February 2018 – Spinor Info

What makes a differential operator linear?

Feb 272018

Enough of politics and cats. Time to blog about math and physics again.

Back in my high school days, when I was becoming familiar with calculus and differential equations (yes, I was a math geek) something troubled me. Why were certain expressions called “linear” when they obviously weren’t?

I mean, an expression like \(Ax+B\) is obviously linear. But who in his right mind would call something like \(x^3y + 3e^xy+5\) “linear”? Yet when it comes to differential equations, they’d tell you that \(x^3y+3e^xy+5-y^{\prime\prime}=0\) is “obviously” a second-order, linear ordinary differential equation (ODE). What gives? And why is, say, \(xy^3+3e^xy-y^{\prime\prime}=0\) not considered linear?

The answer is quite simple, actually, but for some reason when I was 14 or so, it took a very long time for me to understand.

Here is the recipe. Take an equation like \(x^3y+3e^xy+5-y^{\prime\prime}=0\). Throw away the inhomogeneous bit, leaving the \(x^3y+3e^xy-y^{\prime\prime}=0\) part. Apart from the fact that it is solved (obviously) by \(y=0\), there is another thing that you can discern immediately. If \(y_1\) and \(y_2\) are both solutions, then so is their linear combination \(\alpha y_1+\beta y_2\) (with \(\alpha\) and \(\beta\) constants), which you can see by simple substitution, as it yields \(\alpha(x^3y_1+3e^xy_1-y_1^{\prime\prime}) + \beta(x^3y_2+3e^xy_2-y_2^{\prime\prime})\) for the left-hand side, with both terms obviously zero if \(y_1\) and \(y_2\) are indeed solutions.

So never mind that it contains higher derivatives. Never mind that it contains powers, even transcendental functions of the independent variable \(x\). What matters is that the expression is linear in the dependent variable. As such, the linear combination of any two solutions of the homogeneous equation is also a solution.

Better yet, when it comes to the solutions of inhomogeneous equations, adding a solution of the homogeneous equation to any one of them yields another solution of the inhomogeneous equation.

Notably in physics, the Schrödinger equation of quantum mechanics is an example of a homogeneous and linear differential equation. This becomes a fundamental aspect of quantum physics: given two solutions (representing two distinct physical states) their linear combination is also a solution, representing another possible physical state.

Canada’s jury system

Canada, Politics No Responses »

Feb 202018

In an opinion piece, Globe and Mail contributor, lawyer Charles Lugosi argues that calls to reform Canada’s jury system in light of the not guilty verdict in the case of the death of Colten Boushie, who was shot by Gerald Stanley, are unwarranted and deeply harmful.

Mr. Lugosi speaks from personal experience. In 1994, he defended a young indigenous man who, fearing for the safety of his wife and child, struck and killed a white man with a tire iron and then turned himself in to police. Judged by an all-white jury, the defendant was acquitted. Despite the outrage of the victim’s family, no one suggested racism after the verdict.

So why the calls for reforming the jury system now? Lugosi argues that it is simply political intimidation. Having looked at news reports providing details of the case, I am compelled to agree. Lawyers back before the verdict warned that the Crown’s case was tenuous and that a not guilty verdict was very much a possibility. Portrayals of the victim and his friends as harmless teenagers looking for help with a flat tire just do not agree with the facts. In reality, this was a drunk, disorderly, armed (!) lot who already ransacked another farm and were attempting to steal a vehicle from Stanley.

So let me be clear. Mr. Colten Boushie, the victim, was “seen as a menace” not on account of his race but on account of his actions. I have no reason to doubt that the jury reached their conclusions on this basis. For populist politicians to then criticize this decision, even as the jury are legally barred from discussing the case or defending themselves against charges of racism, is unconscionable.

I am sure there are plenty of genuine cases of bona fide racism against indigenous people in Canada. In fact, the behavior of the RCMP that night appears to have been far from impeccable. It may very well be that, as some accuse, the justice system is stacked against indigenous people.

And yes, Mr. Boushie is a victim. A victim of his own and his friend’s actions. A victim of circumstances. I’d go so far as to concede that yes, very probably a victim of a system stacked against him, a victim of still existing systematic racism in Canadian society.

But this jury verdict is not outrageous. It is consistent the facts: very simply, the guilt of the defendant was not proven beyond reasonable doubt. That’s all. It does not mean Mr. Stanley is a nice guy. It does not mean Mr. Boushie deserved death. Calls for a reform of the jury system just because a verdict is politically unpalatable undermines the justice system for all. And fueling tensions with unwarranted, politically motivated charges of racism does a disservice to indigenous and white Canadians alike.

Social safety net in ancient China?

History, Politics No Responses »

Feb 192018

The more I learn about ancient China, the more my respect grows for the Middle Kingdom and its amazing history.

Here is something written by a 9th century Muslim traveler, Abu Zayd al-Sirafi, who visited China during the Tang Dynasty, in 850 A.D. or thereabouts. I found a modern translation that I first came across so incredible, I searched for corroboration. I then came across this 1733 English translation (if I interpret the cover page correctly, by an unnamed English translator using a French translation from the Arabic by Eusebius Renaudot).

Here is how this 1733 text describes China’s social safety net. (And if you are left wondering if perhaps the traveler mistakenly traveled into the future, visiting a late 20th, early 21st century welfare state, you are not alone):

“The Chinese have a Stone ten Cubits high, erected in the public Squares, and on this Stone are graved the Names of all sorts of Medicines, with the exact rates of each; and when the poor stand in need of any Relief from Physic, they go to the Treasury, where they receive the Price each Medicine is rated at.

“There is no Land Tax in China; they only levy so much per Head, according to the Wealth and Possessions of the Subject. When a Male Child is born, his Name is immediately entered into the King’s Books, and when this Child has attained his eighteenth Year, he begins to pay for his Head; but they demand it not of the Man who has seen his eightieth Year; on the contrary he receives a Gratification, by way of Pension, from the public Treasury; and in doing this, the Chinese say, That they give him this Gratification in his old days, in acknowledgment for what they receiv’d of him when he was young.

“There are Schools in every Town for teaching the Poor and their Children to write and read, and the Masters are paid at the public Charge.”

This text was written nearly 1,200 years ago. It took another millennium before public education become the norm in more developed European nations, and at least another century before various forms of social security and public health institutions took root.

I am beginning to appreciate more and more why the Middle Kingdom viewed the period loosely demarcated by the Opium Wars of the 1850s and the Japanese occupation of the 1930s and 1940s as the Century of Humiliation; and why it might appear to many that the recent rise of China as an economic superpower only means a return to the way things were always supposed to be, the way things have been for several millennia preceding the rapid rise of Europe a few centuries ago.

Pipacs kitty

Cats, Personal No Responses »

Feb 152018

Our smallest cat, Pipacs, continues to do well.

Four weeks ago, we thought we were on a death watch: The kitty was severely ill, no longer able to poop as his intestines were blocked by an invasive bone growth. Laxatives, even several enemas were of no use.

We were convinced that the only alternative to a slow, painful death was euthanasia.

But then, he suddenly pooped. (What a relief. Especially for him.) Since then, we’ve been able to manage him with medication.

I don’t know how long he will still be with us, but for now, he is doing okay. The bone growth distorting his pelvis is quite large now, making it all but impossible for him to use one of his hind legs, but it doesn’t appear to bother him too much, except when he is trying to use that leg to scratch an itch. Just watching it is frustrating! (How do multiple amputees scratch themselves when they have to?)

But he is eating well, he is pooping well, and when the other cats aren’t chasing him around, sometimes he chases them instead.