THE BCKLUND TRANSFORMATION AND NONLINEAR SUPERPOSITION FORMULA OF SOLUTIONS FOR THE LIOUVILLE′S EQUATION IN HIGHER DIMENSIONS

In this paper, we show that Bǎcklund transformation derived by Leibbrandt et al. for the Liouville's equation in three spatial dimensions, can be decomposed into several Bǎcklund transformations for the same equation in two spatial dimensions, moreover, the super-position formula which is derived fromthis transformation is actually invalid, thus the discussions based on that formula is incorrect as well. We also considered some results about the Liouville's equation in N spatial dimensions.     

A spectral resolving method for analyzing linear random vibrations with variable parameters

This paper is a development of ref. [1]. Consider the following random equation: Z(t)+2βZ(t)+ω₀²Z(t)=(a⁰+a¹Z(t))I(t)+c in which excitation I(t) and response Z(y) are both random processes, and it is proposed that they are mutually independent. Suppose that a(t) is a known function of time and I(t) is a stationary random process. In this paper, the spectral resolving form of the random equation stated above, the numerical solving method and the solutions in some special cases are considered.     

ROSSBY WAVES WITH THE CHANGE OF β

In this paper, the change of the Rossby parameter β with latitude is considered and the parameter γ≡-dβ/dy=2sinφ/a² is introduced and the β-plane approximation is extended into f=f₀+β₀y-γ₀y²/2 which includes the parameter γ. Such approximation closes further to practice especially in the high latitude regions.We give emphasis to the research of the effect of the parameter γ on the Rossby waves. It is seen that the effect of the parameter γ is remarkable in the high latitude regions. It can produce the Rossby waves caused by the pure parameter γ. And the phase speed formula of Rossby waves with the change of ft is generally given, which is degenerated into the well-known Rossby formula when γ₀=0. The researches also point out that when the change of β is regarded, even if the basic current u is a linear function of y the unstable modes can also take place. However,the parameter γ usually plays a stable part in the Rossby waves and it does affect the longitudinal scale and the structure of constant phase lines(trough-ridge lines)of Rossby waves and slow down the growing or decaying of Rossby waves.     

Nonstationary pandom vibpation analysis of linear elastic structures with finite element method

At present, the finite element method is an efficient method for analyzing structural dynamic problems. When the physical quantities such as displacements and stresses are resolved in the spectra and the dynamic matrices are obtained in spectral resolving form, the relative equations cannot be solved by the vibration mode resolving method as usual. For solving such problems, a general method is put forward in this paper. The excitations considered with respect to nonstationary processes are as follows: P(t)={P_i(t)},P_i(t)=a_i(t)P_i(t), a_i(t) is a time function already known. We make Fourier transformation for the discretized equations obtained by finite element method, and by utilizing the behaviour of orthogonal increment of spectral quantities in random process^[1], some formulas of relations about the spectra of excitation and response are derived. The cross power spectral denisty matrices of responses can be found by these formulas, then the structrual safety analysis can be made. When a_i(t)=l (i= 1,2,…n), the. method stated in this paper will be reduced to that which is used in the special case of stationary process.     

The analytical solution for Helmholtz boundary problem in non-horizontally stratified domains

There are N domains D_j(j=0,1,...,N-1) of different physical parameters in the whole space and their interfaces S_j,i+1 are non-horizontally smooth curved surfaces. The following boundary problem is called Hclinholiz boundary problem:V²H⁽¹⁾+K_jH⁽¹⁾=0 (j=0,1,…,N-1)(H⁽⁰⁾-H⁽¹⁾)=δ(S) (δ(S):generalized function)(H¹-Hⁱ⁺¹)=0 (j=0,1,…,N-2)The analytical solution of the above problem is given in this paper.     

On near-wall hot-wire measurements

A specially constructed hot-wire probe was used to obtain very near-wall velocity measurements in both a fully developed turbulent channel flow and flat plate boundary layer flow. The near-wall hot-wire probe, having been calibrated in a specially constructed laminar flow calibration rig, was used to measure the mean streamwise velocity profile, distributions of streamwise and spanwise intensities of turbulence and turbulence kinetic energy k in the viscous sublayer and beyond; these distributions compare very favorably with available DNS results obtained for channel flow. While low Reynolds number effects were clearly evident for the channel flow, these effects are much less distinct for the boundary layer flow. By assuming the dissipating range of eddy sizes to be statistically isotropic and the validity of Taylor's hypothesis, the dissipation rate ɛ _iso in the very near-wall viscous sublayer region and beyond was determined for both the channel and boundary layer flows. It was found that if the convective velocity U _c in Taylor's hypothesis was assumed to be equal to the mean velocity Uˉ at the point of measurement, the value of (ɛ⁺ _iso)₁ thus obtained agrees well with that of (ɛ ⁺)_DNS for y ⁺ ≥ 80 for channel flow; this suggests the validity of assuming U _c=Uˉ and local isotropy for large values of y ⁺. However, if U _c was assumed to be 10.6u _τ , the value of (ɛ⁺ _iso)₂ thus obtained was found to compare reasonably well with the distribution of (ɛ⁺ _iso)_DNS for y ⁺≤ 15. Received: 31 May 1999/Accepted: 20 December 1999     

BASIC EQUATIONS, THEORY AND PRINCIPLE OF COMPUTATIONAL STOCK MARKET (Ⅲ)-BASIC THEORIES

By basic equations, two basic theories are presented: 1. Theory of stock’s value v^*(t)=v^*(0) exp (ar₂^*t);2. Theory of conservation of stock’s energy. Let stock’s energy Φbe defined as a quadratic function of stock’s price v and its derivative v, Φ=Av²+Bv+Cv²+Dv, under the constraint of basic equation, the problem was reduced to a problem of constrained optimization along optimal path. Using Lagrange multiplier and Euler equation of variation method, it can be proved that Φkeeps conservation for any v,v. The application of these equations and theories on judgement and analysis of tendency of stock market are given, and the judgement is checked to be correct by the recorded tendency of Shenzhen and Shanghai stock markets.     

ON THE CRITICAL POINTS OF THE MAP Fp:X→‖AXB-C‖pp

Suppose A,B and C are the bounded linear operators on a Hilbert space H, when A has a generalized inverse A^- such that (AA^-)^*=AA^- and B has a generalized inverse B^- such that (B^-B)^*=B^-B,the general characteristic forms for the critical points of the map F_p:X^→‖AXB-C‖_p^p(1p=2. Similarly, the same question has been discussed for several operators.     

ON THE STRUCTURE OF SOLUTIONS OF LINEAR PARTIAL DIFFERENTIAL EQUATION sum form i+j≤n aijpiqjφ=0 WITH TWO INDEPENDENT VARIABLES AND CONSTANT COEFFICIENTS

This paper is a continuation of [1]. In this paper, the solutions of the more general linear partial differential equation form i+j≤n a_ijpⁱq^jφ=0 with two independent variables and constant coefficients are discussed in detail. The general solution which can be used in the approximation to the conditions of the definite solution of the practical problems is presented. To illustrate the use of the results obtained in this paper, some practical examples in mechanics are given.     

On the Solution of Elliptic Equation (sum from k-0 to n) akΔkφ=0 and Its Application in Mechanics

In this paper, the solution of elliptic equation (sum from k-0 to n) a_kΔ^kφ=0 is discussedin detail by the method of separation of variables in complex field. The general solution which can be used in the approximation to the boundary conditions of the practical problems is presented. Two practical examples in mechanics are given.     

The strain energy criterion of mixed-mode brittle fracture

In this paper, we proposed a new cr tenon or mixea-moae brittle fracture, i.e., the strain energy criterion, which can be stated as (K_Ⅰ/K_Ⅰc)² +(K_Ⅰ/K_Ⅰc)²+(K_Ⅱ/K_Ⅱc)²=1. This criterion is shown to be in good agreement with known experimental data.In this paper, an experimental criterion:(K_Ⅰ/K_Ⅰc)^m+(K_Ⅱ/K_Ⅱc)ⁿ=1, 1≤_n^m≤2.is also proposed.     

On singular perturbation method of perturbed bifurcation problems

In this paper, the general mathematical principle is over-all explained and a new general technique is presented in order to calculate uniformly asymptotic expansions of solutions of the perturbed bifurcation problem (1.6) in the vicinity of y=0, λ=0,δ=0, by means of singular perturbation method. Simultaneously, Newton’s polygon^[4] is generalized. Finally, the calculating results of two examples are given.     

A MATRIX METHOD OF DISPLACEMENT ANALYSIS OF THE GENERAL SPATIAL 7R MECHANISM

An input-output equation of the general spatial 7R mechanism is derived in this paperby using the method in [1] and applying the rotation matrices.The result is the same as[2],but the process of derivation is simpler.Applying the character of rotation matrices,it is notdifficult to obtain the recurrence formulas of direction cosines of Cartesian unit vectors,tocalculate the scalar products and triple products of these unit vectors,and to derive the 6thconstraint equation.Moreover,an algorithm,which consists of successive applications ofrow transformation and expansion based on Laplace’s Theorem,is given to evaluate the16×16 determinant according to its characteristic.so that the evaluation is muchsimplified.     

On shock layer method for the hypersonic flow problems

Chernyi’s series method[1] is not proper for the case that(γ-l)/(γ+l)<<2/(γ+1)×M²sin²β (γ=c_p/c_v-adiabatic index number, M-Much number, β-shock incidence). In this paper, we only suppose that in the neighbour of the shock, there exists a shock layer in which the density of the gas is very big, but we do not remove the case that (γ-1)/(γ+1)<<2/(γ+1)M²sin²β.     

The effect of the vasomotion on the mass transfer in microcirculation

Detailed structure of the attracting set of the piecewise linear Henon mapping(x, y)→(1- a|x|+by,x)with a=8/5 and b=9/25 is described in this paper using the method of dual line mapping. Let A and B denote the fixed saddles in the first quadrant, and in the third quadrant, respectively. It is claimed that(1)the attracting set is the closure of the unstable manifold of saddle B, which includes the unstable manifold of A as its subset, and(2)the basin of attraction is the closure of the stable manifold of A, bounded by the stable manifold of B, which is in the limiting set of the stable manifold of A.Relations of the manifolds of the periodic saddles with the manifolds of the fixed point are given. Symbolic dynamics notations are adopted which renders possible the study of the dynamical behavior of every piece of the manifolds and of every homoclinic or heteroclinic point.     

Calculation of stresses and deformations of bellows by initial parameter method of numerical integration

By reducing the boundary value problem in stress analysis of bellows into initial value problem, this paper presents a numerical solution of stress distribution in semi-circular arc type bellows based upon the toroidal shell equation of V. V. Novozelov^[8]. Throughout the computation, S. Gill’s method^[1O] of extrapolation is used. The stresses and deformations of bellows under axial load and internal pressure are c-alculated, the results of which agree completely with those derived from the general solution of Prof. Chien Wei-zang^[1-4]. The extrapolation formula presented in this paper greatly promotes the accuracy of discrete calculation.The computer program in BASIC language of Wang 2200 VS computer is included in the appendix.     

THE DIMENSION OF SPLINE SPACE S31(△) ON A TYPE OF TRIANGULATION

A harmonic condition that can distinguish whether the dimension of spline space S₃¹(△) depends on the geometrical character of triangulation is presented, then on a type of general triangulation the dimension is got.     

An analysis of the effects of the prostatic hypertropy on he urinary flow in lower-urinary-tract

In this paper an analytic model corresponding to the collapsible tube for analysing the urinary flow in lower-urinary-tract is set up from physiologic background.By analysing the model it is found that the self-excited oscillations can both occur in the region of negative and positive slope of P_n-Q_n characteristic. So this paper extends the results of Conrad1’1, Griffiths121, Conrad, Cohen and McQueenI3! and others that the self-excited oscillations can only occur in the region of negative slope of P_n-Q_n characteristic. The effects of prostatic hypertropy on the flow parameters in lower-urinary-tract is discussed in detail by numerical calculations. The results show that it is possible to know the conditions of prostatic hypertropy according to the changes of bladder pressure, outlet urinary velocity and other parameters. From these results a theoretical method to detect and diagnose prostatic hypertropy is provided.     

Study on the characteristic of the spray angle in pressure swirl spray atomisation

Based on the suggested atomisation theory for the swirl spray conical film, the formula for the spray angle characteristic of pressure swirl spray atomisation θ=tg^-12·(1-φ) is derived from the relation of acting forces in swirl spray.The spray angle characteristics of swirl spray are worked out with various formulas and compared with actual test data. The results show that the derived formulas for spray angle in this article agree comparatively well with the results from experiments, and that the expressions are simple. They are of definite value in practice.     

On the assumption of Saint-Venant's problem

In this paper we obtain uniquely the solution of Saint- Venant’s problem under theassumption of ∂^m/∂z^mσ_z=0(m≥2) for noncircular prismatic bars.