Wednesday, March 2, 2016

A Brief Glance Backwards

The past year of my life has been unpredictable, to say the least. After graduating from the University of Washington, I moved to upstate New York to start a 5-year PhD program in Mathematics at Rensselaer Polytechnic Institute. Yes, five MORE years of school to go along with the ones I already have. I've been almost a full year in this rigorous program, and I can say with certainty that this is the most rewarding thing I've ever done! I cannot emphasize enough how challenging it has been, how many times I've felt overwhelmed, and how much I've had to think in fundamentally new ways just to keep up. But it is all the more fulfilling, and at the end of the day, I sleep knowing that I am exactly where I belong.

Thursday, September 3, 2015

Calculus of Variations

Suppose we want to find the shortest distance between two points in the Euclidean plane. Common sense tells us that the shortest distance is a straight line, and we have the Pythagorean Theorem that tells us the length of that distance, namely
However, we can think about this a bit differently. In Calculus, we learn that the arclength of a curve u is
Let us suppose that we only know the end points, but not the curve itself. And now we pose the question: "What is the shortest distance between those two points?" Forgetting about the Pythagorean Theorem, we can actually go about finding the function u that minimizes the arclength using Calculus of Variations. With this method, we will show that u is a straight line.

Figure 1
Figure 1 shows our problem setup. We have our end points (a,u(a)) and (b,u(b)). We draw u(x) as a curve that connects these two points. Now suppose we have another function n(x) that is continuously differentiable and zero at the boundaries. Lastly, we have our arclength functional
Suppose that u (whatever it may be) is a function that minimizes the arclength. Consider the quantity
where e is a parameter. It is obvious that for any function n
When is L at its minimum? Since u is the minimizing curve, L is at its minimum when e is zero. Now we do some calculus :)

Calculus tells us that at a minimum, the derivative of the function is zero. So let's differentiate.
The function is sufficiently smooth, so we can swap the integral and derivative operators to get
At this point, we recall that the function is minimized when e = 0. We also know that the minimum occurs when the derivative is zero. Considering these two facts, we get
Integration by parts give us
We can easily evaluate the first integral by the fundamental theorem of calculus. We get
but because n is zero at the end points, the entire expression is zero. Now we focus on the second integral

Let's rewrite it differently
A basic lemma in Calculus of Variations tells us that if n and G are continuously differentiable functions and if n is zero at the boundaries, then the only way for this integral to be zero is if G is identically equal to zero. We sketch out a proof below, but for now, let's use this fact
But this is just an ordinary differential equation. We can solve this by direct integration.
And so we see that the function that minimizes L is a straight line.

Figure 2
So why did we use this round-about way of finding u? Well, for one thing, Calculus is very cool and we should try to use it in our mathematics as much as we can. But the importance of this method becomes more clear when we look at a higher dimensional case. Consider Figure 2. Suppose we wish to find the surface S that minimizes the surface area A(S). We can find such a surface by slightly adapting our method above.

Proof of the basic lemma.
Consider continuously differentiable functions G(x) and n(x), and let n be zero at the boundaries: n(a) = n(b) = 0. Suppose that G(x) is not identically zero in this region. For the sake of argument, suppose that it is positive at some point x'. Because it is continuous, there is a neighborhood around x' that is also positive, say x1 < x < x2. Now consider the integral
It may be that the first and last integrals are zero, for G may be zero in those regions. However, it cannot be that the middle integral is zero for every possible function n unless G is identically zero. Thus, it must be that G is identically zero in all of [a,b].

Monday, August 31, 2015

Vector Calculus

I spent most of last week reviewing for my preliminary exam on Saturday. I realize that in taking so many math courses, I had forgotten the simple joys of Vector Calculus. Vector Calculus is all about finding the little "tricks" that make integrating so much easier. Let's go over some of these.

Fundamental Theorem of Line Integrals

First, we have the Fundamental Theorem of Line Integrals. Performing a line integral is like projecting a vector field onto a curve and summing up those projections over the length of the curve. This is what we mean when we write

Sunday, May 17, 2015

Why Homosexuality Is Not Like Other Sins

Excerpt from a very well written article:
God tells us we’re wrong, that the wages of sin is death, that unrepentant rebellion means judgment, that our rescue required the cursed death of his Son (Romans 3:23; John 3:36; Galatians 3:13). And God tells us we’re loved, that even while we were sinners, Jesus died for us, that while we were unrighteous, Jesus suffered in our place, that though we were destined for wrath, Jesus welcomes us into glory (Romans 5:8; 1 Peter 3:18; Ephesians 2:1–7).
Read the full article here.

Friday, April 24, 2015

Network Models in Epidemiology - Considering Discrete and Continuous Dynamics

In the Fall, I developed a computational model for epidemiology. I modeled infection and recovery using a discrete dynamical system and considered a lattice network for the population. Since then, I have also considered infection for random network populations, and extracted some (hopefully) meaningful information. In December, I wrote that I would be producing a paper soon. Soon seemed to stretch on and on as I did not realize how big of a project this would turn into.

Instead of attempting to produce one huge report, I have decided to split Network Models in Epidemiology into series of smaller articles. And four months after promising a paper, I finally have one! Considering Discrete and Continuous Dynamics serves as an introduction to the project. In it, I outline the ideas and merits of Deterministic Compartmental Models. Considering the flaws in this class of models takes us into the realm of network models. I discuss several benefits and drawbacks when building a network and present the four fundamental types of networks. Finally, I elaborate on the differences in infection and recovery dynamics. It is more natural to think of interactions as discrete events, whereas recovery is naturally continuous. I discuss how to combine the two different dynamics and present the integrator that will be used for simulations.

As I stated above, I am currently working further than the extent of this paper. I have a solid grasp on the lattice network and recently made a breakthrough in the random model. Hopefully, I will have another paper soon. But we all know that soon can mean anything from next week (yea right) to four months (if we could be so lucky). Thanks for your supports, and I hope you enjoy Considering Discrete and Continuous Dynamics. As always, let me know if you have any questions or suggestions.

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