Computing the Approximation Error for Neural Networks with Weights Varying on Fixed Directions

We obtain a sharp lower bound estimate for the approximation error of a continuous function by single hidden layer neural networks with a continuous activation function and weights varying on two fixed directions. We show that for a certain class of activation functions this lower bound estimate turns into equality. The obtained result provides us with a method for direct computation of the approximation error. As an application, we give a formula, which can be used to compute instantly the approximation error for a class of functions having second order partial derivatives.     

Derivation degree sequences of non-free arrangements

In this note, we study the logarithmic derivation module of a non-free arrangement. We prove a generalized addition theorem for all arrangements. This addition theorem allows us to find various relationships between non-free arrangements, free arrangements, and restriction counts. For graphic arrangements, we can use these results to find a lower bound for the maximal degree generator in terms of triangles in the associated graph. We also apply these results to the case of hypersolvable arrangements where we define hyperexponents and use them to find a lower bound for their maximal degree generator.     

T∞-测度分解定理的进一步讨论

在对T∞-测度做进一步研究的基础上,得到了(有限或无限)T∞-测度的Hahn分解定理和Jordan分解定理。同时,用一种新方法证明了有限T∞-测度的Lebesgue分解定理。此外,还得到了一些类似于经典测度的结论。     

Which sets are images of disk-algebra functions?

We give a characterization of those compact sets in the plane with finitely many holes that are images of disk-algebra functions. We also show that the image of the closed unit disk via a polynomial is, in general, not polynomially convex.     

正规六边形和更广泛具有正规结构的Banach 空间

假设Ｘ是一个实Ｂａｎａｃｈ空间，Ｓ（Ｘ）是单位球面。作者引进了一个新的几何参数（Ｈ（Ｘ），讨论了Ｈ（Ｘ）和Ｎｅｕｍａｎｎ－Ｊｏｒｄａｎ常数ＣＮＪ（Ｘ）Ｒ 性质以及Ｈ（Ｘ）和其他几何常数的关系，本文主要结果是：Ｈ（Ｘ）〈２或ＣＮＪ（Ｘ）〈４／５一致正规结构。