Clemson REU – Day 1 and 2

Well, it has begun! I took about a 4 hour drive down to Clemson on Sunday and now I’m here, neck-deep in math. The campus is really pretty, although it is quite spread out. The building I’ve been walking to every day is 3/4 mile, and its not even half way across campus.

The people seem pretty friendly, although almost all are more experienced in mathematics than I am. However, I guess that is to be expected. I’m the only non math major, but I was surprised to find that I was able to find some people that I could still talk about computer science about. I did help my roommate install Sage (the math program we’re using) on her computer, so that was something useful I was able to do.

Since I am one of two people that have cars available, I’ve been doing a bit of chaffeuring. Of course everyone wants to go to Walmart to get groceries and other supplies.

The math has been pretty difficult so far, and this is just the beginning! I’m hoping that I get the hang of it before long. Lots of abstract algebra today, glad I brought my Dover book on Abstract Algebra along.

The daily schedule has been going like this:

• 9am-12pm : Lectures by the professors
• 12pm-1pm : Lunch
• 1pm – 5pm : Work on exercises/problems

After 5pm we usually leave and go eat dinner or something. I’ve taken people to Walmart both evenings so far. I still haven’t unpacked my room very much, having your brain explode with math sure can be tiring.

We’re going over proposed problems this week with each of the professors, and then starting next week and for the rest of the REU we will be choosing a group to work with (2 other people, a professor and ourselves)  and a problem to do research on. I figure its sort of okay that I don’t completely understand everything yet. I can save that for when I need to do the detailed research on one specific problem. I hope I get a good group. I don’t really know what problem I want to do yet, I haven’t understood any of them too well yet, but maybe tomorrow I will find one that I click with.

Stay tuned for some photos, that I will try to take in the next few days. Hello Clemson!

It’s about that time…

Well, I finished up my school exams and such today. I am a free woman! Until the 20th, that is. I’ll be heading to Clemson for this REU (Research Experience for Undergraduates):

http://www.math.clemson.edu/~kevja/REU/

I’m pretty excited about it, but at the same time intimidated. I feel like everyone else that is attending will be smarter than me, and I’m worried about trying to fit in and keep up with everyone. But I suppose these worries are natural, and it will all fall into place when I get down there. And I’ll be posting about my doings here for all to see. Having real research experience is going to be great!

Stay tuned for the mathy goodness…I gotta go learn Python, LaTeX, and all the maths. 🙂

Exam Time – and a bit of an educational rant

So it’s that time at my school…finals week. That’s why I haven’t been posting as much. Lots of studying, paper writing, project finishing, you know — all that good stuff that comes with being a student. But in the midst of all of this, there’s something that bothers me, deep down, about how the university model of education currently works.

You go to class, you sit down, you take notes. You take a test. You write a paper. You do a project. You give a presentation. You do homework. What’s in common with all of these? They’re all graded in some way. I understand that grades are important for evaluation of student progress, but sometimes it seems as if students are working FOR the grade, not necessarily to learn something new. School (and learning) should be enjoyable, and should foster a curiosity and love of learning that creates innovators and creators to shape our world. Unfortunately, most people see school as a chore, and this is unfortunate.

I’m a firm believer that independent projects and research that one is actually interested in is one of the best ways to learn. There should be more student direction rather than professor direction in our courses. Professor feedback rather than a numeric grade would be much more helpful. Of course, I know that’s not necessarily feasible in large classes, but still, I wish that school was more oriented toward learning rather than fulfilling a set of requirements. That’s what it is supposed to be for, right? Learning.

A recent example I remember is in my astronomy class a week or two ago. My professor was talking about relativity, and it was really fascinating stuff. I was enthralled. Someone from the back row (you know, the ones that usually pay half attention or sleep) raised their hand to ask, “Um, is this going to be on the test?” While I suppose that is a valid question, I wish everything didn’t have to be so test-oriented.

I know that there are classes that we all take that we’re less than interested in, and I feel like that’s another flaw in the system, another way to just “check off requirements” so we can graduate. But alas, such is the way things are. There’s not much one can do, I suppose, but fill their silly requirements and try our hardest to keep that curiosity alive, even when our “education” seems dull as hell.

Numbers, Reverses, and Sums Divisible by 11 (Part 2)

Update: As you can see in my previous post A Two Digit Number and Its Reverse Sums to a Multiple of 11, if you have a two digit number, you can

• use the reverse of that number to sum together to a multiple of 11.
• sum the digits to find a factor of the sum.

For example, if one chooses 43 + 34 = 77, one can see that 4+3=7 and 7*11 is 77.

Upon further investigation, I have found that one can use any number whose length is a power of two. This means that not only do 2 digit numbers work in this way, but so do 4 digit numbers, 8 digit numbers, 16 digit numbers, and so on…

However, when one gets to digits larger than 2 (4,8,16), the part about summing the digits to get a factor no longer works. Strangely, that only seems to work on 2 digits.

Fibonacci Squares with Arbitrary Starting Numbers

Consider the Fibonacci Sequence

0,1,1,2,3,5,8,13…

Each number in the sequence is the sum of the two that precede it. The sequence is defined as have 0 and 1 as its first numbers. But what if we chose different numbers to start with?

We’re going to make something I call a Fibonacci Square. It is a little bit of Fibonacci, a little bit of Pascal’s Triangle, and a little bit of my own creativity.

Let’s choose two arbitrary numbers to begin with. Let’s say, 6 and 2. We’ll call 6 f (the first number), and 2 s (the second number). We will now write our “Fibonacci” sequence as normal, but using f and s as our starting numbers instead. For the purpose of our “square” we are constructing, we’ll only write the first four terms for now.

6, 2, 8, 10

Okay, so we’ve got that. Now to continue the pattern we will start a new line below the one we’ve just written, and replace s (the previous second number) with the fourth number generated in our sequence above. This just means that our starting numbers (f and s) are now 6 and 10, respectively. Let’s write a new 4 term line with them.

6, 10, 16, 26

So our “square” so far is:

6,   2,   8, 10
6, 10, 16, 26

We can make more lines as much as we want with this technique. Just take the fourth number and replace it as the second starting number in the next line. The third line in this sequence would be:

6, 26, 32, 58

What is interesting to note is that a formula can be derived to determine what the fourth number on any given line will be. In fact, the formula can be applied to any length lines greater than 4. That means that if we put 5 terms on each line instead of 4, our formula would still work.

Let’s check out the formula:

$(F(t)^n * (s+\frac{F(t-1)}{F(t)-1}f))-\frac{F(t-1)}{F(t)-1}f$

• F(t) is the t’th term of the Fibonacci Sequence (defined as starting at 0)
• t = the number of terms on each line (in our example there are 4)
• f = the number we first start with
• s = the number that we originally had as our “second number”
• n = the line number, starting at one. So for the three lines we calculated, they would have been n = 1, 2, and 3, respectively.

So, for example, we wondered what the number would be on the end of the 20th line of the sequence we started above, but we don’t want to write all that out. We can plug the values into our equation to find the answer.

$(F(4)^{20} * (2+\frac{F(4-1)}{F(4)-1}6))-\frac{F(4-1)}{F(4)-1}6$

If we condense that down (remember that F(4) means find the 4th term of the Fibonacci Sequence, which is 2), we get that the value on the end of the twentieth line in this sequence is 8,388,602. Wow! As you can see, when one creates a sequence in this manner, there is exponential growth.

Anyway, this is one of my pet projects and I hope you will find it interesting. I will probably make another post on other things I have found with this sequence, but this post is getting a bit long for now.

Useful Linux/Programming Snippets – Wave 1

I’ve been keeping a list of things I’ve had to look up for school or work, programming wise, and I thought it might be helpful to list them here. It’s more of a reference than anything, and I know it helps me to look back on it if I forget something little, so maybe someone else will find it useful too.

to execute a premade .sql file
mysql -p -D databasename < file.sql

find/replace in Emacs (discriminately, that is: it will ask you at each instance if you want to replace)
M-x query-replace <ENTER>
oldstring <ENTER>
newstring <ENTER>

•     space – replace
•     n – skip
•     enter – quit

cat a bunch of files together into one
cat file1 file2 > allfiles
or
cat file* > allfiles

to get unix time in bash
date +%s

to get unix time in mySQL
select UNIX_TIMESTAMP(now());

rename a table in mySQL
RENAME TABLE oldname TO newname;

I’ll probably add more as I find them.

Visualizing the Unit Circle

So you’re in math class and you’ve got all these weird values staring at you from the unit circle such as $\frac{7\pi}{6}$ and you’re not really sure what all that means or how in the world you’ll remember what is where. Never fear! Here’s an easy method that only requires basic knowledge of fractions to understand.

Unit circle diagram from wikipedia

Now I know this all looks pretty scary and intimidating, but if you look at the angle measures in radians, and the fractions attached to the $\pi$ in each case, you’ll see it’s not so hard to figure out after all.

We know that

• 0 radians = 0 degrees
• $\frac{\pi}{2}$ radians = 90 degrees
• $\pi$ radians = 180 degrees
• $\frac{3\pi}{2}$ radians = 270 degrees
• $2\pi$ radians = 360 degrees

These will be our starting points. As long as you know where 0, 0.5, 1, 1.5, and 2 are on a number line you should be fine.

Take for an example some random radian measure like $\frac{5\pi}{3}$. How would you figure out where it was on the unit circle without looking at the diagram? Don’t let the $\pi$ scare you off. Let’s just look at the fraction separately.

$\frac{5}{3}$ is almost 2 (aka $\frac{6}{3}$), but not quite. However, it is larger than 1.5 (that would have been $\frac{4.5}{3}$). So from this, we can tell that $\frac{5\pi}{3}$ is in quadrant 4.

That wasn’t so bad, was it? As long as you know where the fraction lies between our 4 “base points”, its easy to figure out the general location of an angle on the unit circle.

Hope this helps someone, let me know if you have questions.