New Data Set

Paul Ginsparg recently provided me with a truly awesome data-set concerning the mathematics papers at the arXiv.org.  I am thinking of ways to put it to use in making new mathematics illustrations and hopefully improving on my previous project.

AMS Council Meeting

I wrote a few articles stemming from my experiences reporting on the AMS‘s graduate student blog to the AMS Council Meeting. I had a bit of writer’s block so it took longer to write them than I expected it would. Here they are:

1. AMS Council Meeting – Part 1
2. AMS Council Meeting – Part 2
3. AMS Council Meeting – Part 3

For the Love of Inequalities

My divagations concerning the Cauchy-Schwarz inequality have not gone unnoticed.

How To Prove It?

A book that I really enjoyed reading was How To Prove It by Daniel Velleman. The main idea of the book is what is called a structured approach to proof. This idea is related to structured programming.

Icon Wrangling

I think most people tend to have a lot of loose icons on their desktop.   There is a neat program that allows you to make fences, which are labeled, semi-transparent boundaries that you can make to group your icons.   Making a fence is as simple as selecting a group of icons while holding down the right-click button.  This brings up a window asking if you want to make a fence.

Formulas for Differentiation of Implicit Functions

Many people who have taken a Calculus sequence of one form or another are familar with the notion that if , then by taking the derivative with respect to x, we find:

which imples that if then

Making Nice Images of Equations in Windows

If you already know LaTeX, there is a quick and easy way to make images that you can paste into many different Windows applications such as PowerPoint.
For the following technique to work, it will be necessary to install MikTeX.

Now, if the details are not what you are interested in and you would like to get on with it,  I have made files available.  To get them to work the idea is simple.   Download the files I have made available.  Put them all in the same directory.  Edit body.txt and double click the batch file (.bat).  The body.txt file that I have provided contains the text:

$\frac{a}{b}$

Double clicking makeLaTeXImage.bat produces the image of the fraction which is called tmp1.png

The code is set up to produce images with a transparent background to make it conducive to pasting into documents with different colored backgrounds.

To change the image generated, change the text in body.txt

One modification to the images produced that I can anticipate an average user needing is a modification in the default resolution.  This can be done by editing the following line of the batch file in your text editor:

dvipng -D 200 tmp.dvi -T tight -bg transparent

Here, 200 means 200 dpi.  You can change this to whatever dpi you need.  For instance,

dvipng -D 600 tmp.dvi -T tight -bg transparent

gives 600 dpi.

MathOverflow Has Siblings

Mathoverflow is a peerless way to get mathematics questions answered at the graduate level and above.

In addition to this website, there are two others that I think would be of general interest to a mathematical audience.

Living & Learning

Things that I have seriously considered doing are:

Painting an entire wall of my apartment with blackboard paint.

Painting an entire wall of my apartment with whiteboard paint.

Making my own transparent whiteboard from plexiglass.

Painting a table top with blackboard paint.

In the Long Run We Are All Dead

Quoting Terry Tao, albeit with small modifications,

There is an island upon which a tribe resides. The tribe consists of 1000 people, with various eye colours. Yet, their religion forbids them to know their own eye color, or even to discuss the topic; thus, each resident can (and does) see the eye colors of all other residents, but has no way of discovering his or her own (there are no reflective surfaces). If a tribesperson does discover his or her own eye color, then their religion compels them to commit ritual suicide at noon the following day in the village square for all to witness. All the tribespeople are highly logical and devout, and they all know that each other is also highly logical and devout (and they all know that they all know that each other is highly logical and devout, and so forth).

For the purposes of this logic puzzle, “highly logical” means that any conclusion that can logically deduced from the information and observations available to an islander, will automatically be known to that islander.

Of the 1000 islanders, it turns out that 100 of them have blue eyes and 900 of them have brown eyes, although the islanders are not initially aware of these statistics (each of them can of course only see 999 of the 1000 tribespeople).

One day, a blue-eyed foreigner visits to the island and wins the complete trust of the tribe.

One evening, he addresses the entire tribe to thank them for their hospitality.

However, not knowing the customs, the foreigner makes the mistake of mentioning eye color in his address, remarking “how unusual it is to see another blue-eyed person like myself in this region of the world”.

What effect, if anything, does this faux pas have on the tribe?

Although, it might not initially be obvious this is a problem about thinking about thinking about thinking about thinking — and so on.   As before, I will proceed with diagrams.  One argument, which is inductive, proceeds in this way: if there were only one blue-eyed villager, he would not know his eyes were blue.  Everyone else would know his eyes were blue.  Because the stranger pointed out there was an individual with blue eyes, an individual our sole blue-eyed villager had never seen,  he would immediately know it was him and kill himself the next day.