joigus

Perturbation techniques is a different idea. It applies to both classical as well as quantum theories. It has to do with studying an interaction that you cannot solve exactly as one that you can solve exactly plus a small deviation from it, or "perturbation," and then expanding the perturbed states as a power series of the small perturbation parameter, and the states of the exactly solvable theory. The closest to what you're saying is what @Halc has explained, AFAIK.

It interacts badly with the round brackets also when you want to show the inline maths tags. It may interact badly with other tags too. I think the safest thing when you want to show code is the code button on the toolbar.

Meanwhile in Nepal... Here's some people paying good money to get a dose of harmful radiation, compounded with hypoxia, frostbite, and perhaps more, all compounded by stress from overcrowding. https://www.bbc.com/news/world-asia-48401491 It is not inconceivable that some of the same people would gladly buy those stickers.

@Anamitra Palit, Maybe you can try some LateX editor, like, https://www.google.com/search?q=wysiwyg+latex+editor+online&oq=wysiwyg+LateX+editor+online and then nest your equations with braces, like this: \[ \textrm{ \[ <LateX code here> \] } \] or with round brackets for inline maths. This would considerably improve the communication of mathematical arguments.

That one's easy, and much less convoluted than the story about the CC. It was Maxwell's equations for electromagnetism. He actually didn't think in terms of 4-dimensional non-Euclidean space for quite a while, until Minkowski came up with his formalism of time as a fourth dimension. For Einstein it was always Maxwell's great unification of electricity and magnetism what was the inspiration, and the experimental corroboration by Hertz. From then on, he always thought the equations must be correct, and must be telling us something very deep about Nature. The equations say the speed of light doesn't depend on the observer, so I'll take it to heart and see what the consequences are. Space and time must be changed to comply with these field equations? It must be it. AAMOF, he conjectured gravitational field equations as kind of an analogue of EM --Maxwell's equations-- for curved spaces. But two corrections about the currently accepted formalism --: Light is not ether. In fact there is no ether, as has been proven to death. There is no lack of an appropriate term: Space-time is the accepted term. It has symmetry properties that explain a great deal of how light behaves. The non-Euclidean character is not a property of the field, but of the space-time continuum. Fields are defined on it, so they inherit this structure.

AKA "overcoming the curse of knowledge." I like to think about it in terms of tying the knot, and untying the knot.

That's nice. Only thing is that you're forced to leave out the braces, which maybe you want to make explicit. But you gave me an idea. Let's see if it works: \[ \textrm{ \[ \frac{a}{b} \] } \] \[ \text{ \[ \frac{a}{b} \] } \] Look at that. It does... You only have to escape the math mode inside the LateX with the \textrm{} or \text{} commands.

Oops! Mmmm. Ok. I have nothing to oppose to that. But, as Swansont has pointed out, I misunderstood. I thought you were talking about arguments in the direction that "something can come from nothing", when you were talking about the argument "anything must come from something". These kind of arguments always give me a headache. Please, carry on.

It must have been Lawrence Krauss: https://en.wikipedia.org/wiki/A_Universe_from_Nothing But it's not a postulate. I would call it a philosophical argument on the periphery of physics.

\[ \frac{a}{b} \] \[\frac{a}{b}\] Oh, so that's how you escape LateX. I thought I'd seen it somewhere. Thank you @Ghideon.

A sphere has no poles. A chart does (a system of coordinates). You can map the sphere in many ways. They all have different poles. So spheres have no special points. Neither does the universe, as far as we know. But it is a mathematical theorem that no matter what the system of coordinates you choose, you must always leave a point out, which would be your pole. But which point it is is up to you. Expanding universes have their problems when cosmology meets quantum mechanics. But that's outside of the scope of what you are proposing here. You seem to be thinking of a classical universe, and we do know already that it's quantum.

I use the backslash + square brackets for maths sections, and the backslash + round brackets for inline maths. I don't know how to escape LateX to show the code. Perhaps a "code" tag... \\[ \\] \\( \\) Simple backslash is a common escape tag. But it doesn't work here.

No. I meant the Euclidean norm of both vectors \( \boldsymbol{\alpha} \) and \( \boldsymbol{\beta} \). That's why I wrote them with boldface characters. If you read carefully, I did introduce this qualification: So I meant different things by plain \( \alpha \) and \( \left| \boldsymbol{\alpha} \right| \). I should have been more explicit --and standard-- though, and written, \[ \left(\alpha^{\mu}\right)=\left(\alpha^{0},\boldsymbol{\alpha}\right) \] I'm sorry about that. Nevertheless, the argument stands.

AFAIK, trans fats are known to be harmful on a number of levels. I'm here really to kindle conversation from the experts and learn more. Take everything I say with a grain of salt. Those studies that I remember mostly showed correlations between trans-fats and atheroschlerosis or cancer. Then I learnt from the MIT courses that part of the problem was the shape of the molecules affecting the melting point. What @CharonY says suggests that more research has further clarified the metabolic connections at the molecular level, which I think is very interesting.

This image from Khan Academy is a bit more realistic: https://www.khanacademy.org/science/biology/properties-of-carbon/hydrocarbon-structures-and-functional-groups/a/hydrocarbon-structures-and-isomers The chain of, eg., -C-C- that is usually represented as a straight line, looks as a gentle zigzag, because the carbon atoms in the main chain are really tetrahedral, not linear. The double bonds make this effect of trans vs cis even more dramatic, as you see in the picture. So they differ considerably from the schematic representation. The effect of both hydrogen atoms being on the same side (cis) amounts to both atoms bending the molecular skeleton to the opposite side. The fatty functional group doesn't affect this feature much, unless they're sitting next to the double bond.

I see. Very interesting. The info that I had comes from an outdated source. Those were lectures from 8+ya --MIT Hazel Sive & also Graham Walker, which are on Youtube. What you tell me suggests that a lot of research has gone into studying them in much more detail. Thanks a lot. It also suggests that TFAs are in part incorporated to the membrane after they're metabolized. Would that be correct?

I don't think they're the same. The static universe is one model; the steady state is a different one. It's explained in the reference you provided. The steady state is better known as an idea by Bondi, Hoyle, and Gold. I didn't know that Einstein had anticipated a steady-state model. He apparently didn't publicize it much. Einstein's hypothesis of the constancy of the speed of light dates back to 1905, while redshift of the galaxies was discovered in 1929. So, no. Edit: Even if you consider Slipher''s observations, as Swansont says, it doesn't fit the timeline. OTOH, so-called recession velocities are actually due related to an expansion factor in the FLRW solution to the Einstein equations, which are quite different from local velocities. So, again, no.

AFAIK, the problem about trans fats --for humans in particular-- is their high melting point. Because they --as John said-- have double bonds in the middle of the chain, they allow for hydrogen atoms to be either both on the same side (cis) or on opposite sides (trans), so that trans give rise to molecules that tend not to coil or bend so much. This results in straight chains that function pretty much as saturated fats when it comes to melting points. Molecules that tend to coil act more "pointlike" than molecules that remain "straightened up", which is what qualitatively explains the difference in melting points. So they tend to clog more easily in your blood. I'm not aware of metabolism having to do with other negative effects, but it could be.

That's not exactly what seems to have happened. Einstein came up with a cosmological constant in order to account for a static universe, which was his, shall we call it, theoretical prejudice. Einstein's field equations do allow for a term which is proportional to the metric tensor, which has so-called covariant derivative (identically) equal to zero. So Einstein realised that adding such a term did not change the auxiliary condition that was the main drive for his conjecture. Namely: A gravitational tensor which had identically vanishing covariant derivative. Einstein knew the right-hand side of his equation had to be conserved, so he guess a left-hand side that was identically conserved so that his equations were some kind of curved-space analogue of Maxwell's equations. was identically conserved --had identically zero covariant derivative. Having identically zero covariant derivative are technical words for expressing a local conservation law in a curved space-time (what disappears inside a volume must do it by crossing its surface.) You can always add this term without changing this property. The problem with Einstein's solution was that it produced an unstable solution, so it didn't make much sense. But also, had he guessed that introducing a positive cosmological constant would require an expanding universe, he would have anticipated an experimental discovery that only came decades later by Penzias and Wilson. He regretted that predicting an expanding universe had been within his reach but he fell short because of this theoretical prejudice that the universe must be static. That's more or less the story. Maybe I can provide some better explanation or bibliography later. Or other users can further clarify.

You're right. I was in doubt about that too. Maybe a ferromagnetic salt could play the role of the "iron particles"? You could package permanent magnetic dipoles, while having the particles be soluble in water... Just a thought. https://en.wikipedia.org/wiki/Ferrofluid Edit: Ferromagnetic salt is probably not the proper term. Paramagnetic salt is more like it. This checks with what I was thinking, although you know much more than I do about this. My impression was that it wouldn't release/absorb much heat.

It's a beautiful experiment, and it seems to me that it's quite doable. But keep in mind that, as long as your jar is not a closed system, it doesn't have to display entropy increase. Same reason why the gas in a refrigerator can be made to cool by expanding adiabatically --not getting or giving heat from/to the outside--, but exchanging work. In this way, the universe as a whole would see its entropy increase, but parts of it --the jar's interior-- would see their entropy decrease. Why don't you try it? Some people here could give you advice. A Dewar flask, a magnet, and a thermometer could do the trick.

Very interesting. The gif didn't work for me. I've found it here, and you can really see the vortices pushing against each other: https://phys.org/news/2021-01-reveals-jellyfish-virtual-wall.html

The book itself in an excellent example of gravity too --6 pounder.

Absolutely. I don't disagree with @HallsofIvy about the specific point. What I meant, of course, was in the context of the general flow of information, which was exclusively unidirectional. I'm seasoned enough not to recognize the pattern of someone being disingenuous on purpose, just for laughs. Making fun of people whom, deep inside, you feel to be more intellectually capable than yourself, and relishing in the cheap thrill you get from pulling it off: "Look at them, they think they're so clever, and they can't tell I'm just having some fun at their expense."

I gave them the benefit of the doubt for a while, way past the reasonable... Just in case. But no. They opened another thread, and then another, and another, without ever addressing anything they were told, not even the slightest trace of a reasonable followup question. Only more questions to new, equally silly, unrelated "problems." The last thread's title was "What's 2+6?", and they said "I'm totally confused," which was the opening statement. Really? Language barrier? I don't think so. Now, you don't have to read between the lines to know this person was pissing on all of us. Good riddance, that's all I can say. I've seen that pattern before and I have my ideas about what can lead a person to act in that sorrily pathetic way, but I would be off base speculating about that here. Then there are people working on social experiments out there. I just know, because I happen to know one scientist who works on that, so... You never know.