joigus

Ok. I'm more trained in statistical mechanics, but from what I remember about real gases, and taken the cue from Studiot's post with the definition of the Joule-Thomson coefficient --which I had forgotten completely, I must confess--. b is the volume of the molecules (or 3x#molecules, I don't remember). So what's playing a role there is the size of the molecules as little hard impenetrable balls. It you take a look at the Van der Waals potential energy, it looks something like this: So it is slightly attractive at long distances but becomes strongly repulsive at short distances. Now, as the temperature from a statistical point of view is the average energy per degree of freedom, large temperatures imply that your molecules get to the range in their collisions that looks like the steep repulsive force you're seeing on the curve. The more average kinetic energy (temperature) the closer you approach the "hard ball" regime, so to speak. That's the only intuitive explanation I can think of. And as the b (the effective size of the molecules in the hard-ball model) is playing a role in your coefficient, I would say that could be the reason. I hope it helps.

I don't think it is, if what one naively thinks is that (weight-normal reaction) are canceling each other when the system is accelerating upwards. They're not. Forces that cancel are (N, -N) N not equaling "minus a weight" but also involving the pulling force (and so augmenting the apparent weight against the table). And that was the whole point. As show in the equations of motion above.

Yes, you're right, but now the system table-book is acted upon by other external forces, so that the normal does not compensate for the weight alone. I think that's what the book probably means. Otherwise I don't understand what it means. And that's why I suggested books sometimes are not clear. Although I have no problem to admit that people with more of an engineering background may find clearer something that, to me, that always tend to break things down in terms of fields and such, become more contorted, actually.. ------ </eqs.> I always think better with eqs. F is the external force that pulls up (the lift.) N is the normal, which equals the reaction -N \[m_{\textrm{book}}\boldsymbol{a}=\boldsymbol{N}-m_{\textrm{book}}\boldsymbol{g}\] \[m_{\textrm{table}}\boldsymbol{a}=\boldsymbol{F}-\boldsymbol{N}-m_{\textrm{table}}\boldsymbol{g}\] The action-reaction between table an book is seen here as compensation between N and -N. If you add both equations you get the motion of total system book+table: \[\left(m_{\textrm{book}}+m_{\textrm{table}}\right)\boldsymbol{a}=\boldsymbol{F}-\left(m_{\textrm{book}}+m_{\textrm{table}}\right)\boldsymbol{g}\] But N does not compensate with the weight of anything. It's just the pair N, -N that compensate each other. So (N, weight) is not a Newton pair. You don't need to write F twice, as it only acts on the table. The book never "knows" of any other force but the normal reaction and its own weight. </eqs.>

Only provided the A-B system is free of external forces acting on it. This is the point. So my advice would be: check your book, the diagram, some sentences, because they may clarify that.

I've heard him say it in some lecture. He may well have changed his mind. People change their mind sometimes. But I'm not sure. I'll check more. He certainly speaks of free will in some sense. I do too, as an emergent concept, as I said. Again, I heard him say that in a public lecture. But I could be wrong. But you still haven't answered my argument that something being undesirable is no good reason for ruling it out as a sufficient reason (now you're going to kick me in my Leibniz, I know.) For example: Darwin's theory must be wrong because a world based on competence for survival is too bleak and unforgiving. Well, you do what you must in order to alleviate the consequences; don't just deny the element of competence for survival and how it partly shapes the evolution of organisms. Here I agree. I'm not sure that for the same reasons. But my definition ignores that the question goes deeper and determinism/free will is more involved. It's tiresome to have too many choices, so constriction of choice (less freedom) actually nurtures a subjective feeling of acting more freely, when really what is happening is that your "internal determinations" act with less "internal friction" so to speak, being more like a weight falling on free space rather than turbulence and friction (agony of inability to decide.) And yet, your subjective impression is that you're doing exactly what you want. You see? Exactly what you want!!! You you can't change it. If constriction of choice leads you to believe that you're acting freer, something's fishy at the core of your instinctive notion of freedom. And... dualistic? I'm nothing that ends with -istic!!!

Ok. Something I didn't understand in the original proposal. Was the table accelerating upwards? Then, of course, it wouldn't be a Newton pair because the system book-table would be accelerating upwards with a total acceleration. Something that wasn't in the premises. Nevertheless I gave Ghideon +1 as the explanation seemed quite thorough.

Are those infinitely many Newton pairs? Contact forces are fields in disguise.

|S| is the cardinality of such hypothetical set sandwiched between aleph naught and aleph one.

Sure. It's, \[\aleph_{0}<\left|S\right|<2^{\aleph_{0}}=\aleph_{1}\] The possibility of a cardinality between the discrete and the continuous. It's one of Hilbert's 23 problems.

On the department of grievances, by the way: This Roman-numerals person who dropped by yesterday with an ill-posed question about energy and atoms, didn't find any of the answers satisfactory, insulted everybody, declared this site as a very, very, very bad site, and has been sniffing around the site for 24 hours non-stop.

OK. But please be aware that sometimes the book that one is using is the problem. Here's my explanation: If the book from the example is on the table, pushing against it by Earth's gravitational pull, and the table is rigid, so that it transmits the equal and opposite, (but minuscule in relation to Earth's mass) gravitational pull that the book exerts on the Earth, and none of them is accelerated with respect to each other, and all other forces into play are compensated so that the table doesn't break apart, deform, etc.; so that they just act as constraining forces; then I can see no reason why that cannot be considered as a Newton pair. If you picture the "binary system" (book) + (Earth and table) in empty space as a two-particle system, as Ghideon is suggesting, I think you will understand what I mean. If you don't find my explanation useful, please pay no attention.

Sorry, I think you may be making a very interesting point here. Just terminology, what do you mean by "collocating function"? By "squiggles" as synonymous of, or implying, "zeros of the function" or the derivative? And calibration points: Fixed points in your approx.? The zeros of solutions to physical problems or their derivatives are another old interest of mine. As to granularity vs continuity, I see an interesting possibility in the fact that an intermediate cardinality between cannot be reached by logic.

Couldn't have put it better myself. You reminded me of a Seinfeld episode titled The Opposite. I agree and sympathize with what I think may be the OP's general intention. It's the bit "everything we know is wrong" that bothers me, and I would have downgraded with "grading logical operators" like "most of" and "not quite true." That worries me quite a bit. Multiverses and pre-big-bang scenarios come to mind. One concept that I find quite interesting is that of hidden assumptions. It is conceivable that in every scientific perspective we have a blind spot for some hidden assumption. Like, e.g., that you can always separate system/environment, or similar. I don't think aliens in another part of the universe would reach very different conclusions from us. Think about it this way: Scientists and mathematicians from different centuries and under very different cultures have reached the same conclusions, theoretical structures, and experimental results, over and over: Madhava and Leibnitz, Hero of Alexandria and James Watt... And Schrödinger, Heisenberg and Dirac stumbled upon the same quantum mechanics from very different lines of reasoning. And Dirac was certainly an alien. Whatever it is science is doing, it doesn't seem to depend on who's doing it.

Maybe this concept is better filled inductively. For many people who end up defined as crackpots their self-assuredness is inversely proportional to their knowledge of the subject. That would be the crackpot's second law. LOL Honest mistake. They're both "g"'s

belief. Sorry I'm glad that you've brought it up. Some so-called scientists of the human psyche have noted that too many choices lead to agonizing over what to do, rather than making you feel the leeway that you're apparently given. If that's any indication at all about whether we've got free will, I think it goes in the direction of saying that we haven't. The question of range is important, but even more is the question of concern, I think. What's at stake. The Libet experiment has been claimed to prove the absence of free will. I'm not so sure about that, because as far as I understand, it had to do with decisions that were irrelevant to the person who was making the choice. Quite a different thing would be to test decisions and see if the machine can guess it right beforehand when there is something really important or valuable at stake for person under scrutiny. There are other criticisms to the Libet experiment, as I've just noticed. I'm no expert on that.

I'd feel more comfortable with 37 % free. Sounds less made up. But for the sake of argument... I see you quote Daniel Dennet in your profile. (I don't know how to do that yet on my profile, by the way.) It's precisely Daniel Dennet who has made really eloquent arguments about how nor human beings nor anything biologically based can be free. One of the most important arguments kind of answers your next objection: Yes, I agree, but: 1) Should we accept anything just on the grounds that it's better for us to believe it, that society would work best, or better? (this argument is not new, nor mine, actually.) 2) Daniel Dennet has said, or perhaps suggested, if I'm quoting him correctly, that not having free will is not that bad, once we realize everyone would be willing to behave properly if they want to be respectable members of this society. You may well get away with saying 'it was my bad youth, your honour," but if you want to be able to be trusted, sign contracts, get a job, etc., you'd better abide by the rules. So there is an external pressure, so to speak that always keeps things in order, to an extent. The analogy of "pressure" is not that far-fetched, actually --see below. The very fact that, even in spite of those pressures, there are people who still break the rules, should give you some pause as to whether we're really free. 3) Another argument, which is my own or maybe some regurgitation I can't remember the origin of, but came to me inspired by Daniel Dennet's words (as an extrapolation of his thoughts, I must confess,) is: OK. Suppose I'm right and we're not free. None of us is actually free; we're all acting based on the script that the molecules in our brain (including enormously complicated interactions: personal history, molecular accidents, so on...) are telling us. There are bound to be people out there who do believe everyone is responsible for their acts, so you'd better behave (abide by the rules,) because the idea of free will, no matter how it has emerged, is operative in the world. So no matter what you believe, the world is acting out the role of responsible beings going about their business knowing all the time what they're doing and being able to do differently if they wanted to. Even if that's just an illusion. The aspect of emergence is very important, I think. And even if society becomes unconceivable scientific and deterministic in their believes (something I think we're very far from, to be honest,) there would still be Dennet's argument that if you want a series of good things in your life, you better behave. So, even if the world is based on just molecules doing their microscopic business, these molecules manage to produce, as an emergent property, this illusion of free will that works very well as a deterring mechanism for us all by trying to to get us out of trouble. Be prepared you too, you don't know who you're messing with, mister. Just joking, of course. A pleasure. I forgot to answer this. By "free will" making any sense I mean being put in the same situation and being able to do otherwise. I know some people use quantum mechanics to argue about free will, but I'm kind of hoping we won't get into that.

Only people who feel really involved with higher levels of certainty can be happy that their intuition is wrong sometimes. And I'm happy too about what you say, yes. But if you won't check, you will never know. Also, only people who feel really involved with higher levels of certainty can be sad that their intuition is right sometimes. Wishful thinking is only too common. Thank you for your words. And as crackpotism is rampant and a danger for the well-being of the human kind, you will always find me there.

Because of quantum mechanics, electrons tend to be in "valleys" of their potential energy well called molecular orbitals. For an isolated atom those would be atomic orbitals. In an isolated atom, the electron is not always better in the valence shell. Depends on the temperature too. On the other hand, rarely are atoms completely isolated, except maybe in a Penning trap. Somebody will correct me if I'm wrong, but I don't think the electron in the valence shell of a Lithium atom would be very "happy" being there. I concur with Studiot that there's something missing in your premises. Maybe you don't want to say an "isolated" atom. We'll work through it, I'm sure. Clearly something's bothering you.

Fair enough. That's why I said 'arguably.' Right now I'm thinking about mass. But I promise I will read carefully your posts and replies on the matter and give you a reply in due time that will meet your high philosophical standards. For the time being, I have a feeling that the molecules in my brain are compelling me to think about mass right now as much as the molecules in your brain are compelling you to think about free will. But in order to take the discussion as far from the realm of opinion as possible, there's a simple experiment you can conduct: If you're right, and I'm wrong, which may very well be, and both you and I are 100 % free, why not dropping the topic no matter how much you want to argue about free will? For the likes of you or I that's harder than passing on a beer, isn't it? It's a pleasure making your acquaintance, Eise. I'm going for that beer, I can't help myself.

unnecessarily adverb /ʌnˈnesəsərəli/ /ˌʌnˌnesəˈserəli/ without any need; in a way that is not needed or is more than is needed I think you misspelled "inconveniently": inconveniently adverb /ˌɪnkənˈviːniəntli/ /ˌɪnkənˈviːniəntli/ in a way that causes trouble or problems, or that makes something more difficult The house is inconveniently situated for local schools. OPPOSITE conveniently My bad. You're right. I brought up the idiom and there's a clear difference.

LOL. I'll watch my step. (my emphasis) Thank you. Maybe some day I'll ask you some questions once I've given them a good deal of thought. Well, you're one of the most impressive amateurs I've ever crossed paths with. I suppose James Hutton was an amateur as you. Get ready for some questions too. Not only about GR. And I agree. I've just run out of points for Markus and you. None of you is here for the points, though.

I agree that there's some indefiniteness in your problem. AFAIKS, e.g., what is at the other end of the hose? 1) An infinite reservoir of water (or another fluid) 2) A finite, or relatively small, and closed, container of water (or another fluid) 3) It is a dead-end hose 4) Another pump, sucking water in whatever direction Some hydrodynamics questions are extremely difficult, because pressure gradients do not distribute homogeneously, neither do they transmit the action as a symmetric reaction at the other end. Part of the pressure would be transmitted to the walls. The detailed answer depends on how the pressure gradients distribute according to differential equations. The most general one is the Navier-Stokes PDE system, which is notable for its difficulty. For a static regime and an incompressible fluid with no vortices (perfect fluid,) I think that your question could be worked out, even if qualitatively, if you gave some more details. People have been racking their brains with similar problems for decades. See, e.g., the Feynman sprinkler. One I particularly like (because I was able to qualitatively solve it, I think, although never to the satisfaction of the OP) was this: Does a fly hovering on a plane really weigh? But then this particular person was extremely difficult to satisfy.

Taeto has answered very proficiently your longer-worded question, which has a different scope than the title really. Namely: "How does one prove that a Fourier transform is well defined?" I don't remember the details, nor can I find them on Wikipedia, but a Fourier transform is well-defined when your function is piecewise-continuous. That means it better not have an uncountable number of discontinuities. But the definition is very solid, in the sense that you can even define it for some non-integrable functions or even temperate distributions (strange objects, like the Dirac delta function, that your garden-variety functions can be integrated against.) I hope that adds significantly to your question.

Thank you. I'm sure any comments you may have to add will be most interesting. Here I meant domain and spectrum.

It's an illusion, clearly. It's an illusion, arguably.