Asymptotic expansions of the Hurwitz-Lerch zeta function

The Hurwitz-Lerch zeta function Φ(z,s,a) is considered for large and small values of a∈C, and for large values of z∈C, with |Arg(a)|<π, z∉[1,∞) and s∈C. This function is originally defined as a power series in z, convergent for |z|<1, s∈C and 1−a∉N. An integral representation is obtained for Φ(z,s,a) which define the analytical continuation of the Hurwitz-Lerch zeta function to the cut complex z-plane C?[1,∞). From this integral we derive three complete asymptotic expansions for either large or small a and large z. These expansions are accompanied by error bounds at any order of the approximation. Numerical experiments show that these bounds are very accurate for real values of the asymptotic variables.     

第二类变型Bessel函数的Chebyshev逼近

第二类变型Bessel函数Kn(z)在自变量趋于无穷时就是指数变小的，使用多项式逼近的方法求解往往误差很大．采用指数变换和J．P．Boyd的有理Chebyshev多项式计算第二类变型Bessel函数，得到了令人满意的在较大范围内有效的解．     

Hypothesis testing for the generalized multivariate modified Bessel model

In this paper, we consider hypothesis testing problems in which the involved samples are drawn from generalized multivariate modified Bessel populations. This is a much more general distribution that includes both the multivariate normal and multivariate-t distributions as special cases. We derive the distribution of the Hotelling's T²-statistic for both the one- and two-sample problems, as well as the distribution of the Scheffe's T²-statistic for the Behrens–Fisher problem. In all cases, the non-null distribution of the corresponding F-statistic follows a new distribution which we introduce as the non-central F-Bessel distribution. Some statistical properties of this distribution are studied. Furthermore, this distribution was utilized to perform some power calculations for tests of means for different models which are special cases of the generalized multivariate modified Bessel distribution, and the results compared with those obtained under the multivariate normal case. Under the null hypothesis, however, the non-central F-Bessel distribution reduces to the central F-distribution obtained under the classical normal model.     

应用无衍射贝塞尔光束的固体线膨胀系数测量系统

测量固体线膨胀系数是工程设计中的一项重要工作。这套非接触固体线膨胀系数测量系统利用激光三角法进行了设计。通过采用线阵CCD作为信号传感器来实现自动化测量。通过采用无衍射贝塞尔光束来提高测量精度和增大测量范围。     

Power expansions for the self-similar solutions of the modified Sawada-Kotera equation

The fourth-order ordinary differential equation that defines the self-similar solutions of the Kaup—Kupershmidt and Sawada—Kotera equations is studied. This equation belongs to the class of fourth-order analogues of the Painlevé equations. All the power and non-power asymptotic forms and expansions near points z = 0, z = ∞ and near an arbitrary point z = z ₀ are found by means of power geometry methods. The exponential additions to the solutions of the studied equation are also determined.      

A complete asymptotic expansion of power means

The complete asymptotic expansion of power means in terms of Bell polynomials is obtained. Some results recently obtained by M. Bjelica are generalized.     

Synthesis and reactivity of enantiomerically pure N-alkyl-2-alkenyl azetidinium salts

A synthesis of stereodefined enantiomerically pure 2-alkenyl azetidines is described using Wittig olefination as key step. The quaternary triflate ammonium salts of these heterocycles were prepared in a stereoselective way and treatment of these azetidinium salts with a base (KHMDS or PhLi) induced a regioselective Stevens rearrangement leading to a 3-alkenyl pyrrolidine. An unprecedented S_N2′ reaction involving phenyllithium as nucleophile and an ammonium as leaving group was observed in one case.     

Reconstructing small perturbations of an obstacle for acoustic waves from boundary measurements on the perturbed shape itself

We derive relationships between the shape deformation of an impenetrable obstacle and boundary measurements of scattering fields on the perturbed shape itself. Our derivation is rigorous by using a systematic way, based on layer potential techniques and the field expansion (FE) method (formal derivation). We extend these techniques to derive asymptotic expansions of the Dirichlet-to-Neumann (DNO) and Neumann-to-Dirichlet (NDO) operators in terms of the small perturbations of the obstacle as well as relationships between the shape deformation of an obstacle and boundary measurements of DNO or NDO on the perturbed shape itself. All relationships lead us to very effective algorithms for determining lower order Fourier coefficients of the shape perturbation of the obstacle.     

Metastable decay rates,asymptotic expansions,and analytic continuation of thermodynamic functions

The grand potentialP(z)/kT of the cluster model at fugacityz, neglecting interactions between clusters, is defined by a power series _n Q _n z ⁿ , whereQ _n , which depends on the temperatureT, is the partition function of a cluster of sizen. At low temperatures this series has a finite radius of convergencez _s . Some theorems are proved showing that ifQ _n , considered as a function ofn, is the Laplace transform of a function with suitable properties, thenP(z) can be analytically continued into the complexz plane cut along the real axis fromz _s to + and that (a) the imaginary part ofP(z) on the cut is (apart from a relatively unimportant prefactor) equal to the rate of nucleation of the corresponding metastable state, as given by Becker-Döring theory, and (b) the real part ofP(z) on the cut is approximately equal to the metastable grand potential as calculated by truncating the divergent power series at its smallest term.     

On the complete solution of ∈y″=y 3

In Ref. 1, the author claimed that the problem y=y ³ is soluble only for a certain range of the parameter . An analytic approach, as adopted in the following contribution, reveals that a unique solution exists for any positive value of . The solution is given in closed form by means of Jacobian elliptic functions, which can be numerically computed very efficiently. In the limit 0+, the solutions exhibit boundary-layer behavior at both endpoints. An easily interpretable approximate solution for small is obtained using a three-variable approach.