darklight03/StackMIAsub huggingface dataset:
In statistical thermodynamics we can prove that the evolution of a system minimises some potential with units of energy (e.g. energy). This can be done purely statistically, by using the first two laws of thermodynamics, and showing that the state where the appropriate potential is minimised is the most likely one. Therefore, minimisation of energy is just an argument about entropy. Is there a similar (or any other) proof of the principle of least or stationary action? I have seen it referred to as an axiom, but is there at least a possibility that there exists an underlying theory from which it can be derived? Can it be shown that systems where action is stationary are the most likely ones? Also, it is interesting that the Lagrangian has units of energy as well. Edit: To clarify on the first paragraph. I was referring to quasistatic systems that satisfy a sepcific set of conditions. Energy is minimised in the stable state of an isolated system (where entropy, volume, and the number of particles is conserved). For a system where the temperature, pressure, and the number of particles are constant, the Gibbs free energy is minimised etc.
I am at a loss to understand the justification behind the following statement being false and would appreciate someone explaining what this particular argument is saying: The set of all finite subsets of a countably infinite set is uncountable. Answer: False. Any countable set has a bijection with the naturals, and one can list the subsets in order of the sum of their values. Every finite subset has a finite sum and therefore will appear in the listing. I am not sure how this argument proves the falsehood of this statement, or what the process of its logic is. Do we not have to say something about how different subsets will have the same sums? I would appreciate a more formal version of the above argument so I could understand what point it is making.
In common parlance, we say "s/he pretends s/he doesn't understand", for people who ignore the context of the conversation and require hyper-specific definitions to keep the conversation going. (hyper-simplified) example of the above, in the form of a dialogue: Q: Hey, how's your day? A: In what sense? Q: Uh, let's say for the sake of argument, how's work? A: What do you mean by that? In terms of what? Q: What are you up to? A: I do lot's of stuff, you have to be more specific - is there something in particular you're interested in? Q: Nope just in general, are you enjoying it? A: Sorry I don't understand, define "enjoying it" in the context of work and so on... Is there a term for what the person (A) is doing?
Based on the double slit experiment we know that in the case of a single particle system the wave function or state vector of position is in a superposition of possibilities before measurement. But does this rule apply in the case of the vacuum state? Is it in a superposition of (non degenerate energy) possibilities before measurement? A physicist has replied to this question, in a personal correspondence, by noting that: It is thought that in quantum field theory, different vacuum states are essentially classical, and don't get superposed quantum mechanically. The reason is their effectively infinite spatial extent, so the distinct vacuum configurations are mathematically very distant. Yet I am not well convinced by his answer. Isn't a quantum state, either a single particle or a vacumm, a nonlocal object in the Hilbert space and independent of physical space? So why the vacuum state is an exception from the general rule of quantum theory that applies to position or momentum quantum state? Why being "mathematically distant" rules out being "superposed" in the case of vacuum. This is not trivial to me. Is there a theorem about this? Should we suppose that there are multiple Hilbert spaces and therefore in a single Hilbert space there is only one quantum state for vacuum? and "by mathematically very distant" we need to consider another Hilbert space with its own vacuum state?
I have the following books about QFT: Peskin-Schroeder Lancaster-Blundell Iliopoulos-Tomaras I would like to know if there is a QFT that is as concise as Dirac's book about general Relativity. What I found useful about Dirac's book was that it developed most mathematical machinery "from scratch" with very little comments. I would like a "quick reference" QFT book that allows one to study the matter from a more mathematical perspective, rather than using a lot of physical analogies as in the books I have read. As an example, I would like to see "This is the QED Lagrangian. Those are the consequences", but a more classical, experiment driven approach is appreciated if it is strictly functional to the development of the theory and if it is kept as concise as possible. Ideally the book should be: More or less "self-contained", allowing a student with all prerequisites to learn introductory QFT from scratch (at least to renormalization) As concise as possible (ideally also in the number of pages) I would also like lecture notes or papers, it does not have to be a book.
I would like to use this template from Overleaf for a PhD dissertation at the University of Aberdeen, offline on TeXnicCenter. (This appears to be a legitimate thing, according to my supervisor.) I have looked at this; it didn't help. My main issue is that I'm not sure how to use the cls file. There might be other problems once that is sorted, so if you could explain to me how to do it, please, I would be most appreciative. I'm not sure what I can offer in terms of context. But here: I did both my MMath and my MPhil dissertations using TeXnicCenter, along with much smaller projects, so I have a fair amount of experience with it. Like I said, I think the issue is with the cls file. I believe it is something like a package, but one that I have to install myself - I don't know . . . I am likely to award a bounty to the first answer that works when I try it!
Three non-collinear points are placed randomly inside a unit circle. Question: What is the probability that if you were to connect these points, forming a triangle, the triangle will have the center of the circle contained inside (including the periferi)? Here are two examples, A and B, illustrating what I mean: where you can see that A works and B doesn't work. What Iv'e tried so far is illustarated in the following image: Two randomly points are obviously always collinear. So we can connect the two. Now let's draw a staright line, starting from each point, through the center and finish on the circumference of the circle. The grey shaded area represents the region where the third point needs to be in order to create a triangle that works. Now, as you can see there can be all kinds of regions depending on where the first two points are located. So I'm simply looking for a method/approach that can answer this question. If you have a another approach than my example feel free to use it. I'm not sure if this is very complicated or if there is a nice solution.
If we are given that propositions P and Q can never be true, is it still accurate to say that P and Q are necessary and sufficient for each other, and why? I am conflicted here, as the statement P iff Q is true here and I have learned that this also means P and Q are necessary and sufficient for each other (for context, I have learnt that P is necessary for Q if P must be true to conclude Q is true, and that P is sufficient for Q if Q must be true to conclude that P is true); however, it seems counterintuitive to stipulate conditions for propositions being true (by deeming other propositions as necessary or sufficient for them) if these are never relevant to the situation, as these propositions are always false anyway. I would appreciate an answer grounded in an explanation of necessity and sufficiency.
I am not well versed in TikZ and PGF Plots, and I am hoping someone would be able to help me with developing the structural code for a Parabola with its Focus and Directrix. I found an image of what I am assuming was created via LaTex and was hoping to recreate this somehow. This was a perfect image of what I am looking for, because the conic sections seem to be so lost to time that I can not find good representations of this particular topic without hand drawing it. I found a sample code for drawing a parabola, and when I attempted to recreate it, it does not let me use a fractional coefficient. If anyone has starter code to this image, that would be greatly appreciated
My understanding is this: the Solar surface becomes speckled with more sunspots near Solar maximum, and these spots tend to form groupings known as active regions; each spot is associated with a given magnetic polarity, and often regions will tend to have spots of both polarities that are typically linked in certain ways, but sometimes the formed regions are much more complex, especially as the polar fields are about to flip and Solar activity tends to increase. Now, my question is this, as the title says: which way does current flow in the plasma streams that form the coronal loops and filaments that are generally "rooted" in the active regions? Is there a way to determine the direction of current solely from the magnetic polarities of the spots involved, or can current flow in either direction?
In a corona discharge, the air around a conductor locally breaks down but remains an insulator further away from the conductor. Therefore, in the case of a positively charged conductor, the free electrons in the locally ionized patch of air can enter the conductor, positively charging the air too. However, corona currents don't exceed a few micro-amps. What is the reason for this? I know that electrically charging the increases the effective size of the conductor lowering the peak electric field. However, can this current be increased to arbitrarily high values is the charged air sheathing the conductor is removed quickly through some other mechanism? In essence, what I am asking is if the current limitation of corona discharge is due to the sheathing effect of the charged air or something intrinsic to the conductor-air system such as the probability of a free electron jumping into the valance band of the conductor being low. Where can I read about the microscopic nature of corona discharge? Any good books would be appreciated. I would like to learn more about the statistical mechanics behind the free electron and valance band interaction of metals.
Perhaps you might want to take this question as some sort of challenge. I'm really just looking for the "most efficient way" of making a table similar to that in the image directly below (in LaTeX, of course): I've attempted several different things, and ended up with a huge mess and a thousand packages in my preamble. (Here "thousand" is a hyperbole for "too much for my brain to keep track of".) Because I provided an image of what I want, I don't feel like any MWEs are necessary here. Whatever looks most like the image I above, and gives the least headache, I'll mark that as the solution. Notice the days of the week are aligned to the horizontal centre of the cells, unlike the rest of the content. Note. The table I provided as a reference was made using LibreOffice writer, and the immortal CMU Serif font. If you have a good eye, you'll have noticed that the content of the list elements are not aligned perfectly, as the second and following lines of all the list text blocks are pushed ever so slightly forward. This is neither intentional nor wanted.
If I want to calculate the length of a string that is wrapped around a cylinder what mathematical equations can I use? The thread is tightly wrapped around the cylinder, creating a spiral that has no gaps between each consecutive loop. Moreover each complete spiral creates one layer. and as a single layer finishes at the end another start to form up to the top creating a second layer, and so on. The catch is, is that each consecutive layer would require a larger length of string than the one before it because the circumference enlarges. Since the yarn ball is manufactured by machine, it leaves no room for error because the machine is in the same constant motion, the gaps between each string, as well as the gaps between each layer, are all the same, as is the tightness of each loop around the centre cylinder that holds the yarn together. So, how would you suggest I calculate the length using only external measurements and observations?
When a thermodynamic system, like an ideal gas within a piston immersed in a heat bath, is subject to changes, such as compression or extension of the piston, then the work that can be extracted from this process is maximal if the process is carried out quasi-statically, that is, at each step of the expansion the gas is allowed to relax to equilibrium. Why is the relaxation to equilibrium at each step related to more efficient energy extraction (or reversibility) ? Here is another way to ask. If I make a large change for the piston, I imagine there will be an abrupt relaxation to equilibrium and this will be related to large dissipation. But If I make many quasi-static changes, then at each change the dissipation should be small, such that at the end more energy is extracted. But the problem is that I do not see any physical picture or equation why the sum of the very small amounts of dissipated energies during the sequence of quasi-static changes should be smaller than the amount of dissipated energy during the one abrupt change. The difference between my question and similar available questions such this one, Is there a quasistatic process that is not reversible? , is that I insist on pictorial or physical answers of the questions.
Its kinda confusing but I will explain. I saw a term a while ago that explained how people can disagree with another. Instead of disagreeing because they do not like their opinion, the person will disagree because they dont like someone or something that also likes the opinion. Example: Person A thinks that driving on the road should require a license. Person B disagrees because a politician Person B doesn't like agrees with Person A, so Person B decides to disagree. You see that the argument Person A gave is reasonable, and Person B would completely agree, but the problem is that because someone Person B doesn't like agrees with Person A, they decide to disagree. This example is something very popular in politics (ex; only voting for a single political party, instead of voting depending on the politician's opinion). Another thing its similar to is contrarianism. You are disagreeing based not on the logic, but for another reason.
How is the background noise of gravitational waves modeled? Is it a thermal model, giving a stochastic distribution of the curvature tensor (field-strength tensor) in ambient space? That is, every binary star, every orbiting planet, every orbiting black hole or neutron star -- anything that accelerates -- is emitting gravitational radiation. The grand-total of all of these sums up to what looks like noise. Is there a "well-known" distribution for this noise? Some power law? Can an argument be made that this noise has a thermal profile? Are there specific equations describing this noise, and what are they? How do they scale? Should one suppose that this varies from galaxy to galaxy, and depends on the local environment? Or can one argue for some generic form that is "typical"? Say, for binary clusters? Somewhat related: what is the order of magnitude strength of this noise, compared to the instrument noise in current gravitational wave detectors? Yes, of course, its frequency dependent, so its a graph, but is this "natural noise" strong enough to be detectable? (Ignoring Earth-bound sources.)
A non-spontaneous change occurs when an external effort is being done to it. Since the external effort is also a natural source , does this mean that there is no truly any non-spontaneous process in nature (universe) ? Example to illustrate my point : An electrochemical process (like in a galvanic cell) can be reversed by providing an external voltage in opposite direction whose magnitude is higher than the cell potential. This external voltage is given by humans. Humans are basically converting some form of energy like mechanical energy of rivers in a dam into electrical energy. Also, humans are doing so because they eat food and they get their energy to do through spontaneous metabolic processes in their body. Also, the conversion of energy in above process is spontaneous because humans are making use of a spontaneous process ( a river flowing down from a height) . So, is there a truly non-spontaneous process ?
I asked my coworker to fix something in a program. When he fixed it, he replied with, "I already fixed it." -- this wasn't intentionally misleading, but was an incorrect translation of "ya". But for a moment I was thinking... "no, you fixed it after I asked". It was a bit jarring. Having learned a decent bit of Spanish when I was younger, I paused and realized there was some confusion when translating between "ya" and "already". BUT...I really struggled to find a clear/cogent explanation to help my coworker--from my English perspective. I finally found an article in Spanish, but honestly it was crazy how many meanings of "ya" there were, many of which I hadn't realized, and it was slow-going to read through it and make sense of it. My Spanish skills are rusty. So nobody else whose primary language was English was likely to realize what happened, had they seen this mistake So, here's my question: How can I help both the ESL coworker, other coworkers, and others in general to be aware of this pitfall, and what is a kind, cogent, and English-centric way to explain it? Thanks. Secondary question: And are there blogs/references or sites where I can learn more patterns of this sort of mis-translation across many languages? As a privileged white guy from a homogenous suburbia in the USA, I probably have a number of ways I might misunderstand cultural differences, and likely need to improve my awareness bit by bit.
I want to use Matex for the figures in my plots which I eventually would like to use in overleaf. I create a plot and use Matex in it and right click on it to save as pdf. I then use this pdf in overleaf. So, when I compile the latex file and download the pdf, the Matex font looks rather shabby compared to other text on the plot (such as numbers on axes for which I didn't use mathematica) when I do not zoom in. . The font on latex can be seen on the description of the figure. The Matex font looks spotty with kinda white gaps in it. The problem with this is that when I print the file onto paper, the font doesn't look good. I don't think the problem is resolution since when I zoom into the pdf file, I get a very sharp Matex font. Does anyone know what's going on?
Consider the sentence: So how can a computer think if it knows nothing of what it means to be a human being. Initially I thought that because "of" in this sentence basically means "about" the latter part of the sentence is not a relative clause as it is not adding any additional information, therefore is the subject of the sentence and cannot use which. Furthermore, as the contents of "to be a human being" is not known, "what" is used opposed to something like "that". However, upon further thinking I'm not sure if this is a relative clause or not as technically it is adding information to the sentence by specifying what the "nothing" the computer knows is. However, none of the resources I looked at concerning relative clauses used "what" in any example sentences. In other words, I'd like to know why this sentence uses "what" and what type of clause/grammar this sentence is.
This is a broad question but it's well documented that GR and QM are very well tested in their own domains but they conflict around black holes. Picture a neutron star slowly accreting matter until it's mass is sufficient to bring about an event horizon. It resists gravity owing to the Pauli exclusion principle and it must surely be comprised of the same 'stuff' as the event horizon forms. Why do we then rely on GR and assume everything collapses to a singularity which seems illogical in nature when the most sensible ('we don't know yet') answer surely should be that there exists a 'black star' under the event horizon? It seems that QM is overshadowed by GR in this instance when GR seems to give more illogical answers. As a thought experiment, if we had a very heavy neutron star and fired one photon at a time at it; I would imagine the surface begins to redshift more and more as time goes on. There would surely reach a point where the miniscule deviations mean that a 'ravine' can no longer emit anything to an observer but a 'mountain' could. It would seem to be on a tipping point of being both a black hole and a neutron star but the mountain is still supported from below from seemingly below the horizon that is forming.
Is there an easy way to draw complex images in LaTeX? By this I mean: Have a look at the following question. Some answers provide incredible pictures, very elaborated. In the (rudimentary) way I can create figures with tikz or pstricks, it would be impossible for me to come up with the correspondings codes. This made me think: Perhaps there are some tools that help write down the code. I am aware of things like tikzcd-editor or LaTeXDraw. I also know about InKscape, although I have never used it myself. Howerver, none of them seem to be appropriate to draw tori or spheres. Yes, with LaTeXDraw one can draw ellipses and create things that resemble those surfaces, but the quality of these draws is as good as the quality of Paint. I am thinking more of some type of image recognition software that identifies the geometric object (as well as some text the figure might have) the user has drawn and produce the corresponding code. Does anything like that exist?
I am seeking a term for what can collectively be referred to as "the leader and/or the most important and powerful roles" in the hierarchy of a society. I have an example from the YA series Warrior Cats by Erin Hunter. Warriors are the generic members of society. Meanwhile Leader, Deputy, and Medicine Cat together are the important and powerful positions. A term, for them together, as a collective, is what I seek so I can quickly an efficiently refer to them instead of some nickname like "the Big Three" every single time. I would like for a term generally applicable to most of societies, as it can be used in many contexts. I've been unable to think of any possibilities, hence I ask. Example Sentence: "The X are foundational to how our clan runs."
Poi are tethered weights used for dancing, which often have battery powered lights in them. Clearly work is being done on the poi, first to accelerate them, then to keep them going at constant velocity despite air resistance. I'm curious whether it would be possible to power the lights via the work done by the dancer rather than needing to replace/recharge the batteries externally. Obviously the amount of energy the dancer needs to exert would go up as well, regardless of the method. One idea that clearly wouldn't be sufficient would be to use current induced by the Earth's magnetic field. Are there any methods that might produce more power? It'd be cleanest if these methods would work while the poi were spun at constant velocity, but methods relying on variations in velocity or tension in the tether would also be interesting.