Stephen Hawking died earlier today.

Hawking was diagnosed with ALS in the year I was born, in 1963.

Defying his doctor’s predictions, he refused to die after a few years. Instead, he carried on for another astonishing 55 years, living a full life.

Public perception notwithstanding, he might not have been the greatest living physicist, but he was certainly a great physicist. The fact that he was able to accomplish so much despite his debilitating illness made him an extraordinary human being, a true inspiration.

Here is a short segment, courtesy of CTV Kitchener, filmed earlier today at the Perimeter Institute. My friend and colleague John Moffat, who met Hawking many times, is among those being interviewed:

Very well, having grown up in a police state of sorts myself, I know I should not take these things lightly. There are plenty of political prisoners in China, and now that Xi declared himself president for life, I expect things to get worse, not better.

Still… I could not help myself, I was laughing out loud as I read the story of an elderly Chinese woman who took revenge of a fortune teller because contrary to his predictions, she did live to see 2018 after all.

According to the South China Morning Post, Mrs. Wang ran into the fortune teller in a park and in her anger, she started vandalizing his stall. But then, “[a] police officer intervened and asked the fortune-teller to apologise to Wang for causing mental suffering.”

Thank you, Mr. anonymous Chinese police officer, for doing exactly what a sensible person should have done. Good to know that common sense is still alive and well in some parts of the world.

There is a very interesting concept in the works at NASA, to which I had a chance to contribute a bit: the Solar Gravitational Telescope.

The idea, explained in this brand new NASA video, is to use the bending of light by the Sun to form an image of distant objects.

The resolving power of such a telescope would be phenomenal. In principle, it is possible to use it to form a megapixel-resolution image of an exoplanet as far as 100 light years from the Earth.

The technical difficulties are, however, challenging. For starters, a probe would need to be placed at least 550 astronomical units (about four times the distance to Voyager 1) from the Sun, precisely located to be on the opposite side of the Sun relative to the exoplanet. The probe would then have to mimic the combined motion of our Sun (dragged about by the gravitational pull of planets in the solar system) and the exoplanet (orbiting its own sun). Light from the Sun will need to be carefully blocked to ensure that we capture light from the exoplanet with as little noise as possible. And each time the probe takes a picture of the ring of light (the Einstein ring) around the Sun, it will be the combined light of many adjacent pixels on the exoplanet. The probe will have traverse a region that is roughly a kilometer across, taking pictures one pixel at a time, which will need to be deconvoluted. The fact that the exoplanet itself is not constant in appearance (it will go through phases of illumination, it may have changing cloud cover, perhaps even changes in vegetation) further complicates matters. Still… it can be done, and it can be accomplished using technology we already have.

By its very nature, it would be a very long duration mission. If such a probe was launched today, it would take 25-30 years for it to reach the place where light rays passing on both sides of the Sun first meet and thus the focal line begins. It will probably take another few years to collect enough data for successful deconvolution and image reconstruction. Where will I be 30-35 years from now? An old man (or a dead man). And of course no probe will be launched today; even under optimal circumstances, I’d say we’re at least a decade away from launch. In other words, I have no chance of seeing that high-resolution exoplanet image unless I live to see (at least) my 100th birthday.

Still, it is fun to dream, and fun to participate in such things. Though now I better pay attention to other things as well, including things that, well, help my bank account, because this sure as heck doesn’t.

Saturday evening, the President of the United States, Donald J. Trump talked to some Republicans behind closed doors. But a recording has been leaked.

Here is what Mr. Trump said, among other things, in response to Xi Jinping turning China back to strongman rule by removing term limits from China’s constitution: “He’s now president for life. President for life. No, he’s great,” Trump said. “And look, he was able to do that. I think it’s great. Maybe we’ll have to give that a shot some day.

Having read, and then heard, these words, I was filled with a combination of dread, disgust, despair and revulsion.

Fuck you, Trump. Fuck you, Republican cheerleaders who sacrifice everything sacred for party loyalty and for power.

Forgive my vulgar outburst but this crossed a line. This REALLY crossed a line. I grew up in a communist state, not on a freaking golf course courtesy of rich daddy, like this megalomaniac asshole who has no idea what he is idiotically blabbering about, cheering on a commie dictator because apparently he, too, would love to be president for life.

I think that with this statement, Trump abdicated the informal position of “leader of the free world”. A true leader of the free world would never cheer on a commie dictator, not even as a tasteless joke. He may be America’s President, but with these words, he has declared himself an enemy of the world that I call free.

To my American friends who believe Trump is doing a good job protecting the US economy: I think it is telling that US stock markets are hurting more today than their Canadian counterparts, despite the fact that Canada is the primary target of Trump’s proposed steel and aluminum tariffs:

What can I say. Be careful what you wish for. And we have not even discussed how Canada (or Europe) might retaliate. But in the end, we will all be losers, no matter what the narcissistic idiot in the White House thinks about the wonderful job that he is doing.

I am playing with JavaScript and HTML5. Three-dimensional transformations.

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CNN
Cable News Network

It's more fun than I thought. And gives a whole new meaning to the word, "spin", as I am listening to CNN.

No, it isn’t Friday yet.

But it seems that someone at CTV Morning Live wishes it was. Why else would they have told us that yesterday, February 28, was a Thursday? (Either that or they are time travelers from 2019.)

Then again, maybe I should focus on what they are actually saying, not on a trivial mistake they made: that even as parts of Europe that rarely see snow are blanketed by the white stuff, places in Canada and Siberia see unprecedented mild weather. A fluke or further evidence of climate change disrupting the polar vortex?

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.

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.

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.

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.

Here is a beautiful picture of the contents of one of our litter boxes:

What makes it beautiful, you ask? Why, it is a thermal infrared image. And it shows some unambiguously fresh poop. Fresh poop produced just a few minutes prior by our kitty Pipacs, whom we were about to write off two weeks ago, as we were certain that his digestive system shut down, him being unable to poop as his bowels are obstructed by an invasive bone growth.

But Pipacs still has a few of his nine little lives left. I don’t know how long he’ll stay with us; we take it one day at a time. But with the medication he receives (a combination of laxatives and stool softeners) he is coping for the time being.

I was surprised by the number of people who found my little exercise about kinetic energy interesting.

However, I was disappointed by the fact that only one person (an astrophysicist by trade) got it right.

It really isn’t a very difficult problem! You just have to remember that in addition to energy, momentum is also conserved.

In other words, when a train accelerates, it is pushing against something… the Earth, that is. So ever so slightly, the Earth accelerates backwards. The change in velocity may be tiny, but the change in energy is not necessarily so. It all depends on your reference frame.

So let’s do the math, starting with a train of mass $$m$$ that accelerates from $$v_1$$ to $$v_2$$. (Yes, I am doing the math formally; we can plug in the actual numbers in the end.)

Momentum is of course velocity times mass. Momentum conversation means that the Earth’s speed will change as

$\Delta v = -\frac{m}{M}(v_2-v_1),$

where $$M$$ is the Earth’s mass. If the initial speed of the earth is $$v_0$$, the change in its kinetic energy will be given by

$\frac{1}{2}M\left[(v_0+\Delta v)^2-v_0^2\right]=\frac{1}{2}M(2v_0\Delta v+\Delta v^2).$

If $$v_0=0$$, this becomes

$\frac{1}{2}M\Delta v^2=\frac{m^2}{M}(v_2-v_1)^2,$

which is very tiny if $$m\ll M$$. However, if $$|v_0|>0$$ and comparable in magnitude to $$v_2-v_1$$ (or at least, $$|v_0|\gg|\Delta v|$$), we get

$\frac{1}{2}M(2v_0\Delta v+\Delta v^2)=-mv_0(v_2-v_1)+\frac{m^2}{2M}(v_2-v_1)^2\simeq -mv_0(v_2-v_1).$

Note that the actual mass of the Earth doesn’t even matter; we just used the fact that it’s much larger than the mass of the train.

So let’s plug in the numbers from the exercise: $$m=10000~{\rm kg}$$, $$v_0=-10~{\rm m}/{\rm s}$$ (negative, because relative to the moving train, the Earth is moving backwards), $$v_2-v_1=10~{\rm m}/{\rm s}$$, thus $$-mv_0(v_2-v_1)=1000~{\rm kJ}$$.

So the missing energy is found as the change in the Earth’s kinetic energy in the reference frame of the second moving train.

Note that in the reference frame of someone standing on the Earth, the change in the Earth’s kinetic energy is imperceptibly tiny; all the $$1500~{\rm kJ}$$ go into accelerating the train. But in the reference frame of the observer moving on the second train on the parallel tracks, only $$500~{\rm kJ}$$ goes into the kinetic energy of the first train, whereas $$1000~{\rm kJ}$$ is added to the Earth’s kinetic energy. But in both cases, the total change in kinetic energy, $$1500~{\rm kJ}$$, is the same and consistent with the readings of the electricity power meter.

Then again… maybe the symbolic calculation is too abstract. We could have done it with numbers all along. When a $$10000~{\rm kg}$$ train’s speed goes from $$10~{\rm m}/{\rm s}$$ to $$20~{\rm m}/{\rm s}$$, it means that the $$6\times 10^{24}~{\rm kg}$$ Earth’s speed (in the opposite direction) will change by $$10000\times 10/(6\times 10^{24})=1.67\times 10^{-20}~{\rm m}/{\rm s}$$.

In the reference frame in which the Earth is at rest, the change in kinetic energy is $$\tfrac{1}{2}\times (6\times 10^{24})\times (1.67\times 10^{-20})^2=8.33\times 10^{-16}~{\rm J}$$.

However, in the reference frame in which the Earth is already moving at $$10~{\rm m}/{\rm s}$$, the change in kinetic energy is $$\tfrac{1}{2}\times (6\times 10^{24})\times (10+1.67\times 10^{-20})^2-\tfrac{1}{2}\times (6\times 10^{24})\times 10^2$$$${}=\tfrac{1}{2}\times (6\times 10^{24})\times[2\times 10\times 1.67\times 10^{-20}+(1.67\times 10^{-20})^2]$$$${}\simeq 1000~{\rm kJ}$$.

Enough blogging about personal stuff like our cats. Here is a neat little physics puzzle instead.

Solving this question requires nothing more than elementary high school physics (assuming you were taught physics in high school; if not, shame on the educational system where you grew up). No tricks, no gimmicks, no relativity theory, no quantum mechanics, just a straightforward application of what you were taught about Newtonian physics.

We have two parallel rail tracks. There is no friction, no air resistance, no dissipative forces.

On the first track, let’s call it A, there is a train. It weighs 10,000 kilograms. It is accelerated by an electric motor from 0 to 10 meters per second. Its kinetic energy, when it is moving at $$v=10~{\rm m/s}$$, is of course $$K=\tfrac{1}{2}mv^2=500~{\rm kJ}$$.

Next, we accelerate it from 10 to 20 meters per second. At $$v=20~{\rm m/s}$$, its kinetic energy is $$K=2000~{\rm kJ}$$, so an additional $$1500~{\rm kJ}$$ was required to achieve this change in speed.

All this is dutifully recorded by a power meter that measures the train’s electricity consumption. So far, so good.

But now let’s look at the B track, where there is a train moving at the constant speed of $$10~{\rm m/s}$$. When the A train is moving at the same speed, the two trains are motionless relative to each other; from B‘s perspective, the kinetic energy of A is zero. And when A accelerates to $$20~{\rm m/s}$$ relative to the ground, its speed relative to B will be $$10~{\rm m/s}$$; so from B‘s perspective, the change in kinetic energy is $$500~{\rm kJ}$$.

But the power meter is not lying. It shows that the A train used $$1500~{\rm kJ}$$ of electrical energy.

Question: Where did the missing $$1000~{\rm kJ}$$ go?

Concerned as we were (and still are) about the health of our smallest kitty Pipacs, it helps to remember that we have two other felines in the house, both in good health as far as we know (fingers crossed and all that.)

The orange tabby on the left, Kifli, will turn 17 in April. He will be old enough to vote in next year’s federal election! He has not stated his political preference yet.

And despite their similarities, the two cats are not related. The cat on the right, Rufus, was a stray. He was approximately one year old when we adopted him in the fall of 2014.

Much to our surprise, not to mention relief, our kittycat Pipacs is a lot better today.

We noticed yesterday afternoon: he began eating again. He seemingly felt better. He groomed himself. He even played with us a little.

And then this morning: a nice, big poop in the litter box.

So his digestive system is functioning again. Probably with difficulty, so he is going to be permanently on a diet of laxatives and stool softeners, but so long as he is able to poop, he should be okay as otherwise, other than the bone growth on his pelvis, he is healthy.

This bone growth is not going to go away, so we’re living on borrowed time. But borrowed time is still a lot better than no time at all.

I already called the vet and canceled the scheduled visit for tonight. Thinking of Murphy’s law, however, I asked them not to process a refund yet, just keep the amount we already paid on our account.

This beautiful creature, who has been our companion for ten and a half years, give or take, is Pipacs (Hungarian for Poppy, pronounced pi-patch or something like, with the ‘i’ as in the word hit, the ‘a’ in the word bark, with emphasis on the first syllable.)

Unfortunately, Pipacs is very ill.

He was diagnosed with a growth on his pelvis three years ago. Even back then, the only treatment option was drastic surgery: Removal of a large part of his pelvis along with the leg on that side.

Pipacs has always been a very skittish cat. He was a stray when we adopted him, probably about a year old in late summer 2007. He is very easily traumatized. This, and the very low risk of such a growth spreading to other organs led us to the decision not to opt for surgery.

One thing we did not anticipate is that the growth, which increased in size rather dramatically in the past few months, would encroach on his digestive system and eventually obstruct his colon.

Which is exactly what happened. Simply put, beyond incidents of explosive diarrhea after receiving enemas, Pipacs cannot poop anymore. And sadly, surgery is no longer a viable option.

Which is why our veterinarian is scheduled to make a home visit tomorrow with her euthanasia kit. And my heart breaks as I am writing these words.

Not too long ago (OK, well, 30 years… it doesn’t feel that long anymore) there was another genius on the news: the Genius of the Carpathians, Romanian dictator Nicolae Ceausescu.

Here is a joke from that era that, no doubt, could be adapted to fit the “very stable genius”, too.

The US president, the General Secretary of the Soviet communist party, the Pope and Ceausescu are flying somewhere on a plane. Suddenly, the pilot enters the compartment and says, “Gentleman, I have bad news. This airplane is about to crash and we only have four parachutes.”

Immediately, the US president yells, “I am the leader of the free world, I must survive!”, grabs a parachute and jumps.

He is followed by the Soviet leader, who yells, “I am the leader of the worldwide socialist revolution. I must survive!”, grabs another parachute and jumps.

Next comes Ceausescu: “I am the Genius of the Carpathians! I must survive!”, grabs the next parachute and jumps.

The Pope and the pilot remain. The Pope looks at the pilot and says, “My son, I am old and lived a full life. Your whole life is ahead of you. Please take that parachute. I’ll pray for your survival.” The pilot responds, “No need to worry, Holy Father, we have two parachutes left. The Genius of the Carpathians grabbed the fire extinguisher.”

When will news portals finally learn that autoplaying a video at maximum volume in the middle of the night guarantees only one thing: that I close the tab in a mad panic while I curse the news site, its creators, editors, their parents and grandparents and just about everybody they ever did business with for scaring me witless and waking up my household?