KuFun人工智能的专栏

声明：部分内容来自于慕课，公开课等的课件，仅供学习使用。如有问题，请联系删除。部分内容来自电子科技大学，北京大学，清华大学，北航，复旦大学等的教材和课件课程视频： B站视频 https://www.bilibili.com/video/BV1V44y1j7vX概述1. 概述   课是围绕微积分的中心而设计的，函数，极限，导数和积分，既有连续的又有离散的。还会学到它们在工程学，物理，生物和社会科学方面是如何发挥作用的2. 章节 2.1 第一章：函数    在第一章内 我们会学习

声明：部分内容来自于慕课，公开课等的PPT课件，仅供学习使用。如有问题，请联系删除。部分内容来自电子科技大学，北京大学，清华大学，北航等的课件数列极限概念引入：求圆的面积 A1,A2,A3,...,An,...⇒A_1,A_2,A_3,...,A_n,...\RightarrowA1​,A2​,A3​,...,An​,...⇒无穷次逐步逼近过程1.数列定义   按自然数编号依...

导数   导数是有一些物理的瞬时速度等的应用中抽象出来的，和平均速度对应。   平均速度=ΔsΔt=s(t0+Δt)−s(t0)Δt=v‾\large\frac{\Delta s}{\Delta t}=\frac{s(t_0+\Delta t)-s(t_0)}{\Delta t} =\overline{v}ΔtΔs​=Δts(t0​+Δt)−s(t0​)​=v   瞬时速度=ΔsΔt...

最优化思路第二步核心steepest descent method: dk=−▽f(xk1.i.e.,dk=−g(xk))d_k=-\bigtriangledown f(x_k1.i.e.,d_k=-g(x_k))dk​=−▽f(xk​1.i.e.,dk​=−g(xk​))Newton method :dk=−H−1(xk)g(xk)d_k= -H^{-1}(x_k)g(x_k)dk​=−H...

最优化思路第三步Steps: k=0, 1, 2,…F(α)=f(xk+α.dk)F(\alpha) =f(x_k+\alpha.d_k)F(α)=f(xk​+α.dk​)xk,dkx_k, d_kxk​,dk​ is fixed∴\therefore∴其实就是一维问题，单变量的最优化问题。我们要做的就是，告诉计算机两件事情：xkx_kxk​点的确定，方向在哪里什么地方停下来...

最优化场景有些场景，变量很多，很大，微积分难于处理。此类问题可用最优化来解决。P1(x)=0P_1(x)=0P1​(x)=0P2(x)=0P_2(x)=0P2​(x)=0...Pm(x)=0P_m(x)=0Pm​(x)=0x=[x1x2...xn]x=\begin{bmatrix}x_1\\ x_2\\ .\\ .\\ .\\ x_n\\ \end{bmatrix}x...

标准差线 SD Line标准差线通过平均点,它的斜率是当x的值增加一个x标准差,y的值增加一个y标准差The SD line is a line that goes through the point of averages.And it’s slope is that it goes up by one SD of y every time . it goes over by one SD ...

概率论概率论是一种处理不确定性的框架(Probability as a mathematical framework for reasoning about uncertainly )Probabilistic models— sample space:1.) “List” (set) of possible outcomes2.) "List"must be :Mutually ...

泰勒级数Center of mass of N+1 blocksX-coordinatePower Series|x|<1 (geometric series)(converge)Resoning incomplete because it requires S existsGeneral Power Series|x|<R(radius of convergence...

反常积分dealing with infinity∞∞→(a=±∞L=±∞)\LARGE \frac{\infty}{\infty}\rightarrow\binom{a=\pm \infty}{L=\pm \infty}∞∞​→(L=±∞a=±∞​) are OK!IF {f(x)→∞g(x)→∞f′(x)/g′(x)→L\left\{\begin{matrix} f(x)\rightar...

参数方程弧长和表面积​​​​斜边长度平方 (hypotenuse)(ΔS)2=(Δx)2+(Δy)2(\Delta S)^2 = (\Delta x)^2 +(\Delta y)^2(ΔS)2=(Δx)2+(Δy)2(dS)2=(dx)2+(dy)2(dS)^2 = (d x)^2 +(d y)^2(dS)2=(dx)2+(dy)2simplify: dS2=dx2+dy2dS^2 =...

部分分式(Partial fractions)P(x)Q(x)=rational  function\large \frac{P(x)}{Q(x)}=rational \ \ functionQ(x)P(x)​=rational  function（有理数）=ratio of two polynomialsSplits P/Q INTO “EASIER”...

反向变量替换，配方sec=1cos   tan=sincos\large sec = \frac{1}{cos} \ \ \ tan = \frac{sin}{cos}sec=cos1​   tan=cossin​csc=1sin   cot=cossincsc = \frac{1}{sin} \ \ \...

数值积分DART BOARD 圆靶Assumption1hits=ce−r2hits=ce^{-r^2}hits=ce−r2r1<r=x<r2r_1<r=x<r_2r1​<r=x<r2​Shells:Part=∫r1r2(2πr)e−r2drPart = \int_{r_1}^{r_2}(2 \pi r)e^{-r^2}drPart=∫r1​r2​​...

微积分第二定理Info about F’ΔF=F(b)−F(a),Δx=b−a\Delta F = F(b) -F(a),\Delta x = b-aΔF=F(b)−F(a),Δx=b−aΔF=∫abf(x)dx (FTC1)\Delta F = \int_{a}^{b}f(x) dx \ (FTC1)ΔF=∫ab​f(x)dx (FTC1)ΔFΔx=1b−a∫abf...

微分方程 differential equationEx:dydx=f(x)\frac{dy}{dx} = f(x)dxdy​=f(x)y=∫f(x)dxy= \int f(x) dxy=∫f(x)dxsolved substitutionEx2:(ddx+x)(\frac{d}{dx}+x)(dxd​+x)为annihilation operator 湮没算符 in quantum ...

Newton’s methodIt works (very)well If |f’| not small and |f’’| not too big and x0 start nearby xand \ x_0 \ start \ nearby \ xand x0​ start nearby xf′(x0)=0...

最值问题f(x)=x+1x+2 , f′(x)=1(x+2)2≠0f(x) = \frac{x+1}{x+2} \ , \ f'(x) = \frac{1}{(x+2)^2}\neq 0f(x)=x+2x+1​ , f′(x)=(x+2)21​​=0no critical pts!Plot pts x=-2f(−2+)=−2+1−2++2=−10+...

导数的应用 Applications of diff线性近似(逼近)Linear Approximationsf(x)≈f(x0)+f′(x0).(x−x0){\color{Red} f(x) \approx f(x_0)+f'(x_0).(x-x_0)}f(x)≈f(x0​)+f′(x0​).(x−x0​)General:ln(1+x)≈xln(1+x) \approx xln(1...

chain rule use隐函数微分(Implicit differentiation)EX:ddxxa=axa−1{\color{Red} \frac{d}{dx}x^a = ax^{a-1}}dxd​xa=axa−1so far,so good! a=0,a=+1/-1,a=+2/-2,…implicit diff allows of any inversef’n provide...

导数例子Ex:Specific f′(x)  (f(x)=xn,1x)f{}'(x) \ \ (f(x)=x^n,\frac{1}{x})f′(x)  (f(x)=xn,x1​)General:(u+v)’=u’+v’(Cu)’ =Cu’ （C为constant）Specific: 根据 difference quotientddxsinx=...

几何观点下的导数:

通路和路径Walks&PathsDef: A walk is a sequence of vertices that are connected by edges.{v1−v2−...−vkv_1-v_2-...-v_kv1​−v2​−...−vk​} length kDef: A path is a walk where all viv_ivi​'s are different环和...

状态机，最大公约数数论Number theorystudy of the integers 0,1,2,3…Def: m | a(m divides a) a能被m整除if ∃k\exists k∃k a=km3 | 6 a=0=0.m m|0

不变式和第二数学归纳法Good proofs are:correctcompleteclearbriefelegantwell-organizedinorderintroduction:Legal Move:slide a letter into adjacent square?ABCDEFHGtoABCD...

反证法和数学归纳法反证法(a proof by contradiction)为了证明命题p为真，So to prove a proposition p is true

证明(proof)是一种确定事实的方法(A proof is a method for ascertaining the truth)物理上和平时：实验和观察(Experimentation & oberservation)抽样和找反例(Sampling & counterexamples)裁判(judge & jury)word of GodBoss内...

行列式公式及其余子式Formula for det A (n! iterms)

行列式及其性质Determinants det A = |A|±\pm± signs行列式的属性(properties)1) det I=1(II=1(II=1(I 为单位矩阵）∣1001∣=1\begin{vmatrix}1 &0\\ 0 &1\end{vmatrix}=1∣∣∣∣​10​01​∣∣∣∣​=12) 交换行： 行列式变符号(Exchange ro...

最小二乘y = C+ Dt最佳总误差最小(optimal = overall error as small as I can)make it{C+D=1C+2D=2C+3D=2\left\{\begin{matrix}C+ D = 1 \\C+ 2D = 2 \\C+ 3D = 2 \end{matrix}\right.⎩⎨⎧​C+D=1C+2D=2C+3D=2​联立无解，但可...

图和网络很多人认为图和网络是应用数学中最重要的模型(Graphs & Networks)离散数学定义其为图(the discrete version is a graph)Graph: Nodes, Edges关联矩阵(Incidence matrix)A=[−11000−110−1010−100100−11]{edge1edge2edge3edge4edge5A=\begi...

四个基本自空间(4 subspaces) A iscolumn space C(A) in RmR ^mRmnull space N(A)row space = all combs of rows=all combs of columns of A = C(ATA^TAT) in RnR ^nRnnull space of AT=N(AT)inRmA^T=N(A^T) in R^m...

线性相关性,基,维数linear independencevectors x1,x2,...,xnx_1,x_2,...,x_nx1​,x2​,...,xn​ are independent if no combination gives zero vector : c1x1+c2x2+...+cnxn≠0c_1x_1+c_2x_2+...+c_nx_n\neq0c1​x1​+c2​x2​+....

求解Ax=b 可解性和解的结构[1222    ∣    b12468    ∣    b236810  ∣    b3]\begin{bmatrix} 1&2&amp...

列空间和零空间subspaces S and Tintersection S∩TS \cap TS∩T is a subspaceColumn space of A is subspace of R4R^4R4 C(A)[112213314415]\begin{bmatrix} 1&1& 2 \\ 2 & 1& 3 \\3 & 1&amp...

向量空间PA = LUP=identity matrix with reordered rowP−1=PT ;     PT.P=IP^{-1} = P^T \ ; \ \ \ \ \ P^T.P = IP−1=PT ;     PT.P=I[122341]T=[124...

乘法和逆矩阵C34C_{34}C34​ = (rows of A).(col 4 of B)=a31.b14+a32.b24= a_{31}.b_{14}+a_{32}.b_{24}=a31​.b14​+a32​.b24​=a31.b14+a32.b24= a_{31}.b_{14}+a_{32}.b_{24}=a31​.b14​+a32​.b24​Gauss-Jordan:[1327][a...

方程组的几何解释2x- y = 0-x+ 2y = 3Matrix:[2−1−12][xy]=[03]\begin{bmatrix} 2&-1 \\ -1 & 2\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}=\begin{bmatrix}0 \\ 3\end{bmatrix}[2−1​−12​][xy​...

一阶方程换元法尺度变换(Scaling)x1=xa;y1=ybx_1=\frac{x}{a};y_1 =\frac{y}{b}x1​=ax​;y1​=by​改变单位(change units)make vars dimensionallyredue # , or simplify constantsDirect Sub:伯努利方程(Bernoulli equation)y′=p...

一阶线性First-order lineara(x)y′+b(x)y=c(x)a(x)y'+b(x)y=c(x)a(x)y′+b(x)y=c(x)linear: ay1+by2=cay_1+by_2=cay1​+by2​=c齐次(homogeneous) (c=0)Standard linear from y’+ p(x)y = q(x)Models温度-浓度模型mixing...