Symmetries Of Equations Research Articles

Pseudo-holomorphic functions at the critical exponent

We study Hardy classes on the disk associated to the equation \bar \partial w= \alpha \bar w for \alpha \in L^r with 2 \leq r < \infty . The paper seems to be the first to deal with the case r=2 . We prove an analog of the M. Riesz theorem and a topological converse to the Bers similarity principle. Using the connection between pseudo-holomorphic functions and conjugate Beltrami equations, we deduce well-posedness on smooth domains of the Dirichlet problem with weighted L^p boundary data for 2D isotropic conductivity equations whose coefficients have logarithm in W^{1,2} . In particular these are not strictly elliptic. Our results depend on a new multiplier theorem for W^{1,2}_0 -functions.

Method of Moving Frames to Solve (An)isotropic Diffusion Equations on Curved Surfaces

As a continuing effort to develop the method of moving frames (MMF) ensuing (Chun in J Sci Comput 53(2):268---294, 2012), a novel MMF scheme is proposed to solve (an)isotropic diffusion equations on arbitrary curved surfaces. First we show that if the divergence of a vector is computed exactly on the surface, the mixed formulations expanded in the moving frames are equivalent to the Laplace---Beltrami operator. Otherwise, the divergence error dominates, but it can be made negligible by either way; the use of a higher order differentiation scheme more than the first order or the alignment of the moving frames. Moreover, the propagational property of the media along a specific direction, known as anisotropy, is represented by the rescaling of the moving frames, not by repetitive multiplications of the diffusivity tensor, without adding any schematic complexity nor deterioration of the accuracy and stability to the isotropic diffusion scheme. Convergence results for a spherical shell, an irregular surface, and a non-convex surface are displayed with several examples of modeling anisotropy on various curved surfaces. A computational simulation of atrial reentry is illustrated as an exemplary use of the MMF scheme for practical applications.

Integrable vector isotropic equations admitting differential substitutions of first order

A symmetry classification of integrable vector evolution equations of third order admitting Miura-type transformations is presented. We obtain the Backlund autotransformation for the new equation as well as differential substitutions relating the solutions of some integrable isotropic equations.

Light propagation in structural anisotropic media in the steady-state and time domains

The determination of the reduced scattering and absorption coefficients of structural anisotropic turbid semi-infinite media and slabs was investigated in the steady-state and time domains. Forward calculations were performed with a Monte Carlo model that considered both cylindrical scatterers aligned in different directions as well as scatterers that were described by a rotationally symmetric scattering function. Analytical solutions of the isotropic and anisotropic diffusion equations were applied to retrieve the optical properties. It was found in the steady-state and time domains that the solutions of the anisotropic diffusion equation have systematic errors compared to the Monte Carlo simulations not only for small distances from the source. However, it is shown that in the time domain it is possible to retrieve useful values for the optical properties using the isotropic and the anisotropic diffusion equations.

Canonical quantization of static spherically symmetric geometries

The conditional symmetries of the reduced Einstein–Hilbert action emerging from a static, spherically symmetric geometry are used as supplementary conditions on the wave function. Based on their integrability conditions, only one of the three existing symmetries can be consistently imposed, while the unique Casimir invariant, being the product of the remaining two symmetries, is calculated as the only possible second condition on the wave function. This quadratic integral of motion is identified with the reparametrization generator, as an implication of the uniqueness of the dynamical evolution, by fixing a suitable parametrization of the r-lapse function. In this parametrization, the determinant of the supermetric plays the role of the mesure. The combined Wheeler – DeWitt and linear conditional symmetry equations are analytically solved. The solutions obtained depend on the product of the two "scale factors".

On the Origin of Ambiguity in Efficient Communication

This article studies the emergence of ambiguity in communication through the concept of logical irreversibility and within the framework of Shannon’s information theory. This leads us to a precise and general expression of the intuition behind Zipf’s vocabulary balance in terms of a symmetry equation between the complexities of the coding and the decoding processes that imposes an unavoidable amount of logical uncertainty in natural communication. Accordingly, the emergence of irreversible computations is required if the complexities of the coding and the decoding processes are balanced in a symmetric scenario, which means that the emergence of ambiguous codes is a necessary condition for natural communication to succeed.

Solutions of Two Nonlinear Evolution Equations Using Lie Symmetry and Simplest Equation Methods

In this paper, we study two nonlinear evolution partial differential equations, namely, a modified Camassa–Holm–Degasperis–Procesi equation and the generalized Korteweg–de Vries equation with two power law nonlinearities. For the first time, the Lie symmetry method along with the simplest equation method is used to construct exact solutions for these two equations.

Quantization of the Schwarzschild geometry

The conditional symmetries of the reduced Einstein-Hilbert action emerging from a static, spherically symmetric geometry are used as supplementary conditions on the wave function. Based on their integrability conditions, only one of the three existing symmetries can be consistently imposed, while the unique Casimir invariant, being the product of the remaining two symmetries, is calculated as the only possible second condition on the wave function. This quadratic integral of motion is identified with the reparametrization generator, as an implication of the uniqueness of the dynamical evolution, by fixing a suitable parametrization of the r-lapse function. In this parametrization, the determinant of the supermetric plays the role of the mesure. The combined Wheeler – DeWitt and linear conditional symmetry equations are analytically solved. The solutions obtained depend on the product of the two "scale factors".

Classification of Plane Symmetric Static Space-Times According to Their Noether Symmetries

In this paper we give a classification of plane symmetric static space-times using symmetry method. For this purpose we consider the Lagrangian corresponding to the general plane symmetric static metric in the Noether symmetry equation. This provides a system of determining equations. Solutions of this system give us classification of the plane symmetric static space-times according to their Noether symmetries. During this classification we recover all the results listed in Feroze et al. (J. Math. Phys. 42:4947, 2001) and Bashir and Ehsan (Il Nuovo Cimento B 123:1, 2008).

Conditional symmetries and the canonical quantization of constrained minisuperspace actions: The Schwarzschild case

A conditional symmetry is defined, in the phase space of a quadratic in velocities constrained action, as a simultaneous conformal symmetry of the supermetric and the superpotential. It is proven that such a symmetry corresponds to a variational (Noether) symmetry. The use of these symmetries as quantum conditions on the wave function entails a kind of selection rule. As an example, the minisuperspace model ensuing from a reduction of the Einstein–Hilbert action by considering static, spherically symmetric configurations and r as the independent dynamical variable is canonically quantized. The conditional symmetries of this reduced action are used as supplementary conditions on the wave function. Their integrability conditions dictate, at the first stage, that only one of the three existing symmetries can be consistently imposed. At a second stage one is led to the unique Casimir invariant, which is the product of the remaining two, as the only possible second condition on Ψ. The uniqueness of the dynamical evolution implies the need to identify this quadratic integral of motion to the reparametrization generator. This can be achieved by fixing a suitable parametrization of the r-lapse function, exploiting the freedom to arbitrarily rescale it. In this particular parametrization the measure is chosen to be the determinant of the supermetric. The solutions to the combined Wheeler–DeWitt and linear conditional symmetry equations are found and seen to depend on the product of the two “scale factors”.

Neutron stars with antikaons: Comparison between two ways of extending the relativistic mean field models

Extending the relativistic mean field (RMF) models either with higher order couplings or with density-dependent couplings has been proven to be successful in explaining several properties of finite nuclei, infinite matter, and neutron stars (NSs) in a unified way. Here, we compare how these two extensions fare while explaining NSs with and without antikaons. Both these extensions soften the equation of state (EoS), but the higher order couplings are more efficient in doing so. Even for a very soft EoS, we find that there is a strong possibility of antikaons being condensed. We find an exception for correlation between NS radius and density dependence of symmetry energy, which gets restored when we introduce antikaons. The implication of a precisely measured 2${M}_{\ensuremath{\bigodot}}$ NS on our results is discussed with four sets of interactions which cover all four combinations of stiff and soft symmetry energy and equations of state.

$$\lambda $$ -Symmetries, isochronicity, and integrating factors of nonlinear ordinary differential equations

We obtain \(\lambda \)-symmetries of some second-order equations of the Painleve–Gambier type and study their relationship with the standard adjoint symmetry equation used for determining the integrating factor of a second-order ordinary differential equation (ODE). This is followed by a brief study of the \(\lambda \)-symmetries of certain special types of third-order ODEs. Finally, we indicate a possible connection between \(\lambda \)-symmetries and the property of isochronicity for the Lienard equation of the second type.

Special Lie symmetry and Hojman conserved quantity of Appell equations for a Chetaev nonholonomic system

A special Lie symmetry and Hojman conserved quantity of the Appell equations for a Chetaev nonholonomic system are studied. The differential equations of motion and Appell equations of the Chetaev nonholonomic system are established. Under the special Lie symmetry group transformations in which the time is invariable, the determining equation of the special Lie symmetry of the Appell equations for a Chetaev nonholonomic system is given, and the expression of the Hojman conserved quantity is deduced directly from the Lie symmetry. Finally, an example is given to illustrate the application of the results.

Pre-stack reverse-time migration based on the time–space domain adaptive high-order finite-difference method in acoustic VTI medium

With the increment of seismic exploration precision requirement, it is significant to develop the anisotropic migration methods. Pre-stack reverse-time migration (RTM) is performed based on acoustic vertical transversely isotropic (VTI) wave equations, and the accuracy and efficiency of RTM strongly depend on the algorithms used for wave equation numerical solution. Finite-difference (FD) methods have been widely used in numerical solution of wave equations. The conventional FD method derives spatial FD coefficients from the space domain dispersion relation, and it is difficult to satisfy the time–space domain dispersion relation of the wave equation exactly. In this paper, we adopt a time–space domain FD method to solve acoustic VTI wave equations. Dispersion analysis and numerical modelling results demonstrate that the time–space domain FD method has greater accuracy than the conventional FD method under the same discretizations. The time–space domain high-order FD method is also applied in the wavefield extrapolation of acoustic VTI pre-stack RTM. The model tests demonstrate that the acoustic VTI pre-stack RTM based on the time–space domain FD method can obtain better images than that based on the conventional FD method, and the processing results show that the imaging quality of the acoustic VTI RTM is clearer and more correct than that of acoustic isotropic RTM. Meanwhile, in the process of wavefield forward and backward extrapolation, we employ adaptive variable-length spatial operators to compute spatial derivatives to improve the computational efficiency effectively almost without reducing the imaging accuracy.

Acoustic VTI modeling and pre-stack reverse-time migration based on the time–space domain staggered-grid finite-difference method

Abstract Reverse-time migration (RTM) is based on seismic numerical modeling algorithms, and the accuracy and efficiency of RTM strongly depend on the algorithm used for numerical solution of wave equations. Finite-difference (FD) methods have been widely used to solve the wave equation in seismic numerical modeling and RTM. In this paper, we derive a series of time–space domain staggered-grid FD coefficients for acoustic vertical transversely isotropic (VTI) equations, and adopt these difference coefficients to solve the equations, then analyze the numerical dispersion and stability, and compare the time–space domain staggered-grid FD method with the conventional method. The numerical analysis results demonstrate that the time–space domain staggered-grid FD method has greater accuracy and better stability than the conventional method under the same discretizations. Moreover, we implement the pre-stack acoustic VTI RTM by the conventional and time–space domain high-order staggered-grid FD methods, respectively. The migration results reveal that the time–space domain staggered-grid FD method can provide clearer and more accurate image with little influence on computational efficiency, and the new FD method can adopt a larger time step to reduce the computation time and preserve the imaging accuracy as well in RTM. Meanwhile, when considering the anisotropy in RTM for the VTI model, the imaging quality of the acoustic VTI RTM is better than that of the acoustic isotropic RTM.

Mass estimates from stellar proper motions: the mass of ω Centauri

We lay out and apply methods to use proper motions of individual kinematic tracers for estimating the dynamical mass of star clusters. We first describe a simple projected mass estimator and then develop an approach that evaluates directly the likelihood of the discrete kinematic data given the model predictions. Those predictions may come from any dynamical modelling approach, and we implement an analytic King model, a spherical isotropic Jeans equation model and an axisymmetric, anisotropic Jeans equation model.We apply these approaches to the enigmatic globular cluster omega Centauri, combining the proper motion from van Leeuwen et al (2000) with improved photometric cluster membership probabilities. We show that all mass estimates based on spherical isotropic models yield $(4.55\pm 0.1) \times 10^6 M_{\odot} [D/5.5 \pm 0.2 kpc]^3$, where our modelling allows us to show how the statistical precision of this estimate improves as more proper motion data of lower signal-to-noise are included. MLM predictions, based on an anisotropic axisymmetric Jeans model, indicate for $\omega$ Cen that the inclusion of anisotropies is not important for the mass estimates, but that accounting for the flattening is: flattened models imply $(4.05\pm 0.1) \times 10^6 M_{\odot} [D/5.5 \pm 0.2 kpc]^3$, 10% lower than when restricting the analysis to a spherical model. The best current distance estimates imply an additional uncertainty in the mass estimate of 12%.

Conformal invariance and conserved quantity of Mei symmetry for Appell equations in holonomic system

For a holonomic system, the conformal invariance and conserved quantity of Mei symmetry for Appell equations are studied. Firstly, by the infinitesimal one-parameter transformation group and the infinitesimal generator vector, we define Mei symmetry and conformal invariance of differential equations of motion for holonomic system, and the determining equation of Mei symmetry and conformal invariance for holonomic system are given. Then, taking advantage of a structure equation that gauge function satisfies, the system corresponding Mei conserved quantity is derived. Finally, an example is given to illustrate the application of the result.

Lie-Mei symmetry and conserved quantities of Nielsen equations for a singular nonholonomic system of Chetaev'type

Using an invariance of differential equations under the infinitesimal transformations of group, we study the Lie-Mei symmetry of Nielsen equations for a singular nonholonomic system of Chetaev'type. We establish the Lie-Mei symmetry equations. The definitions of weak and strong Lie-Mei symmetry are given, and Hojman conserved quantities and Mei conserved quantities are obtained. An example is given to illustrate the application of the results.

Blowup rate of isotropic anti-ferromagnetic equation near the equivariant data

We consider the 2+1 dimensional space–time isotropic anti-ferromagnetic equation (IAF for short) to the 2-sphere for equivariant data of homotopy number N⩾1. Using the method of matched asymptotic expansions, we present an analysis of the asymptotic behavior of singularities arising in this special class of solutions. Specifically, a sharp description of the corresponding blowup rate and the stability are investigated in settings with certain symmetries. We also find out the blowup behavior of two different IAFs are very different. In the end, the blowup results are verified by numerical experiments.

Mei symmetry and Mei conserved quantity of Appell equations for a variable mass holonomic system of relative motion

Mei symmetry and Mei conserved quantity of Appell equations for a variable mass holonomic system of relative motion are studied. The definition and criterion of the Mei symmetry of Appell equations for a variable mass holonomic system of relative motion under the infinitesimal transformations of groups are given. The structural equation of Mei symmetry of Appell equations and the expression of Mei conserved quantity deduced directly from Mei symmetry for a variable mass holonomic system of relative motion are gained. Finally, an example is given to illustrate the application of the results.