Analysis with Programming

More proving, still on Real Analysis. This is my solution and if you find any errors, do let me know. Problems Lebesgue Measure : Let $\mu$ be set function defined for all set in $\sigma$-algebra $\mathscr{F}$ with values in $[0,\infty]$. Assume $\mu$ is countably additive over countable disjoint collections of sets in $\mathscr{F}$. Prove that if $A$ and $B$ are two sets in $\mathscr{F}$, with $A\subseteq B$, then $\mu(A)\leq \mu(B)$. This property is called monotonicity . Prove that if there is a set $A$ in the collection $\mathscr{F}$ for which $\mu(A)<\infty$, then $\mu(\emptyset)=0$. Let $\{E_{k}\}_{k=1}^{\infty}$ be a countable collection of sets in $\mathscr{F}$. Prove that $\mu\left(\displaystyle\bigcup_{k=1}^{\infty}E_{k}\right)\leq \displaystyle\sum_{k=1}^{\infty}\mu(E_k)$ Lebesgue Outer Measure : By using property of outer measure, prove that the interval $[0,1]$ is not countable. Let $A$ be the set of irrational numbers in the interval $[0,1]$. Prove that $...

Another proving problem, this time on Real Analysis. Problem Prove that the Lebesgue outer measure is translation invariant. (Use the property that, the length of an interval $l$ is translation invariant.) Solution Proof . The outer measure is translation invariant if for $y\in \mathbb{R}$, \begin{equation}\nonumber \mu^{*}(A)=\mu^{*}(A+y) \end{equation} Hence, we need to show that Case 1: $\mu^{*}(A)\leq \mu^{*}(A+y)$; and Case 2: $\mu^{*}(A+y)\leq \mu^{*}(A)$. Case 1 : Consider a countable collection $\{I_n\}_{n=1}^{\infty}$, and let \begin{equation}\nonumber W = \left\{\displaystyle\sum_{n=1}^{\infty}l(I_n)\mid A\subseteq\displaystyle\bigcup_{n=1}^{\infty}I_n\right\} \end{equation} Then the outer measure of $A$ is, \begin{equation}\nonumber \mu^{*}(A)=\inf\,\{W\}. \end{equation}

This problem is the continuation of my previous post on Monotonic Sequence. Problem Prove the following: If $A_k$ is monotone, then \begin{equation} \mathrm{P}\left(\displaystyle\lim_{n\to\infty} A_n\right)=\displaystyle\lim_{n\to \infty}\mathrm{P}(A_n). \end{equation} Solution Proof . If $\{A_k\}$ is monotone, then \begin{equation}\nonumber \mathrm{P}\left(\lim_{n\to \infty} A_n\right) = \begin{cases} \displaystyle\mathrm{P}\left(\bigcup_{k=1}^\infty A_k\right)&\text{if}\;\{A_k\}\;\text{is expanding}\\ \displaystyle\mathrm{P}\left(\bigcap_{k=1}^\infty A_k\right)&\text{if}\;\{A_k\}\;\text{is contracting} \end{cases}. \end{equation} So if $A_k$ is expanding, then we can write $\displaystyle\bigcup_{k=1}^\infty A_k$ as disjoint unions, \begin{eqnarray} \displaystyle\bigcup_{k=1}^\infty A_k &=& A_1\cup (A_2\cap A_1^c)\cup (A_3\cap A_2^c)\cup \cdots\nonumber\\ &=& A_1\cup (A_2\backslash A_1)\cup (A_3\backslash A_2)\cup \cdots\nonumber \end{eqnarray}

Analysis with Programming has recently been accepted as a contributing blog on Mathblogging.org , a blogosphere aiming to be the best place to discover mathematical writing on the web. And as a first post, being a member of the said site, I will do proving on the theory of probability. This problem by the way, is part of my first homework on my masteral. This is my solution and if you find errors, do let me know. Problem If $\{A_k\}$ is either expanding or contracting, we say that it is monotone, and for monotone sequence $\{A_k\}$, $\displaystyle\lim_{n\to \infty} A_n$ is defined as follows: \begin{equation}\nonumber \lim_{n\to \infty} A_n = \begin{cases} \displaystyle\bigcup_{k=1}^\infty A_k&\text{if}\;\{A_k\}\;\text{is expanding}\\[0.3cm] \displaystyle\bigcap_{k=1}^\infty A_k&\text{if}\;\{A_k\}\;\text{is contracting} \end{cases}. \end{equation} Prove the above equation. Solution Proof. If $\{A_k\}$ is either expanding or contracting, then for an infinite seque...

$\mathrm{\LaTeX}$ has the capability to draw beautiful graphics. This feature is possible with Ti k Z package. Here is the plot of $f(x) = x$, In $\mathrm{\LaTeX}$, everything has to be coded. From axes, to labels, to points on the $xy$-plane; that explains why four lines of codes, only for single, very simple plot.

There is one package that I've been longing. A package that will give me the power to manipulate and do any 3D stuffs in R. I tried persp and wireframe , but I find them difficult to use especially on complicated mathematical functions, like doing parametric plots. And I am just frustrated about that, since I envy the 3D graphics of Wolfram Mathematica a lot, which I exploited for about a year, before I embrace R. However, that has come to an end after Joseph Rickert introduced the plot3D (authored by Karline Soetaert ) package in his post . And for the moment, we will be playing with the surf3D function. Here is the first one, the Mollusc Shell surface plot: With parametric equations: $$  \begin{eqnarray} x(u,v)&=&\left(1.16^v\right)(1 + \cos(u))\cos(v);\nonumber\\ y(u,v)&=&\left(-1.16^v\right)(1 + \cos(u))\sin(v);\nonumber\\ z(u,v)&=&\left(-2\times 1.16^v\right)(1 + \sin(u));\nonumber \end{eqnarray} $$where $u\in[0, 2\pi],\,v\in[-15, 6]$. B...

One great package in R is the animation made by Yihui Xie . And just for fun , we are going to explore that. Our aim is to create simple animated 2D and 3D plots. Here is the first one, 2D of course The code, It's a piece of cake right? The function we used for wrapping the plot is saveGIF , this function basically collects all the plots made and use these as frames of the GIF file. In other words, the above plot was generated/looped 100 times through the curve function, and in every iteration we increased the limits of the x axis; hence rolling all the generated plots, animates x-axis towards positive values. What about 3-dimensional? Speechless,

Matrix in R is constructed using matrix , rbind , or cbind function. These functions have the following descriptions: matrix - used to transform a concatenated data into matrix, of compatible dimensions; rbind - short for row bind, that binds a concatenated data points of same sizes by row; cbind - short for column bind, that binds a concatenated data points of same sizes by column. Example 1. Consider this matrix, $\left[\begin{array}{ccc} 3&4&5\\ 2&1&3\\ 6&5&4 \end{array}\right]$. Using the matrix function, we can code this as So here's what happened above, first the data was concatenated using the c function into a data.a object. Next, we transformed this into a matrix of compatible dimension, that is $3\times 3$. Below are the description of the arguments: data.a - the data nrow - the number of rows ncol - the number of columns byrow - the orientation of how data is wrapped into a matrix. If TRUE , then it's row-wise,...

R can perform the usual mathematical operations, below are the functions: Arithmetic +     - addition -     - subtraction *     - multiplication /     - division Trigonometry sin       - sine cos       - cosine tan       - tangent asin    - sine inverse acos     - cosine inverse atan     - tangent inverse Linear Algebra +             - element-wise addition -             - element-wise subtraction *             - element-wise multiplication /             - element-wise division %*%         - matrix multiplication t          ...