ON THE COEFFICIENTS OF DIFFERENTIATED EXPANSIONS AND DERIVATIVES OF CHEBYSHEV POLYNOMIALS OF THE THIRD AND FOURTH KINDS

Two new analytical formulae expressing explicitly the derivatives of Chebyshev polynomials of the third and fourth kinds of any degree and of any order in terms of Chebyshev polynomials of the third and fourth kinds themselves are proved. Two other explicit formulae which express the third and fourth kinds Chebyshev expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of their original expansion coefficients are also given. Two new reduction formulae for summing some terminating hypergeometric functions of unit argument are deduced. As an application of how to use Chebyshev polynomials of the third and fourth kinds for solving high-order boundary value problems, two spectral Galerkin numerical solutions of a special linear twelfth-order boundary value problem are given.

THE SHARP ESTIMATE OF THE THIRD HOMOGENEOUS EXPANSION FOR A CLASS OF STARLIKE MAPPINGS OF ORDER α ON THE UNIT POLYDISK IN C^n

In this article, first, a sufficient condition for a starlike mapping of order a f（x） defined on the unit ball in a complex Banach space is given. Second, the sharp estimate of the third homogeneous expansion for f is established as well, where f（z） = （f1（z）, f2（z）,..., fn（z））＇ is a starlike mapping of order a or a normalized biholomorphic starlike mapping defined on the unit polydisk in Cn, and D2fk（0）（z2） /2i= zk（∑l=1^b akzzl）, k = 2t l=1 k = 1, 2,..., n. Our result states that the Bieberbaeh conjecture in several complex variables （the case of the third homogeneous expansion for starlike mappings of order α and biholomorphic starlike mappings） is partly proved.

AN INEQUALITY OF HOMOGENEOUS EXPANSION FOR BIHOLOMORPHIC QUASI-CONVEX MAPPINGS ON THE UNIT POLYDISK AND ITS APPLICATION

In this article,the authors obtain an inequality of homogeneous expansion for f,where f is a quasi-convex mapping(including quasi-convex mapping of type A and quasi-convex mapping of type B)defined on the open unit polydisk in C n .Meanwhile,the authors also investigate its application.

SHARP ESTIMATES OF ALL HOMOGENEOUS EXPANSIONS FOR A SUBCLASS OF QUASI-CONVEX MAPPINGS OF TYPE B AND ORDER α IN SEVERAL COMPLEX VARIABLES

In this article, first, the sharp estimates of all homogeneous expansions for a subclass of quasi-convex mappings of type B and order B on the unit ball in complex Ba- nach spaces are given. Second, the sharp estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in （in are also established. In particular, the sharp estimates of all homogeneous expansions for a subclass of quasi-convex mappings （include quasi-convex mappings of type A and quasi-convex mappings of type B） in several complex variables are get accordingly. Our results state that a weak version of the Bieber- bach conjecture for quasi-convex mappings of type B and order a in several complex variables is proved, and the derived conclusions are the generalization of the classical results in one complex variable.

The (G'/G, 1/G)-expansion method and its application to travelling wave solutions of the Zakharov equations

The （G＇/G, 1/G）-expansion method for finding exact travelling wave solutions of nonlinear evolution equations, which can be thought of as an extension of the （G＇/G）-expansion method proposed recently, is presented. By using this method abundant travelling wave so- lutions with arbitrary parameters of the Zakharov equations are successfully obtained. When the parameters are replaced by special values, the well-known solitary wave solutions of the equations are rediscovered from the travelling waves.

THE INVARIANCE OF STRONG AND ALMOST SPIRALLIKE MAPPINGS OF TYPE β AND ORDER α

The invariance of strong and almost spirallike mappings of type β and order α is discussed in this paper. From the maximum modulus principle of holomorphic functions, we obtain that the generalized Roper-Suffridge operators preserve strong and almost spirallike-hess of type β and order α on the unit ball B^n in C^n and on bounded and complete Reinhardt domains. Therefore we obtain that the generalized Roper-Suffridge operators preserve strong spirllikeness of type β, strong and almost starlikeness of order α, strong starlikeness on the corresponding domains.Thus we can construct more subclasses of spirallike mappings in several complex variables.

The foundation of the theory of the universe dark energy and its nature

Surprisingly recent astronomical observations have provided strong evidence that our universe is not only expanding, but also is expanding at an accelerating rate. This paper pre- sents a basis of the theory of universe space- time dark energy, a solution of Einstein’s cosmological constant problem, physical interpretation of universe dark energy and Einstein’s cosmological constant Lambda and its value ( = 0.29447 × 10-52 m-2), values of universe dark energy density 1.2622 × 10-26 kg/m3 = 6.8023 GeV, universe critical density 1.8069 × 10-26 kg/m3 = 9.7378 GeV, universe matter density 0.54207 × 10-26 kg/m3 = 2.9213 GeV, and universe radiation density 2.7103 × 10-31 kg/m3 = 1.455 MeV. The interpretation in this paper is based on geometric modeling of space-time as a perfect four- dimensional continuum cosmic fluid and the momentum generated by the time. In this modeling time is considered as a mechanical variable along with other variables and treated on an equal footing. In such a modeling, time is considered to have a mechanical nature so that the momentum associated with it is equal to the negative of the universe total energy. Since the momentum associated with the time as a mechanical variable is equal to the negative system total energy, the coupling in the time and its momentum leads to maximum increase in the space-time field with 70.7% of the total energy. Moreover, a null paraboloid is obtained and interpreted as a function of the momentum generated by time. This paper presents also an interpretation of space-time tri-dipoles, gravity field waves, and gravity carriers (the gravitons). This model suggests that the space-time has a polarity and is composed of dipoles which are responsible for forming the orbits and storing the space-time energy-momentum. The tri-di- poles can be unified into a solo space-time dipole with an angle of 45 degrees. Such a result shows that the space-time is not void, on the contrary, it is full of conserved and dynamic energy-momentum structure. Furthermore, the gravity field waves is modeled and assumed to be carried by the gravitons which move in the speed of light. The equivalent mass of the graviton (rest mass) is found to be equal to 0.707 of the equivalent mass of the light photons. Such a result indicates that the lightest particle (up to the author’s knowledge) in the nature is the graviton and has an equivalent mass equals to 2.5119 x 10-52 kg. Based on the fluidic nature of dark energy, a fourth law of thermodynamics is proposed and a new physical interpretation of Kepler’s Laws are presented. Additionally, based on the fact that what we are observing is just the history of our universe, on the Big Bang Theory, Einstein’s General Relativity, Hubble Parameter, cosmic inflation theory and on NASA’s observation of supernova 1a, then a second-order (parabolic) parametric model is obtained in this proposed paper to describe the accelerated ex- pansion of the universe. This model shows that the universe is approaching the universe cosmic horizon line and will pass through a critical point that will influence significantly its fate. Considering the breaking symmetry model and the variational principle of mechanics, then the universe will witness an infinitesimally stationary state and a symmetry breaking. As result of that, our universe will experience in the near future, a very massive impulse force in the order 1083 N. Subsequently, the universe will collapse. Finally, simulation results are demonstrated to verify the analytical results.

The Basic (<i>G'/G</i>)-Expansion Method for the Fourth Order Boussinesq Equation

The (G'/G)-expansion method is simple and powerful mathematical tool for constructing traveling wave solutions of nonlinear evolution equations which arise in engineering sciences, mathematical physics and real time application fields. In this article, we have obtained exact traveling wave solutions of the nonlinear partial differential equation, namely, the fourth order Boussinesq equation involving parameters via the (G'/G)-expansion method. In this method, the general solution of the second order linear ordinary differential equation with constant coefficients is implemented. Further, the solitons and periodic solutions are described through three different families. In addition, some of obtained solutions are described in the figures with the aid of commercial software Maple.

On the Second-order Item Coefficients for ε-starlike Mappings in C^n

Let B n be the unit ball in C n, we study ε-starlike mappings on B n. The upper bounds of second order item coefficients of homogeneous expansion for ε-starlike mappings are obtained.

A novel implementation algorithm of asymptotic homogenization for predicting the effective coefficient of thermal expansion of periodic composite materials

Asymptotic homogenization (AH) is a general method for predicting the effective coefficient of thermal expansion (CTE) of periodic composites. It has a rigorous mathematical foundation and can give an accurate solution if the macrostructure is large enough to comprise an infinite number of unit cells. In this paper, a novel implementation algorithm of asymptotic homogenization (NIAH) is developed to calculate the effective CTE of periodic composite materials. Compared with the previous implementation of AH, there are two obvious advantages. One is its implementation as simple as representative volume element (RVE). The new algorithm can be executed easily using commercial finite element analysis (FEA) software as a black box. The detailed process of the new implementation of AH has been provided. The other is that NIAH can simultaneously use more than one element type to discretize a unit cell, which can save much computational cost in predicting the CTE of a complex structure. Several examples are carried out to demonstrate the effectiveness of the new implementation. This work is expected to greatly promote the widespread use of AH in predicting the CTE of periodic composite materials.

Bohr's Inequality on the Unit Ball B^n

Bohr's type inequalities are studied in this paper: if f is a holomorphic mapping from the unit ball B^n to B^n, f(0)=p, then we have sum from k=0 to∞|Dφ_P(P)[D^kf(0)(z^k)]|／k!||Dφ_P(P)||＜1 for|z|＜max{1／2+|P|,(1-|p|)／2^(1／2)andφ_P∈Aut(B^n) such thatφ_(p)=0. As corollaries of the above estimate, we obtain some sharp Bohr's type modulus inequalities. In particular, when n=1 and |P|→1, then our theorem reduces to a classical result of Bohr.

Lack of mitochondrial genetic structure in the endangered giant clam populations of Tridacna maxima(Bivalvia: Cardiidae:Tridacninae) across the Saudi Arabian coast

The present investigation focuses on population genetic structure analysis of the endangered giant clam species Tridacna maxima across part of the Red Sea,with the main aim of assessing the influence of postulated potential barriers to gene flow(i.e.,particular oceanographic features and marked environmental heterogeneity)on genetic connectivity among populations of this poorly dispersive bivalve species.For this purpose,a total of 44 specimens of T.maxima were collected from five sampling locations along the Saudi Arabian coast and examined for genetic variability at the considerably variable mitochondrial gene cytochrome c oxidase I(COI).Our results revealed lack of population subdivision and phylogeographic structure across the surveyed geographic spectrum,suggesting that neither the short pelagic larval dispersal nor the various postulated barriers to gene flow in the Red Sea can trigger the onset of marked genetic differentiation in T.maxima.Furthermore,the discerned shallow COI haplotype genealogy(exhibiting high haplotype diversity and low nucleotide diversity),associated with recent demographic and spatial expansion events,can be considered as residual effect of a recent evolutionary history of the species in the Red Sea.

基于均匀化理论与上限分析的膨胀土滑坡稳定性分析

在膨胀土和滑坡共同作用下,隧道洞口段施工更容易引发地表开裂甚至滑坡等工程灾害,在隧道内采用微型桩群防治滑坡比抗滑桩具有优势。本文基于均匀化理论与上限分析对某高速公路隧道洞口段膨胀土滑坡的稳定性进行计算,并评价微型桩群和削方卸载不同组合方式的处置效果,计算时将微型桩群及桩周土通过均匀化理论等效为符合摩尔库仑强度准则的等效加固体来,以此提高计算效率,最后通过对现场削方+微型桩群加固处置后的滑坡变形监测来验证计算的合理性,得出如下结论:相较于土的强度参数,等效加固体内摩擦角保持不变,黏聚力从26kPa提高到85.36kPa。处置前后的滑坡稳定性评价结果表明,不做处理时,滑坡滑动面从滑坡上方岩土交界面延续到隧道洞口前;仅采用微型桩群加固时,滑坡安全系数在1.17左右,滑动面从岩土交界面延续到隧道洞口后;同时采用微型桩群加固和削方卸载时,滑坡安全系数提高到1.26~1.28,滑动面上缘由土石交界面前移。现场变形监测表明地表变形与深层土体变形均不超过3mm,该措施能保障滑坡的稳定性,同时也验证了计算方法的合理性,可为同类工程提供参考。

基于径向积分边界元法的周期性复合材料面内等效热弹性参数分析方法

具有人工设计结构的多材料夹杂的复合材料不仅具有可编辑的弹性模量和泊松比,还可以实现热膨胀系数的调控.这些复合材料的等效热弹性参数与其微观结构和材料分布密切相关.文章提出了一种基于径向积分边界元法的周期性复合材料的均匀化热弹性参数的计算方法.该方法基于代表性体积元均匀化方法建立材料的微观结构应力、应变与宏观等效热弹性参数之间的关联.同时,采用径向积分边界元法计算周期变形条件下代表性体积元的应力和应变场.该方法无需内部网格和域积分,仅依赖于边界的位移和面力等信息即可获得材料的等效热弹性参数,具有容易实现参数化建模、容易考虑细小结构特征等优势.另外,在角点处采用非连续元处理,大大简化了周期性边界条件的施加.针对复合材料梁的弯曲问题,通过与直接有限元模拟的结果进行对比,验证了该方法的有效性.然后,还采用该方法对网状复合超材料的热膨胀系数进行了分析.结果表明,通过调整结构的尺寸,可实现结构在热载荷下从正膨胀到负膨胀的转变.

Ginzburg-Landau方程的一种解法

求解Ginzburg-Landau方程在光纤通信中具有重要意义,为此提出一种求解该方程的新方法,即根据齐次平衡原则,利用F-展开法的思想求出其行波解。利用了Riccati方程已知的三角函数和双曲函数表示的解,得到了Ginzburg-Landau方程的多个包络波形式的精确解,并发现了频散系数和Landau系数之间的重要关系。

用(G′/G)-展开法求解Ginzburg-Landau方程

利用最近提出的(G′/G)-展开法,获得了Ginzburg-Landau方程更多的显式行波解,分别以含两个任意参数的双曲函数、三角函数及有理函数表示,当参数取特殊值时,可得到以往文献中相关结果。

一类随机偏微分方程组的均匀化问题

研究一类随机偏微分方程组的均匀化问题,■_(t)y_(ε)(t,x)=Δy_(ε)(t,x)/2+Vε(x)Ayε(t,x),其中(t,x)∈(0,T)×R^(d),d>2.这里的y_(ε)(t,x)=(u_(ε)(t,x),w_(ε)(t,x))^(T),V_(ε)(x)是含参数的高斯过程,当其相关函数满足特定的可积条件时,可以使用Duhamel(迭代)展开和Wick定理,给出以上方程的均匀化收敛性.

微裂均质化处治技术在中洪路改扩建项目中的应用

水泥混凝土路面加铺方案影响改扩建项目的交通组织方案和工程造价。为研究提升路面改建方案的合理性,以中洪路改扩建项目水泥混凝土路面加铺沥青面层设计方案比选为例,通过对原水泥混凝土路面进行病害分析,有效解决反射裂缝等路面病害。从环保、技术、经济、交通疏解等方面比选设计方案,选取微裂均质化处治方案,并确定了微裂均质化处治的施工工艺、相关参数和注意事项,以期为路面改建方案设计和加铺施工提供经验案例。

SHARP ESTIMATES OF QUASI-CONVEX MAPPINGS OF TYPE B AND ORDER α

In this paper,we first establish several sharp inequalities of homogeneous expansion for biholomorphic quasi-convex mappings of type B and order a on the unit ball E in a complex Banach space X by applying the method and technique of complex analysis.Then,as the application of these sharp inequalities,we derive the sharp estimate of third homogeneous expansions for the above mappings defined on the unit polydisk U^n in C^n.

The Sharp Estimates of all Homogeneous Expansions for a Class of Quasi-convex Mappings on the Unit Polydisk in C^n

In this paper,the sharp estimates of all homogeneous expansions for f are established,where f(z) = (f1(z),f2(z),··· ,fn(z)) is a k-fold symmetric quasi-convex mapping defined on the unit polydisk in Cn and Dtk+1fp(0)(ztk+1) (tk + 1)! = n l1,l2,···,ltk+1=1 |apl1l2···ltk+1|ei θpl1+θpl2+···+θpltk+1t k+1 zl1zl2 ··· zltk+1,p = 1,2,··· ,n.Here i = √?1,θplq ∈ (-π,π] (q = 1,2,··· ,tk + 1),l1,l2,··· ,ltk+1 = 1,2,··· ,n,t = 1,2,···.Moreover,as corollaries,the sharp upper bounds of growth theorem and distortion theorem for a k-fold symmetric quasi-convex mapping are established as well.These results show that in the case of quasi-convex mappings,Bieberbach conjecture in several complex variables is partly proved,and many known results are generalized.