Based on this, to avoid using the Green ′ s formula on infinite domain, an infinitely differentiable function with compact subset is constructed and some important results to a especial solution of the inhomogeneous field equation are obtained. 在此基础上,为了避免使用无限区域上的Green公式(不成立),通过构造紧支集上的无穷可微函数,获得满足非齐次控制微分方程的特解的重要结果。
In this paper, we prove a conclusion as following: The necessary and sufficient condition that Banach space is infinite dimension is that it exists nonempty convex subset B without inner point so that B is not on any closed hyperplane. 证明如下结果:设X是Banach空间,则X是无限维的充分必要条件是存在不含内点的非空凸集B,使得B不在任何一个闭超平面上。
Moreover the neural network also can limitlessly approach any continuous mapping, which has limit at infinite place, from limitless close subset of R m to R n. It extends the nonlinear approach ability of traditional feedforward neural network ( BP neural network). 而且,证实该种神经网络可无限逼近Rm上无界闭子集到Rn上的在无穷远有极限的任意连续映射,扩充了传统前向神经网络(BP网络)的非线性逼近能力。
Besides this, there is another difficulty in the infinite dimensional case that a bounded and closed subset in the infinite dimensional state space is not necessarily compact. 另外,无穷维的情形还有一个难点,就是状态空间的有界的闭子集不一定是紧的。
The so-called multifractal, which is defined on a fractal structure, is an infinite set composed by fractal measure of many scaling exponents. It depicts the local scale of fractal subset with different scales and scaling exponents. 所谓多重分形,是定义在分形结构上的由多个标度指数的分形测度组成的无限集合,它刻画了分布在子集上的具有不同标度和标度指数的分形子集的局部标度性。