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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?

It’s the same, each and every Christmas. As Christmas Eve approaches, I remember that famous moment from 49 years ago. The astronauts of Apollo 8 just orbited the Moon. It was Christmastime. These three men were a thousand times farther from the Earth than any human being in history. It was an awe-inspiring moment. Once radio contact with the distant Earth was re-established, the three astronauts took turns reading the first ten verses of Genesis. Frank Borman then closed the broadcast with words that, in my mind, remain the most appropriate words for this evening: “good night, good luck, a Merry Christmas – and God bless all of you, all of you on the good Earth.

So our American friends south of the border have a new tax legislation. I’ve seen many discussions of how it affects people in various income brackets, but all too often, these are more confusing than helpful.

So instead, here is a handy chart that shows how the federal tax rate decreases as a percentage of taxable income for individuals, joint filers and heads of households (based on tables provided by CNN):

While most folks will enjoy a tax break, it is interesting to see that some individuals and heads of households will actually see a tax increase:

Married couples benefit the most (mainly as a result of some arcane tax brackets that were in effect before the new legislation), which is probably a Good Thing; I am not so sure about the 2.6% tax break offered to the wealthiest, however.

Anyhow, I pay my taxes in Ontario, Canada, so it really doesn’t bother me one way or another, I just wanted to understand a little better what actually is going on.

Fair warning: This post contains some adult language.

In case there has been any doubt: Republicans have gone completely bonkers, and it is now official.

According to The Washington Post, the Centers for Disease Control and Prevention, America’s leading public health institute, is no longer allowed to use the following terms in its budget documents:

• vulnerable,
• entitlement,
• diversity,
• transgender,
• fetus,
• evidence-based,
• science-based.

Dear Republicans, dear followers of the orange person in the White House: Are you out of your fucking minds? Have you gone completely bonkers? Truly gone fishing? Are you bat shit crazy? Totally demented?

I mean, what kind of an idiot does this childish shit? You are worse than Soviet political commissars from the Stalin era. OK, you are not sending anyone to the Gulag (yet?) but your retarded drive for ideological purity is like a caricature of itself.

I mean… you are insane. Certifiably. This is not politics anymore. This is not left vs. right, transgender rights vs. traditional families. You are clearly deranged psychopaths.

I am secretly hoping that the news turns out to be false, but despite accusations of “fake news” and the like, I know The Washington Post to be a responsible news organization and I think it is a safe bet that this story was vetted.

The Internet (or at least, certain corners of the Internet where conspiracy theories thrive) is abuzz with speculation that the extrasolar asteroid ‘Oumuamua, best known, apart from its hyperbolic trajectory, for its oddly elongated shape, may be of artificial, extraterrestrial origin.

Some mention the similarity between ‘Oumuamua and Arthur C. Clarke’s extraterrestrial generational ship Rama, forgetting that Rama was a ship 50 kilometers in length, an obviously engineered cylinder, not a rock.

But then… I suddenly remembered that there was another artificial object of extrasolar origin in the science-fiction literature. It is Iilah, from A. E. van Vogt’s 1948 short story Dormant. Iilah is not discovered in orbit; rather, it lays dormant on the ocean floor for millions of years until it is awakened by the feeble radioactivity of isotopes that appear in the ocean as a result of the use and testing of nuclear weapons.

Iilah climbs out of the sea and is thus discovered. It becomes an object of study by a paranoid military, which ultimately decides to destroy it using a nuclear weapon.

Unfortunately, the energy of the explosion achieves the exact opposite: instead of destroying Iilah, it fully awakens it, making it finally remember its original purpose. Iilah then sets itself up for a tremendous explosion that knocks the Earth out of orbit, ultimately causing it to fall into the Sun, turning the Sun into a nova. Why? Because Iilah was programmed to do this. Because “robot atom bombs do not make up their own minds.”

Artist’s impression of ‘Oumuamua

So here is the thing… the Iilah of van Vogt’s story had almost the exact same dimensions (it was about 400 feet in length) and appearance (a rock, like rough granite, with streaks of pink) as ‘Oumuamua.

Go figure.

And since I seem to be posting cat pictures today, here is an “artistic creation” of Google Photos’ AI bot, depicting our youngest cat, Rufus:

I think this picture clearly explains why we often address this feline as Master Rufus.