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Assume I have a hemispherical container of water and an eyedropper. Is it possible to add water to the container one drop at a time, such that each drop continues to add kinetic energy to the water? When a water drop is added, it creates waves. I am wondering if it is possible to keep adding water drops in time with th...
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(I'm South Korean, so non-native question here.) I saw the sentence 'that looks about it'. Does it mean that something is seemingly almost done? I searched this expression on google but I couldn't find the exactly same pattern of the expression but I found at the Cambridge dictionary: that's (about) it : there's no mor...
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In quantum field theory we often hear that particles sometimes behave like waves and sometimes behave like particles. In quantum field theory we say particles are fundamentally fields. Is it correct to say that particles and waves are limits of quantum fields? For example, if we take a quantum field and apply the limit...
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I am no expert in the calculus of variations but I had some lectures about it and attended some talks. I noticed that searching for minima of functionals seems way more popular than searching for saddles. I know about some MiniMax theory or mountain pass theorems that can be used to search for saddles aswell but especi...
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I am once again asking more software-related questions, but I am happy to learn a general answer as well. I'm curious about how the is_infinite function works in SageMath (or in GAP). Does it simply check whether we input something that is already known to be infinite or not? Or does it print "infinite" after a specifi...
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Can you refer me to books that are at the same advanced level of real analysis via measure theory as Folland's and Royden's books? I want to have a solid foundation and I've been referred to books like Cohn's and Axler's, but I don't have deep knowledge of real analysis (I'm still going to start studying introduction t...
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I've always been taught that vectors that form a closed polygon represent an object being at equilibirium, that is there is no resultant force on the object. However, this has never been intuitive to me. How does one prove, that if the vectors form a closed polygon, the resultant force has to be zero? Furthermore why m...
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When reading about the notion of affine connection, the Levi-Civita connection appears naturally as the unique affine connection that preserve the metric and is torsion free. In this case, it is possible to express the connection coefficients analytically as a function of the metric and its derivatives. What I am curio...
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I know what a collocation method is and how it can be useful for the direct transcription of optional control problems, using specific collocation points. I also know what a quadrature rule is and how it can be used to solve integrals numerically, using specific quadrature points. But I'm struggling to see the mathemat...
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I am writing to ask for references. I have been trying for hours to find some paper/ book containing information on cycles in cubic graphs. It is known that to determine whether a given cubic graph contains a Hamiltonian cycle is NP-hard, but since it is not a tree we know that it must contain at least one cycle. But d...
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In some parts of my document math environments, such as "align", are not recognized, e.g. there is no arrow on the left to hide it, there is no automatic indent, and the theme colors are not applied properly. The same happens with "subsection", "section", etc. However, in other parts of the document the environments wo...
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For example, prove that if A AND B and A AND C are logically equivalent. How would I go doing this? I thought of using two truth tables: A | B | A AND B T T T T F F F T T F F T A | C | A AND C T T T T F F F T T F F T Then concluding they are logically equivalent. However, they are from two seperate truth tables. Would ...
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Can you recommend me some papers or lectures on finding the solution of elliptic PDE which is the saddle point of its energy functional. I glance over some methods including mountain pass theorem and min-max scheme, but it's hard to select one paper or lecture to do some intensive readings, so I wonder if you can recom...
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I am currently working through the book Mathematical Logic by Mileti (online version) and just finished the second chapter (induction and recursion). I have to admit that it is quite challenging and whereas I think I can follow through, it really takes a lot of effort and time to thoroughly understand the ideas since t...
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I was always fascinated with pure math, but lately I've been increasingly more interested on applications of math (preferably algebra/topology but other fields would be interesting too) in the real world. I am wondering if there are any books that discuss such applications, as I wasn't able to find any. To be clear, I'...
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I am working on a user manual, compiling it from .rst source files with the pdflatex engine. My user manual has two versions: in English and in Russian. The issue that I have is connected with tables that take more than one page. I have 'continues on next page' and 'continued from the previous page' captions added auto...
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This is a question I came across on Instagram today, and here's the diagram: (Note: The image is NOT to scale) I attempted to solve it first by amending the quadrilateral in various ways, but each of those methods lead to a strange contradiction, likely because the diagram is not to scale. I'm going to post my successf...
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Blum proved that any mathematical theorem can be converted into a graph such that the proof of that theorem is equivalent to proving a Hamiltonian cycle in the graph from Applied Cryptography by Bruce Schneier I'm struggling to understand what the statement means, and I was unable to find anything on it after a few qui...
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Antennas work by accelerating electrons to emit EM radiation. In fact, my understanding is that any accelerating electron will emit EM radiation. But in relativity, no frame of reference is preferred. Therefore, an electron in an antenna can claim that every other electron in the universe is moving around it. So does t...
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I am trying to recreate this diagram in a standalone PDF using TikZ or PSTricks. Two extra requirements: I have to be able to alter the viewing angle arbitrarily I want to use relative coordinates rather than hard-coding the absolute coordinates of every cube. I'd like to be able to define cubes relative to nearby cube...
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I will use an example to explain my question... Transphobia is an irrational fear of, aversion to, or discrimination against transgender people. In the above sentence, is the true definition that only the fear is irrational, or is the aversion and discrimination also irrational? Is the true meaning: Transphobia is an i...
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I've been shown an Out of Office template, and one of the sentences in it asks the sender to contact someone else within the company "in case of need", as the email will not be forwarded. I've never come across that phrasing before. Is it British English? Usually, personally, I'd go with "if/as required", although that...
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I have one cylindrical coordinate system attached with a laboratory device and another cylindrical coordinate system that is attached with an object in it. The object is symmetric about the z-axis in its own coordinate system, which is both displaced and tilted with respect to the laboratory system. How do I express th...
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hello, hello. I am reading the book "Geometry" by Tatiana Alekseyevskaya (Gelfand) and found myself having the question below. An external angle is defined in Wikipedia as an angle formed by one side of a simple polygon and a line extended from an adjacent side. How does that definition applies to the following polygon...
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Is it possible to have one master .cls file in Overleaf that works across different projects? For example, if I make a change in the .cls file while working in one project, it will be updated in all of the other projects that use that .cls file? I know that I could import the .cls file from another project, but I canno...
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I have a particular model with two couplings, let's call it model A, for which I have the set of beta functions and fixed points. Now I am interested in a model where I have two copies of model A, each with a different fundamental field, let's call that model B. So, model B has two couplings for each field plus a fifth...
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I am wondering that whether Kolmogorov spectrum, which describes the energy distribution in turbulence be used to describe the plasma fluctuation in the atmosphere of neutron star. As I know, Kolmogorov spectrum has been used in solar physics to model the small scale fluctuations, and there is also research on the turb...
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I know the evolution of a main sequence star to a giant is a process that takes millions and millions of years, but how fast is the VISIBLE change? Basically, will our sun, for example, slowly grow to envelope the orbits of the inner planets, or will there be a sudden inflation of that star to a new much greater volume...
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All the sources I have read from say that radial component of acceleration of a body in circular motion only accounts for the change in direction . I do not understand how this is possible. Another thing I do not understand is what provides the torque for such a body to rotate. Could you please explain to me the forces...
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Calculus invention happened only few hundred years ago. So I assume that differential equations are being discovered and solved since that time. When I was in school I learnt that most differential equations cannot be solved analytically and are thus solved using numerical methods. The question in my mind is, what diff...
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We have a compactness result, where the "flat limit" of an integral current is itself an integral current. (with some conditions) Now I am curious about submanifolds. I am expecting that an equivalent result does not hold for submanifolds in general. However I am having trouble finding a resource to verify this. Does a...
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so I recently came across the term 'virtual mass' and when I looked up more about it, it just gave me some stuff about fluid mechanics that I dont understand properly. My understanding of virtual mass is that when a body is pushed through a fluid (let's say water) it moves slower than usual and behaves like hoe a big o...
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The harmonics of a theoretically infinitely small diameter string are pure integer multiples of the fundamental frequency. However, a real string has a thickness, and when vibrating in a harmonic, the additional node or nodes should be taken into account when considering the string's length. This should mean that highe...
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In documentation for electrical vehicles there is a common acronym for "state of charge" that is written as "SoC" or "SOC". Are both versions equally ok, or are there rules guiding when to mix case, and when not to mix case? I hope this question doesn't fall under the category "Questions that are primarily opinion-base...
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I am a self learner and I just finished calculus books (Thomas calculus , Larson book ) and I want to learn further topics in integration so I searched for good books on integrals and I heard that the book A Treatise on the Integral Calculus by Joseph Edwards is very good but this book seems very old , so my question i...
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I have been trying to understand how to calculate the kinetic energy of a rotating rod that also has translational movement. I have written my question down on paper because I think it is more clear this way. In case B, I think I have miscalculated the kinetic energy when dividing it as the sum of the rotational and tr...
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Let's say the original source says: "They were at the loud, busy, fair. They played many games there." My adaptation is: "They played many games [at the loud, busy, fair]. Since "at the loud, busy, fair" is taken verbatim from the original source, do I need to write my adaptation as: "They played many games ['at the lo...
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Specifically in relation to meteorology. I was wondering if the angular momentum an object, lets say a parcel of air has due to the roation about the earths axis. Is it conserved if moved to a different lattitude. According to my understanding the conservation of angular momentum applies as long as there is no torque o...
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Hi been trying to solve this question : Show that the following classes are not closed under ultraproducts: (a) the class of torsion groups; (b) the class of simple groups; Thought about showing that they are closed under isomorphism and to show that they are not axiomatizable hence they are not closed under ultraprodu...
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I am looking for resources treating the problem of simultaneous DC Stark and Zeeman Effects for arbitrary directions of the magnetic and electric fields (not necessarily orthogonal or parallel). I am in particular interested in the effect of such simultaneous fields on atoms with hyperfine structure such as alkali. Of ...
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You pull cards out of a shuffled deck until you get a jack and a queen. Given see the jack first, what is the expected number of cards between the jack and the queen? I believe this is a conditional expectation problem. Our goal is to subtract the expected position of the jack from the expected position of the queen. H...
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Abbot et al. "Experimentally probing the algorithmic randomness and incomputability of quantum randomness" remark that "incomputability is a weaker property than Kolmogorov randomness". I understand that a Kolmogorov random infinite sequence is incomputable. The statement implies that there are incomputable sequences t...
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On a fire extinguisher's cover, "In case of fire, break glass" may be written. As you see, in short, instructional contexts, one often finds articles being omitted. Is this grammatically correct? Is there a name for this kind of writing? Must it instead be written as "In case of a fire, break the glass"? This omission ...
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This is a less concrete question I was just curious about. We always talk about straight lines of symmetry and reflections but can non-straight lines of symmetry exist when reflecting something, and if so what would they look like? Logically, I'm not sure if it makes too much sense to ask something like this since I gu...
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I've been playing a game and after completing a mission, one character ask to another the following: How you doing on the feddy? From the context, I assume that it's somehow related to money, but I'm not sure-it's an odd word! The only site I found it was on Urban Dictionary. My question is, what does feddy really mean...
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Would a high-pitched unhearable frequency be heard whilst the doppler effect is in play? For example, when a car uses its horn whilst travelling by, the pitch shifts as it passes - which is the Doppler effect. What would happen if someone were to play a frequency so high that humans couldn't hear it and went past a bys...
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Does there exist a continuous function defined on an open interval and being differentiable in a unique point? I have found examples of real-valued functions which are differentiable in a unique point of their sources, but all of them are not continuous at the points where they are not differentiable. I would like to k...
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Why are pressure hulls machined to be nearly spherical? According to Wikipedia, https://en.wikipedia.org/wiki/DSV_Limiting_Factor has a hull that is machined to be nearly spherical "for enhanced buckling stability". Is this really superior to an unmachined hull in terms of pressure resistance? I can see why it might be...
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So I'm learning data science concepts for analysis of stock price data and I've gone into a rabbit hole. I first came across numpy.triu() in the Numpy library which calculates the Triangle Matrix for a dataset to find large movements in stock price points over a range of time. Then I learned more about the concept and ...
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I was reading the paper on the Hall Effect and found that the initial experiment was performed on a metal strip. The experiment on the metal failed to provide any useful results, and then it was stated that Owing probably to the fact that the metal disk used had considerable thickness, the experiment at that time faile...
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Spatial gradients in the refractive index of a medium have an impact on the wavelength, frequency, and direction of waves propagating through them. What about media whose refractive index changes (spatially uniformly) with time? Is it correct to say that there will be no changes to wavelength or direction of propagatin...
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I was invited to review an article by one of the IOP journals (Journal of Physics Communications) a few months ago in which the author claimed that the EM fields, similar to photons, bend in a gravitational field. The calculations were slightly hard and I had not enough time to follow the calculations, and thus I had t...
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I want to design new characters (mainly ornaments but also mathematical symbols) for the CM modern font family using METAFONT. What is the best way to achieve consistency with the rest of the family? The first idea that came to my mind was to copy paste the cmbase.mf file from https://ctan.org/tex-archive/fonts/cm/mf b...
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I'm aware that in sentences using neither-nor constructions, the verb must agree with the subject closest to it. E.g.: Neither the dog nor the cats have been fed. Neither the cats nor the dog has been fed. However, when it comes to question forms, what should the verb agree with? Does the rule still apply? Has/have nei...
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For a phonon mode to be able to emit electromagnetic radiation, it must have a dipole moment. However, it is not intuitive to me why the LA phonon does not have a dipole moment: suppose all atoms in a unit cell are oscillating in phase back and forth relative to the electronic background, isn't this itself a dipole mom...
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I am confused about how to understand the sheaf on empty set (as an open subset) of the Zariski Topology over the Spec(Z), which is generated by the additive identity zero in Z. The sheaf I have in mind is the structure sheaf sending open sets to local rings. I read related pages and I cannot understand how sheaf over ...
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I've seen these words both used but not interchangeably. My guess is that fascistic refers to 'fascist in nature', e.g if a building is fascistic it is reflective of fascist ideology but not necessarily a product of a fascist regime while fascist means relating to actual instances of fascisism e.g, saying a building is...
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I have read that the Einstein-Cartan theory introduces torsion into general relativity in a way that produces coupling between gravity and the spin of particles. Then, the gravitational field is able cause the precession of this spin axis. But what about spinless particles? Wouldn't there be a difference between the wa...
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Strictly speaking, a stochastic process can characterize a collection of random variables whose index is not time-related. Please, are there examples in the literature where this has been considered? Otherwise, should we refer to those processes as random fields as opposed to stochastic processes, reserving the term "s...
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I watched a video lecture that talked a bit about when proof by contradiction is and isn't a useful approach. This was useful, but it was from a heuristic approach which caused me to wonder is it the case that "P is provable" implies "there is a proof by contradiction of P"? I suspect this may vary by logic system. If ...
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I am helping a friend edit a manuscript for an informal reference book about music. There are several instances where I am perceiving a mixture of tense but I'm not sure how or whether to correct it. Here is an example sentence: If we begin on the first note of the C major scale and skip every other note, the following...
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I read the following text on This Page: Does every polynomial have at least one imaginary zero? No. Real numbers are a subset of complex numbers, but not the other way around. A complex number is not necessarily imaginary. Real numbers are also complex numbers. Now, FTA states that: every non-constant single-variable p...
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Recently I saw two interviews, one with Victoria Beckham, the other one with Elton John. They were talking about their past experiences and that's the phrasing they used: But it wouldn't have happened if we HADN'T HAVE GONE to the Ukraine. (Elton) And I probably would still be that way if I HADN'T HAVE MET the girls. (...
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This is a silly question, but can it be done safely? My friend wanted to have a soup with electrolysis causing it to taste differently on each side of the bowl. I think this is dangerous, because there is table salt in the soup and I suspect that during electrolysis chlorine gas is made. Maybe someone can clarify the s...
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I am studying some results about existence and uniqueness of solutions to some PDEs and many times Banach fixed point Theorem is used. I saw that the ideia is to consider the integral formulation and use the Theorem to prove existence and uniqueness in a ball but I can't understant how the uniquess extends to all the s...
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I have seen that there exists an intimate relationship between electric and magnetic properties. From what I know is that electric field creates electric potential around the source's space which can allow a charged particle to gain some PE and makes it possible for that particle to have translational and rotational mo...
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I've recently found a really old "Philosophy of Math" book in my University library, and in the book it says that the it has been proven that : if non-euclidean geometry is inconsistent, then so is euclidean geometry, and the reverse is also valid. But the book wasn't as helpful as it doesn't prove, or give some refere...
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I am competing in the American Computer Science League (ACSL), and I get problems similar to the following. Look at this directed graph. Now tell me the number of cycles in said graph. These questions need to be completed quickly and by hand. Unfortunately, the ACSL wiki (which provides help and study resources for the...
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The energy of photons is reduced by the expansion of space. Since the photons have been decoupled from matter their wavelengths have decreased from the visible range to the radio range. The same should hold for the neutrinos that emerged from the big bang. They still travel at a speed that's close to the speed of light...
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Every number theory book seems to start with divisibility, congruency, the division algorithm, etc., while completely ignoring the rigorous construction of the natural numbers. It seems like a wasted opportunity to explain to the reader what's actually happening. I feel the same about combinatorics. Permutations, combi...
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I am writing a an essay comparing The Great Gatsby and Pride and Prejudice in English (Australian English style), and to save on word count I wondered if it would be appropriate to refer to them, after the first few uses of the names, as Pride and Gatsby respectively. Is this acceptable to do in a relatively formal con...
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I just built Lyx on Manjaro linux from pamac (there is an entry from the AUR, see https://aur.archlinux.org/packages/lyx). When I try to view a document in pdf using pdflatex an window pops up saying an error has occured in the external program "pdflatex" and that I should see the logs. Unfortunately I can't find the l...
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I need to quote and excerpt a lot of the same passages from a diverse library in my work. I'd like to find an app that's basically a purpose-built database for this kind of thing -- where I can store the excerpted text, the BibTeX source, the page number, a brief description, maybe some keywords/tags/categories. Basica...
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In a nuclear fission reaction the total energy and thus the mass of the products are lower than the total energy of the reactants. I understand the the difference in this energy can be used in a nuclear reactor/atomic bomb. But in the physical sense how does this energy manifest, is the kinetic energy of the reactants ...
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When observed from the ground frame (assuming no friction is present), the only forces that act on the rotating bead are weight and the normal force exerted by the loop towards the center of the loop. I suppose there is no component of the normal force in the direction of the bead's movement, so it shouldn't move upwar...
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In Tao's Fourier Transform preprint, https://www.math.ucla.edu/~tao/preprints/fourier.pdf, they show explicitly that the Laplacian can be viewed as a Fourier multiplier. In the same paragraph, they write ...This identity shows that the Fourier transform diagonalizes the Laplacian Why is the word "diagonalizes" used in ...
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Just a naive question about univalent foundations. As far as I understand, we want to define our mathematical types like sets, groups, categories, etc. such that structurally identical objects are equal in the sense of internal equality. But are we free to choose what kind of isomorphisms we are interested in? For exam...
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I was just thinking about this studying for a biology test as for cells a greater surface area to volume ratio is desirable, so I wondered if there was a theoretically optimal shape that would maximize this ratio. I thought it might be that there is no such shape as you could just add folds to the surface indefinitely ...
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This question is similar to this post, but in my case the actual numbers do not matter, I'm interested in the equations. I have an arbitrary triangle, of which only angle BAC and side AB are known. The sum of sides BC and AC (therefore the perimeter) is known and fixed. I am looking for an equation that gives me the va...
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I've mostly used and seen "utilitarian" in a negative sense: something that only exists for its function, and specifically for those functions. In other words, something that is single-purpose. utilitarian designed to be useful rather than decorative: But what if I wanted to describe something positively, could utilita...
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I am looking into uses of Fourier series. I learned that it can be used to approximate functions (link). However, I think there are ways to approximate functions with polynomials (not limited to Taylor series). What advantage does the Fourier series have over polynomials in approximation? I think there is a disadvantag...
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I was thinking of the trading of kinetic energy during a gravitational slingshot maneuver and wondered if the kinetic energy lost during that process makes any noticeable impact on the orbit of the planet. Since the planet we are performing this maneuver on loses kinetic energy, it is realistically possible to do enoug...
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Taka a standard spinning bicycle wheel with its spokes and place it in a magnetic field that is parallel with the axle. I know that if the spokes were not made of metal but some insulating rods there would be an emf and a corresponding current induced in the rim. My question is what happens when the spokes are made of ...
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Do Boltzmann brain thought experiments suggest literally anything can form randomly? What are the limitations to what random fluctuations can form? Literally any physical, material object? Lastly, I am curious as to how this compares to an object with a high degree of complexity always existing. For example, is a bike ...
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In a scientific abstract, I have a sentence like this: Yet, numerous capture-mark-recapture data sets, across wide taxonomic range, feature transient individuals. I want to emphasize that there are numerous capture-mark-recapture data sets, and that they all together cover a wide taxonomic range. Not each one of them; ...
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I am very new to semiconductors and I'm trying to understand how photodiodes work for imaging sensors. If I am not wrong, a PN junction is required to detect electron/hole pairs that are photo-generated. The electric field in the depletion zone will move the charge carriers and create a current that can be exploited. W...
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The statement reads as follows "If is invertible, there are no special solutions in the null space." According to me, its actually true because if the matrix is invertible, it means that it has full rank, and its null space contains only the zero vector. However, the questions asks you to explain why the statement is f...
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Not that I work in this field, but I have heard the discourse from several people that string field theory is useless. So I would like to ask two simple questions? What makes SFT so difficult? Especially, it seems to me that one can construct an action that satisfies general coordinate transformation invariance and gau...
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A devout Christian, she plays tennis every weekend. The most beautiful of villages, Ullapool has hosted a food festival for the last twenty years. I see this kind of thing a lot in non-fiction writing, particularly things like short bios/blurbs and tourist writing, where people are trying to convey information but also...
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I'm aware that a semantically complete theory can be undecidable. (I believe it's because only logically valid sentences need be provable for a theory to be semantically complete.) But is it possible for a syntactically/negation complete theory to be undecidable? And do "syntactically complete" and "negation complete" ...
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I am having a hard time understanding what eigenvectors and values intuitively represent in an adjacency matrix in a graph problem. I understand in a linear transformation, eigenvectors are vectors that are only scaled after the transform is applied and that the eigenvalues are those scale factors. Is it strictly for c...
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I have these two graphs here: I wish to determine if they are Isomorphic. I know that I need to find a one to one and onto function, however I can't find a way to do it. My questions are: I know that these two are isomorphic. What is the function f ? In general, is there a way to find these functions ? Maybe using matr...
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I've seen diagrams for projectors where condenser lenses focus light directly onto the pupil of the projector lens, after going through the image source. Am I misunderstanding the diagram? If not, why does it focus directly onto the projector lens? I was under the impression that you'd want the condenser lens to form a...
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Being impressed by Barr and Diaconescu paper entitled "Atomic Toposes" (https://www.math.mcgill.ca/barr/papers/atom.top.pdf), I would like to ask whether it makes sense to investigate "Atomistic Categories", namely categories such that for any object the corresponding poset of subobject is atomistic (namely every eleme...
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I am rather rusty and am reading Topology by Graham and Greene. One of the problems refers to a countable complete metric space. The rationals don't seem to be such a space, because the irrationals are adherent to them and one can form a Cauchy sequence of rationals that converges to an irrational. Thus, the rationals ...
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I want to say something along the lines of "We must protect everyone, especially children". But this doesn't exactly work for what I want to say. Because I do not mean that children are MORE deserving of protection in this instance. How can I exemplify a group without raising them up to an elevated status? I thought of...
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In most material I can find on the cosmic distance ladder for a lay-level reader like myself, it never seems to explicitly say what distance its talking about. There is the proper distance when the light was emitted, and the comoving distance, where the light source is now. The light we are receiving however has travel...
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Lenz's law was explained to my class using an example wherein there is a bar magnet passing through a current carrying loop in a gravity free space. Our professor told us that the induced magnetic force is repulsive in nature in order to perform negative work on the bar magnet. The kinetic energy lost by the bar magnet...
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In Landau's book of fluid mechanics, the most general formula for stress looks like this: let's call the eta in picture the first viscosity coefficient, call the zeta in the picture the second viscosity coefficient. I would like to know what are the typical values for these two coefficients for air. I am not sure if my...
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As an Electrical Engineer, I have been studying convex optimization for a while. During my study, I see that most textbook claim that both second order cone programming and geometric programming can be solved effectively with modern solver. However, I cannot find any document that compare them together. So my question ...
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