1. "any representation of 3D rotations using only three values"
That is not representation, that is parametrization. Euler-angle parametrization sometimes fails because it is not a correct parameterization of SO(3) in general by construction, this is why it sometimes fails (essentially, the three consecutive rotations can sometimes effectively collapse into two for certain set of angles, regardless of how you choose your 3 axes, in which case you can't relate 2 independent parameters back to the 3 independent axis-angle parameters). The correct parametrization of SO(3) is the axis-angle parametrization, which can be represented using quaternions or 3D reals matrices.
The "representation", on the other hand, would typically be unit quaternions or 3D orthogonal matrices.
2."it can be proven that any representation of 3D rotations using only three values must contain discontinuities." where is that proof and what discontinuity are you talking about? It sound like he misunderstood what "SO(3) is not simply connected" means. Lie groups are differentiable.
3 parameters are sufficient to represent any 3D rotation. The natural parametrization of all Lie groups, including SO(3), is the axis-angle parametrization, and their elements have the form exp(i θ n.J) where n is a unit vector defining the axis of rotation, θ determines the amount of rotation, and J is a vector of the generators of the corresponding Lie algebra. The "regular" 3D matrix representation in the axis-angle parameterization is obtained with so(3) generators L_x, L_y, L_z in their fundamental representation.
Basis quaternions i, j, k (which can be represented by Pauli matrices) obey the same Lie algebra as L_x, L_y, L_z, but the group that it corresponds to (which is SU(2)) is a double cover of SO(3) (up to a sign), so they can still be used for implementing 3D rotations once you pick a sign.
Also physicist here. I've worked on conventional superconductors, but never on unconventional ones. Last I heard, it was believed to be mediated by magnons (rather than phonons). Who claims it is due to Coulomb interaction?
I'm a physicist. What education are you referring to?
Downplaying the risks of nuclear power plants, which are real, is easy to do, as long as it is far from your home. Most people can't grasp the reality of a nuclear disaster or its aftermath, because they haven't seen it. I have.
Yes, engineering has progressed, yes, there are ever improving safe-guards.
As you're probably aware, in Japan, we had a major disaster 14 years ago. Such disasters are very rare, but they will keep happening in future, and no electrical company that build nuclear reactors can guarantee that the probability is 0%. It is a matter of time (maybe in 50 years, maybe in 500 years), and when it happens, who is going to take responsibility for all the lives and damage? Toden couldn't, and certainly not the armchair (nor actual) nuclear engineers who like to lecture people about how they are uneducated.
>Downplaying the risks of nuclear power plants.... is easy to do, as long as it is far from your home
Please take a look at Wolf Creek, KS, nearby where I lived for many years.
> which are real
> Yes, engineering has progressed, yes, there are ever improving safe-guards
> As you're probably aware, in Japan, we had a major disaster 14 years ago
A 1960s reactor design is still not a valid comparison. In all empathy, I feel sorry this happened in your area; I wish that on nobody, and I understand how hard it is to have to pick up the pieces.
However, it is not just to the rest of the world to put red tape in front of modern designs because of what happened to a 1960s design. Modern designs need to be evaluated on their own merits.
We haven't banned modern cars because of the death traps of the 1960s; instead we iterated upon the designs even though many people experienced personal loss due to unsafe car designs. In fact, more people will die in a car accident this year 2025 than the totality of nuclear deaths from the time we began using nuclear power for power generation, with zero deaths from radiation or sickness at Fukushima.
Nuclear power is expensive for Germany. Uranium would need to be imported and enriched, there is no storage capacity for waste, former solutions have failed massively.
Chernobyl irradiated a huge part of the whole continent of Europe and poisoned parts of the food chain. This is not the same risk dimension as unsafe cars.
And the radiation of Fukushima probably entered the food chain at different points as well. Contrary to popular belief, radiation doesn't homogenously disperse in the ocean. Measuring the negative effects isn't trivial. Sure, water is a very good radiation absorber, so you won't get problems from direct radiation.
I would totally buy the Enron egg for personal use but that is beside the point. And I still would have no means to recycle it, so I would make it a problem of future generations as well.
Of course coal isn't a solution, I am happy that nobody starts with ideas like liquifying it into oil. But I think nobody is disputing that.
You offer condolences, and you say you understand. Thank you, but what do you think you understand?
Your analogy that somehow equates car accidents to nuclear disasters is not even wrong, and the fact that in your mind these can somehow be put into the same analogy is a testament that you don't understand.
1. With a car, you make a decision to use it or not given the risk factors. In contrast, the actual risks of a nuclear reactor built by a corporation in a remote area are not born by the company but rather by the town folks who have been living there and their descendants (an entire population) who have no choice in operating a nuclear power plant in their neighborhood.
And as we also see here, for few who dare to raise their voice, they are shut down by the corporation and local politicians with promises of a bright future or blamed with ignorance by some of their peers. Those same people who are not going to take responsibility when the nuclear disaster happens.
2. A car accident can kill a few people and financial damages are small and can be covered by an ordinary insurance company. A nuclear disaster can kill and make sick an entire population, destroy their livelihoods and poison the environment for generations, and caused damages are so great, well beyond the capabilities of any insurance or electrical company. In practice, the burden of the material damages fell upon the citizens of the entire nation.
As I already said, ever-improving nuclear reactors can result in a nuclear disaster, however "modern" they are. You can guarantee otherwise only in the imaginary world of spherical cows. As long as nuclear power plans are run, Fukushima disaster is not going to be last nuclear disaster the humanity has seen.
And again, the question is not "will another nuclear disaster happen". This is Murphy's law. The question is "when it happens, what will its effect be?"
If a corporation wants to operate a nuclear reactor, they should build it hundreds of kilometers away from any human settlement, with proper containment and cleaning mechanism in place that can be deployed right after the failure.
I looked at Wolf Creek Generating Station on Wikipedia, what of it? I don't see a nuclear disaster happened there. I hope the day won't ever come, but I would be interested in hearing your opinion on the matter after Wolf Creek melts down like in Fukushima ---at which point you will actually understand.
Ferromagnetism has nothing to do with currents, it is due to aligned spins of partially filled shells. Below a certain temperature (Curie temperature of the material), exchange interaction (which penalizes any misalignment, in the case of ferromagnetic exchange interaction) between electrons leads to this alignment.
Spin is a type of intrinsic angular momentum that is not associated with any spatial motion.
The Feynman lecture you linked to is an explanation why currents fail to explain ferromagnetism. You need to read the next chapter, but being a lecture for undergrads, it doesn't go deep into the subject anyway. If you're really interested, any modern book on magnetism would be much helpful.
This is why I said ferromagnetism is circulating current in the sense of "to a first approximation" and "heuristically". Wiktionary defines "heuristic" to be:
> a practical method [...] not following or derived from any theory, or based on an advisedly oversimplified one.
I think that if you ask Feynman, he would probably agree or sympathize with the naive idea of "atomic currents" as a heuristic argument in the introduction of this topic... which is nothing new anyway, and has been a heuristic argument used in electromagnetism for a long time, at least before QM.
In Feynman's own words,
> These days, however, we know that the magnetization of materials comes from circulating currents within the atoms—either from the spinning electrons or from the motion of the electrons in the atom. It is therefore nicer from a physical point of view to describe things realistically in terms of the atomic currents [...] sometimes called “Ampèrian” currents, because Ampère first suggested that the magnetism of matter came from circulating atomic currents.
You said,
> Spin is a type of intrinsic angular momentum that is not associated with any spatial motion.
Yet the concept of spin in quantum mechanics was originally developed using macroscopic rotations as an analogy, although today we know that spin is an intrinsic property of subatomic particles (thus the joke, "Imagine a ball that is spinning, except it is not a ball and it is not spinning.") In the same sense that Ampère's concept of "atomic currents" was developed using circulating electric current as an analogy.
> The Feynman lecture you linked to is an explanation why currents fails to explain ferromagnetism. You need to read the next chapter.
Of course, "The actual microscopic current density in magnetized matter is, of course, very complicated." This is surely explained in the next chapter. I could've mentioned "atomic currents" without citing any link, but I included it to allow anyone who's interested to read the whole thing in context.
To the parent and its sibling comments: There is no atomic or subatomic current that can explain ferromagnetism in any approximation.
You read some Wikipedia pages and Feynman lectures of physics. I'm a physicist who has done well over a decade of research in magnetic materials.
In understanding of ferromagnetism, many incorrect theories have been proposed. By connecting ferromagnetism to circulating currents (i.e, paramagnetism and diamagnetism), you just repeated the same mistake.
You're trying to bend the words to avoid being wrong. Physics is not philosophy or debate club. There is no approximation in physics in which electron is a ball with some radius, or its spin is due to a circulating current in physics. Any such explanation attempt fails spectacularly if you actually try to do the math (which gives an electron surface that is moving faster than speed of light, as Uhlenbeck/Goudsmit who proposed this incorrect idea quickly found out), so it doesn't even work as an approximation of any kind.
> Yet the concept of spin in quantum mechanics was originally developed using macroscopic rotations as an analogy,
Who developed this theory in quantum mechanics, where and when? Pauli, who first introduced it into quantum mechanics and the namesake of spin 1/2 matrices, insisted that it is purely quantum mechanical with no classical analogue. And regardless of who said what over 100 years ago, today, it is well understood that spin has nothing to with electric charges that move or rotate in space.
More importantly, the reason ferromagnetism develops in the first place is due to exchange interaction (as I wrote above) between magnetic moments, which is due to Pauli exclusion principle and also has nothing to do with movement of charges.
Furthermore, such magnetic moments (called magnetic impurities in that context) ruin the superconducting order by breaking the time-reversal symmetry, so trying to make a connection to ferromagnetism in the context of superconductivity is even worse.
> You read some Wikipedia pages and Feynman lectures of physics. I'm a physicist who has done well over a decade of research in magnetic materials.
In the same way that a geodesist navigates using a reference ellipsoid defined by WGS-84, while a city commuter uses Cartesian coordinates on a flat map. The commuter's navigational tool will never work in geophysics research, and it doesn't need to be.
> To the parent and its sibling comments: There is no atomic or subatomic current that can explain ferromagnetism in any approximation. [...] Any such explanation attempt fails spectacularly if you actually try to do the math (which gives an electron surface that is moving faster than speed of light, as Uhlenbeck/Goudsmit who proposed this incorrect idea quickly found out), so it doesn't even work as an approximation of any kind.
I consider "circulating currents create ferromagnetism" to be as true as "an atom's structure is similar to a solar system." Both concepts break down when it's examined in details, so its use by research physicists is obviously unacceptable, but I consider it's nevertheless as an useful mental image in introductory discussions among non-physicists.
Would you consider Rutherford's original atom model to be a first approximation? Can it be considered a very oversimplified but useful heuristic, at least when people who know anything about atoms are first introduced to this concept? Alternatively, would you consider Rutherford's atom to be "an explanation attempt that fails spectacularly if you actually try to do the math (which gives an electron that collapses into the nucleus in picoseconds, as Rutherford's colleagues quickly found out)?
If you believe the latter case, everyone can stop this conversation right now. Because it means the entire disagreement is entirely down to what kinds of "metal images" are acceptable, rather than any factual, like "whether a full quantum treatment of ferromagnetism is necessary to completely explain ferromagnetism (of course it is)." The rest of us who don't solve research problems believe a toy model is still interesting, but don't deny (nor mention) better models. You, as a professional physicist, believe many "what if?" metal models from history are just not legitimate physics, and should not be mentioned at any circumstances to avoid poisoning the minds of youths - an approach known as Whig history, in which scientific progress marches from one victory to another, and all losers be damned - a perfectly valid approach for teaching physics to students who only care about pure physics science, instead of "who said what."
As a side note, I know some engineers who really hate the idea that electric circuits works due to an electron flow. The most extreme one I've seen of wanted to ban this concept in introductory textbooks, calling it a big lie (an explanation attempt that fails spectacularly if you actually try to do the math, which gives the speed of an electron 30 billion times slower than the speed of light in free space). As we all know, the steady-state electron flow was only a result of the transient propagation and reflection of electromagnetic waves in free space or dielectric materials. Thus, they believe the wave model should be the only interpretation in a science textbook, since "they're high-school teachers, I'm a design engineer who work with high-speed digital systems with 20 years of experience, and I know for sure that high-speed circuits and computers can't even be made functional if you ignore fields and transmission line effects." Meanwhile, I believe the electron flow model still works as an introductory mental image (although the field view perhaps needs to be mentioned earlier).
> Who developed this theory in quantum mechanics, where and when? Pauli, who first introduced it into quantum mechanics and the namesake of spin 1/2 matrices, insisted that it is purely quantum mechanical with no classical analogue.
The earlier "electron as a rotating ball" idea was considered by Ralph Kronig and Uhlenbeck-Goudsmit in 1925. Pauli personally never accepted it due to its unphysical flaws. Only in 1927 did Pauli publish a rigorous QM treatment. Thus, "electron spin using classical rotation as analogue" was still an intermediate step before establishing this concept in QM. It was a footnote in history since Pauli was a great physicist and already considered the problem himself earlier and found the solution before everyone else. Otherwise this intermediate step may last longer than 2 years.
> Furthermore, such magnetic moments (called magnetic impurities in that context) ruin the superconducting order by breaking the time-reversal symmetry, so trying to make a connection to ferromagnetism in the context of superconductivity is even worse.
This, in comparison, is a more interesting criticism.
What you say is correct only when you adopt certain specific narrow definitions of the words, which you have not explained.
In its original sense, an electric current is any kind of movement of electric charge. In this wide sense, it also applies to the source of ferromagnetism.
Its meaning can be restricted to refer to the translational movement of electrically charged particles. With this narrower sense, there is still no need to use quantum mechanics to explain ferromagnetism. Even in classical electromagnetism, with the narrower-defined current, the sources of magnetic fields are decomposed into distributions of electric current densities and of magnetic moment densities, where the latter are the source of ferromagnetism. If necessary, it is possible to also use distributions of higher-order moment densities and the series of moments when the "electric current" is used in the narrow sense (of a first order moment) corresponds to the "electric current" used in its original, wide sense.
The isolated sentence "Spin is a type of intrinsic angular momentum that is not associated with any spatial motion" is logically contradictory (because, by definition, angular momentum is a characteristic of moving bodies). It can be correct only when you first specify that by "spatial motion" you mean only a certain kind of spatial motion.
The joke mentioned by another poster "Imagine a ball that is spinning, except it is not a ball and it is not spinning" is just a joke, because there is no doubt that the elementary particles are spinning.
Even when you model the elementary particles in the standard way, as point-like bodies (and it is debatable whether this is a good model), you cannot say that they are not rotating, because this would be the same mistake as saying that a delta distribution has a null value in the origin.
On the contrary, while you cannot say other things about the value of a delta distribution in the origin, what you can say with certainty is that it is not null.
In the same way, while you cannot say anything about characteristics of an electron like radius, mass density, angular velocity, electric current density and so on, you can say with certainty the values of various integral quantities (which integrate the corresponding delta distributions), like mass, electric charge, angular momentum and magnetic moment, so you can say with certainty that any electron is rotating (i.e. it has a non-null angular momentum).
As other commenters have said, whether or not an electron’s magnetic moment is “to do with currents” is a little open to interpretation.
I’ll add that the Dirac equation (governing electron field) correctly predicts magnetic moment given the inputs of charge and mass. * I interpret this as indicating that magnetic moment is a derived phenomenon just as it would be in the classical picture of a spinning ball of charge; I.e. the quantum picture refines but does not totally discard the classical understanding.
* Well, technically, sympathetic vibrations with all the other standard model fields also make tiny contributions to the magnetic moment.
> They have perfectly great existing hardware decoder offloading APIs via the various OS' native APIs for videos
One vendor specific API, not "various OS' native APIs".
Firefox currently supports hardware video decoding with Intel's vendor specific VA-API only on Linux, which is not supported by NVIDIA. (A third-party VA-API to NVDEC translation layer for Linux does exist on GitHub, nvidia-vaapi-driver, but it's not yet reliable as the officially supported VDPAU or NVDEC, and is not included in official linux package repositories.)
Intel has VA-API, AMD has AMF, and NVIDIA has VDPAU which is being replaced by NVDEC/NVENC.
The idea behind Vulkan Video Extensions is to have a vendor independent and cross-platform video API.
> No, Firefox currently supports hardware video decoding with Intel's vendor specific VA-API only, which is not supported by NVIDIA.
Intel is behind VA-API originally, but I don't think it's fair to say it's a vendor specific API anymore. It's supported by the open source drivers for GPUs from all 3 vendors. It's just that the open source drivers for Nvidia cards are not very practical and the proprietary drivers only support vdpau and nvdec/nvenc
You can make the same argument for VDPAU. AMD officially supports it, and there is an unofficial translation layer with limited capabilities for some Intel GPUs. Is VDPAU not a vendor specific API anymore then?
Intel, AMD and NVIDIA have their own vendor-specific video APIs, and even when they provide official support for the API of another vendor, it tends to expose a limited subset of the full functionality (like the list of available codecs and encoding features).
You are free to call these vendor specific APIs for what they are or something else, but the reality has been that there is no single video API officially supported by Intel, AMD and NVIDIA. This changed with Vulkan Video.
But Vulkan Video isn't just about desktop: mobile devices, Raspberry Pi, etc. are expected to get on board with it eventually, just like they did with Vulkan.
> It's supported by the open source drivers for GPUs from all 3 vendors.
Which 3 vendors are you referring to? Intel, AMD, and who?
> It's just that the open source drivers for Nvidia cards are not very practical and the proprietary drivers only support vdpau and nvdec/nvenc
Why are you bringing up open source drivers, and what is not practical? Both official open source drivers (open-gpu-kernel-module) and unofficial open source drivers (nouveau, through binary firmware) support VDPAU. However, NVIDIA's drivers (open source or binary) does not support VA-API.
What of it? So does nvidia-vaapi-driver. They're third party projects, that's very different from "supported by the vendor", and doesn't change the fact that NVIDIA as a vendor offers no support for VA-API.
> Which 3 vendors are you referring to? Intel, AMD, and who?
I either missed some of the other text in your post or it was added after I started to reply.
> Both official open source drivers (open-gpu-kernel-module)
This is not remotely close to being a complete graphics driver. Most of a GPU driver on Linux is in userspace and there is no official open source user space component.
> Why are you bringing up open source drivers, and what is not practical?
nouveau has never been practical for serious use due to poor performance and mediocre hardware support (as you noted). open-gpu-kernel-module is only practical when paired with a proprietary userspace driver.
Anyway, my original point in all this is that describing VA-API as an Intel vendor specific API is unfair given it has been well supported on AMD GPUs for a long time now and on nouveau it's supported as well as VDPAU (i.e. not very well as you note). I did not intend to imply that it was universal. I didn't even intend to imply that VDPAU is a vendor specific API (though as a decode-only API it's not really a complete replacement).
Intel tried to make va-api the standard for hardware encode and decode on Linux, Nvidia tried to make VDPAU the standard for hardware decode on Linux. Neither was entirely successful. By contrast, NVENC/NVDEC, AMF and the Intel Media SDK (and whatever they replaced this with) never had such ambitions.
> One vendor specific API, not "various OS' native APIs".
Incorrect. Firefox uses Windows Media Foundation, which is cross-vendor, on Windows. It uses MediaCodec on Android which is again cross-vendor. Presumably it uses whatever iOS' equivalent is as well.
It only uses VA-API on a single OS, Linux, and that's probably more a reflection on the media qualities (or lack thereof) of Linux as a whole. Maybe Vulkan video extensions will be the savior on Linux. Or maybe it won't because it won't be anyone's focus of investment since it's largely a Linux-only problem in the first place.
What is "incorrect"? The full sentence that you conveniently chose to cut in the middle before quoting (apparently to fit into some pessimistic forecast about the significance of Linux desktop) reads
> Firefox currently supports hardware video decoding with Intel's vendor specific VA-API only on Linux, which is not supported by NVIDIA.
(emphasis added)
You further wrote:
> Firefox uses Windows Media Foundation, which is cross-vendor, on Windows. It uses MediaCodec on Android which is again cross-vendor.
And? None of those APIs are cross-platform. Vulkan Video will eventually allow developers (including Firefox developers) to write a single code path for video to cover a wide range of platforms and vendors (likely with the exception of walled gardens like Apple-land, although someone might find a way to support like via a wrapper like MoltenVk for Vulkan).
> The full sentence that you conveniently chose to cut in the middle before quoting (apparently to fit into some pessimistic forecast about the significance of Linux desktop) reads
What are you talking about? They didn't quote that sentence at all, and didn't cut in the middle of the sentence they quoted.
> And? None of those APIs are cross-platform.
Your original objection, the thing that got quoted, was about whether things are cross-vendor. That question is completely unrelated to whether things are cross-platform.
> You did say that, but it's not the part of your post they were responding to.
So if someone criticizes a portion of your statement which is already countered by your original full statement, you're not allowed to remind your full statement. What kind of logic is that?
My original post says the point of Vulkan Video is it will be cross-platform and cross-vendor. And gives one example of cross-vendor side of things on Linux.
Someone criticizes me by essentially saying "you are incorrect, that's only on Linux. Windows, Android and iOS have their own video APIs...". This "correction" is incorrect because I already said on Linux, and it goes on to actually reinforce the post that he is responding to by highlighting cross-platform side, which also is in the post he is responding to.
So, if you look at the full conversation, the criticism is self-contradictory. This is what I'm pointing out, but you are implying I'm not allowed to do that.
I disagree. When you fragment a statement in a way that changes its meaning and make a straw man out of it, people are justified in responding to it.
> So if someone criticizes a portion of your statement which is already countered by your original full statement, you're not allowed to remind your full statement. What kind of logic is that?
The other stuff in your comment did not "counter" what they said. You made statements about cross-vendor and cross-platform. They chose to only respond to one of those statements. That's not incorrect.
> This "correction" is incorrect because I already said on Linux
The first part of your comment specifically said "not "various OS' native APIs"". That goes beyond Linux. The later part of your comment was about Linux in particular, but your introduction was an overall statement that wasn't true.
> When you fragment a statement in a way that changes its meaning
They didn't. You misspoke and they didn't know what you actually meant.
And from your other post: > Obviously, I meant to say statement, not sentence, but I can't edit it anymore.
That was not obvious. They quoted an entire paragraph, and the subsequent paragraph does not change its meaning the way you're claiming it does.
I'm always annoyed how any Linux media player or encoder needs to bring its own entire media operating system, down to each individual nut and bolt.
On Windows there's Windows Media Foundation and DirectShow that centrally manage everything and also support the "individual nut and bolt" approach. Android has its own central thing (MediaCodec?) that must be used. MacOS and iOS presumably have their own central manager (Quicktime?) too.
But Linux? It doesn't serve as an operating system for media. It's tremendously inconvenient as an admin/user rather than an evangelist.
You don't need to implement every nut and bolt in the application. Lot's of useful things can do the heavy lifting (Pipewire, ffmpeg, libplacebo, Mesa and so on). Linux isn't after calling it all using some uniform "DirectFoo" naming scheme, but tools are there.
Comparison is also invalid. Linux as a whole (not the kernel but OS experience) isn't controlled by some Big Brother who decides what and how it's done single mindedly. So such kind of composite result is somewhat expected.
Those who use it appreciate their efforts. You aren't using it, why are you even complaining especially with complete nonsense comments. Anti Linux shilling should be getting old.
> Android, ChromeOS and WebOS, have replaced most of my needs
Great! I really hope you're happy with that setup! It's your personal computer and by all means, do what works well for you. In the end that's always a personal thing that's different for everyone. Who am I to judge how you use your computer?
But maybe ... stop complaining about Linux desktop then? If you don't like it? This must be like the 3rd time I've seen these types of single-line dismissive "Linux will never win the desktop"-comment from you in the last few days. Just one line, little or no context, or explanation, and IMHO also zero value, and an entire discussion derailed.
This is just becoming disruptive. You don't need to say anything you know. Personally, I rather dislike a number of things, but you don't see me complaining about it with one-liners every chance I get – and when I do say something, at least I make sure it's something of some substance, when I feel it actually contributes. And I sure as hell don't go around complaining people are "children" for disagreeing.
And when someone installs some obscure or outdated and vulnerable codec on these systems, it's then automatically exposed to all sorts of applications to exploit. Maybe Windows sandboxes that these days(?) It was definitely a problem in the past.
No perfect solutions here; both "system-wide codecs" and "every application brings their own codecs" have their own up- and downsides.
Besides, with ffmpeg and gstreamer the system-wide codecs paradigm also works on Linux.
This is one of those "it's different but it doesn't really matter much" type of things. Most people "just" install vlc or mpv or whatnot and things will "just work" for them, not really different from Windows. That it's technically slightly different is almost entirely transparent to the user.
Yeah on UNIX side, NeXTSTEP, Irix, Solaris had their own thing, as graphical workstation UNIXes, and were great.
Ideally that kind of thing would be part of GNOME, or KDE, but then there are those that rather keep using twm like experience, making GNU/Linux really only good for headless experiences, at least the UNIX/POSIX part is always there.
This is a meta-study, touching on that "contrast" already: there is a subsection in the paper dedicated to this, where they claim that
"The major factor in cerebral bleeding however is hypertension, and in an RCT of aspirin based on more than 18,000 hypertensive patients—all of whom were receiving ‘optimal’ antihypertensive treatment—there were no additional cerebral bleeds in patients randomised to aspirin" (Refs 46 and 47).
which seems to be in contradiction with the article corresponding to the news you linked to: https://jamanetwork.com/journals/jamanetworkopen/fullarticle... and strangely doesn't cite or comment on Refs 46 and 47 from the paper of the main thread, possibly because they don't seem to be focusing on older adults.
There is also a subsection on gastrointestinal bleeding.
Intracranial bleeding isn't necessary something that cause permanent damage, or lethal by the way.
Depends on the encoder, this website provides easy-to-visualize data sets for various encoders at various settings
https://arewecompressedyet.com/
AV1 encoders tend to have better VMAF score at a given bits-per-pixel.
That becomes relevant for frequencies that are high enough to break Cooper pairs. But this material is claimed to be in the superconducting phase up to 400K, which corresponds to a superconducting gap of 8.3THz.
1. "any representation of 3D rotations using only three values"
That is not representation, that is parametrization. Euler-angle parametrization sometimes fails because it is not a correct parameterization of SO(3) in general by construction, this is why it sometimes fails (essentially, the three consecutive rotations can sometimes effectively collapse into two for certain set of angles, regardless of how you choose your 3 axes, in which case you can't relate 2 independent parameters back to the 3 independent axis-angle parameters). The correct parametrization of SO(3) is the axis-angle parametrization, which can be represented using quaternions or 3D reals matrices.
The "representation", on the other hand, would typically be unit quaternions or 3D orthogonal matrices.
2."it can be proven that any representation of 3D rotations using only three values must contain discontinuities." where is that proof and what discontinuity are you talking about? It sound like he misunderstood what "SO(3) is not simply connected" means. Lie groups are differentiable.
3 parameters are sufficient to represent any 3D rotation. The natural parametrization of all Lie groups, including SO(3), is the axis-angle parametrization, and their elements have the form exp(i θ n.J) where n is a unit vector defining the axis of rotation, θ determines the amount of rotation, and J is a vector of the generators of the corresponding Lie algebra. The "regular" 3D matrix representation in the axis-angle parameterization is obtained with so(3) generators L_x, L_y, L_z in their fundamental representation. Basis quaternions i, j, k (which can be represented by Pauli matrices) obey the same Lie algebra as L_x, L_y, L_z, but the group that it corresponds to (which is SU(2)) is a double cover of SO(3) (up to a sign), so they can still be used for implementing 3D rotations once you pick a sign.