darklight03/StackMIAsub huggingface dataset:
I recently asked this question about whether there was a "distance" between two galaxies where both the gravitational force and the influence of dark energy would be balanced. The answers and comments seem to indicate that there is indeed such a "radius" around a galaxy. I was very interested in this, so I contacted the authors of this paper about this phenomenon. I asked them if it could be possible to have a satellite galaxy orbiting a bigger one just in the point where there would be a balance between the gravitational attraction of the bigger galaxy and dark energy, so that the satellite galaxy orbit would not decay (through gravitational waves, tidal forces...) and avoiding its fall eventually towards the bigger galaxy. They replied that the answer was basically yes, and that they could keep that orbit as long as there was no external perturbation modifying these orbits. But I had one more question about this scenario. My question is: If that balanced state would be possible, would there still be tidal effects between the two galaxies (So that some of the orbits of planets and stars inside the galaxies could be somewhat modified) but without making the orbits of the galaxies decay over time...? I mean, imagine a satellite galaxy orbits a bigger galaxy just in the radius distance where the influence of gravity and dark energy are balanced out. Is it physically possible (at least theoretically) that the tidal forces between the galaxies may affect some of the planetary systems' orbits in these galaxies (for example changing the orbits of planets around their stars like for example making them orbit their stars further apart)? And would these tidal forces disrupt the satellite galaxy from the distance where gravity and dark energy are balanced out? Or without any external perturbation, it should keep orbiting at that distance (even with these tidal forces between the galaxies or the gravitational waves emitted from the orbits around the bigger galaxy)?
I'm an amateur and this is my first question here, I'm trying to formulate question about a general representation I have in mind after trying to grasp the idea of relativity and the concept of space-time. We always talk about the "speed of light", but it's a bit of a misuse of terminology, since we should really talk about "speed of electromagnetic radiations (EMRs)". Moreover, we recently confirmed that gravitational waves, despite not being of the same nature as EMRs, travel at the same "speed of light". Considering that, according to relativity, an object travelling at the speed of light would not experience time passing, and that it doesn't seem to make sense to envision travelling faster than the speed of light, Would it not be natural to consider the "speed of light" as being the "speed of time", or, let's say the speed at which the present propagates through space ? This way the speed of light and gravitational waves would be actually instantaneous, or at least as much instantaneous as the universe can be, and each point of the universe would "communicate" with the other points at the "speed of light". Usually when trying to [in]validate this idea by reading other questions or other sources, I find complex explanations that do not really help me evaluate the correctness of the broad model I have in mind. I do not know if this is something that seems quite obvious to everyone already, but I couldn't find it expressed this way, and this mental picture really makes sense to me. If it does not correspond to our current state of understanding, could you give me some clue of where my formulation does not fit reality ?
I understand that "laws of physics" is a bit of a misleading term since all they really are is just us applying a logical statement about observed physical phenomena in a way that allows us to predict or understand said physical phenomena. That said, what my question is getting at is whether there are any laws of physics that "hold at all levels?" The idea of the "invariance of physical laws" is, and has been, a key notion for developing new theories and furthering understanding of phenomena. But for a lot of given laws, there seems to be some system or situation, in which the laws must be modified or corrected in order to hold, or is simply not applicable. I'm not concerned with the numerical accuracy of physical laws, which seems to be the focus of similar questions on this forum. An example of this would be Newton's laws of motion, which break down at the quantum scale (despite having analogous principles) or at relativistic speeds. Maxwell's equations have been described as the "solutions" to electromagnetic theory, but they are only correct up to the point of treating magnetism as an unexplained phenomenon and not the result of relativity. Are there any laws that, with current and modern understanding, are always true (I'm not saying that they couldn't be found to not completely the case in the future, but we believe they are currently)? The only one that I could believe fits this description, is the second law of thermodynamics, in that "entropy never decreases." We have entropies for vastly different systems at vastly different scales, from quantum systems to human scales, to galaxies and black holes. However, this may be a little handwavy, as entropy has different definitions for different levels (Von Neumann entropy and the entropy of a black hole for example) and my understanding is that they do not translate from one another.
I am looking for a term to use as the name of a software project that I am working on. The project is a software tool, and this tool aims to be useful in virtually all software, so I am looking for a term that alludes to it being an indispensable, or perhaps even the most indispensable, tool in a profession, as in: the test screwdriver being the most indispensable item for an electrician the monkey wrench being the most indispensable item for a plumber scissors being the most indispensable item for a tailor etc. A different but equally useful direction of meaning would be a term for an item which is guaranteed to be present in a certain line of business or endeavor. For example: In a tire shop they are bound to have lots of tires, so tires are their ___ (fill-in the blank.) If there is one thing that mountain-climbers are guaranteed to use, that is rope, so rope is their ___ (fill-in the blank.) Every priest is bound to have a bible, so the bible is their ___ (You get the idea.) Besides the word "indispensable", other terms near the meaning that I am looking for (but unsuitable) are "essential", "sine qua non", "tool of the trade" and "staple item". I was seriously considering the term "bread and butter", but after reading about it I have formed the impression that it necessarily has a fiscal connotation, while there is none in my case. Please correct me if I am wrong. (Other than that, "BreadAndButter" would be an awesome software project name, despite being unconventional: the modern trend in software project names is strongly towards unconventionality.)
In textbooks on many-body quantum physics (e.g. Fetter and Walecka), Feynman diagrams are typically introduced after formulating the Dyson perturbative expansion of the Green's function using Wicks theorem. Then the Feynman diagrams follow as a way to conveniently represent the resulting integral equations. In most literature however, I have noticed that the language is somewhat different. Typically, after introducing the Hamiltonian a certain quantity of interest will be introduced, typically the single-particle Green's function of self-energy. Then there is often a sentence like: To evaluate this quantity of interest we sum the set of Feynman diagrams shown in Fig. X Then Fig. X will contain the perturbative series already written in terms of Feynman diagrams. It seems that the step where this series is formulated, as I find in the textbooks, is typically skipped. I can think of two possible reasons for this: The actual formulation of the Feynman rules from Dyson's equation and Wicks theorem is seen as trivial, and hence not repeated in typical papers. Even if the system is not a standard system treated in other literature. There is actually a faster or more intuitive way to write down the relevant Feynman diagrams from the Hamiltonian, without having to resort to the perturbative expansion explicitly. If this is the case then I would love to see a textbook where such a procedure is explained. Currently whenever I want to understand a paper I go through the whole perturbative expansion for the respective Hamiltonian, which is a very tedious and time consuming process. I would be very appreciative if someone more familiar with this field could tell me which of these is true. Thanks!
I think this is easier to understand the mapping of injective, surjective and bijective in terms of marraige proposal where men are from set A and women from set B but Is this analogy correct? Injective (One-to-One) Functions: If the marriage process between men from set A and women in set B is injective, this means: Every man proposes to a distinct one and only one woman Some women might not receive any proposals (i.e., remain unmarried), but no woman receives proposals from multiple men. So, in terms of our marriage analogy for injectivity: each man proposes to one woman, and no woman has more than one suitor. Surjective (Onto) Functions: If the marriage process is surjective, it implies: Every woman receives at least one proposal. All men have a marraige proposal and it's possible for a woman to have multiple suitors. For surjectivity: every woman gets at least one proposal Bijective Functions: For a marriage process to be bijective, every man must propose to a distinct woman such that every woman gets exactly one proposal, and every proposal is accepted. No man or woman is left without a partner, and there's no situation where a woman has more than one suitor or vice versa. To sum it up using the marriage analogy: Injective: Every man has a partner (definitely unique since they are not allowed to propose multiple women), but some women might be left without a partner. Surjective: Every woman has a partner and they are allowed to propose multiple men so that no man remains without a proposal Bijective: Every man and every woman have unique partners. Nobody is left out, and no overlaps in pairings.
Trying to solve a problem with a colleague of mine, we prove a theorem that I'm sure someone else must have had to come across but couldn't find anything about it. We needed a way to tell how far away from equilateral was any triangle. A measure for any triangle of its non-equilateral-ness. One idea (that later turned out not to be the best) was this: Let ABC be any triangle (the one we want to know how far away from equilateralness is). Let's take any of its sides, say AB. Let's take the point C' in which ABC' is equilateral, and which lies to the same side of AB as C. Let's measure the distance CC'. Let's take points B' and A' the same way we took C'. What we found is that the distances CC', BB', and AA', are equal. No matter which side of the triangle we start with, the result is the same. Is it known to be found earlier? Edit For the record, the proof goes this way: We have the original triangle ABC and three new points A', B', C' such that ABC' is equilateral and so on. Let's see triangles ABC' and AB'C. Both are equilateral and share the same point A. Let's see the geometric transformation that brings C' into C, B into B' and A into itself. Given that the distances C'A and BA are the same, the distance traveled by C' to go into C must be the same as the distance traveled by B to go into B'. Ergo, distances C'C and BB' are the same. By the same way we prove that the distance A'A is also the same.
I came across this sequence called Digital River, where the next number in the sequence is defined as the sum of the digits of the previous number, plus, the previous number itself. It caught my attention for some reason, and I wanted to analyse it. And I found some curious fractal-like patters. But let me begin by saying I am no mathematician, and I was just doing this recreationally, as I don't have the requisite tools and faculty to unwrap why these fractal-like patterns should appear. So I am posting my analysis notebook here in hopes of finding some answers. Now, I have also come across the summatory Liouville function, and it too has similar fractal-like patterns. So, could it have something to do with, or is related in any way to, digital rivers? Some of the comments say that Liouville has something to do with the Riemann-Zeta function. Could the Riemann-Zeta function also explain why fractals appear in subtraction of digital rivers? If so, could you explain how, in a way that somebody without an undergrad degree in Math can understand the source of these fractal patterns? And in doing so, could we formulate a theory or pattern of what other, similar kinds of sequences can show similar fractal-like patterns? Here are some of the fractal-like patterns I've found, to pique your curiosity to download my notebook: P.S. If this turns out to be an interesting problem that cannot be explained away trivially, and you want to work on analysing it together, then I'm happy to collaborate.
I'm sorry for mistakes if any; English is not my native language, but I'll try to explain myself as thorougly as possible. There is a geomerty topic about construction of different geometric shapes using just a straightedge and a compass (a pair of compasses, more accurately). E.g. a regular pentagon can be constructed under these restrictions, but a regular heptagon cannot. hyperlink My question is: why straightedge and compass but not pencil? I mean, the compass as a drawing tool does contain pencil lead. That's a fact. But using the compass in order to construct straight lines, in my opinion, is not rational. I think nearly all people use a pencil to draw a straight line with the aid of a ruler. And the ruler itself, of course, cannot draw anything on its own. I tried to ask it on a different site. I received two answers. The first answer was very ironical and I considered it rude and impolite. The answerer told me: "If you wish to add a pencil, than you must also add a boy and a girl (who will make your construction)." I'm sorry, but I didn't speak about boys or girls. Just a pencil, that's the plot. A very sad and unsatisfying response! And another answer was quite polite but obscure. I think it needs some clarification. The second person told me that maths never changes anything. The ancient Greeks just used the ruler (straightedge) and compasses, and that's all. So we don't have the right to change it. I think I could agree with the second answer, but I am curious and inquisitive. So why is it so conservative, why have the rules remained unaltered and why didn't anybody try to change anything? That will be my secondary question.
We have hydrogen inside a tube, and we induce a voltage on it; a current passes through it and light is emitted. The frequencies of light correspond to the differences of the eigenvalues of the energy operator, which is the observable in question, so it is customary to give a heuristic explanation that the electric energy produced an energy transition and the residual energy was emitted as light. At what precise moment did the wave function collapse in this experiment, if we try to describe according to the Copenhagen interpretation? How does that description work in this case? Can you maybe direct me to a paper that describes this in detail? Some more words to clarify these questions: I would like to understand if the wave function is supposed to collapse the moment the voltage is applied, or the moment the electronic transition happens, or the moment the light arrives at the spectrometer, or the moment it hits the photographic film. It would be interesting to know what event, in that interpretation, triggers the collapse. A worked-out model of the whole situation, explaining how one describes each component of the system, would be most welcome. Thanks a lot in advance! Edit: This post has been marked as needing more focus, I think by people that did not understand the point of the question, to whom I'm nonetheless very grateful for their feedback (but please if you're one of them, kindly explain better what's going on because I also don't understand your position). The question was phrased as a bunch of different questions in an effort to clarify it, but it boils down to this: what exactly is the role of quantum collapse in the standard quantum theory's description of hydrogen atom gas radiating in a tube? Thanks again.
I have a bit of confusion because when doing QFT and QFT in curved spaces this particular issue seems to be avoided. I have this feeling that when we quantize a theory, we somehow choose a chart and we stick to it. This feeling comes from, for example, the way we deal with Lorentz transformations in QFT, namely via unitary representations. In my head, change of coordinates is something more geometrical rather than algebraic as is done in QFT. I also asked a professor of mine and he told me that the usual way of quantizing things is chart-dependent and then suggested I read TQFT and AQFT papers for which I'm not ready yet. Can someone help me understand? I am searching for a mathematically rigorous construction of the quantization process (in canonical quantization) and if it can be done in a coordinate free way. Hope my question does make sense. EDIT: I think my question was misunderstood: I do believe that, of course, the physics in QFT is Lorentz invariant. But in my understanding of the process of quantization what we are doing mathematically is the following: pick a chart, construct Fock space/Quantize and then model in that Fock space Lorentz transformations via Unitary transformations. In this process, if I take another chart I construct a different (but canonically isomorphic) Fock space. So you see: in QFT (I believe) you don't treat change of chart in a more geometric way, but you model it algebraically. I think that what I'm saying can be seen in Wightman axioms: there is no reference on the spacetime manifold (of course the choice of the Lorentz Group comes from the isometries of the Minkowski metric but one can avoid talking about the metric completely) , it's purely algebraic. So are Lorentz transformations.
What's the word(s) for a feeling of disappointment when you've lost something of financial value? For example, let's say I'd just got an expensive LCD monitor from a raffle, but I accidentally dropped it and it broke, and I now lament its loss. I guess "lament" works okay, but it's not really a colloquial word, and it seems to refer to disappointment in a more general sense (like the loss of a good friend) rather than strictly about valuable possessions. The "value" in question should be strictly financial rather than emotional, meaning that you feel sorry for it being expensive rather than it having been used for a long time. I broke my brand-new LCD monitor. I haven't even gotten a single use out of it. I ... that monitor so much! Edit: Just to drive home the point, there's this word in my language that is very specific, because it's used in contexts where you miss out on something of monetary value, lose something of monetary value. For example, when you almost won a monitor but didn't because of one stupid mistake on a game show; or when you'd just won a monitor, but accidentally damaged it and made it unusable; when you actually owned a monitor, but lost it while moving because the delivery guys dropped it. In all these cases, the disappointment is purely monetary, because the monitor was just too expensive and it'd be hard to ever get another one as good. Even in the case of you having owned the monitor, you still missed it because it was expensive, not because it was with you through thick and thin. As an actual example, I have this tablet that's quite cheap, but if I ever "miss" it, it's because of my emotional attachment to it, not because it's expensive, because it's actually very easily replaceable.
There are a few river crossing problems that I have seen that share some common aspects. The cannibal and missionary problem is typical. All these problems involve moving everyone from one side of the river to the other side by using a boat to cross the river. Denote by complement, an arrangement that is equivalent to what you get at any stage of the problem by swapping the shore that each person is at. For example, the final position of the puzzle is the complement of the initial position. One thing that these puzzles have in common is that it is legal to start at the final position and do the moves in reverse order and in the reverse direction, going from the final position to the initial position. I noticed a relatively simple property of some of these problems that I am having trouble describing formally, let alone proving. In the solution to many of these problems there comes a point where a new position is the complement of the previous one. If you start at this position and perform the previous moves (except for the last) in reverse order, the last position will be the complement of the first, making the sequence of moves a solution. I hope that what I said makes sense. How could I state this in a more formal manner? It should be pointed out that you can't actually do a complementation, but you can achieve the equivalent. For example, in the cannibal and misssonary problem, you go from having two cannibals and two missionaries on the far shore to having two cannibals and two missionaries on the near shore. This is done by having one cannibal and one missionary move from the far shore to the near shore, but could have also been achieved using complementation. I thought that the idea of complementation would simplify proofs. Here is a more compact description. Let the positions be represented as (A,B), where A is the set of people on the near shore and B is the set of people on the far shore. The complement of (A,B) is (B,A). The starting position has the form (S,-) and the final position is (-,S). If we have a sequence of moves (S,-)->...->(W,X)->(Y,Z) then we could perform the moves in reverse order, giving (Y,Z)->(W,X)...->(S,-). Suppose that you can go from (Y,Z) to its complement, (Y,Z)->(Z,Y). I would like to be able to show that we could go backwards (Z,Y)->(X,W)...->(-,S), giving a solution to the problem. Here is an idea of what I would like to be able to say. In going from (W,X) to (Y,Z), the boat ends up on either the near shore or the far shore. In going from (Y,Z) to (Z,Y), the boat ends up on the opposite shore. This means that, in going backwards, it will be able to go from (Z,Y) to (X,W) and generate all the complements of the first set of reversed moves.