Order of Magnitude of Multiple Fourier Coefficients

The order of magnitude of multiple Fourier coefficients of complex valued functions of generalized bounded variations like （∧^1,.. .,∧^N）BV^（p） and r-BV, over [0,2π]^ N, are estimated.

Precise integration method for a class of singular two-point boundary value problems

In this paper we present a precise integration method based on high order multiple perturbation method and reduction method for solving a class of singular twopoint boundary value problems.Firstly,by employing the method of variable coefficient dimensional expanding,the non-homogeneous ordinary differential equations（ODEs） are transformed into homogeneous ODEs.Then the interval is divided evenly,and the transfer matrix in every subinterval is worked out using the high order multiple perturbation method,and a set of algebraic equations is given in the form of matrix by the precise integration relation for each segment,which is worked out by the reduction method.Finally numerical examples are elaboratedd to validate the present method.

Coupling of high order multiplication perturbation method and reduction method for variable coefcient singular perturbation problems

Based on the precise integration method （PIM）, a coupling technique of the high order multiplication perturbation method （HOMPM） and the reduction method is proposed to solve variable coefficient singularly perturbed two-point boundary value prob lems （TPBVPs） with one boundary layer. First, the inhomogeneous ordinary differential equations （ODEs） are transformed into the homogeneous ODEs by variable coefficient dimensional expansion. Then, the whole interval is divided evenly, and the transfer ma trix in each sub-interval is worked out through the HOMPM. Finally, a group of algebraic equations are given based on the relationship between the neighboring sub-intervals, which are solved by the reduction method. Numerical results show that the present method is highly efficient.

High order multiplication perturbation method for singular perturbation problems

This paper presents a high order multiplication perturbation method for sin- gularly perturbed two-point boundary value problems with the boundary layer at one end. By the theory of singular perturbations, the singularly perturbed two-point boundary value problems are first transformed into the singularly perturbed initial value problems. With the variable coefficient dimensional expanding, the non-homogeneous ordinary dif- ferential equations （ODEs） are transformed into the homogeneous ODEs, which are then solved by the high order multiplication perturbation method. Some linear and nonlinear numerical examples show that the proposed method has high precision.

Loops in Digraphs of Lambert Mapping Modulo Prime Powers: Enumerations and Applications

For an odd prime number p, and positive integers k and , we denote , a digraph for which is the set of vertices and there is a directed edge from u to v if , where . In this work, we study isolated and non-isolated fixed points (or loops) in digraphs arising from Discrete Lambert Mapping. It is shown that if , then all fixed points in are isolated. It is proved that the digraph has isolated fixed points only if . It has been characterized that has no cycles except fixed points if and only if either g is of order 2 or g is divisible by p. As an application of these loops, the solvability of the exponential congruence has been discussed.

Integrated unified phase-field modeling(UPFM)

For a long time,the phase-field method has been considered a mesoscale phenomenological method that lacks physical accuracy and is unable to be closely linked to the mechanical or functional properties of materials.Some misunderstandings existing in these viewpoints need to be clarified.Therefore,it is necessary to propose or adopt the perspective of“unified phase-field modeling(UPFM)”to address these issues,which means that phase-field modeling has multiple unified characteristics.Specifically,the phase-field method is the perfect unity of thermodynamics and kinetics,the unity of multi-scale models from microto meso and then to macro,the unity of internal or/and external driving energy with order parameters as field variables,the unity of multiple physical fields,and thus the unity of material composition design,process optimization,microstructure control,and performance prediction.It is precisely because the phase-field approach has these unified characteristics that,after more than 40 years of development,it has been increasingly widely applied in materials science and engineering.

Regional differences of urbanization in China and its driving factors

After more than 30 years of rapid urbanization, the overall urbanization rate of China reached 56.1% in 2015.However, despite China's rapid increase in its overall rate of urbanization, clear regional differences can be observed. Furthermore, inadequate research has been devoted to in-depth exploration of the regional differences in China's urbanization from a national perspective, as well as the internal factors that drive these differences. Using prefecture-level administrative units in China as the main research subject, this study illustrates the regional differences in urbanization by categorizing the divisions into four types based on their urbanization ratio and speed(high level: low speed; high level: high speed; low level: high speed; and low level: low speed). Next, we selected seven economic and geographic indicators and applied an ordered logit model to explore the driving factors of the regional differences in urbanization. A multiple linear regression model was then adopted to analyze the different impacts of these driving factors on regions with different urbanization types. The results showed that the regional differences in urbanization were significantly correlated to per capita GDP, industry location quotients, urban-rural income ratio,and time distance to major centers. In addition, with each type of urbanization, these factors were found to have a different driving effect. Specifically, the driving effect of per capita GDP and industry location quotients presented a marginally decreasing trend, while main road density appeared to have a more significant impact on cities with lower urbanization rates.

MULTIPLICATIVE EXTRAPOLATION METHOD FOR CONSTRUCTING HIGHER ORDER SCHEMES FOR ORDINARY DIFFERENTIAL EQUATIONS

In this paper, we develop a new technique called multiplicative extrapolation method which is used to construct higher order schemes for ordinary differential equations. We call it a new method because we only see additive extrapolation method before. This new method has a great advantage over additive extrapolation method because it keeps group property. If this method is used to construct higher order schemes from lower symplectic schemes, the higher order ones are also symplectic. First we introduce the concept of adjoint methods and some of their properties. We show that there is a self-adjoint scheme corresponding to every method. With this self-adjoint scheme of lower order, we can construct higher order schemes by multiplicative extrapolation method, which can be used to construct even much higher order schemes. Obviously this constructing process can be continued to get methods of arbitrary even order.