Author: samjshah

Generating Fibonacci: Part I

Yes, this is a YAFPOTW (Yet Another Fibonacci Post On The Web).

People have talked about and analyzed the Fibonacci sequence to death. So I’m clearly not going to be doing something new and novel for everyone. But I remember one stunning way to look at Fibonacci is through Generating Functions. You only really need precalculus to understand it, as long as you allow for infinite degree polynomials. And the best part is: you can apply this method to any simple recurrence relation.

To refresh your memory, the Fibonacci sequence is: 1,1,2,3,5,8,13,…, where the nth Fibonacci number is the sum of the two preceeding Fibonacci numbers:

$F_n=F_{n-1}+F_{n-2}$ .

That recursive formula, with the initial conditions ( $F_0=1$ and $F_1=1$ ), defines the entire sequence.

Let’s create a polynomial function from these Fibonacci numbers:

$g(x)=F_0+F_1x+F_2x^2+...+F_{n-2}x^{n-2}+F_{n-1}x^{n-1}+...$

We call this function $g(x)$ the generating function of the sequence $F_0, F_1, F_2,...$ .

So what if we add $g(x)+xg(x)$ ? First we find $xg(x)$ :

$xg(x)=F_0x+F_1x^2+F_2x^3+...+F_{n-3}x^{n-2}+F_{n-2}x^{n-1}+...$

and we add, term by term, to get:

$g(x)+xg(x)=F_0+(F_0+F_1)x+(F_1+F_2)x^2+...+(F_{n-3}+F_{n-2})x^{n-2}+...$

Excellent! Do you see where this is going? HUZZAH We know those coefficients! They use the recursion formula that defines the Fibonacci numbers! So we simplify:

$g(x)+xg(x)=F_0+F_2x+F_3x^2+...+F_{n-1}x^{n-2}+...$

And we know $F_0=1=F_1$ , we replace that $F_0$ above to get:

$g(x)+xg(x)=F_1+F_2x+F_3x^2+...+F_{n-1}x^{n-2}+...$

Notice the right side of the equation is equal to:

$\frac{1}{x}[-F_0+F_0+F_1x+F_2x^2+F_3x^3+...+F_nx^n+...]$

which immediately shows us a $g(x)$ ! So we can rewrite that to as the much simpler:

$g(x)+xg(x)=\frac{1}{x}[-F_0+g(x)]$

And we’re almost there! Replacing $F_0$ with it’s value of 1, multiplying both sides by $x$ , and rearranging, yields: $(x^2+x-1)g(x)=-1$ . So we get:

$g(x)=\frac{1}{1-x-x^2}$

This is our generating function! The coefficients of the polynomial expansion will give you the Fibonacci sequence! But this method of generating functions will prove a nice and general way to find explicit solutions for any basic recursive relation. (What if we had the Lucas Numbers, for example? Or some recurrence like $M_n=2M_{n-1}+M_{n-3}$ ?)

In a later post, I’ll write about how to get an explicit formula for the nth Fibonacci number from this. In case you need a refresher or never knew:

$F_n=\frac{1}{\sqrt{5}}((\frac{1+\sqrt{5}}{2})^n-(\frac{1-\sqrt{5}}{2})^n)$

Where clearly we see some popping up of the golden ratio and its “opposite”! And in our generating function, we see an $1-x-x^2$ which has zeros which are the golden ratio and its “opposite.”

My (Incorrect) Monday Math Madness Solution

This week’s Monday Math Madness was a bit of a brain strainer. I didn’t get the ideal solution, but hey, whatever. I still enjoyed working on it. The solution that I proposed is after the fold.

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I grow old, I grow old

I got an emergency email from a good friend from college (now in grad school at MIT) who is going to soon be meeting with a professor to talk about Number Theory (honestly, I have no idea why she needs to meet him; she does material science!).

She needs a crash course in it, and wanted a recommendation for a textbook. Of course, I go digging around online to find out what book I used, to no avail. I distinctly remember the cover, but I can’t find that on Amazon either. So I finally go to my last resort… digging through my 30 or so binders (from high school (!) and college) looking for the syllabus from that class.

I find the binder, but the syllabus doesn’t have the name of the textbook on it, even though it has 3 pages of assigned problems from the textbook.

My friend is, sadly, out of luck.

But here’s the horrible, horrifying part. Once I started looking through the old binder, I felt dumb. Like really, really dumb. For a number of reasons which I will enumerate here:

1. My three test scores (I don’t know what I got on the final) were 28/40, 29.5/40, and 32/40. [1]

2. One of the tests has a depressing, terse note from the professor “1 [point] for effort”

3. My class notes are completely beautiful but, for me today, totally incomprehensible. I mean, I don’t even remember most of the basic terms that I’m writing about.

Hensel’s Lemma: Let $f(x) \in Z[x]$ . $f(a) \equiv 0 \mod p^j$ , and $f'(a) \not\equiv 0 \mod p$ . Then there exists a unique $t \mod p$ such that $f(a+tp^j) \equiv 0 \mod p^{j+1}$ .

Did I ever understand that? Yeah, I know. If I had a month and the textbook, I would be able to figure out what that meant again. Which is (very) slightly heartening. But how sad it is that all my hard work in college is so fleeting? I wish it weren’t all so temporary.

PS. The subject line is from this poem.

[1] Okay, here I’m being a bit disingenuous. I remember that for at least one of them I scored the highest grade in the class. There were only 10-15 students in the class, and the teacher used to write up on the chalkboard the highest score, the lowest score, the mean, the median, the mode, and the standard deviation. Yikes. Those were scary moments, after the professor wrote that on the board, but before he returned the exams. But somehow, after teaching high school, scores in this form look bad. Even though they were probably fine. (I ended up with an A in the class.)

At the last Math Club meeting…

At the last meeting of Math Club, one of the members presented the solution to a problem he posed the week before.

There is a prison with 100 prisoners. They are all taken into a room and the warden says:

“In the room next to this one, there are 100 boxes in a row. Each of these boxes contains a slip of paper in it, and on these 100 slips of paper are your names. You may strategize as much as you’d like now, but once you have a strategy, you are each going to be sequestered in individual cells. One by one, you will be led to the room with the boxes, and you will get to open only 50 boxes in an attempt to find your name. Then you leave all room, with all the boxes exactly as you found them. You cannot change anything; you cannot take a slip from the boxes, not even your own. Then go you back to being sequestered, and the next prisoner will be led to the room. At the end, after all 100 of you have had your chance, you must tell me which box has your name in it. If all of you get your box right, then you all can leave the prison. If any of you get your box wrong, none of you can leave.”

What is the best strategy for the prisoners?

The dumb “every prisoner randomly picks 50 boxes” method will give them a chance of $\frac{1}{2^{100}}$ . You can beat that. In fact, the prisoners can come up with a strategy that will give them a pretty good chance of leaving. Not 100% (obviously), but pretty good.

And in fact, I had no idea where to even begin. But it turns out that another mathclubber came up with the right strategy. Which is awesome!

UPDATE: If you want the solution, it’s well explained on this blog post. Saves me the time of typing out the answer!

To feel good, only to be dashed cruelly to the ground, and break in a million little pieces

The beautiful New York weather and three day weekend inspired to go out on this Saturday night. So I shaved, dollied myself up, donned my newly purchased tote bag, and left the apartment.

I felt pretty good about everything as I was having dinner with a friend, followed by a postprandial drink at a local coffeeshop. When I left, the barista on his cigarette break asked me if I was a designer. I, feeling good about how I put together my outfit, glowed, even though I played it cool. [1]

We made our way down three blocks to go to a concert at a local bar. The next person to speak to me was the bouncer, who looked me up and down. His first pronouncement arrived in question form: “Did you just get out of bed?”

[1] When I was in Paris, my friend Tom and I had our pictures taken by an aspiring fashion student for her work on “street fashion.” I felt good then too.

We Will Create A Black Hole Which Will Swallow The Earth

The NYT has recently run two articles (one, two) on the fear that exists (among some) that the Large Hadron Collider — when started — will destroy the earth through the creation of a black hole. Interestingly, this was exactly the topic my undergraduate thesis (in STS) was on, except instead of being about the LHC, it was about the particle accelerator being built at Brookhaven National Laboratories.

But when you delve into the history of it it, it isn’t all that surprising. The guy who is making the black-hole-possibility claim about the LHC is exactly the same guy who made the black-hole-possibility claim at Brookhaven: Walter Wagner. The newspaper stories take the same tenor now that they did back then too.

Although my thesis isn’t awesome (I’ve read some of my friends’ undergraduate theses and I know mine doesn’t stack up) nor well written, I can say it’s timely, so I’m uploading this PDF version of it. I was just trying to find a choice paragraph to lob off and stick up here, but I figured: hey, why not? Unfortunately, in this particular electronic version, some of the websites cited in the footnotes have disappeared (“ERROR!”), and this is the only electronic copy I have. (One of many consequences of the many Great Computer Crashes which have befallen me.)

As an added enticement for those on the fence, I spend a whole “chapter” talking about the fear that the atmosphere would ignite when the atomic bomb was set off for the first time.

Calculus and Chaos Theory

On the same blog, God Plays Dice, I learned two important things today:

(1) The Napkin Ring Problem, which basically says if you take a sphere, and cut out a cylinder from the center (and take off the two caps at the top and the bottom of the sphere), the volume of the remaining solid will not depend on the radius of the sphere at all!

In totally unrelated news, today we wrapped up our initial foray into finding volumes by slicing, by revolution, and by washers in calculus. I wonder what I’ll be doing with them tomorrow. I dearly wish I had an interesting problem for them to work through. Sigh.

(2) Edward Lorenz is dead.

All those years ago (okay, not so long ago) I was at MIT getting my first taste of chaos theory. I took this course with Daniel Rothman, who is an amazing teacher, and who taught from an amazing textbook (Strogatz). I loved the class so much that when I was asked if I was interesting in grading the problem sets for the following year’s class, I jumped at the chance.

It was in that course that I was first introduced to Edward Lorenz and his paper on weather (!) which launched a whole phenomena: chaos. But I also got so into the subject that I tried to fix a broken chaotic waterwheel that was being left unused in the basement of MIT. (I was unsuccessful, though I still wrote my final paper for a class on a modification of the Lorenz equations.) And when I was in graduate school, I was asked to write a “history of chaos theory” article for an encyclopedia that never really got published (to my knowledge). Also when I was in graduate school, I traveled to Beijing for a summer school (4 weeks) on complexity and chaos, thinking that I might want to do my dissertation around the topic of complexity and chaos in the sciences. And now that I’m teaching in high school, I wanted to introduce some of the budding mathematicians to the topic, so in math club, I gave a 3 part mini-lecture on chaos and the logistic equation.

Needless to say, chaos = cool. When I took that initial class with Prof. Rothman, Lorenz came to give a guest lecture. I’m sad that he’s gone. He is partially responsible for a really interesting part of my life that keeps recurring, no matter where I am or what I’m doing.

If you want to read my encyclopedia entry on the history of chaos theory (which is okay, but I’m sure has a number of problems with it), I’ll post it below. It talks about Lorenz’s contribution, if you don’t know.