oh but plans can fall through as so often they do

well...here i am stuck in salt lake. i was supposed to be off finding fossils today with my cousin, but his car broke down and he has had to rewrite a report for work. so i'm pretty bored until thursday. i'm staying at his mom's house and am trying to help her with a bunch of genealogy stuff and also trying to keep from being bored out of my mind. i did at least make it up to logan to get my bike, and i had the presence of thought to bring my bike shoes (no helmet or gloves though) so once i get some air in the tires i'll probably try to head up city creek canyon. jeff says there are some decent trails up there. if i'm honest i'm a little worried about riding up the canyon though...i haven't been biking in about 6 months. i get a pretty good workout at work...lots of lifting and carrying and dragging and pulling myself up trees...but it isn't really the sustained cardiovascular workout you get from running or riding a bike. but we'll see how it goes. plus the weather is nice and cool for the next couple of days.

the not-so-high seas

Went sailing on Saturday with Keith and a couple of other people. The winds weren't very good...we topped out at about 3 knots. But it was a beautiful afternoon and a nice sunset.

Ospreys.

Looking aft from the front of the boat.

Me with the furled jib behind me.

Sunset.

We sail from a harbor in Annapolis. Unfortunately some of the kids that came along didn't want to be out sailing too late, and since we got a late start we only made it as far as the bay bridge. We'll try again in 3 more weeks.

math is for nerds

Not that anyone is horribly interested in nerdy math stuff...but here's what a spanning tree is.

The image above is what we mathy people call a graph. It consists of a vertex set and an edge set. Vertices are points (zero-cells, if you really want) and edges are lines (one-cells). So you have a bunch of lines and a bunch of dots...and now you play connect the dots. That's a graph.

A tree (in math) is a graph with no closed loops. Here's a tree:

Note that this tree contains no closed polygons. That's what makes it a tree. Just like trees in the real world, generally.

A spanning tree is a subset of a graph. Specifically, a spanning tree contains the vertex set and enough of the edge set to keep all connected subsets of the graph connected. (Connected = you can draw it without picking up your pencil. And you're allowed to cross lines and draw back over lines you already drew.) The above tree is, in fact, a spanning tree for the graph up at the top. Here's a picture of them together.

Note that spanning trees are not unique. Here's another one for the graph at the top.

And again with the original graph superimposed:

On the second spanning tree I decided to go for the "without lifting the pencil, crossing lines, or drawing back over what you already drew" method.

Anyhow, that's what a spanning tree is. They play a significant role in Bass-Serre theory and graphs of groups. I'm not going to try to explain either. If you really want you can search for Bass-Serre theory on wikipedia. It's on there.

And I'm terribly clever for naming my blog "spanning trees" because I do tree work. Like the leafy things with bark and trunks. I'll get some pictures of that soon.

And I also promise that this blog won't be exclusively about math. Or tree work.