darklight03/StackMIAsub huggingface dataset:
I have been using lists inside multicols for a "condensed" typesetting of lists of short items. However, I do not like what happens if a page break occurs inside, and I couldn't find a way to automatically force a page break before multicols if it cannot fit onto the current page. So I have been thinking about using some kind of a table environment to typeset a single-row table with lists in cells. However, I would like to make the lists in cells spread vertically so that their tops and bottoms be aligned (except when this wouldn't make sense), like columns in multicols. So far I do not see how to make a table whose height is determined by the maximal natural height of its cells, and where I could make the contents of any cell spread vertically to align top and bottom baselines. Is it possible? How? This might look like not a very natural thing to do, but I am used to the behaviour of lists in multicols, and I would like to avoid situations where, despite having the same number of items, the bottoms are not aligned because of accumulation of minor height differences.
In quantum field theory (e.g. lattice QED), perturbation theory can "break down" when interactions become too strong. Can something like that happen in classical non-linear optics? Can there be any combination of material (absurdly non-linear) and light (attosecond pulses, petawatt lasers, etc.) which can (somehow) be described by classical electrodynamics but the usual "expansion of polarization density inside the material in powers of electric field" (maybe later adding magnetic field as well...) breaks down and doesn't work? I.e., where no finite expansion order gives a reasonable description? Alternatively, if the "describable clasically" assumption is weird in this context, is there a regime where some very strong coupling leads to the perturbative non-linear optics approach breaking down for a stable bulk material that can be produced on earth? I mostly care about "shooting light at something and describing the light that comes out". I've read claims about high harmonic generation (HHG) being such a phenomenon but it's not really clear to me how/why the perturbation theory (at any finite order!) does not apply. You can do it in bulk crystals but I guess the "classical system" restriction would not apply there, as it is usually described using some semi-classical models? For example, Wikipedia on "nonlinear optics" writes about some series "not converging" but it's not quite clear what is meant there. Some articles on HHG (e.g. this) use the phrase "non-perturbative" but what exactly do they mean by that?
Plasma and fusion physics experts, help me with this one: Suppose we have a D-T plasma with net positive electrical charge inside a positively charged metal sphere. As the containment sphere's net positive charge increases, my ignorant intuition would expect the plasma to be compressed and its internal pressure to increase. Sputtering would likely occur as electrons from the plasma escape and are drawn towards the containment sphere, releasing atoms from the containment metal into the plasma and cooling it - but ignore that for a sec. Suppose we could both arbitrarily increase the temperature of the internal plasma and the charge of the containment sphere. Would this be a viable means for nuclear fusion containment? Help me shoot down this idea. What are some practical limitations that would prevent this from working? For starters, at arbitrarily high positive charge, the containment medium itself would disintegrate due to the electrostatic repulsion of atoms in the containment vessel, correct? Is there any existing means of calculating charge distribution in a plasma or solid at very high positive charge? Or is there a means of calculating the theoretical spatial gap between the plasma and containment vessel in such an arrangement?
Consider a horizontal long rod that is undergoing free fall. Consider the torque about an axis through the rod (perpendicular to the rod and to the direction of gravitational force), that is a little to the left of the rod's centre of mass. Clearly, there is a net torque about this axis, but we know that the horizontal long rod will only translate downwards. I suspect it arises from somehow this reasoning being inapplicable for some cases of translating axes about which torque is taken, but in the case of a cylinder rolling down a rough inclined plane, the same paradox is not seen by taking torque about the axis passing through the centre of mass of the cylinder, even though this axis is translating downslope. What's going on?
Is there a distinction that we make between what we call "elements" and what we call "sets" in Von Neumann type set theories? Two textbooks I have used for an introductory study of NBG both made distinctions between sets and proper classes (the prinipal primitive undefined notion being that of "class"). The distinction was ofcourse made to avoid various logical paradoxes, in particular, Russell's Paradox. However, what wasn't clearly stated in either textbook is whether the concepts of "set" and "element" are identical. Both did state that a "class" is considered an element if and only if it belongs to another class. Would it be safe for me to identify the notions of "set" and "element" completely interchangeably? I don't see an elementary distinction between the two. Then the only two distinctions we should make are of "sets" (elements) and of "proper classes" with the general notion of "classes" referring to either one of the two distinctions. Thank you in advance.
When studying supersymmetric QFT's, it is very common to compute the moduli space of the theory by solving all F-term equation (derivatives of the superpotential). More precisely, one should also quotient by the complexified gauge group. Here are three fact that I believe to be true about moduli space and IR flow: Let us consider a supersymmetric QFT, dumbed theory A. Let us now take flow towards the IR and obtain an affective theory, called theory B. The moduli spaces of theory A and theory B are not necessarily the same. More generally, the moduli space of a theory changes with the flow. Now, on the contrary, a moduli space can be seen as the space of theories, i.e. the set of all possible vev's configurations for all the scalars. Choosing a point in the moduli space therefore introduces a scale, and implies that we are at a certain point in the flow of the theory. Certain regions of the moduli space can then correspond to strongly coupled or weakly coupled regimes. The moduli space of a theory, i.e. its space of vacua is the most low-energy thing you can do, no fields can be excited so they all take their minimum values, i.e. their moduli. So the moduli space could be seen as the bottom of the IR flow. Those three pictures are clearly in contradiction. What is the problem with this way of thinking ? I would be very interested by references about the link between RG flow, low energy-limits, and the moduli space.
I had a guess from three years ago that I couldn't prove It is impossible to find a polyhedron whose perimeter (represented by the sum of the lengths of its edges) is equal to its area (represented by the sum of the areas of its faces) and numerically equal to its volume It is easy to find a similar shape with any shape such that two of these are equal to each other, but the question is whether all three are equal to each other. All I have been able to prove is that this shape cannot be a Rectangular cuboid, I also proved that it is impossible for it to be a Platonic solid but I do not know if this would hold in the general case. Here is a proof of the condition of the Rectangular cuboid by arabic language
The infinite square well (and variations) are some of the best-studied systems in quantum mechanics and are often used as the starting point for any quantum mechanical education, as the Schrodinger equation is easy to solve for such a system. I'm looking for specific research papers that describe experimental results that affirm the particle-in-a-box theory because, while the theory is beautiful, it is nearly always presented to students without any discussion of experimental confirmations. Obviously, any real system is going to be more like the non-infinite square well. I'm well acquainted with the other experimental results that affirm other introductory aspects of quantum mechanics, such as the double-slit for affirming particle-wave duality, the stern-gerlach experiment for spin, and various interferometers for superposition, but have yet to come across any for particle-in-a-box theory. The best that I am aware of is Bose-Einstein condensate experiments, but those are typically focused on affirming that bosons can indeed occupy the same state and are not so much what I am looking for. The simplest practical system that comes to mind to demonstrate what I'm looking for would be something along the lines of a Jaynes-Cummings model single-atom cavity experiment where the atom is repeatedly found in the most likely predicted locations. Is this question misguided? Are there actually no such experiments and we only know the theory to be true because more sophisticated developments of the theory align with experiments and extrapolate that the basics of the theory must therefore also be true?
I am aware that this series is incomplete, but it has a large body of existing content, and I am also aware that it is written to be "accessible to non-specialists", but that is obviously quite vague. All that really tells me is that I don't have to be a CFSG scholar, and on the other end that I can assume it's probably not, yknow, an undergraduate text. But what is the actual background needed to follow it? Group theory is a pretty vast field, and it is unclear to me whether GLS is written for people on the level of "grad students who took abstract algebra courses", "PhD candidates in group theory", or what. Specifically the kind of answer I want is describing what the curriculum might look like for a hypothetical course whose purpose is GLS reading prep.
When a tiny spherical charged particle (having a conductive surface) moves at a relativistic speed, the Lorentz transformation for EM-fields predicts that its electric field increases at the top and bottom of the charge whether or not it has been made of something conductive, whereas the field decreases at the left- and right-hand sides along the motion direction. On the other hand, we know that as the tiny spherical charge moves at a considerable portion of the speed of light, its geometrical shape shifts to a spheroid. Taking account of the fact that the charge density is no longer uniform and it increases at the top and bottom of the spheroid, I want to know whether it is the new arrangement of the charged particles that implies the electric field to be like that shown in the figure for the relativistically moving charge, or the Lorentz transformation predicts the EM-field independent of the charge distribution. Remember that the electric field of a charged conductive spheroid is very similar to that predicted by the Lorentz transformation, however, I do not know how far they match.
My only familiarity with topology is the basic theory of mereotopology, so apologies is this sounds strange. An open region is a region without a definite boundary--that is, there is no way to draw a boundary including only points in the region such that every point in the region is inside that boundary. I think of it as the interior of, for example, a polygon without the edges that define the polygon. A closed region includes a definite boundary. The complement of a region is all of the points that are not in the region, so in a normal topological space the complement of a closed region is an open region and vice versa. This strikes me as related to continuity in the real numbers. If you imagine the real number line as a topological space where a range of numbers is viewed as a region, then an open range is an open region and a closed range is a closed region. In this case, the statement that the complement of an open region is a closed region seems equivalent to the statement that every range has a least upper bound and a greatest lower bound--which is equivalent to continuity of the real numbers. Does this make sense, and am I right about it?
I have combed through various sources on the internet and I don't have a definitive answer for the above question: The best that I can come think is the following: Because when I remove synthetic clothes, the body gets charged one way, the clothes the other way. Then as I continue the process of removing the piece of clothing off my body, the charge build up is significant, owing to which the potential difference between the cloth and my body is greater than the breakdown potential of the air. Thus, electrons flow between my body and the piece of clothing I am removing. The air heats up, gets ionized, energy is dissipated in the form of heat, light and sound; in other words, as sparks. Also one more follow up question: Do the sparks flow "inwards" between my body and the piece of clothing? Or just dissipate "outwards" into the air, towards nothing in particular? Is the above reasoning sound? Also, does the same reason hold when I comb my hair? However, if this were true, why do I NOT SEE any sparks when I comb my hair but only hear a crackling sound?
Consider two free-falling observers in the Earth's gravitational field, A and B, who meet at point C, where A orbits the Earth at a constant radius from the Earth's center, and B falls towards the Earth's center along a constant radial line: Can either detect the other accelerating locally at C? As an attempt to answer this myself: it's worth noting that I can detect the acceleration of an object dropped onto the surface of the Earth where I am, and that the Earth is free falling along a geodesic within the solar system. However, I don't think the Earth's surface is a geodesic because I can drill a small hole from the surface towards the Earth's center so that a small mass will accelerate towards the center where, at the center, it will experience a zero acceleration. Hence an observer at the Earth's center can't measure the acceleration of another moving mass there, and on another geodesic. So I'm inclined to believe that neither A and B can detect one another accelerating at C locally when they meet, and that more generally: observers on different geodesics can't detect one another accelerating wrt one another locally when they meet.
I am using the word 'avatar' in my work in the sense that an avatar is the physical representation of some higher entity. We could say that a particular person, animal or object is the avatar of a deity, or that a symbol or picture is the avatar of a person, or that a player-character in a computer game is the avatar of the player. For example, my Stack Exchange avatar is a red 'W' on a blue and green field. I am looking for a single word, or failing that, a short descriptor that is a generic term for the person or entity that has one or more avatars. To use an example from mythology with the correct semantic context for my question, we have the sentence: Rama says, 'I am an avatar of Vishnu'. I'm looking for a word (or short phrase) to fit in the paraphrased sentence from the example above: Rama says 'Vishnu is my _____'. Where _____ is the relational antonym of avatar. Edit: 'opposite' changed to 'relational antonym'.
Nondeterministic refers to a system or process that does not have a single predictable outcome. In other words, when a system is nondeterministic, it means that multiple outcomes are possible for a given set of inputs or conditions. Given that it can be associated with randomness. For physics nondeterministic is the lack of predictability (am I right?). The unpredictability in classical regime (say the motion of classical atoms in a box) is given because of existence of many variables that is hard to keep track of and also the ignorance of the human with regard to the system. The quantum mechanics has nondeterministic behavior only in the measurement process. My question is that other than the measurement which is nondeterministic (in the sense that the outcome would be randomly distributed over a probability distribution), is there any other process that has such property (in any other regimes such as relativistic)? If yes what and if no, why quantum measurement seems to be special in this case?
I've seen the notion of the models in the title a lot in the context of automorphic forms and representations, but I wonder if there's any nice reference for the definition of them for general reductive groups, with some motivations for their namings. I'm pretty sure that the name Whittaker model comes from the Whittaker functions that give Fourier-like expansion of Maass wave forms, and somehow generalize the notion of Fourier coefficients for the general automorphic forms. But I have no idea with the rest of them, although I've seen the definition of them for some classical groups in many papers, especially related to Gan-Gross-Prasad conjectures. My naive guess is that Bessel model should be motivated from the classical Bessel functions but don't know how. Also the name Jacobi in Fourier-Jacobi may come from theta functions, since all the Fourier-Jacobin periods I've seen are defined in terms of Weil representations and theta functions.
I want a term for someone who generally doesn't care about their individual popularity or how other's view them and as such is willing to break social expectations about behaviors and do their own thing, even hen doing so may affect how other's view them. So something close to the phrase "Marches to their own drum", but I'd prefer something more concise and possible less colloquial. I don't want a term that implies one is incapable of understanding societal expectations, I prefer the implication be a choice or lack of desire to try to fit expectations rather then an inability to do so. I also do not want to imply a willingness to harm other's as a result of ignoring social expectations, via causing someone embarrassment, hurting feelings, or general rudeness for example. The term should focus on situations where ignoring of expectations would mostly only affect the person ignoring them, not when doing so is likely to harm others. I'm okay with, and may even prefer, if the term comes with a implication that the person is less popular, less understood, or even mildly ostracized as a result of their ignoring social expectations.
I have some idea of what the main concepts of calculus are about but I have never actually taken a calculus class or studied myself. My general understanding is that before the concept of limits were formally defined by later mathematicians, the first people that actually invented calculus explained the idea of derivatives and integrals using the hyperreal numbers, something for which they were actually criticised by some for the concept being more intuitive rather than mathematically provable/valid. I also believe that modern, standard calculus does not deal with hyperreals at all but rely solely on real numbers in explaining all of its concepts. So I was thinking whether it would be a good idea to get the gist of what hyperreals are and how they relate to limits as a foundation before actually taking on calculus, or would it be not worth it. I'd appreciate any comment/suggestion you guys would make.
Could you simulate a Turing machine by a sequence of Turing machines each with strictly fewer states than the simulated machine? By a sequence of Turing machines I mean this: the first machine is given the input tape, then executes until it accepts, then the contents of the tape are given to the next machine in the sequence as input, and so on until the final machine accepts. If any machine in the sequence rejects, the simulated machine would reject, and if any in the sequence would fail to halt then the simulated machine would fail to halt. If this is possible, the more interesting question would be whether there is a bound on the number of machines needed to simulate a machine with more states. The motivation behind this question is that Busy beaver numbers are known for very small machines, so I was curious about whether machines with many states could be (reasonably) reduced to machines with fewer states. It seems to me that there must be some error in reasoning or understanding if the number of machines needed to simulate another could be bounded, because that would imply any machine could have a bound on steps needed before halting, but I am unsure where that disconnect is.
I recently came across some discussion on the fact that "no bueno" is not gramatically correct Spanish, and generally not a phrase Spanish-speakers use, unless they find it funny. Of course, "no bueno" is just a literal translation of the common English phrase "no good". Sometimes, the phrase appears with the implied subject/verb attached: "it is no good" or "this is no good" (accompanied by a sad shake of the head, or a downcast expression). One can of course say "It/This is not good" or for short "not good", which is unimpeachably fine grammar, but I'm not so sure on the "official grammatical status" of just "no good". Note: there is the hyphenation "no-good", which can be used as an adjective, but that is not the usage I'm talking about in this post.
sorry if the title seems vague. English is not my first language and I dont know what to search to find the answer to my problem. I will preface this by explaining the purpose of this question. I have a robot that I know the position of (x,y) through odomotery. and I also have an IMU that returns the robot's yaw angle with respect to the positive x axis of the map. In this map are objects, that I need to rotate the robot to face them. I will attach an image of how It would look like. Some sample cases My problem is I cannot find a generalized way to cover all cases of robot's position and yaw with respect to the object and map. If someone can point me in the right direction or tell me how I could solve this I would be very thankful
I have a data set of values each with a different associated error. If I take the mean, I can use standard error propagation to calculate a much smaller error. This will therefore incorporate the individual uncertainties of each data point. However, is there a method to propagate the individual uncertainties when taking the median? If I take the standard deviation this only provides a measure of the data spread and not the individual instrumental errors. I have tried adding the standard deviation and the uncertainty on the actual median value in quadrature. However, as this only takes in to account one data points' uncertainty this is high and so my total error does not get reduced. Is there a way to propagate these errors when taking the median?
What would happen to metals or ceramics (non-porous) when subjected to extremely high pressures (dozens of GPa) ? I feel like the sample will deform elastically only. So a ceramic object (think carbides) wouldn't crack. Typically, we assign a maximal compressive strength to metals and ceramics. After which, they undergo plastic deformation or fracture but these compressive strength values derive from experiments involving the pressing of a sample with an hydraulic press. So in these experiments, there is "space" without stress on the sides where the material could expand. But that isn't the case in an hydrostatic context where stress/pressure is applied evenly on all surfaces. Also, in the case of ceramics, I feel like fracturing would create more volume which seems contradictory to me as the sample is being compressed.(in the sense that work should be done against the pressure if the sample gets more voluminous and that wouldn't minimise the energy) If I am correct, you could have solid matter even at gigantic pressures. Any idea?