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Suppose I have a principal bundle, and some group G acting on the principal bundle. Is it always possible to find a G-invariant connection on the principle bundle? If G is compact, then I can imagine one can find such a connection by picking any connection to start with and then 'averaging' this connection over the gro...
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I recently asked this question about whether or not profinite groups admit maximal subgroups: And indeed, profinite groups admit subgroups of finite index, so taking any minimum index subgroup containing the finite index subgroup gives a maximal subgroup of the profinite group. Now I wonder if it's true that all subgro...
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(There are a lot of questions about entanglement shown in the Similar Questions box. If there's already something about what I'm about to ask, please refer me there.) Suppose the particle paths are arranged to be of different length (e.g., detector A is close to the emitter but detector B is many miles away). And suppo...
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I learned that the possessive apostrophe -s is used for living beings and the "of" form for non living things: The dog's name is Sherlock The color of the flower is yellow However, I have noticed that these rules are often ignored in spoken language: The road's name is Privet Drive The name of the dog is Sherlock Are t...
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I'm wondering if 'calls have grown for X to be Y-ed' is an acceptable substitute for 'there have been growing calls for X to be Y-ed'? To clarify, these are examples of the latter from the first page of Google search results: 'There have been growing calls by some eco-activists and developing countries for powerhouse s...
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I know that loop-level three-photon vertex in QED is zero since the contribution from fermion and antifermion cancel each other. Also, from what I know this has something to do with gauge invariance of the QED. However, my question is "is this true even when the gauge invariance is broken?". If the gauge invariance is ...
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I need a verb that describes asking someone to stop helping you, even though you know they mean well, because it's not helpful. So you ask them in a way that is almost pleading. It's stronger than just asking, but it's not quite pleading. I'm doing an analysis of a play, and one aspect of this analysis is that I have t...
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When a capacitor is charged, electrons flow from the battery terminal to the plate of the capacitor, as the electrons flow through the wire, they must be having some kinetic energy? So, when the electrons finally stop at the plates of the capacitor, what happens to their kinetic energy? We don't normally assume the kin...
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In the case of elastic scattering or absorption between photons and electrons, the incident light is either fully reflected or fully absorbed and hence it either retains its wavelength or ceases to exist. But how does inelastic scattering between them work? Specifically, if light has a wavelength, how exactly does that...
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Let A and B be solenoids with the same pole facing each other. Then will their repulsion cause the set up to jump upwards? Please correct me if I am wrong: Solenoid A exerts a downward force on Solenoid B, which in turn exerts the same force on the ground. Then due to newton's third law, the ground exerts the same forc...
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I am trying to find the minimum energy point of a catenary curve. Can anyone please let me know whether there is a potential energy or energy formula for a catenary curve ? Especially I would like to know when the end points of a catenary changes it's position, I would like to know how the energy of a particular point ...
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I am confused in which category to put this sentence. It's a polite request, so it looks like it's an example of Imperative Sentence. But it does express the desire to have mango (even though not directly with the use of "I wish" or "I desire"), so feels like it may be Optative Sentence as well. And the listener of thi...
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We know that thermal radiation is one of the ways energy can be transferred from one object to another. And we know, objects at any temperature emit electromagnetic radiation. So does that mean molecules and atoms are constantly losing energy? And why isn't this considered in the first law of thermodynamics? How can we...
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I searched a lot on the internet but I couldn't find any helpful answers, so here I am. I'm trying to create a chessboard figure, using xskak package that contains a label for each square. An example of what I'm trying to achieve is shown in the xskak tutorial on overleaf, linked here. I'll also add a screenshot of the...
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I've heard three seemingly contradictory facts, so apparently I'm not understanding at least one of them correctly. "Proper time along a path is the time elapsed for a clock that travels along that path." "Proper time along a null path is zero." "There are no preferred inertial reference frames, so if I travel at the s...
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A jack of all trades is a master of none, but oftentimes better than a master of one is apparently accredited to William Shakespeare. Just to clarify - I mean the FULL quote, not just 'Jack of all trades' part. Is there a book, poem, or work by Shakespeare that proves he is the author? I have searched through the compl...
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Monoids represent maps from some mathematical object to itself. Groups represent the automorphisms of some mathematical object. What do abelian groups represent? One unsatisfying answer would be that they just represent automorphisms of nice enough mathematical objects. However it seems rather uncommon for the automorp...
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Consider the following: "Saranghe" not only means "I love you," but it also means "Let's be together till we die." In Telugu, "Intiki veltunnava?" means "Are you going home?" What I'm wondering is whether I should capitalize the i in the transliteration of "intiki veltunnava? In other words, which one should it be? In ...
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The Novikov self-consistency principle prevents a paradox in which a billiard ball is sent to its past through a closed timelike curve (allowed under general relativity) such that it collides with its past self and prevents it from entering. Imagine a setup where the ball, as soon as it exits in the past, hits a switch...
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Momentum is the defined as the product of mass and velocity and can be thought as measuring how much motion something has. However, it is not clear to me why we need momentum and why force is not sufficient to explain nearly all the phenomena. Consider a larger mass moving at the same speed as an object of smaller mass...
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I got a mouse which has wireless charging capabilities a while back. I would like to take advantage of this. However, I currently have an extremely nice mousemat from grovemade I grabbed in a sale a while back https://grovemade.com/mouse-pads/ which I would like to keep using. My idea is simply place a wireless chargin...
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This post, currently closed, asks for the intuition behind Liouville's theorem, which states that every bounded entire function is constant; I find the answers unsatisfactory and hand-wavy. I am looking for a more focused, precise description of what is going on. Is there a geometrical understanding of what is going on...
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There has been some recent buzz around generalized/higher form/categorical(?) symmetries in the physics community. I understand Seiberg's papers are a popular resource, and I am aware of McGreevy's review on the subject in the context of condensed matter. Nonetheless, what are some other resources I might look at if I ...
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I'm reading Billingsley's Probability and measure. I'm reading about extremal distributions, and I'm at where he is proving that Gumbel is one of the extremal distributions. I'm having a hard time understanding the proof and I'll appreciate it if someone helps me figure out the proof. The text highlighted in yellow is ...
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I'm new to optimization and I'm reading a paper that they reformulate an ILP problem into QUBO using penalty method. Let's say we use a solver to solve above QUBO, and the global optima of QUBO obtained from the solver is infeasible to the original ILP. (infeasible from my understanding is the QUBO solution does not sa...
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If someone sets up a table at the edge of a conference room selling their wares, it could be described as a "stall" or maybe a "kiosk". But what if someone instead sets up something like a few chess boards that people can play on? The word "stall" doesn't really seem appropriate to describe this ("a chess stall"?) but ...
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When using "too much", it is put after the verb & before the uncountable noun e.g. I drink too much coffee. However, this sounds strange in the sentence: I use too much social media. Instead I use social media too much sounds more natural. Can anyone explain why this one doesn't fit the rule? So far it seems that it on...
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The Gauss-Bonnet theorem for compact regular surfaces is often only enunciated for totally regular surfaces, with no singular points. But what if I wanted to state it on a surface with a finite number of singular points but otherwise regular, like a cube? How is the Gauss-Bonnet modified? I know somehow the curvature h...
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Assuming "phreak" is a portmanteau of "phone" and "freak," giving us the term "phreaking" to describe the illicit act of hacking telephone networks, and "Phishing" pays homage to it by misspelling "fishing" to describe the practice of tricking someone into revealing sensitive information on computer networks, now we ha...
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In one of his lectures, L Susskind stated that he cannot make sense of a metric with more than one timelike dimension. I also have trouble imagining it, but is there a good mathematical or physical reason why this is not possible? Let us assume all extra timelike dimensions are compactified, so we cannot directly obser...
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I have seen several questions regarding the size of the absolute smallest black hole, the smallest stable black hole and similar. These made me wonder; what is the smallest stable black hole if it is immersed in a standard atmosphere? I assume there is some maximum rate at which the black hole can consume matter, and t...
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Imagine, there is a truck of mass 'm' accelerating with an acceleration 'a', this means that the truck definately carries some force 'F'(let). Now, imagine the truck collides with a man standing stationary on the middle of the road. So, the truck will hit the man with force 'F' and due to Newton's third law the man als...
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We know that nothing including light can escape the gravitational pull of black hole. Now special relativity says that nothing travels faster than speed of light. Then how can effects of gravity due to the black hole be felt outside the black hole as nothing can escape black hole? This seems contradictory as anything i...
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I've seen that the Leibniz formula of determinants can be reduced to the Laplace expansion through consolidating terms. What I haven't been able to find online is a connection between any of those and the geometric interpretation of determinants, which is the volume (or measure) contained between the parallelopiped for...
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I have just learned that linear frame dragging exists in General Relativity. I have also seen simulations where a periodically accelerated and decelerated mass causes a sort of gravitational dipole radiation, dragging nearby masses in the direction of the acceleration. My problem with this is that in my whole life I wa...
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I am trying to understand the jump operators in the Lindblad equation. Specifically, if there is any condition of boundedness we need to impose on them. I ask this because, as was pointed out in this post, there should be no assumption on the norm of the jump operators. But for example in this review, the jump operator...
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I think my formulation in the title may be a bit confusing. I mean a behaviour similar to "gaslighting" yourself into seeing/perceiving your own opinion from someone/something, even though that person/thing doesn't actually share your opinion. I've witnessed this a few times on social media when people interpreted sati...
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In textbooks on Category Theory, monoids pop-up all over the place, we several easy examples such as integers, lists and so on. I was then wondering about comonoids. What are some "simple" examples of comonoids appearing in mathematics? I've tried searching, but most texts either only mentions the concept of a comonoid...
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A few days ago was the first time I ran into the word "Nigerien" meant to mean "something related to the African country of Niger" as opposed to "Nigerian" meaning "something related to the African country of Nigeria". Since this is very easily confused, for example by my spelling checker which didn't like "Nigerien" a...
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If there is a circular metal loop and a changing magnetic field perpendicular to the loop, there is an induced electric field that causes the electrons to move and cause a current. But my question is why is there even a current? Why don't the electrons just move and rearrange themselves to cancel out the induced electr...
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I am looking to replace the idiom "kangaroo court" in the following sentence: Class followed its usual script. The professor took center stage, exposing the deep racism, sexism, colonialism and homophobia of a previous generation and like well-rehearsed actors, us students assumed our roles as moral arbiters in a kanga...
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We know that a capacitor and an inductor both offer some opposition to the flow of current through an AC circuit and their reactance is a measure of the amount of Ohmic opposition due to these circuit components. I am aware of how these quantities arise mathematically. However, my question is, physically why do these c...
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Suppose you have a faucet that expels water at a rate r Liters/second. Will the rate at which water flows through some ring beneath the faucet be greater than r or equal to r? On one hand, if the rate at which water flows through the ring were greater than the flow rate of the faucet, wouldn't that violate conservation...
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I hope my question does not break the rules. I can't understand how this statement can be true in the book which I've mentioned: I cannot undestand why it's true because if I have two point charges with same sign and same magnitude then the electric field should be zero at point P: But according to gauss's law it's not...
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I am doing a lab experiment (Franck-Hertz) at uni and am supposed to find the temperature, at which electrons are emitted from a barium oxide cathode (without the necessity of an external E-field that provides additional energy). I am attempting to calculate the temperature T = W/kB from the work function W (i.e. the i...
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I am interested to see a derivation of Kepler's laws from Newtonian mechanics, which would be a suitable exercise or demonstration for a first-semester course in calculus. Therefore I'd like to avoid any use of the cross-product since this is usually only introduced much later in a calculus curriculum. I've looked at N...
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Given a container filled with identical objects, how could I find the number of objects in the container accurately purely by analyzing the sound produced on shaking? More specifically, which acoustic parameter should I focus on? Some ideas I have had so far are: Looking at the number of amplitude "spikes" since the mo...
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I have just switched to writing in LaTeX from markdown. I like to proofread by hand, and so in the past I used a template that increased the line spacing and changed the typeface to something monospaced just for proofreading. Now I cannot figure out how to change an entire document to being monospaced. I found several ...
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Circle k and a line p that intersect at points R and Q are given. Inside the circle point A is given. Construct all the circles that touch the line p and the circle k and pass through the point A. I tried to solve this problem using homothety, but it didn't work out so I guess that's not the proper way to solve it. I w...
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In the double-slit experiment, the interference pattern shows the probability of finding the particle (photon or electron) at a given volume. In a given experimental setting, this probability distribution doesn't change with time, so this probability distribution belongs to a stationary quantum state. I think that this...
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Phonons are typically used to describe quantised vibrations in solids. However, is it legitimate to talk about phonons for e.g. a sound wave propagating in air? Contrary to photons that are particles that can exist both in free space as well as inside an optical cavity, it seems like phonons are quasiparticles defined ...
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I've heard that because light takes time to travel from one place to another, we see objects in distant galaxies as they were when they released the light. new and advanced telescopes are able to see further and further into space, and therefore closer back in time to when the big bang happened. I've also heard that th...
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what would happen if I took a toroidal core, generated a static toroidal magnetic field in the core and then rotated the core around it's symmetry axis within a larger stationary toroidal winding around the same core? Just for simplicity imagine a toroidal transformer with a rotating core inside. Would I get DC current...
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I haven't found satisfying answer yet. Beside those typical examples like how many colourings of a cube is there up to rotations or how many graphs are there on n vertices up to isomorphism etc., is there any "real" usage of Burnside's lemma in advanced math? All examples I could find seem to me more like funny riddles...
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If I have n cards and n stacks of cards, how many ways can I split the cards between the stacks if the order of the cards in the stack is significant but the order of the stacks is insignificant and stacks are allowed to be empty. And is there an algorithm to find them. So for example, with two cards (a, b) and two sta...
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Suppose I filled a bucket of water with a hose, now when it's filled, I remove the hose from the tap's end, while the other end is still submerged in water. Now if I throw the other end on the ground, it sucks all the water from inside the bucket spilling it out the ground till it's empty. Why does this happen? I assum...
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I am trying to first construct a fractional linear transformation, to map the sector to a half or full circle. From there it seems straightforward using rotations, translations, and standard mappings between circles to half-planes to create a map that takes the sector to the right-half plane. My question is, how do I c...
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Good day all, I am looking for a way to estimate the composition of a thermal plasma formed by flowing gas between two charged electrodes. I understand that when plasma is formed this way, it would be composed of a number of species. Please is anyone able to point me in the right direction of finding an equation/correl...
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I remember my high school teacher explicitly saying that it took me a long time to do x was the correct form and I took a long time to do x was not. In recent years, I seem to remember hearing the latter quite often in common language, though. How applicable is the latter phrase compared to the former (a) in everyday s...
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I was discussing with some friends about English grammar, and we ended up confused about the accuracy of the sentences below "I know the motive for your rescheduling the class" "I know the motive for your rescheduling of the class" "I know the motive for you rescheduling the class" "I know your motive for rescheduling ...
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When I was young, a workmate convinced me that there was indeed a word which described the behaviour of someone who would congratulate themselves for behaving the way a person normally ought to, anyway. I don't want to write that word here, because some people might think part of it seems racist. So instead, I'll ask a...
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Imagine a bicycle wheel level with the ground able to rotate around its center. Centered on one of the wheel "spokes" is a heavy marble which can slide from the center of the wheel to rim of the wheel and vice-versa. If the wheel is spun once very hard by hand, the marble will move from the center to the rim and return...
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This is a screenshot from a Khan Academy video on friction at the nanoscale. I understand how the force of friction on the purple block acts opposite to the direction of the force applied explained by the nanoscale diagram, but I don't understand how the Newton third law pair is explained by the diagram. Wouldn't the m...
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The motivation I was given for algebraic topology is to assign some algebraic objects as invariants to topological spaces. This way we can show that two spaces are not homeomorphic if they are assigned a different invariant. However it seems to me like in algebraic topology the invariants are always up to homotopy equi...
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In the book General Relativity by M.P Hoson, the author tries to calculate the velocity of a particle radially infalling in Schwarzschild spacetime. The velocity is measured by the observer sitting at rest at infinity. For the same, he takes the ratio of proper distance and proper time of the particle, both measured by...
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Provided everything remains constant, does the fringe separation, that is the distance between adjacent fringes become further apart for higher order maxima? Consider the above diagram. From the diagram it appears that the distance between adjacent maxima is getting larger with distance from the central maxima. My intu...
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The below question was asked in national English exam in Turkey: Choose the best word or expression to fill the spaces in the passage. As Henry Hill, the actor Ray Liotta gives a complex portrayal of an Irish-Italian kid who succeeds in his youthful ambition ....... popularity as a wealthy gangster, only to have his po...
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I'm studying X-Ray reflectivity at interfaces and the book I'm reading reads: ... The ideally flat, but graded interface, and the ideally sharp, but roughened interface, will be considered... I'm struggling to understand the difference between the two kinds of interfaces mentioned. Please consider that English is not m...
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Imagine a system with a parallel plate capacitor with holes in the middle where the plates are charged before being disconnected and insulated. Neutral atoms are introduced between the plates and are subsequently ionized in some way. The created ions will gain momentum and travel through the hole after which their kine...
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Assuming we can't block neutrinos, and most of them pass through the earth, how do we know that the change in the neutrino detector is not just happened randomly? Is the detection of neutrinos more frequent when earth is closer to the sun, or are we just always seeing the same amount of neutrinos in the detectors becau...
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I'm doing mathematical analysis and I'm stuck with the following question: Give an example of a nowhere dense closed set of irrational numbers with no isolated points (that is, a perfect set of irrational numbers) I read this: Is this a perfect set? so apparently we can find such a set through specific enumeration of r...
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I came across this interview question online: Suppose you sit on the road side and observe cars driving by. Assume the distribution of cars driving by is according to an exponential distribution. Now you observe a first car after sitting for x mins, a second car after sitting for y min. Given those observations, can yo...
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I begin a PhD in coding theory next fall, and it concerns coding over certain types of rings (e.g. finite chain rings) instead of fields, and in consequence submodule codes. I have a fair amount of knowledge in comm. algebra and have had a standard undergraduate coding theory over finite fields course, so I was wonderi...
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If I understand correctly, massive objects cause time dilation, and so time seems to pass more slowly for observers closer to a massive object than those who are farther away. Do cosmologists take the local gravitational effects of earth, the sun, and the Milky Way into account when calculating the expected speeds of s...
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I am an undergraduate, studying physics. I have studied maths courses like Groups, Linear Algebra, Real analysis, Differential geometry and probability. I wish to get into mathematical physics, similar to what Arnold's book has to teach. But i find it a really difficult read since it's a graduate text in mathematics. C...
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Now I am studying elementary number theory, I am interested in arithmetic function, I have studied Burton's Number Theory but I can't find Dirichlet Convolution as a particular topic, I will be highly glad if anyone suggest some good book to study arithmetic function where I can find Dirichlet Convolution in details wi...
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I am trying to understand the concept of regular conditional distribution. I found the definition quit difficult to grasp . This is what i understood . A regular conditional distribution also called a probability kernel is a two variable function (x,A) -> K(x,A) x a scalar and A a set that verifies the following. when ...
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The uncertainty principle holds for pairs of certain observables, such as position and momentum. All these observables have a relation to spacetime. Other particle properties, by contrast, such as mass or electric charge can be measured at arbitrary precision. Quantum theories do not even model them as observables, but...
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I am currently using a software called "SCOUT" to thin film interference simulation, and I would like to know if there is a better software to do this type of simulations. My goal is to vary the thickness of a thin film in a multilayer, in order to create interference between the various thin films, and then create a s...
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I've got a Literate Agda document, which contains a large number of unicode characters. It builds nicely with XeLaTeX and life is good. The problem is that arxiv only supports pdflatex, and I'd like to upload the file to arxiv. Are there any tools/scripts that can process a latex document written for xelatex and add th...
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Hi, I'm taking a course of Electrical and Computer Engineering in my university. I don't understand why the partial circuit on the right side of the switch is ignored after the switch is closed. As you can see, the book says the network becomes (c) from (a) via the short circuit. What happens to the short circuit and t...
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I have two non-orthonormal basis vectors, and I want to represent a third vector as a pair of coordinates using the aformentioned basis vectors. How would I do that? The dot product, which usually transforms a vector into coordinates in each basis vector, doesn't work for non-orthonormal bases. How would I get coordina...
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If Alice is located just beyond the event horizon of a black hole, and Bob is at some other point in space with de minimis gravitiational effect, what, if any, effect would it have on the "spooky action at a distance" between these two entangled entities? I'm a wannabe sci-fi writer looking to exploit untested but poss...
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I am struggling with prooving convergence for an optimizer which uses adaptive step-size with heavy ball algorithm for convex and non-convex functions. In some literature, I could find a regret bound analysis/proof for convex functions and proving that the estimated gradient at t -> inf goes to zero. Could anyone pleas...
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I need help understanding how particles do what they do and maintain the structures they maintain if so many of them exist for such a short time? In the case of the nucleus and pions, pions only exist for mere nanoseconds. So if they decay, and their properties change, what is holding that nucleus together? Are the nuc...
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I really like the greek letters of the default LaTeX font but I want the rest of the characters to be in Times New Roman using the Mathptmx font. I find the greek letters of the mathptmx package rather ugly since they're disproportionately bolder than the usual text. So may I know how can I change the greek characters ...
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When observing interference fringes one often uses a lens after the diffraction grating to focus the fringes on a screen like this: At first sight, one might think that after the lens the path difference between the rays is altered. I think this is not the case, but don't see a clear argument why. Maybe one could apply...
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I had a debate with a buddy about this. He said you could get a chord by drawing the triangle formed by the two points and the center of the sphere and that chord corresponds to a single great circle arc. I can get to the chord but then projecting that chord on to a path on the surface of the sphere seems non obvious. ...
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As the diagrams of categories can be represented as graphs of objects and morphisms, I was wondering if (the diagrams of) higher-dimensional categories could be represented as hypergraphs, and if not why? And in the more general sense, what would be the best west to represent higher-dimensional categories (thinking of ...
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We know a way of measuring energy of a electromagnetic wave is the Poynting vector, which is independent of the frequency. But let's say we want to make two different electromagnetic waves, with different wavelengths and so different frequency, but with the same amplitude (so same Poynting vector). In my opinion the gu...
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Two people, A and B, starts from two different points and move in a perfectly straight line in an infinite plane. When they move they leave a visible trace after them. Question: What is the probability that their path (of traces) will intersect at some point regardless of where they start? What I've tried so far is to ...
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I don't see why or how this could be argued against or if it really matters, but that's why I ask the forum. I am just trying to understand the nature of the fields in the universe. Am I correct in thinking that the entire universe is just a few fundamental fields laid over each other, getting influenced by the differe...
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In a nuclear reaction, protons and neutrons are exchanged and the resulting elements have a number of different protons and neutrons that uniquely define the element that we are studying. However, can we say that a hydrogen atom without any electrons (what we would usually call an ionised atom of hydrogen) is really hy...
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My textbook says: Similarly, non-polar (i.e., covalent or organic) compounds like naphthalene, anthracene etc. are soluble in non-polar (i.e., covalent or organic) solvents like benzene, ether, carbon tetrachloride, etc. and are very little soluble in water. My question is that if we have a non polar compound, then it ...
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What is a word to describe how a cat looks when it's groomed? My husband described it as nappy, which I don't think would be the correct word. To give an image of what I am trying to describe, think of a medium-haired cat that licked its fur and looks messy, with wet fur sticking up a bit. It's a weird request and may ...
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I just read a quote in which someone, on behalf of herself and her spouse, wrote "Our heart goes out for ....". I Googled to know if when it comes to speaking about heart not as a body organ, is it possible to use it in the singular form, but I could not find any reliable source for that. So, what is the correct form f...
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If I shred a book into numerous pieces, and scatter them, so that the original text is completely unreadable, then for me, as a reader, the book has lost all its information. Its entropy has greatly increased. As a reader, the book's information rested upon a state of minimal entropy. However, my (limited) understandin...
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In the image above, we have the principle of induction cooking. An alternating current is run through the coil, which causes a change in flux. This change in flux induces eddy currents in the conductive pan, and by Joule heating/resistive heating (P=VI), this causes the pan to heat up. So the only thing that you need i...
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Often charge is defined as a measure of interaction with the electromagnetic field. However, electromagnetic field is often defined as a field produced by moving electrical charges, or relates to charges in some way or the other. Due to this circular definition, I am not able to understand what really is charge and its...
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There is a word in German, ausgezeichnet which vaguely sounds like the English phrase "out of sight" but that is usually translated as "excellent". I could see some non-German speaker hearing it and, knowing that it meant "very good" from context, deciding to use it and approximating its sound with "out of sight". Note...
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