I started fiddling around with R again, and ended up playing with a zipcode database.

So, first I downloaded the zipcode database at Mapping Hacks, and unpacked the zipfile in my working directory.

Then, I loaded the data into R

> zips <- read.table("zipcode.csv",sep=",",quote="\"",header=TRUE)

> names(zips)

[1] "zip" "city" "state" "latitude" "longitude"

[6] "timezone" "dst"

So, now I have an R frame containing a lot of US cities, their geographical coordinates, and their zip codes. So we can start playing with the plot command! After rooting around a bit, I ended up settling on the smallest footprint plot dot I could make R produce, by setting the option pch=20 in the plot options. Hence, I ended up with a command basically like this:

This post is to give you all a very swift and breakneck introduction to Gröbner bases; not trying to be a nice and soft introduction, but much more leading up to announcing the latest research output from me.

Recall how you would run the Gaussian algorithm on a matrix. You’d take the leftmost upmost non-zero entry, divide its row by its value, and then use that row to eliminate anything in the corresponding column.

Once we have the matrix on echelon form, we can then do lots of things with it. Importantly, we can use substitution using the leading terms to do equation system solving.

The starting point for the theory of Gröbner bases was that the same method could be used – with some modification – to produce something from a bunch of polynomials that ends up being as useful as a row-reduced echelon form.

I have been painfully remiss in keeping this blog up and running lately. I wholeheartedly blame the pretty intense travel schedule I’ve been on for the last month and a half.

To get back into the game, I start things off with a letter from a reader. Rodolfo Medina write:

Hallo, Michi:

surfing around in internet, looking for an answer to my question, I fell into

your web site.

I’m looking for an answer to the following question:

my intuitive idea is that a one-dimensional connected topological submanifold

of a topological space S should necessarily be the codomain of a curve (if we

define a curve to be a continuous map from an R interval into a topological

space).

Conversely, the codomain of an injective curve, defined in an open R interval,

should necessarily be a one-dimensional topological submanifold of S.