darklight03/StackMIAsub huggingface dataset:
By equivalence principle, one can find a local inertial frame at every point of spacetime. Then this is usually used to introduce the general spacetime metric, as a back-transform of the Minkowski metric. The transformation from Minkowski to general metric depends only on the first derivatives of the coordinate transformation. This means that locally we can get an inertial frame by applying a linear coordinate transform. It is puzzling because in motivating examples we always get accelerating transforms, such as a falling elevator. However, it seems that the acceleration does not play role in the final formula. What is happening? Is this because linear transform is enough to diagonalize the metric, but the quadratic part (i.e., the acceleration) is needed to make the Christoffel symbols vanish?
I understand Thomson scattering as: When an EM wave is incident upon a charge causes it to oscillate in turn releasing energy as another electromagnetic wave. In an Electrodynamics lecture we took a different approach by assuming the far-field limit. The EM wave makes the charge oscillate causing it to emulate an oscillating dipole and hence releasing energy as another electromagnetic wave. I understand that these two present the same behaviour in the far field limit. However, I do not intuitively understand how the oscillating charge emulates an oscillating dipole.
i'm writing my first document using LaTex. I want to include good-quality figures, so i opted to use the PDF format. i saved the images from the plots in Matlab as pdf files, and when i included the figure in my document it showed up with extra white borders, i'm not sure how to fix that. Cropping the pdf can be done, but i have many images to include so that would be tedious. any suggestions? or something else i can do? thank you
We use Past Perfect to speak about actions before some moment in the past. What if my speaking partner doesn't know about a moment in the past? I had done everything. (I don't mention it, but in the mind I mean, I had done before some moment, which I know). What if i don't mention an moment, but it's obvious? I had written all documents. (It can be either obvious or unobvious, that I had written all documents before a deadline).
Suppose I have an infinite wire and close by I place a loop which has increasing current. The wire and the loop is fixed . Now an emf will be induced in the infinite wire which in turn induces a current to oppose the changing magnetic field . My question is why this emf is induced in the infinite wire even though the wire is stationary( in order for motional emf to be induced) and also there is no loop formed by the infinite wire? How to define the flux change in this condition for the infinite wire in order to apply faraday's law to calculate emf?
I am currently reading a book about numerical optimization and have reached line search methods. In a section studying the convergence rate of the gradient descent method, I am really struggling to understand how alfa is determined (see below). I really do not understand at all how to even differentiate with regards to alfa, when dealing with vectors and matrices, and I also am very confused as to how the b-vector can just completely disappear in the result. Can anyone explain how the derivative with respect to alfa is determined, and how alfa is isolated from there? Thank you to anyone who may be able to help. Explanation of how alfa is determined
It is common knowledge that submodules of finitely generated modules need not be finitely generated. One can see this by considering a non-Notherian ring and extending this as a module. My question is, how does one think about this intuitively? The way that I think about is: When I restrict myself to a subspace, there is a chance I no longer have access to the elements that finitely generate the whole module. In contrast with Linear Algebra, this can occur when the subspace is non-empty. Could someone let me know if this is a correct way to view this or if there is a more intuitive way of viewing this result?
Suppose I want to cover a whole circle of with hexagons. See the attached photo where coloured hexagons are those required to cover a whole circle. Given the length of the side of the hexagons and the radius of the circle, what's the minimum number of hexagons required to cover such circle? I notice that there are three ways of inscribing a circle in the hexagon tile: the center of the circle lies at the center of a hexagon, at the center of a hexagon side, and at the vertex between three hexagons. Attachment
I know that in the pure algebraic context, connected groupoids are equivalent to groups. Moreover, connected groupoids can be seen as action groupoids. Consequently, any abstract groupoid is essentially a disjoint union of action groupoids. Therefore, is it not more natural to study group actions than study groupoids? Since group actions seems to have a richer structure than groupoids. So, why study groupoids? Does this also hold in a continuous context?
In Information Technology, we often see something that looks like a compound adjective, pairing a number and a noum. Some examples are two-factor authentication, two-step verification and twelve-factor app. I always found it amusing that the noun itself is not plural. Why isn't it named, for example, "two-factors authentication?" Is there a rule for that construct? If so, what's its name? (My theory is that it probably came from multi-factor authentication, for example, in which there is some sense in the format, but I don't know for sure.)
I have the simplistic image of a photon being a disturbance in the photon field caused by the loss of energy of an electron. In this image, I see that the disturbance is essentially a harmonic oscillation and it 'traces out a wave' as it propagates through the field. Also, given this image, it seems that the 'larger' the amplitude of this disturbance, the longer the wavelength and thus the 'lesser' the energy of the photon. This seems counterintuitive. Alternatively, is the 'amplitude' of the disturbance the same for all photons?
While writing a fantasy narration I created this sentence: High among the clouds a castle floated. My American friend tells me it sounds bad to a native ear. I think he sees a problem with the unusual word order. At the same time the Hobbit starts with: In a hole in the ground there lived a hobbit. ...which seems a similar sentence. What is the science behind the word order in an English sentence? UPDATE: The original sentence was actually High among the clouds a castle floated, disregarding the force of gravity. I removed the ending phrase to simplify the problem, but from some of the responses I infer that the phrase may actually matter.
I was just studying statics when I realized that a body can be in both static or dynamic equilibrium at the same time but I am not so sure. My textbook says that an object at rest is in static equilibrium while an object moving with zero acceleration i.e constant velocity is in dynamic equilibrium But.... Motion is relative so if two objects were moving with the same uniform velocity, they would be at rest relative to one another and therefore be at static equilibrium but since they are in motion relative to someone outside the system so they should be in dynamic equilibrium. Can anyone confirm this?
A noon cannon usually has a single convex lens which focuses sunlight onto a fuse. The cannon contains gunpowder. The fuse ignites and the gun discharges at noon. https://en.wikipedia.org/wiki/Sundial_cannon A query has arisen about a noon cannon with TWO lenses -- what is the benefit of having two lenses? Does it provide more heat to ignite the fuse? The noon cannon with two lenses might be larger than usual, therefore needing a lens with a longer focal length. Images below. Thank you for any suggestions. I know quite a lot about sundials but I am not a physicist.
I'm using CircuiTikZ and I came across this issue frequently: every time I want to tweak some options using tikzset, there is no documentation for the available key-value pairs. For example, if I want to change the voltage direction, in the documentation, only value (i.e., RPvoltages) is provided. However, I prefer to use key-value pair in my code for future reference so I was wondering how can I find the respective key name given value name or the other way around.
Sorry if this sounds like a silly question, but what would happen if a scientist observes Schrodinger's cat alive, but is then thrown into a black hole before he has leaked any information to the environment. Then later a second scientist observes the cat - can he observe it as dead this time, since the first scientist and his information can't leave the black hole to contradict? That is, is a wavefunction allowed to show two different faces to two observers so long as they are never allowed to compare notes because of an event horizon? Is exiling information behind an event horizon the same as erasing it?
So bound states have continuous position representation, yet countable discrete energy levels. Doesn't this imply that one of the following operations is invalid: <x|> or <E|>. Because |> cannot simultaneously have countable and uncountable number of elements. And the number of elements in the bra and the ket need to match otherwise the operation doesn't make sense. The ket |> exists independently of whatever representation we choose. So what's going on here?
There are a couple of sentences: The water froze and caused the pipe to burst. If it is a succession of past events it should be Past Simple. But if the first part happened earlier, it should be : The water had frozen and caused the pipe to burst.So, can it be Past Perfect, or this sentence must be written in Past Simple? Year by year the population of the West increased. Should it be increased (Past Simple) or was increasing(Past Continuous) ? Because, I guess, Past Simple does not belong here . I don't know which tense to choose. There is no extra context, just sentences on their own.
Suppose a piston enclosing an amount of air-fuel mixture and then this piston compresses the mixture to the point of auto ignition, if every part of the mixture ignite simultaneously is it safe to say that there isn't a flame front because it can't move because there is no longer a combustible mixture? If there is a flame front are they multiple flame fronts? Also if my reasoning is correct does that type of combustion have a name?
In Arabic, the idiom "You made my neck as small as a sesame seed" is used to mean that someone related to you or a friend, put you in an embarassing situation with others. This idiom is usually used by parents when their children make them anything but proud. People who have long necks are assumed to be proud; hence, 'a neck as small as a sesame seed' in this idiom. Is there an idiom anywhere close to this meaning in English?
I am currently reading electrodynamics from Feynman. When talking about the energy of the electromagnetic fields, he says that the location of the field energy could be known at least theoretically since all energy is a source of gravitational attraction, and if we could measure the gravitational attraction we could comment on the location of the energy in the fields. I have come across formulations where gravitational field is associated with an energy density and is said to be always negative. I wanted to know whether this energy can have an associated gravitational field.
Suppose that, in a homogeneous linear system, the coefficients of one of the unknowns are all equal to zero. Show that the system has a nontrivial solution. Attempt: Note that the rank must be less than or equal to the number of columns in the coefficient matrix. If we had all columns of the augmented matrix different from zero, then, after being reduced to row-echelon form, we would have at most rank = number of columns. As we have a null column, then surely the rank is less than the number of columns. Therefore, we will have at least one free variable. Am I right? Furthermore, is there any way the solution could be more formal than that? (using the same approach)
What comes after "pair"? Maybe "quad" for four? In cards it's "three/four of a kind" but what about contexts beyond cards (like science)? For example, in physics there's a "pair-flip model," but what is the extension of this model to flipping three things? I've also tried the internet, see also: https://en.wikipedia.org/wiki/Tuple Not to be confused with: single, duo, trio, quartet, quintet, ... monad, dyad, triad, tetrad, ... single, couple, triple, quadruple, ... single, twin, triplet, quadruplet, ... Or does "pair" secretly belong in one or more of the categories above? Or is it just an isolated word for, e.g., trousers? Thanks for helping me solve this mystery!
There's a word people around me used to use a lot, phonetically it would spell 'stercus' or 'stircus', I think. It was used in a context meaning excitable, frenzied, frenetic, that kind of thing. Like, you could say that: Kids who eat lots of sugar tend to go stercus. I don't know if it's an actual word and I can't find it because I'm butchering the spelling or misremembering how it was said. I'm a native speaker from Australia, if that helps.
In the double slit experiment, an interaction with a detecting mechanism is needed to know which slit was the one the photon or electron passed through. I read in other questions that polarized filters are placed in the slits, how is it exactly that this helps us know which slit the photon passed through? Do experiments using electrons also use polarized filters? If not, what do they use and how does it work? Are there other types of detecting devices? Is there a detecting device that truly does not change any properties of the photon or electron when interacting with it?
I am using sagemath to compute Einstein tensors of a non-standard spacetime. The output is something horrid and non-diagonal. I need to find the Eigenvalues of this tensor... which is represented as a symbolic matrix. This is proving to be non-trivial with the sagemath documentation. Is there a way to convert my tensor output into a matrix that sagemath can find the eigenvalues for? If a worked example is required, please let me know as I can provide an update easily. Thanks in advance...
uh I'm curious again regarding the theories of reflection. This is known that Metals contain free electrons that absorb energy and vibrate when they come in contact with light. Later, they release the energy back in the form of light. But what if we polish a transparent conductive oxide glass (transparent glass which is electrically conductive). It doesn't seem to be acting like a mirror. So is it really just the electrons which decide the reflection? Do correct me if I'm wrong. I'm just a stupid kid :) My previous post
I know for abelian group, there is a Kronecker decomposition theorem. It said any finite abelian group can be factored as direct sum of cyclic group of prime power order. I want to know is there anything similar with this for the abelian semigroup with identity? As for semigroup, we don't need the inversion, so it seems like it can also support some kind f factorization.
When a spinning top slowly advances and hit a surface (a wall), intuitively one would expect that the top gets bounced mostly along the wall, due to the friction between the top and the wall. But the observed phenomenon is that the top gets bounced mostly in the norm direction of the wall, and in a much higher velocity than when it hits the wall.
Looking for alternative to "opened a can of worms" in the following sentence... It needs to be graphical enough for a young readership to grasp, a readership that won't have heard of Pandora's box or even know what a can of worms is. Having fixed one friend's laptop I wondered if I may have opened a can of worms and now all my friends will expect the same service. Any suggestions?
I have read a book saying that "... backward induction cannot be applied to games of imperfect or incomplete information because this entails cutting through non-singleton information sets." However, in another textbook, it says that backward induction can be applied to games of imperfect information, and we can derive the equilibrium by starting at a historical point and show the subgame perfect equilibrium in that subgame is a equilibrium in the Nash equilibrium. I wonder what are the contexts to use backward induction in a game with imperfect information but complete information?
From what I know, current flows across two points only if there is some potential difference across those two points. If that is the case, why does the current flow through a wire then? See the picture for clarity: As one can see, the potential is same on the left side of the wire. The current should not flow, yet it still does. Why?(I have the same question if the resistor was replaced by a capacitor.)
I know that electric current is a scalar quantity and hence it should not follow vector addition. But I have read that equal currents in opposite directions will cancel out each other so is this kind of addition not a kind of vector addition?... where while adding we are including their directions also. So please explain that even when current doesn't follow vector addition then why the currents which are equal and opposite in direction will cancel out each other?
Is it possible to arrange a set of massive objects such as stars in orbit around a black hole such that objects of lesser mass cannot physically reach and fall into the event horizon of the black hole? This could be similar to how Jupiter may protect the Earth from long period comets. In other words, could there be several star systems with the black hole in their center, such that any possible object on approach to the black hole would be gravitationally deflected away from the black hole?
I need to find the X-coordinate of a vertex in a right triangle. It's been a while since I used trigonometry and I can't seem to wrap my head around it. Image with coordinates The Y-coordinate is always the same as the Y-coordinate of the vertex on the left, under the radius. I've tried using line intersection, but I don't know how to use that here since I don't know where to put the X coordinate of the diagonal line. What method can I use to solve this?
We know that the Mandelbrot fractal contains a countable number of copies of itself. See : Does the Mandelbrot fractal contain countably or uncountably many copies of itself? Where that is explained. Notice that polynomials have a finite amount of zero's and entire functions have a countable amount of zero's. So I started to wonder : Is there a Julia fractal that contains uncountable many copies of itself ? And if so, can they be iterations of entire functions ? What are typical examples ?
During the recent merger of two Neutron stars the lead up to the merger was detected as gravitation waves. This was the merger of two spinning bodies that had very strong magnetic fields and they were orbiting each other. How sure is the community of physicists that the signals detected were not ULF radio waves? Radio waves would have had the same frequency and arrived within the same time frame.
I am not sure why, but when I am working on a multi-file project in TeX with a latexmain file, the compiler included with VIM-LaTeX does not compile multiple times to include newly added citations in my bibliography. This only occurs for multi-file projects. If I work on a project that has only one TeX file the file is compiled multiple times to include new citations. Does anyone have any clue what may be causing this?
Take these sentences: I felt he was mean to do that. We'd be stupid to do something like that. I feel like the "to do that" part in them functions differently syntactically than in sentences like "It is mean to do that" where "to do that" seems to be an infinitive clause functioning as the subject. Is "to do that" a complement to the adjectives "mean"/"stupid"? Comparing "he was mean to do that" and "he was mean for doing that", I think semantically the two sentences seem close but still different, but I can't put my finger on how exactly they are different in meaning. So how are they different? Also do "to do that" and "for doing that" function the same way syntactically? Are they both complements?
In general, knowing the mass distribution of two colliding objects and the exact point of contact, how would one take into account angular motion in the analysis of the dynamics of the individual mechanical energy of the objects and the total energy of the mechanical system? That is, each object may be rotating about it's own axis and in addition moving with some linear momentum. I've looked through my mechanics notes, but I don't quite think I know what happens. Around which point is there going to be torque? Or some angular impulse, I guess. Thanks!
I'm looking for some interesting operators for which I can find invariant subspaces by hand. As most subspaces had some connections to eigenvectors, I'm currently searching for "non-trivial" / interesting (linear) bounded operators on Banach spaces, preferably Hilbert spaces, which do not have any eigenvalues. I know about Volterra and multiplication operators but I can't seem to come up with any of my own. In regards to Lomonosov's theorem, operators without eigenvalues, which commute with compact operators would be even better.
When I jump off of something, even with eyes closed I can be pretty certain that I'm falling due to the associated sensation of falling. However, as gravity should be affecting each of my internal organs equally, surely the sensation must be due only to the air resistance, which causes my outer body to accelerate slower than my organs? As a follow up, assuming the above is correct, would the best way to minimise the sensation of falling be to keep as narrow a profile as possible I.e. keep pencil shape rather than spread arms and legs out flat?
Take a cupboard or just a large wooden box. When pushed it a point above its center of mass, the cupboard topples because there is a net torque due to the friction and the force you apply. When pushed at a point below the center of mass, the cupboard does not as the torque due to the friction and the torque due to the force you apply are in opposite directions (they cancel). Now my question is if the force you apply is equal to the maximum value of static friction, then the box is at rest. However, since the friction applied is at a larger distance from the center of mass than your force, shouldn't there be a net torque?
I asked a similar question before, but it did not give the answer I was looking for, so I will clarify what I actually want in this question. H. Andreka created a formal first-order logic theory that was meant to axiomatize special relativity. I very much enjoyed reading his paper and other papers that axiomatized special relativity. However, I have yet to find a similar text for Newtonian mechanics. What I am looking for is a paper or papers that give a formal first-order theory that axiomatize Newtonian mechanics.
I am reading about monoidal categories and I am not able to think of categories which are non-monoidal. Am I thinking in the wrong direction? Is being monoidal, an additional property like topology? For example, given a set you define a topology on it making it a topological space. Similarly, given a category, you define a tensor product map making it into a monoidal category? Also, given a category does product of two objects from that category always exist? ( I know about the uniqueness but I am not sure about existence). If yes, can a map sending two objects to their product be considered as a tensor product map?