## Archive for the ‘silliness’ Category

### Tonal whiplash

July 26, 2012

People who know me well know that I don’t have any particular objection to targeted advertising. Like most things, it can be used for good purposes or bad purposes, but I think on balance, it’s probably better than untargeted advertising. Either way, it’s more efficient for many advertisers, which is good, because that means they can invest more capital into developing and improving products.

I’m not sure whether YouTube targets the ads that sometimes play before videos, either based on the user or the video. But if so: YouTube, I can’t imagine that there are that many people who want to see something about “The Purina Cat Chow Real Stories Project” when they’re looking for a Nick Cave song.

### Is the Super Bowl a Z/2Z-graded bowl?

February 8, 2010

And if so, what’s just a regular ungraded bowl?

(Hat tip to Vladimir S. for the bad pun; new post coming soon, really, I promise.)

### On bias

January 23, 2010

[From Homer the Smithers, Mr. Burns sends Smithers on a forced vacation and tasks him with finding a temporary replacement]

Smithers: I’ve got to find a replacement who won’t outshine me. Perhaps if I search the employee evaluations for the word ‘incompetent…’

[Computer beeps, screen displays “714 Matches Found’]

Smithers: 714 names?! Heh. Better be more specific. ‘Lazy,’ ‘clumsy,’ ‘dimwitted,’ ‘monstrously ugly.’

[After a couple seconds, computer beeps, screen displays ‘714 Matches Found’]

Smithers: Ah, nuts to this, I’ll just go get Homer Simpson.

Actually, I tried to find the statistical nomenclature for this kind of thing, but couldn’t. Anyone have any idea what this is? (I want to say selection bias, but that’s not quite it…)

New Extremal Toolbox post should be coming later this weekend.

### Someday we’ll find it

January 21, 2010

Apparently it’s used to model some password-protected networks, but I’m still pretty sure that the rainbow connection number of a graph can only have been invented (or at least named) as a joke.

### Excitement!

December 29, 2009

Sorry about the lack of new post; it’s coming. It turned out to be a more interesting problem than I at first thought; look for it around New Years’.

I’m working through some of the holes in my graph knowledge with my shiny new copy of Bollobas’ Modern Graph Theory. Chapter 1, Exercise 19 is a problem I’ve done before, but the way it’s presented makes me want to do it all over again:

Characterize the degree sequences of forests!

Exercise 17 is about the degree sequences of trees, and 18 extends it to forests with a fixed number of components — so this isn’t totally out of the blue. Still, it makes me wonder why more textbooks don’t end problems with exclamation marks.

### Where do graphs live?

December 5, 2009

This post came out of some thoughts I posted (anonymously, but mostly because I didn’t feel like registering) over at nLab. I don’t think it’s a secret that I’m heavily interested in the relationships between category theory and combinatorics, and more generally the ways in which we can use “structured” algebraic objects and “continuous” topological objects to gain information about the unstructured discrete objects in combinatorics. That said, the folks over at the nLab work on some crazy abstract stuff, which seems about as far away as possible from the day-to-day realities of graph theory or set systems. And maybe it is — but I hope it’s not, and as far as I’m concerned, this is a windmill that deserves to be tilted at. (After all, it might be a giant.)

So as my jumping-off point, I’ll take my observation from last time that the relationship between graphs and digraphs is analogous to the one between groupoids and categories. I briefly mentioned something called a quiver, which can be thought of as any of the following:

• Another name for a digraph, which categorical people use when they don’t want us combinatorialists stomping in and getting the floor all muddy;
• A “free category,” i.e., one in which there are no nontrivial relations between composition of morphisms;
• An algebraic object whose representations we want to consider; it’s worth thinking of this way mostly because of the “freeness,” although if you try to define it more formally you’ll probably end up with the previous definition;
• What you get when you take (part of) a category and forget all the rules for how morphisms compose.

This last point is the most interesting one for our purposes, since it’s clearly an algebraic object but isn’t as restrictive as “free category,” and thus has a chance of capturing the unstructured behavior of the combinatorial zoo. But it’s tricky to turn this into a rigorous definition that actually includes everything we want to be a quiver… so we’ll just use “quiver” as a fancy name for “digraph.” However, there’s an important philosophical lesson to be learned from the final point, so I’ll set it off:

Philosophical lesson. The edges of a quiver shouldn’t carry any information except for the vertices they are incident to; more generally, paths in a quiver shouldn’t carry any information except for their sequence of vertices.

### In which I am late to the MaBloWriMo party

November 4, 2009

Hey, guys. Been a while, hasn’t it? Sorry for not blogging more; I blame having to “learn stuff” in “classes.” It’s very silly.

I didn’t find out about Charles Siegel’s MaBloWriMo until almost November 2 (and then there were more of the aforementioned “classes”), but it sounded cool, and gave me a reason to get back to blogging. So to make up for the late start, I’m going to try to write at least 30 total posts for the month of November, which should hopefully average in the range of 1000 words in length.

So, again because of school, I’m supposed to be spending November writing a final (expository) paper for introductory algebraic geometry. This means that I’ll probably be posting bits and pieces of drafts of said paper for MaBloWriMo, which means that there’ll be some unifying theme to some of the posts. The bad news is, I don’t actually know what it is yet, since I haven’t figured out my final paper topic, since I’ve been procrastinating horribly.

So instead I’ll talk about my usual craziness; over the next few days, I’ll ask the question: Is there a higher category theory for graphs? What is it?

So if you’re a combinatorialist, the easy way to think about categories (at least at first) is as special sorts of directed graphs with some extra structure (specifically, a way to glue together edges to make new edges.) To make this rigorous, we can think of every (small) category as having an “underlying digraph” where we just draw a directed edge from X to Y if there’s a morphism $X \rightarrow Y$.

It turns out that this construction’s functorial: it gives us a forgetful functor from the category of small categories to the category of digraphs. (A morphism between digraphs, by the way, is exactly what it “should” be; it’s a function between the vertex sets that respects the adjacency structure.) (more…)

### Terrible joke #379

July 19, 2009

I’m still in a bit of a holding pattern research- and writing-expository-stuff-wise, but I wanted to update. Since you’re kind enough to read this, I’m repaying you by letting you suffer through my horrendous math-related puns.

Q. What do you get when you identify the original topological spaces in the product topology really angrily?