Friday, 29 August 2014

Monotonic Sequence

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

  1. 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

  1. Proof. If $\{A_k\}$ is either expanding or contracting, then for an infinite sequence $A_1,A_2,\cdots$ one can define two events from $\displaystyle\lim_{n\to \infty}A_n$, i.e. \begin{equation} \label{eq:limAn} \lim_{n\to \infty} A_n = \begin{cases} \displaystyle\lim_{n\to\infty}\sup_{k\in [n,\infty)}\{A_k\}\\[0.3cm] \displaystyle\lim_{n\to\infty}\inf_{k\in [n, \infty)}\{A_k\} \end{cases}. \end{equation} Now the $\displaystyle\sup_{k\in [n,\infty)} \{A_k\}$ and $\displaystyle\inf_{k\in [n,\infty)} \{A_k\}$ are defined as follows:

Monday, 14 July 2014

LaTeX: Using gnuplot for Plotting Functions

$\mathrm{\LaTeX}$ has the capability to draw beautiful graphics. This feature is possible with TikZ 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.

Thursday, 22 May 2014

R and Python Meetups, Philippines

There will be upcoming meet ups for R User Group Philippines and Python Philippines (PythonPH) Community. Below are the details:

R Meetup

topic: R for SAS users, and planning of RUG activities 


date: Thursday, June 19, 2014
         7:00 pm

outline:
  • Introducing R to SAS users;
  • common SAS functions used at PPD - c/o Mark Javellosa;
  • group discussion on equivalent packages in R; and,
  • Sharing of experiences of actual SAS converts.
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