Category Archives: Mathematics

A Map of Math?

There has been some recent discussion at Reddit of an attempt I made to turn some data from arXiv.0rg into a visualization of the relationships between research areas in mathematics.  I was never completely pleased with the dataset. 

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Best Proximate Value

My Oxford English Dictionary describes proximate as meaning,

Closely neighbouring, immediately adjacent, next, nearest (in space, serial order, quality, etc.); close, intimate

In this article, a proximate value will mean a value p which we shall use as a stand-in for an unknown value x.  The unknown variable x is known to have the property that

a \leq x \leq b

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The Cauchy–Schwarz Inequality IV

The Cauchy-Schwarz inequality,

\displaystyle\sum_{i=1}^n{a_i^2}\sum_{i=1}^n{b_i^2} \geq \left(\sum_{i=1}^n{a_ib_i}\right)^2,

holds for any pair of lists of n numbers, {a_i} and {b_i}, for i=1,2,\ldots,n. I am about to discuss a fourth proof of the Cauchy-Schwarz inequality. I have previously presented a first, second and third proof.  The fourth proof involves some algebraic pyrotechnics where we measure the difference between the expressions on each side of the Cauchy-Schwarz inequality. Continue reading

Inside the Putnam

An article at the blog, The Accidental Mathematician, reflects on the experience of being on the committee that chooses the questions for the William Lowell Putnam Competition. Several rejected questions are discussed and the reasons for their rejection are also discussed.

[Hat tip: The Accidental Mathematician]

Planarity

I recently discovered the game, Planarity. The goal of the game is to take a graph and put it in a form where none of the edges cross. That is to say, to prove that it is planar. If you don’t know what I’m saying; or you think it sounds boring; or you think it sounds complicated; go play it anyway. It’s lots of fun.

The Cauchy–Schwarz Inequality III

I have previously written about the Cauchy-Schwarz Inequality here and here.  To recap, the Cauchy-Schwarz inequality,

\displaystyle\sqrt{\sum_{i=1}^n{a_i^2}}\sqrt{\sum_{i=1}^n{b_i^2}} \geq \sum_{i=1}^n{a_ib_i},

holds for any pair of lists of n numbers, {a_i} and {b_i}, for i=1,2,\ldots,n.    Now, I have given two proofs so far, and here is a  third.  We will proceed by induction on the number of variables.   Continue reading

LaTeX2WP, a LaTeX to WordPress converter

If you write a lot of mathematics on WordPress, then you might find it useful to look at Luca Trevisan’s LaTeX2WP. As you might have guessed, it converts LaTeX to something you can paste into a WordPress blog editing environment. It seems to use python which should be easy to use on a Linux-based system and would probably require something like Cygwin on a Windows machine.

Incompleteness from the Properties of Algorithms

I somehow missed some great notes on recursion theory and incompleteness by Jeremy Avigad, written about on Ars Mathematica. As noted in the article, approaching incompleteness from the point of view of algorithms is a really accessible way of doing it. If you have some time, you should check it out. Continue reading

The Cauchy–Schwarz Inequality II

I have previously written about the Cauchy-Schwarz Inequality.

\displaystyle\sqrt{\sum_{i=1}^n{a_i^2}}\sqrt{\sum_{i=1}^n{b_i^2}} \geq \sum_{i=1}^n{a_ib_i}

Here is another proof of this fun inequality.  We start at a similar point to the previous proof Continue reading

The Cauchy–Schwarz Inequality

I recently read The Cauchy-Schwarz Master Class by Michael Steele which, in addition to teaching the reader about inequalities, has a lot of fun and varied things to say about the Cauchy-Schwarz inequality.

First things, first, the Cauchy-Schwarz inequality in it’s simplest form says the following:

\displaystyle\sqrt{\sum_{i=1}^n{a_i^2}}\sqrt{\sum_{i=1}^n{b_i^2}} \geq \sum_{i=1}^n{a_ib_i}

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