darklight03/StackMIAsub huggingface dataset:
I hope this question hasn't already been asked, but I looked and couldn't find a question with a similar title. It is my understanding that Maxwell's equations and the Lorentz force law form the foundation of classical electromagnetism. These are fundamental because (in the classical limit) they are always obeyed, in contrast to Ohm's law, which is still a valid empirical law, but is restricted to a specific scope. As I understand them (correct me if I have made a mistake), these describe basic assumptions used to create a model of electromagnetism: the net outflow of electric field lines through a closed surface is proportional to the amount of and polarity of the charge inside the net outflow of magnetic field lines through a closed surface is zero a magnetic field that changes with time will create a circulating electric field an electric field that changes with time will create a circulating magnetic field, taking into account the polarization current the electromagnetic force on a charged particle depends both on the electric field and the magnetic field I've encountered some equations pertaining to phenomena described by QED, but I was wondering if there is a set of fundamental equations like Maxwell's equations and the Lorentz force law that describe all of the core requirements for accurate predictions in its model. In other words, can I summarize QED the same way as I have done above for classical electromagnetism?
I am currently reading "No-Nonsense Quantum Field Theory" by Jakob Schwichtenberg and in several derivations in the book it'll pull a partial derivative operator out of an integral, or change the order of full and partial derivatives on a function, etc. without explanation. So my question is: Are the differential and integral operators commutative? Is this always true? And if not, what are the conditions for it? Furthermore, in the more general sense of abstract algebra, if/when are unary operators commutative? A proof to accompany a given answer would be much appreciated, or at least the citation of one which is free and public. It's certainly intuitive that this is true, at least in the special case of the calculus and physics in this book, but I'm not sure why it is in a more fundamental, mathematical sense and I'm also curious about the general case of all unary operators. This is the kind of thing I would love to try to solve and research on my own but I simply don't have the time right now and I'm having trouble finding a straightforward answer online, so any help would be greatly appreciated. Thank you!
I would like to learn Mathematics for understanding GR, Differential Geometry, Riemannian Geometry and related research papers rigorously. I would like to carve out a clear path to understand these topics by listing out all the necessary prerequisites. I have undergrad Math under my belt such as: Real Analysis, Algebra, Topology and ODEs. I am missing intro to PDEs at this point. I have also created a diagram of the prerequisites in which each bubble represents a subject along with textbooks written in blue. Please take a look at the attached image/file. UG means "Undergrad" in the diagram. Specifically, I need help with the following questions: Is my goal (the center bubble) well defined? I know it may not be specific enough yet, but I have tried to list down some topics I am interested in RED color. Have I listed all the subjects? Am I missing any subject? Is Lie Groups worthy of mention here? Or, would it just fit under Algebra? Is Hyperbolic Geometry worthy of mention here? Is it relevant? How do I learn it? Any textbooks for it? Would anyone please help me break down the following subjects into specific topics that are necessary for my goal: Manifolds, Riemannian Geometry, Real Analysis (grad version), PDEs (grad version), Algebra (grad version).
I was explaining the tidal locking phenomenon to a friend. First I started with the formation of solar system and how at the beginning the planets were actually like balls of magma-esque rocks. And then how the force of gravity would eventually affect their shape and mass distribution and consequently, moment of inertia. But I myself wasn't satisfied with this abstract argument and tried to come up with a more intuitive one. So I said: imagine a fresh egg. Its yolk is at its center (almost) and the mass is evenly distributed. Then after a few days, the earth's gravity pulls the yolk down and moves it closer to the shell. This changes its moment of inertia. But I didn't know how to continue and pull a tidal locking explanation out of this. So I said never mind!! There are large-scale forces and objects involved in tidal locking that I cannot scale them down to something more accessible and easier to demonstrate. Now this has gotten on my nerves. I thought a lot about it and also searched [fruitlessly] quite a lot to find an intuitive, accessible way to understand this phenomenon. We may not be able to demonstrate the effect of gravity on a small scale, but can we use another force that behaves like gravity (e.g. static electricity) to design an experiment for tidal locking?
I've recently started using Texmaker to edit on my LaTeX documents (up to now, I mostly used Overleaf or TeXworks), so I am still new to this program's shortcuts. I have been looking for a shortcut in specific that I can't seem to find, and I would like to know whether this is something this specific LaTeX editor has. I am working on a lengthy file that has several chapters saved as separate .tex documents, which are all called by a single main.tex document. As such, any time I want to edit a specific part of the compiled .pdf file, I need to know in which line of which of my subfiles that specific part of the document was encoded - which is always a bit annoying to find. In Overleaf, this is made simple, since double-clicking on the corresponding line of the .pdf I wanted to edit would link me to the corresponding code line in the document. Does Texmaker also support a similar shortcut? And if so, what is it?
This is a question that a high school student asked me and I couldn't give him a satisfactory answer. He started by saying that An object appears red because the energy corresponding to a "red" (don't take it literally) photon isn't equal to that of the energy gap of the elctronic energy levels of the object. And now if I take the same object in complete dark and shine a red laser from above why don't I see a red beam coming out from the bottom surface the object ? My first reponse was that 'since the object is opaque it reflects most of the red light falling on it' but he wasn't convinced and I think that he is basically trying to ask what causes an object to be more reflective than refractive from the atomic point of view . Can someone help me with this ?
I would like to find the distance function for a curve comprised of all points within a lune that are of equal perpendicular distance to both circular arcs. In other words, what's the minimum distance by which a given point inside of the lune would need to move in order to become equidistant to both arcs? My use case would require both arcs to be of equal radius, if this helps. Constructing a signed distance function for the lune itself is fairly straightforward, but this particular problem seems considerably harder. I've been mulling over it for a few days without any real idea of how to approach it. Can anyone point me in the right direction? Edit For clarity, the curve marked in red is what I would like to find the distance function of:
I am searching for a book on group theory that follows in the style of textbooks written for math students (since I was one), but that covers all (or at least most) topics that would be needed in physics, such as in QFT for example (Think of the topics in Zee's Group theory in a nutshell). My problem with Zee is that it is too verbose and not as mathematically rigorous as I'd prefer (I can understand the concepts better when they are presented in a formal mathematical way). And my problem with other standard group theory books for math is that they usually do not cover all the topics needed for physics (Lie Algebras, Poincare groups...) A single or many-book-combination recommendation that would fill in these criteria is welcome.
Please correct me if I am wrong. Electron is fundamental and is zero dimensional (probably made up of strings). Electron is bound to the nucleus. Electron in an atom has velocity and position which are statistically distributed. Electron mainly interacts electromagnetically. Temperature is the measure of statistical kinetic energy of particles. In the nucleus of an atom, proton and neutron have temperature because they are a bag bound by color forces , valence quarks and a sea of quarks and anti-quarks. Quarks interact electromagnetically. Saying that protons and neutrons have temperature means that there is a randomness in the kinetic energy of the constituents of proton and neutrons. Electrons are bound to the constituents of protons and neutrons through electromagnetic force. Therefore if there is randomness in the kinetic energy of the constituents of protons and neutrons then the electrons must be experiencing wobblyness in its orbit, that is , it must be experiencing some kind of little randomness in its motion around nucleus. And therefore the electron must have a little temperature in the atom. My question is : Does the electron have a temperature in the atom?
I am a veteran of the wars in Iraq and Afghanistan. When talking about the people opposing us, I have always referred to them as the "enemy." Now, as I get a little older, and a little more aware (specifically, as I became a determinist), I have lost any hostility I feel towards them, and I feel that calling them the "enemy" demonizes them in a way I no longer wish to do. So, I was hoping there was some neutral term applied to enemy combatants, even if it's historical or in another language. I think opponent comes close, but it's not quite there. I was hoping there was a word like "interlocutor" meant for combatants of war. I have checked the OED (paywall, I think, I get it through my library), and the words I think that come closest are two obsolete words "contrary" and "adverse." (obviously those words are still in use today, just not in the noun sense of a contrary or adverse being someone who opposes you). Does anyone else have any better matches? One note, some may suggest I call them by the name they go by, such as the Taliban. The problem is that the Taliban is one of dozens of different groups of fighters and we rarely knew who belonged to which. So, I was hoping for a neutral term I could apply to all of them.
I recently received a newsletter from an entity I previously thought to be credible which is embarking on a brand/campaign around the concept of ambition which makes me wonder if my understanding of the word is "off". Can the following be examples of Ambition? Starting a new hobby. Getting out for a walk on the beach. I would have thought that while the above are goals/things a person may strive for - they are not ambitions because they are not substantive enough to qualify as an ambition. Similarly I see an ambition is a goal you strive to reach, while the "Ambition" campaign in question is about the journey (and indeed with no expectation of completing the journey) In other words I think "Planting a new garden" would qualify, but "Planting some plants in my garden" would not, or "Completing a marathon" would qualify, but "getting out for a walk on the beach wouldn't". Do I misunderstand the meaning of the word?
I got a bit confused about independent clauses,so I decided to ask ChatGpt, which has given me three different answers for the same sentence I think he is getting too old, suffering from Alzheimer's disease. The sentence is: I bring a natural positivity and a can-do attitude, allowing me to effectively solve challenging problems even under pressure. Here are my questions about that : First of all "allowing me to effectively solve challenging problems even under" is that a gerund clause functions as an adjective for the main clause describe it Now regarding the clause I have two possible analyses The first one is that: "allowing me to effectively" is a gerund phrase that acts as a subject. "solve" is the main verb. "challenging problems" is the direct object. "even under pressure" is a prepositional phrase that acts as an adverb The second one would be: . "allowing me" is a gerund phrase to start the caluse "to effectively solve" is an infinitive phrase "challenging problems" is the direct object to solve "even under pressure" is a prepositional phrase that acts as an adverb Does anyone have a clue which analysis is right and why?
I was reading a paper on black-hole information loss and it mentioned backreaction. I had never heard the word before so I googled it and was surprised to find no cohesive definition that I could understand. The best I found was this wikipedia article which gives a definition akin to "a backreaction is a field that is often used to calculate forces in quantum systems." However, it gave no strict definition or real physical justification. I was wondering if someone who understood the topic could explain them to me better, because right now they seem to me to be more like quantum epicycles than anything. P.S. I also found this article talking about back-action in quantum measurement which used the term back-reaction once. I could't quite figure out precisely how they were related though.
I am currently studying Linear Algebra Done Right (LADR) by Sheldon Axler and also How to prove it by Daniel Velleman. Currently I am in the chapter of Bases in LADR and I already read the chapter on equivalence relations in How to prove it. I tend to be somebody who looks for analogous things and something that I wanted to verify was whether there is a resemblance of Vector Subspaces and Equivalence Relations: Can one think of subspaces being equivalence classes? What I would suggest would be to say that vectors in a subspace form an equivalence class if they can be written as a linear combination of the basis vectors of that subspace. Moreover, there is a theorem which says: Every subspace of V is part of a direct sum equal to V. Wouldn't then the vector subspaces be analogous to a partition? Am I thinking along the right lines? Doesn't that mean that there is a sort of universal structure of mathematical topics?
A common construct for comparison (making parallels) is "the X of A is similar to that of B". Quoting an example here ... the animals' situation is similar to that of the plants. which is equivalent to the situation of the animals is similar to that (i.e. the situation) of the plants. I often wondered whether this construct can be extended to make parallels in more complex scenarios, where one needs to compare or contrast two relationships (between two pairs of objects). A contrived example is as follows: "The difference between werewolfs and vampires is similar to that (i.e. the difference) between wolfs and bats." Here, what I am trying to compare are two relations: the one between werewolfs and vampires and the one between wolfs and bats. As the two relations are each between a pair of objects, should one say "... is similar to that between X and Y"? If not, what is the proper way to express such comparisons/parallels? (I thought about using such patterns from time to time, but haven't seen it used. Hence the question here.)
I have the following diffraction pattern produced by a fabry-perot etalon The red represents the center of that pattern. Taking the mean intensity of the pattern as a function of the radius from the red dot and applying a cut-off radius results in This clearly shows a superposition of at least two diffraction patterns. Is there any way to automatically decompose the patterns? I originally tried Fourier analysis but that resulted in a complete mess as the peaks appear to be Gaussian, not sine functions. I can fit Gaussian to all the individual peaks. But then my question is if there is an algorithm or method that would allow me to determine which gaussian belongs to which diffraction pattern. I have quite a few more diffraction patterns some of which are more complicated and I wanted to see if there is a way to automate this?
Two up quarks in a proton lead to an imbalance, which results in the proton having the ability to attract electrons. Two down quarks in a neutron lead to balance in the electromagnetic force, leading to no interactions happening with electrons. The strong force and electromagnetic force appear to be directly related. Unless I am misunderstanding something, it appears that a particular arrangement of quarks leads to a specific kind of baryon that has a specific kind of electromagnetic behavior - meaning the strong force is indirectly leading to the behavior of the electromagnetic force. I don't understand why the electromagnetic force is considered to be a separate force than the strong force for the following reasons: Quarks clump together to form baryons due to the strong force, not the electromagnetic force. I understand that quarks have electromagnetic charges, but I don't understand whether the cause for this is due to it "fitting into the math" or it actually being a fundamental property of the quarks. I understand that the strong force overpowers the electromagnetic force, but how do we know for certain that quarks even have charges? Any insights or references would be greatly appreciated! Thanks for reading. So, my question is: why is the electromagnetic force considered separate than the strong force?
I am trying to simulate the reflection of a sound ray, that goes from a sound source, bounces off a wall, and is received by a microphone. The wall has a an absorption coefficient, and a specular reflection coefficient, both of which vary by frequency. Thus, the sound reflected by the wall specularly can be characterized by a certain frequency response curve. LTI filters are characterized by a frequency response and a phase response. Thus, we can treat the contribution of the wall's specular reflection as a LTI filter (applied to the source signal) if we know the correct phase response. The reflection path (shown in red above) corresponds to a time delay proportional to the length of the path. If we assume a constant time delay across the frequency spectrum we get a linear-phase filter, that is symmetric about the time-delay corresponding to the reflection path length. However, this filter clearly has "anti-causal" components: the filter is nonzero before the red line. Thus, the filter begins to have an effect on the signal before the length of the reflection path would suggest that it should. It seems that either the assumption of constant time delay across the frequency spectrum must be wrong then? If so, I wonder what the correct phase response of the filter is.
I have a couple questions on the mechanism for how solar projection through a telescope works: I recently took a small telescope focused at infinity, aimed it at the sun, and held a sheet of paper behind the eyepiece. It projected a focused image of the sun onto the paper. When I changed the focus away from infinity in either direction, the picture became less focused. I'm confused as to why this works since I always thought a focused telescope produces a virtual image that can never be projected onto a screen (and your eye turns it back into a real image for your retina). How can one focus setting for the telescope make an image that your eye can see and that can also be projected onto a paper? If the image coming out of a telescope can be projected onto a paper, I assume it is a real image. However, I find that no matter how far or close I move the paper, the image remains in focus. I always thought that a real image is projected onto a single focal plane at a fixed distance from the lens, so how does this work? Any help would be greatly appreciated.
Suppose we have two bodies at different temperatures, and we let them interact thermally in such a way that the process is not quasistatic (e.g. two different metal spheres touching). Do we arrive at the same final temperature as if the same spheres had equilibrated via a quasistatic process? If so then, rigorously, why? Of course I expect the answer to be yes but I can't convince myself of why in the context of thermodynamic theory. Edit: I think I see that since I can establish that in a final equilibrium, they have equal temperatures, then I can imagine a quasistatic process taking them to this final state? It seems thus that in the case of non-quasistatic heat transfer, ceteris paribus, we have the same entropy change as for the corresponding quasistatic heat transfer?
Clarification: the word "important" is ambiguous. Here, I use "important" to mean that we want to know the value of a quantity accurately. For many optimisation problems, it looks as if we are more interested in finding the minimum value, rather than finding the argmin. For instance, in machine learning models, we care about how well the model can predict, and we do not care about the internal argmin inside the black box that makes this possible. For a linear programming problem, say, for an airport, we do not really care about the exact way to schedule flights to get the maximum efficiency. We just want to come up with a plan which can be completely different from the optimal one, but have similar efficiency in terms of saving time. This is due to the existence of many equally good solutions. Note: one might argue that we do need to know a solution. Of course we do. But we don't care which solution we get, as long as it is a good solution. So it is unimportant to know the solution itself accurately; only the objective value is important. Nevertheless, argmin is discussed a lot in literature (like parameter estimation in statistics). However, I have not seen anywhere why this is useful. Could anyone give an example application where the argmin is more important than the min?
Some English nouns are identical to their verbs (and their adjectives) both in spelling and pronunciation, for example: "This is fake"; "to fake"; "this is a fake" "To tear"; "a tear" "To parody"; "a parody" "A misfire"; "to misfire" There are some examples where the (disyllabic) noun is spelt exactly like its verb but the noun is pronounced with the opposite stress (at least, in the standard accents which I hear day-to-day in the UK), for example: "An escort"; "to escort" "My recall of the event is faulty"; "I cannot recall what happened" "To repeat"; "a repeat" (this last one is a bit weak since people use either stress pattern in different contexts) This phenomenon is observed with some verb-adjective pairs too: "To perfect"; "this is perfect" In all three cases, I would be interested in: The corresponding formal linguistic term (if there is one) Examples! Although I know there are plenty more than I've shown here, they elude me. What's worse is that there's a fair few I've thought of but can no longer recall. Bonus: are there any examples of noun-adjective pairs that are pronounced with opposite stress?
Imagine a situation where I'm moving at the same velocity as the electrons in a conducting wire. In this scenario, from my frame of reference, the electrons appear to be stationary and thus there is no current from my POV. Now, for a brief moment, these "stationary" electrons come into contact with (or pass right next to) a light bulb in the vacuum of space and closes the circuit for a brief moment, causing it to illuminate, and then we all continue moving together. (water this down as much as possible to reach "spherical cows in a vaccuum" state for the purposes of this problem) Question: From my perspective, the light bulb lights up even though the electrons (which I observe as stationary) aren't "flowing" in the traditional sense. How is this phenomenon explained? Would I effectively be seeing a light bulb illuminating without any observable current thus creating something out of thin air (obviously no). Or rather, what do i have to "give up" for the laws of physics to be consistent.