The local great-gain control system is a feedback system with high order derivatives. It can be used to restrain the perturbance, nonlinearity and parameter variation.

Both the derivative and high order derivatives of F(s) are very useful for finding the function \%F(s)\% and its inverse transformation, so, it is indispensable to cerify the existence and derivability of the laplace transformation of \%f(t)\% and try to find a rule of operation for it.

而 F ( s)的导数及 F( s)的高阶导数在求 F ( s)及其逆变换时是很有用的 ,因此 ,弄清函数 f ( t)的 Laplace变换存在性、可导性及其运算规律是非常必要的 .

The Local Great-gain Hierarchical Decentrilezed control problem was disscused. The Local Great-gain Control is a feedback system with high order derivatives. It can realize decouple system.

This decomposition yields, for an appropriate gauge fixing, a Skyrme Faddeev like Wilsonian action and confirms the presence of high order derivatives of a color unit vector at the classical level.

On the differences in determining the time and vertical velocity characteristic scales and analyzing high order derivatives suggested by certain authors, discussions are made and preliminary conclusions are drawn, namely,τ~L/U, W≤ MUH/L and nf/x n～F/L n,n>1 for a single scale motion.

High order combined geometric chromatic aberrations were calculated with Differential Algebra,a method which is capable of directly evaluating arbitrarily high order derivatives with accuracy up to the upper limit of the computer to be used.

In this paper a novel nonlinear method is proposed to meet the precision demand based on the separation technique in frequency domain. After the features of high order derivatives of signal and noise in observed data are analyzed the problem of separating signal from noise is converted into an extremum problem, with the high order difference of stationary time series invariant to the variance.

In the process of system dynamic trajectory simulation and integration of sensitivity dynamic equations,Taylor series expansion technique is used to calculate the high order derivatives of rotor angles,speed and sensitivity variables with respect to time. Therefore,bigger simulation time step can be employed,which can improve computation speed while keeping normal calculating precision.

The proof consists in studying high order derivatives of the pressure pγ(h), which is related to the free energy fγ(m) by a Legendre transform.

Given an algorithmUL that finds all the solutions of linear homogeneous differential and difference equations in some linear space L, we describe two algorithms that construct all solutions whose high order derivatives (or differences) are in L.

In this paper the uniform convergence of high order derivatives of ?n to the corresponding derivatives of ? is proved.

Comparing the exact element method with the general finite element method with the same degrees of freedom, the high convergence rate of the high order derivatives of solution can be obtained.

Reconstruction of high order derivatives from input data

In this paper we discussed mixed problems for the second order quasi-linear hyperbolic partial differential equations, which involve small parameter in the higher order derivatives. As ε=0, the original equations are degenerated into the lower order differential equation, and the part of boundary conditions is losed. The problems of this type are called a singular perturbation problems. We constructed an asymptotic solution of the presented problem and invstegated the asymptotic behaviour of...

In this paper we discussed mixed problems for the second order quasi-linear hyperbolic partial differential equations, which involve small parameter in the higher order derivatives. As ε=0, the original equations are degenerated into the lower order differential equation, and the part of boundary conditions is losed. The problems of this type are called a singular perturbation problems. We constructed an asymptotic solution of the presented problem and invstegated the asymptotic behaviour of the solution.

In this paper we discussed mixed problems for the second order quasi-linear hyperbolic partial differential equations, which involve small parameter in the higher order derivatives. As ε=0, the original equations are degenerated into the lower order differential equation, and the part of boundary conditions is losed. The problems of this type are called a singular perturbation problems. We constructed an asymptotic solution of the presented problem and invstegated the asymptotic behaviour of...

In this paper we discussed mixed problems for the second order quasi-linear hyperbolic partial differential equations, which involve small parameter in the higher order derivatives. As ε=0, the original equations are degenerated into the lower order differential equation, and the part of boundary conditions is losed. The problems of this type are called a singular perturbation problems. We constructed an asymptotic solution of the presented problem and invstegated the asymptotic behaviour of the solution.

This paper deals with the near-tip singular fields for Mode-Ⅲ crack growing quasi-statically and steadily in a power hardening medium. In Section 2 the basic equations are set up and in Section 3 the contiguity conditions for neighbouring domains (3.1), (3.7) and also the supplementary conditions (3.8) for the unloading boundary and (3.15) for the reloading boundary are derived. As shown in Fig. 1, the near-tip region (upper half-plane) is composed of the active plastic zone Ⅱ, the unloading wake zone Ⅲ, and...

This paper deals with the near-tip singular fields for Mode-Ⅲ crack growing quasi-statically and steadily in a power hardening medium. In Section 2 the basic equations are set up and in Section 3 the contiguity conditions for neighbouring domains (3.1), (3.7) and also the supplementary conditions (3.8) for the unloading boundary and (3.15) for the reloading boundary are derived. As shown in Fig. 1, the near-tip region (upper half-plane) is composed of the active plastic zone Ⅱ, the unloading wake zone Ⅲ, and the reloading plastic zone Ⅳ, subtending angles θP, (π - θp - θs) and θs respectively.In Sections 4 and 5 the stress fuction (?)is solved in the form of a series (4.1) in [In (A/r)]-1, and the near-tip stress and strain fields are obtained to within a parameter, say the parameter 8 ( = cFn-1G, n, c being material constants, G the shear modulus and F the intensity of stress singularity), which varies in a range Str≤S≤ Sb, corresponding to the segments MnNn on the S versus θP curves in Fig. 4. θs versus θp curve is given in Fig. 3. Mn state corresponds to a trivial solution (5.16), with the same values of angles θP,tr and θs,tr as for elastic perfectly-plastic medium. Nn state corresponds to the upper bound θb of the angle θP. θb is the natural boundary for the solution of the differential equation (5.11), i.e. θb is determined by the condition that the coefficients of the highest-order derivative f1*"(θ) in (5.11) should vanish at θ = θb (see (5.17)). For elastic perfectly-plastic material (n→∞), MnNn in Fig. 4 shrinks to a point (θP = θp,tr S = 0.5), which corresponds to the solution given by Chitaley and McClintock. For several values of n, the near-tip stress and strain distribution is shown in Fig. 5 and 6. It is seen from these Figures that the difference between the extreme states Mn and Nn is only slight. It is expected that a inner boundary-layer type of solution exists in the neighbourhood of the unloading boundary Γa (Fig. 1). The consideration of this boundary-layer solution will be necessary for the determination of the unique true solution (corresponding to some state on the segment MnNn in Fig. 4), with vanishing plastic strain rates on the unloading boundary, λ|ΓB = 0. The determination of this unique true solution will be given in a separate paper.