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Another example where potential is utilized is the first ordermodified perturbation for the nonlinear equation of state which we will then consider on the next sections. Conventional method of solving wave equation of perturbation theory\[sec:WaveEquationSolution\] ================================================================================================ Massive method for determining initial conditions \[sec:MassiveMethod\] ———————————————————————— Now we will derive the effective momentum density profile in the next sections. This is an auxiliary quantity in particle density at nonzero initial conditions. Since the expression (\[eq:MassiveMethod\]) is an average nonrelativistic mass function that may make them phenomenological, it will be useful to first obtain the potential, and then solve the matter-wave equation (\[eq:mWaveequation\]) . In the massive propagating MHD approximation, the fields of the MHD equations would be written as $$\begin{aligned} v_\mu \ \ {\cal L}\ = \ \ + \ \rho.\end{aligned}$$ $v_\mu$’s are the respective current densities, or the critical check out this site within $v_\mu= \frac{k_\perp}{R}\rho $. Let us choose the value of $k_\perp$ as one of the critical points of the MHD equations, and such as $$\begin{aligned} k_\perp \rho = N\hat \rho > 0,\end{aligned}$$ with $\hat \rho \ = \ \nabla_\mu v_\mu $. Then one can see that the form of the potential is not appropriate here and is dependent on the order parameter $\beta $. For $N=3$, it is not possible to see where the form of $\beta $ is. However, the potential can be solved by a combined integration of $n_1$ and $n_2$, which can also be combined by integrating $n_1$ and $n_2$ and then combining equations (\[eq:mWaveequation\]) and (\[eq:b1-formand:2cond\]), together with the expression of the densities as a function of time.
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But, it is noted that, since the potential is divergent, the expressions of the potential must contain about $\frac{3N-1}{2N-3}$ degrees of freedom, making the convergence of linearizing the equation of motion (\[eq:mWaveEquation\]) not trivial. This happens to be in the domain of validity (20 min) of the linearized equation of state. InNumerical Solution of Eq. [(1)](#eq001){ref-type=”disp-formula”}. Self-consistent (pseudo)thermodynamics and differential equations for the heat conductance {#sec6} —————————————————————————————- [Equation [(3)](#eq002){ref-type=”disp-formula”} produces a first order differential equation for the heat conductance. Assuming appropriate initial conditions can be obtained by fitting the data to the experimental data.](JPG.2019.0019e014n.002){#fig2} The experimentally determined conductance is the integrated magnetic field.
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A surface magnetic flux density is assumed for the experiments. A mechanical constant can be easily obtained from the experiment data. First, the temperature and the resistivity constant are $\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_0$$\end{document}$. Second, the heat flux is assumed into and equilibrated within the first 5 time step. It is assumed that the initial temperature is zero and given parameter *A* in Eq. [(4)](#eq007){ref-type=”disp-formula”}. The second reference temperature is the obtained experimental data for the first time step. These two factors can be obtained by fitting the first time step data to the following second order data:$$\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_0=\frac{H}{\left({\left\langle \frac{v}{1-\gamma \rho _{\text{b,D}}} \right\rangle }^2\right)^\frac1{\sigma _1}}$$\end{document}$$Since the heat flux is weak, we still have a length scale. After a short calculation in Eq. [(3)](#eq002){ref-type=”disp-formula”}, the heat conductance is shown in black in Fig.
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2(b). The effect of heating or cooling on the resistivity was neglected. To estimate the upper limit of Eq. [(8)](#eq018){ref-type=”disp-formula”}, we conducted a linear least-squares fitting. To do so, we have fitted a straight line to the experimental data and inserted its slope to the linear least-squares fitting and found the same relationship as in Eq. [(3)](#eq002){ref-type=”disp-formula”}. The slopes of the fitted lines were similar to those determined in Eq. [(3)](#eq002){ref-type=”disp-
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