May 122022

Dear Russia,

In the wake of Finland’s imminent decision to join NATO, you threaten again. You present yourself as the victim of aggression by Nazis.

So let’s take stock. Who did what in Europe since 1945?

In 1953, you used your military force to crush an uprising against hardline communist tyranny in East Germany.

In 1956, you did the same thing in Budapest, inflicting severe damage on my city of birth, still recovering from the devastations of WW2.

In 1968, you crushed the Prague Spring with tanks.

Kudos to you: You refrained from the direct use of military force in Poland in 1980, though you supported the regime that imposed martial law.

After the Soviet collapse, as the newly independent Russian Federation you supported separatists in the Transnistria region of Moldova.

You waged not one but two wars in Chechnya in the 1990s, with tens of thousands killed and cities like Grozny leveled.

In 2008, you launched a war against Georgia, seizing territory and creating two phony mini-republics.

In 2014, you launched another war, against Ukraine, seizing the Crimea and parts of the Donetsk and Luhansk regions, creating a permanent war zone, even shooting down a civilian airliner that flew over the area on a scheduled route.

And now you attempted a full-scale war, hoping to decapitate Ukraine, Blitzkrieg-style, and resorting to horrific, genocidal tactics of purposefully targeting civilians when your ineptitude and Ukrainian resistance thwarted your plans.

But you are the good guys. I get it. Meanwhile, what did evil, imperialist, Nazi NATO do? How many times were you attacked by NATO nations? What territories were seized by NATO?

Oh, I get it. NATO bombed Belgrade. Never mind that the goal was not to seize territory or even change a regime, simply to stop the (now well-documented) ongoing genocide in Kosovo. Because, I get it, that’s what Nazis do: they stop genocide. And you, the good guys?

This video speaks for itself.

Still wondering why Finland is keen on joining NATO?

They don’t want to end up like this hapless car dealership owner and his security guard.

Killed by Putler’s Russist thugs.

 Posted by at 5:57 pm
May 092022

It’s now Monday, May 9, 2022. And it is an anniversary of sorts.

No I am not talking about Putin and his planned “victory” parade, as he is busy desecrating the legacy of the Soviet Union’s heroic fight in the Great Patriotic War against a genocidal enemy.

I am referring to something much more personal. This sentence:

I watched The Matrix, for the first time. I’ve seen Dark City, and I loved it. I have heard all sorts of bad things about The Matrix, so I had low expectations. I was pleasantly surprised. Maybe not as well done as Dark City, it was nevertheless a surprisingly intelligent movie for a blockbuster.

Not very profound or insightful, is it.

But it happens to be my first ever blog entry, written when I still refused to call a blog a “blog”, calling it instead my “Day Book”, in the tradition of the late Jerry Pournelle.

So there. Will I be around twenty years from now? Perhaps more pertinently, will the world as we know it still be around?

What can I say? I am looking forward to marking the 40th anniversary of my blog on May 9, 2042, with another blog entry, hopefully celebrating a decent, prosperous, safe, mostly peaceful world.

 Posted by at 3:01 am
May 072022

There are things I am learning about the Roman Empire that I never knew. What a magnificent society it really was.

For instance, the Romans were not only master builders of sewers and roads, but also invented traffic management. Julius Caesar restricted the use of private vehicles to the last two hours of daylight. Business deliveries had to be made at night.

The Embattled Driver in Ancient Rome

What blew me away, however, was their use of… plumbing? I mean, aqueducts are one thing, but to have bona fide valves, water faucets in businesses and homes? Now that blew me away.

And speaking of sewers, the Romans might have been master sewer builders but it appears some form of a sewer system may have existed almost ten thousand (!!!) years earlier in a settlement in Anatolia. That’s just… wow.

 Posted by at 11:49 pm
May 062022

A beautiful study was published the other day, and it received a lot of press coverage, so I get a lot of questions.

This study shows how, in principle, we could reconstruct the image of an exoplanet using the Solar Gravitational Lens (SGL) using just a single snapshot of the Einstein ring around the Sun.

The problem is, we cannot. As they say, the devil is in the details.

Here is a general statement about any conventional optical system that does not involve more exotic, nonlinear optics: whatever the system does, ultimately it maps light from picture elements, pixels, in the source plane, into pixels in the image plane.

Let me explain what this means in principle, through an extreme example. Suppose someone tells you that there is a distant planet in another galaxy, and you are allowed to ignore any contaminating sources of light. You are allowed to forget about the particle nature of light. You are allowed to forget the physical limitations of your cell phone’s camera, such as its CMOS sensor dynamic range or readout noise. You hold up your cell phone and take a snapshot. It doesn’t even matter if the camera is not well focused or if there is motion blur, so long as you have precise knowledge of how it is focused and how it moves. The map is still a linear map. So if your cellphone camera has 40 megapixels, a simple mathematical operation, inverting the so-called convolution matrix, lets you reconstruct the source in all its exquisite detail. All you need to know is a precise mathematical description, the so-called “point spread function” (PSF) of the camera (including any defocusing and motion blur). Beyond that, it just amounts to inverting a matrix, or equivalently, solving a linear system of equations. In other words, standard fare for anyone studying numerical computational methods, and easily solvable even at extreme high resolutions using appropriate computational resources. (A high-end GPU in your desktop computer is ideal for such calculations.)

Why can’t we do this in practice? Why do we worry about things like the diffraction limit of our camera or telescope?

The answer, ultimately, is noise. The random, unpredictable, or unmodelable element.

Noise comes from many sources. It can include so-called quantization noise because our camera sensor digitizes the light intensity using a finite number of bits. It can include systematic noises due to many reasons, such as differently calibrated sensor pixels or even approximations used in the mathematical description of the PSF. It can include unavoidable, random, “stochastic” noise that arises because light arrives as discrete packets of energy in the form of photons, not as a continuous wave.

When we invert the convolution matrix in the presence of all these noise sources, the noise gets amplified far more than the signal. In the end, the reconstructed, “deconvolved” image becomes useless unless we had an exceptionally high signal-to-noise ratio, or SNR, to begin with.

The authors of this beautiful study knew this. They even state it in their paper. They mention values such as 4,000, even 200,000 for the SNR.

And then there is reality. The Einstein ring does not appear in black, empty space. It appears on top of the bright solar corona. And even if we subtract the corona, we cannot eliminate the stochastic shot noise due to photons from the corona by any means other than collecting data for a longer time.

Let me show a plot from a paper that is work-in-progress, with the actual SNR that we can expect on pixels in a cross-sectional view of the Einstein ring that appears around the Sun:

Just look at the vertical axis. See those values there? That’s our realistic SNR, when the Einstein ring is imaged through the solar corona, using a 1-meter telescope with a 10 meter focal distance, using an image sensor pixel size of a square micron. These choices are consistent with just a tad under 5000 pixels falling within the usable area of the Einstein ring, which can be used to reconstruct, in principle, a roughly 64 by 64 pixel image of the source. As this plot shows, a typical value for the SNR would be 0.01 using 1 second of light collecting time (integration time).

What does that mean? Well, for starters it means that to collect enough light to get an SNR of 4,000, assuming everything else is absolutely, flawlessly perfect, there is no motion blur, indeed no motion at all, no sources of contamination other than the solar corona, no quantization noise, no limitations on the sensor, achieving an SNR of 4,000 would require roughly 160 billion seconds of integration time. That is roughly 5,000 years.

And that is why we are not seriously contemplating image reconstruction from a single snapshot of the Einstein ring.

 Posted by at 4:01 pm
May 032022

I am beginning to wonder if the American political system is truly broken beyond repair.

I wonder what this means for Canada. No change? Will we become a safe haven for refugees from Gilead, as in The Handmaid’s Tale? Or will we be this new America’s Ukraine?

I am afraid that we will find out.

 Posted by at 12:36 am
Apr 272022

Someone reminded me that 40 years ago, when we developed games for the Commodore-64, there were no GPUs. That 8-bit CPUs did not even have a machine instruction for multiplication. And they were dreadfully slow.

Therefore, it was essential to use fast and efficient algorithms for graphics primitives.

One such primitive is Bresenham’s algorithm although back then, I didn’t know it had a name beyond being called a forward differences algorithm. It’s a wonderful, powerful example of an algorithm that produces a circle relying only on integer addition and bitwise shifts; never mind floating point, it doesn’t even need multiplication!

Here’s a C-language implementation for an R=20 circle (implemented in this case as a character map just for demonstration purposes):

#include <stdio.h>
#include <string.h>

#define R 20

void main(void)
    int x, y, d, dA, dB;
    int i;
    char B[2*R+1][2*R+2];

    memset(B, ' ', sizeof(B));
    for (i = 0; i < 2*R+1; i++) B[i][2*R+1] = 0;

    x = 0;
    y = R;
    d = 5 - (R<<2);
    dA = 12;
    dB = 20 - (R<<3);
    while (x<=y)
        B[R+x][R+y] = B[R+x][R-y] = B[R-x][R+y] = B[R-x][R-y] =
        B[R+y][R+x] = B[R+y][R-x] = B[R-y][R+x] = B[R-y][R-x] = 'X';
        if (d<0)
            d += dA;
            dB += 8;
            d += dB;
            dB += 16;
        dA += 8;

    for (i = 0; i < 2*R+1; i++) printf("%s\n", B[i]);

And the output it produces:

             XXX         XXX             
           XX               XX           
         XX                   XX         
        X                       X        
       X                         X       
      X                           X      
     X                             X     
    X                               X    
   X                                 X   
   X                                 X   
  X                                   X  
  X                                   X  
 X                                     X 
 X                                     X 
 X                                     X 
X                                       X
X                                       X
X                                       X
X                                       X
X                                       X
X                                       X
X                                       X
X                                       X
X                                       X
 X                                     X 
 X                                     X 
 X                                     X 
  X                                   X  
  X                                   X  
   X                                 X   
   X                                 X   
    X                               X    
     X                             X     
      X                           X      
       X                         X       
        X                       X        
         XX                   XX         
           XX               XX           
             XXX         XXX             

Don’t tell me it’s not beautiful. And even in machine language, it’s just a few dozen instructions.

 Posted by at 1:21 am
Apr 252022

I am reading an article in The Globe and Mail, with an attention-grabbing headline: Unvaccinated disproportionately risk safety of those vaccinated against COVID-19, study shows.

Except that the actual study shows no such thing. Nor does it involve actual vaccines or actual people.

What the study shows is that the simple (dare I say, naive) epidemiological model known as the SIR (Susceptible-Infected-Removed) model, when applied to a combination of two (vaccinated vs. unvaccinated) populations, indicates that the presence of the unvaccinated significantly increases the chances of infection among the vaccinated as well.

D’oh. Tell us something we didn’t know.

But the point is, this is a purely theoretical study. The math is elementary. The results are known (in fact, it makes me wonder why this paper was published in the first place; it’s not that it is wrong per se, but it really doesn’t tell us anything we didn’t know simply by looking at the differential equations that characterize the SIR model.) Moreover, the actual nuances of COVID-19 (mutations, limited and waning vaccine efficacy, differences between preventing infection vs. preventing hospitalization, death, or “long COVID” — in other words, all those factors that make CODID-19 tricky, baffling, unpredictable) are omitted.

So what will such a study accomplish? The authors’ point is that the model’s simplicity is a strength, as it also offers transparence and flexibility. I think in actuality, it will just muddy the waters. Those who are opposed to vaccination, especially mandatory vaccination, will call this study bogus and naive, an example of ivory tower theorizing with no solid foundations in reality. Conversely, some of those who are in favor of vaccination will present this paper as “proof” that the unvaccinated are selfish, irresponsible extremists who should be marginalized, ostracized by civilized society.

The Globe and Mail article is already evidence of this viewpoint. A quote from one of the study’s authors serves as a representative example: “In particular, when you have a lot of mixing between vaccinated and unvaccinated people, the unvaccinated people actually get protected by the vaccinated people, who act as a buffer – but that comes at a cost to the vaccinated.”

Does anyone think that pronouncements like these will help convince those who are ideologically opposed to vaccination or bought into the various nonsensical conspiracy theories about the nature, dangers, or efficacy of COVID-19 vaccines, especially mRNA vaccines?

 Posted by at 5:50 pm
Apr 202022

Came across a question tonight: How do you construct the matrix


Here’s a bit of Maxima code to make it happen:

(%i1) M(n):=apply(matrix,makelist(makelist(mod(x-k+n,n)+1,x,0,n-1),k,0,n-1))$
(%i2) M(5);
                               [ 1  2  3  4  5 ]
                               [               ]
                               [ 5  1  2  3  4 ]
                               [               ]
(%o2)                          [ 4  5  1  2  3 ]
                               [               ]
                               [ 3  4  5  1  2 ]
                               [               ]
                               [ 2  3  4  5  1 ]

I also ended up wondering about the determinants of these matrices:

(%i3) makelist(determinant(M(i)),i,1,10);
(%o3) [1, - 3, 18, - 160, 1875, - 27216, 470596, - 9437184, 215233605, - 5500000000]

I became curious if this sequence of numbers was known, and indeed that is the case. It is sequence number A052182 in the Encyclopedia of Integer Sequences: “Determinant of n X n matrix whose rows are cyclic permutations of 1..n.” D’oh.

As it turns out, this sequence also has another name: it’s the Smarandache cyclic determinant sequence. In closed form, it is given by

$${\rm SCDNS}(n)=(-1)^{n+1}\frac{n+1}{2}n^{n-1}.$$

(%i4) SCDNS(n):=(-1)^(n+1)*(n+1)/2*n^(n-1);
                                      n + 1
                                 (- 1)      (n + 1)   n - 1
(%o4)               SCDNS(n) := (------------------) n
(%i5) makelist(determinant(SCDNS(i)),i,1,10);
(%o5) [1, - 3, 18, - 160, 1875, - 27216, 470596, - 9437184, 215233605, - 5500000000]

Surprisingly, apart from the alternating sign it shares the first several values with another sequence, A212599. But then they deviate.

Don’t let anyone tell you that math is not fun.

 Posted by at 1:36 am
Apr 142022

I’ve been finding parallels between the events in Ukraine and 1914 (when Austria-Hungary attacked Serbia), 1938 (when the Western world acquiesced to Hitler’s annexation of the Sudetenland in the hope that it would bring “peace for our times”, to use Chamberlain’s words), 1939 (Hitler’s attack on Poland) and 1943 (Hitler’s historic defeat at Stalingrad.)

But perhaps it’s 1904-1905, when the Russian Empire waged a naval war against the Japanese Empire.

The specifics are different (for starters, the Russo-Japanese war was started by Japan), but the similarities abound: The general expectation was that a major European power, Russia, will easily prevail over the perceived inferiority of an “Asiatic” empire. Instead, Russia was humiliated, and the political backlash directly led to the 1905 revolution, itself a precursor to the 1917 Bolshevik revolution.

And here we are, in 2022, with Russia humiliated again by an enemy that was widely perceived as inferior. Beyond the tactical defeats, the strategic, political blunders are palpable: threats of Russian aggression have not only breathed fresh life into the NATO alliance, but also prompted traditionally (and fiercely) neutral Sweden and Finland to take concrete steps towards NATO membership.

And now, the Moskva. I am not sure but I am beginning to think that this 12,500 ton guided missile cruiser is the largest warship sank by enemy action since WW2. I mean, if this does not reek criminal recklessness and incompetence on behalf of the Russian leadership, both political and military, I don’t know what does.

Does this mean that a revolution is in the works? Are Putin’s days numbered? Perhaps, but I wouldn’t bet on it. And the nuclear wildcard is… there. Perhaps Putin will resort to nukes if all other options fail. Or perhaps a palace revolution widens into civil war, and the world may yet witness the consequences of the first ever nuclear civil war. Whatever happens, I doubt it will be pretty.

 Posted by at 10:37 pm
Apr 132022

Acting as “release manager” for Maxima, the open-source computer algebra system, I am happy to announce that just minutes ago, I released version 5.46.

I am an avid Maxima user myself; I’ve used Maxima’s tensor algebra packages, in particular, extensively in the context of general relativity and modified gravity. I believe Maxima’s tensor algebra capabilities remain top notch, perhaps even unsurpassed. (What other CAS can derive Einstein’s field equations from the Einstein-Hilbert Lagrangian?)

The Maxima system has more than half a century of history: its roots go back to the 1960s, when I was still in kindergarten. I have been contributing to the project for nearly 20 years myself.

Anyhow, Maxima 5.46, here we go! I hope I made no blunders while preparing this release, but if I did, I’m sure I’ll hear about it shortly.

 Posted by at 2:26 am
Apr 102022

Last week, Hungary’s Orban won a resounding victory in Hungary’s elections.

I was not happy about it, to be honest, and many of my liberal-minded friends were bitterly disappointed.

But a few of them began to look more deeply into the reasons behind this outcome. A recent article* by my friend László Mérő, a Hungarian mathematician and publicist, was illuminating. László decided to volunteer in the elections process as a scrutineer. He ended up in a rural electoral district, where he had a chance to talk to many of the voters who cast their ballots, including both the roughly 30% who voted for the united opposition, but also the majority who chose Orban’s government instead.

His conclusion? There was no fraud. The process was conducted professionally, transparently, and cleanly. The government may dominate traditional media (indeed, this is one of the cardinal sins of Orban’s autocratic government) but the voters were not uninformed. They were aware of the opposition, its message, and its goals. They didn’t favor the government because they were brainwashed. They favored the ruling party because those candidates did a much better job on the ground. They were known to the voters, unlike the fly-by-night opposition.

And anyone who thinks Orban represents some fringe must think again. Just days after his resounding victory (he not only retained, he even increased his two thirds parliamentary supermajority) we were reminded by press releases that CPAC, the Conservative Political Action Conference, was already set to hold its next meeting in Budapest. Orban will be the keynote speaker.

Wait, CPAC? The conference series held by one of the most respectable American conservative organizations, the American Conservative Union, dedicated to the foundations of conservatism, including personal liberty, freedom, and responsible governance?

That’s no fringe.

And, I strongly suspect, at least one of the reasons why Trump, Orban and other “illiberal” leaders and opinion-makers managed to hijack conservatism is that liberalism, in turn, was hijacked by the “woke” culture: a culture that is ready to “cancel” anyone who disagrees with whatever the canonical view of the day happens to be on race, gender, and related issues. I mean, J. K. Rowling? For real?

The fact that it all happens in a geopolitical context, allowing the likes of Putin to use his propaganda machine to try and divide the West even as his murderous horde of ill-equipped, badly led soldiers is rampaging in Ukraine, behaving like the worst of the Nazis (and prodded by Kremlin propaganda outlets to erase Ukrainian national identity) while pretending to “denazify” a country led by a Jewish president, is just the icing on this proverbial cake.

Elections are under way in France today. A possible Le Pen victory might have dramatic consequences.

*Magyar Nemzet, 2022.04.06 — In Hungarian

 Posted by at 1:20 pm
Apr 052022

The news we get from liberated areas of Ukraine are getting grimmer every day.

Army general Mark Milley, the Chairman of the Joint Chiefs of Staff of the United States, America’s highest-ranking military officer, told us in his Congressional testimony today that “we are witness to the greatest threat to the peace and security of Europe and perhaps the world in my 42 years of service in uniform”.

I made up my mind. If we want to preserve at least a tiny chance of keeping this world both peaceful and decent, we must take the Churchillian step of openly and boldly confronting evil.

If I were in charge… I would make Ukraine unconditionally a full-fledged member of NATO, effective immediately, with all the protections of NATO’s Article 5 implied. I would at the same time announce that a large contingent of NATO troops are entering the country, in cooperation with Ukraine’s legitimate, elected government, to guarantee the country’s original (!) borders, yes, including Crimea and Donbass.

And let’s do this while the Russian military is led by the same nincompoops who thought that the most important item for an invading army to carry is dress uniforms, or that digging in the Red Forest, in the world’s most contaminated soil with radioactive fallout, is a smart idea.

Mr. Putin, then, it’s your move. You push your button, we push ours, and the world as we know it ends. But I am betting that you are, in the end, just a worthless coward. Waste of skin.

 Posted by at 1:06 pm
Apr 052022

His followers probably dismiss these as “fake news”, in the wonderful tradition of Joseph Göbbels, Donald Trump and the rest of their heroes.

But we know that these are not fake news. Like this image of a dog not leaving the body of his dead master.

Or how about distressed parents writing their names and contact information on their children’s backs, in case they get separated or killed? When I saw this image, I must admit I felt like I was viciously kicked in the gut.

Putin must answer for these crimes. His followers, his enablers, his supporters must bear responsibility.

Whatever it takes.

Until and unless that happens, this world will not be whole again.

 Posted by at 2:05 am
Apr 032022

Hungary’s elections are coming to a close. All indications are that not only did Viktor Orban win, he won big: it appears he will retain his two -thirds constitutional supermajority in Hungary’s parliament. Yey-ho, illiberal democracy!

And they had the audacity to campaign as the party of peace: by confronting Putin, they argued, the opposition wants war and it is the government’s cowardly attitude that will somehow save the country from getting bogged down in conflict. (Because, you know, this attitude worked so well in the last major war in Europe, which witnessed Hungary as Hitler’s last ally, deporting over 600,000 of its own citizens to Nazi death camps, and having much of the country destroyed when the frontlines finally reached it in 1944-45.)

Meanwhile, Putin’s popularity in Russia is through the roof: The “Nazis” of Ukraine must be crushed, say people on the streets, and perhaps then Poland is next!

In America, Trump’s followers are regrouping, hoping to take back the House, the Senate, and eventually the White House, lining up behind their authoritarian leader to whom the rule of law means nothing unless it serves his personal interests.

Even here in Canada, all in the name of democracy of course, hundreds of unruly truckers blocked my city for nearly a month, and what is worse, their effort was (and remains) supported by millions. Perhaps still a small minority but still: It is a minority that is supporting a movement that is openly unconstitutional and insurrectionist.

And the sad thing is, we’ve all seen this before. The world went through something eerily similar a century ago. The Bolsheviks were popular in Soviet Russia. Mussolini was popular in Italy. Franco was popular in Spain. Hitler was popular in Germany. Even in places like the US and the UK, the likes of Charles Lindbergh or Oswald Mosley had considerable following.

I could ask the pointless question, why? Social scientists and historians probably offer sensible answers. But that doesn’t help. So long as people, even well-educated people, are willing to believe pseudoscience, ridiculous conspiracy theories, half truths, insinuations, and above all messages of hate: the compulsion to hate (or at least fear or distrust) someone, anyone, be it Ukrainians (or Russians), hate “migrants”, hate liberals, hate homosexuals, hate “others” however their other-ness is defined…

Screw you, world. I’m going back to physics. Just leave me alone, please.

 Posted by at 5:30 pm
Apr 012022

With apologies to the late Dr. Seuss (indeed I have no delusions about my talents as a cartoonist), I felt compelled to make a few, ahem, creative adjustments to one of his classic caricatures, as the message appears just as relevant today as it was more than eight decades ago, when some overly “patriotic” Americans were trying to be BFFs with another murderous tyrant.

The original was published on June 2, 1941. The lessons of history should not be forgotten.

 Posted by at 12:22 am
Mar 312022

Very well. I have now become convinced that Putler’s army in Ukraine is led by criminally incompetent clowns whose expertise is limited to cruelty and wholesale murder.

So they decide to occupy the Chernobyl Exclusion Zone. It’s an Exclusion Zone for a reason. Not because Ukrainians are fond of the Strugatsky brothers’ amazing novelette, Roadside Picnic, which describes similar Zones (albeit zones left behind by visiting, “picnicking” extraterrestrials.) They might be; the story really is good (and it formed the basis for Tarkovsky’s film Stalker, as well as the amazing S.T.A.L.K.E.R. series of video games. Scenes from which, incidentally, now bear alarming resemblance to the ruined cities of Ukraine.)

No, it is called an Exclusion Zone because it contains, you know, radioactive fallout from the world’s worst nuclear disaster: the explosion at reactor 4 of the Chernobyl Nuclear Power Station that happened in 1986.

Fallout that, until now, was mostly safely buried under fresh soil, causing relatively little harm. That is, until Russian conscripts, led by the aforementioned nincompoop murderers, dug it up with trenches. Soldiers who are now ill with signs of radiation sickness. With generals like the ones leading them, they need no enemies. They are perfectly capable of defeating themselves even without Ukrainian help.

 Posted by at 2:34 pm
Mar 292022

For the first time in my life, I exercised my right as a Hungarian citizen and voted.

Before I left Hungary, voting was pointless: my choice was the Hungarian Socialist Workers’ Party or else. The “or else” meant nothing.

More recently, I didn’t feel it kosher to participate in elections in a country where I do not reside and do not pay taxes. Then again, thanks in part to the current government, a great many Hungarian citizens who do not reside in Hungary or pay taxes there have gained the right to vote… so perhaps it’s not unethical for me to do so as well.

So I did.

Incidentally, the Parliament building in Budapest is quite an impressive edifice.

 Posted by at 3:28 pm