一个关于素变量的四元加性方程

用[x]表示不超过x的最大整数.本文证明了对于充分大的正整数N以及任意的1<c<14/11,关于素变量P_(1),P_(2),P_(3),P_(4)的加性方程[p^(c)_(1)]+[p^(c)_(2)]+[p^(c)_(3)]+[p^(c)_(4)]=N有无穷多组解.

Hot deformation behavior of homogenized Al-3.2Mg-0.4Er aluminum alloy

The hot deformation behavior of the homogenized Al?3.2Mg?0.4Er aluminum alloy was investigated at 573?723 K under strain rates of 0.001?1 s?1. On the basis of compression experimental results, an accurate phenomenological constitutive equation that coupled the effects of strain rate, deformation temperature and strain was modeled. Furthermore, a kinetic model of dynamic recrystallization and processing map were also presented. The results show that the flow stress of the studied Al?3.2Mg?0.4Er alloy can be predicted accurately using the proposed constitutive model. The evolution of microstructure and the volume fraction of dynamic recrystallization can be described exactly in terms of S-curves with the proposed kinetic model. Moreover, the processing maps for hot working at different strains were constructed, suggesting the optimum processing conditions for this alloy are 573 K, 0.001 s?1 and 723 K, 0.001?0.1 s?1.

Hot deformation behavior and workability of pre-extruded ZK60A magnesium alloy

The hot deformation behavior and workability of pre-extruded ZK60A magnesium alloy were investigated by compression tests in the temperature range of 250-450 ℃and the strain rate range of 0.001-10 s 1. The constitutive equation for the pre-extruded ZK60A alloy can be described by hyperbolic sine function. Processing maps were constructed from true strains of -0.2 to -0.8. The alloy experienced complete dynamic recrystallization （DRX） and showed good workability in the temperature range of 300-400 ℃ and the strain rate range of 0.01-0.001 s-Z, where hot working in pre-extruded ZK60A, such as forging, can be carried out. For large deformation to true strain of over -0.5, strain rates above 0.1 s-1 are not recommended at all temperatures, where flow instability such as local strain concentration, twinning deformation, abnormal grain growth, micro-cracks, and shear fracture were observed. Climb-controlled dislocation creep dominates both the plastic deformation and nucleation of DRX of the pre-extruded ZK60A magnesium alloy.

A method of determining the permeability coefficient of coal seam based on the permeability of loaded coal

This study developed the equipment for thermo-fluid–solid coupling of methane-containing coal, and investigated the seepage character of loaded coal under different working conditions. Regarding the effective pressure as a variable, the variation characteristics of the gas permeability of loaded methane-containing coal has been studied under the conditions of different confining pressures and pore pressures. The qualitative and quantitative relationship between effective stress and permeability of loaded methane-containing coal has been established, considering the adsorption of deformation, amount of pore gas compression and temperature variation. The results show that the permeability of coal samples decreases along with the increasing effective stress. Based on the Darcy law, the correlation equation between the effective stress and permeability coefficient of coal seam has been established by combining the permeability coefficient of loaded coal and effective stress. On the basis of experimental data, this equation is used for calculation, and the results are in accordance with the measured gas permeability coefficient of coal seam. In conclusion, this method can be accurate and convenient to determine the gas permeability coefficient of coal seam, and provide evidence for forecasting that of the deep coal seam.

Possible Linear Superpositions for Special Solutions of the Fifth-Order KdV-Type Equations

Using Lou and Ni's deformation and mapping idea in nonlinear equations to a set of fifth order KdV type equations, it is found that some types of solitary wave solutions and periodic solutions with special velocities can be linearly superposed to new exact solutions.

Constitutive equation and processing map of equiatomic NiTi shape memory alloy under hot plastic deformation

In order to describe the deformation behavior and the hot workability of equiatomic NiTi shape memory alloy (SMA) during hot deformation, Arrhenius-type constitutive equation and hot processing map of the alloy were developed by hot compression tests at temperatures ranging from 500 to 1100 °C and strain rates ranging from 0.0005 to 0.5 s?1. The results show that the instability region of the hot processing map increases with the increase of deformation extent. The instability occurs in the low and high temperature regions. The instability region presents the adiabatic shear bands at low temperatures, but it exhibits the abnormal growth of the grains at high temperatures. Consequently, it is necessary to avoid processing the equiatomic NiTi SMA in these regions. It is preferable to process the NiTi SMA at the temperatures ranging from 750 to 900 °C.

Analysis of a Failed Port Slope and Stabilization

A nonmetal transportation port with soft soil foundation of Tianjin New Harbor failed during construction in 1997. The soil properties and geological changes before and after landslide were in- vestigated by in situ test means. The stability of the slope was estimated with several calculation methods, such as the simplified Bishop method and the non-circle analysis method. According to the investigation and analysis results, the factors that may cause the slope failure can be deter- mined as follows: excessive excavating, violating the normal construction procedure, excessive pore water pressure during earlier piling and sudden water-level fluctuation in front of the slope. Measures to improve the soil foundation for reconstructing the port include strengthening the failed slope with vibro sand piles, filling up the crushed stone layer, dividing the excavation into four steps and driving piles by larger intervals. The port now has been successfully reconstructed and begun operation.

Static characteristics of new type externally pressurized spherical air bearings

In order to provide some theoretical guideline for the structure design of the new type externally pressurized spherical air bearings,the static characteristics and the factors affecting the static characteristics of the air bearings were analyzed.A finite volume method was adopted to discretize the three-dimensional steady-state compressible Navier-Stokes equations,and a modified SIMPLE algorithm for compressible fluid was applied to solve the discretized governing equations.The pressure field and velocity field of the air bearings were obtained,and the factors and rules affecting the static characteristics were analyzed.The results show that the pressure of near air intakes can reach above 80% of air supply pressure,and there is a pressure steep fall around the air intakes.When the film thickness is greater than 20 μm,the bearing capacity rapidly decreases as film thickness increases.As the air supply pressure increases from 0.2 to 0.6 MPa,the maximum static stiffness increases by more than three times.The calculation method proposed well fits the general principle,which can be extended to the characteristic analysis of other air bearings.

Modified Lindstedt-Poincaré Method for Obtaining Resonance Periodic Solutions of Nonlinear Non-autonomous Oscillators

A modified Lindstedt-Poincar＆#233; （LP） method for obtaining the resonance periodic solutions of nonlinear non-autonomous vibration systems is proposed in this paper. In the modified method, nonlinear non-autonomous equa-tions are converted into a group of linear ordinary differential equations by introducing a set of simple transformations. An approximate resonance solution for the original equation can then be obtained. The periodic solutions of primary, super-harmonic, sub-harmonic, zero-frequency and combination resonances can be solved effectively using the modi-fied method. Some examples, such as damped cubic nonlinear systems under single and double frequency excitation, and damped quadratic nonlinear systems under double frequency excitation, are given to illustrate its convenience and effectiveness. Using the modified LP method, the first-order approximate solutions for each equation are obtained. By comparison, the modified method proposed in this paper produces the same results as the method of multiple scales.

Simulation of Reciprocating Compressor Start-up and Shut-down under Loaded and Unloaded Conditions

During the start-up and shut-down phase of reciprocating compressors, the loads on all components of driven train system are very high. In this paper a method for calculating the forces on coupling, e-motor, crank shaft as well other components of the system will be described. The modelling of the electrical induction motor, coupling, crank shaft, damper as well as the compressor resistance torque are extremely important in simulating start-up and shut-down of reciprocating compressor. Furthermore the switching torque of the electrical motor and the instantaneous moment of inertia of the reciprocating compressor crank gear are important as well. The transient start-up and shut-down process under loaded and unloaded conditions is described using a non-linear differential equation for driven train system： E-motor--coupling--flywheel--reciprocating compressor--damper. Shaft torsional moments on the drive train and especially on the coupling, whether elastic or stiff, can then only be calculated using numerical simulation. This paper will describe some of the key elements in modelling, simulating and measurements of drive train start-up and shut-down carried out on already operational piston compressor units.

Modeling of Bubble Motion in Two Dimensional Space

The equations of motion of a bubble, expanding adiabatically through an incompressible viscous fluid, are deduced when the centre of the bubble moves in a vertical plane in the presence of gravitational acceleration, acting vertically downwards. The non-linear equations of motion obtained are solved numerically for different values of the various parameters of the problem. The path traced by the centre of the bubble and velocity of the centre, the change of radius R with time, and the influence of the buoyancy force, which is experienced by the expanding bubble for different values of the gravitational acceleration on these quantities, are investigated. The radius R（t） of the bubble is found to vary periodically with time when the acceleration due to gravity is small. But when the acceleration due to gravity increases, this periodicity in the value of R（t） with t is lost. The influence of viscosity in determining the periodicity of the bubble motion is also investigated.

On a Linear Equation Arising in Isometric Embedding of Torus-like Surface

The solvability of a linear equation and the regularity of the solution are discussed. The equation is arising in a geometric problem which is concerned with the realization of Alexandroff's positive annul in R3.

NONLINEAR GALERKIN METHODS FOR SOLVING TWO DIMENSIONAL NEWTON－BOUSSINESQ EQUATIONS

The nonlinear Galerkin methods for solving two-dimensional Newton-Boussinesq equations are proposed.The existence and uniqueness of global generalized solution of these equations, and the convergence of approximate solutions are also obtained.

On a Quasilinear Elliptic Equation with Superlinear Nonlinearities

This work is devoted to studying a quasilinear elliptic boundary value problem with superlinear nonlinearities in a weighted Sobolev space in a domain of R^N. Based on the Galerkin method, Brouwer's theorem and the weighted compact Sobolev-type embedding theorem, a new result about the existence of solutions is revealed to the problem.

A Study of the Behavioral Controllability for Superposed Kuznetsov–Ma Soliton of the (3+1)-Dimensional Generalized Nonlinear Schr?dinger Equation

Abstract The （3＋1 ）-dimensional variable-coetfficient nonlinear SchrSdinger equation with linear and parabolic traps is studied, and an exact Kuznetsov-Ma soliton solution in certain parameter conditions is derived. These precise expressions indicate that diffraction and chirp factors influence phase, center and widths, while the gain/loss parameter only affects peaks. By adjusting the relation between the maximum accumulated time Tm and the accumulated time To based on maximum amplitude of Kuznetsov Ma soliton, postpone, maintenance and restraint of superposed Kuznetsov-Ma solitons are investigated.

Time Optimal Feedrate Generation with Confined Tracking Error Based on Linear Programming

In this paper,the problem of time optimal feedrate generation under confined feedrate,axis accelerations,and axis tracking errors is considered.The main contribution is to reduce the tracking error constraint to constraints about the axis velocities and accelerations,when the tracking error satisfies a second order linear ordinary differential equation.Based on this simplification on the tracking error,the original feedrate generation problem is reduced to a new form which can be efficiently solved with linear programming algorithms.Simulation results are used to validate the methods.

STOCHASTIC DIFFERENTIAL EQUATIONS AND STOCHASTIC LINEAR QUADRATIC OPTIMAL CONTROL PROBLEM WITH LEVY PROCESSES

In this paper, tile authors first study two kinds of stochastic differential equations （SDEs） with Levy processes as noise source. Based on the existence and uniqueness of the solutions of these SDEs and multi-dimensional backward stochastic differential equations （BSDEs） driven by Levy pro- cesses, the authors proceed to study a stochastic linear quadratic （LQ） optimal control problem with a Levy process, where the cost weighting matrices of the state and control are allowed to be indefinite. One kind of new stochastic Riccati equation that involves equality and inequality constraints is derived from the idea of square completion and its solvability is proved to be sufficient for the well-posedness and the existence of optimal control which can be of either state feedback or open-loop form of the LQ problems. Moreover, the authors obtain the existence and uniqueness of the solution to the Riccati equation for some special cases. Finally, two examples are presented to illustrate these theoretical results.

Coupled modified Kd V equations, skew orthogonal polynomials, convergence acceleration algorithms and Laurent property

In this paper, we show that the coupled modified Kd V equations possess rich mathematical structures and some remarkable properties. The connections between the system and skew orthogonal polynomials,convergence acceleration algorithms and Laurent property are discussed in detail.

From Dislocation Motion to an Additive Velocity Gradient Decomposition, and Some Simple Models of Dislocation Dynamics

A mathematical theory of time-dependent dislocation mechanics of unrestricted geometric and material nonlinearity is reviewed. Within a "small deformation" setting, a suite of simplified and interesting models consisting of a nonlocal Ginzburg Landau equation, a nonlocal level set equation, and a nonlocal generalized Burgers equation is derived.In the finite deformation setting, it is shown that an additive decomposition of the total velocity gradient into elastic and plastic parts emerges naturally from a micromechanical starting point that involves no notion of plastic deformation but only the elastic distortion,material velocity, dislocation density and the dislocation velocity. Moreover, a plastic spin tensor emerges naturally as well.