Please correct me if I am wrong. Almost all materials are made up of atoms with electrons. If we rub two different materials, like plastic scale and oily hair on the head, then static charge develops on the surface of scale. This charge can attract tiny bits of paper. Also during lightening tiny droplets of water and ice crystals rub against each other to create static electricity. However if I rub my hands on the wall then no static charge develops on the hand. I checked it, as my hand did not attract tiny bits of paper , which means no charge developed on my hand due to rubbing. My question is : On which atomic factors of the materials does the phenomena of static charge depends ?


In mathematics, scalars are defined as elements of a field. Are scalars in mathematics invariant under coordinate transformation? In physics, the way they are defined, I know they are. Since in physics, they are defined to be quantities having only magnitude and no direction, they are invariant under rotations or translation. Some sources say that in mathematics, scalars are, by definition, invariant under coordinate transformations. Can anyone please explain how defining scalars as elements of a field would intrinsically assume invariance under coordinate transformation? If not, can you give an example of a field whose elements may not be invariant under coordinate transformation?


I am currently studying Analysis on manifolds for my thesis. I am looking for some references about the McKean-Singer conjecture and Patodi's proof. I read with much attention to detail his paper: Curvature and the eigenforms of the laplace operator. However I am having difficulty over some of the arguments. I would like to kindly request some references about this specific paper, for example some extra details or maybe if this is contained in a book. Thanks in advance.


I am a chemist currently trying to dive in to details of random motions. I have been studying Einstein-Smoluchowski treatment, and want to learn further (Fokker-Plank's equation and Langevin treatment etc). Although I am comfortable with mathematics some of the textbooks I came across include lot of symbols I am not aware of, and hence it gets overwhelming. Could someone please recommend me textbooks which provide detailed conceptual and mathematical treatments, assuming I am not aware of all the symbols or denotations?


Most objects in the universe cycle around their centre of mass: earth, sun, galaxies. Even galaxy clusters. However, the whole universe doesn't possess an angular momentum. It can't: If it had one, the centre would be a special point (which isn't allowed) and the velocities of the outer regions would easily become infinite. How does the angular momentum vanish with increasing scale? Shouldn't the laws of nature that lead to cycling for smaller objects also hold for larger objects?


The Doves Press was one of the major players in the early British private press movement, hand-printing books in Hammersmith that were known for the absence of any illustrations and a focus on the aesthetics of beautiful type. Below is the famous first page from the Doves Press Bible. Can anyone provide some guidance on how to replicate this "I" that extends downwards to the height of the text? I find this quite interesting because there are several letters that could elegantly be extended vertically by stretching the vertical strokes of the letterform while leaving the proportions of other parts unstretched (I, J, H, E, P, L, F, T, Y, U, M).


I read the paper Example of indeterminacy in classical dynamics. I understand the paper: Because the differential equation does not satisfy the Lipschitz condition, its solution is not unique, and thus there is indeterminacy. But is it really possible in physics, as in the example described in the paper, for the ball to rest at the origin to suddenly deviate to the left or right from the origin, which violates the law of causality? Or does the physical world not follow the law of causality?


Are electric and magnetic multipole transitions are classified based on the nature of change in angular momentum of an electron during a transition between two states? In electric multipole transitions, the electron changes its orbital angular momentum, leading to transitions between different electron shells or orbitals within the same shell. In magnetic multipole transitions, the electron changes its spin angular momentum, resulting in transitions between different spin states.


I'm looking for examples of dramatic typos - where a minor edit can dramatically change the meaning of a phrase, and would like to know if there's a name for this phenomenon. Paronyms are a similar concept, but seems to incorporate pronunciation - "love" and "loathe" might be paronyms but they're not especially similar in spelling. For example, "symptomatic" and "asymptomatic" are one letter apart but have opposite meaning, "heroin" and "heroine" have very different meanings. By edit distance, I mean the similarity of the two words - the number of single-letter additions or substitutions to turn one into the other.


I did find the proof of pseudo torque about the center of mass of a body in combined rotation and translation being zero, and thus I'm convinced that in this frame of reference, we can conserve angular momentum. However, although the net torque about the point of contact in rolling is zero, I don't understand how we can conserve angular momentum about the point of contact in rolling motion, even though this axis itself is not fixed and constantly translating. Any help is appreciated.


I have a question which asks me to compute the mapping matrix, but I have no idea how to do this. I understand the concept, but I am unable to understand how I actually solve problems. Would appreciate if someone gave me a step by step explanation of how I would tackle problems like these. Would it be the same method for finding a change of basis matrix as well? Here is the image of the question: Mapping matrix question


If the axis of rotation is not passing through the centre of mass but some other point of a rigid body then how do we define the direction of angular parameters that is angular velocity, angular acceleration and angular displacement. For example : This is a sphere with a void in between rolling on a flat surface.C is the centre of mass and O is the axis of rotation. So basically my question is, will the angular velocity of the point A be perpendicular to OA or CA? Also will the angular acceleration of A be parallel or antiparallel to the angular velocity and correspondingly the centripetal acceleration would be perpendicular to the angular velocity right? I am confused about this.


In the article about Coulomb's law There is following sentence: Coulomb's law is an experimental law of physics that calculates the amount of force between two electrically charged particles at rest. However in the Griffith's Introduction to electrodynamics Introduction before explaining Coulomb's Law, there is following sentence: To begin with, we shall consider the special case of electrostatics in which all the source charges are stationary (though the test charge may be moving). What i'm confusing is: Wikipedia applies to two charges at rest but Griffith's book applies to stationary source charge. They collide with each other. Which is correct one?


I tried to implement Kruskal's minimal spanning tree algorithm and I found that the output is essentially a set of the edges... The output is not a tree at all. In my opinion, a tree can be represented by a parents array, and each node associates a key which is its parent. For a tree we need to know which node is its root at least I think. But the output of Kruskal's algorithm are just edges. How to fill this gap?


All ferromagnets have domains in which the magnetic atomic dipoles are alligned. However, soft ferromagnets like iron respond quickly to the external magnetic fields while the hard ferromagnets(permanent ferromagnets) don't. In atomic view, what causes this difference? My fist guess is that in the hard ferromagnets the atomic dipoles are stronger and their density is higher so that their greater interaction keeps them more alligned compared to the soft ferromagnets. So,the hard ferromagnets are not affected by external magnetic field as easily as soft ferromagnets? I would appreciate if anyone please help me with that.


Given two positive semidefinite matrices X, and Y. My question is, can we use von Neumann or the log-Determinant divergences to measure the distance between the two matrices. Most Manifold measure distance require two positive definite (PD) matrices. In this paper, the authors used the log-Determinant divergence with two positive semidefinite matrices (PSD). However, from what I know, the two divergences are for measures involving positive definite matrices (PD). Any insight please.


Consider a mass rotating in a plane around a fixed point O at constant speed. In such case the angular momentum with respect to O is constant and its time variation is zero according to the fact the torque is zero.But if you change the point to which you evaluate torque and angular momentum, the time derivative of angular momentum is not necessarily zero. So?


I have been brushing up on real analysis recently and, knowing nothing about topology but having seen some very general YT videos on the subject, I have developed a vague feeling that it somehow is very closely related to real analysis. For instance they both seem very concerned with continuity and continuous maps. Could someone please elaborate on the contact points between these branches of math?


I was taught in my image processing class that when a mean filter is applied infinite times on a given image, the intensity of each pixel reaches the same value. I understood this that time entirely on intuition. But looking back, I tried to come up with the formal proof of this result but couldn't make any progress. So is there a way to actually proof this notion. Any help would be appreciated.


My question is on the nature of oscillating electric field. From Maxwells equation, we know that the solution of the electric and magnetic field for an empty space are always oscillating fields. I learned that oscillating fields are connected to oscillating charges. However, what about the case when the the charge is not oscillating but rather, lets say, just linearly accelerating over a given distance. The field gets disturbed once the particle accelerate suddenly but what I don't understand is why this would still cause an oscillating field rather than just a locally distribution of the field. Hope my question is clear.


I would like to know whether bounded linear integral operators (defined on a separable Hilbert space of functions) always have a countable point-spectrum. Or if not, what would be a practical counterexample. The following situations for instance already lead to a countable point spectrum: self-adjoint (or symmetrizable) operator kernel integral operator with bounded kernel But what about bounded integral operators in general? I couldn't find any helpful theorem.


I'm trying to ad some probabilities on my model and I'm wondering which conditions are needed on my initial space. I'm starting with a metric space X and consider the set of operators from X to itself equipped with the topology of compact convergence. Let's consider the Borel sigma algebra of this space. Does there always exists a Probability on this measurable space ? Do I need some stronger assumptions on X ? EDIT : I forgot to specify that I'm asking about the existence of a probability with a "large support", equal to the space or infinite Thanks !


I'm trying to find models that allow me to calculate dropped objects, specifically, a model where I can consider lateral displacement due to wind force and vary the weight and resistance force of the object. This is crucial because I'm trying to calculate with utmost precision the point of impact of a tool falling from a height at a construction site, to determine the safety zone that must be established below to prevent injury to anyone. I appreciate any information in advance.


It is my understanding that sets are said to contain their elements but also a set is said to contain its subsets. However, I feel like this might lead to a confusion since while a set contains its subsets, its subsets are not necessarily elements of it. It seems to me the word "contain" must be restricted to only one of these meanings and there must be another word for the other. Am I missing something here or is this supposed to be understood from the given context?


As far as I understand a material is a better thermal conductor if its thermal diffusivity is higher. The thermal diffusivity is inversely proportional to density, thus denser materials have higher thermal diffusivity all else being equal. Intuitively I would have expected the opposite to be true! Atoms close together interact more often with each other, transporting thermal energy more easily. In my mind that would explain why gases (and 'zero density' vacuum) are good insulators. Why exactly low density improve thermal diffusivity?


I believe anyone who has driven a car without power steering has noticed this effect. I am not able to find a justifiable argument online. My thought process suggested it must have to do something with the static friction in the case of a moving car which provides it a centripetal acceleration, but I am not sure. A reasonable and complete argument will be much appreciated.


It is easily shown that in proving compactness, we can assume the arbitrary open cover is given by a basic open cover. However, the same proof does not work when proving countable compactness. The reason is that not every open set can be written as a countable union of basic open sets. So my question is: in proving countable compactness, is there a counterexample showing that we cannot assume the cover is by basic elements? Note that such a space cannot be second countable. The trivial example of a discrete uncountable space is not a counterexample.


In A_Treatise_of_Human_Nature, of David Hume, it is written: For if truth be at all within the reach of human capacity, it is certain it must lie very deep and abstruse: and to hope we shall arrive at it without pains, while the greatest geniuses have failed with the utmost pains, must certainly be esteemed sufficiently vain and presumptuous. I gathered that the usage seems to be to lie deep instead of to lie deeply. Why does an adjective follow the verb lie instead of an adverb? Is lie here a linking verb? Would the phrase It is certain it must lie very deeply and abstrusely. be incorrect?


There is a book that is called embeddings in manifolds that studies topological embeddings and how they relate to each other (by homeomorphisms). I was wondering if there is a study of isometrically embedded polytopes in manifolds. "Isometric" here would probably mean conservation of the metric (as in metric spaces) rather than the Riemannian structure. So I would like to ask if there is anything I can find on the topic, if there is such topic, preferably a book but papers will do as well?


I understand that the steady-state Kalman filter is more computationally efficient because the noise in a measurement equation and the shock in a state equation have a constant variance over the time period. However, what model should I use if I want to study the effect of a noise shock, which refers to a large shock that was not imaginable? Should I use a time-varying Kalman filter that allows for the variance of noise to vary? I am faced with this question because I cannot exactly tell the differences between these models and their respective advantages and disadvantages. Please help me out.


I need to know the grammar and syntactical explanation of a phrase in the following sentence: The protest was reportedly linked to members of the group objecting to the replacement of the group leader. Is this phrase "members of the group objecting to the replacement of the group leader" an absolute phrase? If so, can absolute phrases act as the objects (noun clauses) of verbs?


The back side of a coin is called reverse, the left or back side of a bound manuscript is the verso, but what is the back of a piece of paper such as a letter (not the envelope) called? Dictionaries give a wide selection of options (rear, back, reverse, backside, flipside, back page, etc.), and I don't understand which of these a native speaker would most likely use here. I would like to direct the reader of a formal letter (which in this case is a legal document) to further information on its back, e.g. "bla bla (see the back of this letter)".


I found this https://en.wikipedia.org/wiki/Harmonic_oscillator in my search results when I search for "electrical oscillator Hamiltonian" and some other things too. But none of them answer my question about the form of the kinetic and potential energies of an electrical oscillator. Searching for the kinetic energy and potential energy does not show up fruitful either such as here. I am specifically looking for an electrical oscillator Hamiltonian. Any help would be appreciated.


We are learning the rational root theorem right now. It is pretty clear how to prove a number is irrational when we know how to construct a polynomial that only includes integer and has the irrational number as a root (rational root theorem). However, if it comes to sum of two number with different power, it is pretty hard to construct the polynomial. Is there a general way to construct polynomial? Or is there any alternative ways to prove this question?


If we have a smooth and frictionless pulley with two different masses attached at the ends of the rope (meaning they will accelerate) and the pulley is attached to the roof by a string, why cant we consider everything as one closed system and calculate the wieght of the that system to find the tension on the string attaching the pulley to the roof. After thinking for some time I feel that somehow allowing the masses to accelerate in one direction is relieving some of the force on the pulley. But why? Why can't we consider it a single system and find its wieght?


I think it is computing the fine-grained entropy. However, I am confused by the case that when there is a black hole in the bulk. The Ryu-Takayanagi surface may include the horizon of the black hole in this case. However, the Bekenstein-Hawking entropy is a coarsed-grain entropy. I have also heard that the RT formula is computing the coarse-grained entropy of the complementary region. What does this mean?


I am working on a model of a Fe square planar complex with nitrogen and oxygen given below is a monomer of that polymer. While constructing the hamiltonian matrix of this monomer I'm confused as to if I should take the orbitals of Nitrogen and Oxygen as normal atomic orbitals or hybridised orbitals? I tried to search up about it but couldn't find any method that used hybridized orbitals


While solving this problem, we arrived at the solution in a form of discontinuous function of electric field. Intuitively it doesn't seem right. What is a physics explanation behind this discontinuity? Or is the mathematical approach wrong? Or is it just so idealised problem that it doesn't exist in reality anyway, so discontinuity doesn't have any physics meaning? Tried solution (electric field as sum of electric fields of all thin rings):


I am a mathematician and a while ago I started studying quantum physics. I know the concept of superposition mathematically, but my question is regarding a famous experiment that Aaron O'Connell performed. Link to the TED talk that describes the experiment: https://youtu.be/dvYYYlgVAao He says in this video, "The small piece of metal was vibrating and not vibrating at the same time." I did not understand this phrase because it seems illogical. Can this be explained in the experiment and is his statement correct or is it just to stir up the audience?


I a m a high school student who has spent a lot of time learning about limits but still do not understand how they work completely. What I have understood is that sometimes a function does not have a y-value/s for one/(or more?) x-value/s however, since we know the y-values of the x-values close to the x-value with an undefined y-value we can approximate where the y-value for that x-value will be. To express this we use a concept called limits. My question is, can we use limits for functions where there is a y-value for every X-value? If yes,why would we do that? To show the behaviour of the points of a function as they get closer to a certain point? Thank you in advance!


(I know this is going to sound odd just please consider it) What if you were to throw an indestructible ring straight into a black hole with a hole larger than it's singularity? Would its trajectory be straight through the black hole missing the singularity? If this is the case then wouldn't it reach escape velocity and as result break the speed of light? Think of it like a ball rolling down a hill and being able to speed up enough to roll back up a hill of the same height. I understand that a black hole achieves it's event horizon through distortion of space time so what I am truly pondering is if it would have a straight and un-impeded trajectory through the black hole.


I found this problem on Facebook from a Mathematics group. Apparently it had appeared as one of the problems on a college entrance exam. The person who shared it on Facebook did not disclose the correct answer. I'm going to post my own approach below, please let me know if my answer is correct or if there are any other methods possible. (I even encourage solutions that may typically go beyond pre-college level mathematics.)


Every well-ordered set is linearly ordered. However, is there a notion of well-orders that generalizes to partial orders? Maybe, the generalized definition will be that every non-empty subset has a minimal element, not necessarily a least element. I would be very interested to hear about such a definition, and more importantly, I would like to know about research on this topic about these "generalized" well-orders.


I use VS code with the Latex workshop on Mac and it works perfectly fine when writing a paper. But I don't understand "pdflatex", "mactex", "texlive" or something other people are saying What are the essential components to compile a paper using Mac? How can I check what is working behind my VS code? (I don't remember what I installed so far to make my environment possible )


I was reading in a patent that a rotating spherical body in space will experience a curvilinear translation, due to the conservation of angular momentum. This confuses me, as per my understanding if there is no friction, or any other force applied, conservation of angular momentum will keep the same rotating speed but I do not see why it will make it move/translate from its location. On the other hand, I have seen a video of a spinning object in the ISS changing direction alternatively, and I also do not know how it happens, in that case there is no translation that can be observed, but there is still friction with the air. Anyone can give some insights? Thanks!


I am just wondering how do we transfer IMU readings to Body frame? Especially, linear speed and acceleration. In following diagram, the IMU is mounted on one of the surfaces of the body (you can imaging it to be a bot, vehicle, drone, or etc.). The center of the vehicle is at Body (green) point in this diagram, and it is on a different surface of the body. Both IMU and Body point are on a rigid body which moving in an inertial frame. Any hint is highly appreciated. I'd really appreciate if there are some references. Thanks!


I am wondering about the correct verb form to use here: This article limits itself to consider... or This article limits itself to considering... For this example both sound dubious to me (possibly because of the presence of 'to'), and those I have asked are also unsure, including a native English speaker. And what about the passive counterpart: This article is limited to consider... This article is limited to considering...


Earlier I was using texstudio with MikTex. In MikTex, I am facing some issue with its updates. So, I downloaded MacTex. How to configure texstudio with MacTex? When I am running texstudio, it is taking MikTex compiler. The error is shown below. Sorry, but /usr/local/bin/pdflatex did not succeed. The log file hopefully contains the information to get MiKTeX going again: /Users/bp/Library/Application Support/MiKTeX/texmfs/data/miktex/log/pdflatex.log pdflatex: major issue: So far, you have not checked for MiKTeX updates.


I am confused by the definitions of a gauge transformation, a coordinate transformation and a diffeomorphism. In particular, should observables in GR be fundamentally invariant under gauge transformations, coordinate transformations or diffeomorphisms? For example, in these notes by Bertschinger, observables should be gauge invariant, and he defines a gauge transformation as an infinitesimal diffeomorphism. This appears standard and in line with e.g. Sean Carroll's GR textbook. On the other hand, there is this paper by Yoo and Durrer, which claims that cosmological observables are invariant under diffeomorphisms but not gauge transformations in general. Their definition of a gauge transformation appears to refer to simply "an equivalent" coordinate transformation - the sort which Carroll disagrees with being strictly correct in his textbook.


Is it correct to say "None can stop me?" I feel like "No one can stop me" is the one I've seen used most often, but I'm not sure if "none" works too? Does there have to be a specific context where there are several people: e.g. "There are many who will try, but none can stop me." And in other contexts (without specifically mentioning other people) "none" doesn't work?


I'm having trouble reading documentation from packages. It seems like there's something wrong with the way I'm reading them. How should I read them, in order to learn how to use the packages correctly and efficiently? Also is there a place where you can learn typography related topics in LaTeX like glue and boxes? I don't find it talk all that much, and if they do they don't teach me why that's necessary. I haven't learn the details of typography other than fiddling with MS Word. And I would like to learn how to make is so that the fonts and font size aren't too long for the average characters per line.


On the average Semi-trailer truck, assuming an average load, can the engine overcome the brakes and continue spinning the wheels despite the operator applying the brakes fully? This would be assuming the brakes are in good operating condition, and are not overheated or failing in some way. Additionally, would this also be the case if the tractor unit had no trailer on it, and no weight? How about fully loaded?


I am having a conceptual difficulty understanding the following issue regarding space-time: it is clear to me why a full description of coordinates requires three spatial dimensions plus time. However, Linear Algebra also teaches that dimensions needs to be independent from one other, such that my movement along one dimension has no bearing on my movement along another dimension. This does not seem to hold in space-time, given the effects of time dilation and space contraction, which is essentially a rescaling transformation of one dimension based on the object's velocity along another. This seems to preclude such independence between said dimensions. Can somebody clear up my confusion?


In the case of big freeze, space expansion will be accelerating and there appears to be a lot of different phenomena occurring. However, in the case of big rip, the expansion is super-accelerating so that light speed will be reached, ripping apart every structure. How come this speed is not reached in the case of big freeze. If the speed of expansion is accelerating, it should reach light speed at some moment. How does the evolution of scale factor in this case look like?


So far I have not needed to insert jpg images in my LaTex files, but now I want to write a text that will include some pictures. I would like a format where the picture doesn't take the full width of the page but be rather set to the right or the left and the remaining space on its side filled with text. I have tried to use the figure environment from the graphicx package together with a minipage environment, but failed. What would be the best way to proceed? Is it safe (or even possible) to change from a one column format to a two columns format in the middle of a page?


It's on the tip of my tongue. The great thing about this word is that there's a connotation that a person it describes is not just smart or curious, but that they find a childlike joy in learning. This word is paired with the word "intellectually", as in "intellectually curious." But when it's on its own, it's more likely to be describing someone who's cheeky or sassy, particularly a child. But in that usage, there's the connotation that their cheekiness or sassiness belies gifted intelligence. I am pretty sure it starts with a p, or even maybe 'pro,' but I could be wrong about that.


In indirect speech some modal verbs usually change. can -> could He said "I can ride a bike" = He said that he could ride a bike may -> might/could He asked "May I use the bathroom?" = He asked whether he could use the bathroom But what about dare? It can be a modal verb. "No one dare go there" he whispered = He whispered ...


I'm from South America and I've been fluent (at least in my opinion) in English for several years as of now. With this in mind it has really called my attention lately that I've heard Afro-American people say, for example, "I don't like no cats" instead of "I don't like cats". With double negation. And the other one is, for example, "Look at them kids" instead of "Look at the kids". With them instead of the. It is not really confusing tbh but I'm just curious about why this happens?


Would it be feasible to extend the range of an EV using a generator? I would guess that F=ma and perhaps a myriad of other factors would make a generator placed on an EV not work very well. I think of things like a wind generator to capture the wind power but the drag would slow the car down thus canceling the added energy. Or how about a generator in the rear wheels of the car to add back some of the loss? I could imagine something like that working in a downhill scenario but i can't see it adding much if any in the whole driving scheme. Seems Elon would have figured this out by now if any such thing would be feasible. What you guys think?


This question is not similar to what I have read so far in this site. I am just trying to understand what electricity actually is. I have come to know that when charges move or vibrate in a wire by the supplied voltage, the electrons drift velocity is very slow. And the bulb glows instantly because energy travels through EM field, which is perpendicular to electric and magnetic field. Now, comes the question, suppose if I create a circuit, and connect it to the battery. So, the bulb will glow, but what if I put another unconnected wire near the circuit connected with another bulb. Will the bulb glow? Because electricity travels through EM field. Shouldn't the bulb glow? Which actually doesn't happen. I am just confused.


Let's assume a galaxy has super massive black hole in it's center. We can't see what's inside the black hole because light cannot escape from it. In simple terms, black hole is a hole in the space and also assume that this black hole goes down all the way somewhere, like a straw/pipe that we cannot see. So now imagine a a spaceship is traveling exactly beneath the supermassive blackhole (but far away from event horizon where the blackhole can't pull it inside). And now what happens if the spaceship hits the blackhole straw/pipe? Just trying to understand is there any science that we can use to see where this blackhole straw/pipe goes to from outside. Or what happens if a spaceship or a planet hits this pipe?


The s-procedure or s-lemma tries to solve a system of quadratic inequalities via a linear matrix inequality (LMI) relaxation. There are many different enunciations and if you are not sure what i mean you can check here. I have seen it be used in papers where the inequalities are not quadratic, for example here, but I cannot find a form of the theorem that seems to apply. Can anyone help me? Thanks in advance.


I remember hearing at some point that pertubation QFT using Feynman diagrams can be thought of as certain limits of lattice QFT. Is there a precise statement of this fact? Or is it just a heuristic with no mathematical formulation/justification? Alternatively, is there a way to do calculation on lattices, that gives the same result as calculating Feynman diagrams, taking into account things like Pauli Villars/dimensional regularizations and renormalizations?


I have read so many posts about the implication, but I am still confused. The statement "if A, then B" is always true when A is false. But I think that when A is false, nothing can be concluded. Moreover, I do not understand the example, "If today is Saturday, then tomorrow is Monday." As today is Wednesday, both sentences are false so the implication should be true, but it seems in daily life that this implication is false.


In the Vlasov variant of Boltzmann equation, the Coulomb pair forces are included via the force term that appears as the prefactor of the derivative of the density with respect to velocity. In that case, one is effectively calculating a mean force or a "force field" by averaging over all the pair forces. The average is obtained by an integral over the pair potential weighted with the density. Since this Vlasov approach allows to have good analytical insight into systems mainly governed by Coulomb forces (i.e Landau damping and other), is it possible to apply the same approach for Lennard Jones pair forces ? I could not find references and I do not understand why this is the case.


I'm junior student in physics major, and i didn't study QFT or other graduate course yet, just undergraduated QM. Recently I take brief explanation class about SUSY, and in the class I heard that simple harmonic oscillation (SHO) is important because it is unique potential which is invariant under some gauge transform therefore it doesn't change some kinds of scattering properties (reflection coefficient, etc). Is SHO only example which is invariant under the gauge transform in SUSY QM? In the description, what kind of gauge transform is meant? How does the gauge transform relate to SUSY?


I have a bit confusion of understanding. The central limit theorem in statistics states that, given a sufficiently large sample size, the sampling distribution of the mean for a variable will approximate a normal distribution. But the definition of sampling distribution is the probability distribution of a statistic that is obtained from many random samples of a specific population. I don't get it and see a contradiction of how getting a sample distribution from one large sample as CLT states while sampling distribution definition says it needs more than one sample?


I am trying to understand the hall effects and have a few problems with them. So let's consider the classical Hall effect. We know that we consider a sample, where the electrons flow, we apply the perpendicular magnetic field, and the electron's motion is curved so they reach the sample's edge. There appears a "new" electric field. On the other hand, the quantum Hall effect sometimes showed that electrons move in a closed circular orbit, but not in the edge where they can go through "skipping orbits". So my question is, why in classical hall effect the electrons teachers the sample's edge and don't just move in a circle? What is more, do they move also along the edge like in the quantum hall effect?


I want to say that a particular painting never gets old because you notice something new each time you look at it. What is a noun to describe this quality? An example sentence would be: "Due to the xxxx of the painting, you see something new each time you look at it." An example to the opposite of this word would be a pop song that you get bored of after a few listens. Compound words are fine as long as they are nouns. The answers to this question come close to what I'm looking for but don't fit exactly since in my situation it's not the object that changes but it's so complex that your perception of it is different each time.


Leaving school seems like such a huge event in our lives. There's this single summer where you have more freedom than you've ever had before with a group of people you've grown up with before you all go off in different directions in life: work, uni, travel, whatever. It might be the last chance to spend time with these people before you ever see them again, and so people want to make the most of this time, moreso than other summer breaks. It therefore seems strange to me that I've never heard that summer be given a special name, even in slang. Whether it's a formal term, colloquialism, slang, or TikTok hashtag, has anyone ever heard a name for this liminal period straight after compulsory education?


In my understanding, light works as follows: every point in space where there is light, this light works as a point source. When we progress in time, the light spreads out from there in all directions (spherically). The new locations where the light is now, act as new point sources. From this it follows that the light from a laser beam would go in all directions, not just in the direction of a laser. However, we see in practice that (most of) the light only goes in the direction of the laser. How is this possible?


I'm studying about method of undetermined coefficient. But i can't make sure when method of undetermined coefficient can be used. I know that the differential equation need to be non-homogeneous and the non-homogeneous terms need to speical form such as polynomials, trigonometric functions, or exponential functions. what I'm wandering is following: To use the method of undetermined coefficient, the coefficient need to be constant? and the differential equation need to be linear??


According to the dictionary, the "e" is optional when a word ends in "-dg(e)ment". Dictionary examples: oxfordlearnersdictionaries.com: "acknowledgement" (also "acknowledgment") oxfordlearnersdictionaries.com: "judgement" (also "judgment", especially in North American English) oxfordlearnersdictionaries.com: "abridgement" (also "abridgment") ...And some other similar examples from the internet: ledg(e)ment, lodg(e)ment, grudg(e)ment. Of these, only judgment is explicitly said to be found "especially in North American English"; regional distinctions don't seem to play a role in any of the other options. Is this statement verifiable? What is the evidence, and how widespread is the variation? Is there a historical reason for judgment being unique in this sense? (If it even is unique...)


Given two points and a line in the plane, is it possible to construct a circle tangent to the line which passes through the two points using ruler and compass? For those who don't know, these are the rules of ruler and compass constructions: you start with a set of initial objects (points, lines, etc.), at every step you can do one of the following: Construct a line or segment between two points in the set Construct a circle with a segment within the set being its radius Intersect two curves within the set Each of these moves add the outcoming curve/point to the set.


If I understand correctly, the intensity of light is proportional to the number of photons hitting a certain area. If we then look at a single photon when described as a transverse wave, is its own energy equal to the square of the amplitude of that photon's wave? If we have a high energy photon this would mean the amplitude would be large, and the converse for a low energy wave.


For example a car with a known speed and mass crashes with a completely unyielding wall. The car has a crumple zone, (and you know the modulus of the crumple zone) so it doesn't stop immediately, but how can you calculate the time it takes for it to stop? I've been reading a relevant article (Analytical approach for vehicle body structures behavior under crash at aspects of overloading and crumple zone length) but their equations are beyond me. First question, had a hard time finding any help with this. Thanks! The journal article in question: Analytical Approach for Vehicle Body Structures Behaviour Under Crash at Aspects of Overloading and Crumple Zone Length


I am a Physics Major. I have done an undergraduate level course of differential geometry. I want to get into symplectic and Riemannian geometry, but I would like to start over from differential one again. I found this book that seems to cover what I want, but I am not sure if it is a good book or not. Can someone recommend a book or other sources (or even roadmap of multiple books) that start from differential geometry and teaches till (or gives proper introduction to) Riemannian and symplectic geometry (physics perspective is preferred but not necessary).


Consider that I have a piece of paper, and I do a drawing on such paper. Now, consider the set of all possible "transformations" I can apply to this piece of paper that do not tear it. For example, I can rotate it, I can do an origami, I can make it into a cylinder... Is there a name for the set of all of such transformations? Could I say that all these transformations are within the conformal transformations, i.e. since conformal transformations also have scaling, could I say that the set of such transformations that do not tear the paper are within the set of conformal transformations?


I've known that the distance between a compact set and a closed set is zero if and only if they are not disjoint, and I can also find counterexamples in which two disjoint closed set has zero distance. But how about a closed set and a bounded closed set? Does there exist a closed set and a bounded closed set with zero distance between them, and they are disjoint? The counterexample should not be found in Euclidean spaces though.


I use xsim to create exams and it works very well. I kove the way it handles exercises and solutions. Now I'm starting to create a script and I was wondering if there is a way to hide the numbering in exercises so I can just write an unnumbered exercise and hide the solution for the students but leave a blank space for them to write in.


I have an experiment where I have to find the damping coefficient of a compound pendulum situated in a clock. It swings at a very low amplitude, so I think I can make a small angle approximation. Because if I assume it has SHM, I can simply use logarithmic decrement to find the coefficient, which would make my work a lot easier. I cannot find much literature on whether a compound pendulum can even display SHM.


This is a naive question. I am studying the surface-adsorbate systems using DFT calculations implemented in VASP. By setting LVHAR = True, I obtain LOCPOT which contains information about the ionic and electrostatic potential. I would like to clarify whether this information is actually the electrostatic potential or the electrostatic potential energy. My source of confusion is that VaspWiki states that the electrostatic potential is written in terms of eV, which indicates it should be an energetic quantity.


The first pair of these sentences with perfect participles, shows a reason while the second shows a time, and this is how the teacher taught me their meaning, so I want to know why one is reason and the other is time. How can I know that one describes reason and the other time? Please convince me. He has failed four times already, he doesn't want to appear in the examination again. (Reason) Having failed four times, he doesn't want to appear in the examination again. (Reason) He has a lot of money in business, he doesn't want to risk his money any more. (Time) Having lost a lot of money in business, he doesn't want to risk his money any more. (Time)


I am a person who gets bored all the time, so I found that counting something in my head helps occupy my time. I started counting in binary, but after a while this started getting old as I was doing it in the back of my mind, so I got bored again. So, my question is, can you count up prime numbers in your head using a "mind-friendly" formula, or is there something different you can count that keeps your mind busy?


In the adjoining table, a dot indicates the combination of chemicals that cannot be placed in the same location as the other because of safety issues. If a company is dealing with the chemicals a,b,c,d,e,f,g,h,i,j listed in the table, what is the minimum number of storage locations needed by the company? My idea of the solution is something like this: I start with 'a'. I can collocate it with c,d,f,g,h,i, but not with b,e,j. But then, I note that d and f cannot be collocated. How/where do I go from here?


I'm learning the grammar of relative pronouns; I thought all relative pronouns should be at the start of a clause, and then I learned that pronouns can be put after a preposition, as in "The bed on which I was lying...", but again, I came across some other examples that showed that we can put more words before the relative pronoun, as in "The bed, the owner of which we had seen previously, ...", or "The bed, lying on which was a small cat, ..." Does anyone know what are the rules? Can we put the relative pronoun anywhere in a clause? What about these sentences: "The bed, we had seen the owner of which previously, ..."?, are they grammatically correct?


A ball is bouncing elastically after its collision from ground so can we call it SHM as it is oscillating and also it is periodic and it seems similar to other cases of SHM? If yes then where is it's mean position? But the thing is in SHM force always opposes direction of motion but here it is always in the same direction that is downward force(due to g). Also in SHM force is zero at the mean position but here force is never zero. There can be some flaws in my understanding of SHM as I am new to this. But all the answers are welcome!!


I have generators for my group in GAP. Additionally, I possess a specific unitary element and wish to determine whether this unitary element arises from the Lie algebra of my group's generators. Initially, I considered a brute-force approach, intending to compute the matrix exponential of a particular generator of the group. However, I couldn't find any matrix exponential function in GAP (e.g., expm, ExpMat, MatExp, etc.). Now, I have two questions: How can I compute the matrix exponential in GAP? Is there a systematic method to compute the Lie algebra of the group generators, allowing me to identify my specific unitary element, and subsequently terminate the computation in GAP?


I am doing a review for an article where I need to write some equations myself and want/have to stay anonymous. So far I plan to just LaTeX my notes, compile them with latexmk -view=pdf and send them in. My fear is that latexmk/pdflatex adds metadata that will break the anonymity of the review process. I checked the result with pdfinfo, and could not find info that would identify me in this case. But in general I would like to not only rely on some defaults of pdflatex, but to know if there is a cheap way (preferably via CLI options) to guarantee, that there is no identifying metadata.


I'm making a roster in a spreadsheet of all the current employees, volunteers, sports coaches, and other workers at a school. What is the best way to label the column showing if someone is Full-time, Part-time, Volunteer, or On-call (like substitute teachers)? What word represents this? I already have a Status field denoting Current vs. Former. Should I just merge this unnamed field so that Current becomes more precise as Full-time/Part-time and then Former? It's in the same logical realm, I think.


I am looking for an algorithm that could determine adjacent extreme rays of a polyhedral cone (extreme rays that share a face), given the set of inequalities defining the cone and an extreme ray for which the adjacent rays are to be determined. I was hoping there exist a python library that might already have coded up, but I could only find pycddlib, which seems to need to determine all extreme rays first. This seems inefficient to me: I was hoping there would be a method that doesn't neccesitate computing all extreme rays first. Thank you for help!


I want a certain path that I generate to remain hidden (so I use the path command without actually drawing it), so that I can use that path to compute other points of interest such as intersections, but I don't want that hidden path to affect the overall scaling of my shown figure. Is there a way to generate such a path that does not affect the overall figure scaling?


I am writing an article about data mapping, and discussing the various types of challenges in mapping data elements from one system to another. One of these challenges is assuming that a data element called X in one system is the same as a data element named X in a different system, simply because they are named the same. I am looking for a classification for words that have multiple, disparate meanings. My mind keeps jumping to polynomial, but I need that type of word that would relate to the meaning of words. Poly-meaning-al?


I am trying to understand the following to defintions of being "intrinsic". They are from Differential Geometry of Curves and Surfaces by Tapp. Question: In what way are these defintions consistent? To call quantity intrinsic if it can be expressed completely in terms of the first fundamental form makes sense. To me it seems that requiering invariance w.r.t. isometries captures different properties that I would expect from intrinisc properties. For example that deforming a surfaces should not alter the length of a curve on that surfaces. It seems to me that there are two types of intrinisc properties and that the "intrinisic geometry" consists out of these. Many thanks in advance for any hints or explanations!


If we consider a charge in space then we can draw infinite number of spheres (taking the distance between the spheres equal) around the charge and these spheres act as equipotential surfaces. Now, As the distance of all points on the sphere are same from the charge then why does equipotential surface gets farther apart when electric field decreases??i mean my questions is what is the problem with taking the distance between equipotential surface equal??


When someone tends to hesitate to acknowledge a problem and avoid involving trouble even if needed, is it ok to say in a negative nuance that He/She always has a "don't rock the boat" attitude. I found the below definition at Collins Dictionary and considering that, it seems that not rocking the boat doesn't necessarily have a negative meaning: If you say that someone is rocking the boat, you mean that they are upsetting a calm situation and causing trouble. (www.collinsdictionary.com I'd like to describe a person who usually avoids troubles and never speaks up regarding existing issues such as social issues with a negative implication. I'd appreciate it if you tell me appropriate phrases to describe such a person.


I'm struggling to explain it, but some examples are (mention does not imply endorsement!): Sir Keir Starmer -> Sir Kid Starver (criticism of his stance on child benefits) Benedict Cumberbatch -> Butterscotch Candybatch (surreal humour) Financial Conduct Authority -> Fundamentally Complicit Authority (critical of regulatory capture) Saddam Hussein -> Sadman Insane (general derision) These are nicknames, i suppose, but they are a specific kind of nickname.


I was attending lectures om holography where the lecturer kept on mentioning that a QFT lives on a Cauchy slice. What does that mean? Is it such that each point of the slice is associated to a unique vector in the Hilbert space? Can this be generalized to QM? Does that mean that the harmonic oscillator lives on an integer lattice? Is this geometric correspondence the motivation behind the operator state correspondence?


I am just a high school student. I had already heard that we can't determine the exact location of big bang. So now i have a question can't we approximate it? Like say divide the observable universe in many grids and you take the Average age of each grid and do this for all the grids. I assume intuitively that on average the parts of the observable universe which were formed earlier would be closer to the point of origin of the big bang right? I know that there are other effects to consider such as expansion of space but still it is not possible so my question is why?


Let (B, S) a multi period binomial model that is arbitrage free. Prove that the unique time for an American call is the maturity T. I would like to prove that the unique optimal exercise time for an American call option is the maturity time T. My idea is to prove this result by contradiction. I want to find an exercise time t<T, such that there exists an arbitrage opportunity, so that the value of the portfolio at time t is greater than zero, starting with an initial value equal to zero. Is this a good idea, and how can I find such strategy explicitely?


Let's say a small magnet is attached with its South pole on to the north pole of a bigger magnet vertically.There is no friction between the magnets so this means the normal force increased due to the attraction doesn't have any effect on its tendency to fall.My question is that : Does the small magnet slide down OR Do the magnetic field lines have such a property to pull the magnet up. Also, what changes are seen if the small magnet is attached vertically to a iron bar (frictionless again).


I'm attempting to self-study real analysis and was looking at the popular textbook recommendations. So far, I've seen Abbott and Cummings for beginners. With Rudin, Stein, Bartle, and Zorich for more advanced books. My question is : What book is the sort of "Bible of Real Analysis" or reference book, where if one knows the book from cover to cover he would have a colossal grasp on the subject. To clarify my question, I mean a textbook that would cover the whole surface of analysis after learning it from a more elementary book such as Abbott or Zorich.


I understand that if there are different capacitors of different capacitance then they can store same charge at different voltage differences only. But I wanted to understand that why the above happens when the capacitors are arranged in series...I want to know the mechanism behind that if we arrange different capacitors in series then they store same charge at different potential differences.....why they store same charge only that is flowing in the whole circuit due to net potential difference of the circuit....why they don't store different charges whose sum is equal to the net charge flowing in the circuit?


I am currently studying the method of characteristics. So far I understand what I read in literature. The problem is that I didn't find a good explanation for imposing boundary conditions anywhere. Also, in the textbook problems, only the initial conditions are given. Please recommend me some literature where I can study how to impose boundary conditions on the solutions obtained using this method. A hyperlink to the literature would be of great help.


Are there any public stats that demonstrate how much literature is produced in LaTeX as opposed to Microsoft Word, Adobe InDesign, etc? When I search "latex market share" every single result talks about rubber/pillow/mattress companies and not the typesetter. From what I have heard LaTeX is heavily used in Math & Science theses, and I would like to find specific numbers to back that claim. Does LaTeX stand a chance against other software outside the Natural Sciences, its stronghold?


I have read a recently published paper on arXiv where the scientists claim to have developed a superconductor material working at room temperature and pressure. While it is clear that the results - if proven true - are preliminary, and that a lot of refinement would be necessary anyway, I am wondering what technological advances can we expect from such a discovery? Or, what technologies are currently impractical, impossible or unexplored due to the impracticality of currently industrially available superconductors?


Our teacher told us that objects store potential energy in them according the position of the object above the ground because of gravitational force. Like we are holding a box up and it stores potential energy, and if we drop it, its potential energy gets converted to kinetic energy. My doubt is, If, we send a rocket outside the Earth, then after some point, the influence of Earth's gravitational field nearly comes to an end, Then where does the potential energy of the rocket go if we stop providing it velocity (Like dropping the box)?


I'm studying a math book. There is a definition as seen below. A category is a class C of objects (denoted a,b,c ...) together with (i) a class of disjoint sets, denoted hom(a,b), one for each pair of objects in C... (ii) ... I want to ask which one of the following meanings should be deduced from this statement regarding the grammer rules; hom(a,b) is one of the sets of a class. hom(a,b) is a class of disjoint sets. The whole thing may confuse you so please just consider the part saying: " a class of disjoint sets, denoted hom(a,b), one for each pair of objects in C..."


Please see here to see question & my solution Here are question & solutions I tried. How should one approach problems where both s-shifting and t-shifting are required? Is there a specific method or sequence that is recommended? I have attempted to solve these problems by applying the first shifting theorem (s-shifting) followed by the second shifting theorem (t-shifting). However, this method seems to lead to incorrect results. Answer said that t-shifting first, s-shifting later. Could you explain why this approach fails? What is the significance of the sequence in which the first and second shifting theorems are applied? When both shifting theorems are necessary, which should be applied first, and how does the order affect the final result? Thank you for your time and assistance.


I have read about wave propagation and dipole antennas here. I do not understand how a single electromagnetic wave (i.e. the electric field in particular - I can come up with the magnetic field later) propagates when its electric field of a dipole antenna is shown below. For example, the iconic picture of an electric field of an electromagnetic wave looks like the figure in the bottom left corner in red below. How does it look in the bottom right diagram? I tried to plot it below again in red, but it doesn't make sense to me. For example, the amplitude of the wave would get bigger. So, I would like to know how a single electromagnetic wave propagates in this field.


In the following sentence is "painted" being used transitively or intransitively? The sunset painted the sky with a tapestry of fiery colors. I asked ChatGPT about this and it gives reasons for both... For intransitive that it's being used metaphorically or poetically; the focus is on the action of sunset creating a visual effect, not the application of paint. When I reminded it that "the sky" certainly seemed to be the direct object it agreed at first that it was actually transitive. Yet when asked to confirm, it stated it was intransitive. Then it finally stated it's a matter of interpretation. Personally, I think it's transitive. Am I right?


I want to make a program that will solve polynomials, but I don't want to just look at another program that already works as I think that would defeat the purpose of the exercise. And that's why I'm asking the question here instead of on Stack Overflow. I think it would be very difficult to implement Alpha-Beta Theory/Veita's Formula, and guessing using Factoring doesn't seem like a neat solution to me, so I'm wondering if there is in fact an easier implementation. I would also like this to be scalable for higher degree polynomials if possible. Thanks.


I'm reading the famous paper by Haah: Local stabilizer codes in three dimensions without string logical operators. In the last sentence of the introduction, he wrote: A logical operator is a Pauli operator possibly with infinite support that commutes with every generator. Since each generator is local, the commutation relation between the generator and an arbitrary Pauli operator is well-defined. Why the locality guarantees the commutation relation is well-defined? Are there examples where removing the locality condition yields a not well-defined commutation relation between the logical operators and the generators of the Hamiltonian?


I'm studying nonlinear control systems and I have some questions about the nature of this optimal control given by PMP: Traditionally, control methods based on Pontryagin's Minimum Principle (PMP) and Euler's method (or Runge Kutta) have been extensively used to address nonlinear system control problems. Is there a potential to get trapped in local minima resulting in suboptimal control outcomes? If there is one, why do we call it "optimal"?


Solving Fresnel integral for vortex beam diffraction is necessary and difficult. Without the necessary integral formula, this isn't easy to derive. Therefore, I need some means to calculate it numerically quickly. I have learned some methods, such as angular spectrum method, fast Fourier algorithm. But I still can't understand its principle and implement it through matlab programs. I think I may need some correct codes or related books to learn, and I also hope to get a good explanation so that I can understand them better.


Just a quick informal Question. I was wondering if the boundary of an LCF Manifold is also LCF ? By definition a LCF manifold is one in which, for each point, there exists a neighborhood around that point that can be conformally transformed to a flat Euclidean space. Since the boundary is a submanifold the answer to the above question should be, yes ! Do I have to be careful to which EUclidean Space i send the conformal transformation? i.e should boundary points only be conformally mapped to an euclidean half space? Thanks for clarification!


Most articles show the plural form of equivalence as equivalences. The wikipedia on logical equivalence uses this form. However, I feel like I have seen equivalencies used in contexts like mathematics as well, like in this paper Equivalencies in Mathematical Structures: An Exploration through Model Theory and Groups Is the choice region specific (American vs European), domain specific (mathematics versus general conversation or other technical fields), or just whatever the writers preferred?


Let's consider a glass slab ABCD. Now a light ray is incident on the near end of the air-glass interface AB and is emerging from the glass-air interface BD. In such a manner: Is there something wrong how I have made my figure. Can light emerge in such a manner from a glass slab. This is my first question and English is not my first language I hope you understood my doubt. I feel like the figure I have drawn does not look like the conventional ray diagram of light entering a glass slab. Also I am not an expert I am just a student who wants to know the answer to my question.


I have a huge .bib file that contains all my references. Very few of them currently have a doi field. Given access to an online database that includes many of the same references, is capable of generating .bib entries, and includes dois, is there any more or less automatic way of using the database to add dois to as many of the references in the .bib file as possible? Note that the citation keys in my .bib file and those generated by the database are not identical, so matching would have to be done by e.g. title and author. Answers using e.g. Jabref or Zotero are fine. It feels like one of those tools ought to be able to do this. I just don't know how.


A cumulative distribution function (cdf) has a countable set of discontinuity points. They need not be isolated. Let us call non-isolated points 'accumulation points' of this set of discontinuity points. Is it possible that a cdf admits also countably infinite accumulation points in its set of discontinuity points? If so, is this something as pathological as requiring a singular distribution or it can also happen with discrete distributions? Thanks for any clarification, it would be nice to have concrete examples.


A long time ago, I had a teacher who said you could write "lit." to mean "read this as". The typical example is to provide a tongue-in-cheek translation from English to English, such as: The power company charges an administrative (lit. bogus) fee on every bill. The author of the sentence is telling the reader to read past the euphemism ("administrative") and understand the subtext that the power company really just wants to take their money for made-up reasons. I just looked this up now, and couldn't find any reference material saying this was an accepted usage of "lit.". Maybe my memory isn't completely accurate?


Given a table with x and f(x) values, g(x) a non-linear least squares approximation, and p(x) the Lagrange interpolating polynomial that passes through all of the points in the given table. What would be the best approximation on f, for a point x = k by only using p(x) and g(x)? Intuitively I thought about calculating p(k) and g(k), then taking the middle point of both, however I have no idea if this is reasonable to think of, nor if it is the best approximation for f(k). Is there a visual way to think of this problem? Or will I have to use the properties of both p and g to find the optimal approximation?


Suppose there are two seeds kept at equal distance from a light source which emits a photon each on either directions. Seed germinates when a photon falls on it. According to rest frame both the seeds receive photons simultaneously but according to moving frame one of the seeds receive a photon earlier than the other seed. The second seed dies off due to delay, so there is only one plant in the moving frame but two plants in the rest frame. Is my line of reasoning correct? What happens when I bring the moving observer to rest so that he should agree with whatever observations made by the observer who was at rest?


I've been asked by a friend taking a TOEIC exam to get an explanation of why "wipe" should be used in the following sentence, as opposed to "wipes": It was mandatory that each person wipe[s] off equipment after use with provided spray and wiping towel. Is it because there is an implied (should) before "wipe"? Or is it related to the fact that it is "each person" - meaning, of course, multiple people and perhaps substitutable with "they" - i.e. you couldn't say "It was mandatory that they wipes"? Or is it something completely different? This one has got me extremely curious, really looking forward to some of your insightful responses.


Is there a word for a decision that's already been made despite the relevant processes not having been followed / analysis done. Foregone conclusion feels similar to what I'm after; but that implies that the facts are so obvious as to make the decision process moot; whilst I'm after a word for where the conclusion is based on one person's perspective without them having consulted relevant stakeholders / investigated the various options / collected evidence to support or dispute their arguments / etc.


Assume that we have positively charged two solid conducting spheres of the same radius and material. It is evident that the surface charge is uniform on them to eliminate the internal electric field everywhere inside the spheres. However, how does the charge distribution change if we bring them close to each other? Recall that we place them close (not in contact) to each other. Does the exerted Coulomb's force upset the uniformity of the surface charges on both spheres? How can we find the new charge distribution so that the internal electric fields remain zero?


I understand it's not possible to transmit information using entangled particles due to the randomness of the measure results. But what if the act of measuring itself IS the information? Let's say there's a constant flow of entangled particles received either side by Alice and Bob Bob measures every other particle. Is it possible for Alice to know which particle has been measured on the other side, and which didn't? I'm assuming unmeasured particles with superposition can show an interference pattern and measured ones don't Sorry if this is a silly question, I'm not a physicist and I fell in the quantum rabbit hole quite recently :)


I was reading about the Pascal barrel bursting experiment, and it almost makes sense to me, save for one bit. I know that pressure corresponds to the number of collisions in the fluid. So my q is: why does adding water in the tall, narrow pipe in the experiment increase the number of collisions down in the barrel drastically? What is actually going on if we zoom in on a molecule of water in the barrel when we add all the extra water at the top? Thank you!


I am not an expert in languages and this is my first question here... To my understanding, most of the languages were spoken before writing was discovered/evolved. What would cause a language to have same/similar sounding words with different meanings when the language was spoken? Wouldn't it cause more confusion when spelling could not distinguish the different words as there was no writing? I can think of some other examples such prey/pray or ate/eight which sound similar but their meanings aren't related. Did these words evolve/change in pronunciation over time, or were they used with similar meanings even a long time back, since before writing was a thing?


For the operation of the transistor, the base-emitter junction is forward-biased and the Base collector junction is reversed-biased. Due to the concentration gradient, electrons from the N side (Emitter) go towards the P side (Base). Since the base is lightly doped and thin and due to the reverse biasing of the base-collector junction(electric field exits between the base and collector), the electrons coming from the N side (Emitter) go towards the N side (Collector) without much recombining with the hole in the P region (Base). Since we have a concentration gradient for electrons to flow from emitter to base then why do we need the base supply voltage in a transistor circuit?


I would like to know some tips or hints to find a way to recognize whether "as" is using a meaning of "when" or a meaning of "because" when "as" connects two sentences. For example, the sentence below which currently confuses me. The third point, where it is in our interests to engage directly with China in a constructive and positive way to shape China's actions and behaviors as they impact on the wider world. In the above sentence, does "as" mean "because"? addtion: this sentence was from an interview script, so I cannot know if there is komma before "as". But thanks for all comments.It seems this "as" doesn't mean "because".


Trying to work out whether to use 'is' or 'are' in the following sentences (aware they might differ): How many grams of flour is/are on the scale? How many centilitres of cough syrup are/is in the measuring cup? How many kilograms of fruit are/is on the scale? 'Are' seems most natural to me when speaking, but have read that units of measurement are usually counted as singular amongst the scientific community. Looking for UK English rules :)


I was reading a preprint on archives (not yet published), and in the introduction the authors say "Following X in [ref]..." and go on to set up the problem. When I went to [ref], it turns out the authors of the preprint actually take three paragraphs word for word from X and include them in their introduction. Is this appropriate? Just saying "Following X...." makes it okay? To me, it still looks like plagiarism.


Suppose we have a finite group G, and we had a configuration of points, lines etc. and lets say that the group acts faithfully and primitively on the set of points. Will the group be guaranteed to be flag-transitive on the entire figure or can G fix some proper subset of lines, for instance? My intuition says that shouldn't be possible, but I can't figure out how to prove it.


Electromagnetic radiation phenomena exhibit a temporal asymmetry: we observe radiation coherently diverging from a radiating source, such the light emitted by a star, but we do not observe radiation coherently converging into a source, unless we delicately set up such a system. What can explain this asymmetry? And how is the asymmetry related to the causal asymmetry, on the one hand, and the thermodynamic asymmetry, on the other?


Next year I have choose a few optative courses from a big list, but syllabi are not available yet, so I have to make a choice based on names only. I am thinking about taking one titled "Groups and representations". I have really enjoyed my Abstract Algebra courses so far, specially the part that covered Group theory. However, we never got into Representation theory, and I ignore what mathematical interest it might have. I was wondering if someone could provide me with a text explaining what is representation theory about and what is the mathematical interest or motivation behind it. I have found some books on the topic, but they all just start giving definitions and theorems and never give a good introduction.


What makes quantum computers faster in certain problems than normal computers? Does quantum computing means that many solutions are explored simultaneously instead of one at a time due to quantum superposition of qubits? Is the complexity of solutions explored proportional to the number of qubits? That's my impression and my understanding of quantum computing. Any books or articles that exactly pinpoint how superposition or any other quantum phenomenon makes computation faster?


A grammatically correct way to ask someone a question would be: Why are you still here? If I want to make a statement (instead of a true question), such as, You've been here too long or Your comments are annoying, or You're either a troll or a quarrelsome person, would "Why you are still here" (meaning- why do you continue arguing with everyone) be acceptable in a conversational English? Do you know why you're still here? is grammatically correct, but I don't want to know if he knows, I just want to express an annoyance with the person.


Has anyone (in general) experimented with simulating moon lighting? It means the following. A gray ball is illuminated with light with a brightness equal to that of the sun. And at the corresponding angular distance from it, the level of illumination by the light reflected from the ball surface is measured and visually assessed. The goal is to compare the illumination intensity of real moonlight and that simulated in the experiment and to check the reliability of the calculated values.


Cambridge says that the idiom home and dry is British English, which explains why I hear it used around me. It means: to have successfully finished something but I have heard it used also literally. Then I understand it as meaning to return home safe and sound after a travel/adventure of some sort. I wonder if this idiom has originated among sailors. I understand home but why dry? Is dry here a synonym of safe (and sound)?


I'd like to know a word to describe a person who loves celebrities, gossips, trends or anything that is popular and is easily affected by them. She said she had moved to Paris because she loved "Emily in Paris" on Netflix. It seems that she is such a " "! I came up with "sheep"(https://www.urbandictionary.com/define.php?term=sheep), but I'd appreciate it if you could tell me other words to describe such a person better or more accurately.


I'm trying to get a deeper understanding of the classification problem for finite groups and in particular the extension problem. I'm looking for some advanced books on the theory of finite group extensions which extensively and deeply covers the topic. Maybe an all-in-one book doesn't exist, in case I'd like also suggestions on books covering related and more specific topics such as Shur's multiplier and Group cohomology.


Is there any book for a Mathematics student who can learn Machine learning in the aspect of Banach space geometry? Or, one can understand the connection between Geometry of Banach spaces and Machine learning? Any suggestions in terms of article, note, book regarding the development of such study is always welcomed. It will be very much helpful if anyone can suggest a good book which contains the discussion about the topic of Geometry of banach spaces that is required for Machine learning, Neural network such things. Thank you in advance.


The professor asked me to write a small computer program that takes double integrals of rapidly varying functions. He did not mention which functions should be integrated but he said that I must use a numerical method with variable step size. I hope someone can recommend me a method that will do the job. It would be nice to have a link where this method is adequately described.


There are many ways to draw a simple flowchart, using packages like tikz. But recently I read a paper and it seems that their complex flowchart is not a picture, but actually generated by latex. Here is what it looks like: I really would like to know how to make a chart like this, if there is a minimal example. EDIT: From what I can see on this picture, it is a combination of table and flowchart, which I don't know how to. So my actual question is, how could we combine a table with a flowchart? Or maybe is it just a complex Tikz (or other packages) implementation?


If the expansion of the universe happens uniformly, how does this affect small objects? For example, are the distances between protons and neutrons inside a nucleus actually expanding? Is the nucleus constantly pulling itself together so its diameter stays constant? What about say between the carbon atoms in a diamond? If I can observe said diamond forever, will its atoms eventually disassociate with each other or will it stay in one piece due to its covalent bonds counteracting the expansion of space?


I have a problem concerning a cylinder, cut by an arbitrary plane which is perpendicular to an axis, z. I know dz, the distance from centre of the cylinder to the plane. I know all dimensions of the cylinder. I know theta, the angle of rotation from the axis z. My poor drawing How do I calculate the volume of the cylinder above the plane? Is it possible? I've tried some trigonometry to get an integral based on the changing size of a circle segment... but it's getting very convoluted.


Say I have some random variables X,Y that are jointly distributed according to some probability distribution (not necessarily independent). Is it possible to construct a Z = f(X,Y) (i.e. Z is solely dependent on both X and Y with no other source of randomness) such that X is independent of Y given Z? I believe this holds for all Z : Z = g(Y) or Z = g(x) for some deterministic function g, although I struggle to prove this formally. One straight forward example is in the binary case with Z = not Y. However, is there any formulation on how the conditional independence requirement restricts the function f?


As far as I understand, the branes of brane cosmology are lower-dimensional "sub-manifolds" of some space. It was hard to imagine for me how such structure could exist and be physical. But then I learned of topological defects, which are essentially that. So my question is: Question: Are the branes of brane cosmology supposed to be topological defects of some sort? If not, how else could a "sub-manifold" be something physical? I also posed a related question which received little attention. Please note that I have only an undergraduate understanding of physics, but are fine with advanced mathematics.


i'm taking a qft theory class, and we are doing a little bit of group theory, today the professor introduced us to Lie groups, but his definition is a little be weird he said that lie groups are groups whose elements are function of n parameters. This kinda confuses me and i can not see how this definition is equivalent to the one where a Lie Group is defined as smooth manifold with some other proprieties like the operation of the group be differentiable. Anyone can help me to clarify and get how these two definitions are equivalent? Thanks


I am struggling to answer this question, and I was hoping for some assistance and/or help, it would be greatly appreciated. This link is a screenshot of the question because it does include diagrams: https://i.stack.imgur.com/wMVod.png The question includes a definition that is used to answer the question; if anyone knows how to answer or go about this, you would be helping me out a lot.


I have some simulated data from a model of an IMU's gyroscopes. I am trying to use Allan deviation to verify that we have correctly modelled various sources of error in the sensor data. I was able to verify angular random walk, and have now moved on to bias instability. I'm following a tutorial on mathworks which describes finding a critical point in the Allan deviation curve, depicted as such: However, my sensors do not output this single, easily spotted critical point. My simulated data looks like so: Is this normal? If so, are there any techniques to choose the best critical point to get an instability reading? If not, what might cause it?


By equivalence principle, one can find a local inertial frame at every point of spacetime. Then this is usually used to introduce the general spacetime metric, as a back-transform of the Minkowski metric. The transformation from Minkowski to general metric depends only on the first derivatives of the coordinate transformation. This means that locally we can get an inertial frame by applying a linear coordinate transform. It is puzzling because in motivating examples we always get accelerating transforms, such as a falling elevator. However, it seems that the acceleration does not play role in the final formula. What is happening? Is this because linear transform is enough to diagonalize the metric, but the quadratic part (i.e., the acceleration) is needed to make the Christoffel symbols vanish?


I understand Thomson scattering as: When an EM wave is incident upon a charge causes it to oscillate in turn releasing energy as another electromagnetic wave. In an Electrodynamics lecture we took a different approach by assuming the far-field limit. The EM wave makes the charge oscillate causing it to emulate an oscillating dipole and hence releasing energy as another electromagnetic wave. I understand that these two present the same behaviour in the far field limit. However, I do not intuitively understand how the oscillating charge emulates an oscillating dipole.


i'm writing my first document using LaTex. I want to include good-quality figures, so i opted to use the PDF format. i saved the images from the plots in Matlab as pdf files, and when i included the figure in my document it showed up with extra white borders, i'm not sure how to fix that. Cropping the pdf can be done, but i have many images to include so that would be tedious. any suggestions? or something else i can do? thank you


We use Past Perfect to speak about actions before some moment in the past. What if my speaking partner doesn't know about a moment in the past? I had done everything. (I don't mention it, but in the mind I mean, I had done before some moment, which I know). What if i don't mention an moment, but it's obvious? I had written all documents. (It can be either obvious or unobvious, that I had written all documents before a deadline).


Suppose I have an infinite wire and close by I place a loop which has increasing current. The wire and the loop is fixed . Now an emf will be induced in the infinite wire which in turn induces a current to oppose the changing magnetic field . My question is why this emf is induced in the infinite wire even though the wire is stationary( in order for motional emf to be induced) and also there is no loop formed by the infinite wire? How to define the flux change in this condition for the infinite wire in order to apply faraday's law to calculate emf?


I am currently reading a book about numerical optimization and have reached line search methods. In a section studying the convergence rate of the gradient descent method, I am really struggling to understand how alfa is determined (see below). I really do not understand at all how to even differentiate with regards to alfa, when dealing with vectors and matrices, and I also am very confused as to how the b-vector can just completely disappear in the result. Can anyone explain how the derivative with respect to alfa is determined, and how alfa is isolated from there? Thank you to anyone who may be able to help. Explanation of how alfa is determined


It is common knowledge that submodules of finitely generated modules need not be finitely generated. One can see this by considering a non-Notherian ring and extending this as a module. My question is, how does one think about this intuitively? The way that I think about is: When I restrict myself to a subspace, there is a chance I no longer have access to the elements that finitely generate the whole module. In contrast with Linear Algebra, this can occur when the subspace is non-empty. Could someone let me know if this is a correct way to view this or if there is a more intuitive way of viewing this result?


Suppose I want to cover a whole circle of with hexagons. See the attached photo where coloured hexagons are those required to cover a whole circle. Given the length of the side of the hexagons and the radius of the circle, what's the minimum number of hexagons required to cover such circle? I notice that there are three ways of inscribing a circle in the hexagon tile: the center of the circle lies at the center of a hexagon, at the center of a hexagon side, and at the vertex between three hexagons. Attachment


I know that in the pure algebraic context, connected groupoids are equivalent to groups. Moreover, connected groupoids can be seen as action groupoids. Consequently, any abstract groupoid is essentially a disjoint union of action groupoids. Therefore, is it not more natural to study group actions than study groupoids? Since group actions seems to have a richer structure than groupoids. So, why study groupoids? Does this also hold in a continuous context?


In Information Technology, we often see something that looks like a compound adjective, pairing a number and a noum. Some examples are two-factor authentication, two-step verification and twelve-factor app. I always found it amusing that the noun itself is not plural. Why isn't it named, for example, "two-factors authentication?" Is there a rule for that construct? If so, what's its name? (My theory is that it probably came from multi-factor authentication, for example, in which there is some sense in the format, but I don't know for sure.)


I have the simplistic image of a photon being a disturbance in the photon field caused by the loss of energy of an electron. In this image, I see that the disturbance is essentially a harmonic oscillation and it 'traces out a wave' as it propagates through the field. Also, given this image, it seems that the 'larger' the amplitude of this disturbance, the longer the wavelength and thus the 'lesser' the energy of the photon. This seems counterintuitive. Alternatively, is the 'amplitude' of the disturbance the same for all photons?


While writing a fantasy narration I created this sentence: High among the clouds a castle floated. My American friend tells me it sounds bad to a native ear. I think he sees a problem with the unusual word order. At the same time the Hobbit starts with: In a hole in the ground there lived a hobbit. ...which seems a similar sentence. What is the science behind the word order in an English sentence? UPDATE: The original sentence was actually High among the clouds a castle floated, disregarding the force of gravity. I removed the ending phrase to simplify the problem, but from some of the responses I infer that the phrase may actually matter.


I was just studying statics when I realized that a body can be in both static or dynamic equilibrium at the same time but I am not so sure. My textbook says that an object at rest is in static equilibrium while an object moving with zero acceleration i.e constant velocity is in dynamic equilibrium But.... Motion is relative so if two objects were moving with the same uniform velocity, they would be at rest relative to one another and therefore be at static equilibrium but since they are in motion relative to someone outside the system so they should be in dynamic equilibrium. Can anyone confirm this?


A noon cannon usually has a single convex lens which focuses sunlight onto a fuse. The cannon contains gunpowder. The fuse ignites and the gun discharges at noon. https://en.wikipedia.org/wiki/Sundial_cannon A query has arisen about a noon cannon with TWO lenses -- what is the benefit of having two lenses? Does it provide more heat to ignite the fuse? The noon cannon with two lenses might be larger than usual, therefore needing a lens with a longer focal length. Images below. Thank you for any suggestions. I know quite a lot about sundials but I am not a physicist.


I'm using CircuiTikZ and I came across this issue frequently: every time I want to tweak some options using tikzset, there is no documentation for the available key-value pairs. For example, if I want to change the voltage direction, in the documentation, only value (i.e., RPvoltages) is provided. However, I prefer to use key-value pair in my code for future reference so I was wondering how can I find the respective key name given value name or the other way around.


Sorry if this sounds like a silly question, but what would happen if a scientist observes Schrodinger's cat alive, but is then thrown into a black hole before he has leaked any information to the environment. Then later a second scientist observes the cat - can he observe it as dead this time, since the first scientist and his information can't leave the black hole to contradict? That is, is a wavefunction allowed to show two different faces to two observers so long as they are never allowed to compare notes because of an event horizon? Is exiling information behind an event horizon the same as erasing it?


So bound states have continuous position representation, yet countable discrete energy levels. Doesn't this imply that one of the following operations is invalid: <x|> or <E|>. Because |> cannot simultaneously have countable and uncountable number of elements. And the number of elements in the bra and the ket need to match otherwise the operation doesn't make sense. The ket |> exists independently of whatever representation we choose. So what's going on here?


There are a couple of sentences: The water froze and caused the pipe to burst. If it is a succession of past events it should be Past Simple. But if the first part happened earlier, it should be : The water had frozen and caused the pipe to burst.So, can it be Past Perfect, or this sentence must be written in Past Simple? Year by year the population of the West increased. Should it be increased (Past Simple) or was increasing(Past Continuous) ? Because, I guess, Past Simple does not belong here . I don't know which tense to choose. There is no extra context, just sentences on their own.


Suppose a piston enclosing an amount of air-fuel mixture and then this piston compresses the mixture to the point of auto ignition, if every part of the mixture ignite simultaneously is it safe to say that there isn't a flame front because it can't move because there is no longer a combustible mixture? If there is a flame front are they multiple flame fronts? Also if my reasoning is correct does that type of combustion have a name?


In Arabic, the idiom "You made my neck as small as a sesame seed" is used to mean that someone related to you or a friend, put you in an embarassing situation with others. This idiom is usually used by parents when their children make them anything but proud. People who have long necks are assumed to be proud; hence, 'a neck as small as a sesame seed' in this idiom. Is there an idiom anywhere close to this meaning in English?


I am currently reading electrodynamics from Feynman. When talking about the energy of the electromagnetic fields, he says that the location of the field energy could be known at least theoretically since all energy is a source of gravitational attraction, and if we could measure the gravitational attraction we could comment on the location of the energy in the fields. I have come across formulations where gravitational field is associated with an energy density and is said to be always negative. I wanted to know whether this energy can have an associated gravitational field.


Suppose that, in a homogeneous linear system, the coefficients of one of the unknowns are all equal to zero. Show that the system has a nontrivial solution. Attempt: Note that the rank must be less than or equal to the number of columns in the coefficient matrix. If we had all columns of the augmented matrix different from zero, then, after being reduced to row-echelon form, we would have at most rank = number of columns. As we have a null column, then surely the rank is less than the number of columns. Therefore, we will have at least one free variable. Am I right? Furthermore, is there any way the solution could be more formal than that? (using the same approach)


What comes after "pair"? Maybe "quad" for four? In cards it's "three/four of a kind" but what about contexts beyond cards (like science)? For example, in physics there's a "pair-flip model," but what is the extension of this model to flipping three things? I've also tried the internet, see also: https://en.wikipedia.org/wiki/Tuple Not to be confused with: single, duo, trio, quartet, quintet, ... monad, dyad, triad, tetrad, ... single, couple, triple, quadruple, ... single, twin, triplet, quadruplet, ... Or does "pair" secretly belong in one or more of the categories above? Or is it just an isolated word for, e.g., trousers? Thanks for helping me solve this mystery!


There's a word people around me used to use a lot, phonetically it would spell 'stercus' or 'stircus', I think. It was used in a context meaning excitable, frenzied, frenetic, that kind of thing. Like, you could say that: Kids who eat lots of sugar tend to go stercus. I don't know if it's an actual word and I can't find it because I'm butchering the spelling or misremembering how it was said. I'm a native speaker from Australia, if that helps.


In the double slit experiment, an interaction with a detecting mechanism is needed to know which slit was the one the photon or electron passed through. I read in other questions that polarized filters are placed in the slits, how is it exactly that this helps us know which slit the photon passed through? Do experiments using electrons also use polarized filters? If not, what do they use and how does it work? Are there other types of detecting devices? Is there a detecting device that truly does not change any properties of the photon or electron when interacting with it?


I am using sagemath to compute Einstein tensors of a non-standard spacetime. The output is something horrid and non-diagonal. I need to find the Eigenvalues of this tensor... which is represented as a symbolic matrix. This is proving to be non-trivial with the sagemath documentation. Is there a way to convert my tensor output into a matrix that sagemath can find the eigenvalues for? If a worked example is required, please let me know as I can provide an update easily. Thanks in advance...


uh I'm curious again regarding the theories of reflection. This is known that Metals contain free electrons that absorb energy and vibrate when they come in contact with light. Later, they release the energy back in the form of light. But what if we polish a transparent conductive oxide glass (transparent glass which is electrically conductive). It doesn't seem to be acting like a mirror. So is it really just the electrons which decide the reflection? Do correct me if I'm wrong. I'm just a stupid kid :) My previous post


I know for abelian group, there is a Kronecker decomposition theorem. It said any finite abelian group can be factored as direct sum of cyclic group of prime power order. I want to know is there anything similar with this for the abelian semigroup with identity? As for semigroup, we don't need the inversion, so it seems like it can also support some kind f factorization.


When a spinning top slowly advances and hit a surface (a wall), intuitively one would expect that the top gets bounced mostly along the wall, due to the friction between the top and the wall. But the observed phenomenon is that the top gets bounced mostly in the norm direction of the wall, and in a much higher velocity than when it hits the wall.


Looking for alternative to "opened a can of worms" in the following sentence... It needs to be graphical enough for a young readership to grasp, a readership that won't have heard of Pandora's box or even know what a can of worms is. Having fixed one friend's laptop I wondered if I may have opened a can of worms and now all my friends will expect the same service. Any suggestions?


I have read a book saying that "... backward induction cannot be applied to games of imperfect or incomplete information because this entails cutting through non-singleton information sets." However, in another textbook, it says that backward induction can be applied to games of imperfect information, and we can derive the equilibrium by starting at a historical point and show the subgame perfect equilibrium in that subgame is a equilibrium in the Nash equilibrium. I wonder what are the contexts to use backward induction in a game with imperfect information but complete information?


From what I know, current flows across two points only if there is some potential difference across those two points. If that is the case, why does the current flow through a wire then? See the picture for clarity: As one can see, the potential is same on the left side of the wire. The current should not flow, yet it still does. Why?(I have the same question if the resistor was replaced by a capacitor.)