Proof of the Day (POD): Rings, Cancellation, and Zero Divisors

I’m going to try and start a new feature here, the “proof of the Day”.  It’s based on the crossfit idea of the Workout of the Day where they post a new workout everyday.  In this case, I’ll try (and maybe some of my colleagues here will also) and post a new proof (not EVERY day, but often) on a particular topic, without much but the proof itself.

While I’ll also try and post on these topics generally in other articles, the POD’s will consist of simply interesting, important, or fun proofs for their own sake.  Proofs are pretty.

A ring has the cancellation property if and only if it has no zero divisors.

Proof:
Part I.

Suppose R is a ring that has the cancellation property.  Then for a,b,c\epsilon R, ab=ac implies b=c.  Now, let a\epsilon R such that a\neq 0.  Further, suppose a is a zero divisor.  Then, there exists a b\epsilon R such that b\neq 0, but ab=0. Then, ab=a0.  But, since R has the cancellation property, then this implies that b=0. This is a contradiction.  Therefore, if R has the cancellation property, R has no zero divisors.

Part II.

Suppose R has no zero divisors.  Let a,b,c\epsilon R, and let ab=ac, where a\neq 0.  Then, ab-ac=0.  This implies that a(b-c)=0.  Since a\neq 0, then (b-c) must equal 0, which implies b=c.  So, R has the left cancellation property.

Similarly, if ba=ca, and a\neq 0, then (b-c)a=0 which implies b-c=0, so b=c.  So, R also has the right cancellation property.  Therefore, if R has no zero divisors, then it has the cancellation property.

\Box

Winter Quarter

And so it begins again.

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Take two…

Well- not much happened this fall…  OK, a lot happened, but we were all too busy to tell anyone about any of it!

The biggest lesson I learned was not to try teaching two classes while taking a full load of graduate credits. And buying a house.  And moving…  Yes, I took on a bit too much, and my posts necessarily suffered.

Winter quarter will be different! Continue reading

Brace For Impact!

The Fall Quarter is about to start here at PSU!  I’ll be teaching two sections of Calc I; learning Genetics, Stochastic Analysis and Partial Differential Equations; and attempting to have a weekly seminar on Mathematical Biology (Tuesday afternoons for anyone who’s interested).

I’m sure we’re all feeling that there is something big about to hit – let’s channel the Viking spirit to turn a face it head on!

ex animo–

Felicis

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.

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Biologists Need to Clean Up Their Math

I ran across this post from 2004 on jobs in Mathematical Biology.  In it they remind us that the onus is not only on we math-folk to bridge the gap between the two fields.  Biologists need to start cleaning up their math:

But the onus isn’t all on mathematicians. Biologists could use an infusion of mathematics as well, says Iya Khalil, vice president of R&D at Gene Network Sciences Inc. in Ithaca, New York. Biologists frequently run experiments that generate large amounts of data, but the usefulness of the data will likely depend on the design of the experiment. “In the realm of high-throughput experiments, often times [mathematicians] figure out that if the biologist had done the experiment in a particular way, it would have improved the statistics of the analysis by an order of magnitude,” she says. By then it’s too late.

“Biologists have to pick up the mathematics,” agrees Laubenbacher. “If you want to use your data to make a mathematical model, then you need to take the modeling method into account when you design new experiments. Different modeling methods will require different kinds of data.”

Proof that Humans Are Evil

The movie WΔZ was apparently inspired by Price’s equation:

Reed A. Cartwright goes into it here.

It describes how the change in trait with phenotypes is related to the phenotypes’ fitnesses, . Note that the genetics of the trait (mutation, ploidy, etc.) is contained in the second term. See Wikipedia for more details.

Now, of course, that doesn’t mean that we’re all going to hell. But, the movie which was inspired by a book on Darwin may make it seem that way:

“It featured a mathematical equation—W Delta Z—formulated by American population geneticist George R. Price,” he explains. “It supposedly shows that there’s no real altruism in nature; no such thing as selflessness. Price was so upset by his findings that he ended up giving away all his possessions to the poor and, eventually homeless himself, committed suicide with a pair of nail scissors in a filthy London squat.”

Now that’s what I call dedication to your work!