I'm building the Proximal Policy Optimization algorithm from scratch (well, using PyTorch). I've been studying it by my own, but I'm a little bit confused in the optimization phase, here is the thing. From what I know... First, we initialize a Policy Network with random parameters. Second, we start with policy rollouts. At each time step (t) of the episode, we compute the Value Functions (functions approximators) in order to get the Advantage Function A(s,a), also, we compute the clipped surrogate objective (J) of each time step (t). At the end of the episode, we sum all the clipped surrogate objective values, this will give us the expectation of the entire episode of the expected cumulative rewards. In the paper of PPO the equation contains the expected value of all time steps (t). Once we have the value of the expected clipped surrogate objective (sum of all clipped surrogate objective values) at the end of the episode, we compute the Stochastic Gradient Descent (SGD). In order to compute SGD we need a loss function, we have our expected clipped surrogate objective function, so we just do -(expected clipped surrogate objective function) and that will represented the loss, that is the same as computing the Stochastic Gradient Ascent in order to maximize the expected cumulative reward or the objective function. Now my confusion comes in here... I thought that the clipped surrogate objective function was computed at each individual time steps (t) and then at the end of the episode, we sum all of them in order to optimize it (compute SGD). The thing is that some authors say that the optimization process (SGD) is done at each time step (t) of the episode instead of at the end of the episode, but why? doesn't the clipped objective function equation in the paper contains an expectation symbol? if the computation is done at each time step (t) then the expected value symbol in the equation is redundant, isn't it? Also, they say that in order to compute SGD we need a loss, I thought that by setting the clipped surrogate objective function to negative (-) we can get the gradient of it and then minimize it using SGD that it would be the same as maximizing it, but the paper shows another equation as being the loss function that the optimization phase uses, how is this? So my questions are... When and how is the clipped surrogated objective function computed (at each time step (t) or at the end of the episode), is my implementation of the computation correct?. When and how is the optimization phase computed (at each time step (t) or at the end of the episode)? My thoughts about the loss function correct? or what does the paper means with this loss function they show? Thank you in advance:)


I am currently very confused about the "topological" prerequisites of Lee's Riemannian Geometry (RG) book, An Introduction to Riemannian Manifolds. I have heard that this is a "truly introductory" text for beginners in RG. But as for prerequisites, in his preface Lee states that his other two books on Topological and Smooth Manifolds are a sufficient preparation for his RG text. But from several online forums I've come to the realizations that this would be an "overkill" just for studying RG (e.g., one need not study all of his Topological Manifolds text to study most of his Smooth Manifolds text!). So, in general, his preface is confusing to me when discussing prerequisites. I'm confident with my analysis background (inverse/implicit function theorem, etc.) As for the differential geometry stuff, I'm thinking about going over doCarmo's Curves & Surfaces text. I'm a beginner in DG and I prefer E. Kreyszig's Differential Geometry as it introduces tensors early on (something I'm highly interested in learning about). I'm also aware of another "standard" text on RG by doCarmo as well. So, here's what I need guidance on: Which of the books among Lee and doCarmo on RG is preferable if one wants to get a decent exposure to RG so that they can pursue research in related areas? How much topology does one need to know in order to tackle the texts by Lee and/or doCarmo on RG? For differential geometry, is Kreyszig's DG book sufficient to prepare for Lee and/or doCarmo's RG text(s)? Or is it preferable to study do Carmo's DG before tackling RG? My ultimate goal is (broadly) mathematical physics, in particular, Relativity and Quantum Gravity. Moreover, I consider myself more of a "math"-person and so would not prefer texts that ruin mathematical rigor.


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.


I saw other posts such as this one but I don't think it's quite the same question, or even if it is, the answer employs the operator formalism and I'm not sure I follow it. I'm wondering, if you have two multiparticle states - a multiparticle state being, in my mind, a complex probability amplitude for each possible configuration of particles, as a function of time - then, is a normalized linear combination of these two states still a valid multiparticle state? Keep in mind they are both functions of time that obey the equations of QFT, so the linear combination is also a function of time, and I'm asking if it still obeys the equations of QFT. I'm trying to think about it in terms of Feynman diagrams. In particular, I'm pretty sure you can linearly combine two multiparticle states at one time with no problem - you just get a superposition rather than a pure state. And since the amplitude for a final configuration is essentially the sum of all propagators from the initial state, well, this sounds linear enough to me. I think you would just sum up the independent contributions from all the pure states of which the initial state was composed. What else could it be? What throws me for a loop is that I've seen several posts here and elsewhere talking about the inherent non-linearity of QFT. But I think they might be talking about linearity with respect to combination of single particle states. I'm not worried about that, however, since I can see that multiple identical particles really form a single mathematical entity (a product rather than a sum, e.g. the Slater determinant for fermions), so linearity wouldn't have much meaning in this context anyway. Still, the whole thing appears rather murky so I'd really like to clear up this point. To put it another way, I know that there is interaction within the evolution of a pure multiparticle state, and this leads to entanglement, which, mathematically, is just the inability of the final state to be factored into a single tensor product of one-particle states. But, is it fair to say there is no interaction between the pure multiparticle components of a mixed multiparticle state? (at least, as long as we don't classify summation of complex amplitudes as "interaction")


Context: I ask this as a school teacher reaching past the boundaries of my expertise. A colleague was talking about the standard model with an advanced student, explaining how particles interact by exchanging gauge bosons, asking the student to imagine gauge bosons as little spheres. Of course they got to the issue that this mental model doesn't add up, since in classical mechanics such a process always leads to repelling forces, while the forces in electron-positron-scattering are attractive. He asked me on my opinion on how to rectify his explanation in such a way that a student could still create a mental image of the process. This is what I came up with, and while I'm sure that it can't do the actual physics full justice, I would like you to point out precisely in what ways this is inaccurate, so that I can improve it in such a way that an advanced student can still have a mental image of the process. The mental model: In classical physics, waves typically obey the superposition principle, that is, two waves don't interact, but can pass through each other without influencing each other. This is true for electromagnetic waves, but also all other waves the students might have encountered in class. It is not always true for water waves, however. For instance, two tall waves crashing into each other won't pass through each other unchanged since they lose energy due to turbulences. The correct description of their interaction requires an additional term besides just the sum of the two wave functions. We can imagine something similar to be true for wave functions in QFT: As a particle like an electron is described as a wave function, the interaction of, say, two electrons is similar to that of two water waves in the sense that they do actually interact, yielding more complicated wave effects. A main difference is that unlike classical fields, which don't interact with other fields, the electron field interacts with the electromagnetic field as well, and the nonlinear effects come from one electron wave packet interacting with the em. field, which then interacts with the other electron wave packet. And since photons are electromagnetic wave packets, we can think of the em. part of the interaction as being comprised of photons. So what do you think? How (in)accurate is this mental model of interactions mediated by gauge fields.


I am currently thinking through evaporation over lakes, specifically the Laurentian Great Lakes (a complex subject, I know). Particularly, I am trying to wrap my head around why evaporation peaks in the fall and winter. Based on what I have read, this fact is due to the vapor pressure gradient that exists between relatively warm water and dry air (dry in an absolute sense because the air is cold) and the high winds which continually replace that dry air over the water. I have also read that this evaporation from the Lakes has a cooling effect on the Lakes themselves, causing a temperature decrease in the Lakes. I have seen it insinuated that over-lake evaporation is synonymous with a latent heat-flux. Based on what I have read, latent heat is an exchange of energy to a substance without a change of temperature of the substance. For water transitioning from a liquid to a gas, the required amount of energy to cause this phase change is called the enthalpy of vaporization or the latent heat of vaporization. When the air is colder than the water, where is the energy coming from to supply the latent heat of vaporization that causes evaporation (assuming its a cloudy day)? If the vapor pressure gradient is the main driver of the rate of the evaporation, how can the evaporation still be said to be a latent heat flux, i.e., how is the transfer of particles based on a pressure differential called a transfer of heat, albeit a latent transfer of heat? Does a latent transfer of heat mean that the water particles which evaporate do not change temperature, though the liquid water which they leave behind decreases in temperature? Sorry, that's more like three questions! Hopefully they are not stupid ones :) it seems this is a complex topic, and my initial intuition that evaporation is always higher when the air is warmer than the water (thus transferring heat energy to the water, increasing the energy of the molecules, leading to increased evaporation rate since the water molecules are now higher energy) is simplistic and even wrong, as it depends on so many more factors than that. Our world is not a simple one, physically at least!


As is well-known, classical conserved systems have conserved quantities by virtue of continuous symmetries, which can be derived from Lagrangian mechanics. For example, two masses on a spring can swap momentum between the two, but translational invariance ensures that the total momentum is conserved. But what if we introduce damping, specifically viscous damping, into the equations? If we consider a fluid medium, our two-mass system surely loses momentum to the fluid. But suppose that the spring itself is viscoelastic, so that it generates equal and opposite forces on the two masses proportional to their relative velocity. Then, the system is translationally invariant in some sense and conserves momentum, but is not Lagrangian (at least not in a simple way). Is there a theory of how to calculate conserved quantities based on symmetries in damped systems, analogous to how it is done in undamped systems? I can do this for specific cases, but I'm not sure how to do it in general. A few attempts: Constructing a complicated Lagrangian consistent with damping Declaring that damping is a purely phenomenological effect coming from many microscopic degrees of freedom behaving conservatively, and so concluding that the system should have the same conserved quantities as if there were no damping (problem here is that some of the conserved quantities, like energy, must leak into the unobservable microscopic degrees of freedom). Considering the corresponding conservative system and showing that it has conserved quantities. Then arguing that this is true for each frequency even if the "spring constants" are complex and frequency-dependent, and so concluding that damping doesn't really change anything. Again, though, this seems to prove too much, since energy shouldn't really be conserved. By way of motivation, I'm working with a system that has an unusually large and complicated number of symmetries, and I'm trying to determine how worried I should be about damping killing the conservation in a real experimental system.


Imagine we have a hollow metallic toroid, with copper wire winded around it, which carries electric current. That implies we have magnetic field inside the hollow toroid. The toroid has vacuum inside. We have a set up of a high voltage supply and an electron gun that takes the free electrons from the metallic toroid and shoots them inside it. The velocity of electrons shot inside the toroid is low enough so the magnetic field will bend its trajectory in a complete circular path within the boundaries of the toroid. Now the electrons are flying in circular trajectory inside the toroid. But this situation can't be hold forever, as the electrons are centripetally accelerated the irradiate photons and thus loose kinetic energy. As they loose kinetic energy and velocity, at some point they will stop orbiting and will stay still. But unless they still exactly in the geometrical center of the section of the toroid (which is highly unlikely), they will be attracted to the boundary of the toroid due to Coulomb forces. And as they move towards the boundary, they will regain velocity and start orbiting again. Resuming, the electrons will loose energy due to their accelerated motion, then regain energy, and then loose it again. Apparently this cycle will repeat endlessly, meanwhile they will radiate photons as they loose kinetic energy. If my analysis is correct, how the energy conservation principle will be applied here? Radiating photons endlessly means giving endless energy. The second part of my question is as follows. Suppose the electrons doesn't radiate photons, due to some arbitrarily stated postulate (like the Bohr's explanation on why electrons doesn't fall on the nucleus of an atom). Apparently there is no obstacle to hold an infinite amount of electrons inside the toroid. The only limiting factor will be the amount of the voltage applied to fire new electrons inside the torus, as previously fired electrons will create repelling Coulomb force for the new incoming electrons. But there will not be such thing as "dielectric rupture" as in the case of an ordinary capacitor, so hypothetically an infinite amount potential difference can be set between the hollow toroid and its inside. Is this assumption correct?


I'm having trouble finding a good phrasing to describe a component of a system that is too important -- in a sense that it distracts a person from all the other components, that should have been, or were formerly, important aspects of the experience. The case where I'm trying use it is something like this: We tried adding a new super-queen to our variant chess game that could move anywhere -- but we found that it was ______, and that none of the other pieces were important to the gameplay as a result. Another example, possibly from academia: We recommend that teachers not offer students extra-credit work during a semester. We have found this to be _____, in that the option routinely distracts students from the main body of work and preparation for exams. The key thing I'm trying to highlight is that participants in the system become hyper-focused in their attention on this one component, to the detriment of the other parts. (E.g., while the chess example indicates an overpowered piece, that's not what I want to emphasize here; it could also be the result of a super-weakness that becomes the only thing you'd want to attack, say.) I could also imagine this being used in a work of art: Say, an element of a painting that is too bright or oddly placed. Or a breakout character in a TV show who takes over what was meant to be an ensemble story (i.e., they "may... overtake the other characters in popularity, including the protagonist"). Is there a good word or phrase to indicate this state of being too important, and attracting too much attention?


Scenario Consider an empty universe with just the Earth, Moon and the Sun. The Earth and the Sun will rotate about their center of mass, which is inside the Sun. The Earth and the Moon will orbit their center of mass, which is inside the Earth. The Moon and the Sun will orbit about their center of mass, which is inside the Sun. Assumptions Since the Sun has a center of mass with respect to both the Earth and the Moon, which are moving separately, the Moon pulls on the Sun too, just like the Earth does. This results in the Sun not having a closed loop orbit around their common center of mass. I assume the third body affects the two other bodies in this manner: The center of mass of the Earth-Sun system rotates in the same manner the Earth-Moon system rotates about their center of mass The center of mass of the Moon-Sun system rotates in the same manner the Earth-Moon system rotates about their center of mass These two effects describe the same motion when viewed from different frames (Earth-Sun and Moon-Sun systems) The above scenario is more pronounced if you consider Jupiter instead of the Moon, but I'm looking at the minute details. Question The question is whether all three bodies always revolve around their common center of mass, which also accounts for the motion of all bodies with respect to one another? If I'm wrong, how exactly does motion happen in this three body system? (Explain using simple words) Can this be generalized for an n-body problem? (Although probably not useful for computation but seems good for a mental image) I think this can be consolidated into a two-body problem by considering the Earth-Moon system as a single body at their center of mass, resulting in a single stationary point as the center of mass of the whole system. That is, I'm taking the center of masses of the Earth-Sun system and the Moon-Sun system, and taking their center of mass, and I expect it to be the center of mass of the Earth-Moon-Sun system. Or is it just that bodies only revolve around their common center of mass, and the center of mass of the three-body system has no say in describing the motion, aside from being a static point? References The motion of the Earth and the Moon around the Sun for a month (from a WIRED article) The Solar System's barycenter with respect to the Sun (from the Wikipedia article on Barycenter)


Initially, I was looking for how centripetal force is produced on the surface of the rotating earth for a mass kept at any latitude. I went through the following threads - Which force provides the centripetal acceleration that makes objects on earth's surface rotate about Earth's axis of rotation? Is the normal force equal to weight if we take the rotation of Earth into account? Question about the Normal Force exerted by Planet Earth in relation to centripetal force If the ground's normal force cancels gravity, how does a person keep rotating with the Earth? From there, I understood that the resultant of normal force (N) and gravity (mg) is the required centripetal force. But what is bothering me now is HOW? According to the answers, the normal force is slanted such that it is not exactly opposite to gravity. Thus, they don't cancel out, resulting in a horizontal centripetal force. I'm still confused and have the following questions : Why is normal force slanted in the first place ? (Is it because of the earth's bulge, friction or centrifugal force?) I think there's also a vertically-upwards component of the resultant, why is that? (the resultant of normal force and gravity) This is what I see, what is the reason behind this? Source of the image - https://en.wikipedia.org/wiki/Equatorial_bulge Edit - I've gone through this question Is the normal force equal to weight if we take the rotation of Earth into account? but this doesn't clear my doubt regarding the upward component of the resultant of gravity and the normal force. I posted this question because I wanted some more insight into that poleward force and its relation to the bulge of the earth, which isn't emphasised in that question. kindly reopen my question.


This Q&A did not answer my question. The voltage of a circuit is the difference in each Coulombs potential energy at the negative pole, compared to the positive pole. At the negative pole, there's a whole wire for the electrons to pass through under the influence of the Coulombic forces; a whole wire with atoms between which they can accelerate, before transmitting their kinetic energy into the atoms. The longer the wire, the more durations of acceleration (thus, the more kinetic energy is produced). But, the longer the wire, the more resistance there is, thus, the lower the amperage. A lower amperage means less acceleration between each collision (assuming constant wire diameter). So, as the wire gets longer, there's more durations of acceleration, but the acceleration is lower. The accepted answer to this Q claims that this is the explanation. But I doubt these two factors cancel each other out such as to leave the voltage unchanged with change in wire length. That is, despite this, I still think a longer wire would mean more kinetic energy is produced. More kinetic energy means there must have been more potential energy that was transmitted into it; thus, the voltage must have been higher. But it isn't, so what gives? There's the distance factor; the longer the wire, the further away from the charged poles of the battery the electrons get. However, this would then make the voltage dependent on how close you lay the wire to the poles, which is again contrary to the assertion that the voltage only depends on the battery.


I'm now in the process of writing a report on a lab work i did with an Abbe refractometer. In all sources i found the working principle of this refractometer is described as such: "Light shines into a illuminating prism who's side that contacts the sample is roughed so light scatters uniformly in all direction into the sample. The sample is held between the illuminating prism and the refraction prism (RP). Rays that hit the sample/RP interface at an angle large than the critical angle of the interface suffer total internal reflection (TIR) and thus do not penetrate into the refracting prism. This creates a zone in the refracting prism where there is a shadow. The angle of that shadow depends on the refractive index (RI) of the sample, and it can be measured by adjusting the light/shadow line". Additionally, some sources explicitly say that the RI of both prisms must be larger than the RI of the sample. Critically, i did not find any source that said otherwise. If this is the case, how is it possible for there to be TIR in the sample/RP interface if the RP's RI is larger than the sample's? Furthermore, i used https://phydemo.app/ray-optics/simulator/ to simulate the apparatus. I found that a shadow zone only formed when the condition: "prism's RI > sample's RI" was met. Of course in this situation there was no TIR. In fact, forcing the situation where there is TIR, absolutely no shadow zone was created. Can someone explain to me how this actually works then? Is there something i am getting wrong? I'm almost sure there is because even manufacturer pages say it works with TIR... But still, it is very confusing!


Whenever I Google to try to find an actual formal statement of the first incompleteness theorem (as opposed to all the oversimplified explanations that talk about "true but unprovable theorems" rather than theorems independent of the axioms), the definitions that don't just mention something like a system with "strong enough to do arithmetic on the natural numbers" mention a system with an "a sufficiently expressive procedure" for enumerating theorems, which is reminiscent of terminology used in explanations I've seen of Turing machines, and so I thought perhaps it meant that the incompleteness theorems apply specifically to Turing complete systems. However, when I posted about this on Quora I got responses saying that Turing completeness has nothing to do with it, whereas, on the askcomputerscience subreddit, I got responses saying that yes, the type of systems the incompleteness theorems apply to are mathematically equivalent to Turing complete models of computation. So which is correct? Is a system with a "a sufficiently expressive procedure for enumerating theorems" just a Turing complete complete formal language/model of computation? If not, what exactly does that terminology mean? To be clear, I understand that the first incompleteness theorem states only that a formal language with "a sufficiently expressive procedure for enumerating theorems" must be either inconsistent or incomplete (i.e. must either imply contradictions or contain theorems that are probably true in some models of the language and provably false in such models), so I don't need an explanation of what the theorem says, so much as clarification on what types of formal languages it actually applies to. I did see this answer to a related question, which seems to imply that the types of languages/systems the incompleteness theorem applies to are systems that are "recursively axiomatized" systems, but it's not entirely clear what that means. Based on other simplified explanations I've seen of the the theorem, I guessing it refers to something like "a system sufficiently powerful to talk about itself" which would intuitively seems to match up with the concept of using Godel numbering to construct a statement like, "This theorem is unprovable", but that still seems rather vague and nonrigorous. Surely, there has to be a specific and rigorous definition/description of the sorts of systems the theorem applies to though, otherwise how would it be useful?


I have been using '[...]' to indicate skips in the middle of sentences, '[...].' to indicate that a single sentence has been skipped (or that a middle of sentence skip '[...]' is at the end of a sentence and so is punctuated), and '[....].' to indicate that two or more sentences have been skipped (or to indicate that an end of sentence skip '[...]' bleeds into a '[...].' or '[....].'): [...]. Because of this qualitative simplicity of negation that has returned to the abstract opposition of nothing and ceasing-to-be to being, finitude is the most obstinate of the categories of the understanding; [...] finitude is negation fixed in itself and, as such, stands in stark contrast to its affirmative. [....]. The determination of finite things does not go past their end. [....]. But since it looks clunky, I was wondering if this is correct instead: [...] Because of this qualitative simplicity of negation that has returned to the abstract opposition of nothing and ceasing-to-be to being, finitude is the most obstinate of the categories of the understanding; [...] finitude is negation fixed in itself and, as such, stands in stark contrast to its affirmative. [....] The determination of finite things does not go past their end. [....] I have not been able to find anything on the topic of skipping entire sentences within block quotes that incorporate entire paragraphs. I know it would be preferable to avoid this situation in the first place by paraphrasing or excluding parts (some style guides even say that ellipses should never be used at the start or end of quotes), but I can't do that here for multiple reasons.


In the past two months, I had found a wonderful pdf that went through a derivation of the determinant with calculating the area of a parallelepiped as its starting point. The document did not get into the weeds of calculating the determinant given a matrix or even focus explicitly on matrices at all; it was probably the single best description of why the determinant tells us about area I had found and I just really found the document's approach to determinants extremely useful. I'm trying to relocate this pdf to no avail and was really helping someone could help me find it. What I remember is this: it begins with calculating the area of a parallelogram as its starting point. It describes how we want the area to change when a side is scaled and when the image is sheared (I very distinctly remembering these two things being bulleted in the first page of the paper) and goes onto describe how these properties would manifest in a general "area function," that is a function that takes in the vectors as inputs and returns some type of unsigned area. It proves several facts about these functions that begin to show how the determinant "appears" from answering this question naturally. After this point, they expand their scope to signed area functions and, building upon the results in the first part, show that any function which returns the signed area of a parallelogram is simply a multiple of the determinant. The paper rounds out with a discussion about expanding these ideas to higher dimensions; it describes the problem of measuring the area of a higher dimensional parallelepiped with regards to measuring the area of its shadow. One of the more distinct parts of this description is that it begins with a hypothetical scenario where you find a floating parallelepiped outside of your window one day and cannot interact with it but wish to find its area. My attempts at googling with these bits of information I remember have not been fruitful, so any help finding this pdf would be appreciated.


Motivated by question Can IC engines be modeled as Carnot engines?. I am wondering whether/how Carnot theorem could be generalized to other kinds of devices performing "useful work", such as, e.g.: Motor (or generator) fed by a battery Nuclear power generators Solar cells Water wheels I think that the theorem must be generalized in at least three ways: Operating media is neither a gas nor a liquid - that is the reasoning based on isothermic adiabatic expansions might not apply. Generalizing the concept of temperature (introducing "effective temperature"?) - e.g., in case of a batter or a water wheel, we do not have two reservoirs with different temperatures to properly speak of, but rather two reservoirs with different (chemical) potential. Generalizing the concept of useful work - solar cell and water wheel are not really transferring the energy between two reservoirs - the energy already flows, and the device simply diverts a part of this energy into work. But, since the energy flows anyway, it is not clear whether/how the part of it that is diverted is useful: e.g., how is the current generated by a solar cell is more useful than the heat generated in the surface illuminated (which may be also "useful" in everyday sense.) Perhaps, there is not much left of Carnot theorem with all these generalizations, and we simply need to consider it as limited to a particular class of phenomena? If so, are there other upper boundaries on converting energy to work (that would be applicable to the devices cited in the beginning?)


The Earth, effectively a non-inertial frame of reference, is where Newton concluded his laws of motion. However, Newton's first law only holds in an inertial frame of reference. In the process of inventing the Newton's laws of motion, since almost all (I suppose) the experiments were done in a non-inertial frame of reference i.e. the Earth, why were people confident enough to believe that Newton's first law is true (to some extent, I am not talking about relativity etc.) in an inertial frame of reference? I am not trying to say that Newton's laws of motion are lies. I just had a logical question: since the research was done in a non-inertial frame of reference, how can we invent the laws regarding motion in an inertial frame of reference? I suppose that's because the Earth can be approximated to an inertial frame of reference as the effect of the self-rotation (which causes the Earth to be a non-inertial frame of reference, in my opinion) of the Earth on the objects on the Earth is quite small. And, therefore when people did the experiments, the uncertainty caused by the self-rotation of the Earth is too small to be significant (or, maybe they didn't even find/realise such an uncertainty!). And therefore, by inference (or guessing?), we can invent the laws of motion regarding objects in an inertial frame of reference. In theory, if we do the same experiment in a true inertial frame of reference, we will effectively get the same/similar results. (This is just my explanation, which can be very debatable!) I want to hear how other people think about this question. Thank you!


Let me try to illustrate what I mean. Consider e.g. a Solar radiation storm (Solar particle event) where high-energy protons are hurled at Earth from Solar flares. I've tried to illustrate my conception of this (I know the protons will typically not follow straight paths out from the flare due to the Parker spiral, but it's a simplification): So the protons get captured by the field (given sufficiently low velocities perpendicular to the field, as far as I understand it) and are then led to the poles due to their drift velocity, as they will almost always have some velocity component that's not perpendicular to the field. Now, to me it seems (and the same applies to the plasma in coronal loops as far as I can tell) that there's a current along the field lines themselves due to the drift of the protons, in the direction they're traveling. This should itself induce a magnetic field surrounding the imaginary magnetic field lines at the centers of the helical proton motion as if the magnetic field lines themselves are current-carrying wires, should it not? Something like this: Am I correct in thinking about it roughly in this manner? If so, does that mean that these new magnetic fields could potentially themselves partially trap particles (although I assume the stronger original field would overwhelm it) and induce new magnetic fields around them in turn? Is there a limit to this "fractal" process of magnetic field lines acting as currents inducing magnetic field lines acting as currents, and so on?


My question, right up front, is: what is the term for a modifier that behaves this way? But "this way" takes some explanation, and that is the rest of the question. I am a mathematician, and my question makes the most sense in a context where words are formally defined anyway, but you can freely substitute "foo" and "bar" for any technical jargon. A ring is defined to be a set with two operations that satisfy certain properties. Among these properties, there is not universal agreement: should, or should not, the ring be required to admit a multiplicative identity ('unit')? For a mathematician who does not require this, it is easy to indicate when they wish temporarily to impose the hypothesis: they can just refer to a "unital ring". A mathematician who will almost always consider only unital rings might decide to make that part of the definition of a ring, so that they can say "ring" where a more permissive mathematician would say "unital ring". (This lightweight controversy is discussed in the Wikipedia article.) However, this latter kind of mathematician, on making the rare encounter of a ring that does not have a unit, must either make up an entirely new term for it, or call it a "non-unital ring". This usage is almost universally understood, but somewhat puzzling: while it makes sense to understand, e.g., a "commutative ring" as being a structure that it is a ring, and also satisfies the requirement that its multiplication be commutative, there is no way that one can so interpret "non-unital ring" if a ring, by definition, has a unit. So, what's happening here is that "non-unital" is not refining "ring" by adding conditions, but changing the meaning of "ring" by dropping existing conditions. That is, you know something about a commutative ring even without cracking open the definition of "ring" (namely, that it has a commutative operation); but, to know something about a non-unital ring, you must not only crack open the definition of "ring", but recognize which among the properties one is expected to remove. I'm looking for a word describing the behavior or function of modifiers that behave like "non-unital", in the sense outlined above.


So, I was reading some books by Stephen King, S.D. Perry, and a couple authors I really love. I notice they'll use pronouns or certains words twice in the same sentence. When I read it, it's pleasant and doesn't sound weird in my head at all. I assumed that if something is published under an author like Stephen King or Neil Gaiman, it must be grammatically correct to use something like "he" or "her" twice in the same sentence. I'm not sure if this is why it doesn't sound strange in my head when I read it, but I've provided an example below. Here it is (bolded words are the repeated pronouns): My example: Martin stared at the ceiling above and grimaced, counting the number of bumps and flecks embedded in its expensive paint. He gave up after some number he couldn't remember, tossed to his side, and yawned. The smell of curing lacquer and stain drifted off a nearby nightstand, burning his nose. And here is another example from It, by Stephen King: When he grinned, there was a ghost of the handsome man he would become in the lines of his face. There examples in multiple Stephen King novels of names repeated in the same sentence as well. I'm using Stephen King as an example, but note that almost every published author I've read does this in wildly different genres and styles. Anyway, is using a pronoun more than once in the same sentence grammatically correct? I know sometimes things are grammatically correct, but discouraged or frowned upon, but that's not my question, here. tl:dr; Is it grammatically correct to repeat a pronoun in the same sentence in any circumstance, and if not, why is there published material that violates this rule?


In my limited knowledge of statistics, I am confused between the meaning of Latent Variables and Nuisance Parameters. Here is my current understanding: Sometimes the variance parameters in a regression model (or the overdispersion parameters) are considered as Nuisance Parameters because we are not directly interested in estimating them In a Gaussian Mixture Model, the weights of each Gaussian Mixture are called Latent Variables because we do not observe them directly (although this point somewhat confuses me because we dont observe any parameter directly, latent or non-latent ... all we observe is data) From an estimation perspective: We remove the nuisance parameters (via marginalization, factorization) because they can bias the estimates. I don't fully understand this, but I think it has to do with degrees of freedom, MLE produces biased variance estimates (in complex regression models, we can not a priori know the correction factor needed to remove the bias). Later, if required, we estimate the variance parameters using a more complicated form of MLE called RMLE We remove the latent variables (again via marginalization) ... because they make the optimization problem easier? (i.e. reduce the dimensionality of the function being optimized) Is this the only reason we remove latent variables? Or is there some other reason ? (ex: the estimation of latent variables is biased?) From here, it looks like these concepts are almost interchangeable: A nuisance parameter is latent and a latent variable is a nuisance. They both complicate the estimation process for similar reasons and we try to remove them through clever math tricks (e.g. marginalization). I feel I am missing something here - any one have hints?


My partner frequently asks me questions that, when read literally, are questions about the past, but in intent and intended response are actually conditional questions: Did you have any thoughts about dinner? Did you want to have coffee? ...where the intent is not merely to inquire about the preferences I have established prior to the time of questioning, but also about what my preference might be at the time of questioning. "Would you have any thoughts..." or "Did, or would, you have any thoughts..." might also capture intent, but awkwardly. I'm quite literal so I've had to train myself out of interpreting "did" as a past tense question of fact, but it occurred to me that it might be a regionalism or other speech pattern where there's a kind of merger going on between "did" and "would", or other shift where "did" is used conditionally in casual settings, maybe in cases where there's the possibility that the answer might have been determined in the past, but also might not have. Even in my examples above, "would" does feel a bit stuffy and formal by comparison, but I've never used "did" this way myself. Does this interpretation make sense idiomatically outside of the individual context of my partner? Does anyone know if there is any regionalism in play? My partner comes from a family that has lived in Connecticut for generations. My family is mostly from the midwest and south. Please feel free to correct me if I am misapplying "conditional" or can otherwise better describe the usage here, I will update as I can! Also, unfortunately I don't have the gift of search terms on this one, so I haven't been able to find any discussion of it elsewhere, but let me know if I can add any research or context.


Double-slit experiment image source: Wikipedia The double-slit experiment can be regarded as a demonstration that light and matter can display characteristics of both classically defined waves and particles. It also displays the fundamentally probabilistic nature of quantum mechanical phenomena. In a double-slit experiment using an electron beam an interference pattern is formed after experimenters record a large amount of electron detections. I have seen this answer by "anna v" which states an electron never travels through both slits only one slit per electron and the pattern formed is only a statistical probability distribution for the entire accumulation. But assuming in an experiment in which electrons travel one after the other and each electron travels only through one slit then how could the pattern on the screen be different from the one when we close one slit interchangeably and send electrons only through one slit at a time. (Actual experiments have shown patterns are different indeed) I think an electron through double-slit as a superposition of probabilities of spatial distribution. Like this picture below: But according to anna v picture that come to my mind is below (several electrons illustrated): So I have two related questions: Is the so-called wave nature of particles only a mathematical model or is there some physical nature (properties) to the probabilistic wave that passes through the double slit? Is stating whether the electron passes through both slits or it only passes through one slit just a personal opinion/interpretation that cannot be proven or disproven by observations? Edit: Evidence supporting simultaneous two (path) position: Using a Mach-Zehnder Interferometer to Illustrate Feynman's Sum Over Histories Approach to Quantum Mechanics One particle on two paths: Quantum physics is right by Vienna University of Technology Double-slits with single atoms: Selective laser excitation of beams of individual rubidium atoms by Andrew Murray, professor of atomic physics, University of Manchester, UK


I am working on a problem where I need to extract a connected tree of nodes based on certain attributes while optimizing for the minimum number of nodes. Some attributes of the nodes are known in advance, such as resource capacity. However, I do not know in advance how many nodes will be selected in the final tree, especially the intermediate nodes that ensure connectivity and may act as task-forwarding nodes. The goal is to formulate a mathematical optimization model with a set of constraints to select an ordered tree with the minimum number of nodes, where each node hosts specific tasks. Selecting individual nodes is straightforward, but I am struggling to write the constraints to ensure that: All selected nodes are connected. The hosted tasks follow a desired ordered sequence. For example, if nodes host tasks A, B, C, and D, task A should be in a node that comes before a node hosting task B, and so on. Multiple tasks can be placed in the same node if they do not exceed the resource capacity of the node and do not violate the ordered sequence of tasks. A node with task D may act as a root node in the tree. Most of the existing literature I found online assumes either the total number of nodes in the final tree or the total number of edges in the tree is known beforehand. In my case, it is possible that only a single node may satisfy the required attributes and host all the required tasks, or a set of constraints may be needed to satisfy the required attributes. Any hints or suggestions regarding how to formulate these constraints are highly appreciated.


Sorry if this equation is not phrase in precise mathematical form. I am open to suggestions to improve the explanation, and I have tried to formulate the problem as precisely as I could. I was talking to a friend--postdoc in PDEs--today and he was asking me about obtaining some Yahoo data on stock prices for his students. He explained that he had some students working on stock price data and he was looking for software packages/libraries for time series analysis--like ARIMA, GARCH, ARMA, ETS, etc. I told him that as far as I remember, most many models to stock prices or asset prices use stochastic differential equation like the Black-Scholes models. So I was asking if he actually needed like a stochastic differential equation solver, rather than a time series package. He said something that surprised me. He said that many people used to use Black-Scholes models, but implied that they are not used as much any more. He basically said that many people lost money using these models. I myself don't work in mathematical finance or in banking at all. However, I was unaware of any major critiques of stochastic differential equations and their applications. From a formal perspective, I imagine that these SDE models have a set of parameters (means, variances, etc.) that users can tune against some data using an optimization method. I am not sure what optimization would make sense, I can imagine approximate bayes or something, could work, but there are probably a million choices. So is the claim of a bad fit, mean that the error between the model predictions and the ground truth data too high, or perhaps changing over time, etc. Also, are there a different set of finance models being used as replacements for Black-Scholes?


I have seen that it is possible to approximate the metric in the presence of a gravitational field by the Rindler metric: Does a uniform gravitational field exist? Is there any acceleration in a uniform gravity field? Applying the principle of equivalence to an accelerated frame Rindler Coordinates and homogeneous Gravity Field Gravitational field strength and Horizon in Rindler coordinates Now as some answers in the links have pointed out, this doesn't quite describe a uniform gravitational, because the acceleration described by the coordinates depends on one of the spatial coordinates. My question is, how can we refer to this as a gravitational field at all? The Rindler metric is derived from a coordinate transformation on inertial coordinates on Minkowski spacetime, so we know a priori the Riemann curvature tensor will be exactly zero in the Rindler metric. In doing this "Rindler approximation" to the gravitational field, say, near the surface of the Earth, we started out with a nonzero Riemann curvature tensor (indicating spacetime curvature exists), and then we obtained a situation in which the curvature tensor vanishes everywhere in the region we're approximating. Doesn't this render the approximation invalid? Even if you argue that the region of approximation is small (which makes sense), there is no sense in which we can make the curvature tensor outright vanish (because the Ricci scalar, which is a contraction of the curvature tensor, is supposed to be invariant). Accompanying this strange change in the curvature, objects that "fall into the Earth" followed geodesic paths prior to the approximation, and under the new approximation, the same objects are now undergoing proper accelerations, meaning they are no longer following geodesic paths. Is there a physical interpretation or the mathematical reasoning behind this change? It seems like we are replacing spacetimes outright as opposed to approximating them. Is this understanding correct?


This question was inspired by the interesting discussion here: Why isn't the T in "relative" flapped? It seems like the adverb already and the two-word phrase all ready should be pronounced differently, but as far as I can tell, both sound exactly the same. For comparison, consider the other phrase/adverb pair all ready and already, where there is a difference: "The suitcases are all ready" doesn't sound the same as "The suitcases are already..." because the stress pattern is different ([all ready] as opposed to [al ready]). Additionally, the "l" sound at the end of "all" in the phrase goes on for longer than the "l" sound in the adverb, corresponding to the separation between the two words. If someone said the second sentence (fragment) out loud, it would leave the listener asking "already what?" In contrast, I wouldn't be able to hear the difference if someone incorrectly substituted the adverb altogether for the phrase all together. They both have the same sequence of stressed and unstressed syllables ([all to ge ther] vs. [al to ge ther]). More surprisingly, the adverb also keeps the aspirated "t" sound found in "together," even though that t is between a stressed and an unstressed syllable. This seems very exceptional, since that context usually requires the substitution of the "flapped t" for the "aspirated t." But, if this were the case, then "altogether" could be spelled "aldogether" without changing its pronunciation, which it can't. ("Aldogether" just sounds like I have a stuffy nose.) This also contrasts with "relative"/"reladive" like in this question, where the two pronunciations are interchangeable (the flapped t one being more common when it's said faster). "Relative" is also not an exception to the rule cited in the accepted answer, since there the t comes between two unstressed (or at least only tertiary stressed) syllables. The cited paper states that flapping the t is optional in that case. I'm interested in what the context (either historical/usage-related or some less well known phonological rules, or something else?) could be behind the pair not being differentiable by sound, and only by spelling, but none of the answers to the other question mention the word "altogether." No one seems to have asked this specific question yet, as this one is about meaning and not pronunciation. Here is a reference for the pronunciation of "altogether," confirming it always has a distinct "t" and not a flapped t/d sound: https://en.wiktionary.org/wiki/altogether. This proves that I haven't just been mishearing and mispronouncing the word the entire time!


Euclid based much of his geometry on a theory of magnitudes that looks roughly like this: A general theory of whole and part and how they are related in size (eg: the whole is greater than the part). A general theory of the properties of magnitudes (eg: equals added to equals are equal). A basic rule that allows one to determine in specific cases that one magnitude is equal to another (eg: all radii of a circle are equal; all right angles are equal). Euclid applied this theory to lines, angles, and figures (meaning the area). From these simple foundations, he is able to prove the equivalence of all sorts of things that are not related by the basic rule alone. I've been looking for a modern development of this theory, and though there is some interest in it (Robering), modern work on mereology and mereotopology that I've been able to find online all does things that seem to violate the spirit of Euclid's work; in particular: They treat points as parts of lines and lines as parts of the plane (Robering comments on this). Euclid did not do that. In Euclid the parts of a line are lines and the parts of a plane figure are plane figures. They assume a single universal whole of which everything is a part. This is related to the first issue. Obviously there isn't a single universal line of which all lines are a part. Similarly for angles. Also related to the first issue: the works on mereotopology make use of the concept of an interior part, which basically means, for example, the part of a line segment not including the endpoints or the plane figure excluding the boundary. This doesn't makes sense when points aren't parts of lines and lines aren't parts of plane figures. The works on mereology assume that any two objects form a whole. Euclid only considered connected wholes. They assume completed infinities. Euclid avoided those. Robering treats an angle as the infinite plane section enclosed by the rays of the sides of the angle. I assume this is to turn angles into parts of the plane so they can be part of the universal whole. They assume that there is no maximum-sized whole, but this doesn't work for angles if you treat them as Euclid does as their own kind of magnitude. Can anyone suggest a body of work that I can find online that deals with some of these issues?


The direction of polarization of a transverse wave is defined to be the direction perpendicular to the direction of propagation of wave or the direction of oscillation of wave, right? But in the case of Electromagnetic waves, a class of transverse waves, there are two directions of oscillations that are perpendicular to the direction of propagation of the wave, the direction in which the electric field oscillates and the direction in which the magnetic field oscillates. But while referring to the direction of polarization of an Electromagnetic wave, only the direction of oscillation of Electric field is called the direction of polarization. But why isn't defined to be the direction of Magnetic field? For this I thought like this: "For transverse waves like waves on a string, to determine the direction of polarization, just put a plane with slit in different directions and the direction perpendicular to the direction of slit in which the wave completely gets absorbed is its polarization direction." Now, if I apply the same logic to the Electromagnetic waves by replacing the plane-with-slit with a material that can interact with the electric or magnetic components of the electromagnetic field, i.e., a polarizer, then I can define exactly what the polarization direction of EM wave is. First let us introduce a polarizer that can interact with electric field. Generally these polarizers contain some long chain linear molecules placed in the same direction. If an EM wave is allowed to fall on this polarizer, with the normal of plane of polarizer being in the direction of propagation of EM wave, in different directions each time, then the electric component of EM wave gets blocked if the direction of electric component and the axis of the long chain linear molecules are parallel. Hence the polarization of EM wave is along the direction perpendicular to the axis of such long linear molecules or simply the direction of Electric field. Now if I conduct the exact experiment with such a polarizer that can interact with magnetic field then the polarization of the EM wave comes out to be the direction of oscillation of magnetic field. But when direction of polarization of EM wave is referred, only the direction of oscillation of Electric component of EM wave is considered. Why is that so? Don't the polarizers that can interact with magnetic fields exist? Or between the both directions of polarizations only the direction of Electric component of EM wave is considered conventionally because of simplicity? Or am I missing something?


Firstly: having a lot of difficulty figuring out how to articulate this question due to lack of general math knowledge. There are multiple questions posed below, but I feel like if I knew more they could be condensed into a single question and am hoping someone can suggest and edit to the effect of the below. Thank you in advance for your consideration and assistance on this point! Background I'm taking a course which includes descriptive statistics. In that course they describe the method of calculating covariance and provide that equation. I find myself wondering - why did they choose to define it as they did (using multiplication instead of addition between the terms in the numerator - not even sure if that description is accurate). My question? Are equations for things like covariance derived from looking at phenomenon and 'cracking the code' of how those phenomenon could be described mathematically via a proof? Or does one, at some level of mathematical skill, say: I want to model this phenomenon and I want that model to have these features and qualities to its output, therefore, I choose this particular structure to achieve that goal and then I prove that functionality through a proof? If the latter case - what progression of mathematical learning develops that skillset? Is the same skillset used in both cases? Why I ask If I understand the motivation of the creator of the covariance equation, I could compare it to my own motivation and perhaps come up with a different approach to the same problem that better fits my own goals because maybe our goals are similar but not the same... Thank you again for any advice on how to simplify this..


I'm trying to understand how electromagnetic radiation is created and can propagate through the void. I do understand the concept of an electromagnetic field. But I don't understand how we get from a "field" to a "wave". I'm not really interested in detailed mathematics of how this happens, rather I'm looking for a complete high-level answer that to the extent possible: Explains every step of how we get from having nothing to having an electromagnetic wave that propagates through space (e.g wifi signal) and Provides clear and intuitive justification for any point/law/fact it uses. Preferably all of this should be included within the answer but I do appreciate inclusion of helpful links and references for additional context. Below is my current understanding along with some more specific questions. To create electromagnetic radiation: You make some charge move (e.g through an antenna by creating an oscillating dipole). This moving charge creates fluctuating electric fields and magnetic fields around it All good so far as this is what you expect a charge to do, however these fields will become weaker proportional to inverse square of the distance and you would expect them to basically disappear at a distance. Electric and magnetic fields somewhat magically interact and now you have a self-perpetuating wave that doesn't fade out like the field. I don't understand this part, e.g how we get from a "field" to a "wave" and unfortunately most of the resources I tend to skip over why. I believe there should be a better explanation. For example, according to Wikipedia, Faraday/Len/Lorentz laws have to do with this. However all these laws/theories require a conductive "circuit" that we don't have in the air in the vacuum of space (I do understand how these laws explain how your antenna would receive an electromagnetic signal). Considering magnetic and electric fields/forces will act on charged particles, this raises a few questions: Does the magnetic field produced by the antenna somehow create charge in the air surrounding it? If the whole wave propagation is based on interactions between the fields and charged particles, then how can a wave propagate through the void of space when there is no charge? Thanks!


I'm an undergrad with an electronics repair background trying to find an explanation for one of the fundamental aspects of a transformer. Every explanation I've found of a transformer's basic operation insists that there is essentially no power loss; that is that VxI in the primary = VxI in the secondary. This suggests to me that if there was effectively no current in the secondary (i.e. it was an open circuit or there WAS no secondary) that current would more or less stop in the primary as well. My instructor adamantly insists that this is not the case, and that current in the primary is constant no matter what might be happening in the secondary which boggles my dang mind as it seems to fly in the face of every explanation I see, and I have not been able to get a clear explanation out of him. If this were the case, then a device with a transformer would consume equal power regardless of whether it was operating or not! Right? It seems to me from an intuitive and practical perspective that the EMF on the secondary should be relatively constant with respect to the voltage across the primary, and that the currents through each should be proportional at any given time, so that a changing load across the secondary should somehow influence the current in the primary, but I can't work out or find an explanation of that... mechanism. I feel like it must have to do with the secondary influencing the magnetic field in the core... Please help! I've been wondering about this for years!


In QFT, many mathematical issues arise. Setting aside renormalization, these deal with rigorous constructions of objects underlying QFT: i) In the canonical quantization approach, the main issue comes from trying to multiply (operator-valued) distributions. My understanding is that mathematicians have formalized some settings in which this makes sense, but you have to be very careful ii) In the path integral approach, the main issue comes from defining the path integral (both in terms of defining a sensible measure on paths, as well as making the integral well-defined despite the presence of oscillatory integrals). My main question is: Are the two issues (one for the canonical approach and the other for the path integral approach) related? If so, intuitively (from a purely mathematical perspective) how is the problem of defining products of distributions related to the problem of defining path integrals? I'm particularly curious as to whether there's some intuition to be gained from the usual proof of the equivalence (between the two approaches) in non-relativistic QM (which begin's with Schrodinger's equation, inserts of bunch of intermediate states, and removes operators one by one). In the case of non-relativistic QM, my understanding is that the canonical approach can be made fully rigorous, while the path integral approach isn't quite so (one can use Wick rotation to compute the integral using the Weiner measure, then rotate back using some analytic continuation argument, but I have read that this is only justified in some cases). Given this, do we expect that the issues of rigor aren't quite the same in the QFT land as well? Disclaimer: Apologies for any inaccuracies in my characterization of anything, as I'm still a beginner grasping with many aspects of QFT I am aware that decades of work have gone into formalizing QFT rigorously, and have addressed many of of i) and ii) for different variations of QFTs. What I'd like to understand here in particular is, if these approaches have given insight into how the different issues I outlined above are related. EDIT: I edited the original question to focus on just one question. Originally, I also asked about renormalization (which is what one of the comments addresses)


Would the positive and negative charges line up on either end of the wire? Or would it induce a current? Or would the wire be unaffected by the magnet? This was deleted for being a homework question. I'm not aware of any homework questions like this. I thought this up in my own head. (I drew the diagram in a program called Biorender, which I use regularly.) This IS a question about the underlying nature of physics. The question I have is this, "If simply moving a straight wire through a field causes motional emf, why then does a loop of wire require an area and a certain angle to create a current?" Why wouldn't moving a loop create a current if moving a straight wire creates a potential difference? All of the explanations I can find out there separate the two concepts from each other. "Because Lenz's law states..." How are they tied together? It's still a wire through a field. Physically, what happens to the charges when a loop moves through a field as opposed to when a straight wire moves through a field? But before I can even get to how the two situations are the same or different, I had to clarify some things about how a current can be induced in a straight wire. The following image was meant as the answer to my question, but Stack Exchange seemed to feel it went in the question. Now it may not make sense why I needed an answer when the answer is below. But that's why the answer is in the question itself in case anyone is confused about that.


I have been trying to learn about lattice path integrals. Unfortunately, majority of the literature on this topic is in regard to Lattice Quantum Field Theory and Lattice Quantum Chromodynamics. That is fine, however what I desire is literature on evaluating plain, standard quantum-mechanical path integrals on a discrete spatial-lattice, ideally in continuous-time. This could be a simple chain of atoms or a crystal lattice, but it would be a system where the lattice actually exists and is not just used as a regularisation tool for a continuum quantum field theory. For example, what is the "path integral" (I imagine this would instead be a summation now instead of an "integral") for a free particle on a lattice? Can it be evaluated in that it has a closed-form solution? Even if an ideal closed-form expression cannot be obtained, how far can one go? (E.g. reduce to discrete Gauss-sum or a Jacobi-theta function) In the continuum case, this is just a Gaussian functional integral, which results in a Gaussian. What would it be if space is discretised? Then what is the path integral for a harmonic oscillator on a lattice? What if you add a force term? What about a general quadratic Lagrangian? I have struggled to find mention of these problems in the literature. The standard approach is to introduce the typical time-sliced path integral derivation, and then jump to the Klein-Gordon equation. Are there any resources that do not jump to field theory and instead just investigate these lattice problems for simple quantum mechanical toy models? As a slight aside, I was wondering also if you could start from a typical continuum space path integral, for say a free particle, and add a delta-functional constraint to restrict the continuous path to lattice sites? This would enter the action then as a Lagrangian multiplier and may be evaluable. Thanks!


I've started to study in details quantum optics and I find difficulties in linking the concepts of coherence and correlation among fields, especially because I'm building right now a background on classical optics aven if I already have a strong background in QFT. As far as I have understood, roughly speaking, coherence is meant to be some kind of measure on how similar the properties of one or more waves are at two different spacetime points. To describe with a quantity this feature, we make use of correlation functions to build the coherence factor (I'm thinking about the second degree of coherence): the closest the factor to one, the greater coherence my wave(s) shows between the two points. Furthermore, correlation functions usually arise in photodetection experiment, stemming for the importance of coherence itself in any process regarding the measurement of EM field (I know there are alternative ways to takle detection, but I'm sticking the question to the very basics of quantum optics). Now, this means that correlation functions give me the "degree" of dependence between the photodetection probabilities at two different points. This means that they have more or less the same role that in any other QFT. But now, this seems to me conflicting with the nature of coherence itself: more specifically, in (free) QFT we have the so called "miracolous cancellations" where spacelike correlation functions vanish due to commutation relations, but in the optical theory spacelike correlation functions are at the grounds of spatial coherence measurements, and thus are usually non-zero. I don't get what am I missing, probably I'm misunderstanding some basic stuff regarding coherence itself, and I'm hoping to get enlightned about it. Thanks for the help.


After reading the first two answers to this question, I've become interested in understanding the concept of (co)tangent complex as a way to get some intuition about homotopical algebra, being somewhat more used to the algebro-geometric framework than to the algebro-topological one. More specifically, I'd like to understand this concept in order 'to do basic geometry - this time calculus - on a singular variety', as stated in the first answer (whatever this means), but not 'mechanically', instead trying always to keep an organizing point of view like the one described in the second answer. Also, I'd like to do so following the shortest possible path from basic algebraic geometry and basic category theory directly to the subject matter, with the smallest possible amount of detours, but comprehensively including all the needed basics. (I've studied some scheme theory and homological algebra before, including derived categories, and also ventured a little bit more deeply in the categorical world, but never dealt professionally with these topics and will have to recall a lot before being sufficiently at ease with them.) In this context, what I'm looking for is a double list of topics, one from algebraic geometry and the other from category theory, both ordered by degrees of complexity, designed to be studied in a parallel manner, showing the highest possible level of correspondence since the very beginning, and if possible accompanied with the most up-to-date literature available for this purpose. I'd be very grateful if someone would spend some time thinking about this and writing a nice answer.


There is a particular Twitch streamer from a video game I played, MermaidonTap. If you subscribed and follow her, not all but most of her public streams, she uses "fuck" and a lot of the word "cunt." I am a fellow American citizen, but was not born in US but in an Eastern Asian country, and am trying to learn English. I'm just wondering, is it normal for women to use the word "cunt" normally? I took university English and finally passed it on the fifth time growing up in America. So I'm trying my best to understand how the words in English come about and how certain words passed down to certain generations. As I have researched, the word, "cunt" is a derogatory term towards women and as well as another meaning as vagina. It just seems like, it's derogatory when men call women cunts. But what about women calling other people "cunts." I still find it derogatory, vulgar and I find it hard on my ears to hear words like "cunts" or "fucks" from women like her. Also, some of my teachers and some friends and my own mother, they both say that people who curse a lot and can NOT control their word usage, means they don't know any other language but vulgar words. They didn't go through college and university. They're not educated was what I heard and been told. Do you think this is also true or not? Overall, I'm just trying to understand the American English literacy. I don't mean to offend anyone but rather understand the meaning of the word being used at a person or in a public audience, like her on her public Twitch stream, MermaidonTap.


As far as I know, the first statement of the correspondence is between two formal theories named simply typed lambda calculus and intuitionistic propositional logic, which maps types to formulas and terms to proofs. We also have other statements for higher order logics and type theories. But it is also famous to replace the word "term" with "program" when people try to express the correspondence informally (like Wikipedia). I think that if we assume that programs and terms are same things here, then we can conclude that writing a program, which means expressing a program in a possibly Turing-complete language, is actually proving a mathematical theorem intuitionistically (without the use of law of excluded middle and other proof techniques that are banned for an intuitionist!) Is this true? I think that replacing the word "term" with "program" is misleading here! Because in any statements of the correspondence that I am aware of, we don't have a Turing-complete type theory, and I don't think that type theories are computational models like Turing machines and untyped lambda calculus. Also I think the fact that untyped lambda calculus is equivalent to Turing machine is also misleading to think that a term in a type theory is equivalent to some Turing machine, while as far as I know the equivalence between untyped lambda calculus and Turing machines is not necessarily a bijection, and even if it is, what does it have to do with simply typed lambda calculus?! All in all, don't you think that using the word "program" in the statement is wrong and misleading?


A textbook I'm using to refresh some basic grammar states that indirect objects can be identified by it's answering of questions such as 'to whom', 'to what' etc. (fair enough) and they always come before direct objects in a sentence (this raises questions for me). So the text would identify the pattern in: The teacher gave the students homework. as: S - TV - IO - DO but the pattern in: Tim kicked the ball to Ken. as: S - ITV - Prep. It's been a while, but I was taught the direct object received the action of the verb and the indirect object received the direct object, and also that a verb's classification of transitive or intransitive arises from how it is used in the sentence (i.e. it's not intrinsic to the word itself). So that I would have identified the second example's pattern as: S - TV - DO - IO because the preposition is receiving the DO and therefore is the indirect object. Since 'kicked' can be used with or without an object (i.e. The baby kicked.) I let it pass thinking the text and I could both be correct. But a third example from the text has me questioning how transitivity is assigned: Problems led to desperation. the text again gives the pattern as: S - ITV - Prep. But, 'Led' is almost never intransitive - not unless it's the answer to a question or given some additional context. And so this classification seems more forced to me. The text seems to be implying that the role the verb plays in the sentence depends on how you classify the thing it is acting on and that prepositions can not be indirect objects (despite receiving the direct object as well as answering the question 'to what'). Could someone please clarify, illuminate, or otherwise help me make sense of this?