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Background: Isolated gastrocnemius tightness (IGT) is a prevalent condition linked to various foot pathologies. In a previous quantitative gait analysis study, we identified an increase in knee flexion during the midstance phase in IGT patients compared with controls. Although stretching and eccentric exercises (the Stanish protocol) are commonly used for IGT management, their impact on gait parameters remains poorly understood. This study aimed to assess the influence of a Stanish protocol on gait parameters in bilateral IGT subjects. Methods: We enrolled 10 asymptomatic bilateral IGT subjects and 10 controls. Quantitative gait analysis and dynamic baropodometry were carried out on each subject. A Stanish protocol was applied for 4 weeks (five sessions/week) by the IGT group, followed by a similar gait analysis. The ankle and knee range of motion and foot pressure distribution were assessed during the midstance phase of the gait. Results: An increase in knee flexion was initially present in the IGT group compared with controls (8.9 +/− 4.6 vs. 3.4 +/− 2.3 degrees, p < 0.001). There was no difference in the ankle range of motion and foot pressures between the groups at that time. Significant reductions in knee flexion during gait were observed in the IGT subjects after the Stanish protocol (8.9 +/− 4.6 to 3.7 +/− 2.3 degrees, p < 0.001) with a normalization of this parameter (3.4 +/− 2.3 in controls vs. 3.7 +/− 2.3 degrees in IGT, p = 0.72). There was no change in ankle range of motion and foot pressure after the Stanish protocol. Conclusions: Our findings support the effectiveness of the Stanish protocol in reducing knee flexion and normalizing gait in IGT subjects. This protocol not only offers a noninvasive approach for IGT-related issues management but could also enable prophylactic care in asymptomatic cases. [ABSTRACT FROM AUTHOR]

This study examined the validity and compared the precision and accuracy of a distance-time linear model (DTLM), a power law and a nomogram to predict the distance running performances of female runners. Official rankings of French women ("senior" category: between 23 and 39 years old) for the 3000-m, 5000-m, and 10,000-m track-running events from 2005 to 2019 were examined. Performances of runners who competed in the three distances during the same year were noted (n=158). Mean values and standard deviation (SD) of actual performances were 11.28±1.33, 19.49±2.34 and 41.03±5.12 for the 3000-m, 5000-m, and 10,000-m respectively. Each performance was predicted from two other performances. Between the actual and predicted performances, only DTLM showed a difference (p< 0.05). The magnitude of the differences in these predicted performances was small if not trivial. All predicted performances were significantly correlated with the actual ones, with a very high correlation coefficient (p< 0.001; r> 0.90), except for DTLM in the 3000-m, which showed a high correlation coefficient (p< 0.001; r >0.895). Bias and 95% limits of agreement were acceptable because, whatever the method, they were≤–3.7±10.8% on the 3000-m, 1.4±4.3% on the 5000-m, and -2.5±7.4% on the 10,000-m. The study confirms the validity of the three methods to predict track-running performance and suggests that the most accurate and precise model was the nomogram followed by the power law, with the DTLM being the least accurate. [ABSTRACT FROM AUTHOR]

This study examined the validity, precision and accuracy of the predictions of distance running performances in female runners from three nomograms. Official rankings of French women for the 3000-m, 5000-m, and 10 000-m track-running events from 2005 to 2019 were examined. Only female runners who performed in the three distance events within the same year were included (n=158). Each performance over any distance was predicted using the three nomograms from the two other performances. The 3000-m, 5000-m and 10 000-m performances were 11min17 s±1min20 s, 19min29 s±2min20 s, 41min18 s±5min7 s, respectively. No difference was found between the actual and predicted running performances regardless of the nomogram (p>0.05). All predicted running performances were significantly correlated with the actual ones, with a very high correlation coefficient (p<0.001; r>0.90). Bias and 95% limits of agreement were acceptable because, whatever the nomogram, they were less than or equal to − 0.0±6.2% on the 3000-m, 0.0±3.7% on the 5000-m, and 0.1±9.3% on the 10 000-m. The study confirms the validity of the three nomograms to predict track-running performance with a high level of accuracy. The predictions from these nomograms are similar and may be used in training programs and competitions. [ABSTRACT FROM AUTHOR]

Exact moments of the Boltzmann collision operator are calculated in the irreducible Hermitian moment expansion written in terms of the random-velocity variable of each species. The formulas are presented in closed, algebraic form and can be straightforwardly implemented in computer algebra systems. They are valid for two arbitrary masses, temperatures, and flow velocities, and hence include all other existing results derived for distribution functions expanded with respect to reference states of one temperature and flow velocity. In comparison, the Landau collisional moments are good approximations for large Coulomb logarithm and small relative flow velocity, but they fail to predict the correct behavior of most collisional moments for large relative flow even for weakly coupled plasmas. [ABSTRACT FROM AUTHOR]

The magnetized resistivity and electrothermal tensors when substituted into the induction equation lead to electrothermal magnetic field generation, resistive magnetic diffusion, and magnetic field advection due to resistivity gradients, temperature gradients, and currents. The advection terms driven by the temperature gradient and current have cross field components (perpendicular to both the magnetic field and the driving term) that depend on significantly modified versions of Braginskii's transport coefficients [S. I. Braginskii, in Reviews of Plasma Physics, edited by M. A. Leontovich (Consultants Bureau, New York, 1965), Vol. 1, p. 205]. The improved fits to Braginskii's coefficients given by Epperlein and Haines [Phys. Fluids 29, 1029 (1986)] and Ji and Held [Phys. Plasmas 13, 042114 (2013)] give physically incorrect results for cross field advection at small Hall parameters (product of cyclotron frequency and collision time). The errors in Epperlein and Haines' fits are particularly severe, giving increasing advection velocities below a Hall parameter of one when they should decrease linearly to zero. Epperlein and Haines' fits can also give erroneous advection terms due to variations in the effective atomic number. The only serious error in Braginskii's fits is an overestimate in advection due to perpendicular resistivity. New fits for the cross field advection terms are obtained from a direct numerical solution of the Fokker–Planck equation and Ji and Held's higher order expansion approach that are continuous functions of the effective atomic number. [ABSTRACT FROM AUTHOR]

Integral (nonlocal) closures [J.-Y. Ji and E. D. Held, Phys. Plasmas 21, 122116 (2014)] are combined with the momentum balance equation to derive electron parallel transport relations. For a single harmonic fluctuation, the relations take the same form as the classical Spitzer theory (with possible additional terms): The electric current and heat flux densities are connected to the modified electric field and temperature gradient by transport coefficients. In contrast to the classical theory, the dimensionless coefficients depend on the collisionality quantified by a Knudsen number, the ratio of the collision length to the angular wavelength. The key difference comes from the proper treatment of the viscosity and friction terms in the momentum balance equation, accurately reflecting the free streaming and collision terms in the kinetic equation. For an arbitrary fluctuation, the transport relations may be expressed by a Fourier series or transform. For low collisionality, the electric resistivity can be significantly larger than that of classical theory and may predict the correct timescale for fast magnetic reconnection. [ABSTRACT FROM AUTHOR]

Ion parallel closures are obtained for arbitrary atomic weights and charge numbers. For arbitrary collisionality, the heat flow and viscosity are expressed as kernel-weighted integrals of the temperature and flow-velocity gradients. Simple, fitted kernel functions are obtained from the 1600 parallel moment solution and the asymptotic behavior in the collisionless limit. The fitted kernel parameters are tabulated for various temperature ratios of ions to electrons. The closures can be used conveniently without solving the kinetic equation or higher order moment equations in closing ion fluid equations. [ABSTRACT FROM AUTHOR]

Electron parallel closures for the ion charge number Z=1 [J.-Y. Ji and E. D. Held, Phys. Plasmas 21, 122116 (2014)] are extended for 1≤Z≤10. Parameters are computed for various Z with the same form of the Z=1 kernels adopted. The parameters are smoothly varying in Z and hence can be used to interpolate parameters and closures for noninteger, effective ion charge numbers. [ABSTRACT FROM AUTHOR]

According to analytical calculations of the ion collision operator, the ion-electron collision terms could be larger than the ion-ion collision terms. In the previous work [J.-Y. Ji and E. D. Held, Phys. Plasmas 20, 042114 (2013)], the ion-electron collision effects are diminished by the ion temperature change terms introduced from unlikely assumptions. In this work, the high-collisionality closures for ions are calculated without the temperature change terms. The ion-electron collision terms significantly modify existing closure coefficients. [ABSTRACT FROM AUTHOR]

Electron parallel closures for heat flow, viscosity, and friction force are expressed as kernelweighted integrals of thermodynamic drives, the temperature gradient, relative electron-ion flow velocity, and flow-velocity gradient. Simple, fitted kernel functions are obtained for arbitrary collisionality from the 6400 moment solution and the asymptotic behavior in the collisionless limit. The fitted kernels circumvent having to solve higher order moment equations in order to close the electron fluid equations. For this reason, the electron parallel closures provide a useful and general tool for theoretical and computational models of astrophysical and laboratory plasmas. [ABSTRACT FROM AUTHOR]

Mathematical formalism to solve a system of general moment equations [J.-Y. Ji and E. D. Held, Phys. Plasmas 13, 102103 (2006); 16, 102108 (2009)] for magnetized plasmas is presented. A series of ordered moment equations are written using a perturbative expansion based on large cyclotron frequency. For the most general solution, formulas for homogeneous and particular solutions are obtained. These formulas generalize the CGL [G. F. Chew et al., Proc. R. Soc. London, Ser. A 236, 112 (1956)] and non-CGL [C. T. Hsu et al., Phys. Fluids 29, 1480 (1986)] tensors, respectively, from rank-2 to arbitrary rank. The parallel moment equations to determine parallel moments in the homogeneous solution are derived. The formalism can be applied to plasmas of general collisionality and magnetic geometry with accurate collision operators provided. [ABSTRACT FROM AUTHOR]

Parallel closures are obtained by solving a linearized kinetic equation with a model collision operator using the Fourier transform method. The closures expressed in wave number space are exact for time-dependent linear problems to within the limits of the model collision operator. In the adiabatic, collisionless limit, an inverse Fourier transform is performed to obtain integral (nonlocal) parallel closures in real space; parallel heat flow and viscosity closures for density, temperature, and flow velocity equations replace Braginskii's parallel closure relations, and parallel flow velocity and heat flow closures for density and temperature equations replace Spitzer's parallel transport relations. It is verified that the closures reproduce the exact linear response function of Hammett and Perkins [Phys. Rev. Lett. 64, 3019 (1990)] for Landau damping given a temperature gradient. In contrast to their approximate closures where the vanishing viscosity coefficient numerically gives an exact response, our closures relate the heat flow and nonvanishing viscosity to temperature and flow velocity (gradients). [ABSTRACT FROM AUTHOR]

The full Coulomb collision operator and its moments including nonlinear terms are analytically calculated in the moment expansion. In coupling nonlinear terms, the product formula which expresses a product of two harmonic tensors as a series of single harmonic tensors is derived. The collision operators and moments are written in explicit formulas for arbitrary moments and for arbitrary temperature and mass ratios. These expressions easily reduce to formulas for the small mass-ratio approximation or for like species. [ABSTRACT FROM AUTHOR]

In the moment approach, a parallel electron heat flux density is obtained for arbitrary collisionality in the transport ordering. The parallel moment equations are derived from the drift kinetic equation with the exact linearized Landau operator and analytically solved for the heat flux in integral form. Quantitative analysis of the integral heat flux for sinusoidal temperature profiles shows that the number of moments required for convergence increases as collision length increases. The integral heat flux well agrees with the Braginskii heat flux and the collisionless heat flux for high and low collisionalities, respectively. Incorrect application of the Braginskii heat flux in moderately collisional or nearly collisionless plasmas often leads higher heat flux than the integral closure. This fact is consistent with numerical studies of electron heat confinement in the sustained spheromak physics experiment [Hooper et al., Nucl. Fusion 39, 863 (1999)] which show a modest (6%) increase in core temperature when the integral heat flux is adopted. [ABSTRACT FROM AUTHOR]

The general moment equations for an electron-ion plasma are established. The distribution functions for electrons and ions are expanded in terms of orthogonal polynomials of random velocity variables in contrast to the total velocity variables [J.-Y. Ji and E. D. Held, Phys. Plasmas 13, 102103 (2006)]. The moments of the streaming part of the kinetic equation are explicitly written with simple formulas. A simple version of the exact linearized Coulomb collision integrals is presented for like species. The electron-ion and ion-electron operators that conserve momentum and energy are also calculated with a small mass-ratio approximation. It is shown in the relaxation theory that the Lorentz operator, as a replacement of the like-species operator, is acceptable only for the high-order harmonic moments. [ABSTRACT FROM AUTHOR]

In the moment expansion, the Rosenbluth potentials, the linearized Coulomb collision operators, and the moments of the collision operators are analytically calculated for any moment. The explicit calculation of Rosenbluth potentials converts the integro-differential form of the Coulomb collision operator into a differential operator, which enables one to express the collision operator in a simple closed form for any arbitrary mass and temperature ratios. In addition, it is shown that gyrophase averaging the collision operator acting on arbitrary distribution functions is the same as the collision operator acting on the corresponding gyrophase averaged distribution functions. The moments of the collision operator are linear combinations of the fluid moments with collision coefficients parametrized by mass and temperature ratios. Useful forms involving the small mass-ratio approximation are easily found since the collision operators and their moments are expressed in terms of the mass ratio. As an application, the general moment equations are explicitly written and the higher order heat flux equation is derived. [ABSTRACT FROM AUTHOR]

Systems of algebraic equations for a high-collisionality electron-ion plasma are constructed from the general moment equations with linearized collision operators [J.-Y. Ji and E. D. Held, Phys. Plasmas 13, 102103 (2006) and J.-Y. Ji and E. D. Held, Phys. Plasmas 15, 102101 (2008)]. A systematic geometric method is invented and applied to solve the system of equations to find closure and transport relations. It is known that some closure coefficients of Braginskii [S. I. Braginskii, Reviews of Plasma Physics (Consultants Bureau, New York, 1965), Vol. 1] are in error up to 65% for some finite values of x (cyclotron frequency × electron-ion collision time) and have significant error in the large-x limit [E. M. Epperlein and M. G. Haines, Phys. Fluids 29, 1029 (1986)]. In this work, fitting formulas for electron coefficients are obtained from the 160 moment (Laguerre polynomial) solution, which converges with increasing moments for x≤100 and from the asymptotic solution for large x-values. The new fitting formulas are practically exact (less than 1% error) for arbitrary x and Z (the ion charge number, checked up to Z = 100). The ion coefficients for equal electron and ion temperatures are moderately modified by including the ion-electron collision operator. When the ion temperature is higher than the electron temperature, the ion-electron collision and the temperature change terms in the moment equations must be kept. The ion coefficient formulas from 3 moment (Laguerre polynomial) calculations, precise to less than 0.4% error from the convergent values, are explicitly written. [ABSTRACT FROM AUTHOR]

The kinetic equation for a single-component uniform plasma in a magnetic field is analytically solved by the moment method. The linear system of ordinary differential equations for the moments is decomposed into subsystems of lower dimensions by a geometric method. The eigensystem of each subsystem shows that parallel moments decay monotonically, but perpendicular lth harmonic moments decay while oscillating with the l,l-2,… ,-th harmonics of gyrofrequency. A generalization to a multicomponent plasma is discussed.