Modular Transformations of Ramanujan's Fifth and Seventh Order Mock Theta Functions

In his last letter to Hardy, Ramanujan defined 17 functions F(q), where |q| < 1. He called them mock theta functions, because as q radially approaches any point e ^2ir (r rational), there is a theta function F _r(q) with F(q) – F _r(q) = O(1). In this paper we obtain the transformations of Ramanujan's fifth and seventh order mock theta functions under the modular group generators + 1 and –1/, where q = e ⁱ. The transformation formulas are more complex than those of ordinary theta functions. A definition of the order of a mock theta function is also given.     

Legendre Transformation for Regularizable Lagrangians in Field Theory

Hamilton equations based not only upon the Poincaré–Cartan equivalent of a first-order Lagrangian, but also upon its Lepagean equivalent are investigated. Lagrangians which are singular within the Hamilton–De Donder theory, but regularizable in this generalized sense are studied. Legendre transformation for regularizable Lagrangians is proposed and Hamilton equations, equivalent with the Euler–Lagrange equations, are found. It is shown that all Lagrangians affine or quadratic in the first derivatives of the field variables are regularizable. The Dirac field and the electromagnetic field are discussed in detail.     

Some Geometry of the Cone of Nonnegative Definite Matrices and Weights of Associated X2 Distribution

Consider the test problem about matrix normal mean M with the null hypothesis M = O against the alternative that M is nonnegative definite. In our previous paper (Kuriki (1993, Ann. Statist., 21, 1379–1384)), the null distribution of the likelihood ratio statistic has been given in the form of a finite mixture of ² distributions referred to as X² distribution. In this paper, we investigate differential-geometric structure such as second fundamental form and volume element of the boundary of the cone formed by real nonnegative definite matrices, and give a geometric derivation of this null distribution by virtue of the general theory on the X² distribution for piecewise smooth convex cone alternatives developed by Takemura and Kuriki (1997, Ann. Statist., 25, 2368–2387).     

应用具有手征流关联函数的光锥求和规则计算B(Bc)→Dlν过程的形状因子

在QCD光锥求和规则(LCSR)框架内应用具有手征流关联函数计算B(B_c)→Dlν衰变过程的弱形状因子. 所获得的形状因子的表达式仅依赖于D介子的主导级分布振幅(DA). 应用了三类D介子的分布振幅计算了形状因子F_B→D(0)和F_Bc→D(0). 在速度迁移1.142=0附近算符乘积展开(OPE)得以有效的情况下所计算的形状因子行为在误差范围内与B(B_c)→Dlν过程实验数据相一致. 在大反冲区域1.35B→D(0)是与微扰QCD(pQCD)结果相一致的. 所以本文的计算在联接格点QCD, 重夸克对称性和pQCD之间起桥梁作用，有助于进一步对B(B_c)→Dlν跃迁过程的理解. 计算 使用了在端点具有指数压低的分布振幅行为, 对F_Bc→D(0)的预言与其他方法获得的结果是可比的, 有利于具有库仑修正的三点求和规则(3PSR)方法所得的结果.     

The second Born approximation in electron-helium scattering in a Nd-YAG laser field

The dynamics of laser-assisted elastic collisions in helium is studied using the second-order Born approximation. Detailed calculations of the scattering amplitudes are performed by using the Sturmian basis expansion. Differential cross sections for elastic scattering with the net absorption/emission of up to two photons are calculated for collision energies of 5 eV, 10 eV, and 20 eV. We discuss the influence of the low-energy electrons on the differential cross section (DCS) as a function of the scattering angle for selected choices of the laser frequency and the number of photons exchanged between the external field and electron-helium system.     

数字书法生成方法

将传统书法艺术用数字技术在电子屏幕上呈现,对于书法艺术的传播有重要意义。提出了一种数字书法生成技术,首先,通过在屏幕上采集毛笔书写的轨迹信息,根据不同的速度,分别使用卡尔曼滤波和二次样条曲线插值实现书写轨迹平滑;然后,使用基于真实书写的笔触形态还原方法,呈现具有较高还原度的书法笔迹;最后,为了更好地实现书法中飞白的效果,采用基于规则的方法对书法笔迹进行修饰。     

On Form Invariance, Lie Symmetry and Three Kinds of Conserved Quantities of Generalized Lagrange's Equations

In this paper, the form invariance and the Lie symmetry of Lagrange's equations for nonconservative system in generalized classical mechanics under the infinitesimal transformations of group are studied, and the Noether's conserved quantity, the new form conserved quantity, and the Hojman's conserved quantity of system are derived from them. Finally, an example is given to illustrate the application of the result.     

Supersymmetric invariance and universal central extensions of lie supertriple systems

In this article, we discuss some properties of a supersymmetric invariant bilinear form on Lie supertriple systems. In particular, a supersymmetric invariant bilinear form on Lie supertriple systems can be extended to its standard imbedding Lie superalgebras. Furthermore, we generalize Garland's theory of universal central extensions for Lie supertriple systems following the classical one for Lie superalgebras. We solve the problems of lifting automorphisms and lifting derivations.     

和式∑k=0^nμ^kf（k）的发生函数算法

利用普通幂级数发生函数方法，通过对发生函数进行xD算子，得到和式∑k=0^nμ^kf（k）的计算公式，并计算该类和式＋