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I see that the Oxford dictionary has a plural dice for "a small cube with a different number of spots on each of its sides, used in games of chance". However, there is no plural listed for "a block with a special shape that is used for shaping pieces of metal" (both definitions summarized). Researchgate has a photograp...
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For a new book I am writing with two friends, I like to include three biographies (mine as well as the other two friends) after the preface. Ideally, we like to keep three photos (A.jpg, B.jpg and C.jpg) on the left and our biographies on the right.. The book is in single column format, and I don't know how this biogra...
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I have a pretty strict requirement for a document (I'm currently using memoir), which must: use biber/biblatex in the usual way based on a primary bib file (e.g., main.bib), and sort the resulting list of refs by author then year. at some other point in the text, I want to print out a list of all the refs of type artic...
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This particular example comes from a peer-reviewed publication with authors who seem to be native speakers: This trend is both popular and has presented a variety of challenges I wonder if this is grammatical? If this sentence didn't have "both", we would parse it as follows: This trend ((is popular) and (has presented...
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How or where can I find the Absorption cross-section of Gaseous Ions? I have looked over the internet and can only find it for neutral atoms. Does it increase or decrease for ions? I have found formulas but they include the imaginary part of the refractive index of the specific medium and I can't find how to calculate ...
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Let ABCD be a convex quadrilateral with AD = BC. Show that AD and BC determine congruent angles with the line passing through the midpoints of sides AB and CD... MY IDEAS MY DRAWING As you can see i noted some points. Okey, so, i thought of similar or congruent triangles. But i don't know where to start. Hope one of yo...
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There is literature on the lie symmetries of quantum harmonic oscillator differential equation. The generators satisfy certain lie algebra. On the other hand, we have ladder operator method. The relevant operators satisfy some lie algebra. I was interested in knowing if there might be some connection with the two appro...
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In the Einstein field equations, the only tensor that shows up is the Ricci tensor and the metric tensor, together with the Ricci scalar. The Weyl tensor though is a tensor that is a part of the Riemann curvature tensor as well (the off-diagonal part). It represents tidal forces, which are non-local. Why doesn"t the We...
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I had recently come across "Russian for the Mathematician" by S.H. Gould, and saw it as a wonderfully useful resource. I have very little experience in Russian (only skill is ability to read Cyrillic), but I saw it as a useful introduction to the language, in the sense that I would be able to decipher Russian language ...
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I hear this kind of false implications pretty often, e.g.: If you're smart you'd agree with me People who understand the situation would all agree that ... Anyone who says something else must be trying to confused us These often come up in social deduction board games, where the good team is trying to gain information ...
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Is it possible to use magnets to lessen the force of a pull that is associated with Cervical Dystonia (spasmodic torticollis) which is causing an involuntary pull of the head over one shoulder. Having the magnets attached to clothing on one side such as a shoulder pad and more on a collar. Would the strength of magnets...
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Imagine a tank that accelerates towards a relativistic speed. As the caterpillars are at rest with respect to the ground, the caterpillars will show no length contraction there. But the tank body above it will. The upper part of the caterpillars, on the other hand, will show more contraction than the tank body does. Wh...
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This question may have been asked a bunch of times, but I have not found exactly what I was looking for. I say calculus, but mathematical/real analysis works as well as long as it does not have calculus as a prequisite. I have been looking for books that take an inquiry based approach, where the theory and everything e...
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According to MathWorld, the probability distribution of the length of lines between randomly selected pairs of points inside a cube has a very complicated form. In particular it is not uniform. This distribution relies on the fact that the pairs are chosen uniformly and independently. My question is - is there a way to...
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If I use a transformer the get a high voltage, the power of the transformer remains the same, as the current is reduced proportionally. However, according to ohms law, the current through a resistor connected to that secondary side of the transformer will be the voltage divided by the impedance of the resistor. But tha...
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Yesterday I heard this: suppose you have a rubber hose, in which you inject helium, which replaces the air which was there before. Now, because there is no helium in air, so the partial pressure is zero there, the helium escapes through the rubber into the surrounding air and creates vacuum in the hose (insofar it is p...
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I have found a sentence that goes like this: "Serenity may be the name of the company, but it doesn't describe the company." Someone translated this sentence to my language as if "may be" meant there was a possibility that Serenity was the name of the company (and a possibility that it wasn't), but I feel like the sent...
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[A partial groupoid (half-magma) is a set S equipped with a (single-valued) partial binary operation, as in Bruck's Survey of Binary Systems.] This question may be nonsensical, given that the duality between Cayley graphs and groups is deeply related to the mutual complete-ness of each structure. If that is the case, p...
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There is a medical specialty called obstetrics and gynecology. In the UK I have only heard this truncated to something like the first syllables separated by a truncated "and". This could perhaps be written as "obs 'n gy-nee". I have multiple times recently heard this specialty referred to by Americans by spelling out p...
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While solving inverse function problems, I got confused in a part, like for any Inverse function to be defined, it must be one-one and onto, then in many questions why the codomain is given more than the Range as if we know that the codomain must needs to be equivalent to the Range for the Inverse function to be valid ...
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So I understand that matter emits EM waves when hot. And that the higher the temperature, the shorter the wavelength, so cooler flames start off orange and the hotter flames reach light blue and white. What I'm wondering is if, hypothetically, you could take that blue-white flame and could make it considerably hotter, ...
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I am interested in modeling the populations of low-lying, long-lived "dark" electronic states after excitation of an atom in the ground state with a laser. Using just a regular rate equation solver, I get nonsensical results. Provided there are no interactions between the low-lying states and the branching ratios to th...
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I downloaded a nice latex class, but I'm having trouble with one aspect of it - every page's header gets marked with either the chapter name or section name. As you can see, it makes the page look really cluttered. The code for the class is here. I've tried for a really long time to solve this, but I can't. I searched ...
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If a block of mass m is attached to one free end of a spring and is performing simple harmonic motion of some amplitude A . Now if a wall is placed at some distance x (x < A ) from its mean position . It collides elastically, how is its amplitude going to change ? [ Will it be different in the side with the wall or uni...
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We know that sound travels through a medium through vibrations in the form of longitudinal waves. An example of it is here: We also know that particles of any medium vibrate when we give them more energy in the form of heat which gives them more kinetic energy. From this a logical inference can be drawn: heating the me...
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I am taking the second law of thermodynamics in the 'useful energy minimized' sense, although I am not sure if that is necessarily the most correct. Take the example of two electrons: electron A and electron B. Electron B has a wavefunction. Can we say that electron B's wavefunction will not have high probabilities clo...
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I know that the work done by gravity in going from A to C, and from A to B to C is the same. I also know that the work done by an applied force in going from A to C is the same as negative of the work done by gravity if there is no change in kinetic energy. But what is the work done by an applied force in going from A ...
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It is assumed that a valid QFT, must have states/observables in a Hilbert space, transform according to the projective unitary representations of the Poincare group. I can understand why the observables must transform according to unitary representations of the Poincare group, but don't see any compelling reason for th...
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Exactly the question above. I've recently posted a question, and the more I look at it, the more it seems like a bug to me. The upstream code base on GitHub seems pretty inactive too (for about two years), having many open issues. I'm missing some experts who could help me or even the maintainer. Do you know whether th...
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Imagine spreading double-sticky tape all over the surface of a car or a plane. Would there more significantly more aerodynamic drag as a result of the adhesive 'sticking' to air molecules and slowing down? This would certainly cause more resistance in solid mediums but would this cause more resistance in air or water? ...
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In finding this article, I questioned my understanding of the word 'gadget'. My current understanding is the word is (and please let me know in your answer if I'm right or wrong on this one) synonymous with 'accessory'. The only difference really being that accessory somewhat precedes the technological age. Hence, 'gad...
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A lever and a fulcrum is able to to multiply the input force to the output force and trade moving distance for increased force. The same happens in automobile transmissions, taking advantage of high motor RPMs. What exactly is the physical mechanism behind this? I am very aware you can mathematically prove it, for inst...
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I will have to teach a first course in differential equations. A good motivator might be to promulgate modelling with differential equations but I have seen some teachers have made polemic against modelling. Are there any really good resources on modelling with differential equations? I want something which will have a...
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What are local laws? I was reading this line in a book... Newtons second law is a local law. This means that it applies to a particle at a particular instant without taking into consideration any history of the particle or it's motion. What does this mean in simpler terms? Are their non local laws as well? Please expla...
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It might well be that universal constants, say the speed of light, are only constant as far as we can tell in our chunk of the universe - in the same way that the Earth looks flat in the area you live. Are there any ways to tell that the speed of light is actually the same not just in our "neighbourhood" but in all the...
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Take a deck of cards, cut it in three, and rearrange the three packs in any order (for instance, the middle goes on top, the top in the middle, and the bottom stays). Repeat as many times as you want with different (and possibly uneven) cuts. What permutations can you generate? Motivation: given a set of symmetric matr...
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I may be misunderstanding the general idea and the following specific example, but I was told that Dirac predicted the existence of the positron based on a formula for the motion of the electron(?) and then feeding in a negative value or imaginary value and somehow this suggested antimatter being a real thing. Sure eno...
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If we consider the addition of orbital sngular momentum L and spin angular momentum S to produce the total angular momentum vector J, then J is a member of the vector space that is constructed by taking the tensor product of the vector spaces associated with L and S. Why don't we take direct sum of the vector spaces of...
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From my textbook, A noun phrase is headed by a noun. Modifiers include articles, adjectives and demonstratives. Qualifiers include prepositional phrases and relative/adjectival clauses. Given the following: Your bag is in your locker What is the grammatical name? I know that its grammatical/syntactic function is the co...
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Recently I learned about Markov chains. Consider a simple two-state Markov chain: This looks like a categorical diagram, with states as objects, transition probabilities as morphisms/arrows, and of course, identities as the same state transition probabilities. Although I don't know much, I also came across connections ...
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Today i studied about mirror formula, its derivation and sign convention used while deriving it, a question which i have is, why do i have to use sign convention while doing the derivation ? focal length, object and image distance all are distance so what is the point of using sign convention ? and while solving proble...
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When thinking about a "random" atom in space (say a Hydrogen atom), should I assume that the electron is in a ground state (or any particular excited eigenstate) or is it in a superposition of all energy levels before I measure the energy of the atom? On the one hand, If the latter is the case that would mean that ther...
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I'm not sure where else to ask this since I'm not very familiar with this area. What is the backdrop of a theatre play act called? Like when the actors are behind the curtains and their silhouettes are instead flashed on the white screens or curtains. The play or act is all going on in the background just like a puppet...
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Baum-Welch is an instantiation of the more general Expectation-Maximization (EM) algorithm. But recently when I learned about it I couldn't figure out where the Maximization step is used. Because the Baum-Welch algorithm give me a sense of just iterating between computing some intermediate variables according to old pa...
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Consider a vessel completely filled with water and a vertically oriented capillary. If the water vessel was open at the top, you would expect capillary action to occur, lowering the water level in the vessel until a force equilibrium is reached. However, if we seal the vessel, the water would have to increase its volum...
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Order the following functions in terms of little oh notation: where: Im actually confused on how to do it. I think its the functions that confuse me not the method on how to do it. I know that: But either my mathematical skills are failing me or there's a different way to do it, that I don't know. Any help is appreciat...
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I just wanted to bring up some discussion about an apparently essential concept for some fields in mathematics as so as for some in physics, as already mentioned in the title, I'm referring to the word "symmetry". I'm currently studying about Lie symmetry method to search for first integrals of ODE's, and already saw t...
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In an infinite universe composed of single point masses which can be simplified as a uniformly distributed mass density, what is the equation for the gravitational field intensity in Newtonian gravity? I assume that: The gravitational field intensity is constant through space since the matter density is constant. The n...
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Is there a word that means "lost thing"? Ir perhaps a word for "forgotten thing/ something that has been forgotten"? Similar to how requirement means "that which is required" or necessity is "something necessary"? I'm ideally looking for a noun or other words that are constructed to from an adjective describing somethi...
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I don't know much about this topic, but I read something saying that gravity is not a force using an example of inertial observation. I started thinking about the topic again when I was researching photon spheres and thought why does gravity (if it is a force) curve light when photons have no mass. The only possible an...
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I don't understand why you can't invert a hessian matrix using gauss jordan method. Can't you integrate or differentiate an entire row (because they are linear operators) and then subtract/add/swap? To clarify, I mean inverting the generic hessian formulation before it is applied to any function. In this way, I can get...
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Can we calculate the total observed time dilation on a spaceship and a planet by combining the planet's relativistic spin velocity and the spaceship's approach or recession velocity from the planet it came from and where it's heading? Would the spaceship see both planets as time-dilated due to their combined relativist...
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As I understand, it is perfectly possible to explain the standard double slit experiment with a laser beam using only classical physics. On the other hand, this experiment is one of those experiments which is mentioned the most when quantum theory is introduced. So at what point of the experiment -- or through which va...
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In Baby Rudin a subset E of a metric space X is defined to be connected if "E is not a union of two nonempty separated sets." This may be nit picking, but should this not be E is not a union of two or more seperated sets ? Wikipedia agrees. To be super nit picking, by Rudin's definition (and taking it super literally) ...
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According to the argument in this post, a theory of Quantum Gravity should not be compatible with the notion of time evolution. This is also called "The Problem of time". However, the target spacetime formulation of string theory does have a notion of universal background time given by the target spacetime. So it seems...
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I heard this dialog in a Western film: Ruffian: Are you Josey Wales? Josey: That['d] be me. Methinks Josey responded That be me and not That'd be me. Is the former even correct? Is That be me. also a sentence in subjunctive mood, just as That'd be me is. If That be me is correct, what would be the difference in meaning...
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The intended meaning is that 'it', described simply, is 'everything'. Would I write: It is in short...everything. Or: It is, in short...everthing. Instinctively, I believe that a comma is needed before "in short". But the confusion arises here: Is it a common enough phrase that the comma is not needed? Is it a publishe...
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In one of our informal math research club sessions we are thinking of hosting a math meme making competition. Our idea is that math memes will trigger curiosity and interest among the participants. This would not only help in questioning and understanding the mathematical ideas behind the meme but the participants will...
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What is the meaning of the phrase, 'as thus construed'? I was writing an essay and felt that phrase was appropriate, but I looked it up with quotes on google and found only references in some legal documents and academic papers. Is there a similar phrase that I am thinking of, or does this make sense by itself? For exa...
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so I am looking for good text books that treat signed measures, integration thereof and total variation etc. In particular also for finitely additive signed measures. I have found Dunford and Schwartz a little bit too general in that they dont necessarily consider (signed) measures but more generally set functions. I w...
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I am in charge of deciding the name of a skill in a role-playing game that serves as a catch-all for abilities/skills that are otherwise unlisted in our handbook (think skills that will help players flesh-out their character's personality, quirks, and life). Such a skill could be a profession type, a craft (e.g.,whittl...
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I am currently studying a paper on Wasserstein Fair Classification. Several places they mention the Wasserstein barycenter, weighted barycenter distribution or the Wasserstein barycenter distribution. What is it? How is it defined? I can only find very deep measure-theoretic explanations, which are both a bit out of my...
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I was wondering is it possible to try and do optimisation of an ellipsoid given the formulae for its volume and surface area? I would be trying to keep the volume constant whilst minimising SA. This is for one of my maths investigations, basically trying to minimise packaging of a kinder egg, so if this is somewhat of ...
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I happened upon this video purportedly showing a coin on a horizontal table receding away from a lens would disappear from bottom up. The explanation via diffraction resolution limit is wrong. However, I would like to find the correct explanation of the phenomenon. When the camera zooms in and focuses on the coin, the ...
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My question is this: Why can't electric potential energy be defined at one point. Surely a point charge has some inherent electric potential energy cause by force applied by the source field. The applied force would cause the charge to accelerate. The acceleration approaches zero as you get farther from the field, so t...
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I have questions about the inner product and coordinate transformation matrix. For F inner product space V, consider vector a and b in V and its inner product <a,b> then the inner product is invariant about changing the Basis of V? And let B and B' be Basiss of V. Then can I think that the coordinate transformation mat...
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Electrons have an e-field, and a moving e-field causes a b-field. These fields are defined in such a way that it makes calculating things like forces much simpler. Is this just a model and there aren't actual 'waves' traversing around us, but rather mere forces caused by a movement of an electron? If the force caused b...
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I have always been told that if you press the brake pedal fully and fast it can make the wheels lock themselves, and that you shouldn't do so because it would take longer to stop the car than if you push the brake pedal progressively, why so? I can imagine that when done progressively the car won't start doing jumps du...
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The existence of dark energy leads negative pressure. Is the pressure can be anisotropic? What situation can cause anisotropy of pressure? Further, Is the pressure can be negative in some directions and be positive in other directions? Such as in spherical coordinates, the radial pressure is negative and the tangential...
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I want to determine the largest area ellipse inscribed in a rectangle, in order to write a code that can determine the lengths c and d given a and b (see image here). I know the obvious answer would be c and d are halves of a and b, but I want the derivation (and also the maximum area) so I can stop thinking about this...
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I'm quite close to release my first game, and I can't decide which is grammatically correct on my first user facing screen. My game will surely have some grammatical mistakes but I would appreciate it if the first screen would be more or less accurate ;P So the title wants to welcome the user into my game but I dunno w...
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i would like to find a way to study physics online and get academic credentials in it and be able to contribute to the research and publish. I know some topics are experimental and require physical attendance, however there must some subfields in physics that can be studied online. Now there are many online masters in ...
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The Wikipedia page for locally constant function says that a locally constant function is constant on each connected component, but that the converse only holds if the space is locally connected. What would be an example of a function that is constant on each connected component but not locally constant? I first though...
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I need to prove in graph theory that if L(G)=G then G is a combination if simple cycles. I already did it assuming G is connected, so feel free to use this fact as given. My only issue is that it may be possible that in a non-connected graph not every one of the connected components will satisfy L(G)=G. For example, if...
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Suppose there is constant diameter pipe where a liquid flows through it, and if pressure of a liquid is due to external force or its weight when it is constrained or semi constrained (according to the accepted answer on this question), how can someone visualize how does pressure drop occur due to frictional losses? Doe...
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I am studying the spontaneous symmetry breaking in the mean-field Ising model and it's clear to me the necessity of taking first the thermodynamic limit and then the zero-field limit to see the phase transition, otherwise we only see zero magnetisation even under Tc. I was wondering if there was a phase space approach ...
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I went through a quite a lot of posts here but did not find the information I wanted. I'm looking for a beginner / introductory (yet mathematically rigorous) book recommendation for differential geometry / topology required for General Relativity. I'm perfectly fine (in-fact I like) the so-called dry mathematics books ...
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Is there any way one can modify a known functional so as to get a solution for another quantity related to it? Because this is too general I would like to make it more clear with an example. Given the classical EM Lagrangian, we acquire the equations of motion for a particle in an EM field. Starting from this, is there...
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To understand the attractive and repulsive force between two currents, I am looking at two electrons moving parallel and antiparallel to each other. How can an attractive (or less repulsive) Lorentz force between two charges and its directional dependence be explained? I would like to understand this in terms of dynami...
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Pramod Achar's book on perverse sheaves has a quick reference in the back that looks like this. How can I achieve something similar? The important things to me are: the general spacing (although maybe I would enjoy less margins on the outside), the two columns, the numbering and labeling, the horizontal line separators...
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Consider this fact that we all know from school mathematics: There is a one to one correspondence between real numbers and points of a line. But the problem is I have never seen a rigorous proof of this fact. This is what Apostol writes. And similar is the case with other modern analysis books. So my question is, where...
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I'm bit confused about the current flow in capacitors. My question is, let's assume those two electric circuits: As shown in the picture, the only difference is only the capacitance. I'm wondering about the current flow magnitude. In which of the circuits the current flow will be bigger? or will it be the same? To star...
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it is said that the time period of a simple pendulum in space is infinite. it is because the gravity in space is zero and anything divided by zero is equal to infinity.how can time period be infinite? also, in space, everything just floats around. how can a pendulum in space do oscillations if there is no restoring for...
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I was wondering if it is posssible to write my own environment for writing redox equations in form of: O: ... R: ... ___________________________ S: ... O stands for oxidation, R for reduction and S for the summary equation. Also, there is line that should separate O and R from S. I have never written any environment in...
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In surface integrals the process is to use the dot product of the vector field with the unit normal vector first and multiply that with the area element. In the line integral the process is to use the dot product of the vector field with the length element. Why isn't the unit tangent vector used in the line integral co...
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While deriving the Planck's radiation formula, why do we use MB statistics when we calculate the average energy of oscillators? Shouldn't we use BE? Is this because temperatures concerned are very high and quantum stat goes to classical at high temperatures? But again, in a similar calculation, while calculating the la...
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The sentence is as follows: "Isn't it evident who/whom they have in mind for the position?" Depending on how you try to reword it to figure it out, the answer seems different. --Is it not evident that he is the one they have in mind for the position? (This seems to be a case for "who" here.) --Is it not evident that th...
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I know multiple singular subjects have been addressed here and singular compound subjects here. However, I came across a sentence with multiple subjects, one single paired with one plural. I could not find this question addressed. For example Does a table and four chairs sound like sufficient furniture for a breakfast ...
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An intuition that I thought of about a year ago, but my level does not allow me to prove it. I think it needs advanced tools to prove it, or just a counterexample to refute it. Who can help, please, thank you. The locus of the point of intersection of the perpendicular tangents of a convex curve is the convex curve Of ...
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Using Hugyen's principle and wave nature of light, refraction, diffraction are relatively simple to explain but I have been visualing a mental model for a couple of days where the photons on reach the boundary of two media, experience a difference in the structure of the medium of propagation and hence some sort of col...
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how one would know that the given frame of reference will be inertial or non inertial, without the use of 'accelerometers'. earlier physicst has defined this concept and I am sure that they define it without the help of accelerometers. I just want to know about the logic behind the whole concept. And I will be not gett...
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How come there is no loss due to coupling between the superconducting channel and the single particle channel? I imaging the two channels being two parallel wires, so the magnetic field from the current running in the superconducting state, should fully inclose the single particle channel generating a electric field, t...
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I posted about monoid coproduct to try to understand the correct way for constructing them. In the process and from the answer I got, I am further confused whether the product and coproduct of plain monoids compare free monoids, what are the subtle differences in terms of what they look like. I understand in both copro...
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If we add a constant to both sides of an equation, we will get a new equation with identical solutions. This is mostly true for multiplication and square roots, but there are some minor complications/exceptions. Taking the derivative of both sides will result in entirely different solutions. Is there a formal way to sh...
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This is a question about grammar, in particular about a specific language syntax. In the german langage it's legit to write negative prefixes in braces like "Die JSON (De-)Serialisation wurde implementiert." Is this valid in english? "The JSON (de-)serialization has been implemented." As a software developer I need to ...
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I wonder whether there is a space filling curve that looks like the following if the width of the domain has an odd number of cells and looks like the following if the width of the domain has an even number of cells. I would like to fill a three dimensional grid none of whose sides (lengths) are guaranteed to be a powe...
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How is it paradoxal that a set of all sets exists in set theory? Russel's paradox is about the set of all sets that do not contain themselves cannot exist, that I understand. But what about the set of all sets, in a unrestricted manner? Is it the fact that it contains itself (which may be what allows to define the para...
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Is the result of a contour integral around a closed loop path unique? I'd think it'd be similar to a definite integral, which is unique. This came up regarding the proof of uniqueness of a Laurent Series - if the contour integral is unique, then the definition of the Laurent Series can be used to prove its uniqueness. ...
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Is it true that if a curve is increasing, Euler's method will always underestimate an actual solution? So if a curve is either increasing and concave down, or increasing and concave up, we can simply ignore concavity and justify the answer based on the fact that it's increasing? Euler's method will therefore underestim...
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There are two machine learning models, A and B. Model A classifies samples into 'O' and 'G'. Model B classifies the samples determined to be 'O' into 'S' and 'C'. Each model shows the probabilties of classification. Input samples pass through model A first, and if the result is O, then they enter model B. (If not they ...
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