Noether Method Research Articles

Remarks on propagating waves in non-linear vacuum electrodynamics

Using the quadratic expansion in the photon fields of Euler–Heisenberg (EH) non-linear electrodynamics (NLED) Lagrangian model we study relevant vacuum properties in a scenario involving the propagation of a photon probe in the presence of a background constant and static magnetic field, mathbf{B_e}. We compute the gauge invariant, symmetric and conserved energy–momentum tensor (EMT) and angular momentum tensor (AMT) for arbitrary magnetic field strength using the Hilbert method under the soft-photon approximation. We discuss how the presence of magneto-electric terms in the EH Lagrangian is a source of anisotropy, induce the non-zero trace in the EMT and leads to differences between EMT calculated by the Hilbert or Noether method. From the Hilbert EMT we analyze some quantities of interest such as the energy density, pressures, Poynting vector, and angular momentum vector, comparing and discussing the differences with respect to the improved Noether method. The magnetized vacuum properties are also studied showing that a photon effective magnetic moment can be defined for different polarization modes. The calculations are done in terms of derivatives of the two scalar invariants of electrodynamics, hence, extension to other NLED Lagrangian is straightforward. We discuss further physical implications and experimental strategies to test magnetization, photon pressure, and effective magnetic moment.

Numerical investigation on the thermal response of an unsteady magnetohydrodynamics straight porous fin

Fins are frequently used in many heat transfer applications and also for the control and prevention of thermal damage in electronic and mechanical equipment. Thus, the current investigation focuses on estimating the thermal performance, efficiency, and effectiveness of MHD straight porous fin with temperature-dependent heat transfer coefficient, thermal conductivity, and emissivity. The finite-difference approximation is utilized to solve the governing nonlinear differential equation and numerical solutions are endorsed by analytical solutions of the steady-state model using the partial Noether method, Adomian decomposition method, and homotopy perturbation method. From the model predictions, it is noticed that the rise in geometry parameter, magnetic parameter, Rayleigh number, time, and Biot number rises the heat transfer rate of the system. In addition, fin efficiency and effectiveness are directly correlated with boiling parameter, thermal conductivity parameter, and porosity.

Existence of wormholes in f(𝒢) gravity using symmetries

The current study examines the geometry of static wormholes with anisotropic matter distribution in the context of modified [Formula: see text] gravity. We consider the well-known Noether and conformal symmetries, which help in investigating wormholes in [Formula: see text] gravity. For this purpose, we develop symmetry generators associated with conserved quantities by taking into consideration the [Formula: see text] gravity model. Moreover, we use the conservation relationship gained from the classical Noether method and conformal Killing symmetries to develop the metric potential. These symmetries provide a strong mathematical background to investigate wormhole solutions by incorporating some suitable initial conditions. The obtained conserved quantity performs a significant role in defining the essential physical characteristics of the shape-function and energy conditions. Further, we also describe the stability of obtained wormholes solutions by employing the equilibrium condition in modified [Formula: see text] gravity. It is observed from graphical representation of obtained wormhole solutions that Noether and conformal Killing symmetries provide the results with physically accepted patterns.

Sample size calculation for Mann-Whitney U test with five methods

<p><strong>Background: </strong>Precise sample size estimation plays a vital role in the planning of a study specifically for medical treatment expenses that are expensive and studies that are of high risk.</p><p><strong>Methods: </strong>Among a variety of sample size calculation methods for the nonparametric Mann-Whitney U test, five potential methods are selected for evaluation in this article. The evaluation of method performance is based on the results obtained from high precision Monte Carlo simulations.</p><p><strong>Results: </strong>The sample size deviations (from the simulation ones) are performance indicators. The sum of the squared deviations over all scenarios is used as the criterion for ranking the five methods. For power comparisons, the percentage errors (relative to the simulated powers) are used. The effect size and target power both have large impacts on the minimum required sample sizes.</p><p><strong>Conclusions: </strong>Based on the ranking criterion, Shieh's method has the best performance. Noether's method always overestimates the minimum required sample sizes but not too severe.</p>

Exact conservation laws for gauge-free electromagnetic gyrokinetic equations

The exact energy and angular momentum conservation laws are derived by the Noether method for the Hamiltonian and symplectic representations of the gauge-free electromagnetic gyrokinetic Vlasov–Maxwell equations. These gyrokinetic equations, which are solely expressed in terms of electromagnetic fields, describe the low-frequency turbulent fluctuations that perturb a time-independent toroidally-axisymmetric magnetized plasma. The explicit proofs presented here provide a complete picture of the transfer of energy and angular momentum between the gyrocentres and the perturbed electromagnetic fields, in which the crucial roles played by gyrocentre polarization and magnetization effects are highlighted. In addition to yielding an exact angular momentum conservation law, the gyrokinetic Noether equation yields an exact momentum transport equation, which might be useful in more general equilibrium magnetic geometries.

(3 + 1)‐Dimensional time‐fractional modified Burgers equation for dust ion‐acoustic waves as well as its exact and numerical solutions

In this paper, the theoretical model of dust ion‐acoustic waves in collisionless, unmagnetized dust plasma is analyzed. According to the (3 + 1)‐dimensional nonlinear dynamic propagation equations with the electron distribution in the presence of trapping particles and the kinetic equation of charge of dust particle, based on multiscale analysis and perturbation method, a new (3 + 1)‐dimensional modified Burgers equation is constructed. Because of the space property of dust plasma, (3 + 1)‐dimensional modified Burgers equation is more suitable than low‐dimensional equation to describe the dust ion‐acoustic waves. Furthermore, applying the semi‐inverse method and the fractional variational principle, time‐fractional modified Burgers equation is given, which owns potential value for comprehending the propagation feature of dust ion‐acoustic waves in three‐dimensional space. Following, the conservation laws of the time‐fractional modified Burgers equation are discussed by using the Noether method. Finally, the shock wave solutions of the new model is obtained with the help of the G′/G‐expansion method; meanwhile, the radial basis function method is also used to get the numerical solutions of the (3 + 1)‐dimensional time‐fractional modified Burgers equation.

Formula omitted][formula omitted] Lifshitz-Wess-Zumino model: A paradigm of reconciliation between Lifshitz-like operators and supersymmetry

By imposing the weighted renormalization condition and the (super)symmetry requirements, we construct a Lifshitz-like extension of the three-dimensional Wess-Zumino model, with dynamical critical exponent z=2. In this context, the auxiliary field F plays a key role by introducing the appropriate Lifshitz operator in the bosonic sector of the theory, avoiding so undesirable time-space mixing derivatives and inconsistencies concerning the critical z exponent, as reported in the literature. The consistency of the proposed model is verified by building explicitly the susy algebra through the Noether method in the canonical formalism. This component-field Lifshitz-Wess-Zumino model is in addition rephrased in the Lifshitz superspace, a natural modification of the conventional one. The one-loop effective potential is computed to study the possibility of symmetry breaking. It is found that supersymmetry remains intact at one-loop order, while the U(1) phase symmetry suffers a spontaneous breakdown above the critical value of the renormalization point. By renormalizing the one-loop effective potential within the cutoff regularization scheme, it is observed an improvement of the UV behavior of the theory compared with the relativistic Wess-Zumino model.

Perturbative variational formulation of the Vlasov-Maxwell equations

The perturbative variational formulation of the Vlasov-Maxwell equations is presented up to the third order in the perturbation analysis. From the second and third-order Lagrangian densities, the first-order and second-order Vlasov-Maxwell equations are expressed in gauge-invariant and gauge-independent forms, respectively. Upon deriving the reduced second-order Vlasov-Maxwell Lagrangian for the linear nonadiabatic gyrokinetic Vlasov-Maxwell equations, the reduced Lagrangian densities for the linear drift-wave equation and the linear hybrid kinetic-magnetohydrodynamic (MHD) equations are derived, with their associated wave-action conservation laws obtained by the Noether method. The exact wave-action conservation law for the linear hybrid kinetic-MHD equations is written explicitly. Finally, a new form of the third-order Vlasov-Maxwell Lagrangian is derived in which ponderomotive effects play a crucial role.

THERMAL PERFORMANCE ANALYSIS OF CONVECTIVE-RADIATIVE FIN WITH TEMPERATURE-DEPENDENT THERMAL CONDUCTIVITY IN THE PRESENCE OF UNIFORM MAGNETIC FIELD USING PARTIAL NOETHER METHOD

In this paper, thermal performance of convective-radiative straight fin with temperature-dependent thermal conductivity in the presence of uniform magnetic field is analyzed using partial Noether method. The exact analytical solution is used to investigate the effects of magnetic field, convective, radiative, thermo-geometric and thermal conductivity (non-linear) parameters on the thermal performance of the fin. The results reveal that as the magnetic, convective and radiative parameters increase, the temperature of the fin decreases rapidly and by implication, the rate of heat transfer through the fin increases. The study provides a platform for comparison of results of any other method of analysis of the problem with the results of the exact analytical solutions in this paper. Also, such an analytical tool is valuable as a design and optimization approach for large scale (not necessarily in size) finned heat exchangers where each fin/row are analytically analyzed and where the surrounding fluid is influenced by a magnetic field.

The gravitational wave stress–energy (pseudo)-tensor in modified gravity

The recent detections of gravitational waves by the advanced LIGO and Virgo detectors open up new tests of modified gravity theories in the strong-field and dynamical, extreme gravity regime. Such tests rely sensitively on the phase evolution of the gravitational waves, which is controlled by the energy–momentum carried by such waves out of the system. We here study four different methods for finding the gravitational wave stress–energy pseudo-tensor in gravity theories with any combination of scalar, vector, or tensor degrees of freedom. These methods rely on the second variation of the action under short-wavelength averaging, the second perturbation of the field equations in the short-wavelength approximation, the construction of an energy complex leading to a Landau–Lifshitz tensor, and the use of Noether’s theorem in field theories about a flat background. We apply these methods in general relativity, Jordan–Fierz–Brans–Dicky theoy, and Einstein-Æther theory to find the gravitational wave stress–energy pseudo-tensor and calculate the rate at which energy and linear momentum is carried away from the system. The stress–energy tensor and the rate of linear momentum loss in Einstein-Æther theory are presented here for the first time. We find that all methods yield the same rate of energy loss, although the stress–energy pseudo-tensor can be functionally different. We also find that the Noether method yields a stress–energy tensor that is not symmetric or gauge-invariant, and symmetrization via the Belinfante procedure does not fix these problems because this procedure relies on Lorentz invariance, which is spontaneously broken in Einstein-Æther theory. The methods and results found here will be useful for the calculation of predictions in modified gravity theories that can then be contrasted with observations.

Variational principle for the parallel-symplectic representation of electromagnetic gyrokinetic theory

The nonlinear (full-f) electromagnetic gyrokinetic Vlasov-Maxwell equations are derived in the parallel-symplectic representation from an Eulerian gyrokinetic variational principle. The gyrokinetic Vlasov-Maxwell equations are shown to possess an exact energy conservation law, which is derived by the Noether method from the gyrokinetic variational principle. Here, the gyrocenter Poisson bracket and the gyrocenter Jacobian contain contributions from the perturbed magnetic field. In the full-f formulation of the gyrokinetic Vlasov-Maxwell theory presented here, the gyrocenter parallel-Ampère equation contains a second-order contribution to the gyrocenter current density that is derived from the second-order gyrocenter ponderomotive Hamiltonian.

Variational formulations of guiding-center Vlasov-Maxwell theory

The variational formulations of guiding-center Vlasov-Maxwell theory based on Lagrange, Euler, and Euler-Poincaré variational principles are presented. Each variational principle yields a different approach to deriving guiding-center polarization and magnetization effects into the guiding-center Maxwell equations. The conservation laws of energy, momentum, and angular momentum are also derived by Noether method, where the guiding-center stress tensor is now shown to be explicitly symmetric.

Confidence intervals for ratios of proportions: implications for selection ratios

SummaryThe selection ratio (proportional resource use divided by proportional availability) is often used by ecologists to measure the degree to which individuals and populations are selective in their food sources and habitats. Yet, confidence interval approaches for this metric are scarce and poorly evaluated.In this paper, we compare 13 methods that can be used to construct simultaneous confidence intervals for selection ratios. Seven of the methods are applicable when availabilities are unknown. These are bootstrapping and six methods adapted from relative risk (Katz‐log, adjusted‐log, Bailey, inverse hyperbolic sine, Koopman and Noether). The other six approaches are applicable when availabilities are known. These are bootstrapping, two existing methods for relative risk (Wald‐adjusted, Noether‐fixed) and three straightforward new methods (fixed‐log, Agresti–Coull‐adjusted and Bayes‐beta). None of the 13 approaches have been previously evaluated in the context of selection ratios.In simulations with unknown availabilities, the Koopman method performed best, and the currently recommended Noether method performed worst. In simulations with fixed availabilities, the new Agresti–Coull‐adjusted, fixed‐log, Bayes‐beta methods all outperformed the currently recommended Wald‐adjusted method. In the context of real ecological data sets, the Noether and Wald‐adjusted methods produced anomalous results that putatively would alter management decisions. We note that our findings, including those for new methods, are directly conferrable to relative risk, allowing extension of our work to the many branches of biology that rely on this measure.The poor performance of the Noether and Wald‐adjusted methods is troubling because these are currently the most widely used procedures for calculating confidence intervals for type I resource‐selection designs. Based on our findings, we recommend that the Noether method be replaced with the Koopman method, and the Wald‐adjusted method be replaced with the Agresti–Coull‐adjusted method.

Exact momentum conservation laws for the gyrokinetic Vlasov-Poisson equations

The exact momentum conservation laws for the nonlinear gyrokinetic Vlasov-Poisson equations are derived by applying the Noether method on the gyrokinetic variational principle [A. J. Brizard, Phys. Plasmas 7, 4816 (2000)]. From the gyrokinetic Noether canonical-momentum equation derived by the Noether method, the gyrokinetic parallel momentum equation and other gyrokinetic Vlasov-moment equations are obtained. In addition, an exact gyrokinetic toroidal angular-momentum conservation law is derived in axisymmetric tokamak geometry, where the transport of parallel-toroidal momentum is related to the radial gyrocenter polarization, which includes contributions from the guiding-center and gyrocenter transformations.

Higher spin interactions with scalar matter on constant curvature spacetimes: conserved current and cubic coupling generating functions

Cubic couplings between a complex scalar field and a tower of symmetric tensor gauge fields of all ranks are investigated on any constant curvature spacetime of dimension d>2. Following Noether's method, the gauge fields interact with the scalar field via minimal coupling to the conserved currents. A symmetric conserved current, bilinear in the scalar field and containing up to r derivatives, is obtained for any rank r from its flat spacetime counterpart in dimension d+1, via a radial dimensional reduction valid precisely for the mass-square domain of unitarity in (anti) de Sitter spacetime of dimension d. The infinite collection of conserved currents and cubic vertices are summarized in a compact form by making use of generating functions and of the Weyl/Wigner quantization on constant curvature spaces.

Noether derivation of exact conservation laws for dissipationless reduced-fluid models

The energy-momentum conservation laws for general reduced-fluid (e.g., gyrofluid) models are derived by Noether method from a general reduced variational principle. The reduced canonical energy-momentum tensor (which is explicitly asymmetric and has the Minkowski form) exhibits polarization and magnetization effects associated with dynamical reduction. In particular, the asymmetry in the reduced canonical momentum-stress tensor produces a nonvanishing reduced intrinsic torque that can drive spontaneous toroidal rotation in axisymmetric tokamak plasmas.

Lagrange–Noether method for solving second-order differential equations

The purpose of this paper is to provide a new method called the Lagrange–Noether method for solving second-order differential equations. The method is, firstly, to write the second-order differential equations completely or partially in the form of Lagrange equations, and secondly, to obtain the integrals of the equations by using the Noether theory of the Lagrange system. An example is given to illustrate the application of the result.

On higher spin interactions with matter

Cubic couplings between a complex scalar field and a tower of symmetric tensor gauge fields of all ranks are investigated. A symmetric conserved current, bilinear in the scalar field and containing r derivatives, is provided for any rank r>0 and is related to the corresponding rigid symmetry of Klein-Gordon's Lagrangian. Following Noether's method, the tensor gauge fields interact with the scalar field via minimal coupling to the conserved currents. The corresponding cubic vertex is written in a compact form by making use of Weyl's symbols. This enables the explicit computation of the non-Abelian gauge symmetry group, the current-current interaction between scalar particles mediated by any gauge field and the corresponding four-scalar elastic scattering tree amplitude. The exact summation of these amplitudes for an infinite tower of gauge fields is possible and several examples for a definite choice of the coupling constants are provided where the total amplitude exhibits fast (e.g. exponential) fall-off in the high-energy limit. Nevertheless, the long range interaction potential is dominated by the exchange of low-spin particles in the low-energy limit.

Noether's and Poisson's methods for solving differential equation xs(m) = Fs (t,xk(m-2), xk(m-1))

This paper studies integration of a higher-order differential equation which can be reduced to a second-order ordinary differential equation. The solution of the second-order equation can be obtained by the Noether method and the Poisson method. Then the solution of the higher-order equation can be obtained by integrating the solution of the second-order equation.

The Dyon Charge in Noncommutative Gauge Theories

We construct a dyon solution for the noncommutative version of the Yang‐Mills‐Higgs model with a ϑ‐term. Extending the Noether method to the case of a noncommutative gauge theory, we analyze the effect of CP violation induced both by the ϑ‐term and by noncommutativity proving that the Witten effect formula for the dyon charge remains the same as in ordinary space.