Symmetries Of Equations Research Articles

Particles dynamics around time conformal quintessential Schwarzschild black hole

In this paper, the effect of time is explored on the dynamics of neutral and charged particles around the Schwarzschild (Sch) black hole (BH) environed by quintessence. We introduce a general time conformal factor [Formula: see text] to the quintessential Sch spacetime, where [Formula: see text] is a time ([Formula: see text])-dependent function and [Formula: see text], a small parameter, that causes perturbation in the corresponding spacetime. The Noether symmetry equation is obtained with the help of Lagrangian corresponding to time conformal Sch spacetime environed by quintessence. Consequently, we get a 19 partial differential equations (PDEs) system. The solution of this system gives Noether symmetries, Noether approximate symmetries, and the corresponding conservation laws. The dynamics of neutral/charged particles are studied as well as graphically analyzed at the foot of these invariants.

Migdal-Eliashberg theory of multi-band high-temperature superconductivity in field-effect-doped hydrogenated (111) diamond

We perform single- and multi-band Migdal-Eliashberg (ME) calculations with parameters exctracted from density functional theory (DFT) simulations to study superconductivity in the electric-field-induced 2-dimensional hole gas at the hydrogenated (111) diamond surface. We show that according to the Eliashberg theory it is possible to induce a high-Tc superconducting phase when the system is field-effect doped to a surface hole concentration of 6×1014 cm−2, where the Fermi level crosses three valence bands. Starting from the band-resolved electron-phonon spectral functions α2Fjj′(ω) computed ab initio, we iteratively solve the self-consistent isotropic Migdal-Eliashberg equations, in both the single-band and the multi-band formulations, in the approximation of a constant density of states at the Fermi level. In the single-band formulation, we find Tc≈40 K, which is enhanced between 4% and 8% when the multi-band nature of the system is taken into account. We also compute the multi-band-sensitive quasiparticle density of states to act as a guideline for future experimental works.

A uniformly exponentially stable ADI scheme for Maxwell equations

A modified alternating direction implicit scheme for the time integration of linear isotropic Maxwell equations with strictly positive conductivity on cuboids is constructed. A key feature of the proposed scheme is its uniform exponential stability, being achieved by coupling the Maxwell system with an additional damped PDE and adding artificial damping to the scheme. The implicit steps in the resulting time integrator further decouple into essentially one-dimensional elliptic problems, requiring only linear complexity. The convergence of the scheme to the solution of the original Maxwell system is analyzed in the abstract time-discrete setting, providing an error bound in a space related to H−1.

A non-balanced staggered-grid finite-difference scheme for the first-order acoustic VTI equation modeling

Numerically solving wave equation is widely used to seismic modeling, migration and inversion. It is important to improve the simulation efficiency while still preserving high accuracy. Different from the balanced staggered-grid finite-difference (SGFD) scheme for the first-order acoustic vertical transversely isotropic (VTI) equation modeling, we propose a non-balanced FD scheme which uses the second-order SGFD operator for part of the spatial derivatives in the first-order VTI wave equation to improve the simulation efficiency. Meanwhile, we compute the SGFD coefficient from the normal-grid FD coefficient. Compared with the balanced SGFD scheme, the non-balanced SGFD scheme could save simulation time. The efficiency and accuracy of the proposed SGFD scheme are demonstrated by the dispersion analysis and numerical simulation.

Some Connections between Classical and Nonclassical Symmetries of a Partial Differential Equation and Their Applications

Essential connections between the classical symmetry and nonclassical symmetry of a partial differential equations (PDEs) are established. Through these connections, the sufficient conditions for the nonclassical symmetry of PDEs can be derived directly from the inconsistent conditions of the system determining equations of the classical symmetry of the PDE. Based on the connections, a new algorithm for determining the nonclassical symmetry of a PDEs is proposed. The algorithm make the determination of the nonclassical symmetry easier by adding compatibility extra equations obtained from system of determining equations of the classical symmetry to the system of determining equations of the nonclassical symmetry of the PDE. The findings of this study not only give an alternative method to determine the nonclassical symmetry of a PDE, but also can help for better understanding of the essential connections between classical and nonclassical symmetries of a PDE. Concurrently, the results obtained here enhance the efficiency of the existing algorithms for determining the nonclassical symmetry of a PDE. As applications of the given algorithm, a nonclassical symmetry classification of a class of generalized Burgers equations and the nonclassical symmetries of a KdV-type equations are given within a relatively easier way and some new nonclassical symmetries have been found for the Burgers equations.

The Physical Significance of Time Conformal Minkowski Spacetime

The Minkowsiki spacetime is flat and there is no source of gravitation. The time conformal factor is adding some cuvature to this spacetime which introduces some source of gravitation to the spacetime. For the Minkowski spacetime the Einstein Field equation tells nothing, because all the components of the Ricci curvature tensor are zero, but for the time conformal Minkowski spacetime some of them are non zero. Calculating the components of the Ricci tensor and using the Einstein field equations, expressions for the cosmological constant are cacultaed. These expressions give some information for the cosmological constant. Generally, the Noether symmetry generator corresponding to the energy content in the spacetime disapeares by introducing the time conformal factor, but our investigations in this paper reveals that it appears somewhere with some re-scale factor. The appearance of the time like isometry along with some re-scaling factor will rescale the energy content in the corresponding particular time conformal Minkowski spacetime. A time conformal factor of the form () is introduced in the Minkowski spacetime for the invistigation of the cosmological constant. The Noether symmetry equation is used for the Lagrangian of general time conformal Minkowski spacetime to find all those particular Minkowski spacetimes that admit the time conformal factor. Besides the Noether symmetries the cosmology constant is calculated in the corresponding spacetimes.

Symmetries of stochastic differential equations using Girsanov transformations

Aiming at enlarging the class of symmetries of a stochastic differential equation (SDE), we introduce a family of stochastic transformations able to change also the underlying probability measure exploiting Girsanov theorem and we provide new determining equations for the infinitesimal symmetries of the SDE. The well-defined subset of the previous class of measure transformations given by Doob transformations allows us to recover all the Lie point symmetries of the Kolmogorov equation associated with the SDE. This gives the first stochastic interpretation of all the deterministic symmetries of the Kolmogorov equation. The general theory is applied to some relevant stochastic models.

ON THE NATURE OF A CLASSICAL PSEUDODIFFERENTIAL EQUATION

The work is devoted to the study of the general nature of one classical parabolic pseudodi- fferential equation with the operator M.Rice of fractional differentiation. At the corresponding values of the order of fractional differentiation, this equation is also known as the isotropic superdiffusion equation. It is a natural generalization of the classical diffusion equation. It is also known that the fundamental solution of the Cauchy problem for this equation is the density distribution of probabilities of stable symmetric random processes by P.Levy. The paper shows that the fundamental solution of this equation is the distribution of probabilities of the force of local influence of moving objects in a nonstationary gravitational field, in which the interaction between masses is subject to the corresponding potential of M.Rice. In this case, the classical case of Newton’s gravity corresponds to the known nonstationary J.Holtsmark distribution.

Continuing Results for Effervescent Aerosol Salt Water Spray Nozzles Intended for Marine Cloud Brightening

Marine Cloud Brightening (MCB) by effervescent spray atomization of mixed sea water brine with air is a candidate for solar radiation management to compensate for global warming. We discovered that the flow from mixing tee nozzle described earlier had occasional unstable slug flow. A new design that adding rotational swirl to the salt brine as it is mixed into the air stabilized the nozzle flow and no longer showed slug flow in spray pictures. Flow equations were developed for the relatively low speed of sound of a choked flow mixed brine and air nozzle. Experimental mixed flow measurements with 300b pressure and a 200 μm diameter nozzle and calculations using perfect gas, and isotropic processes equations compared well with the chocked flow equations. Analysis in EXCEL of particle sizers measurements from both a scanning mobility particle sizer (SMPS) and an aerodynamic particle sizer (APS) showed production of many nanometer sized particles estimated as usable for MCB. A small number of micron sized particles were also always present but with about 90% of the sprayed mass. This is a first report with good data over the complete size range. The micron sized particles measured were similar to the measurements of earlier reports which reported no nanometer sized particles. We hypothesize that many nano-particles are always produced by liquid-air effervescent sprays, but earlier, were not observed because SMPS instruments were not available. The presence of the large mass percentage of large particles in the spray may cause problems by evaporative cooling preventing the rise of the MCB particles. We suggest future systems design with an impactor filter to remove the large particles. Calculations combining increased brine concentration, lower pressure, and larger nozzle area showed that significant reductions in required power and number of nozzles could be realized. An EXCEL model is developed to calculate flow from experimental analysis equations and compare with mixed choked flow equations. Solving with the model predicted the power required and the number of nozzles required to produce 1015 particles/s. The model showed that increasing brine concentration strongly lowered total power. Lowering pressure decreased power and increased number of nozzles. Increasing nozzle area lowered the number of nozzles. This model predicted that, at 300b pressure and 200μm diameter nozzle as the experiment but using an increased brine concentration of 0.1 instead of 0.032 would require only 115 nozzles instead of 358 and power of 146 kw instead of 493 kw. Combining increased brine concentration, lower pressure, and larger nozzle area, the model predicted that with a 1 mm diameter nozzle at 30b pressure and salt concentration of 0.2, the nozzle count and power required would drop to only 24 nozzles and power of 28 kw. Whether extending the model to these conditions is valid is not known but suggests further development should be investigated. Filtering out and reusing the 90% or greater large particles mass sprayed combined with the lower power advantage of higher brine concentration is suggested for future systems.

Nonautonomous symmetries of the KdV equation and step-like solutions

We study solutions of the KdV equation governed by a stationary equation for symmetries from the non-commutative subalgebra, namely, for a linear combination of the master-symmetry and the scaling symmetry. The constraint under study is equivalent to a sixth order nonautonomous ODE possessing two first integrals. Its generic solutions have a singularity on the line t = 0. The regularity condition selects a 3-parameter family of solutions which describe oscillations near u = 1 and satisfy, for t = 0, an equation equivalent to degenerate P5 equation. Numerical experiments show that in this family one can distinguish a two-parameter subfamily of separatrix step-like solutions with power-law approach to different constants for x → ±∞. This gives an example of exact solution for the Gurevich—Pitaevskii problem on decay of the initial discontinuity.

Augmented Lévy–Michell equations for flexural plates

A new sixth order, isotropic elastic and flexural plate theory is presented based on the fourth order equations of Lévy (1877) and Michell (1900). The lack of surface loading in these Lévy–Michell equations is overcome by including the dominant flexural component of Dougall’s (1904) solution for a point load on the surface of a three dimensional isotropic, elastic layer. The augmented plate equations are arranged to fit the form of a consistent sixth order system of isotropic plate equations which include the well known work of Reissner and static equations of Mindlin. Numerical applications to two three dimensional problems with analytical solutions show good comparative results.

Classification of integrable vector equations of geometric type

A complete classification of isotropic vector equations of geometric type that possess vector isotropic higher symmetries is proposed. New examples of integrable multi-component systems of geometric type are found.

Conserved Quantity and Adiabatic Invariant for Hamiltonian System with Variable Order

Hamiltonian mechanics plays an important role in the development of nonlinear science. This paper aims for a fractional Hamiltonian system of variable order. Several issues are discussed, including differential equation of motion, Noether symmetry, and perturbation to Noether symmetry. As a result, fractional Hamiltonian mechanics of variable order are established, and conserved quantity and adiabatic invariant are presented. Two applications, fractional isotropic harmonic oscillator model of variable order and fractional Lotka biochemical oscillator model of variable order are given to illustrate the Methods and Results.

The dynamics of particles around time conformal AdS-Schwarzschild black hole

Abstract In this article, we explore the effect of the cosmological constant ( Λ ) on the dynamics of neutral as well as charged particles in vicinity of the time conformal Schwarzschild black hole. The general time conformal term e ϵ f ( t ) is introduced in the AdS-Schwarzschild spacetime, where f(t) is an arbitrary function of time t and ϵ is a small parameter which causes perturbation in the spacetime. The corresponding Lagrangian is used in the Noether symmetry equation which gives a system of 19 partial differential equations (PDEs). The solution of the 19 PDEs give us the require time conformal AdS-Schwarzschild spacetime along with approximate Noether symmetries and conservation laws. The dynamics of neutral and charged particles are discuss over the fabric of time conformal Ads–Schwarzschild spacetime.

Effects of nuclear symmetry energy and equation of state on neutron star properties

We study the effects of nuclear symmetry energy on the mass-radius relation and tidal deformability of neutron stars, considering the self-consistency of the equation of state (EOS). We first construct a set of unified EOSs based on relativistic mean-field models with different density dependence of the symmetry energy. For the description of pasta phases appearing in the inner crust of neutron stars, we perform a self-consistent Thomas-Fermi calculation using the same nuclear interaction as that for the uniform matter in the core. To examine possible effects from the self-consistency of the EOS on neutron-star properties, we separately investigate the impacts of crust and core segments. By matching the same core EOS to different crust EOSs, it is found that neutron-star radii are significantly affected by the crust segment. On the other hand, the neutron-star radii are also strongly dependent on the core EOS. However, the correlation between the radius and the symmetry energy slope of the core EOS is opposite to that of the crust EOS. It is likely that the nuclear model with a small slope parameter is favored by recent astrophysical observations.

Группы движений собственно гельмгольцевой трехмерной геометрии и симплициальной трехмерной геометрии III типа

Основными задачами теории феноменологически симметричных (ФС) геометрий (геометрий локальной максимальной подвижности) являются их полная классификация, вывод уравнения феноменологической симметрии и нахождение групп движений для каждой из них. ФС геометрия задается на многообразии функцией пары точек. Феноменологическая симметрия трехмерных ФС геометрий состоит в наличии функциональной связи между значениями функции пары точек для всех пар из пяти произвольных точек. Их классификация была впервые построена В.Х. Левом и позже дополнена В.А. Кыровым симплициальной геометрией III типа. Методами установления групповой симметрии ФС геометрий являются метод решения функциональных уравнений на множество движений, разработанный для двумерных и некоторых трехмерных ФС геометрий, и метод экспоненциального отображения.Методом экспоненциального отображения для собственно гельмгольцевой и симплициальнойIII типа трехмерных ФС геометрий находятся явные выражения групп движений. Данные вычисленияпроизводятся с использованием аппарата комплексного анализа и формулируются в виде отдельной теоремы. Группы движений этих геометрий являются действиями группы Ли SL2(C)R в пространстве R3.

Lie symmetry and invariants for a generalized Birkhoffian system on time scales

Abstract The Lie symmetry and invariants for a generalized Birkhoffian system on time scales are studied, which include exact invariants and adiabatic invariants. First, the generalized Pfaff-Birkhoff principle on time scales is established, and by using Dubois-Reymond lemma the generalized Birkhoff’s equations on time scale are derived. Secondly, the determining equations of Lie symmetry for the generalized Birkhoffian system on time scales are established. We prove that if the Lie symmetry satisfies the structural equation, it leads to a conserved quantity, which is an exact invariant of the system. Again, the perturbation of Lie symmetry under the action of small disturbance is considered, the determining equations and the structural equations of disturbed system are established, and the adiabatic invariants led by the Lie symmetry perturbation for the generalized Birkhoffian system on time scales are given. Because of the arbitrariness of selecting time scales and the generality of the generalized Birkhoffian system, the results of this paper are of universal significance. The results of this paper contain the corresponding results for Birkhoffian system on time scales and classical generalized Birkhoffian system as its special cases. At the end of the paper, an example is given to illustrate the validity of the method and the results.

Relativistic self-energy decomposition of nuclear symmetry energy and equation of state of neutron matter within QCD sum rules

Properties of the relativistic nucleon self-energy decomposition of the symmetry energy as well as the equation of state (EOS) of pure neutron matter (PNM) are explored systematically within the QCD sum rules (QCDSR). Our main conclusions are as follows: (1) The five self-energy decomposition terms of the symmetry energy according to the nucleon Lorentz structure are carefully studied, leading to the conclusion that the symmetry energy increases as the nucleon sigma term ${\ensuremath{\sigma}}_{\mathrm{N}}$ increases and the contributions to the symmetry energy due to the momentum dependence of the self-energies in symmetric nuclear matter (SNM) are very small compared with those from other decomposition terms. (2) A smaller strange quark mass is found to generate a larger symmetry energy, and this correlation is useful for understanding the origins of the uncertainties on the (nucleonic matter) symmetry energy from quark level. (3) The EOS of PNM at low densities can be effectively approximated by ${E}_{\mathrm{n}}(\ensuremath{\rho})\ensuremath{\approx}{E}_{\mathrm{n}}^{\mathrm{FFG}}(\ensuremath{\rho})+(M\ensuremath{\rho}/2{\ensuremath{\langle}\overline{q}q\ensuremath{\rangle}}_{\mathrm{vac}})[(1\ensuremath{-}\ensuremath{\xi})({\ensuremath{\sigma}}_{\mathrm{N}}/2{m}_{\mathrm{q}})\ensuremath{-}5]$ which depends only on several physical quantities such as ${m}_{\mathrm{q}},{\ensuremath{\sigma}}_{\mathrm{N}}$ and ${\ensuremath{\langle}\overline{q}q\ensuremath{\rangle}}_{\mathrm{vac}}$, and this formula already has predictive power and the results are found to be consistent with those from other celebrated microscopic many-body theories at low densities. (4) The higher order density terms in quark condensates are shown to be important to describe the empirical EOS of PNM in the density region around and above nuclear saturation density, and these higher order density terms are also found to hinder the appearance of chiral symmetry restoration in PNM at high densities. (5) The symmetry energy is shown to depend strongly on the five-dimensional condensate ${\ensuremath{\langle}{g}_{\mathrm{s}}{q}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\sigma}\mathcal{G}q\ensuremath{\rangle}}_{\ensuremath{\rho},\ensuremath{\delta}}$, providing a useful approach to explore the symmetry energy through knowledge on the condensates which can be extracted from hadronic physics. (6) The twist-4 four-quark condensates are shown to have significant effects on the EOS of both SNM and PNM but have minor effects on the symmetry energy, and combined with the analyses on the effects of the higher order density terms in the chiral condensates, three parameter sets of QCDSR are constructed and they are shown to be able to describe the EOS of PNM and the symmetry energy within a wide range of densities. Our results in the present work demonstrate that the QCDSR approach can provide a useful way to understand the properties of dense nucleonic matter from nonperturbative QCD vacuum.

A Potential Constraints Method of Finding Nonclassical Symmetry of PDEs Based on Wu’s Method

Solving nonclassical symmetry of partial differential equations (PDEs) is a challenging problem in applications of symmetry method. In this paper, an alternative method is proposed for computing the nonclassical symmetry of PDEs. The method consists of the following three steps: firstly, a relationship between the classical and nonclassical symmetries of PDEs is established; then based on the link, we give three principles to obtain additional equations (constraints) to extend the system of the determining equations of the nonclassical symmetry. The extended system is more easily solved than the original one; thirdly, we use Wu’s method to solve the extended system. Consequently, the nonclassical symmetries are determined. Due to the fact that some constraints may produce trivial results, we name the candidate constraints as “potential” ones. The method gives a new way to determine a nonclassical symmetry. Several illustrative examples are given to show the efficiency of the presented method.

Mixed bounce-back boundary scheme of the general propagation lattice Boltzmann method for advection-diffusion equations.

In this work, a mixed bounce-back boundary scheme of general propagation lattice Boltzmann (GPLB) model is proposed for isotropic advection-diffusion equations (ADEs) with Robin boundary condition, and a detailed asymptotic analysis is also conducted to show that the present boundary scheme for the straight walls has a second-order accuracy in space. In addition, several numerical examples, including the Helmholtz equation in a square domain, the diffusion equation with sinusoidal concentration gradient, one-dimensional transient ADE with Robin boundary and an ADE with a source term, are also considered. The results indicate that the numerical solutions agree well with the analytical ones, and the convergence rate is close to 2.0. Furthermore, through adjusting the two parameters in the GPLB model properly, the present boundary scheme can be more accurate than some existing lattice Boltzmann boundary schemes.