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## And Speaking of Biologists and Mathematics

Well, I see that Nick has been a little more active this last month.  Thus, it is with great guilt that I try to step beyond my laziness and post something!

First, some announcements:

• I am going to try keeping the Mathematical Biology Seminar going this Fall quarter, though the meetings will be every other week.  My plan is to focus on Neural Network modelling and the use of NEURON software to implement some simple network models.
• I would also like to keep the old CAMG meeting going, assuming there is any interest.  Drop a comment is you’d like to join us in exploring $\LaTeX$, as well as some other software.

## Mathematical Biology Seminar

Felicis and I are running a Mathematical Biology Seminar here at PSU. Felicis has been helping to bring us all up to speed on some of the basic Neuroanatomy of Hearing. (We’re starting with a paper co-authored by Lars Holmstrom of PSU entitled: Responses to Social Vocalizations in the Inferior Colliculus of the Mustached Bat are Influenced by Secondary Tuning Curves)

I’ll be talking next week about some basic modeling techniques in relation to the topic above.

This is a big deal for both Felicis and I, as we’re both primarily interested in doing work in Mathematical and Theoretical Biology here in the mathematics department. Felicis’s core area of interest is in Neurology, and mine is in Ecology and Evolution.

We’re meeting on Wednesdays at 3:30 pm. If you are a mathematics student (or for that matter a biology student) interested in mathematical biology, feel free to stop by the new conference room at that time on the 3rd floor of NewBurger Hall

For those totally new the Idea of mathematical biology, here’s the Wikipedia page.

And here are a few more links to wet your appetite:

Some equations from EqWorld

Why is Mathematical Biology so Hard? from the Notices of the AMA

Getting Started in Mathematical Biology, by Frank Hoppensteadt and the AMS

## CAMG: First Meeting Notes

OK, so it’s the weekend after the second meeting… Better late than never. I was not able to make it to the second meeting, which is a shame since Saij tells me it was quite interesting! I’ll see if I can get Eric to post what he covered…

As for the first meeting, well…

It was my job to talk about $\LaTeX$, and so we gathered around and went over the pre-processing stuff, also known as the ‘preamble’ to a document. If you are interested, you can see more here. While chatting, we went into a discussion of proof-verification systems using Coq and Mizar, and their connection to $\LaTeX$. Interestingly, I have heard that Mizar is able to produce $\LaTeX$ output so that one can immediately produce the typeset article (although I cannot now re-find that reference, so perhaps I am incorrect…). This led us to discuss the general inhospitality of such formal mathematics towards human readers. We then talked about the possibility of an additional program to take a verified proof from the formal setting and render it in a more human-friendly way, such as what might be required by an instructor.

Perhaps we could ‘Erdmanize’, or ‘Jiangize’ a formal text, rendering it into something either of these instructors (who ask their students to produce very different mathematical prose) would accept. Even more important, I think, would be to have the produced text include examples and perhaps counterexamples – something completely unnecessary to a formal proof checker, but very important for human understanding.

Tim is very interested in computer verifications systems, and may take us on a tour of Coq, which he is actually using in another course. Eric has already started us out on Sage, and I will be continuing the $\LaTeX$ journey next week.

Speaking of which- we continued our $\LaTeX$ discussion with a brief introduction to the fancy headers package (available here) and elaboration on the different environments for arranging text on the page in a ‘tabular’ form: lists, tables, arrays, et cetera. Next week, I am planning on talking about how to adjust margins and the appearance of the page, along with a better discussion of the different ‘environments’ available to us. If there is anything in particular you would like to hear about, drop a comment my way and I’ll either add it in this week, or at least have it for the future!

We finished up with a discussion of what we might cover in the future: the Haskell programming language (With a book suggested by Tim), Discrete Mathematics, Computer Proof Verification, computer modeling, SAGE, and $\LaTeX$. Tom expressed the desire, that if we do nothing else, focusing on SAGE and $\LaTeX$ would be a good start for this quarter.

Thus endeth the first meeting. Hope to see you at future ones!

ex animo-

Felicis

## The equivalent definitions of ‘topology’…

From Armstrong:

On page 13, we have definition 1.3:

Given a set $X$ and for each point $x$ in $X$ a nonempty collection of subsets of $X$, called neighborhoods of $x$. These neighborhoods are required to satisfy four axioms:

1. $x$ lies in each of its neighborhoods.
2. The intersection of two neighborhoods of $x$ is also a neighborhood of $x$.
3. If $N$ is a neighborhood of $x$ and $U$ is a subset of $X$ which contains $N$, then $U$ is a neighborhood of $latex x. 4.$If $N$ is a neighborhood of $x$ and $\mathring{N}$ denotes the set $\{z\in N|N\text{ is a neighborhood of }z \}$, then $\mathring{N}$ is a neighborhood of $x$. (The set $\mathring{N}$ is called the interior of $N$.)

In which case we can call $X$ (along with the specified neighborhoods) a topological space.

Whew! That’s quite a bit, but it simply gives us a firm standing for the intuitive idea of a neighborhood, and then declares that when we have a set such that every point has at least one neighborhood (the “nonempty collection” requirement above), then we have a topological space.

Does it seem to you that it might be a bit difficult to verify that we have a topological space? It’s not enough that we verify the axioms for some point in $X$, we have to do so for every point! Perhaps there is an easier way…

Enter definition 2.1 on page 28:

A topology on a set $X$ is a nonempty collection of subsets of $X$ called open sets such that: any union of open sets is open; any finite intersection of open sets is open; and both $X$ and $\emptyset$ are open. A set together with a topology on it is called a topological space.

Well, that seems a little simpler, but does it describe the same structure as the first definition? (Hint: yes, it does.) If we read carefully the beginning of chapter 2 (pp. 27-8), Armstrong shows the logical equivalence in both directions.

To show that 1.3 implies 2.4, we just describe a set as open if it is a neighborhood of each point it contains. The properties of 2.4 then follow from 1.3 fairly easily.

To go the other way, we take any point $x \in X$ and then say that a subset $N$ of $X$ is a neighborhood of $x$ if there exists an open set $O$ such that $O \subseteq N$. It is then a matter of a little work to show that all of the axioms of neighborhoods are then satisfied.

So- that’s it for the moment. Have a great weekend, and we’ll see how our first classes go on Monday!

ex animo-

Felicis

## Let’s Study Linear Algebra

For anyone interested in getting a jump ahead for MTH 444/544: Advance Linear/Multilinear Algebra, we’ll be meeting at 10 a.m. on Monday, 17 December in the Neuberger Atrium. Read the first four sections of the text (up through page 33) and be ready to discuss all the homework problems. That is – take a look at them and either do them or at least be sure that you can do them on the fly. There’s only 26 total for the four sections and some of those are duplications (the same kind of problem, just with different parameters – several induction proofs, for example…)

## LaTeX Follow Up

So we only had three people show up, but that’s better than none! If anyone else is interested, post a comment, and we’ll try to set up another session.As for what we discussed — see below!

## LaTeX Get-Together

When: 10 a.m. Tuesday, 11 December 07

Where: 3rd Floor Atrium, Neuberger Hall, PSU Campus

What: Installing and Learning (Basic) LaTeX. Bring your laptops (if you have one), or whatever computer on which you plan to do your homework. Extra laptops (as loaners) and power strips would probably be helpful. Also have some mathematics you want to render so we can get you up to speed as fast as possible! Continue reading