This post summarises different ways of constructing continuous functions, which are introduced in Section 18 of James Munkres "Topology". Constant function. Inclusion function. N.B. The function domain should have the subspace topology relative…

In general differential calculus, we have learned the definitions of function continuity, such as functions of class \(C^0\) and \(C^2\). For most cases, we only take them for granted as for example, we have memorized the formulations of Green identi…

One of the most striking facts about neural networks is that they can compute any function at all. That is, suppose someone hands you some complicated, wiggly function, f(x)f(x): No matter what the function, there is guaranteed to be a neural network…

1346. Intervals of Monotonicity Time limit: 1.0 secondMemory limit: 64 MB It’s well known that a domain of any continuous function may be divided into intervals where the function would increase monotonically or decrease monotonically. A number of in…

colah's blog Blog About Contact Neural Networks, Manifolds, and Topology Posted on April 6, 2014 topology, neural networks, deep learning, manifold hypothesis Recently, there’s been a great deal of excitement and interest in deep neural networks beca…

1.Introduction 2.First-order Differential Equations Exercise2.1. Find solutons of the following intial-value problems in $\bbR^2$: (1)$2u_y-u_x+xu=0$ with $u(x,0)=2xe^{x^2/2}$; (2)$u_y+(1+x^2)u_x-u=0$ with $u(x,0)=\arctan x$. Solution: (1)Since $(-1,…

Understanding Convolution in Deep Learning Convolution is probably the most important concept in deep learning right now. It was convolution and convolutional nets that catapulted deep learning to the forefront of almost any machine learning task the…

Search是数据结构中最基础的应用之一了,在python中,search有一个非常简单的方法如下: 15 in [3,5,4,1,76] False 不过这只是search的一种形式,下面列出多种形式的search用做记录: 一.顺序搜索 顺着list中的元素一个个找,找到了返回True,没找到返回False def sequential_search(a_list, item): pos = 0 found = False while pos < len(a_list) and not fo…

许久以前,我读到了侯捷先生于<深入浅出MFC>一书中所写的“勿在浮砂筑高台”这句话,颇受警醒与启发.如今在工科领域已摸索多年,亦逐渐真切而深刻地认识到,若没有坚实.完整.细致的数学理论作为基石,任何技术工作与所谓的“唯象”理论研究也都无异于浮沙筑台,经不起举一反三的推敲与寻根究底的质疑.而这也像是不系安全带的无照司机驾车在高速公路上疾驰,终究逃不掉翻到沟里的命运.拓扑学作为数学理论大厦的一块基石,既起到了稳固上层建筑的作用,又提供了不同于我们通常的眼光和感性直观的视角与思路,值得细心学习和反复…

Example 1 Let \(X\) be the subspace \([0,1]\cup[2,3]\) of \(\mathbb{R}\), and let \(Y\) be the subspace \([0,2]\) of \(\mathbb{R}\). The map \(p: X \rightarrow Y\) defined by \[ p(x)=\begin{cases} x & \text{for}\; x \in [0,1],\\ x-1 & \text{for}\;…