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Saturday, March 30, 2013

Mathematical Intro to Data Science

I just came across Yuan Yao's lecture notes/book titled A Mathematical Introduction to Data Science.  Unfortunately, the last few chapters are presently empty.

Friday, March 29, 2013

Non-linear Additive Maps


While teaching an introductory linear algebra course, a colleague noticed that most of the examples of additive maps he gave turned out to be linear.  He asked whether I could think of a map which was additive, but not linear.  In a general context, the question was to find a ring $R$, $R$-modules $ V$ and $ W$, and a map $ f \colon V \to W$ such that $ f $ is a group homomorphism, but not an $ R$-module morphism, i.e, $ f(x+y)=f(x)+f(y)$ for all $ x,y \in V,$ but there is some $ r \in R$ and $ z \in V$ such that $ f(r \cdot z) \neq r \cdot f(z)$.

Symmetries of the Naturals


This was a little problem I thought about while backpacking a few years ago.

We define a symmetry of the natural numbers to be any bijective function $ f \colon \mathbb{N} \to \mathbb{N}$.  Let $ S$ denote the set of symmetries of $ \mathbb N$.

Claim.  $ S$ has the cardinality of the continuum.

Sums of Consecutive Integers

A fun fact I recently learned about is the following.

Claim . A natural number can be written as the sum of at least two consecutive positive integers if and only if it is not a power of 2.

All Horses Are the Same Color


I was fortunate to TA UCSC's introductory proofs course three times.  One of the exercises I like to pose to my students is the following fallacious induction proof that all horses are the same color.

The "Proof"

As a base case, it's clear that any horse is the same color as itself.  Assume as an inductive hypothesis that any collection of n horses are the same color, where n is some fixed natural number.  Given any collection of n+1 horses, we may select a subset consisting of all but a single horse.  By the inductive hypothesis, all the horses in this subset are the same color.  Now regroup the horses and select another n horse subset, but this time we pick the horse that left out of the first selection.  Again all the horses in this subset are the same color, so all the horses in the original set are the same color.

Code Snippets

Presently, I am using Google's Code-Prettify to include code snippets in my posts.  Getting this up and running was fairly painless.

1.  Navigate to Design>Template>Edit Html.
2.  After the <head> tag insert <script defer="defer" src="https://google-code-prettify.googlecode.com/svn/loader/run_prettify.js?autoload=true&lang=tex&skin=sons-of-obsidian"> </script>. 3. Within the <body> insert onload='prettyPrint()'. The tag should now look similar to: <body expr:class='"loading" + data:blog.mobileClass' onload='prettyPrint()'>
4.  Consult the Google pages linked to above to see how to employ code-prettify.

One must be careful to encode the snippet so that it is HTML ready.

Thursday, March 28, 2013

MathJax

A few days ago I heard about MathJax, "an open source JavaScript display engine for mathematics that works in all modern browsers."  As this is a blog dealing with mathematics, I require that my host supports some form of mathematical markup, preferably LaTeX.  WordPress.com allows basic LaTeX, but its engine renders images, which do not scale well on mobile devices.  MathJax renders the markup in CSS and Web fonts, which scale nicely even in a mobile setting.