Quirky Quintet

Quadrature rules on the interval is pretty well-documented in many different textbooks (see for example Cheney and Kincaid). By transformations, we can easily integrate across the whole real line spectrum. Going into two dimensions, we can take tensor products to integrate rectangles easily. On the other hand, integration on the 2D simplex (i.e. triangles) is a bit harder. There really exists two types of quadrature rules for the triangle:

1. Tensor products + transformations
2. “Symmetric quadrature rules”

The latter offers better convergence with smaller points, and has better spread-out points than the tensor product version, but offers less technical support through software packages; we refer the reader to the following paper. We will discuss the first, and exposit a little more; I will be borrowing heavily from Karniadakis and Sherwin’s fluid dynamics textbook.

Suppose our triangle at (-1, -1), (1, -1), and (-1, 1) (so called reference triangle) uses as it’s coordinates. If we use the change of coordinates , it’ll map the triangle to the square defined by (-1, -1), (1, -1), (-1, 1) and (1,1). Hence, we can express the integral using a change of variables (details omitted)

.

We note that the weird factor is simply the change of variables Jacobian. Now it’s easy to use Gaussian quadrature on this! But there are better version, as we have the Jacobian term; looking up the quadrature rules, we see that Gauss-Jacobi fits our bill. We refer the reader to page 144 of the Sherwin textbook to see the full-details.

A simple implementation is below (this is far from the best). I guess I’ll talk about this at the end of the week.

from scipy.special import roots_jacobi as rj
def quadtriangle(f, w00, w10, x00, x10):
    total = 0
    for i in range(len(w00)):
        subsum = 0
        for j in range(len(w10)):
            subsum += w10[j] / 2 * f([(1 + x00[i]) * (1 - x10[j]) / 2 - 1, x10[j]])
        total += w00[i] * subsum
    return total
def f(x):
    return x[0]**2 + x[1]**2
p = 4
x00, w00 = rj(p + 1, 0, 0)
x10, w10 = rj(p + 1, 1, 0)
print(quadtriangle(f, w00, w10, x00, x10))

Before you read on, watch the following clip from the Hateful Eight:

Of course, there are no more hangmen left in the civilized world; they have been outsourced by giant pharma companies or the electric chair. But crimes of passion carried out under the banner of revenge will always live on. How true is the quote “justice delivered without dispassion is always in danger of not being justice”?

I bring this up because it’s been a few weeks since I found out that my landlord (hereby denoted as CB) was murdered recently. It was definitely a crime of passion from the context, which requires us to go back in time. In 1999, a police report stated that neighbors accused CB of being a pedophile but no charges were filed.

Later, CB lost his chiropractor license due to having sex with a patient in 2003. His license was reinstated sometimes after; we know this because his license was once again revoked in Sept. 2017 by a complaint from a patient. Four months later, CB was found dead in his home.

The alleged killer of CB is a 21 year old college student who filed the complaint that led to CB losing his license. The charges were “violating the professional boundaries of the chiropractic physician-patient relationship.” Whatever that means, one can only guess, but with the recent case against Nassar, I find myself unable to think of other alternatives which could lead to a stabbing. What other infractions could lead to such a visceral response of targeted stabbing? Was our alleged killer ignored when he cried out for help?

CB was not a good man by all accounts it seems. His former neighbor is quoted to have said ‘it didn’t come as a surprise to him.’ I don’t know who this neighbor is, but finding out someone is murdered should be a huge fucking surprise… unless the neighbor believed that CB is a terrible person.

In the end, is this justice for the alleged killer? He will have to face trial for murder, and possibly end up in prison for the rest of his life. If he lived an average life span, that’s over 60 years in jail for a crime committed against a varmint. And CB does not deserved to die such a painful death also; no matter how heinous his past deeds, CB should have walked through the gates of a court. It seems justice was left at the wayside.

In the end, I feel this should indicate that the system is tarnished, and justice in this case will never be reached. CB should have never had his license again after the first complaint. The alleged murderer should have had avenues of help that could have dealt with this. The fact that vigilante justice was involved at all is upsetting, but one can hope the recent activism will legitimately change our system in the future.