Viking Metal Meets Viking Math!

Hammerheart, by Bathory. The earliest Viking Metal album.

Wikipedia is a wonderful place.  First the definition of Viking Metal.  And then a list of bands.

Here’s one of Bathory’s videos (only video?) entitled One Road to Asa Bay.  Go Vikings!

Proof of the Day: The Kernel’s Injective Fried Chicken

Let be a linear map between vector spaces. Show that T is injective if and only if .

Proof:

Suppose first that T is injective. Then implies , so .

Now suppose . Then implies which implies , and . So, T is injective.

Greatest Jobs: Mathematician

JobsRated.com has come out with a listing of top jobs in America and Mathematician is number 1!

Proof of the Day: Homomorphisms and Abelian Groups

Let be a homomorphism of Abelian groups.  Show that for each .

Proof:

Since f is a homomorphism, and since , an Abelian group, then:

(by associativity)

(f is a homomorphism)

The last line is true because (but that’s another proof).

God Created the Integers, Man Created God

Michael Atiyah gives a presidential address on Mind, Matter, and Mathematics (good alliteration).  In it he discusses the difference between mathematical philosophy and natural philosophy.  It’s an interesting read throughout.

But, near the end he says:

Mathematical physicists believe that there are indeed simple and beautiful mathematical equations that govern the universe, and that the task of the scientist is to search for them. This is an article of faith.

An alternative faith is to believe in a God who created the universe and kindly provided us with laws or equations that we would be able to understand.

He touts that these are compatible philosophies.  As faiths, they are similar (I take issue with the first idea:  Mathematical equations do not “govern” the universe, they are just really good at representing it.)

But, more importantly, I disagree with the idea that belief in God is always compatible with science (an implication I think he was making).  In physics, it’s an easier sell.  There is nothing alive in physics.

A harder sell is in biology.  Belief in God is one thing, but belief in a soul is problematic.  If one believes in a soul, that every human (homo sapien) is singled out from among God’s creatures as different (better), then all of biological evolution (and what it can tell us about who we are) falls apart.

If it is true that humans ARE totally and fundamentally different than all of the other creatures on earth (and potentially on other planets), and if this is due to our having a soul, then we MUST abandon much of what we believe to be true in biology.

If biology is right, then we are not different in any fundamental way than other species.  Unique, sure.  (So is the norwhal.)  But, not totally different.

I am hesitant to say that a belief in a God-given soul is compatible with biological science, and from there science generally.

(HAT TIP:  Noncommutative Geo)

Only 3 Classes

Well, my class-load this term is not as large as Felisis’s.  I’m only taking 3: Abstract Algebra, Topology, and Multi-linear Algebra (with Erdman, for those who know what that means).  I’m also still working (as a private weightlifting coach).

I’m gonna try and post a few times a week, including POD’s and other tidbits.

Four Classes:

That is what I have to contend with this quarter.  One class, Calculus II, I’m teaching; the other three: Stochastic Processes and Probability Theory; Graph Theory; and Partial Differential Equations; I’m taking.

I had hoped to be posting something weekly about all of my classes, but after getting started on this post last weekend and not taking it any further, I’ve decided to just make one math-related post a week.  Later today or tomorrow I am going to put up an explanation of l’Hospital’s rule, which gives us a way to calculate limits that would otherwise give rise to what are called ‘indeterminate forms’.  Sometime over this week I’ll throw in a note about where Bessel functions come into play, and next weekend I’ll catch up with some measure theory from the Stochastic Processes class.

ciao for now!

Felicis

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 , implies .  Now, let such that .  Further, suppose a is a zero divisor.  Then, there exists a such that , but . Then, .  But, since R has the cancellation property, then this implies that . 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 , and let , where .  Then, .  This implies that .  Since , then must equal 0, which implies .  So, R has the left cancellation property.

Similarly, if , and , then which implies , so .  So, R also has the right cancellation property.  Therefore, if R has no zero divisors, then it has the cancellation property.

Winter Quarter

And so it begins again.

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!