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.

Special Lie symmetry and Hojman conserved quantity of Appell equations in a dynamical system of relative motion

Special Lie symmetry and the Hojman conserved quantity for Appell equations in a dynamical system of relative motion are investigated. The definition and the criterion of the special Lie symmetry of Appell equations in a dynamical system of relative motion under infinitesimal group transformation are presented. The expression of the equation for the special Lie symmetry of Appell equations and the Hojman conserved quantity, deduced directly from the special Lie symmetry in a dynamical system of relative motion, are obtained. An example is given to illustrate the application of the results.

Lie symmetry and its generation of conserved quantity of Appell equation in a dynamical system of the relative motion with Chetaev-type nonholonomic constraints

Lie symmetry and conserved quantity deduced from Lie symmetry of Appell equations in a dynamical system of relative motion with Chetaev-type nonholonomic constraints are studied.The differential equations of motion of the Appell equation for the system,the definition and criterion of Lie symmetry,the condition and the expression of generalized Hojman conserved quantity deduced from Lie symmetry for the system are obtained.The condition and the expression of Hojman conserved quantity deduced from special Lie symmetry for the system under invariable time are further obtained.An example is given to illustrate the application of the results.

Lie symmetry and Hojman conserved quantity of a Nielsen equation in a dynamical system of relative motion with Chetaev-type nonholonomic constraint

The Lie symmetry and Hojman conserved quantity of Nielsen equations in a dynamical system of relative motion with nonholonomic constraint of the Chetaev type are studied. The differential equations of motion of the Nielsen equation for the system, the definition and the criterion of Lie symmetry, and the expression of the Hojman conserved quantity deduced directly from the Lie symmetry for the system are obtained. An example is given to illustrate the application of the results.

Mei symmetry and Mei conserved quantity of the Appell equation in a dynamical system of relative motion with non-Chetaev nonholonomic constraints

The Mei symmetry and the Mei conserved quantity of Appell equations in a dynamical system of relative motion with non-Chetaev nonholonomic constraints are studied.The differential equations of motion of the Appell equation for the system,the definition and the criterion of the Mei symmetry,and the expression of the Mei conserved quantity deduced directly from the Mei symmetry for the system are obtained.An example is given to illustrate the application of the results.

THE MAPPING RELATION AMONG SOLUTION OF SOME NONLINEAR EQUATIONS

Among the solutions of three kinds of nonlinear equations in one dimensional systems, cubic nonlinear Klein-Gordon (including Φ~4), Sine-Gordon and double Sine-Gordon, some mapping relations exist. When a solution of any one equation is known, so are the other two.

Practical Method for Determining All the Minimum Solutions of Fuzzy Matrix Equation and Transitive Closure of Fuzzy Relation

In this paper,the new theory frame and practical method for determining all the minimum solutions of Fuzzy matrix equation and transitive closure of Fuzzy relation is described,and it has been carried out on the miero-computer quickly and accurately.

Canonical Characteristic Relation for Two Dimensional Euler Equations

We propose a new way of rewriting the two dimensional Euler equations and derive an original canonical characteristic relation based on the characteristic theory of hyperbolic systems. This relation contains the derivatives strictly along the bicharacteristic directions, and can be viewed as the 2D extension of the characteristic relation in 1D case.

Alternative constitutive relation for momentum transport of extended Navier–Stokes equations

The classical Navier–Stokes equation(NSE)is the fundamental partial differential equation that describes the flow of fluids,but in certain cases,like high local density and temperature gradient,it is inconsistent with the experimental results.Some extended Navier–Stokes equations with diffusion terms taken into consideration have been proposed.However,a consensus conclusion on the specific expression of the additional diffusion term has not been reached in the academic circle.The models adopt the form of the generalized Newtonian constitutive relation by substituting the convection velocity with a new term,or by using some analogy.In this study,a new constitutive relation for momentum transport and a momentum balance equation are obtained based on the molecular kinetic theory.The new constitutive relation preserves the symmetry of the deviation stress,and the momentum balance equation satisfies Galilean invariance.The results show that for Poiseuille flow in a circular micro-tube,self-diffusion in micro-flow needs considering even if the local density gradient is very low.

Development of Granular Fuzzy Relation Equations Based on a Subset of Data

Developing and optimizing fuzzy relation equations are of great relevance in system modeling,which involves analysis of numerous fuzzy rules.As each rule varies with respect to its level of influence,it is advocated that the performance of a fuzzy relation equation is strongly related to a subset of fuzzy rules obtained by removing those without significant relevance.In this study,we establish a novel framework of developing granular fuzzy relation equations that concerns the determination of an optimal subset of fuzzy rules.The subset of rules is selected by maximizing their performance of the obtained solutions.The originality of this study is conducted in the following ways.Starting with developing granular fuzzy relation equations,an interval-valued fuzzy relation is determined based on the selected subset of fuzzy rules(the subset of rules is transformed to interval-valued fuzzy sets and subsequently the interval-valued fuzzy sets are utilized to form interval-valued fuzzy relations),which can be used to represent the fuzzy relation of the entire rule base with high performance and efficiency.Then,the particle swarm optimization(PSO)is implemented to solve a multi-objective optimization problem,in which not only an optimal subset of rules is selected but also a parameterεfor specifying a level of information granularity is determined.A series of experimental studies are performed to verify the feasibility of this framework and quantify its performance.A visible improvement of particle swarm optimization(about 78.56%of the encoding mechanism of particle swarm optimization,or 90.42%of particle swarm optimization with an exploration operator)is gained over the method conducted without using the particle swarm optimization algorithm.

The Dispersion Relation of Internal Wave Extended-Korteweg-de Vries Equation in a Two-Layer Fluid

To understand the characteristics of ocean internal waves better, we study the dispersion relation of extended-Korteweg-de Vries (EKdV) equation with quadratic and cubic nonlinear terms in a two-layer fluid by using the Poincaré-Lighthill-Kuo (PLK) method which is one of the perturbation methods. Starting from the partial differential equation, the PLK method can be used to solve the dispersion relation of the equation. In this paper, we use PLK method to solve the equation and derive the dispersion relation of EKdV equation which is related to wave number and amplitude. Based on the dispersion relation obtained in this paper, the expressions of group velocity and phase velocity of the equation are obtained. Under the actual hydrological data, the influence of hydrological parameters on the dispersion relation for descending internal wave is discussed. It is hope that the obtained results will be helpful to the study of energy transfer and other internal wave parameters in the future.

The Equation of the Universe (According to the Theory of Relation)

A new equation is found in which the concept of matter-space-time is mathematically connected;gravitation and electromagnetism are also bound by space-time. A mechanism is described showing how velocity, time, distance, matter, and energy are correlated. We are led to ascertain that gravity and electricity are two distinct manifestations of a single underlying process: electro-gravitation. The force of gravitation arises of electromagnetism—inherently much stronger—divided by the cosmological space-time. The radius of space-time belongs to the family of electromagnetic waves: the wavelength is the radius (1026 m) of the universe and the period (1017 s) is its cosmological age. For the first time, the cosmological time, considered as a real physical object, is integrated into a “cosmological equation” which makes coherent what we know regarding the time (its origin, its flow …), the matter, and space. It sets up a mathematical model allowing us to interpret dark energy (or cosmological constant) as being both “negative” and “tired” energy. After an introduction with a brief history of unifications and the presentation of two roughly equal ratios arising out from Dirac’s large-number hypotheses which relate to the ratio of electric force to gravitational force and the ratio of the age of the universe to the atomic time unit associated with atomic processes, we present in §2 this new equation of quantum cosmology which operates the reconciliation between the macrocosm and the microcosm. In §3 and §4, we discuss the irreversible cosmological time resulting from the equation, as well as the role of the mass (heavy) relative to the gravitational constant G. In §5 we discuss the links that the equation establishes between gravitation (structure of condensation) and electromagnetism (structure of expansion), between relativity and quantum theory. We apply the formula to Planck’s time. We speak of the new essential variable? ?, and briefly of a new principle, the principle of compensation. In §6 we discuss the negative energy solutions banned by physics, and we deplore that half of the universe escapes us. We present the electro-gravitation in §7, from the equation which represents a super hydrogen atom. In § 8 we show that the global mass (gravitational) is variable: it increases during the expansion while the mass of the elementary particles decreases. In §9 we approach the spontaneous symmetry breaking;when it occurs, the arrows of the equation are momentarily reversed: such a mechanism would apply to the Allais effect, also mentioned in §6.4. §10 and §11 deal with the energy linked by the equation to matter through expansive space-time. The equation transforms electromagnetic kinetic energy into a gravitational mass, considered as a potential energy. Entropy increases according to the arrow of time towards the future. In §12 we discuss of the prevailing theory of inflation. We note the similarity between the proclaimed acceleration of current expansion and inflation. Physicists have interpreted the positive cosmological constant in terms of vacuum energy which would be 10120 times higher than the dark energy density deduced from the astronomical measurements. However, the high theoretical value of the vacuum energy (and the cosmological constant) has no observable pending in the cosmos. In §13 we suggest that these several orders of magnitude difference problem are solved by the theory of relation, which indicates a deceleration of the expansion. Finally, in § 14, we close by speaking of a model of cyclic universe and about the object of this paper, a dynamic equation that allows to build a quantum cosmology.

Binary Relations between Magnitudes of Different Dimensions Used in Material Science Optimization Problems Pseudo-State Equation of Soft Magnetic Composites

New algorithm for optimizing technological parameters of soft magnetic composites has been derived on the base of topological structure of the power loss characteristics. In optimization magnitudes obeying scaling, it happens that one has to consider binary relations between the magnitudes having different dimensions. From mathematical point of view, in general case such a procedure is not permissible. However, in a case of the system obeying the scaling law it is so. It has been shown that in such systems, the binary relations of magnitudes of different dimensions is correct and has mathematical meaning which is important for practical use of scaling in optimization processes. The derived structure of the set of all power loss characteristics in soft magnetic composite enables us to derive a formal pseudo-state equation of Soft Magnetic Composites. This equation constitutes a relation of the hardening temperature, the compaction pressure and a parameter characterizing the power loss characteristic. Finally, the pseudo-state equation improves the algorithm for designing the best values of technological parameters.

Depreciation factor equation to evaluate the economic losses from ground failure due to subsidence related to groundwater withdrawal

Subsidence due to groundwater withdrawal is a complex hydrogeological process affecting numerous cities settled on top of fluviolacustrine deposits. The discrete spatial variation in the thickness of these deposits, in combination with subsidence due to groundwater withdrawal, generates differential settlements and aseismic ground failure (AGF) characterized by a welldefined scarp. In cities, such AGF causes severe damages to urban infrastructure and considerable economic impact. With the goal of arriving to a general criterion for evaluating the economic losses derived from AGF, in the present work we propose the following equation: ELi = PVi*DFi. Where PVi is the value of a property “i”, and DFi is a depreciation factor caused by structural damages of a property “i” due to AGF. The DFi is calculated empirically through: . This last equation is based on the spatial relations of coexistence and proximity of property polygons and the AGF axis. The coexistence is valued as the quotient of the affectation area divided by the total area of the involved property;and the proximity to the AGF axis is expressed as the inverse of the perpendicular distance from the centroid of the property polygon to the AGF axis. The sum of these terms is divided by two to determine the percentage that affects the property value (PVi). These equations are relevant because it is the first indicator designed for the discrete assessment of the economic impacts due to AGF, and can be applied to real estate infrastructure from either urban or rural areas.

A Modified Form of Mild-Slope Equation with Weakly Nonlinear Effect

Nonlinear effect is of importance to waves propagating from deep water to shallow water. The non-linearity of waves is widely discussed due to its high precision in application. But there are still some problems in dealing with the nonlinear waves in practice. In this paper, a modified form of mild-slope equation with weakly nonlinear effect is derived by use of the nonlinear dispersion relation and the steady mild-slope equation containing energy dissipation. The modified form of mild-slope equation is convenient to solve nonlinear effect of waves. The model is tested against the laboratory measurement for the case of a submerged elliptical shoal on a slope beach given by Berkhoff et al. The present numerical results are also compared with those obtained through linear wave theory. Better agreement is obtained as the modified mild-slope equation is employed. And the modified mild-slope equation can reasonably simulate the weakly nonlinear effect of wave propagation from deep water to coast.

A RIEMANN-HILBERT APPROACH TO THE INITIAL-BOUNDARY PROBLEM FOR DERIVATIVE NONLINEAR SCHRDINGER EQUATION

We use the Fokas method to analyze the derivative nonlinear Schrodinger （DNLS） equation iqt （x, t） = -qxx （x, t）＋（rq^2）x on the interval [0, L]. Assuming that the solution q（x, t） exists, we show that it can be represented in terms of the solution of a matrix Riemann- Hilbert problem formulated in the plane of the complex spectral parameter ξ. This problem has explicit （x, t） dependence, and it has jumps across {ξ∈C｜Imξ^4 = 0}. The relevant jump matrices are explicitely given in terms of the spectral functions {a（ξ）, b（ξ）}, {A（ξ）, B（ξ）}, and {A（ξ）, B（ξ）}, which in turn are defined in terms of the initial data q0（x） = q（x, 0）, the bound- ary data g0（t）= q（0, t）, g1（t） = qx（0, t）, and another boundary values f0（t） = q（L, t）, f1（t） = qx（L, t）. The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.

New Numerical Scheme for Simulation of Hyperbolic Mild-Slope Equation

The original hyperbolic mild-slope equation can effectively take into account the combined effects of wave shoaling, refraction, diffraction and reflection, but does not consider the nonlinear effect of waves, and the existing numerical schemes for it show some deficiencies. Based on the original hyperbolic mild-slope equation, a nonlinear dispersion relation is introduced in present paper to effectively take the nonlinear effect of waves into account and a new numerical scheme is proposed. The weakly nonlinear dispersion relation and the improved numerical scheme are applied to the simulation of wave transformation over an elliptic shoal. Numerical tests show that the improvement of the numerical scheme makes efficient the solution to the hyperbolic mild-slope equation, A comparison of numerical results with experimental data indicates that the results obtained by use of the new scheme are satisfactory.

Semi-Analytical Solution of the 1D Helmholtz Equation, Obtained from Inversion of Symmetric Tridiagonal Matrix

An interesting semi-analytic solution is given for the Helmholtz equation. This solution is obtained from a rigorous discussion of the regularity and the inversion of the tridiagonal symmetric matrix. Then, applications are given, showing very good accuracy. This work provides also the analytical inverse of the skew-symmetric tridiagonal matrix.

NONTRIVIAL SOLUTIONS FOR SEMILINEAR SCHRDINGER EQUATIONS

The authors prove the existence of nontrivial solutions for the SchrSdinger equation -△u ＋ V（x）u =λf（x, u） in R^N, where f is superlinear, subcritical and critical at infinity, respectively, V is periodic.

A New Coupled KdV Equation： Painlevé Test

联合 Korteweg de-Vries 方程的一种新类型被发现到 bePainleve-integrable。新模型是能被用来与不同分散关系描述二层的液体的一种特殊情况。