The human civilization is founded on the basis of “extended order of human cooperation”, and the market order came from the order expansion, it has an inconceivable creativity.

Furthermore, the sound interaction of market order expansion, growth of specialized exchange organizations and industrial development enables Yiwu to have its unique industrialized and urbanized path.

the existence of optimal solution set W(u) of the U—Lagrangian and the characterization of the associated smooth trajectary x + u + W(u) tangential to U, so that the second order expansion of f can be develped.

UV decomposition theory is an efficient method dealing with the second order expansion of nonsmooth functions. It is obtained via the U Lagrangian that a function f has second order expansions along smooth trajectories.

Characteristics of picosecond pulses generated from synchronously pumped dye laser system——The third order expansion of gain of mode-locked equation and its solution

One more significant difference between the two models is in the first-order expansion of the average field.

The dynamic critical index related to the cross section of scattering on fluctuation modes is calculated in the vicinity of the same point for the second-order expansion in ∈.

Second-order expansion for the expected regret risk in classification of one-parametric distributions

The properties of combined multiplier and penalty function methods are investigated using a second-order expansion and results known for the Riccati equation.

We also give explicit solutions for thermodynamic length along isotherms in the case of first, second and third order expansion

The three-dimensional unsteady second-order non-homogeneous differential equation has been derived by superposition of a small disturbance on a given steady three-dimensional flow. Based on the assumption of high Mach numbers this second-order equation for unsteady flow reduces to a form analogous to that for steady flow. This makes it possible to solve the equation by methods used in the second-order theory for steady flows. In the course of solution the flows are constrained and corrected according to the...

The three-dimensional unsteady second-order non-homogeneous differential equation has been derived by superposition of a small disturbance on a given steady three-dimensional flow. Based on the assumption of high Mach numbers this second-order equation for unsteady flow reduces to a form analogous to that for steady flow. This makes it possible to solve the equation by methods used in the second-order theory for steady flows. In the course of solution the flows are constrained and corrected according to the PLK method, and singularities are thus eliminated. The crucial point in this procedure is to find the correct particular solutions. Two particular solutions are used. One is the approximate three-dimensional particular solution. The other is obtained under the assumption of local two-dimensionality. In addition, the uniform particular solution is given, from which the uniform second-order solutions may be obtained. Then, we have treated the unsteady problem for delta wings with low aspect ratio and supersonic leading edges. The Mach number range for application of the present theory is from supersonic to low hypersonic values with reduced frequencies up to near unity. The theoretical results derived in this work can be used to calculate the unsteady aerodynamic characteristics of wings having arbitrary airfoil sections.As experimental information for similar conditions is not yet available, we can only compare our results with those of Liu D. D. . For this reason, only the derivation for a flat delta wing oscillating at low frequencies has been carried out and an analytical expression is obtained for the first order expansion of the unsteady velocity potential. In the range of Mach numbers 4 to 8, our results are in agreement with those of Lui D. D. .It is also shown that under conditions of three-dimensional thin wings the second-order theory is valid up to Mδ=1.0, while according to application of the second-order theory to bodies of revolution by Van Dyke, the useful upper limit of M5 is only 0.7. Hence, with Mδ=0.7-1 .0, the principal non-linear effects can be calculated by our second-order theory, while for thin wings the third-order terms connected with heat transfer and entropy change can be ignored.

In this paper taking high order expansion proposed from the reference[7], the GGP (general geometric programming) is transformed to solve sequential quadratic programming in the log-log space, which is called as GGP-LOG-SQP algorithm of the GGP. A program of the algorithm is incooprated in micro-computer Cromem-co and is used to solve GGP problems in the structural optimzation. The examples given in the third part of the paper show that the algorithm is quite efficient.

In this paper, we present the third order expansion of gain of mode-locked equation, and solved the equation with Lagrange multiplier method. The results provide) simple analytic expressions for the pulsewidth, pulse intensity and delay (which is the advance of the pump pulse relative to the dye pulse) in synchronously pumped dye laser. They can explain satisfactorily some experimental results[4].