analysis equations of the structure functions. From the earliest stages of the diffusion process we measured the diffusion coefficients for the three salts, in very good agreement with reference measurements, validating the technique and method of analysis for studying free-diffusion in reservoirs targeted for CO2 and energy vectors storage.

Some Differential Equations for the Riemann $\theta$-Function on Jacobians

We prove some differential equations for the Riemann theta function associated to the Jacobian of a Riemann surface. The proof is based on some variants of a formula by Fay for the theta function, which are motivated by their analogues in Arakelov theory of Riemann surfaces.

Development of reference values and equations for the pulmonary function of Nigerian children aged 6–11 years measured with digital peak flow meter and its validation against local and the Global Lung Function Initiative (GLI) equations

Development of reference values and equations for the pulmonary function of Nigerian children aged 6–11 years measured with digital peak flow meter and its validation against local and the Global Lung Function Initiative (GLI) equations

Small-scale Dynamo with Nonzero Correlation Time

The small-scale dynamo is typically studied by assuming that the correlation time of the velocity field is zero. Some authors have used a smooth renovating flow model to study how the properties of the dynamo are affected by the correlation time being nonzero. Here, we assume the velocity is an incompressible Gaussian random field (which need not be smooth), and derive the lowest-order corrections to the evolution equation for the two-point correlation of the magnetic field in Fourier space. Using this, we obtain the evolution equation for the longitudinal correlation function of the magnetic field (M L ) in nonhelical turbulence, valid for arbitrary Prandtl number. The nonresistive terms of this equation do not contain spatial derivatives of M L of order greater than 2. We further simplify this equation in the limit of high Prandtl number, and find that the growth rate of the magnetic energy is much smaller than previously reported. Nevertheless, the magnetic power spectrum still retains the Kazantsev form at high Prandtl number.

Time-dependent electron Boltzmann equation for hypersonic plasmas

The electron kinetics in hypersonic plasmas is modeled by solving the time-dependent electron Boltzmann equation for the electron energy distribution function (EEDF). This plasma is created by strong shock compression of the gas in front of a vehicle moving with hypersonic speed. The main source of energy for the electrons is gas heating due to elastic collisions and second-kind collisions (de-excitation) from vibrationally excited states of N2. We established that the electron energy distribution function is most sensitive to vibrational level populations. At mid-altitudes (tens of kilometers), the electron temperature equilibrates with the vibrational temperature on a microsecond timescale. The electron distribution function reaches steady state on a comparable timescale. Numerical simulations of air plasma showed that the electron energy distribution function is far from Maxwellian and the collision rates differ by orders of magnitude from those computed with a Maxwellian distribution. The two most important parameters for the electron kinetics and the electron energy distribution function are the vibrational temperature and ionization degree.

To plasma electrons motion theory in high-frequency fields

The article is dedicated to the kinetic theory of plasma, where the problem of interaction of high-frequency electric fields with weakly non-uniform plasma is investigated using kinetic equations with binary collision integrals for plasma particles. A new methodology for determining the expression of the averaged highfrequency pressure force is proposed based on solving the kinetic equation and using the method of successive approximations (separation of slow motions and fast oscillations) under the limiting conditions. An expression for the averaged quasi-potential force is derived based on the kinetic equation for the electron distribution function in weakly inhomogeneous magnetoactive plasma, taking into account electrons collisions with fixed ions and the presence of a longitudinally inhomogeneous high-frequency electric field using wellknown methods of theoretical and mathematical physics, such as successive approximations, averaging over the effective field oscillation period, and integration over trajectories. The amplitude of the field is a slowly varying function in both time and space coordinates. The obtained expression allows us to estimate the influence of collisions of plasma particles on the Miller force, and under limiting conditions it coincides with the known expressions for the high-frequency pressure force derived from the equation of plasma electrons motion in high-frequency fields. In all calculations, the contribution of the magnetic component of the electromagnetic field is neglected, which is quite justified for the longitudinal electric field.

Numeral simulation of solution mixing effect and product preparation effect in the process of preparing cerium oxide by microwave heating

Microwave heating has the characteristics of simple control and high efficiency, in which electromagnetic force plays an important role. Using the UDF function and energy equation in FLUENT software, the electromagnetic force model in the tube is established. The mixing process of cerium chloride solution under different heating modes, the mixing effect of Lorentz force and the preparation process of cerium oxide under different inlet material velocities were numerically simulated. CFD-Post and MATLAB software are used to post-process the data of the whole fluid area. Through the above analysis, the best process conditions of numerical simulation and experiment are finally determined. The results show that microwave heating has better mixing effect and lower material loss compared with conventional heating, in which electromagnetic force plays a promoting role; Under the condition of microwave heating, the product conversion rate is high, and the concentration of cerium oxide is high and distributed evenly. The more appropriate mixing method of cerium chloride and preparation method of cerium oxide are microwave heating, and the more appropriate inlet velocity of materials is 0.15 m/s.

A Modification to the Novel Toy Model of the Riemann Zeta Function Roots Equation

It is true that my previous series of papers about the Riemann Hypothesis has raised lots of discussions or the concerns among the mathematics society. Indeed, some professionals may think that the Riemann Zeta function at s = 1 or ∑_(n=1)^∞▒1/n is actually divergent and may tend to an infinity, how it can be acted as a denominator in a proof or something like 1/∞ ? This author’s idea is there may be no need to evaluate such kind of summation as one may consider the sum just as an (imaginary) constant (but NOT the complex number). Or we may only consider such summation as a kind of improper integrals with a black-boxed answer. In other words, the true final value of the summation ∑_(n=1)^∞▒1/n is NOT an concerning serious issue. The function ∑_(n=1)^∞▒1/n^s is in practice a smooth function and equals to log N or the logarithmic function for s = 1 or have the infinite differentiable properties except n = 0, that is why the commercial software Maple Soft can compute the Zeta function’s associated Taylor series successfully. In fact, from the above series together with the harmonic properties, one may calculate the artifical Zeta function’s root model equation such as the 0.5 +/- β*cot(ln(x))/(αx+1)n (will be discussed in my next paper of the series). Indeed, this is NOT the best model. Actually, the true Riemann Zeta non-trivial Root model equation is: {z = x + yI | z∈0.5±(y_1⩽y⩽y_2 )I} where y1= (±cot(k))/ln(k) , y2 = (±tan(k))/ln(k) with 0<k<2π. Certainly, this author cannot avoid mistakes. Some of the defects in the previous paper such as one in the linguistic proof of the RH will be stated and amended. Lastly, this author wants to remind that the present series of paper is first to establish a model equation for zeta function’s roots and the Plank’s like constant. This author then applies the telescopic and logarithmic methods for a 3 cases proof of the RH. Also, we have a Matlab code for the evaluation of RH complex contour integral. Finally, the RH problem will be transformed into a linguistic one and hence we may have solved the Riemann Hypothesis issue completely through a 4 cases of truth table collaborations with the logical inference [7] & [15] to the causality (structural) organization of the sentences [16] or just named as “Logical & Organized Context”.