joigus

You guys are very valuable members of this community. I've been swept away by both of you several times. Yes, you're right, @studiot. But the OP was about the very special, very non-prime number \( 2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23 \), factor of the first 9 prime numbers.

Why wouldn't rational and irrational numbers be summable?

Sorry, the problem reduces to how many numbers there are between 1 and 223'092'869 that are not divided by 1, 2, 3, 5, ..., 23.

Now that I think about it, the product of the first 23 is, as you say, 223092870. Now, the number of ways in which you can sum two (arbitrary) natural numbers to give N is just 111'546'435, because 223'092'870 is even. So the problem reduces to finding how many numbers there are between 1 and 111'546'435 that are not divided by 1, 2, 3, 5, ..., 23 (the 1st 23 primes). The only thing I can say is that's not an elementary problem. How did you get your conjecture @Tinacity? Counterexample: \( \pi \) is irrational; \( 1-\pi \) is irrational too. but \( \pi + 1- \pi = 1 \).

She didn't specify, but I thought it was kind of implied... For arbitrary products of primes, certainly nothing like that can be proved with the present mathematics. But for 223092870, I just don't know.

The maths for predicting that kind of thing is called number theory. I know very little about number theory. It studies connections between numbers. The result that you propose reminds me of some theorems by Fermat. Have you proved it, or is it just an intuition? Maybe @wtf or @Sensei, or @mathematic, or @taeto can help you. @studiot is encyclopedic. Maybe he can help you too. Number theory is not very interesting for physics, AFAIK. And physics and mathematical physics are my turf.

It's a mark-up language; a subset of XML: https://www.w3.org/Math/whatIsMathML.html Sorry what I posted was inline LateX, but you can read it as MathML by right-clicking.

This could be hard proof of tunneling for macroscopic objects...

Thank you, Zapatos. I've found a little bit more of the geological history on Wikipedia: https://en.wikipedia.org/wiki/Kegon_Falls From https://www.visitnikko.jp/en/spots/kegon-falls/ The place is such a tourist magnet that it's kinda difficult to find something more science-oriented about it: Geology, fauna and flora, etc.

Another real-estate agency's hopes have been shattered, as we speak.

I always thought they were laughing.

NASA has found water on the Moon's lit surface. https://www.npr.org/2020/10/26/927869069/water-on-the-moon-nasa-confirms-water-molecules-on-our-neighbors-sunny-surface?t=1603753199805 https://www.nature.com/articles/s41550-020-01222-x#_blank Abstract from Nature Astronomy: Interesting news, though not Earth-shattering, probably.

Testing some inline maths like \( d^{3n}xd^{3n}p\rho\left(x,p\right) \), \( y = x^2 \) or \( \varepsilon_{\alpha\beta\gamma}\varepsilon_{\alpha\mu\nu}=\delta_{\beta\mu}\delta_{\gamma\nu}-\delta_{\beta\nu}\delta_{\gamma\mu} \) should work.

Mating-brilliant. (@ Koti.)

LOL. A noose, that's what it is. <giggle>

I was thrown away from that discussion with the escape velocity.

IOW, why are acids so important, so central to chemistry? They say the basic unit of chemical exchange is the electron. But that's only half the story. Protons are very powerful mediators of chemical reactions too. And the reason is that the size of a hydrogen atom compared to the size of just a proton (ionized hydrogen) is like the size of the Earth compared to the size of an orange. So when you have a substance that is capable of liberating protons, you're liberating myriads of little "positive versions of the electron," so to speak. That's why there is no central concept in chemistry of how easily a substance can liberate any other ion, like e.g. Na+. But liberating H+ is very powerful, very reactive. Protons are elementary particles, small as can be, and move about very freely, especially in aqueous solution. The mitochondria in your cells are powerful proton-pumps.

Then I can trust you. Some people even proactively overreact. I always consider things reactively or retroactively.

I'm reading more info about it and I also made a mistake. It was 5 times. I've read it was 7 somewhere else. But Ian Plunkett's --Twitter's spokeperson-- words are: When someone "proactively implements" something my BS alarms go off. And the adverb "proactively" would really make it worse, not better, if it did happen. OTOH, these claims seem hard to just make up. The guy took screenshots. Are they fake? Why no explanation about the screenshots? And it's true that Trump's Twitter account is small potatoes, but this guy has access to top-security material, I suppose. Can't anyone give him a quick tutorial on safety security procedures? I know you've proactively considered these possibilities, @iNow.

Careful. They could hire you as digital-security expert. Oh, I always think things can be made worse.

The story from a non-political medium: https://techcrunch.com/2020/10/22/dutch-hacker-trump-twitter-account-password/ If seven tries is all it took Victor Gevers to use the psychological method to crack the US president's Twitter account. How bad is that for the Western world? Your thoughts eagerly awaited.

You seem to be overly interested in words and moot points. Saying just "thermodynamics" suggests either classical, chemical or equilibrium thermodynamics; all of them based on equilibrium. There's also statistical mechanics, but that's almost never called thermodynamics. And there's non-equilibrium thermodynamics, but that's such a misnomer (it's not about just heat, temperature, and the like) that everybody referring to it always mentions it by the whole name, "non-equilibrium thermodynamics," only to make clear that it's not thermodynamics (T, Q, etc.) Here are all as covered by Wikipedia: https://en.wikipedia.org/wiki/Thermodynamics#Branches_of_thermodynamics https://en.wikipedia.org/wiki/Non-equilibrium_thermodynamics You tell me which one you're referring to. There's also kinetics, but I'm not interested in climbing the tower of Babel. I meant (and I said) regions of phase space. Regions of phase space are not regions of 3-dimensional space. You're confusing both. I mean volumes in the way of, \[d^{3n}xd^{3n}p\] IOW, regions in a humongous 3Nx3N-dimensional space. It is in that space of huge dimension where sampling is robust. Macroscopic systems are ergodic, meaning that time averages give you a very good idea of phase-space averages (averages to all momenta and all positions) when systems are at equilibrium or going round and round in cycles. The molecules you describe as going up to the outer reaches of the atmosphere have to go back and recycle, participating in the overall thermal and dynamical processes, and exchanging the energy. That's the key. But all this is quite academic and, if pressed, I wouldn't be too sure of anything, the way you seem to be. Here's a much more intuitive explanation of why sampling in this way works even for chaotic systems: Tim Palmer's lecture at Perimeter Institute Lecture: Climate Change, Chaos, and Inexact Computing 11' 40''-16' 37'' (I copied the link starting at about the time when he explains the point.) I'm not an expert in climate change. @Area54 or @Ken Fabian can probably give much more accurate information and point out my excesses. I just wish to argue with my toolkit. And with my toolkit at hand, the arguments about "global warming" (however much of just a catchphrase that is) make a lot of sense to me.

Yes, you're right. My own pessimistic thinking is that we would be so busy at each other's throat that nobody would care about other survivors out there. But your question is very interesting, as well as the answers so far. I do believe with you that attempts at communication with intelligent animals should be a good laboratory for that hypothetical situation. Maybe we can train some animals to speak once a code is "agreed."

This is a very good idea. I just hope when/if they read the message, they don't mistranslate it to "Hey, there's food and other usable energy here."

Thermodynamics is the study of equilibrium. The Earth is not a system in equilibrium. Climate science most emphatically is not sheer thermodynamics. Thermodynamics, e.g., does not allow you to calculate anything statistical, like, e.g., fluctuations. Statistical mechanics does. In (most) statistical mechanical systems, you can see that the temperature is the average kinetic energy per degree of freedom. These degrees of freedom are coded in what I called specific heat. Because systems are ultimately Hamiltonian, they satisfy an interesting mathematical property: The phase-space points can mix all they want (and in chaotic systems they do, which makes averages more robust, not less), but they do not contract in volume, meaning that microscopic systems spread their dynamical information very efficiently. IOW, there is no chance that a small region of phase space can store big quantities of energy making local averages non-robust, as you are suggesting. As to local cooling: If the first statistical moment of the distribution is shifting, an increase in the second moment is exactly what I would expect before the system reaches the next closer-to-stationary stage. If the variance goes up, some places would overheat and others would "overcool." Nothing unexpected there, because the system is "trying to equilibrate." So temperature measurements are significant. But they're not the whole story, as has been pointed out to you over and over. The sea is nearly a perfect absorber of radiation and the ice caps are nearly a perfect reflector (albedos.) There is the question of sea currents too. The ice caps should be building up by now because we are well within a Milankovitch cycle. They're not: It's just the opposite. This will interrupt the circulating flow in the seas. None of these important details seem to have caught your attention, which would have amounted to an interesting conversation. All you're interested in is to not let go of your strawman (the average temperature parameter) and punch its face repeatedly. The average temperature, being significant, is the catchphrase you've chosen to attack. It's your voodoo doll against climate science. Your point does not stand, it's a blurry blob. Your strawman does.