Multiple Darboux–B?cklund transformations via truncated Painleve′ expansion and Lie point symmetry approach

For a given truncated Painleve′ expansion of an arbitrary nonlinear Painleve′ integrable system, the residue with respect to the singularity manifold is known as a nonlocal symmetry, called the residual symmetry, which is proved to be localized to Lie point symmetries for suitable prolonged systems. Taking the Korteweg–de Vries equation as an example, the n-th binary Darboux–Ba¨cklund transformation is re-obtained by the Lie point symmetry approach accompanied by the localization of the n-fold residual symmetries.

Phase field simulation of liquid-solid-eutectoid multiple phase transformations of a Fe-C binary alloy

A new continuous multi-phase transformation field model was established for liquid-solid-eutectoid transformation. Taking Fe-C alloy as an example, the model was used to simulate the evolution of the micro-morphology of the liquid-solid phase transition, and the effects of temperature, solute and free energy on the nucleation of pearlite after the liquid-solid phase transition were analyzed. The micro-morphology of pearlite was simulated. The simulation results show that the austenite structure has hereditary effect on the pearlite, the morphology of pearlite structure was similar to that of the parent austenite. The eutectoid structure at the front of pearlite grows toward the interior of austenite grains in a bifurcation manner and in the spherical coronal shape. In addition, the growth rate of pearlite was related to the shape of concave-convex interface at the nucleation site, and the growth rate at the convex interface was faster than that at the concave interface.

The multiple-parameter fractional Fourier transform

The fractional Fourier transform （FRFT） has multiplicity, which is intrinsic in fractional operator. A new source for the multiplicity of the weight-type fractional Fourier transform （WFRFT） is proposed, which can generalize the weight coefficients of WFRFT to contain two vector parameters m,n ∈ Z^M . Therefore a generalized fractional Fourier transform can be defined, which is denoted by the multiple-parameter fractional Fourier transform （MPFRFT）. It enlarges the multiplicity of the FRFT, which not only includes the conventional FRFT and general multi-fractional Fourier transform as special cases, but also introduces new fractional Fourier transforms. It provides a unified framework for the FRFT, and the method is also available for fractionalizing other linear operators. In addition, numerical simulations of the MPFRFT on the Hermite-Gaussian and rectangular functions have been performed as a simple application of MPFRFT to signal processing.

GRAPHICAL-BASED RELIABILITY EVALUATION OF MULTIPLE DISTINCT PRODUCTION LINES

This paper proposes a graphical-based methodology to evaluate the performance of a manufacturing system in terms of network model.We focus on a manufacturing system which consists of multiple distinct production lines.A transformation technique is developed to build the manufacturing system as a manufacturing network.In such a manufacturing network,the capacity of each machine is multistate due to failure,partial failure,or maintenance.Thus,this manufacturing network is also regarded as a multistate network.We evaluate the probability that the manufacturing network can meet a given demand,where the probability is referred to as the system reliability.A simple algorithm integrating decomposition technique is proposed to generate the minimal capacity vectors that machines should provide to eventually satisfy demand.The system reliability is derived in terms of such capacity vectors afterwards.A practical application in the context of IC card manufacturing system is utilized to demonstrate the performance evaluation procedure.

3-D CONSOLIDATION ANALYSIS OF LAYERED SOIL WITH ANISOTROPIC PERMEABILITY USING ANALYTICAL LAYER-ELEMENT METHOD

Starting with governing equations of a saturated soil with anisotropic permeability and based on multiple integral transforms, an analytical layer-element equation is established explicitly in the Laplace-Fourier transformed domain. A global matrix of layered soil can be obtained by assembling a set of analytical layer-elements, which is further solved in the transformed domain by considering boundary conditions. The numerical inversion of LaplaceFourier trans- form is employed to acquire the actual solution. Numerical analysis for 3-D consolidation with anisotropic permeability of a layered soil system is presented, and the influence of anisotropy of permeability on the consolidation behavior is discussed.

THE MULTIPLICATIVE COMPLEXITY AND ALGORITHM OF THE GENERALIZED DISCRETE FOURIER TRANSFORM(GFT)

In this paper, we have proved that the lower bound of the number of real multiplications for computing a length 2(t) real GFT(a,b) (a = +/-1/2, b = 0 or b = +/-1/2, a = 0) is 2(t+1) - 2t - 2 and that for computing a length 2t real GFT(a,b)(a = +/-1/2, b = +/-1/2) is 2(t+1) - 2. Practical algorithms which meet the lower bounds of multiplications are given.