Dec 13

The World’s Greatest Limerick

Nonsense 12 Comments »

I am short on time with preparing to give and grade finals. (Aww, poor professor. . .) But, I’d thought I’d share with each of you my favorite limerick of all time:

\displaystyle \left(\int_1^{\sqrt[3]{3}} z^2 \; dz \right) \ \cos \left(\frac{3\pi}{9}\right) = \ln \sqrt[3]{e}

TRANSLATION:
The integral z-squared dz
From one to the cube root of 3
Times the cosine
Of three pi over 9
Is the log of the cube root of e.

Nice, eh?

written by SplineGuy

Dec 13

Spiritual Principle #2

Spiritual Principles No Comments »

Discretize Spaces

My last entry mentioned that the next post would focus on why I find approximation theory so interesting. I’ve been chewing on an idea / illustration for some time now that I want to run by my readers. Much of my research in has been spent in using finite dimensional function spaces to approximate much more general spaces of functions. In most cases, the goal is to find a linear combination of basic functions that will be “close” to some function from a larger, infinite dimensional space.

For example, a simple infinite dimensional space is the class of continuous functions over a closed interval. These are generally nice functions to work with but it is still a very “large” space. Given a continuous function over an interval [a,b], let’s say we want to approximate it with some quadratic function, p(x) = ax^2+bx+c. A quadratic function can be thought of as an element of a 3-dimensional function space with basis functions, p_1(x) = 1, p_2(x)=x, p_3(x)=x^2. In approximating a function, we are trying to find the coefficients of these basic functions so that we have a “close” function to the original function. There are many ways to measure this closeness, but the idea is that each basic function contributes additional information that helps the approximating function get closer to the original function. If we chose to use cubic polynomials instead of quadratics, we have one more basis function p_4(x)=x^3. There is a theorem, which I will probably post later as a favorite theorem, called the Weierstrass approximation theorem that implies that for any continuous function over a closed interval, there exists some polynomial that is arbitrarily close to that function. But, the caveat is that the more accurate you want the approximating function to be, the more basis functions you’ll need to contribute information. For polynomials, this translates to higher and higher degrees. As a side note, it turns out that there are much better functions for approximating than general polynomials, such as piecewise polynomials or splines. And don’t get me started on the beauty of splines.

Now, for the spiritual application. I see this as a perfect explanation for the necessity of community in our faith. Let God, himself, be the infinite space that we wish to understand (e.g., obtain an approximate knowledge of). Our minds, being finite, are limited and unable to fully comprehend the character and nature of God. Our experiences throughout our life help to inform our faith but these experiences are also finite and limited to a single individual. I see each person as a basis element in a finite dimensional space. Meaning that to better understand God, we can use other’s experiences and knowledge to contribute to our understanding of who God is. We listen and learn from each other and thus our approximate understanding of God becomes closer and closer to an accurate picture of God. Obviously there exists the possibility that individuals can take away from our understanding thus we must listen and learn with discernment but, in general, we are better able to know God as part of his body, the community of faith.

It was Henry Blackaby, in Experiencing God, who described the four fold way to know the will of God, through the Bible, the Holy Spirit, Circumstances and other believers. Each, in a limited way, serves to better inform us and help us gain and more accurate, but still approximate knowledge of God.

The nice thing about numerical analysis, my chosen field, is that we are able to specifically quantify the error. We can know in some sense a bound on our error, and know how far off we could possibly be in our approximation. Perhaps, the analogy fails here but I’d like to believe that necessarily, the more elements we use to develop our knowledge of the Holy, the more accurate our image of our Father in Heaven.

What do you think?

written by SplineGuy