19世纪上半叶的无穷级数敛散性判别法

对19世纪上半叶欧洲数学家在正项级数敛散性判别方面的工作作了考察和分析.     

射频板条CO2激光器输出光束的光学变换

射频激励板条CO2激光器输出激光近场光斑近似为一条线,远场光斑为O形图样。在垂直于光束传输方向的平面内,光束在一个方向发散很大,无法用于激光加工。根据激光器不同的激光功率级别及光束尺寸,分别采用了圆柱面镜、滤波光阑、扩束望远镜等不同组合的方法,将200 W激光器输出光束变换成远场光束为Φ6、近似TEM00模的圆形光束,500 W激光器远场光束为Φ10的低阶模的圆形光束。成功用于激光切割。     

The directionality of partially coherent beams expressed in terms of the far-field divergence angle and of the far-field radiant intensity distribution

Taking the partially coherent cosh-Gaussian beam (ChG) as an illustrative example, the far-field divergence angle and directionality of partially coherent beams are studied. There are three competing physical mechanisms, i.e., the spatial coherence, diffraction and decentration, which affect the far-field divergence angle of partially coherent ChG beams. Two partially coherent ChG beams may generate the same far-field divergence angle, and partially coherent ChG beams may also have the same far-field divergence angle as a fully coherent ChG beam or as a fully coherent Gaussian laser beam if the three physical mechanisms are appropriately balanced. The consistency of the directionality of partially coherent beams expressed in terms of the far-field divergence angle and in terms of the far-field radiant intensity distribution is examined. Generally, two partially coherent beams with the same far-field divergence angle have not certainly the same far-field radiant intensity distribution. However, under certain conditions, it is possible to achieve the consistency of the directionality expressed in terms of the far-field divergence angle and of the normalized far-field radiant intensity distribution.     

快电子发散角的实验测量

采用飞秒激光辐照铜靶,利用电子角分布仪和LiF热释光探测器测量了快电子发射的发散角.实验结果显示,快电子的发散角与激光入射角密切相关,随着激光入射角增加,快电子的发散角逐渐减小.在相同入射角条件下,加上预脉冲将导致快电子的发散角变小.这个结果为获取较小发散角的快电子束提供了实验参考.     

Computing f-divergences and distances of high-dimensional probability density functions

Very often, in the course of uncertainty quantification tasks or data analysis, one has to deal with high-dimensional random variables. Here the interest is mainly to compute characterizations like the entropy, the Kullback–Leibler divergence, more general $f$-divergences, or other such characteristics based on the probability density. The density is often not available directly, and it is a computational challenge to just represent it in a numerically feasible fashion in case the dimension is even moderately large. It is an even stronger numerical challenge to then actually compute said characteristics in the high-dimensional case. In this regard it is proposed to approximate the discretized density in a compressed form, in particular by a low-rank tensor. This can alternatively be obtained from the corresponding probability characteristic function, or more general representations of the underlying random variable. The mentioned characterizations need point-wise functions like the logarithm. This normally rather trivial task becomes computationally difficult when the density is approximated in a compressed resp. low-rank tensor format, as the point values are not directly accessible. The computations become possible by considering the compressed data as an element of an associative, commutative algebra with an inner product, and using matrix algorithms to accomplish the mentioned tasks. The representation as a low-rank element of a high order tensor space allows to reduce the computational complexity and storage cost from exponential in the dimension to almost linear.