Mar 012021
 

Now it is time for me to be bold and contrarian. And for a change, write about physics in my blog.

From time to time, even noted physicists express their opinion in public that we do not understand quantum physics. In the professional literature, they write about the “measurement problem”; in public, they continue to muse about the meaning of measurement, whether or not consciousness is involved, and the rest of this debate that continues unabated for more than a century already.

Whether it is my arrogance or ignorance, however, when I read such stuff, I beg to differ. I feel like the alien Narim in the television series Stargate SG-1 in a conversation with Major (and astrophysicist) Samantha Carter about the name of a cat:

CARTER: Uh, see, there was an Earth physicist by the name of Erwin Schrödinger. He had this theoretical experiment. Put a cat in a box, add a can of poison gas, activated by the decay of a radioactive atom, and close the box.
NARIM: Sounds like a cruel man.
CARTER: It was just a theory. He never really did it. He said that if he did do it at any one instant, the cat would be both dead and alive at the same time.
NARIM: Ah! Kulivrian physics. An atom state is indeterminate until measured by an outside observer.
CARTER: We call it quantum physics. You know the theory?
NARIM: Yeah, I’ve studied it… in among other misconceptions of elementary science.
CARTER: Misconception? You telling me that you guys have licked quantum physics?

What I mean is… Yes, in 2021, we “licked” quantum physics. Things that were mysterious in the middle of the 20th century aren’t (or at least, shouldn’t be) quite as mysterious in the third decade of the 21st century.

OK, let me explain by comparing two thought experiments: Schrödinger’s cat vs. the famous two-slit experiment.

The two-slit experiment first. An electron is fired by a cathode. It encounters a screen with two slits. Past that screen, it hits a fluorescent screen where the location of its arrival is recorded. Even if we fire one electron at a time, the arrival locations, seemingly random, will form a wave-like interference pattern. The explanation offered by quantum physics is that en route, the electron had no classically determined position (no position eigenstate, as physicists would say). Its position was a combination, a so-called superposition of many possible position states, so it really did go through both slits at the same time. En route, its position operator interfered with itself, resulting in the pattern of probabilities that was then mapped by the recorded arrival locations on the fluorescent screen.

Now on to the cat: We place that poor feline into a box together with a radioactive atom and an apparatus that breaks a vial of poison gas if the atom undergoes fission. We wait until the half-life of that atom, making it a 50-50 chance that fission has occurred. At this point, the atom is in a superposition of intact vs. split, and therefore, the story goes, the cat will also be in a superposition of being dead and alive. Only by opening the box and looking inside do we “collapse the wavefunction”, determining the actual state of the cat.

Can you spot a crucial difference between these two experiments, though? Let me explain.

In the first experiment involving electrons, knowledge of the final position (where the electron arrives on the screen) does not allow us to reconstruct the classical path that the electron took. It had no classical path. It really was in a superposition of many possible locations while en route.

In the second experiment involving the cat, knowledge of its final state does permit us to reconstruct its prior state. If the cat is alive, we have no doubt that it was alive all along. If it is dead, an experienced veterinarian could determine the moment of death. (Or just leave a video camera and a clock in the box along with the cat.) The cat did have a classical state all throughout the experiment, we just didn’t know what it was until we opened the box and observed its state.

The crucial difference, then, is summed up thus: Ignorance of a classical state is not the same as the absence of a classical state. Whereas in the second experiment, we are simply ignorant of the cat’s state, in the first experiment, the electron has no classical state of position at all.

These two thought experiments, I think, tell us everything we need to know about this so-called “measurement problem”. No, it does not involve consciousness. No, it does not require any “act of observation”. And most importantly, it does not involve any collapse of the wavefunction when you really think it through. More about that later.

What we call measurement is simply interaction by the quantum system with a classical object. Of course we know that nothing really is classical. Fluorescent screens, video cameras, cats, humans are all made of a very large but finite number of quantum particles. But for all practical (measurable, observable) intents and purposes all these things are classical. That is to say, these things are (my expression) almost in an eigenstate almost all the time. Emphasis on “almost”: it is as near to certainty as you can possibly imagine, deviating from certainty only after the hundredth, the thousandth, the trillionth or whichever decimal digit.

Interacting with a classical object confines the quantum system to an eigenstate. Now this is where things really get tricky and old school at the same time. To explain, I must invoke a principle from classical, Lagrangian physics: the principle of least action. Almost all of physics (including classical mechanics, electrodynamics, even general relativity) can be derived from a so-called action principle, the idea that the system evolves from a known initial state to a known final state in a manner such that a number that characterizes the system (its “action”) is minimal.

The action principle sounds counterintuitive to many students of physics when they first encounter it, as it presupposes knowledge of the final state. But this really is simple math if you are familiar with second-order differential equations. A unique solution to such an equation can be specified in two ways. Either we specify the value of the unknown function at two different points, or we specify the value of the unknown function and its first derivative at one point. The former corresponds to Lagrangian physics; the latter, to Hamiltonian physics.

This works well in the context of classical physics. Even though we develop the equations of motion using Lagrangian physics, we do so only in principle. Then we switch over to Hamiltonian physics. Using observed values of the unknown function and its first derivative (think of these as positions and velocities) we solve the equations of motion, predicting the future state of the system.

This approach hits a snag when it comes to quantum physics: the nature of the unknown function is such that its value and its first derivative cannot both be determined as ordinary numbers at the same time. So while Lagrangian physics still works well in the quantum realm, Hamiltonian physics does not. But Lagrangian physics implies knowledge of the future, final state. This is what we mean when we pronounce that quantum physics is fundamentally nonlocal.

Oh, did I just say that Hamiltonian physics doesn’t work in the quantum realm? But then why is it that every quantum physics textbook begins, pretty much, with the Hamiltonian? Schrödinger’s famous equation, for starters, is just the quantum version of that Hamiltonian!

Aha! This is where the culprit is. With the Hamiltonian approach, we begin with presumed knowledge of initial positions and velocities (values and first derivatives of the unknown functions). Knowledge we do not have. So we evolve the system using incomplete knowledge. Then, when it comes to the measurement, we invoke our deus ex machina. Like a bad birthday party surprise, we open the magic box, pull out our “measurement apparatus” (which we pretended to not even know about up until this moment), confine the quantum system to a specific measurement value, retroactively rewrite the description of our system with the apparatus now present all along, and call this discontinuous change in the system’s description “wavefunction collapse”.

And then spend a century about its various interpretations instead of recognizing that the presumed collapse was never a physical process: rather, it amounts to us changing how we describe the system.

This is the nonsense for which I have no use, even if it makes me sound both arrogant and ignorant at the same time.


To offer a bit of a technical background to support the above (see my Web site for additional technical details): A quantum theory can be constructed starting with classical physics in a surprisingly straightforward manner. We start with the Hamiltonian (I know!), written in the following generic form:

$$H = \frac{p^2}{2m} + V({\bf q}),$$

where \({\bf p}\) are generalized momenta, \({\bf q}\) are generalized positions and \(m\) is mass.

We multiply this equation by the unit complex number \(\psi=e^{i({\bf p}\cdot{\bf q}-Ht)/\hbar}.\) We are allowed to do this trivial bit of algebra with impunity, as this factor is never zero.

Next, we notice the identities, \({\bf p}\psi=-i\hbar\nabla\psi,\) \(H\psi=i\hbar\partial_t\psi.\) Using these identities, we rewrite the equation as

$$i\hbar\partial_t\psi=\left[-\frac{\hbar^2}{2m}\nabla^2+V({\bf q})\right]\psi.$$

There you have it, the time-dependent Schrödinger equation in its full glory. Or… not quite, not yet. It is formally Schrödinger’s equation but the function \(\psi\) is not some unknown function; we constructed it from the positions and momenta. But here is the thing: If two functions, \(\psi_1\) and \(\psi_2,\) are solutions of this equation, then because the equation is linear and homogeneous in \(\psi,\) their linear combinations are also solutions. But these linear combinations make no sense in classical physics: they represent states of the system that are superpositions of classical states (i.e., the electron is now in two or more places at the same time.)

Quantum physics begins when we accept these superpositions as valid descriptions of a physical system (as indeed we must, because this is what experiment and observation dictates.)

The presence of a classical apparatus with which the system interacts at some future moment in time is not well captured by the Hamiltonian formalism. But the Lagrangian formalism makes it clear: it selects only those states of the system that are consistent with that interaction. This means indeed that a full quantum mechanical description of the system requires knowledge of the future. The apparent paradox is that this knowledge of the future does not causally influence the past, because the actual evolution of the system remains causal at all times: only the initial description of the system needs to be nonlocal in the same sense in which 19th century Lagrangian physics is nonlocal.

 Posted by at 12:48 pm
Feb 182021
 

I really cannot tell which impresses me more: The incredibly complex landing or the fact that there is now a de facto infrastructure in orbit around Mars, in the form of earlier spacecraft that provide communications relay capabilities for real-time tracking of the landing.

Or perhaps the fact that Perseverance also carries the Ingenuity helicopter. If successful, it will be the first drone to fly in the atmosphere of another planet.

The children of future settlers on Mars will be learning about these moments in school.

 Posted by at 4:06 pm
Feb 152021
 

A Dallas-Fort Worth TV station characterized Texas as the energy capital of the world as it asked the rhetorical question: How could this happen?

I have friends in Texas. One of them e-mailed me to let me know that they’ve been without power since 5 AM this morning.

Unlike the US-Canadian northeast with its interconnected power grid, Texas has its own power grid. This means, I understand, that they cannot rely on excess generating capacity in neighboring states to help with the crisis.

And the weather is bitterly cold, much colder than up here in wintry Ottawa. Right now, according to that Dallas-Fort Worth TV station it’s 8 degrees Fahrenheit but it will drop to several degrees below zero on the Fahrenheit scale overnight; that’s -20 C for us folks in metric lands.

This is not a joke. In weather like this, people can die, especially in ill-insulated homes as they struggle with unexpected secondary disasters such as bursting, frozen pipes that may very well happen.

 Posted by at 10:42 pm
Feb 102021
 

Sometimes, simple programming mistakes make for interesting glitches.

Take this image:

No, this is not something that a gravitational lens would produce. Or any lens.

Even-numbered multipoles in a gravitational lens produce images that have fourfold symmetry. This allows me to reduce the amount of computation needed to generate an image, as I only need to generate one quarter; the rest are just copied over.

But this is not true for odd-numbered multipoles. This image was supposed to represent the J3 multipole, which would yield a triangular shape.

Unfortunately, I generated it accidentally by assuming the symmetries of an even-numbered multipole.

I kind of like the result, even though it is of course scientifically worthless. It still looks neat though.

 Posted by at 12:36 pm
Dec 152020
 

A very nice article about our work on the Solar Gravitational Lens was published a few days ago on Universe Today, on account of our recent preprint, which shows quantitative results assessing the impact of image reconstruction on signal and noise.

Because the SGL is such an imperfect lens, the noise penalty is substantial. However, as it turns out, it is much reduced when the projected image area is large, such as when an exoplanet in a nearby star system is targeted.

While this is good news, the Sun’s gravitational field has other imperfections. We are currently working on modeling these and assessing their impact on noise. Next comes the problem of imaging a moving target: an exoplanet that spins, which is illuminated from varying directions, and which may have varying surface features (clouds, vegetation, etc.) Accounting for all these effects is essential if we wish to translate basic theory into sound scientific and engineering requirements.

So, the fun continues. For now, it was nice to see this piece in Universe Today.

 Posted by at 11:08 pm
Dec 022020
 

According to the immortal Douglas Adams, God’s final message to His creation is simple: “We apologize for the inconvenience.”

But there’s also another final message of sorts, the answer to the Ultimate Question about Life, Universe, and Everything: 42.

Recently, a researcher by the name of Michael Hippke analyzed the seemingly random bits that are contained in minute fluctuations of the Cosmic Microwave Background (CMB) radiation. His conclusion: there is no discernible pattern, no appearance of constants of nature, no detectable statistical autocorrelation. The message is random.

I beg to respectfully disagree. In the 512-bit segment published by Hippke, the bit sequence 101010 appears no fewer than eight, er, nine times (one occurrence split between two lines).

Now if we only knew the question to which the answer is 42…

 Posted by at 1:18 pm
Dec 012020
 

The giant Arecibo radio telescope is no more.

Damaged by a broken cable just a few weeks ago, the telescope completely collapsed today.

Incredibly sad news.

Completed in 1963, the telescope was 57 years old, just like me. I hope I will last a few more years, though.

 Posted by at 9:52 pm
Oct 192020
 

Since when is listening to the scientists a bad thing?

Now as to the claim that if he listened to the scientists, he’d have tanked the economy… here is what Mr. Trump’s pandemic response accomplished with respect to the US economy:

Compare this against China, where the totalitarian government, to their credit, actually listened to the scientists:

Even Canada, with a much less spectacular pre-pandemic economic performance than the United States, fared much better than our southern neighbor on which much of our economy depends:

So I suppose, Mr. Trump, you just gave millions of thinking Americans another reason to vote for Mr. Biden. I’d laugh but… neither the state of the US economy nor the number of dead, rapidly approaching a quarter million, qualify as laughing matter. Not to mention the seemingly inevitable march towards an increasingly illiberal, increasingly authoritarian America, driven by structural causes that go back decades and which a Biden Administration is just as unlikely to address as preceding administrations.

 Posted by at 1:21 pm
Sep 032020
 

Tonight, Slava Turyshev sent me a link to an article that was actually published three months ago on medium.com but until now, escaped our attention.

It is a very nice summary of the work that we have been doing on the Solar Gravitational Lens to date.

It really captures the essence of our work and the challenges that we have been looking at.

And there is so much more to do! Countless more things to tackle: image reconstruction of a moving target, imperfections of the solar gravitational field, precision of navigation… not to mention the simple, basic challenge of attempting a deep space mission to a distance four times greater than anything to date, lasting several decades.

Yes, it can be done. No it’s not easy. But it’s a worthy challenge.

 Posted by at 10:54 pm
Aug 302020
 

One thing follows another…

I’m listening to old MP3 files on my computer, one of which contains this once popular song by Paper Lace, The Night Chicago Died.

I once read that the band knew nothing about Chicago’s geography; I checked again on the Wikipedia page dedicated to this song.

The page mentions, among other things, how then Chicago mayor Daley hated the song. Daley? The same Daley who demolished Meigs Field airport, the island airport serving downtown Chicago that was the starting location of Microsoft Flight Simulator for many years?

Indeed. (Well, almost. There were two Daleys, father and son, Richard J and Richard M.) And Wikipedia tells me that the island has indeed since been turned into a park and nature preserve. But there are few pictures, so I figured I’d check it out using Google Maps.

So I typed Chicago into Google Maps and was greeted with this message in response:

I don’t know but this seems… a tad embarrassing isn’t it. Unless of course Chicago actually did die last night, and was promptly removed from Google Maps in response.

But no, Chicago is still there. The Google Maps thing was just a glitch. As is Northerly Island, which once hosted that ill-fated airport, its future as uncertain as it has always been in the past century or so.

 Posted by at 5:21 pm
Aug 082020
 

A popular Internet meme these days is to present an arithmetic expression like, say, 6/3(4−2) and ask the poor souls who follow you to decide the right answer. Soon there will be two camps, each convinced that they know the truth and that the others are illiterate fools: According to one camp, the answer is 4, whereas the other camp will swear that it has to be 1.

In reality it is neither. Or both. Flip a coin, take your pick. There is no fundamental mathematical truth hidden here. It all boils down to human conventions. The standard convention is that multiplication and division have the same precedence and are evaluated from left to right: So 6/3×(4−2) is pretty unambiguous. But there is another, unwritten convention that when the multiplication sign is omitted, the implied multiplication is assumed to have a higher precedence.

Precisely because of these ambiguities, when you see actual professionals, mathematicians or physicists, write down an expression like this, they opt for clarity: they write, say, (6/3)(4−2) or 6/[3(4−2)] precisely so as to avoid any misunderstanding. Or better yet, they use proper math typesetting software such as LaTeX and write 2D formulas.

 Posted by at 5:46 pm
Aug 052020
 

A tragedy took place in Beirut yesterday.

The actual power of the massive explosion is yet to be estimated accurately (probably not quite as large as the largest non-nuclear, accidental explosion that took place in Halifax, Nova Scotia 103 years ago though it comes close), but the images and videos are horrifying.

Reportedly, windows were shattered as far as 25 kilometers away from the epicenter.

The audio on one of the many videos showing the moment of the explosion accurately captures the event: “What the actual fuck?” asks a woman’s voice incredulously.

In light of the scope of the disaster, I expect that the final death toll will far exceed the 78 deceased that we know about for now.

It now appears that it was an industrial accident: welding work setting off a fire that in turn spread to a warehouse where thousands of tons of ammonium nitrate was stored.

Naturally, it didn’t prevent America’s “stable genius” from talking about an “attack”. When asked, he even referred to his “generals” who, according to him, told him that it was likely an attack.

I have no doubt that he made it all up on the spot. But his pronouncement had predictable consequences. It was like pouring oil on the fire, as it gave an excuse for every closet antisemite to come out and spread the conspiracy theory that it was an attack by Israel. Twitter accounts spreading this inflammatory nonsense include a Robert de Niro parody account; for a brief moment, I thought it was the real Robert de Niro, which would have been terribly disappointing.

Wildfires in Australia, locusts in Africa, a global pandemic, widespread racial riots in the United States, “murder hornets” spreading in North America… The Internet was already full of joke calendars for this year with disaster memes, as well as speculation that perhaps that infamous Mayan calendar was misinterpreted, as it referred to 2020, not 2012. In light of this catastrophe in Beirut, I am inclined to ask, what next? Alien invasion? The Yellowstone caldera? Global thermonuclear war? Giant asteroid impact? I won’t even try to guess, just note that we still have nearly five months left of this year.

 Posted by at 1:30 am
Jul 312020
 

A few weeks ago, Christian Ready published a beautiful video on his YouTube channel, Launch Pad Astronomy. In this episode, he described in detail how the Solar Gravitational Lens (SGL) works, and also our efforts so far.

I like this video very much. Especially the part that begins at 10:28, where Christian describes how the SGL can be used for image acquisition. The entire video is well worth seeing, but this segment in particular does a better job than we were ever able to do with words alone, explaining how the Sun projects an image of a distant planet to a square kilometer sized area, and how this image is scanned, one imaginary pixel at a time, by measuring the brightness of the Einstein-ring around the Sun as seen from each pixel location.

We now understand this process well, but many more challenges remain. These include, in no particular order, deviations of the Sun from spherical symmetry, minor variations in the brightness of the solar corona, the relative motion of the observing probe, Sun, exosolar system and target planet therein, changing illumination of the target, rotation of the target, changing surface features (weather, perhaps vegetation) of the target, and the devil knows what else.

Even so, lately I have become reasonably confident, based on my own simulation work and our signal-to-noise estimates, as well as a deconvolution approach under development that takes some of the aforementioned issues into consideration, that a high-resolution image of a distant planet is, in fact, obtainable using the SGL.

A lot more work remains. The fun only just began. But I am immensely proud to be able to contribute to of this effort.

 Posted by at 7:41 pm
Jul 162020
 

I met Gabor David back in 1982 when I became a member of the team we informally named F451 (inspired by Ray Bradbury of course.) Gabor was a close friend of Ferenc Szatmari. Together, they played an instrumental role in establishing a business relationship between the Hungarian firm Novotrade and its British partner, Andromeda, developing game programs for the Commodore 64.

In the months and years that followed, we spent a lot of time working together. I was proud to enjoy Gabor’s friendship. He was very knowledgeable, and also very committed to our success. We had some stressful times, to be sure, but also a lot of fun, frantic days (and many nights!) spent working together.

I remember Gabor’s deep, loud voice, with a slight speech impediment, a mild case of rhotacism. His face, too, I can recall with almost movie like quality.

He loved coffee more than I thought possible. He once dropped by at my place, not long after I managed to destroy my coffee maker, a stovetop espresso that I accidentally left on the stove for a good half hour. Gabor entered with the words, “Kids, do you have any coffee?” I tried to explain to him that the devil’s brew in that carafe was a bitter, undrinkable (and likely unhealthy) blend of burnt coffee and burnt rubber, but to no avail: he gulped it down like it was nectar.

After I left Hungary in 1986, we remained in sporadic contact. In fact, Gabor helped me with a small loan during my initial few weeks on Austria; for this, I was very grateful.

When I first visited Hungary as a newly minted Canadian citizen, after the collapse of communism there, Gabor was one of the few close friends that I sought out. I was hugely impressed. Gabor was now heading a company called Banknet, an international joint venture bringing business grade satellite-based Internet service to the country.

When our friend Ferenc was diagnosed with lung cancer, Gabor was distraught. He tried to help Feri with financing an unconventional treatment not covered by insurance. I pitched in, too. It was not enough to save Feri’s life: he passed away shortly thereafter, a loss I still feel more than two decades later.

My last conversation with Gabor was distressing. I don’t really remember the details, but I did learn that he suffered a stroke, and that he was worried that he would be placed under some form of guardianship. Soon thereafter, I lost touch; his phone number, as I recall, was disconnected and Gabor vanished.

Every so often, I looked for him on the Internet, on social media, but to no avail. His name is not uncommon, and moreover, as his last name also doubles as a first name for many, searches bring up far too many false positives. But last night, it occurred to me to search for his name and his original profession: “Dávid Gábor” “matematikus” (mathematician).

Jackpot, if it can be called that. One of the first hits that came up was a page from Hungary’s John von Neumann Computer Society, their information technology history forum, to be specific: a short biography of Gabor, together with his picture.

And from this page I learned that Gabor passed away almost six years ago, on November 10, 2014, at the age of 72.

Well… at least I now know. It has been a privilege knowing you, Gabor, and being able to count you among my friends. I learned a lot from you, and I cherish all those times that we spent working together.

 Posted by at 2:04 pm
Jul 162020
 

Seventy-five years ago this morning, a false dawn greeted the New Mexico desert near Alamagordo.

At 5:29 AM in the morning, the device informally known as “the gadget” exploded.

“The gadget” was a plutonium bomb with the explosive power of about 22 kilotons of TNT. It was the first nuclear explosion on planet Earth. It marked the beginning of the nuclear era.

I can only imagine what it must have been like, being part of that effort, being present in the pre-dawn hours, back in 1945. The war in Europe just ended. The war in the Pacific was still raging. This was the world’s first high technology war, fought over the horizon, fought with radio waves, and soon, to be fought with nuclear power. Yet there were so many unknowns! The Trinity test was the culmination of years of frantic effort. The outcome was by no means assured, yet the consequences were clear to all: a successful test would mean that war would never be the same. The world would never be the same.

And then, the most surreal of things happens: minutes before the planned detonation, in the pre-dawn darkness, the intercom system picks up a faint signal from a local radio station, and music starts playing. It’s almost as if reality was mimicking the atmosphere of yet-to-be-invented computer games.

When the explosion happened, the only major surprise was that the detonation was much brighter than anyone had expected. Otherwise, things unfolded pretty much as anticipated. “The gadget” worked. Success cleared the way to the deployment of the (as yet untested) simpler uranium bomb to be dropped on Hiroshima three weeks later, followed by the twin of the Trinity gadget, which ended up destroying much of Nagasaki. The human cost was staggering, yet we must not forget that it would have been dwarfed by the costs of a ground invasion of the Japanese home islands. It was a means to shorten the war, a war not started by the United States. No responsible commander-in-chief could have made a decision other than the one Truman made when he approved the use of the weapons against Imperial Japan.

And perhaps the horrors seen in those two cities played a role in creating a world in which the last use of a nuclear weapon in anger occurred nearly 75 years ago, on August 9, 1945. No one would have predicted back then that there will be no nuclear weapons deployed in war in the coming three quarters of a century. Yet here we are, in 2020, struggling with a pandemic, struggling with populism and other forces undermining our world order, yet still largely peaceful, living in a golden age unprecedented in human history.

Perhaps Trinity should serve as a reminder that peace and prosperity can be fragile.

 Posted by at 12:52 pm
Jul 092020
 

This may not be an all-time record-breaking day according to Environment Canada (supposedly, the peak temperature today at Ottawa Airport was 34.8 C at 2 PM) but it sure is hot.

You could be forgiven if you thought that this measurement is of the body temperature of a COVID-19 patient with mild symptoms, not the outdoor temperature on our balcony, measured in the shade:

As I said… really hot. Praise be to air conditioning.

 Posted by at 4:25 pm
Jun 082020
 

Long overdue, but I just finished preparing the latest Maxima release, version 5.44.

I am always nervous when I do this. It is one thing to mess with my own projects, it is another thing to mess with a project that is the work of many people and contains code all the way back from the 1960s.

 Posted by at 2:11 am
Jun 032020
 

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.

 Posted by at 11:59 am
Jun 022020
 

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.

 Posted by at 9:40 pm