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I'm trying to reconcile two seemingly conflicting sources of information about Lorenz force. Let's consider a solenoid in proximity to a permanent magnet with the field pointing in the same direction as the axis of the solenoid. From the right hand rule I conclude that the wire should either be pushed towards the cente...
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I am aware there are similar questions already asked, however, I find none of the answers satisfactory, they either do not contain any mathematics at all, or mathematics of a level I am not capable of comprehending. My doubt is that, if the change in kinetic energy is dependent on frame, then how is the law of conserva...
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I have recently started to study integrals and faced some difficulties of understanding why we need antiderivatives to calculate area under curve. I am trying to visualize it using graphs but it just doesn't make sense to me. If anyone can explain me if it's even possible to understand them the way I try. here is the i...
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I was curious if anyone knows the font used by the Desmos graphing calculator. All the formulas are formatted as LaTeX, so I wanted to know if there was a way to use this font in my own documents. It seems at first glance like standard Computer Modern, but upon closer inspection as you can see the "w" and "v" character...
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What is the interpretation of temperature in the Ising model? Can one describe a theoretical thermometer for the Ising model? As far as I know, in the theory of gases, temperature is interpreted as a quantity proportional to the mean kinetic energy. A thermometer is a device capable of converting the average collisions...
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I already know the term anaphora exists for repeating the same word for emphasis. I'm specifically interested in a term for repeating a word twice without additional clarification to express that something is especially strong or genuine. Examples "Do you like her, or do you like like her?" "Are you making quick bread,...
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Imagine that Alice is the President of Arstotzka. Alice has a lot of enemies but she's generally an upstanding President so her critics have a hard time building a campaign against her. Instead, the critics are patiently waiting in the shadows until Alice makes a minor mistake - perhaps she misspeaks at a conference or...
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Can you compute geodesics by treating it as a problem where you want to minimize the length of a curve through two points on a specified surface while having the constraint that the curve must reside on the specified surface? If so, can you explain how one could do so for a cylinder? Or is the calculus of variations th...
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I am working on my thesis and need to convert my paper to a different format. I'm new to using latex, only having started using it for this project, and am struggling to get the base version of this template working. No error occurs, but compilation appears to halt, judging by the fact that main.log log ceases to updat...
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Show that, given four coplanar points, we can always draw two intersecting circles coplanar with the points, such that two of the given points are diameter endpoints of one circle, and the other two given points are diameter endpoints of the other circle. In this question, "intersecting" means that the circles share at...
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Ive recently been introducing myself to Gaussian Processes. In Bayesian linear regression, one would expect that when adding new features, the likelihood on the training set would weakly increase due to the larger degree of freedom, of course leading to overfitting most likely. I was wondering if this is true in genera...
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I heard that natives use Present Cont. to describe things in the future. As I understood, we use Present Cont. when we have arranged an action or there is at least one person with whom we agreed to have plans. But I saw this: I am teaching English tomorrow. I will be teaching English tomorrow. What is the difference an...
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I work in an office where the airconditioning is not working, so I brought a small mobile airconditioning device: it's a small machine, emitting cooled air. It looks like this: However, according to the laws of thermodynamics, the total amount of heat should always increase in a closed system. Therefore, regular aircon...
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Two congruent circles that touch at point H are given. Let the line p be their common tangent that doesn't pass through the point H. Construct a circle that touches both given circles and the line p. The most intuitive thing was to construct an equilateral triangle NOL (such that L is on p, and N and O are on the given...
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If a set is equipped with total order (reflexive, transitive, antisymmetric, strongly connected), does this necessarily mean there exists an injective mapping from that set to the reals? Alternatively, if a set has larger cardinality than the reals can it not be well-ordered? I'm struggling to think of a counter exampl...
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This question is bugging me from a very long time and I don't know whether it can be answered or not but still I am posting to get other's opinions. Is there any inherent property among all the uncountable sets which make them uncountable ? Or more precisely:- If I ever wanted to "make" an uncountable set, what ingredi...
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Imagine a uniform thin layer of a viscous gel sticking on a vertical wall. A small sphere is submerged in this gel. The gel has a yield stress (Bingham model). How can I calculate the yield stress that will cause the sphere to start sliding with the fluid on the wall? This yield stress will obviously depend on the weig...
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I'm having a problem with the second part of the question, which shows that the sum of the vectors equals zero. I came up with a geometric proof, but I wondered how to prove it by using complex numbers and maybe links to the first part of the question. I was thinking about the equally distributed roots of unity, but do...
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I understand how we can have three coin tosses (or three events if you prefer) such that all three tosses are pairwise independent, but not mutually independent. One trivial example of this would be to just have the third coin be the xor of the first two coins. My question is whether it's possible to have four or more ...
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I am a PhD student working on observational cosmology. My research is based on the experimental side of it (receiver design, etc.) but I would like to get deep into the CMB and CMB statistics. I have found several resources and have read some of the papers but I feel like they are a bit "unconnected". Does anyone know ...
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I am studying blackbody radiation and modelling a cavity as a blackbody. However, I am encountering a number of confusions in this description: Many textbooks mention that the cavity consists of metallic walls that act as perfect reflectors. Light entering through the hole, after multiple reflections, get absorbed. How...
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For example, a carpenter works in carpentry and a plumber works in plumbing. So what trade does an electrician work in? Electrical? I searched the definition for "electrical" and found that it gives the part of speech as an adjective and not a noun like the dictionary does with carpentry or plumbing. This is confusing ...
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Imagine a mass hanging from a string attached to the ceiling of a lift and the lift descends with a certain acceleration. From the reference frame of the ground the tension in the string would be smaller than when the lift remained stationery. However, let's say your friend was inside the lift and when asked about the ...
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For example: It is a thing that works producing stuff. This feels wrong to me, but I can't quite put my finger on what exactly is wrong about it. It seems like it's trying to be a participle phrase, but it's not necessarily modifying the current state of "it", and is, instead, describing what "it" is -- i.e. by working...
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I have been taught that path difference and phase difference are essentially the same thing, except phase difference is measured in an angle unit while path difference is measured in multiples of wavelength. This seems incorrect. I think they are rather different concepts, with path difference representing the differen...
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For example, when referencing a webpage: Visit the "The Performers" page to learn about our musical lineup. In this case, I want to tell someone to visit a webpage entitled "The Performers" to find more information, but the repetition is throwing me off. If I were to remove the first "the" before "The Performers," thou...
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I think something is missing in the definition of homeomorphism I saw. It just said it maps the collection of open sets to the collection of open sets in a bijective way. What exactly makes this preserve topology? I can think of weird situations where each individual open set is mapped to totally disjoint open sets in ...
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The definition of optical path length is the distance that light could have travelled in the same time, in vacuum. So, can we define something analogous for sound waves, like an "acoustical path length", with respect to a specific medium, say air? And if we were to define something similar, how far would it's applicati...
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I know foliations as a particular topic in differential geometry. I understand the definition and basic properties of a foliation from the DG point of view, including the Frobenius theorem. While trying to understand why this concept is important I came across this Wikipedia page on integrability conditions for differe...
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I was going through this paper in which the authors state that : In Landau theory, a continuous phase transition is associated with a broken symmetry. The phase transition in a black hole system can also be characterized by the symmetry and order degrees as in a conventional thermodynamic system. The authors don't expl...
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I am attaching the solution of a school question asking to calculate the inverse of a matrix. However, the solution proposed does not fit in anything I ever seen about calculating inverse matrices. I did not found any property that fits the final part of the solution (the one I highlighted in the image below) although ...
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I read an article that referred to the idea that a double slit experiment near the event horizon of a black hole observed by someone inside the black hole creates a paradox because the inside observer can break the interference pattern, thereby bringing information of their presence out of the black hole. This seems od...
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I have often heard that two physically distinguishable mixed quantum states produce different density matrices. However, how would I prove it? I know that they have to differ on the main diagonal, because these elements correspond to measurement probabilities in the standard basis. Moreover, I have seen someone using d...
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I'm learning some math by myself and I thought of this question that has to do with how useful complex numbers are: I'm looking for a specific example of an equation that has the following properties: The equation is true, i.e., its LHS and RHS are always equal, no matter the values of its variables, if any. The LHS an...
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I want to understand in some detail why superconducting qubits need periodic calibration. The usual, hand wavy explanation is environmental effects that tend to vary from time to time. However, I suspect that the actual picture is more complex than that. I want to understand in particular which effects are purely exter...
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A spanning tree of a graph is a subgraph obtained by deleting only edges of the graph and which is also a tree. Why does one study "spanning tree" in graph theory? What are "spanning trees" real life application? We know the number of spanning trees of a graph is equal to any cofactor of the Laplacian matrix of the gra...
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Hi I was trying to determine the basis of the topology on the following collection in R {[a,b), a <b}. Here is what I tried : Knowing that the basis for a subspace topology is equal to the set of all intersections of {[a,b), a <b} with the basis of the topology it is a subspace to. Now I also know that the basis for th...
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I understand that Berry curvature sinks and sources correspond to Weyl points. However, I'm curious about the identity of points exhibiting a Berry curvature spiral, highlighted by red circles in the figure below. Could these be Dirac points? Is it possible to differentiate between Dirac and Weyl points based on the di...
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I read that a sound wave (a scalar wave) produces monopole radiation, an eletromagnetic wave (a vector wave) produces dipole radiation, and a gravitational wave (a second order tensor wave) produces quadrupole radiation. How do I make sense of multipole radiation physically? The Wikipedia article on multipole radiation...
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first post here. I've been looking into acoustics a bit recently and I'm wondering if for a compressed solid, acoustic impedance should change along different axis in certain cases. Consider a cubic lattice of atoms with pressure being applied to the top and bottom, compressing the lattice. Atoms would be spaced furthe...
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We know that Maxwell's electromagnetic theory was originally an ether theory. Later, the ether was denied. But Maxwell's electromagnetic theory can be transformed into a theory of "field". The theory opposite to Maxwell's theory is the theory of action and reaction. Or it could be the theory of action-at-a-distance (Wh...
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Consider a few sentences: I read a lot. ?I read much. I don't read a lot. I don't read much. Do you read much? This seems to suggest "much" is a negative polarity item, but then we can say things like "Much of it has to do with you not being here." So how much of a negative polarity item is "much"? I am particularly in...
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While reading the CMU FLAC overview, I stumbled upon the above image. While I have seen Greek numerals before, I had no idea that there was an entire algebraic symbology developed in Greek (though this isn't too surprising). To try and find more about this particular notation, I reverse Google Image searched it and fou...
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As a thought experiment, say, for the sake of simplicity, we have a meson. This meson, which is traveling near light speed, is traveling towards a black hole. And skirts the event horizon in such a way where the anti-quark ends up inside it's event horizon, but the quark does not. What would happen? Would this create a...
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The operational principle of frequency combs is that you generate very short pulses (in time domain), and that in the frequency domain (due to Fourier's transform) the spectrum of such pulses is a comb. But this is just maths. And how it works from the physics perspective? Let's say I have a monochromatic continuous la...
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If so, then why? We know that a current carrying closed loop in a uniform magnetic field experiences no force but when it's suddenly pulled by an external force, according to Lenz law a magnetic force will start acting on it to oppose the external force and eventually reach steady state at infinity. How can this be? Ed...
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What does the Stark-Lo Surdo effect consist of in the interaction of the electromagnetic field and the active medium in a laser? I thank those who want to give a clear and concise answer. In my student notes from the seventies, I found the topic I was looking for, the calculation for which I attach. But what happens in...
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Am getting the following error when I try to do tlmgr install package: tlmgr.pl: package repository http://mirrors.rit.edu/CTAN/systems/texlive/tlnet (not verified: pubkey missing) It appears I need to get the keys somewhere, but I haven't been able to find them on CTAN. Where else can I find them? If it's any use, I j...
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Any regular curve may be parametrized by the arc length (the natural parametrization or the unit speed parametrization). But I haven't seen an analogous development for regular parametric surfaces. I hope we can do this at least for orientable surfaces with no umbilical points. For such a surface, there will be two ort...
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When talking about black holes and singularities, most books say that combining relativity and quantum mechanics gives the answer of infinity in some equations. They also say that: Infinity is the answer you get when the universe is trying to tell you that you have made a mistake. Why can't equations have answers of in...
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In the original version of Sequent Calculus by Gentzen for classical and intuitionistic propositional logic there is a structural difference: the classical version admits succedents with multiple formulas but the intuitionistic version admits only succedents with one formula. I'm searching for another formulation of Se...
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It is often said that the achievability proof for Shannon's coding theorem relies on the channel being discrete and memoryless. At the same time, following the classical proof (using random coding and AEP), I can't find any part of the proof that explicitly uses the fact that the channel is memoryless (except arguably ...
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Quick question with probably a simple answer. For context, I am currently in undergraduate classical mechanics studying potential energy. My question is, if a conservative force is one in which is only a function of position, can radioactivity be a conservative force? The example I thought of was if we take two identic...
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I know work done is equal to product of force, displacement and cosine of angle between them. But that formula works only when we assume that the force is constant during displacement and it acts so long the body moves and also when the direction of force and displacement is constant. But how should I calculate work do...
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I have an object that takes a shape similar to a catenoid, which is known for its zero mean curvature. According to Laplace's law, the pressure difference is proportional to the mean curvature. This implies that if the mean curvature is zero, then the pressure difference is also zero. However, I am aware that when ther...
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Reading through Einstein's Brownian motion paper "On the Movement of Small Particles Suspended in Stationary Liquids Required by the Molecular-Kinetic Theory of Heat", it seems the final argument is that he can calculate the Avogadro's constant by using data on the diffusion rates, particle size and fluid viscosity. Bu...
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Is it correct to say: for any family of bounded operators (proper subset of B(H)) acting on a non-separable Hilbert space there is a non-trivial subspace of the Hilbert space that reduces the family? I know that this is true for any operator on the space so, I assume that it holds also for the von Neumann algebra gener...
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If you were to measure a regular hexagon, is there a way to rotate the hexagon so that if there were two lines bisecting the center of the hexagon, both lines would be the same length? Can we then use the lengths of those lines to determine the area of the hexagon? Edit: To clarify, I mean that I intend for the lines t...
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There are four concepts which are studied in Calculus and Analysis: Convergence, Continuity, Differentiability and Integrability. In Calculus, you can define the latter three in terms of the first, but also you can define integrability without convergence using Darboux approach. In topology, you can define convergence ...
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Consider the following diagram Here, the diode is in forward bias, and allows current to flow. However, I am slightly confused why this is the case. A diode is defined to only allow current to flow from the anode ( Positive ) to the cathode ( Negative ). When drawing a diode, the cathode is the line and the anode the b...
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In the derivation of Torriceli's law it is assumed that the pressure at the tank hole is equal to atmospheric pressure. I believe this is true when the water starts flowing through the hole but I do not understand why this should hold during the whole purge procedure. Do you have ideas about the explanation? CFD calcul...
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The second law of thermodynamics states, that entropy (of the universe) always increases. Entropy can, however, be (locally) reduced when energy is provided. At the phase transition from a relatively unordered to a more ordered state (e.g. condensation of vapor to a liquid), the entropy is lowered. At the same time, en...
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I know that the Heisenberg Uncertainty principle states that the position of an electron is uncertain, however, if an electron is created due to beta decay, then at what location is it more likely to begin its movement? Is it right inside the proton? Is it the outer edge of the proton? Is it adjacent to the proton? Is ...
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According to Fermat's principle, light should take the least time between two points. Therefore, is it correct to say that the angle of refraction is solely dependent on the difference between speed of light in two mediums? Surely the the position of the observer is a key factor too, given the findings of the double-sl...
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I've been reading more category theory as a prerequisite to understanding some more complicated theorems, and for the first time I'm running into arrows that are distinctly colored. Examples include these diagrams from Wikipedia for natural transformations and universal morphisms: The Wikipedia articles do not readily ...
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I understand that the word spook is a racial slur that rose in usage during WWII; I also know Germans called black gunners Spookwaffe. What I don't understand is why. Spook seems to also mean 'ghost' in German. Did the Americans call them spooks because the Germans did? If so, why did the Germans call them that? Or, if...
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When researching the reflexive, symmetric, transitive, addition, subtraction, multiplication, division, and substitution properties of equality, the sources I found said they were true "for all real numbers". Are any or all of them not true for complex numbers? i.e., is the "for all real numbers" a necessary caveat, or...
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I am trying to calculate the determinant of the following matrix. I literally have no idea if there's a general approach for solving such strange looking determinants, but I decided to subtract the first row from each row after the second. I don't know what to do next... May you show me what we're supposed to do and ho...
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Assume a Gaussian surface in a non-uniform electric field that is directed along X-axis. Say the field is getting weaker as we go along positive X direction and it's constant along Y and Z directions. Then, if the Gaussian surface is a cube and charge enclosed is zero, the electric flux coming into the surface is more ...
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The usual semantics of first-order theories involves models who have a structure-set. Hence it is required to use some sort of set theory as the metatheory. However it is also widely known that one can do logic with a minimal metatheory, say a fragment of Peano. If we are using such a metatheory, what will the semantic...
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Here is the problem, I thought it the I've been thinking for a long time, but I still don't have any ideas. Problem: Let A(S) be the group of all bijective functions on S with composition as its binary operation. then A(S) is finitely generated if and only if S is finite set. I know that proving from the right to the l...
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I am typesetting my thesis in latex overleaf. I am struggling with one issue. I want to highlight my section with blue rectangular background with white color for heading as can be seen in picture. For subsection I don't want a background rectangle I simply want subsection of a blue color as can be seen in attached pic...
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I was given an honors project to solve for the equations of motions of a pendulum with an oscillating fulcrum. I (somewhat) understand the procedure on how to solve it with lagrangian mechanics and the Euler-Lagrange equation, but I am stuck at a question. How do we know this cannot be solved with Newtonian mechanics? ...
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I am having trouble understanding why is temperature coefficient for avalanche breakdown is positive? The explanation I found online says as the temperature increase the mean free path decreases making collision and generation of carrier pair easier. But the electric field require to accelerate charge carriers to point...
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The Coriolis force predicts that winds in the northern hemisphere should be deflected in a clockwise pattern and winds in the southern hemisphere should be deflected in an anti-clockwise pattern. Why is it that in the case of cyclones however, the cyclones spin anti-clockwise in the northern hemisphere and clockwise in...
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What is it called when someone says no, but actually doesn't mean that? Imagine this situation: when two lovers have had an argument then one of them is trying to apologise but the other one (usually the girl) says "no, I don't forgive you." but she isn't really angry. She just want him to insist more. She says no in a...
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I understand why all non-degenerate energy states can be chosen to be real up to an overall phase (as is highlighted here Is a non-degenerate wavefunction real or complex?). However I've been told the argument holds in the opposite direction -- (that all energy eigenstates that can be represented as purely real are non...
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I know that as the kernel parameter for the RBF kernel increases, the Gaussian function becomes less peaked and broader. The reach of the points become larger meaning that farther datapoints have more weight. Intuitively, it makes sense that a larger reach means a smoother decision boundary. However, since I don't have...
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I came across this question while studying for the SAT. What the humanities teach are valuable skills. These include the ability to think critically; to construct, analyze, and respond to arguments; to see both sides of an issue; and to understand the processes that have brought us to where we are today. (A) NO CHANGE ...
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This stems out of my personal curiosity and it's not related to any homework of sort. Suppose I have a table made of some uniform substance (like plastic), and then I strike some point of the table with a hammer. Will the disturbance/wave I produce travel always the speed of sound in that medium, or will there be cases...
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I have a weighted directed graph representing the intensity of movements of individuals between locations, with weights representing the mobility flow. I have another dataset consisting of particular paths that are taken by specific individuals. For example, for a person named Alice, I know how she moved from node A to...
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I just discovered Lean and using computers for stating and proving theorems. The first question that came to my mind, can we write any proof with Lean? Or are there limitations (something that you can write a proof for on paper but not in Lean)? I'm not talking about practicality or feasibility. But does the Lean syste...
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So I'm confused with The Second Law of Thermodynamics. It says that heat flows from the high temperature to low temperature. For example, if we put a glass of cold water on the dining room, the cold water will eventually have the same temperature as the dining room, because heat will flow from the air in the dining roo...
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Geometrically, what is the difference between a diagonizable matrix and an orthogonally diagonizable matrix? I understand the difference algebraically, as explained here and many other places. But I'm struggling to see the difference geometrically: both will not rotate (assuming the field is the reals), but will stretc...
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I'm a physics graduate that is joining the master degree program of my University. One of the areas of research that i'm very interested is Quantum Field Theory over curved surfaces, but the point is that i don't know how to begin to start studying these subjects. I imagine that i should star from classical field theor...
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There is a word for animals like horses and cows that defecate wherever they happen to be when the need strikes them, versus animals like dogs and cats that seek out one place or another to do their business. It may be that all such animals are herbivores but "herbivore" is not the word I'm looking for as it relates to...
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I am getting confused as I study the AdS/CFT correspondence, so I ask this question. CFT is given on the conformal boundary of AdS, which can be derived from Poincare coordinate patch to AdS. Would this mean that CFT has its AdS dual only on the subspace covered by this Poincare coordinate patch? Or does CFT has its Ad...
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This Wikipedia's page says that: David Hume's problem of induction demonstrates that one must appeal to the principle of the uniformity of nature if they seek to justify their implicit assumption that laws which held true in the past will also hold true in the future. For which I'm confused about how to interpret the m...
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I just came across this sentence while studying for the SAT and wonder if it makes sense. The women soon disperse, SOME entering the shallow waters at the beach, OTHERS venturing out onto the rocks to access deeper waters. My first question is, it sounds a little awkward to add some additional fragments beginning with ...
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I am taking a graduate-level mechanics course right now, and we are working with the continuous limit of coupled harmonic oscillators. My professor mentioned that he prefers the "bloch wave method" to the determinant method for finding the wave equation, but I do not know what this is. There is no mention of it in Gold...
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For measuring the electric potential difference between two oppositely charged plates A and B, a test charge q is moved from plate A to plate B. My textbook says that the charge is kept in electrostatic equilibrium such that an equal and opposite force to the electrostatic force is applied on the charge to move it oppo...
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One of the postulates of Copenhagen interpretation of quantum mechanics is that every observable has a Hermitian operator A. With these operators we can then find eigenvalues of observables, make a time-evolution of a wave function and, in general, calculate the properties of the system. However we also introduce anti-...
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I've used both Polish and Reverse Polish (RPN) notation. I get that using them one doesn't have to deal with varying evaluation order. While the benefits to automatic computation (as in software) is clear, it's unclear how this aided human reasoning. Can anyone explain how Polish Notation (or even RPN) simplified reaso...
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I have a Hotwheels track set. It has a looped structure such that at some point in the car's trajectory, it becomes upside down. But in such a situation, it doesn't fall to the ground. I understand this has something to do with its fast speed. Can someone please explain as to why exactly this happens? Why it doesn't fa...
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A decimal representation of a rational number is necessarily either terminating or non-terminating and repetitive. This comes from the fact that when you are dividing there are only a finite set of possible remainders (remainder is always less than the divisor). As you go on dividing for decimal representation, the rem...
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(Disclaimer: I'm not all that familiar with any of the two topics) Consider the Ising model on some graph with, lets say, two heavily inter-connected components that are sparsely connected between each other. This "bottleneck" would imply (or would be implied by?) that the Cheeger constant of the graph (and thus the fi...
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I'm trying to compile a document which needs the package minted, but it throws the same error over and over: ! Package minted Error: You must have `pygmentize' installed to use this package I can't seem to make it work. Other forums say to install the package "Pygments," which is a Python package, but MikTeX doesn't se...
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So, when we solve the optimization problem using Lagrange Multiplier method, I know that lambda can be positive or negative. Lambda is simply the rate of change in the optimal value when the constraint changes. So, I understand that lambda can be positive or negative. Now, my question is when we have inequality constra...
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I have the following list of points : I'm trying to find the best regression model to fit these points. The logistic regression is not close enough to the points : I guess I need something closer to a spline, but I don't know how to compute a regression model based on a spline, all I can find are interpolation models. ...
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How to punctuate a quotation which is an exclamation, at the end of a sentence which is a question? The quotation is "Fire!" The sentence is: Did he really shout "Fire!"? OK, this is simply the shortest example I can think of to illustrate the question, the actual quotation of concern is longer, and so is the question ...
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