How Classical, Quantum-Mechanical, and Relativistic Wave and Field Equations Are Uniformly Generated by Velocity-Field Divergence Equations

We expand previously established results concerning the uniform representability of classical and relativistic gravitational field equations by means of velocity-field divergence equations by demonstrating that conservation equations for (probability) density functions give rise to velocity-field divergence equations the solutions of which generate—by way of superposition—the totality of solutions of various well-known classical and quantum-mechanical wave equations.

Space-Energy Duality Generalized 4-Index Einstein Field Equation

The introduction of a new concept of space-energy duality serves to extend the applicability of the Einstein field equation in the context of a 4-index framework. The utilization of the Weyl tensor enables the derivation of Einstein’s equations in the 4-index format. Additionally, a two-index field equation is presented, comprising a conventional Einstein field equation and a trace-free Einstein equation. Notably, the cosmological constant is associated with a novel concept that facilitates the encoding of space and energy information, thereby enabling the recognition of mutual interactions between space and energy in the presence of gravitational forces, as dictated by Einstein’s field equations (EFE) and Trace-Free Einstein Equation (TFE).

On Quasi-Einstein Field Equation

In this paper some properties of a symmetric tensor field T（X,Y） = g（A（X）, Y） on a Riemannian manifold （M, g） without boundary which satisfies the S quasi-Einstein equation Rij-S/2gij=Tij＋bξiξj are given. The necessary and sufficient conditions for this tensor to satisfy the quasi-Einstein equation are also obtained.

On the Field Equations of General Relativity

In this work, we examine the geometric character of the field equations of general relativity and propose to formulate relativistic field equations in terms of the Riemann curvature tensor. The resulted relativistic field equations are also integrated into the general framework that we have presented in our previous works that all known classical fields can be expressed in the same dynamical form. We also discuss a possibility to reformulate the field equations of general relativity so that the Ricci curvature tensor and the energy-momentum tensor can appear symmetrically in the field equations without violating the conservation law stated by the covariant derivative.

CONSTITUTIVE RELATION OF UNSATURATED SOIL BY USE OF THE MIXTURE THEORY(Ⅰ)—NONLINEAR CONSTITUTIVE EQUATIONS AND FIELD EQUATIONS

The nonlinear constitutive equations and field equations of unsaturated soils were constructed on the basis of mixture theory. The soils were treated as the mixture composed of three constituents. First, from the researches of soil mechanics, some basic assumptions about the unsaturated soil mixture were made, and the entropy inequality of unsaturated soil mixture was derived. Then, with the common method usually used to deal with the constitutive problems in mixture theory, the nonlinear constitutive equations were obtained. Finally, putting the constitutive equations of constituents into the balance equations of momentum, the nonlinear field equations of constituents were set up. The balance equation of energy of unsaturated soil was also given, and thus the complete equations for solving the thermodynamic process of unsaturated soil was formed.

CONSTITUTIVE RELATION OF UNSATURATED SOIL BY USE OF THE MIXTURE THEORY (Ⅱ)—LINEAR CONSTITUTIVE EQUATIONS AND FIELD EQUATIONS

The linear constitutive equations and field equations of unsaturated soils were obtained through linearizing the nonlinear equations given in the first part of this work. The linear equations were expressed in the forms similar to Biot's equations for saturated porous media. The Darcy's laws of unsaturated soil were proved. It is shown that Biot's equations of saturated porous media are the simplification of the theory. All these illustrate that constructing constitutive relation of unsaturated soil on the base of mixture theory is rational.

STUDY OF EXPLICIT ANALYTIC SOLUTIONS FOR THE NONLINEAR COUPLED SCALAR FIELD EQUATIONS

By using two different transformations, several types of exact analytic solutions for a class of nonlinear coupled scalar field equation are obtained, which contain soliton solutions, singular solitary wave solutions and triangle function solutions. These results can be applied to other nonlinear equations. In addition, parts of conclusions in some references are corrected.

Bach-Einstein Gravitational Field Equations as a Perturbation of Einstein Gravitational Field Equations

The Bach equations are a version of higher-order gravitational field equations, exactly they are of fourth-order. In 4-dimensions the Bach-Einstein gravitational field equations are treated here as a perturbation of Einstein’s gravity. An approximate inversion formula is derived which admits a comparison of the two field theories. An application to these theories is given where the gravitational Lagrangian is expressed linearly in terms of R, R2, |Ric|2, where the Ricci tensor Ric = Rαβdxαdxβ is inserted in some formulas which are of geometrical or physical importance, such as;Raychaudhuri equation and Tolman’s formula.

Unified Field Equation Generated by Longitudinal Electromagnetic (LEM) Waves

According to special relativity,the relationship of electromagnetic conversion in a linear moving vacuum and the relationship formula between the magnetic vector potential/scalar potential and the LEM(Longitudinal Electromagnetic)waves,it is inferred that the spherical vacuum space we are in undergoes outward helical motion at the speed of light following the right-hand screw rule,accompanied by a radial space expansion motion far less than the speed of light.Based on this space basis,we derive a unified field equation indicating that the gravitational field might be equivalent to the acceleration field of the radial expansion motion of our vacuum space,the strong nuclear force field presumably is generated by the light-rotation angular velocity of our space,the weak nuclear force field is most probably produced by its radial expansion motion and the electromagnetic field is undoubtedly produced by the radial linear motion of our space at the speed of light.We have also demonstrated both theoretically and experimentally that the LEM waves can generate artificial gravitational fields,and the LEM waves are the material basis of the unified field theory.Essentially,on Earth,time is the result of the relativistic length contraction effect caused by the radial space expansion speed,which leads to the rate of change of distance in the radial dimension on the unit radial space expansion speed.Moreover,based on the length contraction effect in special relativity,the time and space generated by the outward helical motion of space at the speed of light can be expressed as zero.This indicates that such motion not only does not affect the seemingly perpetually stationary space that we can constantly perceive but also enables the gravitational field formula to remain unchanged in our space.They constitute the spatio-temporal basis of the unified field theory.Based on our unified field theory,we have also discussed some forward-looking perspectives,such as motion at the speed of light,anti-gravitation fields,and interstellar travel.

The boundary value problem for the mean field equation on a compact Riemann surface

Let(Σ,g)be a compact Riemann surface with smooth boundary■Σ,Δ_(g) be the Laplace-Beltrami operator,and h be a positive smooth function.Using a min-max scheme introduced by Djadli and Malchiodi(2008)and Djadli(2008),we prove that ifΣis non-contractible,then for anyρ∈(8kπ,8(k+1)π)with k∈N^(*),the mean field equation{Δgu=ρhe^(u)/∫∑he^(u)dv_(g)in∑,u=0 on■∑has a solution.This generalizes earlier existence results of Ding et al.(Ann Inst H PoincaréAnal Non Linéaire,1999)and Chen and Lin(2003)in the Euclidean domain.Also we consider the corresponding Neumann boundary value problem.If h is a positive smooth function,then for anyρ∈(4kπ,4(k+1)π)with k∈N^(*),the mean field equation{Δgu=ρhe^(u)/∫_(∑)he^(u)dv_(g)-1/|∑|in∑,■u/■v=0 on■∑has a solution,where v denotes the unit normal outward vector on ■Σ.Note that in this case we do not require the surface to be non-contractible.

An Explanation of the Temperature-Dependent Upper Critical Field Data of H3S on the Basis of the Thermodynamics of a Superconductor in a Magnetic Field

Excellent fits to a couple of the data-sets on the temperature (T)-dependent upper critical field (Hc2) of H3S (critical temperature, Tc ≈ 200 K at pressure ≈ 150 GPa) reported by Mozaffari, et al. (2019) were obtained by Talantsev (2019) in an approach based on an ingenious mix of the Ginzberg-Landau (GL), the Werthamer, Helfand and Hohenberg (WHH), and the Gor’kov, etc., theories which have individually been employed for the same purpose for a long time. Up to the lowest temperature (TL) in each of these data-sets, similarly accurate fits have also been obtained by Malik and Varma (2023) in a radically different approach based on the Bethe-Salpeter equation (BSE) supplemented by the Matsubara and the Landau quantization prescriptions. For T TL, however, while the (GL, WHH, etc.)-based approach leads to Hc2(0) ≈ 100 T, the BSE-based approach leads to about twice this value even at 1 K. In this paper, a fit to one of the said data-sets is obtained for the first time via a thermodynamic approach which, up to TL, is as good as those obtained via the earlier approaches. While this is interesting per se, another significant result of this paper is that for T TL it corroborates the result of the BSE-based approach.

Time-periodic solutions of the Einstein's field equations Ⅲ:physical singularities

In this paper we construct a new time-periodic solution of the vacuum Einstein's field equations, this solution possesses physical singularities, i.e., the norm of the solution's Riemann curvature tensor takes the infinity at some points. We show that this solution is intrinsically time-periodic and describes a time-periodic universe with the "time-periodic physical singularity". By calculating the Weyl scalars of this solution, we investigate new physical phenomena and analyze new singularities for this universal model.

Solutions without a maximum mass limit of the general relativistic field equations for neutron stars

We have investigated the general relativistic field equations for neutron stars.We find that there are solutions for the equilibriummass distribution without a maximum mass limit.The solutions correspond to stars with a void inside their centers.In thesesolutions,the mass density and pressure increase first from zero at the inner radius to a peak and then decrease to zero at the outerradius.With the change of the void boundary,the mass and particle number of the star can approach infinity.Neutron stars withlarge masses can remain stable and do not collapse into black holes.

Solitary Wave Solutions of Delayed Coupled Higgs Field Equation

This paper is devoted to the study of the solitary wave solutions for the delayed coupled Higgs field equation{vtt-uxx-αu+βf*u|u|^(2)-2uv-τu(|u|^(2))x=0 vtt+vxx-β(|u|^(x))xx=0.We first establish the existence of solitary wave solutions for the corresponding equation without delay and perturbation by using the Hamiltonian system method.Then we consider the persistence of solitary wave solutions of the delayed coupled Higgs field equation by using the method of dynamical system,especially the geometric singular perturbation theory,invariant manifold theory and Fredholm theory.According to the relationship between solitary wave and homoclinic orbit,the coupled Higgs field equation is transformed into the ordinary differential equations with fast variables by using the variable substitution.It is proved that the equations with perturbation also possess homoclinic orbit,and thus we obtain the existence of solitary wave solutions of the delayed coupled Higgs field equation.

A mean field equation with symmetry on torus

Let T^2 be a flat two-dimensional torus with fundamental cell domain [-1/2,1/2]×[-1/2,1/2],h（x） a positive smooth function satisfying the symmetric property （8） on T^2.In this paper we give some sufficient condition under which the mean field equation △u = 16π - 16πe^u, has a smooth solution.

Mean Field Equations for the Equilibrium Turbulence and Toda Systems on Connected Finite Graphs

In this paper,we study existence of solutions of mean field equations for the equilibrium turbulence and Toda systems on connected finite graphs.Our method is based on calculus of variations,which was built on connected finite graphs by Grigor'yan,Lin and Yang.

Equation of Motion of a Mass Point in Gravitational Field and Classical Tests of Gauge Theory of Gravity

A systematic method is developed to studY the classical motion of a mass point in gravitational gauge field. First, by using Mathematica, a spherical symmetric solution of the field equation of gravitational gauge field is obtained, which is just the traditional Schwarzschild solution. Combining the principle of gauge covariance and Newton＇s second law of motion, the equation of motion of a mass point in gravitational field is deduced. Based on the spherical symmetric solution of the field equation and the equation of motion of a mass point in gravitational field, we can discuss classical tests of gauge theory of gravity, including the deflection of light by the sun, the precession of the perihelia of the orbits of the inner planets and the time delay of radar echoes passing the sun. It is found that the theoretical predictions of these classical tests given by gauge theory of gravity are completely the same as those given by general relativity.

Multiple Non-Axially Solutions to a Mean Field Equation on S^2

We study the following mean field equation■,whereρis a real parameter.We obtain the existence of multiple non-axially symmetric solutions bifurcating from u=0 at the valuesρ=4 n(n+1)πfor any odd integer n≥3.

The Kinematics and Field Equations for Porous Media

With a porous medium regarded as an immiscible mixture of multiphase and each phase as a miscible mixture of multi constituent, a systematical research on the kinematics and field equations for porous media is carried out from the point of view of mixture theory. It is shown that the motion of each phase is the mathematical average of the motions of all constituents in the phase, and that the motion of porous media may be described as the motion of the skeleton and the relative motion of each phase with respect to the skeleton. The influence of mass exchange between different constituents in each phase and the influence of mass exchange of same constituent between different phases in porous media are considered in field equations which are self consistent in theory. All the field equations in the references are special cases of the equations proposed in this paper.

A NEW CLASS OF SOLUTIONS OF VACUUM EINSTEIN'S FIELD EQUATIONS WITH COSMOLOGICAL CONSTANT

In this paper, a new class of solutions of the vacuum Einstein＇s field equa- tions with cosmological constant is obtained. This class of solutions possesses the naked physical singularity. The norm of the Riemann curvature tensor of this class of solutions takes infinity at some points and the solutions do not have any event horizon around the singularity.