In case anyone doubted that modern birds are descendants of dinosaurs, here is a reminder: the shoebill.
These amazing creatures are apparently quite docile with humans, but eat baby crocodiles for lunch, which they kill by decapitating them.
They really look like survivors of the K-T asteroid impact. They are… I think they are beautiful.
Amidst all the tension that has been unleashed in the United States, there is this small ray of hope.
A black flight attendant on a Southwest flight initiated a conversation with a white passenger, who was reading the book White Fragility: Why It’s So Hard for White People to Talk About Racism, by Robin DiAngelo.
The white passenger’s remark, “It’s our fault. We have to start these conversations,” caught her by surprise. A short conversation followed. Then the big revelation: The unassuming gentleman happened to be Doug Parker, CEO of American Airlines.
I can already hear some of my friends objecting: “It’s not our fault!” Do not misconstrue Parker’s words (perhaps they weren’t even quoted verbatim.) He of course didn’t mean, I am sure, that every white person must bear personal responsibility for every vile act of racism that happens in America or elsewhere.
Rather, what I read into those words is an acknowledgement of a simple reality: In an unequal relationship, the dominant party has the power to make change for the better. In America, this means whites.
The fact that the CEO of a company as large as American Airlines recognizes this is, well, a ray of hope. As is the fact that he traveled, unassumingly, as an ordinary economy passenger on a competitor’s flight. As rising inequality between the super-wealthy and the stagnating middle class plagues Western societies, the US in particular, as disadvantaged minorities fall even further behind, it is nice to know that at least some folks in positions of power recognize that their wealth and status also come with a huge responsibility. Especially if the nice thoughts are also followed by deeds.
I don’t always like commercial publishers. Some of their textbooks are prohibitively expensive, yet often lacking in quality. (One persistent exception is Dover Publications, who published some of the best textbooks I own, as low-cost paperbacks.)
Last night, however, I was very pleasantly surprised by Springer, who made several hundred textbooks across a range of disciplines available for free, on account of COVID-19.
I did not get greedy. I didn’t download titles indiscriminately. But I did find several titles that are of interest to me, and I gladly took advantage of this opportunity.
Thank you, Springer.
Still playing with some COVID-19 maps, so here is another one: This one ranks countries by the number of COVID-19 cases per 1,000 square kilometers.
What’s the point, you might ask? Well, when there are lots of people confined to a small area, even a few cases can mean trouble; in contrast, when you have many cases but spread over half a continent, you may never come across a single infected person in your travels.
Again, the color legend remains a little whacky; it is logarithmic, but I don’t know how to convince this R package to display it nicely.
The numbers are pretty high. It goes without saying that densely populated microstates like Monaco win the contest, but then there is Qatar (3494), Belgium (1866), the Netherlands (1324), the UK (1051)… numbers that are way too high. For comparison, the USA is at 180, Hungary at 41, Russia at 20 (no surprise there), China at 9 (!), and so is Canada.
Maps that show the spread and mortality rate of COVID-19 are worrisome, to say the least.
So here is something slightly more encouraging: the day-to-day growth rate of COVID-19 infections, in the world, the United States, Canada, and the province of Ontario where I live.
What this plot shows is that the growth rate has been consistently and steadily decreasing (note that the vertical axis is logarithmic). Furthermore, the overall behavior of the worldwide, US, and Canadian plots is remarkable similar.
Of course this plot is also a reflection of the fact that the cumulative number of cases increases, so even if the number of new cases remains steady, the growth rate will indeed decrease. Nonetheless, it demonstrates that at least for now, the pandemic’s exponential spread has been arrested. This can, of course, change for the worse quite dramatically if we lose our collective patience and start relaxing too soon.
I suppose the most important thing to know about COVID-19 is not so much the total number of cases or deaths, but the number of cases or deaths per million people. A number like 1,000 deaths is huge in a country of one million, a drop in the bucket in a country like India or China.
In that vein, I produced two maps using my rather rusty R programming skills. I am sure I could have done a better job labeling the legend, but there is only so much time I wanted to spend on this. The color bar legend is logarithmic between its two end values.
First, the cumulative number of cases per million people (May 22 data):
Next, deaths per million:
Not a pretty picture. It appears that the countries that are the most disproportionately affected (or the most likely to offer accurate reports?) are in Europe (including Russia), North and South America and the Middle East.
But now, here is perhaps the scariest map of all:
Yes, in many countries more than 10% of the confirmed cases result in death. The global average is 6.4 per 100 confirmed cases. The fact that COVID-19 is deadly in countries with underdeveloped health care systems is perhaps understandable, but the fact that the reddest parts of the map are places with the most sophisticated health care systems in the world is food for thought.
I am one of the maintainers of the Maxima computer algebra system. Maxima’s origins date back to the 1960s, when I was still in kindergarten. I feel very privileged that I can participate in the continuing development of one of the oldest continuously maintained software system in wide use.
It has been a while since I last dug deep into the core of the Maxima system. My LISP skills are admittedly a bit rusty. But a recent change to a core Maxima capability, its ability to create Taylor-series expansions of expressions, broke an important feature of Maxima’s tensor algebra packages, so it needed fixing.
The fix doesn’t amount to much, just a few lines of code:
It did take more than a few minutes though to find the right (I hope) way to implement this fix.
Even so, I had fun. This is the kind of programming that I really, really enjoy doing. Sadly, it’s not the kind of programming for which people usually pay you Big Bucks… Oh well. The fun alone was worth it.
One of the most fortunate moments in my life occurred in the fall of 2005, when I first bumped into John Moffat, a physicist from The Perimeter Institute in Waterloo, Ontario, Canada, when we both attended the first Pioneer Anomaly conference hosted by the International Space Science Institute in Bern, Switzerland.
This chance encounter turned into a 15-year collaboration and friendship. It was, to me, immensely beneficial: I learned a lot from John who, in his long professional career, has met nearly every one of the giants of 20th century physics, even as he made his own considerable contributions to diverse areas ranging from particle physics to gravitation.
In the past decade, John also wrote a few books for a general audience. His latest, The Shadow of the Black Hole, is about to be published; it can already be preordered on Amazon. In their reviews, Greg Landsberg (CERN), Michael Landry (LIGO Hanford) and Neil Cornish (eXtreme Gravity Institute) praise the book. As I was one of John’s early proofreaders, I figured I’ll add my own.
John began working on this manuscript shortly after the announcement by the LIGO project of the first unambiguous direct detection of gravitational waves from a distant cosmic event. This was a momentous discovery, opening a new chapter in the history of astronomy, while at the same time confirming a fundamental prediction of Einstein’s general relativity. Meanwhile, the physics world was waiting with bated breath for another result: the Event Horizon Telescope collaboration’s attempt to image, using a worldwide network of radio telescopes, either the supermassive black hole near the center of our own Milky Way, or the much larger supermassive black hole near the center of the nearby galaxy M87.
Bookended by these two historic discoveries, John’s narrative invites the reader on a journey to understand the nature of black holes, these most enigmatic objects in our universe. The adventure begins in 1784, when the Reverend John Michell, a Cambridge professor, speculated about stars so massive and compact that even light would not be able to escape from its surface. The story progresses to the 20th century, the prediction of black holes by general relativity, and the strange, often counterintuitive results that arise when our knowledge of thermodynamics and quantum physics is applied to these objects. After a brief detour into the realm of science-fiction, John’s account returns to the hard reality of observational science, as he explains how gravitational waves can be detected and how they fit into both the standard theory of gravitation and its proposed extensions or modifications. Finally, John moves on to discuss how the Event Horizon Telescope works and how it was able to create, for the very first time, an actual image of the black hole’s shadow, cast against the “light” (radio waves) from its accretion disk.
John’s writing is entertaining, informative, and a delight to follow as he accompanies the reader on this fantastic journey. True, I am not an unbiased critic. But don’t just take my word for it; read those reviews I mentioned at the beginning of this post, by preeminent physicists. In any case, I wholeheartedly recommend The Shadow of the Black Hole, along with John’s earlier books, to anyone with an interest in physics, especially the physics of black holes.
Heaven knows why I sometimes get confused by the simplest things.
In this case, the conversion between two commonly used cosmological coordinate systems: Comoving coordinates vs. coordinates that are, well, not comoving, in which cosmic expansion is ascribed to time dilation effects instead.
In the standard coordinates that are used to describe the homogeneous, isotropic universe of the Friedmann–Lemaître–Robertson–Walker (FLRW) metric, the metric is given by
$$ds^2=dt^2-a^2dR^2,$$
where \(a=a(t)\) is a function of the time coordinate, and \(R\) represents the triplet of spatial coordinates: e.g., \(dR^2=dx^2+dy^2+dz^2.\)
I want to transform this using \(R’=aR,\) i.e., transform away the time-dependent coefficient in front of the spatial term in the metric. The confusion comes because for some reason, I always manage to convince myself that I also have to make the simultaneous replacement \(t’=a^{-1}dt.\)
I do not. This is nonsense. I just need to introduce \(dR’\). The rest then presents itself automatically:
$$\begin{align*}
R’&=aR,\\
dR&=d(a^{-1}R’)=-a^{-2}\dot{a}R’dt+a^{-1}dR’,\\
ds^2&=dt^2-a^2[-a^{-2}\dot{a}R’dt+a^{-1}dR’]^2\\
&=(1-a^{-2}\dot{a}^2{R’}^2)dt^2+2a^{-1}\dot{a}R’dtdR’-d{R’}^2\\
&=(1-H^2{R’}^2)dt^2+2HR’dtdR’-d{R’}^2,
\end{align*}$$
where \(H=\dot{a}/a\) as usual.
OK, now that I recorded this here in my blog for posterity, perhaps the next time I need it, I’ll remember where to find it. For instance, the next time I manage to stumble upon one of my old Quora answers that, for five and a half years, advertised my stupidity to the world by presenting an incorrect answer on this topic.
This, incidentally, would serve as a suitable coordinate system representing the reference frame of an observer at the origin. It also demonstrates that such an observer sees an apparent horizon, the cosmological horizon, given by \(1-H^2{R’}^2=0,\), i.e., \(R’=H^{-1},\) the distance characterized by the inverse of the Hubble parameter.
So here I am, reading about some trivial yet not-so-trivial probability distributions.
Let’s start with the uniform distribution. Easy-peasy, isn’t it: a random number, between 0 and 1, with an equal probability assigned to any value within this range.
So… what happens if I take two such random numbers and add them? Why, I get a random number between 0 and 2 of course. But the probability distribution will no longer be uniform. There are more ways to get a value in the vicinity of 1 than near 0 or 2.
And what happens if I add three such random numbers? Or four? And so on?
The statistics of this result are captured by the Irwin-Hall distribution, defined as
$$f_{\rm IH}(x,n)=\dfrac{1}{2(n-1)!}\sum\limits_{k=1}^n(-1)^k\begin{pmatrix}n\\k\end{pmatrix}(x-k)^{n-1}{\rm sgn}(x-k).$$
OK, so that’s what happens when we add these uniformly generated random values. What happens when we average them? This, in turn, is captured by the Bates distribution, which, unsurprisingly, is just the Irwin-Hall distribution, scaled by the factor \(n\):
$$f_{\rm B}(x,n)=\dfrac{n}{2(n-1)!}\sum\limits_{k=1}^n(-1)^k\begin{pmatrix}n\\k\end{pmatrix}(nx-k)^{n-1}{\rm sgn}(nx-k).$$
How about that.
For what it’s worth, here is the Maxima script to generate the Irwin-Hall plot:
fI(x,n):=1/2/(n-1)!*sum((-1)^k*n!/k!/(n-k)!*(x-k)^(n-1)*signum(x-k),k,0,n);
plot2d([fI(x,1),fI(x,2),fI(x,4),fI(x,8),fI(x,16)],[x,-2,18],[box,false],
[legend,"n=1","n=2","n=4","n=8","n=16"],[y,-0.1,1.1]);
And this one for the Bates plot:
fB(x,n):=n/2/(n-1)!*sum((-1)^k*n!/k!/(n-k)!*(n*x-k)^(n-1)*signum(n*x-k),k,0,n);
plot2d([fB(x,1),fB(x,2),fB(x,4),fB(x,8),fB(x,16)],[x,-0.1,1.1],[box,false],
[legend,"n=1","n=2","n=4","n=8","n=16"],[y,-0.1,5.9]);
Yes, I am still a little bit of a math geek at heart.
My lovely wife, Ildiko, woke up from a dream and asked: If you have a flower with 7 petals and two colors, how many ways can you color the petals of that flower?
Intriguing, isn’t it.
Such a flower shape obviously has rotational symmetry. Just because the flower is rotated by several times a seventh of a revolution, the resulting pattern should not be counted as distinct. So it is not simply calculating what number theorists call the \(n\)-tuple. It is something more subtle.
We can, of course, start counting the possibilities the brute force way. It’s not that difficult for a smaller number of petals, but it does get a little confusing at 6. At 7 petals, it is still something that can be done, but the use of paper-and-pencil is strongly recommended.
So what about the more general case? What if I have \(n\) petals and \(k\) colors?
Neither of us could easily deduce an answer, so I went to search the available online literature. For a while, other than finding some interesting posts about cyclic, or circular permutations, I was mostly unsuccessful. In fact, I began to wonder if this one was perhaps one of those embarrassing little problems in combinatorial mathematics that has no known solution and about which the literature remains strangely quiet.
But then I had another idea: By this time, we both calculated the sequence, 2, 3, 4, 6, 8, 14, 20, which is the number of ways flowers with 1, 2, …, 7 petals can be colored using only two colors. Surely, this sequence is known to Google?
Indeed it is. It turns out to be a well-known sequence in the online encyclopedia of integer sequences, A000031. Now I was getting somewhere! What was especially helpful is that the encyclopedia mentioned necklaces. So that’s what this problem set is called! Finding the Mathworld page on necklaces was now easy, along with the corresponding Wikipedia page. I also found an attempt, valiant though only half-successful if anyone is interested in my opinion, to explain the intuition behind this known result:
$$N_k(n)=\frac{1}{n}\sum_{d|n}\phi(d)k^{n/d},$$
where the summation is over all the divisors of \(n\), and \(\phi(d)\) is Euler’s totient function, the number of integers between \(1\) and \(d\) that are relative prime to \(d\).
Evil stuff if you asked me. Much as I always liked mathematics, number theory was not my favorite.
In the case of odd primes, such as the number 7 that occurred in Ildiko’s dream, and only two colors, there is, however, a simplified form:
$$N_2(n)=\frac{2^{n-1}-1}{n}+2^{(n-1)/2}+1.$$
Unfortunately, this form is for “free necklaces”, which treats mirror images as equivalent. For \(n<6\) it makes no difference, but substituting \(n=7\), we get 18, not 20.
Finally, a closely related sequence, A000029, characterizes necklaces that can be turned over, that is to say, the case where we do not count mirror images separately.
Oh, this was fun. It’s not like I didn’t have anything useful to do with my time, but it was nonetheless a delightful distraction. And a good thing to chat about while we were eating a wonderful lunch that Ildiko prepared today.
The other day, I ran across a question on Quora asking why Einstein didn’t support his country, Germany, during the Second World War. Thinking about this question reminded me of an old Star Trek episode and one of the root concepts (or, at least, my reading of it) of the Abrahamic family of religions.
In answering the question, I pointed out the difference between supporting a country vs. supporting a regime. I argued that Einstein, though not even a citizen of Germany at the time (he gave up German citizenship after Hitler’s rise to power in 1933, and became a naturalized US citizen in 1940), did, in fact, support his country of birth, precisely by the act of following his conscience and opposing the despotic, murderous Nazi regime.
And that takes me to the Star Trek episode Bread and Circuses from 1968. In this episode, the USS Enterprise encounters a planet governed by a regime not unlike the Roman Empire, but with 20th century technology, broadcasting gladiatorial matches by analog television. In due course, the crew of the Enterprise gets into trouble and link up with a group of rebellious Sun-worshippers. When at the end of the episode, after the conflict is resolved and the good guys prevail as usual, Spock expresses surprise over the fact that such a primitive religion could have survived on this planet into its modern era, Uhura corrects him by clarifying that they were, in fact, worshippers of the son of God. In other words, this planet’s version of early Christianity arrived two thousand years later than on the Earth.
Christianity borrows its creation mythology from Judaism, including the notion of the Garden of Eden and the Tree of Knowledge, the fruit of which let Adam and Eve understand the difference between good and evil. In my reading, this is what it really means when the Bible proclaims that humans are created in God’s image: that just like God, humans are free agents with a conscience, capable of acting independently, not robots blindly executing a predetermined divine script. They even have the capacity to act against God’s will.
Think about this, just what a revolutionary, what a deeply subversive concept this really is even today, never mind ancient times. The Book of Genesis is probably about 3,000 years old if not older. Egypt, in its third intermediate period, was ruled by pharaohs, seen as intermediaries between gods and ordinary people, whose words must be obeyed. Whether or not the Egyptian captivity happened (there do appear to be reasons to doubt), it’s no wonder Egypt’s rulers didn’t look kindly upon these pesky Jews and their subversive religion that claimed that it is more important to listen to your conscience than to blindly follow the orders of your divine ruler.
Despots can claim whatever they want: They can claim to represent the state, they can even claim to be the earthly representative of a divine power, like the pharaohs of old, but you have something over which they have no power: your conscience, which allows you to defy the will of any ruler, even God’s will, just as Adam and Eve have done back in the Garden of Eden.
And this is precisely what Einstein did when he lent his support, for instance, to Leo Szilard’s letter to Roosevelt that arguably launched the Manhattan project: Instead of slavishly following a despot claiming to represent the country of his birth, he listened to his conscience.
Just as things are beginning to look ever so slightly hopeful with infection rates at least stabilizing, conspiracy theorists are now having a field day.
In case you are wondering, we “know” that SARS-CoV-2 was manufactured (or at least released) by that Wuhan “bioweapons” lab, as its intended purpose was to weaken China’s strategic opponents, in particular the US military, so that they can mess with Taiwan. And even more blatantly, they did so using in part funding from the United States, according to information that was “just revealed”. And all in the service of some demonic Chinese plot to achieve some nefarious goal, such as the subjugation of the renegade province of Taiwan.
Like all good conspiracy theories, this nonsense is also based on a carefully picked set of selected facts. So much so that it reminds me of those old Soviet-era Radio Yerevan jokes.
Meanwhile in the real world…
None of this matters to conspiracy theorists, of course. If they want to believe that it was all some evil Chinese plot, they will do so, damn the evidence. (Sadly, I think it is a safe bet that they get at least some help from Russian troll farms. Unfortunately Putin’s regime has not stopped spreading disinformation about health care, epidemics, pandemics, vaccines and now, the origins of COVID-19.)
So to all the conspiracy theory fans out there, please keep this in mind: Conspiracy theories serve a single purpose. They turn us: individuals or entire nations, against each other. They are the means to sow discord. Just when the world has to act in unison more than ever, they pull us apart. Conspiracy theories are weapons: dangerous in the wrong hands even when used unintentionally, but deadly if used purposefully.
Please… listen to the actual science. Don’t believe every piece of garbage you hear.
I haven’t blogged in two weeks. In my excuse, I was rather busy. The good kind of busy, that is, busy with paying work, busy with scientific research, not busy with illness or anything on that front.
Anyhow, even though I haven’t blogged, I’ve been keeping track of the numbers. And for the past few days, a ray of hope began to emerge.
To make a long story short, there are significant signs that mitigation measures are working. Here, this chart shows the doubling rate of COVID-19 infections worldwide:
Infection rates doubled every 5-6 days back in late March; now, the doubling rate is over 15 days and rapidly rising.
Perhaps the world data are manipulated. But then, here is the doubling rate for Canada. The data are much more noisy (the population is much smaller, so this is to be expected) but the similar trend is unmistakable:
But there is another sign that things just might be working. I’ve been following US data in more detail, and lately, the simple SIR model’s predictions began to match the data rather well. If the model is to be believed, we may not be out of the woods quite yet, but we may be surprisingly close:
What a difference a few weeks make: Back in late March, the same model predicted catastrophic numbers. Now, it seems to tell us that we are mere weeks away from life gradually beginning to return to normal.
I dare not believe it just yet, but it is a ray of hope.
While I applaud the fact that there are very few partisan voices in Canada, and that governments at all levels constructively cooperate with each other, I cannot help but wonder if the measures taken are sufficient to fight this demon of a virus, COVID-19.
Take these two notifications that I received on my phone from the Radio Canada app yesterday:
My question is… why exactly do we still have non-essential travel within Canada? And why exactly are the interprovincial borders still open?
This virus will not be beaten with half-measures. If we are not able to bring down the infection rate, soon an extremely large number of people will become simultaneously sick, completely overwhelming the intensive care capacity of our health care system. Which means that a great many people who could survive with adequate medical care will die.
SIR model prediction based on US data as of March 28, 2020.
See this simple simulation of the US situation that I put together, using the simplest epidemiological model. If its predictions come true, at one point in late April more than 20% of America’s population will be sick. If 5-10% of these patients require intensive care, that is up to 6 million people or more, only a small fraction of which will receive the care that they will need to stay alive. The rest will die. The same thing can happen here in Canada if we don’t take the necessary measures.
Ottawa looks like a ghost town these days. Here are a few images from this morning’s “rush hour”:
The one good thing about this is that when you actually have to go somewhere, it has never been this easy.
Oh, and gas is cheaper than… well, pretty much cheaper than it has ever been in my experience, since I moved to Ottawa in 1987.
Two weeks, or to be precise, fifteen and a half days ago, I was walking the streets of downtown Vienna, enjoying a bright late winter day, eating a bit of authentic Viennese street food and a fabulous slice of cake in a Vienna coffee house. The next day, I boarded a flight at a busy Vienna Airport. To be sure, some signs were already present that not everything was normal. The plane had fewer passengers than usual, especially in business class. There was news of Lufthansa grounding all their A380 superjumbos, and when I asked our pilot about this, he just shook his head, not knowing what the future would bring. But all this felt distant; the world around us, by and large, still felt normal, busy as usual, with people lining up at checkpoints, roadways busy with traffic, airplanes landing and departing at regular intervals.
Today, fifteen days later, we visited our favorite deli store in a nearly completely deserted Byward Market in downtown Ottawa. I literally could have parked in the middle of the street. The store was open (we phoned ahead to make sure) but deserted as well. All the good food there… will it ever sell? Will they at least get a chance to donate some of it, e.g., to the Food Bank or to a nearby shelter? Will they be able to stay open? Will they be able to stay in business?
I don’t know what hit me more, this store or the Web site of Vienna Airport. You know, the same airport where I stood in line, two weeks ago, to go through customs and security.
Not much of a chance of a lineup today.
How will our world recover from this?
My wife and I went on a shopping spree.
No, we didn’t win the lottery. But apart from our desire to support our local economy in times of crisis, we were also rather worried that our favorite deli store in the Byward Market may be forced to close for an indefinite period of time.
So we stocked up on things. That said, I hope they are able to stay open. I hope they are able to stay in business. Other deli stores have shut their doors. I hope Continental remains open and that the owner and employees stay healthy.
In the meantime, I thank them for serving us.
Working from home is easier for some than for others.
Members of a symphony orchestra have to get a little more creative than most of us, but that didn’t stop members of the Danubia Symphony Orchestra of Óbuda, from Budapest, Hungary:
Nicely done!
I am reading this article in Mother Jones, worrying about the United States following the fate of the Western Roman Empire, leading to its collapse in 476 AD.
But… Empire?
I think it speaks volumes about America that even a left-wing outlet, like Mother Jones, worries about the end of an Empire… instead of worrying about the end of a Republic.
For these are not the end times for the American Empire. Not even the beginning of the end. It is, to put it plainly, just the beginning. If the analogy with Rome has any validity (and I suspect that it might), what we are witnessing is not the end of an Empire, but its birth.
What we see is not the weakening of the American political entity, quite the contrary. But we do see a transition, as republican values erode, as liberal democracy is abandoned, and the United States inches ever closer to an imperial presidency.
I expressed my concerns about this before. There are certain unmistakable parallels between the history that unfolds in the United States in the present day vs. the history of Rome some 2100 years ago.
The fact that even a Mother Jones commentator misses this point and thinks of his nation as an Empire only reinforces my concerns.