If we are moving a block on a rough surface extremely slowly (quasi-statically) with the help of an external horizontal force, then is it the case that no heat will be produced, but the work done by friction force will still be equal to the work done by the external force for any given displacement? The reason this question comes to my mind is, in thermodynamics, if we say that a process is reversible when it is slow, then there should not be any heat production despite having frictional/viscous forces in the system when the process is slow. So, this implies that the existence of a reversible process rests on the idea that friction should not produce any heat. So, it is correct to assume that friction can do work without producing heat when the process is extremely slow? Because if in all the examples we see around, we see that work by friction is always accompanied by heat production. Kindly help.


Suppose we have a system of electrons a very tightly confined space (like a tiny magnetic trap). Let's say we continually increase the degree of confinement such that the electrons are confined into a smaller and smaller space, to the point where the degeneracy pressure reaches a scale comparable to the masses of the other charged leptons. Does there come a point where it's favourable for an electron to "decay" into a heavier lepton due to degeneracy pressure (producing neutrinos and antineutrinos in the process)? My thought is that this process would be somewhat analogous to the formation of a neutron star (or hypothetical quark star). What would happen if we kept confining the particles into a smaller and smaller space to increase the potential further? Would even heavier particles be produced?


Consider a simple experiment, such as boiling water in a pot in your kitchen, is it possible to estimate the time needed for the water to boil based on elementary properties of water ? In the physics and chemistry literatures we find many computational and theoretical works attempting to predict the boiling temperature of water, from elementary properties, such as the molecular bonding strength or angles. But what about the boiling time ? The basic parameters for the boiling time should the heating power, the heat conductivity of the pot, the heat capacity of water, and possibly other properties that characterize the transfer of energy from the environment to the water. My question is about the time needed from the moment where the water temperature is close to boiling point, to the moment where water has left its initial liquid state. It is mentioned in many lectures and textbooks that nucleation is a limiting step in the formation of a new phase, in particular for the boiling of water. But I was never able to find a critical comparison of such statements. So I would appreciate any references or indications. Which kind of computations do you expect to be necessary ? What kind of empirical parameters enter in the calculation ? Was this done in a reference?


Suppose we place a current carrying loop of wire in a magnetic field. Of, course its easy to see that for a uniform field the wire will experience uniform force in all directions and so it'll get stretched out into the shape of a circle. But I wanted to ask if we could make any comments about the shape of the wire, in any random magnetic field without knowing the exact function describing the field. At places where the field is stronger, the wire will tend to get stretched out more so I suppose the local radius of curvature there would be less as compared to a point where the magnetic field strength is lower. But thats just a qualitative rough way of looking at things. Is there some exact way to find out the answer to this


I just started reading about the conduction mechanism in polymer. From what i read, polarons are used as method of charge transportation in non-degenerate polymer. While for degenerate polymer, both charged soliton and polaron will do the job. As i go a bit deeper, polaron can be defined as: A polaron can be thought of as a bound state of a charged soliton and a neutral soliton of which the midgap energy states hybridize to form bonding and antibonding levels. A Quasiparticle that is created through the interaction of charged particles ( electron or holes ) with Phonon. If i understand it correctly, the definition of the polaron contradicts with its existance in non-degenerate polymer as non-degenerate polymer doesnt have soliton due to its topological reason. Also, as i am quite new to all these terms (polaron, bipolaron, soliton, charged soliton and phonon), i tend to get confuse and couldnt differentiate each of them precisely in layman terms as most of the literature are mostly explained in what could say too "academic" for a starter like me. Would someone be able to help me clear my confusion? Any help, advice and guidance are really appreciated. Thanks in advance


Basically, as the title says. Maybe this is trivially true or false, but I have not enough intuitions about topological surfaces or aperiodic tilings. To make it a bit more precise - I mean the kind of aperiodic tilings such as Penrose tilings or the tiling generated by the recently found hat tile. By "gluing" the patch into a torus (or some other surface) I mean essentially just like you fold a net into some geometric shape by connecting edges. And the edges should be glued in such a way that all matching rules of the tiling are satisfied. If this is possible - would the resulting "tiling of the surface" also be aperiodic, or could it be made to be? Here I would consider "aperiodic" to mean that there are no non-trivial symmetries between any two tiles on the surface.


I saw a question asking "If the Moon was moved to the surface of the Earth, from how far away on the Earth's surface would it be visible?". (The question says to ignore any other factors like gravity making it smash into the Earth's surface and effectively destroying both bodies). Basically, it's this, I think: We know AB, which is the radius of the Earth, and we know CD, which is the radius of the Moon. AC is the sum of these two. Even though it doesn't really look like it in my sketch, I think that the lines CD and AB are parallel, and that ABE and CDE are both right-angle triangles. We need to work out the angle at A, which we can use as a ratio of the circumference of the Earth (which obviously is another known quantity), to work out the distance around the circle to B. Are my assumptions correct? Is it possible, from those givens, to calculate the angle at A?


EDIT: Completely rewritten because of the 'needs clarity' tag and some useful related questions appearing in the side-bar. I hope this is clear now This answer gives a long list of properties of particles whose value differs by a minus sign when comparing a particle to its antiparticle. We know that anti-particles exist, so apparently for every particle there is a particle where the value of all the properties in this list are 'flipped': i.e. the same magnitude but of opposite sign. My question is: given a particle, say an electron, does there exist a different particle where some of the properties in the list in the linked answer are flipped and some are not? If the answer is no, why is this not possible? If the answer is yes, what is an example of such a pair of particles?


I am beginning to learn some very basic electronics. I was learning how and why lightbulbs light up. It turns out it happens because they have a very thin filament which makes the passage very narrow for electrons so they lose a lot of energy colliding with the molecules making up the filament instead of using it to drive the passage of the current. This causes three things. Voltage drop Decrease in intensity Transfer of the electrons' energy to the atoms making up the filament so they vibrate causing it to heat up, and the energy is also used to excite electrons to a higher energy level momentarily where they release the energy in the form of photons where the lower the wavelength, the brighter the light color and the greater the energy. So the filament also glows and lights up. I am interested in the latter and I have a couple of questions. Why don't the electrons remain in the higher energy levels? Why do they choose to release the energy as photons?


I've been struggling with proposed answers to the twin paradox. I know that an object traveling at relativistic speed ages slower than a stationary object. This must be caused by some interaction with spacetime that leads to a slowing of time. But how does this work when we are looking at relative time dilation between two or more individuals. Imagine a huge crowd of relativistic rockets moving inertially and crossing each other in all directions at different velocities. How can one pilot claim to be younger than another, while the other is younger than a third who is younger than the first? Then it came to me that this all might just be the same as relativistic length contraction where it is only occurs while they are in motion, and all effects stop once they stop moving. Once they stop the only age effects are measured against an earth clock. We know that for kinetic length contraction, the man on the train station platform will see the relativistic train length contracted while the main on the train will see the train station platform length contracted. But the instant the train stops moving, all measurements return to normal. Is it the same with the twin paradox? That any age difference as seen by one pilot over another disappears once the motion stops? And the only real aging is when measured against an earth clock?


Exactly what the title asks. "Diode" comes with it the ideas of depletion layers and forward/reverse biasing and electron-hole recombination, but SPAD physics doesn't seem to be dependent on any of that. You just put a giant electric field on, and as soon as a photon comes and promotes a charge to the conduction band, it gets so much energy that it can knock other electrons out of their atoms, creating the avalanche. In other words the fact that it's "reverse-biased" relative to the doping of the semiconductors seems irrelevant, as does the fact that the reverse-bias is above the breakdown voltage. Could I take a piece of undoped silicon and use it as a SPAD if I put a sufficiently large voltage on it? Or else why is it important that it's a diode?


I have figures which I use in several places, some of them in dark context and some of them light (for example, a my obsidian vault is dark but a PDF of my thesis is light). As such, some of the figures are colored to work in a dark environment, and some in light. But sometimes I want to take a figure from a dark context and use it in a light context, or vice versa. So far I have created a copy of the figure for the light/dark mode by simply inverting the colors. I use color schemes in which this looks mostly fine. But manually inverting the colors and saving a copy is exhausting, and inefficient. I want to do this automatically, and possibly without saving a copy of the original figure. Is there a way to invert the colors of a figure while compiling the output of a latex document? Thanks a lot! Edit: I saw this question, but the answers were basically "you can't". Since the answers are unsatisfactory and the question is VERY old, I thought it would be best to open a new one. But if this is not the custom here, please let me know and I'll ask there instead.


Noether's theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. Energy, however, is not conserved in an expanding spacetime because there is no symmetry on the time axis. This enables dark energy to exist and is the reason that general relativity does not conserve energy (except in symmetric situations). My question is about how this translates to theories of quantum gravity that predict the existence of spacetime that becomes spatiotemporally isolated near the center of a black hole. Examples are white holes, fecund universes, and collapsing wormholes. I know that black holes conserve energy, at least for an outside observer. So can energy be created in a disconnected region of spacetime at the other side of a black hole center for certain theories of quantum gravity? Or does Noether's theorem prevent this? (Bing insists that the theorem applies here, but I don't trust the answer.)


I was taught that there are four types of nouns: singular countable: journey, sheep, child plural countable: journeys, sheep, children singular uncountable: travel, water, fruit plural uncountable: groceries, customs, thanks Some words are only used with: countable nouns: one, two, three, many, number, few uncountable nouns: much, little, good/great deal, quantity, amount Am I right? What about the word 'cattle'? You can say neither 'three cattle' nor 'much cattle'. (According to Practical English Usage) What type of noun is it? If it is an uncountable noun why I can say 'many cattle' but not 'much cattle'? Isn't 'many' used only with countable nouns and 'much' only with uncountable nouns? Also, you might consider 'staff' or 'jeans'. You can say for example 'four staff' but not 'a staff'. (According to Practical English Usage)


While trying to understand General Relativity, I'm struggeling with disentangling coordinates and curvature. The metric tensor contains information on both: coordinates as well as curvature. Curvature is a physical characteristic of spacetime, while the coordinates can be chosen completely arbitrary. Is there a method to disentagle coordinates and curvature? (Stephan Hawking must have had one as he managed to show that the singularity at the event horizon of a black hole is only a coordinate singularity, not a physical one...) I fistly thought that one could simply use cartesian coordinates - and then would be left with only the curvature (in this question: Determinant of metric tensor in Cartesian Coordinates constant in vacuum) However, I was taught there, that cartesian coordinates can only be used in flat spacetime. Now, I'm totally clueless again how to disentangle coordinates and curvature. I'm looking for a tensor/measure/metric/field, where only the physical curvature is in - and no coordinate curvature. Where the coordinates are flat like in a cartesian grid. Where the metric tensor is simply that of a flat spacetime in cartesian coordinates, if the spacetime is flat (Minkowski spacetime).


We know that a complex manifold with a hermitian metric h is Kahler if and only if the corresponding Kahler form (which can be obtained from Hermitian metric) is closed.The complex manifold has an underlying real smooth manifold whose Riemannian metric can be found from the Hermitian metric of the corresponding complex manifold (Re(h)). We also know that a Riemanian manifold is Kahler if and only if there exists a smooth almost complex structure on it which is metric compatible and whose covariant derivative is zero. Now, the question is: if a complex manifold is Kahler then does this imply that the underlying real smooth Riemannian manifold will be also Kahler? I think that it should be Kahler as if the manifold is already Kahler in complex case so there should be no obstruction to the existence of compatible almost complex structure with vanishing covariant derivative.


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?


I'm interested in writing a document following not only the Chicago Manual of Style in principle but also very literally the style used by the University of Chicago Press in their publishing. In other words, I'm not just trying to follow the style guidelines outlined in the manual, but also directly emulating the full style used in their publications. Here is a sample page: Obviously their style has a large number of rules (many more than used on that particular page) and I may not need to use all of them, but even a small subset would be painful to implement manually. So what I'm wondering is, is there an existing (la)tex template that has already implemented most or all of their style? When I search, I only find results for templates for their bibliography style, and that's more about following their guidelines rather than directly reproducing their formatting.


Suppose we have an arbitrarily well-isolated empty region which we initialize with a cold, diffuse, homogenous photon gas, such that the only energy density gradients are random thermodynamic fluctuations and the only massive particles are those that are created after initialization. Thor pokes a tiny hole in the arbitrarily isolating boundary, puts his magic omni-detector on the hole, and measures an interaction event every time a particle - typically a photon, but sometimes perhaps a massive particle or antiparticle - reaches the omni-detector. Magic omni-detectors always interact with every kind of particle. However, Loki has secretly tampered with Thor's omni-detector so that it records everything as if time was proceeding many times faster - not just the time stamp on each measurement, but also the values measured, as if the SI Second had been re-defined to be some multiple of its current value with all the downstream consequences on other units and constants. Is there any way Thor can tell, by looking at a subset of his data (i.e. not just comparing the total run time of his experiment to a clock that hasn't been tampered with), that he was looking at a large, diffuse, cold region, and not a small, dense, hot region?


I have been revising some maths and in particular I have been focusing on vector calculus. Today I have just finished looking at surface integrals. I do understand how we parametrize a surface integral starting from a region R, but I am having troubles actually understanding how these integrals are put in context when having a scalar function or a vector function. For instance, a popular example is that you have a surface (for instance a metal sheet) and then you have a function which gives the density of each point of this metal sheet (if I were to plot this function, would it have the same shape of the surface?) Further, I dont understand why calculating the surface integral of the density function gives me the mass (isn't the definition of density mass/volume and hence I would need some kind of volume?) Thank you for whoever can explain this in simple terms


Something got messed up in my lyx settings and now the editor window previews the text in a very ugly and informal looking font. The pdf output is still the regular latex font. How can I reset the editor window font? This would probably be fixed by a complete uninstall and reinstall of lyx, but I can't figure out how to do that either. Here are some screenshots of what the editor window and pdf output look like For context, this happened after I tried to add a lyx -> markdown converter (https://wiki.lyx.org/Tips/ConvertMarkdown). I couldn't get it to work, and quickly gave up. But not my lyx font is hideous! Things I've tried: Uninstalling and reinstalling, but all my previous settings are maintained, including this font issue. I tried looking for any saved files in the indicated directory here, but don't see any files. How can I completely uninstall lyx? Tweaking the screen fonts options, but not exhaustively Running "tools/reconfigure", which produces an error


My question is both naive and subtle. Naive because I don't know much more than the layman about physics and in particular quantum physics. Subtle because physics is an attempt to model the world, and as a computer scientist, with a strong interest in machine learning but also formal logic and models, a model is just that, a model. Not necessarily reality. It is not because a model fits reality that the model is the truth about reality. From my understanding, we know that: the quantum physics model has not been contradicted on the notion of superposition that it introduces there are experiments that can be explained using the quantum model where the classical model fails Am I correct to say that this only proves that the classical model, is just that, a model, and therefore incomplete? And that for those cases quantum mechanics has a better predictability power. Now the question: Has it been somehow proven that a physical entity can a some point in time and space have a dual state (independently of the model)? Or is it only that quantum mechanics is the only known model that allows us to explain things we otherwise couldn't? I would like to know if objects of our world can be in two states at the same time or that it is just more practical for predictability purposes to model things this way.


I live in an arabic speaking country, and while I was sitting in a food court today, I noticed that the sound of the crowd was the same as the one I see in the stereotypical sound of the crowd. I have seen many threads discussing this on different forms, eg: Quora , reddit, Metafilter, and on this site, there is this tangentially related question. The problem with all of these is that, none of them is concrete, in sense of not having any calculative evidence. Going down the reddit rabbithole and reading the metafilter thread gives me some idea on how a calculation should look like, but I don't feel so confident in my abilities to do it, could someone provide a fermi type calculation of this? Else, link to a source which does?


Recently I was on an airplane on a sunny day. The sun was shining on the other side of the plane and noticed a bright patch on the ground following beside us. Eventually I noticed a dark centre to this bright patch, the plane's shadow, which became more distant as the plane descended. When the plane flew over a city road signs in this bright patch lit up brightly because of their retro-reflective paint. My question is, what was acting as a retro-reflector to produce this bright patch when it wasn't passing over a road sign? When flying over water the bright patch disappeared, or was very faint. I could see it over forests, more clearly over cut grain fields and, at least faintly over a wide range of terrain.


For a general linear map, there are infinitely many matrix representations. You choose a basis of the vector space, and this gives you a certain matrix for the linear map. You choose a different basis, and this gives you another matrix different from the first. For a linear map that does nothing however, we only get the identity matrix no matter what basis we choose. It isn't hard to see that the identity matrix will always do what we want from this linear map mechanically. But I feel there is some deeper intuition for why the matrix for this linear map shouldn't change with the basis, without thinking mechanically about the matrix. On a side note, is there any other linear map where the choice of basis doesn't change the matrix representation or is this the only one?


In quantum mechanics, unbound states will tend to spread out in space over time according to the Shrodinger equation. So it seems to me there is a degree of freedom for "wild" particles (i.e. those have not been prepared in the lab) as to how much they are spread in space and momentum. So my question is: do we have any knowledge of how disperse "wild" particles (i.e. those which have travelled a long way such as photons from the Sun or cosmic rays) are? Is there any way to tell, or are we completely unable to extract this knowledge? Fundamentally we can't for an individual particle, but I mean statistically. I know that the spread must satisfy the Heisenberg uncertainty---my question is what do we know beyond that?


Why can't we see light from beyond the observable universe? I've done a lot of research on this and all I've found is unsatisfactory answers and straight up nonsense. Some claim that the universe "expands faster than the speed of light" beyond the observable universe. Such a claim doesn't even make sense because the units of speed are m/s and for expansion are Hz. That's like saying "the area of this square is larger than the volume of this cube". All that the expansion can do to light (as far as I know) is redshift it. And light doesn't have a minimum possible frequency or energy value. So even if the expansion of the universe is very rapid, why does the light of distant objects "never reach us". Surely it still would, just extremely redshifted. In this case it does still reach us, and yet the claim is that it cannot. We often detect redshifted light, and that light has not been slowed down. When we detect it, it still goes at c, even though (in fact a better word than "even though" here is "because") it is redshifted. Light is always propagating at c no matter the reference frame. More precisely: does the light really never reach us, or can we just not detect it? If it never reaches us - why? If we cannot detect it although it does reach us - why?


Let's suppose the earth is perfect sphere and let's ignore its rotation and movement. What would happen if i would be in the center of the earth? Would the gravity be zero in any direction so i wouldn't feel any gravity force? Or would there be 'pulling' forces (of half of the gravity force on the surface) that would attract me in every direction. Or in other words - If i would be in the center of the sun, would i be in a weightlessness state or would i be torn to a pieces (except of having a serios sunburn of course)? I know there are similar questions here: Effect of gravity at center of Earth Would you be weightless at the center of the Earth? but they are ot answering whether there would be no force or there would be forces to all direction. Some answers implies that the forces would be the same but in every direction so they would 'cancel' each other. But i'm not sure what does it mean. If I'm dragging a paper to different directions the forces are not zeroed, the paper is torn apart.


If I'm trying to advertise that you can scroll through this webpage to find additions that go in/on your home, would it be... "Find new additions for your home." "Find new additions to your home." Does "find for your home" technically mean that your home itself is literally looking for new additions and you're looking on its behalf? And therefore "to" is correct? The problem is, what if it were something like advertising gift-giving, and said... "Find shoes for your friend." In that case, it could be read as finding shoes on behalf of your friend, but also finding shoes that suit your friend. Certainly, "find shoes to your friend" is absurd. So, why does it work in the home example (assuming "to" is correct in the home one)? Thank you :)


I was once asked the question: What French word is commonly used in English for which an English word is commonly used in French? The answer was respectively rendezvous and date, which I found very unsatisfying. So, does such a situation exist, in which a French loanword is used in English to mean something for which an English loanword is used in French? Criteria: Both loanwords should be fairly common in the language that borrow them. By that, I mean that it is expected of the general population to know this word. A more precise definition of this criterion would be the absence of the "specialised" tag in the Cambridge dictionary. Both words should ideally be actual loanwords, meaning that their spelling hasn't changed, except for any punctuation. ("Gastronomy" or "beef" wouldn't count.) Research done: I've looked at various lists of French words used in English and did not find a match, as well as this related question. ChatGPT was no help.


I am working with optically active nanomaterials (quantum dots, perovskites), that have pretty large exciton binding energies and can form multiexcitonic complexes, e.g. biexcitons, relatively easily. It has been well established that they also exhibit biexcitonic lasing/ASE under above threshold excitation, so here I have a question. To have lasing behavior/ASE, stimulated emission process should occur which requires population inversion between upper (biexciton) and lower (exciton) states. However, in the current literature when this effect is reported for any new material, population inversion requirement is barely ever mentioned and I wonder why. Is it for some reason biexcitonic recombination is a "special" transition, and therefore this requirement is lifted, or population inversion indeed takes place? If any of these options is correct, could you, please, elaborate why that happens? What is the mechanism lasing or inversion build-up? Thank you!


Massive compact halo objects ("MACHOs") include a wide variety of hardly detectable bodies such as brown / white / black dwarfs and black holes, to name a few. If we take into account the inevitable end of all stars into either a white dwarf, a neutron star or a black hole and we compare the average lifespan of a star with the time star formation has been active since the Universe started, then we can tell galaxies already contain many dead stars which still have mass and can therefore interact gravitationally with the rest of the bodies in said galaxy. Furthermore, stars tend to accumulate in the galaxy's bulk (primarily) and in the galactic disc as well, so if dark matter happened to be MACHOs, which are distributed in said proportions, then this would account for the fact that models such as the dark matter halo need to have a density such that it decreases the further away we get from the centre of the galaxy. Finally, MACHOs are bodies we know to exist for a fact, whilst particle dark matter (e.g. WIMPs) has not been found. Is there a reason for which MACHOs are not the most likely candidate for dark matter?


I have recently watched a couple of lectures in order to revise some of the notions that I will have to tackle in the coming months, and as I did so I have stumbled over something which baffled me a tad. Now, within these lectures the lecturer described a relation as being a function solely if it maps one element from the domain to one sole, unique element from the codomain (or range assuming it is a surjective function we are dealing with); I found this a tad confusing since, ever since I can remember we considered functions to map a value from the domain to one sole value from the codomain/range, and if no value from the domain is mapped multiple times to the same value within the codomain/range then it is injective, hence abiding to the lecture's definition of a function. Have I been missing something, do people automatically discard non-bijective, or at least non-injective functions for whatever reason?


According to Archimedes' principle, the buoyancy force on a submerged object is given by the product of the mass of the displace fluid and the gravitational acceleration. Effectively, it is determined by the net pressure difference of the fluid between the bottom and the top of the object, or in other words the weight difference of the corresponding columns of fluid. However, if we have a weight placed on a scale for instance, this weight will change if we accelerate the scale upwards or downwards. As scale in freefall for instance will register zero weight. Now as the buoyancy force will accelerate an object initially at rest, should it not be reduced as a result of this (the buoyant object can be considered the scale here and the fluid column the weight being measured)?


I want to start learning some optimization theory (mainly to get into ML in the future but I'm also just very interested in mathematics that intersects heavily with engineering) and I would like to get some recommendations to start tackling the subject. I found two books: Bierlaire's Optimization book and Convex Optimization by Boyd and Vandenberghe but the former seems to lack convex optimization which I've heard is quite foundational and the latter seems to target more practitioners of optimization algorithm where I want to get into theory to understand exactly why the algorithms work and both mention that they target engineers in their preface and I'm not sure if that's a good or a bad thing. So are there any other introductory references to the subject that target mathematicians which I'm missing?


Very roughly, dark energy tends to cause space to expand and mass tends to cause space to contract. If nothing gets in the way, the math on that contraction breaks down when it forms a black hole singularity. So the question is, can enough energy halt the contraction before it forms a singularity, but after it forms a black hole (i.e. an event horizon)? Are there any theories that have tried to use this to avoid the the math breaking down? Edit: after some digging, I have found that the Penrose-Hawking singularity theorems gives conditions where a singularity must form, but I've yet to find a description of what happens (or could happen) if those conditions are not met. Namely, it seems like "interesting" thing might happen if negative energy became part of the conditions inside a black hole.


For each segment of a piecewise Lyapunov function that exhibits asymptotic stability, we can utilize the LaSalle-Yoshizawa theorem and solve it using a differential inclusion. This allows us to merge all the piecewise Lyapunov functions and demonstrate that the entire system is asymptotically stable. Now, if each segment of my piecewise Lyapunov function achieves finite-time convergence, specifically in the form of square root, how can I utilize the differential inclusion? Can you explain whether the overall system is finite-time stable? Upon searching through some resources, I found that these studies generally assume each segment to be asymptotically stable and then prove that when the function obtained through the differential inclusion exhibits square root convergence, the overall system is finite-time stable. However, if my current equations do not exhibit asymptotic stability for each segment but instead demonstrate finite-time stability, how can I prove it?


I'm not a physicist in any way, but I'm curious enough to watch and attend some pop-science lectures. Let's imagine the following situation - there is a free-standing unbound electron. It has its wave-function describes probabilities to find it in particular position. Also there is incoming photon with a proper wave-length to be absorbed. How the electron "decides" to absorb it, from what distance? If it's probabilistic too then electrons can absorb photons from centimeters away? And we shoud be able to verify it statistically by observations of many photons and electrons. Since the photons are stretched out by cosmological expansion, then I assume that they have the length in physical space. Is the absorption instant or does it take time to absorb the "whole" photon? If the absorption is not instant then it's possible to make a trick: make the electron annihilate with positron - after it started to absorb photon, but right before it finished. What will happen to "unabsorbed part" of photon? What is a correct theoretical answer to my questions?


Was all the energy that is the universe today, present at that femtosecond theory has pushed back to or was there an influx of energy for a period of time? My question was: is the big bang like a pin prick in an infinite balloon of water? Initially, you have very high pressure, then as the hole expands the pressure drops but the flow doesn't stop. In the case of the universe is the big bang still pouring gluons and muons in everywhere at about the same rate as it always has, it's just that over this much larger volume it is less dramatic than the first few centuries. Does our universe create a back pressure at its various stages that would explain the changes in rate of expansion over time? It is equally likely I am fundamentally misunderstanding what it means to be in an expanding universe. I suspect it's fairly obvious to those that might answer that I've watched more science documentaries than is healthy but fail to comprehend most of the wikipedia pages describing the physics and maths that support them. I expect you see lots of these stupid questions and I suspect this has been asked before, but I couldn't find it.


I was interested in electromagnetic induction in loops, since ground loops and such tend to be a problem in electronic system engineering. To me, it seems like induction in loops may tend to cancel out. For example, if the plane of a simple wire loop (that is just a ring of wire and nothing else) is perpendicular to an electromagnetic wave, using the normal rules for induction in wires, the E-field of the electromagnetic wave would be parallel to the top and bottom of the loop, and induce a current in the same direction in both the top and bottom. The sides, would be perpendicular to the E-field, and have no electromotive force on them. I asked about this on Quora, and was told that a "counterclockwise circulation" would be induced per Lenz's law. I know inserting a north pole into a wire loop does that, but, that was not what I asked. I found this question here on Physics Stack Exchange, EM waves from AC current in a loop?, and it seems to suggest cancellation of forces for a radiating wire loop. A receiving wire loop seems like it might have similar behaviour. So my question is, what current is induced in a wire loop whose plane is perpendicular to an electromagnetic wave, when the electromagnetic wave passes over it.


We, as humans, given our height and size, view the world from the same general perspective. An ant, on the other hand, will understand the same world in a completely different way, given how limited their knowledge and experiences can be. When we spend a couple seconds taking a step out onto our front yard, it might have taken a microbe years to travel that same distance (assuming it can survive and all that). To the ants living in your front yard, your back yard might just be the "edge of the Universe" for them, let alone the other side of the city or continent. For an organism smaller than even Planck's length, assuming such thing exists, then Planck's length might just be the limit of their world, and who knows what's going on down there. Let's reverse that thinking and apply it to something that's bigger than us. Now, we are playing as the "ants" or the "organisms smaller than Planck's length". A bear, for example, can travel farther than a human given the same amount of time, and what we view as "further apart" might seem "closer" to the bear. To something as big as a planet, the Earth might be its "front yard" and the moon its "back yard". For something much much bigger, the size of a galaxy, lets say, what would be its "other side of the city"? By "taking a step", that thing would travel a distance worth of many light years in just a few seconds, right?


I am very poor at English, so I used ChatGPT to translate my question. My question is here (Is my idea right?): Is it possible to conceive concepts like a coordinate system in a completely empty space? I believe that without some reference, even establishing an origin becomes impossible. In a space where there is just one object (please consider the object is very small), it is possible to consider an origin. You can regard the location of that object as the origin. However, I think it's not possible to conceive axes in this context. In a space with exactly two objects (please consider the objects are very small), it is possible to establish an origin and one axis. You can choose a location of one object as the origin and define an axis with a positive direction towards the other object's location. Nevertheless, I believe it's not possible to conceive the other two axes. It is only when there are three objects (please consider the objects are very small) in a space that one can begin to consider a coordinate system.


I have been looking up at available database that had dielectric constants for metals since I need them for my research and wanted to get an idea of what the values were before I focus on actually calculating them numerically if ever it is needed for my research. Thankfully I was able to find this website https://refractiveindex.info/?shelf=main&book=Ti&page=Johnson However I noticed that there are two values for the relative permittivity when it came to metals and I was wondering why this is the case. From what I know so far, metals should have a negative permittivity, so I was confused as to why it shows that Titanium also has a positive permittivity. Can someone enlighten me as to why this is the case for the permittivities that I found on the website. Apologies as well for the question since I dont know a lot yet about the properties of materials when it comes to permittivity and permeability.


If a quantum particle is described by a certain wave function and we express it as superposition of for example its possibles energy states, after we measure it the wave function collapses and we observe the quantum particle as having a certain energy. I have a couple questions. No matter what happens to that particle or what I do with it everytime that I observe it again it's going to be in that same energy state? If after measuring energy I now try to measure something else like position could the energy state change when I measure it again? In general I'm trying to understand what it means that the wave function collapses because by what I've studied it sounds like it's undoable but that doesn't make much sense to me. So far I thought that when we observe a particle's energy for example we measure a certain energy but then if we measure it again right after we could measure any of the other possible energy states but I'm told this is wrong. How does it actually work then?


Reading Harris' Modern Physics, specifically the classic example of Anna holding two lightbulbs on a moving train. I find myself confused about this passage: Before moving on, we emphasize a vital point: The preceding arguments had nothing at all to do with Bob's specific location in his frame or the time it takes light to reach his eyes. The issue here is not which beam reaches Bob first, which does depend on where he stands, but rather which begins first, which has nothing to do with where he stands. Attempting to blame the effect on optical illusions is the comfortable yet incorrect way out of accepting a counterintuitive notion. The truth is more challenging. Bob could be standing right next to one flashbulb when it flashes and have an assistant, Bob Jr., standing next to the other flashbulb when it flashes. Each records the time of his flash and neither has to wait for the light to reach his eyes. They record different times. Anna, in her frame, causes the bulbs to flash simultaneously, but Bob and Bob Jr., from their frame, determine that she causes the bulbs to flash at different times. I guess my understanding of the relativity of simultaneity was dependent on thinking about the path that light took to travel to Bob. If Bob/Bob Jr can be right next to the lightbulb, I am failing to see how that's any different then them being on the train with Anna.


Context: I would like to talk about how a book questions the reader's traditional, conservative values about family structures. I want to say we are introduced to a shared household wherein there isn't a typical nuclear family structure, the children are not trained in table manners and everything is loud, lively and disorganised. This contrasts the presumed reader (at the time)'s traditional beliefs that a house should have a nuclear family, children should be taught table manners, etc. The sentence I want to say is, "The reader is positioned to feel [insert word], as their beliefs are subverted and challenged." Words I've tried so far: Unsettled Uneasy Disturbed These are the closest words I could find but I feel like they have a more negative association, and creates a malaise in the reader. I feel that the effect of the book is more subtle and just lightly 'shocks' the reader, positioning them to just take notice of their preconceived beliefs, rather than evoke a negative reaction. Destabilise Bewildered These words suggest that the representation of these households are completely weird and confusing, however, that is not the case. It's not so weird that a reader wouldn't comprehend it, these households are just different to the mainstream households. Source for definitions: Google dictionary


Background: I tile lidar and photogrammetry products as part of my job. Project areas are typically single polygons outlining an area, and can be quite irregular in shape. Overall there is a lot of data and I am required to achieve a maximum file size per tile by creating square tiles that cover the project area. I know from experience what this tile size is but it differs by product. Although tiles are square, they are allowed to hang over the edge of the polygon. I'd like an algorithm to take a polygon and a square tile and return the tiling which has the fewest number of tiles that completely cover the polygon. The square tiles must not be off-grid or rotated, they must be orthogonal. If possible I'd like actual code, preferably in python and utilizing arcpy, however I'm willing to accept any answer that provides the algorithm I'm after. Bonus - multiple polygons representing disjoint areas need to be tiled using one tiling scheme. Tiles that don't cover any project area aren't needed, but all tiles must be on the same grid.


Levitated pits were introduced after after solid pits. In this design the tamper is separated from the fissile with an airgap. From the Nuclear Weapon Archive: The original Fat Man pit design used a Christy solid plutonium core, surrounded by a close fitting natural uranium tamper. The Sandstone devices all replaced the contiguous tamper-core approach with a "levitated core" in which the core was suspended within a larger hollow space within the tamper so that a gap existed between them. The collision between the tamper and core would create more efficient compression of the core than the explosive-driven shock in the wartime design. Other vague descriptions I found: The first improvement on the Fat Man design was to put an air space between the tamper and the pit to create a hammer-on-nail impact. Efficiency of the implosion can be increased by leaving an empty space between the tamper and the pit, causing a rapid acceleration of the shock wave before it impacts the pit. I recently started studying shockwaves and I don't understand this. How does leaving an empty space increase schokwave acceleration? If I remove the airgap, the mass of the core won't change but now I can use the extra volume for more explosive. Inside the pit, compression of the fissile is the result of work performed by the explosive charge, so how is it possible that I can increase compression by reducing explosive volume ??


In table tennis it is often desirable to produce as much spin on the ball as possible using a glancing contact. (The rubber covering of a table tennis bat is typically highly elastic and has a high coefficient of friction.) Anecdotally, the view is sometimes expressed that acceleration at the point of contact is more important than speed. I take this to mean that when the ball is hit with the bat at the same angle and travelling in the same direction, it is possible for a bat at a lower velocity but with some acceleration to impart more spin on the ball than a bat travelling at a constant higher velocity (higher than the maximum speed of the accelerating bat) without acceleration. Is this true? If so, why, and how can the optimal contact (in terms of speed and acceleration) to produce maximum spin be determined?


One English rule is to hyphenate two or more words when they come before a noun they modify and act as a single idea, called a compound adjective. This is the most common use of the hyphen I've seen. For example: A non-cloud platform Some cloud-based platforms What happens when there are three adjectives before a noun? Should I hyphenate all three? This example looks awful, but maybe that's because it's joining two compound adjectives that we usually see by themselves: Non-cloud-based platform That compound adjective would look better without the two hyphens. For most groups of three or more hyphenated words, they tend to form expressions that behave like compound adjectives, e.g. up-to-date, hit-or-miss, trial-and-error, etc. but they would also look alright as phrases by themselves. The three-word compound adjective in my example doesn't fall into that category.


I heard that Lagrange mechanics can be derived from Newtonian mechanics, and Newtonian mechanics can be derived from Lagrange mechanics. I've heard many times that they have equal explanatory power. But I encountered that there is a tricky point in deriving the law of conservation of angular momentum in Newtonian mechanics. On the other hand, theories that use the principle of least (or stationary) action derive each conservation law from the Noether's theorem by assuming each symmetry. So, is my understanding appropriate? That is, while Newtonian mechanics itself derives conservation of linear momentum without any additional assumptions, and requires some manipulation to derive conservation of angular momentum, but Lagrangian mechanics cannot derive both without the assumption of symmetry, or derive both if the assumption is made. Then, it seems that it can be explained by Lagrangian mechanics, but there is something that is not correct in Newtonian mechanics, or it seems that more assumptions are made in Newtonian mechanics than in Lagrange mechanics. Which of the two makes sense? I'd also like to ask if Newton's third law conflicts with Lagrange mechanics, and if it's possible to find a set of Newtonian laws that have the same explanatory power as Lagrange mechanics. However, I want to know (even short) to just the above two questions.


I have problems in understanding the terminology used in Sylvester's Criterion about the "sign" of a matrix. I got the "positive-definite", the "negative-definite", the "indefinite" and "non-definite" (the last one is when the determinant is zero, whilst the penultimate refers to two eigenvalues with different signs). What it's really unclear are those two: positive semi-definite: "all its eigenvalues are non negative" positive semi-definite but non positive definite: "there is one zero eigenvalue and the rest are non negative". I would really understand why one has to complicate the terminology in such a horrible way. Why shall we use "non negative"? Just say "Positive". Also, what's the difference between "all its eigenvalues are non negative" and "there is one zero eigenvalue and the rest are non negative"? I don't get this. If zero is counted as "neither negative nor positive" then wouldn't positive semi-definite mean positive definite? It's all so messy.


My current level of maths does not allow me to understand any of the proofs I was able to find online for the Fundamental Theorem of Algebra. I find it very unsettling having to continue learning about polynomials without being able to grasp such a fundamental property of them that is there exists a complex root for any polynomial with complex coefficients. My question is, is there a way I can reach the same conclusion of the Fundamental Theorem of Algebra for polynomials with real coefficients instead of complex coefficients. In other words, to prove that at least one complex root exists for a polynomial with real coefficients. Does restricting the coefficients to be real instead of complex make things any easier? If the answer to my question above is no, then I would appreciate it if you could provide any sources that I could've missed, that explain the Theorem by using the least amount of mathematical notation and advanced concepts as possible. If I cannot figure this out, I am going to have to accept the existence of a root as an axiom going further, which is something I really don't want to do.


I've been thinking about the critical point of water, which has three distinct and specific properties: critical temperature, critical pressure, and specific critical volume. However, when I draw a PV and TV graph I can't seem to think of any way in which to bring the water to its critical point because in order to get one property to its final state you have to change one of the others. For instance, if you already have the water at its critical temperature and pressure, you would have to decrease or increase one of those values to get it to its critical volume. To this end, I'm wondering whether it's even possible to bring water to its critical point - because once you achieve two of the three intended properties, it seems impossible to me to get the third property to the correct state without changing one of the other two.


I am reading up on Christol's theoreom and an important part is that k-uniform transducers (where k is somehow related to prime numbers) preserve the algebricity of a formal power series (taking the coefficients as a sequence). So basically, we start with a formal power series that has roots (my friend told me that algebraic means there are roots or zero crossings), run the coefficients through a k-uniform transducer and, when we use the values that come out as coefficients for a power series, that power series is again algebraic. There is a lot I still need to understand here clearly (like, what is a "formal" power series vs a "business casual" power series...), but I'd like to start by understanding what is so special about a power series having roots. Just for context, I have nearly no formal background in mathematics but came across Christol's theorem and thought it was very cool and maybe related to my area of work (computer science, temporal logics, and automata). Any help is appreciated!


Is there a conceptual problem in formulating of Liouville's theorem and the BBGKY Hierarchy for classical field theories? I always see treatments of Lioville's theorem only in the context of classical mechanics. I also know that most references treat linear response theory only for classical mechanics and do not touch upon linear response theory aspects of classical field theories such as Kubo formula. One can definitely take the continuum limit of a classical mechanical system and obtain some sort of Lioville's theorem or BBGKY hierarchy for the continuum field theory. I would like see if this is discussed in some rigorous way in any reference. In particular, I'm looking for references which discuss aspects such as the phase space for classical field theories, BBGKY and Green-Kubo Formula. I'm not a mathematician so a good systematic treatment I'm comfortable with for classical mechanics is David Tong's lecture notes and the text by Evans and Moriss. I'm looking for similar treatments for classical field theories.


When we calculate the excess pressure on the concave side of the meniscus of the liquid surface formed in a capillary tube, we balance the force by the atmospheric pressure, force by the pressure on the the side of the fluid , and the force due to surface tension, to derive the expression for the excess pressure on the concave side of the meniscus, but, my question is, when we balance the forces acting on a body, we balance the net external forces on a body for it to be in equilibrium, but here the surface tension force is an internal force on the surface, applied by the surface molecules themselves, then why do we include it in the equation for balancing the net external force on the meniscus?


I'd like to get some clarity about how nice the problem of decomposing semigroups using the wreath product is compared to the problem of decomposing groups, and where this apparent difference in niceness comes from. Looking at the Krohn-Rhodes theorem for finite semigroups, one can decompose a finite semigroup as a wreath product of finite simple groups and really basic "reset automata" semigroups. If I understand the Krohn-Rhodes theorem correctly, if one applies it to a finite group, you don't get any "reset automata" semigroups, so it gives us a decomposition of our group as the wreath product of simple groups. Is this correct? On the other hand, Meaning of factorization of groups and looking into the extension problem for finite groups leads me to doubt that the case for finite groups is that simple. So is it not as simple as "there's some kind of wreath product-like construction that allows us to reconstruct a group from its factors"? I'd greatly appreciate some clarity about this.


I have been trying to define the notion of a product of second-order classes using (finitary) second-order and if needed third-order logic. It seems to be possible to define the product of finitely many classes, because I can just express this using a finite second-order sentence. The problem is that, when I try to define an infinite product, I seem to need to be able to refer to an indexed family of classes. More specifically, so that I can use an infinite index class. But something tells me this should not be possible to do in higher-order logic, because intuitively, it seems to involve some circularity. Do you know if it is possible? Also, someone suggested me that type theory should help me with this problem, and if so, do you know why?


Im reading make: electronics, and I have investigated a little bit about displacement current and how a change in electric flux can create a magnetic field in a vacuum. A can understand that, however there is one effect which I don't comprehend the physics behind it. The effect that I'm talking about can be resumed in the next quote from the book: "A sudden change in voltage on one plate in a capacitor induces an equal change in voltage on the other plate, as if is reacting in sympathy". An image from the book showing the voltages in both plates of the capacitor in respect to time. My theory: So my theory is that since there is a current displacement there is also a change in magnetic flux, and this change in magnetic flux can generate a similar voltage on the other plate, since the plate has thickness and thus inside the plate there is an area that is perpendicular to the direction of the magnetic field, which will lead to a magnetic flux, and if you change the magnetic flux a voltage is created (according to Lens and Faraday). However, I don't know if this is right, and at the same time trying to find information on the internet about it is impossible. If anybody has an answer I will be delighted.


I am seeking answers from experts in mathematical logic about the amount (if any) of university mathematics I need to know in order to understand mathematical logic and later hopefully do meaningful (independent) research on the subject in general and Godel's Theorems in particular. I am proficient in high school math and have a bachelor's degree in Physics. I have also recently taught myself some calculus, linear algebra, and parts of real analysis as I assumed you must need at least undergrad math to eventually get proficient in a certain math discipline. Earlier I had decided to learn up to grad level math but after I glanced through some logic books it appears they make close to zero use of even undergrad math. Also, I have come to know that philosophers too do research in mathematical logic, and as far as I know, they don't study any university math. So, my question is should I first teach myself undergrad (and grad math) or just dive into mathematical logic as I don't want to later find myself in a position where I have to study all that university math before I can make further progress in logic? If that is the case I would consider enrolling myself in a math program first and doing the research later in the conventional way.


My understanding is that the word means a rephrasing of something the person has said in expressing the same sentiment or idea. But I think people use the word to mean essentially, say something in a similar way to mean something else. A simple example: Marge is talking to a bullied Bart: "If they beat you up, I do not think they are your friends." Someone who wants to borrow this sort of phrasing to apply to the situation where someone has more extremely pushed an acquaintance off a roof and this is reported in the news as "Woman's friend pushed her from roof" and says, "To paraphrase Marge Simpson, 'If someone pushes you from the roof, I don't think she is your friend.'" In the above, is "paraphrase" being misused and if so, what is a better word for such borrowing/reuse with modification of a phrase?


I'm currently working as a (undergrad) TA for a mutlivariable calculus class and while covering the topic of multivariable differentiability, one of our students suggested that it'd be nice to have some actual functions we could use as examples to navigate how the theorems can be used to both establish differentiability and to flip them back around to deduce things about the partial derivatives if they're not differentiable. In particular, we were discussing a sufficient condition for differentiability which, roughly, states that if all of the partial derivatives of a scalar field are continuous at a given point, then the field is also differentiable at that point. But more specifically I was warning them about how the contraposition of this statement does not forbid a function from being differentiable while having discontinous partial derivatives, as this beautiful example shows. The previous example has all of its partial derivatives be discontinuous, but then a student asked if I knew any scalar fields which only had one or two discontinuous, while the rest remained continuous at the given point. Here's where I ask for your help :) I have looked around in some of the literature but haven't been able to find good enough example of this, although in principle this should be possible (from what I understand). Also, if you have other interesting examples that show how differentiability can be wrinkly or deceiving I'd really appreciate it (and our students would too)!


I have read this question (no answer, just comments): Light, including pulses of light, consists of many photons. A very short pulse has a wide range of frequencies. See The more general uncertainty principle, regarding Fourier transformsA single photon can also have a long or short duration, and a narrow or wide range of frequencies. I have never heard of single photons with a wide rage of frequencies. Each photon is always supposed to have a single quantified energy/frequency, otherwise the quantum effects would not occur. Can you refer to studies that show single photons with a range of frequencies? Relation between attosecond light pulses and photons? And this: This is very fast for a pulse of light, and it is actually so fast that the pulse of light is no longer a periodic electric-field oscillation, and instead it lasts only for a few cycles. But it is still not fast enough. What is an "attosecond pulse", and what can you use it for? Now we have had single photon emitters for a while and the photon is the smallest amount (quantum) of EM energy. But then the question remains, why are attosecond pulses better (for example to track electron orbitals) then single photon emissions? Very naively thinking, if you shoot single photons at the electron (atom), you are using the smallest quantum of energy to do so. Then why is attosecond pulse generation superior to single-photon emitters? Question: How can an attosecond pulse be shorter than a single photon (quantum) emission?


In practical engineering we are limited in the upper temperature of thermodynamic cycle due to the material properties. So, after the max temperature is fixed, we want to make our cycle as close as possible to a Carnot cycle with the same max temperature. It is clear that the compression and expansion should be adiabatic and we are technically close to the realization of this. Also, we have to cut lower right angle of the Carnot cycle, because each next turbine stage should be larger in diameter, and building them becomes economically or technically impractical at a certain point. But the process of heat addition is a bit more tricky to understand. Consider the case of Humphrey and Brayton cycles in the same range of temperatures: The heat addition process is more vertical in case of faster heat addition, but according to Carnot process it should be more horizontal to be more efficient (i.e. close to isothermal). However, we do know that faster heat addition is more efficient (detonation engines are more efficient), so Humphrey cycle should be more efficient. Can't solve this puzzle...


I have viewed the definitions of the Hubble Sphere and related cosmological concepts, as well as various explanations, yet Im still struggling to comprehend a full visualisation of this, which I would prefer instead of taking its word for it. To phrase the question that would most help me, the following follows: so from the moment that a celestial object crosses the hubble sphere boundary, as in going out of our hubble sphere scope, the images of that object will never reach us, the observer, from that point on, and we are left with a developing and cascading history of prior images only just reaching us now. If what I said is accurate, and requires no correction, my question now is how will the developing and changing history of past images for this celestial object play out in our view for the next millions of years until that moment in the old present where it crossed the Hubble Sphere boundary? How would it appear to us, the observer, over such a long term period? My intuition is telling me that such a celestial object will continue to age in our view of its past images, but also becoming increasingly redshifted until phoof, it is no longer visible due to being outside of the Hubble sphere.


I understand that infalling objects reach the speed of light at the event horizon. Any associated clock would be observed to stop, according to SR. But isn't the key here 'observed'? An infalling observer at the event horizon would observe a remote clock to stop. Who is to say which clock stops. No clocks actually stop in SR, because it is symmetrical. But there is gravitational time dilation. In this case clocks really slow down. But you would expect the force of gravity to be infinite at the singularity. It is here you would expect gravitational time dilation to be infinite and clocks would really stop, not just be observed to stop. The force of gravity is nowhere near infinite at the event horizon, so why would clocks stop here rather than just slow down? Do properties at the singularity transfer themselves to the event horizon? Perhaps because there is no physical singularity?


This paper has the following abstract: We theoretically consider a graphene ripple as a Brownian particle coupled to an energy storage circuit. When circuit and particle are at the same temperature, the second law forbids harvesting energy from the thermal motion of the Brownian particle, even if the circuit contains a rectifying diode. However, when the circuit contains a junction followed by two diodes wired in opposition, the approach to equilibrium may become ultraslow. Detailed balance is temporarily broken as current flows between the two diodes and charges storage capacitors. The energy harvested by each capacitor comes from the thermal bath of the diodes while the system obeys the first and second laws of thermodynamics. Can someone explain the importance/meaning of the claimed result for a layperson? In particular, how does this not violate the second law, if it's "thermodynamic fluctuations" being harvested?


I am working on this problem. A hexagon is inscribed in a circle. The length of five of its sides is a, while the length of the six side is b. My question is : a. How to find the area of this hexagon? b. how to find the each of the diagonals's length? What I have in mind is that, since only one side is different from all other sides, the figure is still symmetric. Then I can calculate the area of the two trapezoids. I think I should use the condition that it is inscribed in a circle. However, I am stuck with how to find the angles or what other fomula should be used here. Then for the length of the diagonals, I am thinking that once I got the two trapezoids, I got them. Anyway, I think I lack of the knowledge of some intermediate theorems. Thank you for the help in advance!


This question closely ties into a question I had about verbless clauses. However, I am writing a new one at the suggestion of a user. Polarity-sensitive aspectual-related words are those such as 'still' and 'already.' I have established that modifiers like 'obviously,' which I believe to be evaluative in nature, can function as modifiers in an adjective phrase, as determined by the answer to my question linked above, where I confused them to be part of verbless clauses. But can 'still' and 'already' also be modifiers in an adjective phrase? Below is an example similar to the one from my previous question, using 'still' instead of 'obviously.' I have highlighted in bold what I believe to be the complete adjective phrase. He was crying, still sad because of the passing of his father.


According to my current understanding of special relativity, photons do not experience the passage of time. It is as though the universe is completely 'paused' for them. I know that objects with mass cannot accelerate to the speed of light. But isn't the whole point of relativity to predict the motion of an object from a different frame of reference? The universe also tends to increase its size with time. In that case, consider the following thought experiment: Let's go back in time to when the universe was relatively young. There was once a photon which, when created, observed the universe to be completely stationary as it didn't experience the ticking of time. As the time does not tick, the universe doesn't expand (relative to the photon) and instantaneously, it exits the universe. So my question is, what part(s) of the above statements is false?


I made the claim on a manuscript that all manifolds possess an acceleration field. The referee rejected the idea saying "The nature of this acceleration field has not been seriously discussed and the theory lacks justification in physics". This confused me as I understand that a field, in physics, is a region of space for which each point is associated with a specific physical quantity. Velocity is a physical quantity. If we impose an evolution parameter, then don't all points on this manifold have a velocity (vector) quantity? Is that not sufficient to claim the manifold has a velocity field? Can we not also take the second derivative of the position with respect to our evolution parameter and say that every point on a manifold has an acceleration (vector) quantity? Does that not also qualify as a field? Is there an argument to be made that all smooth, differentiable manifolds do not possess an acceleration field?


I am looking for a word that describes something whose meaning is instantly recognizable, that is so simple as to be offensively lazy, and yet completely apropos. Such a word might describe Kazimir Malevich's painting Black Square: Chef's kiss conveys artfulness and minimalism, but not to the point of laziness or offense. I like sophomoric for its literal translation "wise fool", but the word is not used to suggest brilliance. Cunning has the right connotation of artfulness, but its connotations of subtlety, and further, deception, make it inappropriate. Stupidly/offensively obvious almost works, but is boring. Unsubtle is ok; blunt better. Both don't convey the artfulness that I'm looking for. I love the divine feel of manifest; I'm wondering if this is the best I can do. I'm tempted to coin kazimirean, but of course that's too indulgent.


Let's bring a positively charged rod near a conductor. Now since some electrons in outer shells are not strongly bound to atoms,they will get to the side near the rod. But why does it mean that dipole is formed in that conductor? We take the vertical cross sections of the conductor. Now,in each vertical layer,there will be atoms. So electrons from each vertical layer will get to the side of the rod. If that happens,except from the region closest to the rod,there will be positive charge in every cross section. Hence,apart from the very region towards the rod,the entirety of the conductor will be positively charged since electrons have left from each of the cross sections. As we can see dipole(positive at one side and negative on another side) hasn't been created,rather negative at a very small region and positive charges throughout the rest of the conductor. The logic that the rod will push the positive charges at the extreme end also doesn't work since atoms are firmly immovable. So why is it still said that dipole is created as a result of electrostatic induction whereas we have proved that it isn't the case?


In basic physics lectures, the teacher or professor in my class never explains the behavior of rotating two or many body particles. In my experience and intuition doing physics, two-particle or many-body systems tend to rotate about their center of mass. For example, the earth-sun rotates around its center of mass, or two black holes rotate around its center of mass; also in ideal conditions, without air friction, a stick when a force is applied to it at the end edge, stick will rotate around its center of mass. Can anyone explain the physics of a system of mass that tends to rotate around its center of mass? I tried to solve this problem and read some reference books, but I don't have any conclusion. most of the books I read just explain what rotation or torque is. Or, is my intuition wrong?


I'm hoping maybe this is the right spot for this question, if not, I would love for someone to suggest a better spot I am attempting to use SOLIDWORKS flow simulation to simulate the airflow through a ram air duct. Think like a venturi tube on an airplane, but the convergent divergent duct is embedded in the structure rather than being a protrusion. The issue I'm having, is that for the area ratio of the entrance to the throat, I don't get the velocities I expect at the throat when doing an external flow sim. When I build just the duct itself and put lids on it and do an internal sim I get the numbers I expect. Is there any deficiency in the Flow Simulation when using the external simulation and the flow must through a completely bounded flow area? Or is the simulation giving me true results and the flow through the duct is that degraded by being embedded in the larger structure moving through the air?