On the divergence of the means of double Walsh-Fourier series

In 1992, Móricz, Schipp and Wade proved the a.e. convergence of the double means of the Walsh-Fourier series () for functions in ( is the unit square). This paper aims to demonstrate the sharpness of this result. Namely, we prove that for all measurable function we have a function such as and does not converge to a.e. (in the Pringsheim sense).

     

Mean convergence of orthogonal Fourier series of modified functions

We construct orthonormal systems (ONS) which are uniformly bounded, complete, and made up of continuous functions such that some continuous and even some arbitrarily smooth functions cannot be modified so that the Fourier series of the new function converges in the -metric for any 2. $"> We prove also that if is a uniformly bounded ONS which is complete in all the spaces , then there exists a rearrangement of the natural numbers such that the system has the strong -property for all 2$">; that is, for every and for every and 0 $">there exists a function which coincides with except on a set of measure less than and whose Fourier series with respect to the system converges in      

High spatial intensity 10 W-CuBr laser with hydrogen additives

High beam quality output has been demonstrated from a single medium-scale (2 cm dia. × 50 cm) CuBr laser with hydrogen additives. The addition of hydrogen leads to beam divergence reduction and laser power lift-up finally yielding 5-fold increase in laser power spatial intensity compared with the case of no hydrogen added. With a simple positive-branch unstable resonator practical (pulse-average) divergence of 80 rad is achieved. The easy-feasible sealed-off version of CuBr laser makes it an attractive source for many high-quality laser beam applications.     

刻螺纹对Nd：YAG晶体棒输出功率影响的实验研究

针对Nd：YAG棒刻螺纹是否能改善激光器的输出，理论上分析了Nd：YAG晶体棒内热传导过程及影响其冷却效果的因素。给出了表面刻上螺纹后，Nd：YAG晶体棒表面积的变化结果和其对表面冷却换热方式的改变。对使用螺纹棒的激光器输出进行了实验研究，与普通棒的实验结果对比发现：用螺纹棒比用普通棒时平均功率大了23．2％，发散角小了3mrad。说明激光棒的表面刻槽可以提高冷却效果，有效地提高了激光器的输出。     

Fractional integration associated with second order divergence operators on R<Superscript>n</Superscript>

The classical Hardy-Littlewood-Sobolev theorems for Riesz potentials (−Δ)^−α/2 are extended to the generalised fractional integrals L ^–α/2 for 0 < α < n, where L=−div A∇ is a uniformly complex elliptic operator with bounded measurable coefficients in ℝⁿ.     

Flow-invariant closed sets with respect to nonlinear semigroup flows

A general result on the flow-invariance of a closed subset with respect to a differential equation associated with a nonlinear semigroup generator on Banach spaces is given. Applications to the flow-invariance of controlled flux sets (including the Enstrophy and Helicity sets) with respect to semilinear parabolic equations and Navier-Stokes equations are given.     

Statistical Asymptotic Theory of Active Learning

We study a parametric estimation problem. Our aim is to estimate or to identify the conditional probability which is called the system. We suppose that we can select appropriate inputs to the system when we gather the training data. This kind of estimation is called active learning in the context of the artificial neural networks. In this paper we suggest new active learning algorithms and evaluate the risk of the algorithms by using statistical asymptotic theory. The algorithms are regarded as a version of the experimental design with two-stage sampling. We verify the efficiency of the active learning by simple computer simulations.     

On Pointwise Divergence of Trigonometric Series of General Type

Mathematical Notes -     

任意规范下有限温度二圈QED热力学势及其交叉发散

采用有限温度下维数正规化的方法在任意协变规范下精确地计算了双圈ＱＥＤ的热力学势及其交叉发散．计算表明交叉发散彼此相消且所得的结果与规范的选取无关．