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Saturday, June 9, 2012

Ah... summer

For those of you in college, you can understand the mess which has taken over my desk.  My once organized, clean, and productive workstation is now a pile of various notebooks, folders, and textbooks.  Digging deeper you can see that even my notes from the beginning of the semester are thorough and easy-to-read, and as the dates near the end of the semester they are anything but thorough or easy-to-read.

The busy schedule of a college student leaves time for working, exercising, and sleeping, but at the cost of properly absorbing the course material.  When I say "properly absorbing" I mean that I have a recollection of the concepts and main ideas of the courses I've survived, but I certainly couldn't explain an elaborate proof without re-reading at least a chapter on the matter.  The last thing I want is to forget everything I have learned, and although I have a degree in mathematics and not a foreign language, the saying goes "If you don't use it, you'll lose it."

Although I'm excited to begin my graduate school experience, it is not August yet.  So, I will be giving myself a refresher course of abstract algebra.  I will actually have the time to read the book, and several times if necessary.  I will complete the problems at the end of each lesson with out the pressures of an unreasonable deadline.  Finally, I will have the time to share this review with you.

The book is Abstract Algebra, Third Edition, by I.N. Herstein.

Chapter One: Things Familiar and Less Familiar
     Section One: A Few Preliminary Remarks

There wasn't anything I took note of and only three problems in this section.  I'd say this was a nice section to get me thinking in the realm of abstract algebra.

Problems: 1-3

Sunday, May 20, 2012

Stewart's Hoard: Magic Circle



Arrange the numbers 1, 2, 3, 4, 5, and 6 in the small circles so that the sum of the numbers on each large circle is fourteen.

First, notice that each small circle is intersected by two large circles.  If I give each circle its own color, we have the following.




For each small circle we have three options,
Red/Green
Green/Blue
Blue/Red
and each of these appears twice in the diagram





In order to solve this problem I first began with C(6;4) to find all the four-number combinations of the numbers 1-6.  For those of you who don't know, C(6;4) = $\dfrac{6!}{4!(6-4)!} = 15$. 
Here are the fifteen combinations:

1234     1235     1236     1245     1246    
1256     1345     1346     1356     1456
2345     2346     2356     2456     3456

Now find those that have a sum of fourteen:

1234     1235     1236     1245     1246    
1256     1345     1346     1356     1456
2345     2346     2356     2456     3456

There are three combinations and three circles and now our task is complete.  As you can see, each of the combinations shares two digits with another combination, and the other two digits with the remaining combination.  This allows us to fulfill the intersections of the circles.
Source of Problem: Professor Stewart's Hoard of Mathematical Treasures by Ian Stewart



Saturday, April 21, 2012

Great News

Do you know that feeling?  When you push yourself to jump through hoops and it actually pays off?  I totally have that feeling.

For those of you who have read my past posts about the GRE, you know I've had my sights on Graduate School.  Let me tell you, the GRE is the first hoop (and the second if you are also required to take the subject test).  Then once you've decided which schools to apply to, the application process is just as brutal.  Each University wants a different version of a Statement of Purpose, and some want a separate essay concerning who you are as a person.

If you're looking for a better option than student loans to cover the high cost of Graduate School, especially if applying out-of-state, then you'll need to be aware of the extremely early deadlines for the applications.  If you are seeking a Fellowship or an Assistantship, it may not be enough to simply meet the early deadline; you may be required to submit a special form stating that you are seeking one or both of the above.  Considering schools in any state except Michigan, I opted for this route, and once each school received my application I felt relief, but I knew it still wasn't over.

Most schools require three letters of recommendation, usually from professors or instructors you've had the pleasure of working with during your undergraduate career.  Although I had previously asked three professors in the mathematics department, who all gladly agreed, to compose letters on my behalf, the applications I submitted were not complete until their letters were in the hands of those who would decide my fate.  I sent many e-mails, and even made some phone calls, checking in to ensure my applications would make the deadline.

Some were late and some on-time, but it was out of my hands.  I can say these last few months of waiting have been difficult.  I do not know the process my applications underwent, but they must have been read by many eyes.  Out of the five schools I applied to, one school replied with a, "Thanks, but no thanks," three schools said, "Congratulations!  We'd love to have you attend," and one school has yet to say anything.

Focusing on the three possibilities, this week the decision has been made.  If you knew me, then you'd know that I've been spending much of my free-time researching the three schools more closely.  Looking into the cities in which they reside, including the community, climate, and future opportunities.  Yet somehow none of these were the deciding factor.  In fact, all that time I spent researching was almost a waste.  You see, I didn't choose a school, a school chose me.

I was officially offered a Graduate Teaching Assistantship at Montana State University.  Who could pass up an offer like that?  I'm extremely happy I pushed myself and made it to this moment, knowing that it was all worth it.  Now it's just a matter of working out the details that come along with moving to a place I've never been.  I think I'll put that off for a few days, and simply bask in the glory.