Intuitionistic logic can't prove as many sentences as classical logic, for example: Peirce's Law; Reductio Ad Absurdum; Double Negation Elimination; and Tertium Non Datur, which are all equivalent in classical propositional calculus and they can't be proved in intuitionistic systems. I can prove their equivalences in some famous system, but I can't prove that we can't prove all of them in intuitionistic proof system especially without using semantics. I would like to prove this fact only using syntactic properties, i.e. proving these formulas are not in the set of provable sentences or proving that there aren't any proof of them in the set of proofs of intuitionistc proof system, without interpretation into some semantics: Kripke models or Heyting algebra. Where can I read this topic? is there any great book for this proof?


I'm using pandoc in my website to allow my content creators to use Word, which would simplify developmnent immensely. The only issue I have is that Pandoc seems to output very little information about document styling. It's possible for me to manually re-add these with CSS later, but that somewhat defeats the pupose of using Word. Hence my question. Can I get pandoc to spit out more information about how my document is structured, such that I can apply more generic styles and have the document looking comparable to how it was written. Example, I've got a table with user-defined centre-aligned text (the person centred the text in Word). When I export my document to HTML using the following command: pandoc --from docx --to html --embed-resources --reference-doc ./reference.docx --section-divs. (I'm streaming in the contents of the file via stdin, as it isn't guaranteed to be in a consistent location)`. When the HTML comes back, the contents of the table elements are simply paragraphs. I'd like them to be wrapped in <center> tags or the equivalent - just something to identify that they need to be centred. There's a number of examples like these that I'd like to introduce, but momentarily, centring text is a priority. Thanks for any pointers


When there is wind at a surface, and a rotating cylinder is placed there, mounted in a way that it is free to reorient itself, can it be predicted if the cylinder will align itself in a certain way? Perhaps it is easier to first consider the same scenario but without a surface. So, simply, a rotating cylinder in wind. Can it be predicted if it will prefer to orient itself in a certain way? A cylinder with its axis perpendicular to the wind direction, will for example experience lift. Is there anything to suggest a rotating cylinder, free to orient itself in any direction, will prefer to orient itself perpendicular to the wind (its axis perpendicular), such that it also develops lift? And, what if a surface is then added? May be possible to make an analogy to how cylinders rotating in the same direction repel while those rotating in opposite direction attract. In this case, the surface with wind over it may "appear" similar to the surface of a rotating cylinder.


It is very common in physics, when we refer to the most diverse theories, on the most diverse length scales, we also refer to their energy scale. It is through the energy scale that we classify a theory as being classical, quantum, or relativistic, for example. Likewise, on such a scale, we know whether we are dealing with sub-atomic or sub-nuclear particles. Thus, when constructing a theory, it seems to be of vital importance for the physicist to understand where such a theory lies on the underlying energy scale. However, for a young physicist student, it is not always easy to make such a correlation and find available material that deals with the subject in a didactic way. So, the question of how to determine the energy scale of a physical theory and classify it in the energy scale of physical phenomena remains somewhat nebulous, without many students realizing the importance of knowing how to classify the energy regime of a given theory. From this, as the question in the title of this topic already delivers, I would like to know how, given any physical theory, one can determine its energy and classify it according to the cosmic energy scale (I don't know if I can call it so: "cosmic energy" scale.). I would like, if possible, indications of references that deal with the subject.


I was walking by the sidewalk during the night when I noticed a swarm of flies circling a street lamp. It was difficult to see at first, so I tried looking in different angles. The security gate came in by pure chance between me and the streetlight. Now the insects became crystal clear. I tried to capture this in camera, and the same effect can be observed. Now the object between me and the streetlight doesn't matter. My umbrella, or even my hand also works. I would like to know why exactly the blocking of the dominant light source allows the less dominant ones to be seen. The light from the less dominant ones were still present before blocking the light. What roles does blocking play here? Many a times me and my friends have to abandon astrophotography camps in the night due to an overly bright moon or street lamp nearby. Can this same method be utilized to capture a clear image of stars in the night sky?


My teacher gave us the assignment to find the moment of inertia of any shape you want. So I decided to find the moment of inertia of our milky way galaxy. I found out that our galaxy is shaped like a warped disk, not a flat. After I found out that, I looked for the equation of the shape and density distribution of the Milky Way to find its I. But, to me, Nasa's and some high university papers had so many big wards that I could bearly understand what they were talking about(I'm Korean.. ^^). I'm sure that I can understand the equation if I had it, but I just can't find any information about them. I would really appreciate it if someone could tell me what you know about this equation(shape & density distribution) Thank you! (Img)Dorota M. Skowron, A three-dimensional map of the Milky Way using classical Cepheid variable stars


I was watching this video on Hall effect, and to demonstrate that it is not electric fields that are bending the electron beam, the presenter puts a metal plate between the magnet and the beam. So, my question is, say there was a charge instead of the magnet. How will the metal plate shield the electric field produced by a charge? My reasoning till now was, say, you put a positive charge in front of the metal. Since the electric field inside the metal is zero, negative charges will develop on the face near the charge, and to maintain charge conservation, a positive charge will develop on the other face of the metal. So, in effect, the plate has done nothing. There are still positive charges pointing toward the beam, and the beam will still perceive a force from the charge. So, how is the metal plate acting as a shield? ( If I assume that the plate is grounded, then I can say that the plate will serve as a shield. But in the video, it is not grounded.)


Some visuals are so obvious that you would think proofs are not needed. But then trying to proof them rigouresly is a whole other kettle of fish. I was stumped by the following puzzle I made for myself: (really it is no homework question) How do you proof that the axis of an ellipse are perpendicular? Yes you can see it, it is obvious but seeing in itself is no proof. Yes you cannot construct a countermodel, but again that is no proof. I am really stumped with this one, it is so obvious, and easy to see, but a proof? As definition of an ellipse I want to use: (reused from Wikipedia) An ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the points is a constant. As definition of the axis i want to use: (all made up by myself, so maybe incorrect, the second one, defining the minor axis, was a real struggle ;) The first axis of the ellipse is the line containing the longest segment possible between two points on the ellipse. The second axis of the ellipse is the line containing the midpoint of the two focal points and the shortest segment possible between two points on the ellipse,


Good afternoon everyone, I am currently resetting my mac and reinstalling everything in a clean way. I would love to have guidance here if possible. I currently installed TexLive and TextDist but now i am guiding myself with this : https://github.com/James-Yu/LaTeX-Workshop/wiki/Install If anyone could explain to me this section : Setting PATH environment variable After installing TeX Live, you must add the directory of TeX Live binaries to your PATH environment variable except on Windows. See the official document. LaTeX Workshop never touches the variable. If VS Code cannot find executables of TeX, it means that the setting of your system is broken. For the ways of setting environment variables on Windows, see link or link. On macOS and Linux, see the documentation by the rbenv dev team. Very detailed information is also available on stackoverflow for macOS. Notice that you have to restart VS Code and the operating system after changing the variable. If you can not fix the setting of your system, you can also override PATH with the env property of LaTeX tools in LaTeX recipes. Notice that, to set the PATH environment variable for VS Code Remote Development, you usually have to edit .bash_profile or .profile instead of .bashrc. See the document for WSL and an issue for Remote SSH. If you want to know about environment variable itself, please read Wikipedia and stackexchange. I do not really understand how it is important and what to do if anyone could help here !


I often have to translate sentences such as: The Department of Environment has offices everywhere in the country, and we would love for you to join us [us as in "the whole department, and not a specific team]. Or: The Police Department is assigning as many resources possible to the case, but we [the Police Department] cannot counter these threats alone I've been told quite a few times that a switch from the third person singular to the first person plural like that is to be avoided in French (supposedly because "we" could refer to anything and cause confusion, even in simple cases like these), and was wondering if the same kind of rule applies in English. I couldn't find any grammar or style guide to help me with this. All help is appreciated, thanks.


I have a great interest in synthetic geometry since I was in senior high school. Although I have graduated from university I still like them now. When I was in high school, I read many textbooks and exercise books about synthetic geometry of mathematical competition and learned many interesting things like harmonic range and inversion transformation. But I think people must have invented something new. I am really curious about what new techniques synthetic geometry has developed other than I learned in old textbooks and I want to know if there are any reference on modern synthetic geometry. Also I want to learn how synthetic geometry studies other curves such as conic curves, especially ellipses. I have read the book Geometry of Conics, and the Conics Books by Apollonius, but I hope I could read more such books with deeper results, like those ordinary mathematical competition books with various theorems and interesting and difficult exercices. On a Chinese forum I find out that there are many people discussing modern synthetic geometry and writing short essays on various topics, but I hope I can read some systematic reference books. Thank you in advance.


I have a problem I fail to research properly, so I hope you may at least push me in the right direction (or maybe even provide me an answer right away?). I know how linear regression works, that it attempts to find a linear curve such that the sum of all residual errors to the square of the given data points is minimal (least squares). Now, how does one go about, when data points have deviations attached? I have a set of data points, where every point has a unique deviation. The linear regression should still work as usual, but how do I get the resulting error bar of the linear regression curve? Also, since deviations fluctuate, I'd assume some sort of weighted method would make more sense in order to prioritize those data points with less deviation relative to those with larger deviation. What weighted method would make most sense in my case? Thanks a lot for your reply!


I've read other answers about how vinyl records reproduce sound, but they don't quite address a main thing I'm curious about. Play Middle C on a piano or a Clarinet, its the same note that can be played at the same volume because the main note is the same. You can tell the instruments apart because there is a different timbre in the instruments due to the components of the other harmonics present in the sound that differ between the instruments. You can break this down using Fourier Analysis. Part of the reason for these differences is the natural frequency of the materials involved. Strike metal with a small hammer, you typically get a higher pitched sound than if you strike a similar volume of wood. A vinyl record has its own natural frequency, yet it can sound just like instruments neither of which sound like each other. Somehow the natural frequency of the vinyl doesn't matter. What's going on there?


If an isolated conductor has a net zero charge and is not in an external field, would the free charges still move to the surface of the conductor eg. a conducting sphere? Wouldn't this create an electric field inside the conductor pointing radially outwards as there would be positive ions within the conductor and negative charges on the surface? And then wouldn't the electrons move back to join the positive ions? [I understand how free charges will move in a conductor to cancel any external field, cancelling the external field inside the conductor, but then wouldn't there still be a field pointing radially outwards?] I also don't understand this in terms of energy - why is it a lower electric potential energy position for the charges to go to the surface - wouldn't it be lower energy for the electrons to go back to the positive ions? Gauss's law talks about there being no charges inside a conductor, but wouldn't there still be all the positive ions left there if the conduction electrons have gone to the surface? My guess is that people are usually talking about any NET charge going to the surface of a conductor. But wouldn't an isolated conductor at equilibrium (as in the general textbook questions on this) be net neutral and so then not have charge on the surface?


I understand my question sounds stupid but hear me out. I wondered if protons or any charged particle could generate photons and I found this wonderful answer that says yes: Does shaking an atom produce photons? The issue now is that the more I thought about it the more I became confused and here is why. If any shaking charged thing can make a photon, well, which way does the photon choose to go into? I don't have a clear understanding of photons so maybe that's the problem, so to better answer me I'll give my understanding. Photons are packets of energy that represent an oscillating electric and magnetic fields. Please correct me on that if i'm wrong, I'm mostly using the visual model of a photon when i describe it. How does a photon decide in which direction it moves in? Is it just completely random?


I've gotten some contradictory answers on where the definition of the element within a set comes from. From a quick Google search, I got this strange idea that the definition of the element comes from the definition of a set. I reached this conclusion because some websites state simply that the element is A member of a set But I feel this results in some type of circular logic (At least I think that's what this logical fallacy is called). Since the definition of a set is A set is a collection of objects whose contents can be clearly determined. The objects in a set are called the elements of the set. (Robert blitzer college algebra) I am almost certain that the issue here lies in the definition of an element I found, But I can't seem to find a definition of the element that is independent of the set so to speak. I hope you guys can help clear this up for me thanks


Suppose the situation where an object undergoes linear motion at a constant velocity on a frictionless surface. This motion is often introduced as inertial motion, which is the motion described with Newton's first law. On the other hand, suppose the situation where a person applies a force to an object on a surface with friction, adjusting the force to balance the kinetic frictional force and causing the object to undergo linear motion at a constant velocity (here assuming no torque). In terms of the fact that net force exerted to the object is zero, I know there has been recognition that the motion of the latter case can be interpreted as inertial motion according to Newton's first law. However, is such motion really an example of inertial motion? Such motion seems to be rather different from the motion of the former case (close to Galileo's original thought experiment in which the essence of inertial motion is highlighted in idealized situation with no friction). Or is it just a problem of how we define inertial motion?


I have an understanding of electrical circuits, however I am very interested to know more about electromagnetic waves radiation. In particular I want to know how an oscillating voltage causes the electrons move inside a conductive antenna. Voltage is only a difference in electrical potential in our local circuit, it is only difference between electrical potential of two nets (ex ANT and GND) so how the ANT net can itself cause movements of electrons inside an antenna? (because in some circuits the antenna is only connected to ANT.) I want to know which physics model explains this phenomenon? as far as I know circuit theory doesn't explain this behavior. so It should be something else. I know there are some formulas to model electromagnetic radiation, however I want to build an intuition around this, I don't need formulas to solve problems. Also I know that speed of the light has relation to design of the antenna, so I think the theory should include speed of the light as well. I prefer to not get very deep in physic subjects, so I am looking for the most higher level theory/model which is able to fully explain this :)


This is a discussion I was having with a friend: Its very likely that the temperature of the water you place a tea bag in while change the flavor of the tea (although I don't know why exactly from a physics perspective) But suppose I have water at a certain temperature: If I let the teabag naturally sit and soak in the hot water.... compared to taking a spoon and pressing/squeezing the teabag as soon as it lands in the hot water: will the flavor of the tea be different? From a diffusion perspective, I know that squeezing the tea bag in the hot water changes the color of the water much faster compared to letting the tea bag rest. But based from a physics perspective, looking at the interaction of the water and the tea molecules - will accelerating the diffusion process of the tea by pressing/squeezing change anything about the tea? (e.g. intensity, flavor). Are there any physics related equations (e.g. statistical mechanics, thermodynamics) that can explain this (if this hypothesis is true)? Thanks!


I downloaded a latex template for Wiley journal articles, and can open the template file but cannot seem to get Texstudio/Miktex to find the .cls file. The template comes with a latex file which uses the class as an example, and if I open that right in the extracted ZIP folder, all is well; however, if I save-as anywhere else, I get a "cannot locate name.cls file". I have made sure that I saved the extracted ZIP folder within a directory listed in the MikTex path (have tried two different locations) and I have refreshed the filename database (multiple times) but still the same error occurs. This also happens if I open a blank file and attempt to use the class. I have closed and opened TexStudio a number of times, but I am unsure of what else I can try.


During the last few days I have been interested in the gravitational hierarchy problem and the different explanations for it/solutions to it. Among the most "concrete" (insofar as anything this complex can be concrete) explanations are several realisations of the so-called "brane cosmology". In these, whether they be the original ADD model or some string theory model, the standard model particles/modes are confined to a brane (either due to them being open strings or some other effect) while gravity/gravitons can freely propagate into the so-called "bulk", therefore allowing gravity to "leak" away from "our" spacetime thereby explaining why it's so weak. Common to these theories is that while these dimensions can be taken to be either compact or non-compact (but often are assumed to be compact) they are "large" extra dimensions. Now in your "garden-variety/lift pitch"-string theory the necessary extra dimensions are assumed to be very, very small and compact. My questions now revolve around the parallels of these Kaluza-Klein compactified theories and the brane cosmology ones in regard to the hierarchy problem: Can standard model particles/modes access the extra dimensions? I seem to recall that this should only be possible at "high enough" energies but I am unsure. Can gravity/the graviton access the extra dimensions even at low energies? If so, why? What is the mechanism/explanation for this? If neither of the above yields a similar solution to the hierarchy problem as in the brane cosmology scenario, how is the hierarchy problem solved in these "classical ", small-extra-dimensions Kaluza-Klein types of theories?


So imagine the classic science experiment where you take an empty aluminium can and a rubber bar that you charge with fur. At first, that can is electrically neutral. Two different cases now that I want to discuss: If I put the can on a desk made out of wood, the can will be attracted by my charged rubber bar due to influence. So the electrons in my rubber bar push away the electrons being in the can as far away as possible (the other side) and so the rubber bar only "sees" the protons. If I put the can on a desk made out of a metal which is grounded, what happens then? Still influence? Or is the electrically neutral can now positive due to electrons escaping to ground? Would love to hear some thoughts. Best regards


Can we determine the truth value of the statement "if A then B" knowing only that B is definitely false whenever A is true, for example, "If you live in Paris, then you live in London." From what I understand about truth tables, this on the face of it is indetermined because it could be true or false based on whether you live in Paris and not in London, or you don't live in Paris. But at the same time, the fact that we know that whenever A is true B has to be false, seems to go against the purpose of implication, that whenever A is True B has to be True, and from that alone, the conditional statement can't be anything other than false? The book I'm reading says that the statement in the above example is false.


The existing English language term needs to refer to a robot that can navigate environments and it incorporates a human who is present virtually inside the robot from a remote location. The human navigates the robot and communicates with the environment in real-time. The closest example are the so-called telepresence robots being tested for use in classrooms or offices, but they are not exactly what I am describing. I feel like there is a term for such a machine vessel that 'embodies' a human, or that there should be such a term. The terms robot or cyborg do not describe this vessel. I can imagine in the near future a humanoid robot that is not driven by AI but by a human from a remote location using VR with ability to sense the environment (e.g. sense of touch). For example, this would benefit persons with significant mobility disabilities by allowing them to be present and interact with environments, the idea behind the current telepresence robots but in a much more integrated way between the remote human and the machine, and more interactive with the environment. So, to recap, is there a word for such a machine vessel virtually occupied by a human?


In the picture below, mountains farther away blends into the color of sky, this is "atmospheric perspective", caused by atmospheric scattering. This effect is also called "fog". But dark blue appears in the middle of the [green -> light blue] gradient somehow. Why? Is there a physically-based model to explain that? In many video games, they used linear gradient to simulate the fog. The color what we see is a linear interpolation between the color of the object and the color of sky. It looks like the gradient in the right side of the picture. But that isn't realistic because of the missing "dark blue" color, so mountains in video games usually looks too grey. My guess: The dark blue color is caused by Rayleigh scattering. The white color at the farthest mountain is caused by Mie scattering. Somehow combine them makes the fog both blue and white. But I have no idea how.


In a room at normal room temperature, certain materials, such as metal, glass, ceramic, or rock, will feel cold to the touch, but others, such as wood or plastic, do not so much. Which physical properties do the former materials have in common that cause them to be cold to the touch but the others lack? The fact that wood is porous? Solid plastic pieces still do not get/feel as cold even though they are not porous. Most sources point to thermal conductivity, but how can that alone be the answer? Rock and rock-like materials (such as glass and ceramic) have a very poor thermal conductivity whereas metal has a very high thermal conductivity. I know that because rocks take a long time to heat up and cool down in a fire, but metal takes not very much time. Which physical properties (might be more than one) determine whether a material at ambient temperature will be perceived as cold or not?


As a forward, I'm no a physicist or a student of it. In fact I'm pretty ordinary. So if I mischarecterize some concepts, bear with me. So I was reading up on some of the new technologies and then I had a moment. I am wondering if you could use quantum entanglement to transmit data across a vast distance, theoretically even light years, in an instant. I was debating this in a youtube clip where there was a whistle blower from the antarctic and they were speaking on this. My "opponent"(I guess?) Stated that measuring any entangled particle removes the connection, but quantum computers can use gates that flip an electron spin without disrupting the connection. Another objection was multi light year transmission, but I thought that should be fine since according to the theory, it's instant, no matter the setting. So basically I'm wondering with the current tech that we have, technically couldn't we set up an experiment to send a byte of data via quantum entanglement by changing the state of an electron, using a gate to measure it on the other end, and translating that to a binary digit? In other words, do we have Faster-Than-Light communication on the horizon?


My understanding: electrocaloric effect relies on polarized molecules lining up to an external electric field, which lowers the local entropy and thus heat is given off to the surroundings. When the field is removed, the molecular dipoles again arrange randomly, which increases the local entropy, thus heat is taken in from the surroundings. I ask about a macro level system. Here I have micron sized particles, each particle made of two hemispheres using different materials from the triboelectric series. These serve as my "molecules". Upon initial shaking, the surfaces rub and so each particle has a dipole. Before shaking: After shaking: Now I turn on an electric field and vibrate the container a little so the particles loosen and align themselves to the field. This vibrating is kind of like of how they used to hammer the living hell out of heated iron in a strong magnetic field, so the magnetic domains align as the object solidified. Now I can't think of any reason why this won't display the electrocaloric effect. And since here we don't rely on the molecule shapes, crystallization, polarization and the intermolecular forces that prevent the dipoles from completely aligning themselves, I think this could display interesting levels of cooling and heating. Of course, I'm a nobody, so to you experts I hand over my question. Will this work, and if yes, can we quantify the effect using any equations?


I'm looking for a word that effectively conveys an attitude showing a preference for having a child related to oneself ... ie, a lineally/genetically-related child. In particular, a word that would fit the following example: Michael had a _____ attitude, he wanted children of his own and / or possibly a noun form: Michael was a _____ unlike Sam who was a non-_____ which would be helpful in distinguishing between people that prefer to have their own children and those that are happy to adopt, foster, etc. I had considered "autogenic" ... "auto" meaning self, and "genic" meaning "produced by" ... and even "progenic" (or "progenetic" possibly) which might work here, "pro-", I take to mean "following" or "in favour of" which in combination with "genic" / "genetic" relates to genetic lineage or heritage. However, I'm not sure I've constructed them appropriately, or whether they're right or not.


Scientocracy (to my understanding and as my intended meaning) is government by results, i.e., eschew policy debates to instead argue and then agree on metrics, run an experiment and then transition to whichever solution produces the best outcome as determined by the agreed upon metrics (of course, if cultural, climatic or other considerations make a new policy ineffective in a certain place, the policy should be revoked or adapted there). My question is what is the 'ism' associated with this? Scientocracism isn't a word, and ends with racism, so that doesn't really work. Technocracy (a related, if more loaded term) has the same problem. An example sentence would be: I believe in representative democracy, but I also believe in [] because while I think that we should be able to elect people to represent our interests - and fire them when they fail - I also believe in results.


In the preface to Landau and Lifshitz's Statistical Physics, they comment the following In the discussion of the foundations of classical statistical physics, we consider from the start the statistical distribution for small parts (subsystems) of systems, not for entire closed systems. This is in accordance with the fundamental problems and aims of physical statistics, and allows a complete avoidance of the problem of the ergodic and similar hypotheses, which in fact is not important as regards these aims. After explaining that they will consider (macroscopical) subsystems of the whole closed system, they claim A fundamental feature of this approach is the fact that, because of the extreme complexity of the external interactions with the other parts of the system, during a sufficiently long time the subsystem considered will be many times in every possible state. But this is very similar to the ergodic hypothesis, is it not? From my Stat Physics lecture notes, Given a sufficiently large time, any closed system will approach arbitrarily closely any point in phase space I don't understand how considering subsystems overcomes the need for this hypothesis, neither what advantage it has. It seems to me that the authors are using the complexity of the interactions between the subsystem and the rest of the system to grant credibility to the claim that the subsystem will eventually have populated the entire phase space. But why isn't this equally credible to stating the ergodic hypothesis?


Im not necessarily talking about emf (and also not considering rotation) , but rather the potential difference created due the induced electric field to balance the lorentz force on the electron. For example if the disk is moving toward the right and the magnetic field is into the screen then the lorentz force is acting on an electron downwards pushing it down. now due to the excess of electrons downwards an electric field is created top to bottom which increases until the lorentz force is equal to the force due to the electric field on the electron, resulting in an electrostatic condition. So my question is does this actually happen? if not why dosent it happen? Im a bit bothered by this because according to faradays law there should be no emf induced in the disk but according my reasoning there should be a potential difference. Is there a difference between emf and potential difference?


As I understand it, the (real) index of refraction is given by the ratio of speed of a monochromatic light wave in vacuo versus speed in medium. When it comes to the question of propagation of a light pulse in a (dispersive) medium I know that I need to calculate the group refractive index to obtain the propagation speed of the pulse. My question specifically now is, if it is possible to use radiative transfer theory to calculate light pulse propagation in a scattering environment embedded within a medium (not vacuum)? If the former is possible, can that speed be said to be the effective group velocity of the mixture of the scatterers and medium? Further can this effective group velocity be used to calculate an effective group refraction index? Background: Although I have no working experience with radiative transfer theory I can imagine that it would be possible to use it for calculation of attenuation related values such as the imaginary part of the index of refraction, but I cannot see how it would be possible to get at the real part of the index of refraction.


I've recently watched this video about an electromagnetic ring accelerator. I get how it works: it uses controlled electro-magnetic coils to accelerate metal balls' Passing trough one of the coils has two stages: when the coil passes enters the coil, the coil is powered on, generating an electro-magnetic field, accelerating the ball towards the center as the particle passes trough the origin of the coil, the coil is quickly powered off and the ball continues forward What I am curious about is exactly what forces act on it. Searching online there appear to be some inconsistent answers, but it always either is the magnetic force or the Lorenz force law. From what I understood the magnetic force is purely for describing magnetic attraction, either between two magnets or a magnetic (ferromagnetic?) object and a magnet, while the Lorenz force law is the combination of the magnetic and electric laws. Could somebody explain which of these two is and what exactly their formulas are in the context of the balls passing trough the coils?


For example, ZFC and ZF. I have come across the notion of pure and applied mathematics, and how the development of the former can (and is usually intended to) lead to the furtherance of the latter. In this case, how do we know that the axioms we do pure maths with are "sound"? For me, there is no point in doing something as rigorous as maths unless we are certain it could at least possibly provide a practical use to us, however subtle that use may be: otherwise, it is just an elaborate (and hugely unproductive) mind game. I suppose my question boils down to why we do maths and what it is. If its a way of abstractly describing physical natures and phenomena (like Plato thought), I'd have thought you could test axioms with the scientific method, in which case there would be no confusion surrounding, for example, the millennium problems. There is so much nuance about axioms and proofs, truth, falsity and their natures that really interests me, and I would like to know more about, but I don't know where to start. I have heard of "metamathematics" as related a field of study, which is why it's included in the title, but I honestly don't know much about it. I also appreciate that ZFC and ZF may be bad examples, in which case please point out why. Thank you!


In Slavery in Massachusetts, Thoreau writes: But it chanced the other day that I scented a white water-lily, and a season I had waited for had arrived. It is the emblem of purity. It bursts up so pure and fair to the eye, and so sweet to the scent, as if to show us what purity and sweetness reside in, and can be extracted from, the slime and muck of earth. I think I have plucked the first one that has opened for a mile. What does for a mile mean in this sentence? I looked up for in a dictionary, and methinks the author might mean the first lily that he came across, and that was opened, after walking a mile. Or does Thoreau mean something else here?


Shouldn't acoustic oscillations created by primordial matter anisotropies create multiple peaks and troughs when those oscillations are frozen by decoupling ? It's not unthinkable to imagine that the exact calculations of the expected amplitudes of each will lead to a result where the first peak is much more significant than the rest, but do those other peaks and troughs theoretically exist, just at very small currently non measurable amplitudes ? In other words, the BAO peak corresponds to the first peak in the CMB power spectrum, but am I wrong in assuming the other peaks and troughs are theoretically also present in present matter distribution, just at negligible amplitude ? As a bonus, if the answer is indeed yes, why are the other peaks so negligible ? I understand that the first one has the greatest amplitude, but why are the remaining ones so vanishingly small as to remain undetected in current matter distribution, despite being nowhere near insignificant in the CMB power spectrum ?


A current-carrying loop of wire is placed in a uniform external magnetic field as shown. If the current in the wire is traveling counterclockwise in the picture, what do you predict the loop will do when released? The answer provided for this question: From the RHR for loops, place the heel of the right hand in the plane of the loop so that the fingers are curled in the direction of the current. The extended thumb points in the direction of the B-field. Since it is anti-parallel to the external B-field, the loop will flip over and then remain at rest. I don't understand what causes that. When you apply the Right-Hand Rule or Flaming Left-hand Rule I prefer, the force induced is inward around the loop. How come it makes it flip over? I must be missing something deep here. What is the underlying concept that causes the wire to flip over?


If I take a pair of classical particles with some energy interacting via Lennard-Jones potential, then the motion of the two particles will be in such a way that the distance oscillates around the equilibrium point (if the energy is not too high compared to barrier depth). If I consider a large number of Lennard Jones particles packed into a crystalline lattice, then intuitively I expect that each pair of particles should have a distance close to the equilibrium distance of the pair potential. In other words the "lattice constant" should be close to the equilibrium of the pair potential. But is it actually the case ? Can we consider a pair of neighbors in a Lennard-Jones lattice to be roughly an equilibrium of the pair potential with small corrections of further particles ?


The situation: I'm trying to generate my resume with Latex, and get a pdf with all the text - no problems here. When I copy all of this text into a plain .txt file though, the formatting transforms all whitespaces (newlines etc.) into a single space, it seems. For example, lists are not vertically placed anymore, but placed sequentially. Does this maybe have something to do with how Latex files are built up (compiled) differently than ordinary text documents, causing it not to have any classical newlines characters, for example? In that case, would there be some way to insert those, allowing for a proper copy-and-paste functionality? I tried to make use of the pachage cmap, which assists with Unicode mapping, but could not get it to work. Anyone who knows a fix to this, possibly with cmap, or possibly another package?


I'm working on performing a linear stability analysis on a system with fluid undergoing applied vibrations. Now, usually these systems are analyzed with no fluid velocity. Under these conditions, it can be shown that the system reduces to the form of a Mathieu equation (or a modified Mathieu equation in the case of viscous fluid). For Mathieu equations, it's known that there are discrete regions where the system response is either Harmonic or Subharmonic resonance, which is in line with the Faraday instability that's observed on the fluid interface. However, what happens when the system equations cannot be reduced to the form of Mathieu equations (for example, for complex fluids or fluids with flow)? Can the system allow resonance at frequencies outside of Harmonic and Subharmonic, or is there something more fundamental that says that these are the only allowable options? Should I also consider the possibility that the response might not even be periodic? Analysis procedure for inviscid Newtonian fluids Analysis procedure for viscous Newtonian fluids


All explanations of jet propulsion that I've seen are formulated as "due to conservation of momentum, air with momentum coming out of one end means the rocket must gain momentum in the opposing direction". However it can't be the case that particles simply exiting some area with a momentum means all of a sudden something else must gain opposing momentum. If we had a straight pipe open on both ends, particles suddenly exiting one side (maybe a chemical exploding inside or the like) does not mean the pipe will all of a sudden start moving in the opposing direction. You need one side of the pipe to be closed in order for the pipe to move opposingly. Similarly it can't be the case that particles suddenly exiting an inflated balloon or rocket means the object must propel in the opposing direction due to some 'rule'. I'm assuming what is actually happening is a case of 'particle collision averages'. In the case of a balloon the air inside is constantly bouncing off the walls in all direction. So when I let go of an inflated balloon I have disturbed this average. There are now particles hitting a side of the balloon with no opposing force, as the particles opposing that wall are now simply exiting the balloon. Assuming my above formulation is correct, does this mean all statements regarding conservation of momentum are simply a rule of thumb for this above 'particle collisions average' explanation?


I was trying to find some references for modelling the equations of motion of a simple dynamical system (say a pendulum on a moving mass) when I realized that the very vast majority of the material you find online or even in textbooks suffer from the following problems: Reference frame poorly defined, wildly assigned or even implicit. No idea which point is considered to be origin, which direction positive, etc. Free body diagram completely unprincipled and/or confusing. Confusion between magnitude of a vector (e.g., Force) and the vector itself (not mentioning switching back and forth) No general form for basic laws of motion. Sometimes when the authors apply Newton's second law, for instance, they are applying it implicitly along some dimension. Constants are stated and used without derivation or motivation, e.g., moment of inertia are mysterious constants that you would look up from a table And many more problems. Does there exist any textbook on modeling classical dynamical systems (pendulum, masses, multi-link robot) from scratch?


While reading the Wikipedia article on infinite sets I found the following quote: A set is infinite if and only if for every natural number, the set has a subset whose cardinality is that natural number. If the axiom of choice holds, then a set is infinite if and only if it includes a countable infinite subset. This raised an interesting question; without AC, is it possible to have a set where, given any natural number, you can find a subset with cardinality great than that number, but not necessary one equal to it? For example a set where all definable subsets have even cardinality? Edit to clarify the question: are there sets that satisfy the intuitive meaning of infinat but not the definition as quoted above? FWIW, id be interested in both the case where "subsets larger than n exist" requires those subsets be finite and where they don't. (Though I suspect the second case is uninteresting as most of the interesting properties would be trivially true.)


Let's say the origin is an equilibrium position for a particle. If we have a case such that a slight displacement of the particle in x direction makes it return to the mean position (stable equilibrium) and a slight displacement of the particle in y/z directions makes it go further away from the mean position (unstable equilibrium). The directions of displacement for which particle is in unstable equilibrium are more than directions of displacement for which particle is in stable equilibrium. I have seen this question but the answer didn't clearly state whether there will be an equilibrium type defined for that position or will that position have no type of equilibrium (and the comments on that answer about Unstable equilibrium "winning" over Stable equilibrium confused me even more). So what will we actually call the equilibrium type at the origin? Or will we just define its equilibrium associated with direction (like stable in x direction and such)?


My understanding from popular science articles is that the boundary is a field theory that has no gravity and has one less spatial dimension than the bulk. However, I am not sure I understood this picture correctly. I just read a recent article at quanta magazine that states: "A solar system in the central anti-de Sitter region, for instance, can be described as a collection of particles scattered around the boundary that obey only quantum theory and have no sense of gravity or space-time at all". So my question is, did the quote mean no curved spacetime, or there is not even a minkowski spacetime associated with the physics in the boundary? I dont even understand how it is possible to have a quantum theory of particles without either space or time.


I am trying to understand Turing's halting problem proof by applying the same paradox to a similar problem where, instead of determining if a given code will halt, you instead determine if it will return True or False (assuming it will always return eventually). Obviously such a machine can exist, as it is simply a standard compiler. However, when run through the paradox: We assume h is a function that determines if the return is True or False We construct a machine P around h in the following way: def P(i): return not h(i, i) P(P) As we can see, if P returns True then P returns False, and vice versa. Thus compilers cannot exist, as a compiler is an example of h. Can someone please help me find the error in reasoning?


According to this and this answer, and as far as I understand these answers, dark matter halos cannot collapse to a black hole because, due to uncoupling from the EM field, they are unable to radiate their kinetic energy, and hence, getting closer to some gravitational center point also means that they get faster and so they resist further "collapse". But what about the motion perpendicular to the galactic plane? I would naively expect the dark matter to gravitationally fall down on the galactic plane on both its sides, until it concentrates there. Depending on whatever the type of dark matter may be, this may cause other forces (e.g. weak interaction) to take over (possibly at nuclear distances) or it may oscillate until it becomes spherical again. One way I imagine that this might happen is a small-scale alternating velocity variation in the dark matter field, so that dark matter is alternately falling to/moving away from the mid plane from location to location, and these regions simply pass by one another infinitely. If this alternating pattern in the velocity field exactly balances, the dark matter halo is able to maintain a spherical shape. But even the slightest imbalance might result in a global oscillatory motion between spherical and disc-like. Have the available observations been investigated with respect to these possible oscillations of the dark matter halos? And isn't it likely that such oscillatory motion (if it existed) would eventually stop due to second order dissipation (dark-matter to ordinary matter to radiation).


I am working in an algorithm to order a bed of close-packed spheres. In the case where I have got four spheres, I understand that the fifth sphere position and radius is determined by the positions and radii of the four other spheres. It seems that there would be different solutions: one that produces a sphere that is similar in size to the other four, and a second solution in which the four spheres are encompassed in a bigger sphere. What interests me is the first solution. Any ideas what equation system solves this? The input data for the systems would be the positions of the original four sphere centres and their respective radii, and the output would be the position and radius of the fifth sphere. It is also important to notice that the four original spheres can but do not need to be tangent between them. Thanks a lot in advance!


From what I understand, each particle has an energy called kinetic energy. When we consider a system of particles, in addition to their individual kinetic energy, there is an energy associated with the systems which depends on their relative configuration which we call the potential energy. Mathematically a system of particles can be analysed at the individual particle level by assuming that system of particles induces a field and associate potential energy for each particle and then sum up over particles to get the potential energy for the system of particles. So I thought that potential energy is some sort of energy associated with the field (which is a mathematical construct). But later I studied that fields are real and they have their own energy. So is the potential energy the same as field energy, if not how are they related? Additionally, just as the energy of a particle is decomposed as potential and kinetic energy, can we do the same for the field, i.e., potential field energy and kinetic field energy? If so, then how is the potential energy of the particle related to the potential energy of the field? Also, potential energy in my mind is related to the shape a system of particles takes. Changing shape releases or absorbs energy. Entropy is a concept which is also related to the structure/order of the system. So how does the entropy relate to the potential energy?


Can we answer the question: "What are all the types?" i.e. can we recursively generate all the types in a given type theory according to a certain set of rules? Pardon for asking such a weird question. Question/personal background: I'm leaving this question intentionally vague with regards to what I mean by "type theory" because I guess I'm just interested in the status of this question for type theories in general. My background is pure math and I am comfortable with proofs in ZFC and FOL but I can't write a single line of code in any programming language. I hope to understandthe basics of using type theory for proof like Lean does, and started reading about type theory yesterday in preparation, although I've heard snippets about type theory in passing elsewhere.


Let's say you're looking at your keyboard on your phone.. let's take the letter g. Light from g is going in every direction.. we know this because we can see g from all angles above the phone. So light from g is also hitting your eye where you would see all the other keys,screen and background objects however we don't see g in those areas.. just wherever g is located. What happened to the images of g everywhere else but where we see it? We should be seeing g in all places but something happened and we don't.. if you're getting what I'm saying I you should be as perplexed as I am.. also knowing this now think about how a camera can take a picture with light from all objects being scattered in all directions.. it shouldn't be able to take a picture like it does.. how on earth was it even made???? It shouldn't be able to work but it does!!! Does this prove something else about life?


In the same vein/spirit as the UIPAC books for standardising systematic names and formulae of chemicals, the SI units for units of physical measurements, or the INN for generic drug names, have there been efforts to write every statement in mathematics (eg. axiom, postulate, result, theorem) in a standardised language? This standardised language might be a formal logic like those touched upon in mathematical logic textbooks or monographs, or it might simply be a set of standardised terms or statements in a natural language like English. (I'd like to emphasise that a standardisation effort need not necessarily be one in a formal language.) I'd like to know if there existed, or there exists currently, a consortium, an organisation, an initiative etc. like this.. I tried searching for this on google but couldn't find one so far... Ermm, do put aside the debate on whether such an effort is desirable or not.. I just like to know if there is/was such an effort, not whether it is a good/bad thing.. I've come across TPTP.org, but not sure if this is the largest/most well-known/most followed/most active effort so far in mathematics.. Do let me know if there's any better..


In google translate, the word "bizarre" means "very strange or unusual, especially so as to cause interest or amusement.". But I believe that this description is more suited for "eccentric". For example, the character of Jack Sparrow from Pirates of the Caribbean is eccentric. On the other hand, AI chat GPT says that "bizarre" usually means something odd and unsettling. And I have seen people make use of the word "bizarre" for criticizing something, rather than an expression of fascination. In Meriam-Webster site, "bizarre" means odd or eccentric in stye. This means "bizarre" and "eccentric" share the same meaning? But "Jack Sparrow is such a bizarre person" and "Jack Sparrow is such an eccentric person" don't seem to resonate. I got different vibes from them. Or is it just my feeling? Both means "unconventional"? What are their subtle differences, and can I have one example for each that would enlighten me about their differences?


Almost every text on Category theory uses categories such as Ab, Grp, and so on as examples to work with but can category theoretic methods actually help us understand the structures better? In particular, does Category Theory aid us in proving some significant abstract algebraic results that are otherwise tedious to prove? Furthermore, is there any structural insight about abstract algebraic objects that is not readily apparent from algebra itself but becomes crystal clear with a category theoretic approach? I do not have a specific format for the answer in mind. As category theory is quite general, the answer may also include ways to say use results from other branches such as analysis or topology, so as to prove an abstract algebraic result which was tedious to prove using an algebraic approach. In this light, any insight about the relationship between category theory and abstract algebra is welcome.


I find it amusing why the infimum norm or supremum norm defined on function spaces work actually? It's difficult for me to reach to an intuition where I would see the infimum or supremum as a natural way to represent distance between two functions. Is there any intuition which can motivate the norms defined on function spaces ? How do we make sure that two functions having the same infimum or supremum don't deviate too much from each other and that the function which is a limit under such a norm satisfies the properties the functions in the sequence have ? What is the norm measuring in function spaces I went thorugh the above question but could not find an intuitive way to address the last question in the first paragraph of my post . Disclaimer : I am not a pure maths student , please pardon my immaturities . I was just trying to find an intuition here.


Im working on a nonlinear control based on Lyapunov theory and its working really well. I am able to implement it on a dynamical model of the system in simulink. However I think it has a really big limitation and that is: I cant guarantee that the real system will have the exact parameters that I am modelling. I want to make my lyapunov controller into a more robust controller by adding parameter estimation too it. I see some texts based around system identification, but I do not need to go that deep, I know what the system looks like. I just need some kind of method to zero in on the drift between my modelled and real components. Is this possible? I am controlling an DC to AC power converter. Physical System model: Model: Controller:


Often metaphors are likenesses where there's a direct connection. For example on the news somebody describes a crash/ earthquake/ explosion as It was like a bomb going off. What about where the two things are not similar. Her painted toe-nails were an explosion of colour. (Exploding feet anyone!) This poem Cherry Ripe by Thomas Campion is full of insinuations rather than direct likenesses. There is a garden in her face Where roses and white lilies blow; A heavenly paradise is that place, Wherein all pleasant fruits do flow: There cherries grow which none may buy Till 'Cherry-ripe' themselves do cry. Those cherries fairly do enclose Of orient pearl a double row, Which when her lovely laughter shows, They look like rose-buds fill'd with snow; Yet them nor peer nor prince can buy Till 'Cherry-ripe' themselves do cry. Her eyes like angels watch them still; Her brows like bended bows do stand, Threat'ning with piercing frowns to kill All that attempt with eye or hand Those sacred cherries to come nigh, Till 'Cherry-ripe' themselves do cry. My question is: Are there technical terms that distinguish between: (a) Direct likeness. eg sky-blue eyes (b) Not really like at all. eg eyes like angels (c) Over-arching 'big metaphor'. eg garden in her face


I am using TeXstudio with LaTex on Windows I also know about Git. In fact, I use Git to collaborate on RStudio but on TeXstudio I cannot. While typing on TeXstudio on my own without collaboration, I had my first problem saving different versions of .txt files on the same project, which sometimes confuses me. When I was writing my PhD thesis, I would send the PDF to my professor, which he could not edit. How do I solve these two problems? WHAT I WANT I have read different posts on reproducibility on TeXstudio, but I cannot figure out my foot on any. I want a step-by-step guide on achieving reproducibility and collaboration on TeXstudio using Git. I want to be able to revert to my previous changes without saving multiple documents.


Goldblatt gives a brief overview of adjunctions in his "Topoi", and one of the exercises asks to characterise the partial arrow classifier in terms of some universal arrow. Well, I gave it some thought, and I'm not even sure where to go from there. The main problem I have is that in the "there exists" part I have only one arrow (the PAC itself), and in the "such that for every" part I'm given two arrows (the "top-left" components of the PAC pullback), and I'm just not sure how to construct a category and a functor into/from it that'd allow for both. I tried something along the lines of the diagonalization functor used for (co)products derivation via adjoints, but it didn't get me far (or anywhere, frankly). So, how would I do that? Again, my exposition to adjoints is limited to several pages of "Topoi", so I likely miss some otherwise well-known results.


In statistical thermodynamics we can prove that the evolution of a system minimises some potential with units of energy (e.g. energy). This can be done purely statistically, by using the first two laws of thermodynamics, and showing that the state where the appropriate potential is minimised is the most likely one. Therefore, minimisation of energy is just an argument about entropy. Is there a similar (or any other) proof of the principle of least or stationary action? I have seen it referred to as an axiom, but is there at least a possibility that there exists an underlying theory from which it can be derived? Can it be shown that systems where action is stationary are the most likely ones? Also, it is interesting that the Lagrangian has units of energy as well. Edit: To clarify on the first paragraph. I was referring to quasistatic systems that satisfy a sepcific set of conditions. Energy is minimised in the stable state of an isolated system (where entropy, volume, and the number of particles is conserved). For a system where the temperature, pressure, and the number of particles are constant, the Gibbs free energy is minimised etc.


I am at a loss to understand the justification behind the following statement being false and would appreciate someone explaining what this particular argument is saying: The set of all finite subsets of a countably infinite set is uncountable. Answer: False. Any countable set has a bijection with the naturals, and one can list the subsets in order of the sum of their values. Every finite subset has a finite sum and therefore will appear in the listing. I am not sure how this argument proves the falsehood of this statement, or what the process of its logic is. Do we not have to say something about how different subsets will have the same sums? I would appreciate a more formal version of the above argument so I could understand what point it is making.


In common parlance, we say "s/he pretends s/he doesn't understand", for people who ignore the context of the conversation and require hyper-specific definitions to keep the conversation going. (hyper-simplified) example of the above, in the form of a dialogue: Q: Hey, how's your day? A: In what sense? Q: Uh, let's say for the sake of argument, how's work? A: What do you mean by that? In terms of what? Q: What are you up to? A: I do lot's of stuff, you have to be more specific - is there something in particular you're interested in? Q: Nope just in general, are you enjoying it? A: Sorry I don't understand, define "enjoying it" in the context of work and so on... Is there a term for what the person (A) is doing?


Based on the double slit experiment we know that in the case of a single particle system the wave function or state vector of position is in a superposition of possibilities before measurement. But does this rule apply in the case of the vacuum state? Is it in a superposition of (non degenerate energy) possibilities before measurement? A physicist has replied to this question, in a personal correspondence, by noting that: It is thought that in quantum field theory, different vacuum states are essentially classical, and don't get superposed quantum mechanically. The reason is their effectively infinite spatial extent, so the distinct vacuum configurations are mathematically very distant. Yet I am not well convinced by his answer. Isn't a quantum state, either a single particle or a vacumm, a nonlocal object in the Hilbert space and independent of physical space? So why the vacuum state is an exception from the general rule of quantum theory that applies to position or momentum quantum state? Why being "mathematically distant" rules out being "superposed" in the case of vacuum. This is not trivial to me. Is there a theorem about this? Should we suppose that there are multiple Hilbert spaces and therefore in a single Hilbert space there is only one quantum state for vacuum? and "by mathematically very distant" we need to consider another Hilbert space with its own vacuum state?


I have the following books about QFT: Peskin-Schroeder Lancaster-Blundell Iliopoulos-Tomaras I would like to know if there is a QFT that is as concise as Dirac's book about general Relativity. What I found useful about Dirac's book was that it developed most mathematical machinery "from scratch" with very little comments. I would like a "quick reference" QFT book that allows one to study the matter from a more mathematical perspective, rather than using a lot of physical analogies as in the books I have read. As an example, I would like to see "This is the QED Lagrangian. Those are the consequences", but a more classical, experiment driven approach is appreciated if it is strictly functional to the development of the theory and if it is kept as concise as possible. Ideally the book should be: More or less "self-contained", allowing a student with all prerequisites to learn introductory QFT from scratch (at least to renormalization) As concise as possible (ideally also in the number of pages) I would also like lecture notes or papers, it does not have to be a book.


I would like to use this template from Overleaf for a PhD dissertation at the University of Aberdeen, offline on TeXnicCenter. (This appears to be a legitimate thing, according to my supervisor.) I have looked at this; it didn't help. My main issue is that I'm not sure how to use the cls file. There might be other problems once that is sorted, so if you could explain to me how to do it, please, I would be most appreciative. I'm not sure what I can offer in terms of context. But here: I did both my MMath and my MPhil dissertations using TeXnicCenter, along with much smaller projects, so I have a fair amount of experience with it. Like I said, I think the issue is with the cls file. I believe it is something like a package, but one that I have to install myself - I don't know . . . I am likely to award a bounty to the first answer that works when I try it!


Three non-collinear points are placed randomly inside a unit circle. Question: What is the probability that if you were to connect these points, forming a triangle, the triangle will have the center of the circle contained inside (including the periferi)? Here are two examples, A and B, illustrating what I mean: where you can see that A works and B doesn't work. What Iv'e tried so far is illustarated in the following image: Two randomly points are obviously always collinear. So we can connect the two. Now let's draw a staright line, starting from each point, through the center and finish on the circumference of the circle. The grey shaded area represents the region where the third point needs to be in order to create a triangle that works. Now, as you can see there can be all kinds of regions depending on where the first two points are located. So I'm simply looking for a method/approach that can answer this question. If you have a another approach than my example feel free to use it. I'm not sure if this is very complicated or if there is a nice solution.


If we are given that propositions P and Q can never be true, is it still accurate to say that P and Q are necessary and sufficient for each other, and why? I am conflicted here, as the statement P iff Q is true here and I have learned that this also means P and Q are necessary and sufficient for each other (for context, I have learnt that P is necessary for Q if P must be true to conclude Q is true, and that P is sufficient for Q if Q must be true to conclude that P is true); however, it seems counterintuitive to stipulate conditions for propositions being true (by deeming other propositions as necessary or sufficient for them) if these are never relevant to the situation, as these propositions are always false anyway. I would appreciate an answer grounded in an explanation of necessity and sufficiency.


I am not well versed in TikZ and PGF Plots, and I am hoping someone would be able to help me with developing the structural code for a Parabola with its Focus and Directrix. I found an image of what I am assuming was created via LaTex and was hoping to recreate this somehow. This was a perfect image of what I am looking for, because the conic sections seem to be so lost to time that I can not find good representations of this particular topic without hand drawing it. I found a sample code for drawing a parabola, and when I attempted to recreate it, it does not let me use a fractional coefficient. If anyone has starter code to this image, that would be greatly appreciated


My understanding is this: the Solar surface becomes speckled with more sunspots near Solar maximum, and these spots tend to form groupings known as active regions; each spot is associated with a given magnetic polarity, and often regions will tend to have spots of both polarities that are typically linked in certain ways, but sometimes the formed regions are much more complex, especially as the polar fields are about to flip and Solar activity tends to increase. Now, my question is this, as the title says: which way does current flow in the plasma streams that form the coronal loops and filaments that are generally "rooted" in the active regions? Is there a way to determine the direction of current solely from the magnetic polarities of the spots involved, or can current flow in either direction?


In a corona discharge, the air around a conductor locally breaks down but remains an insulator further away from the conductor. Therefore, in the case of a positively charged conductor, the free electrons in the locally ionized patch of air can enter the conductor, positively charging the air too. However, corona currents don't exceed a few micro-amps. What is the reason for this? I know that electrically charging the increases the effective size of the conductor lowering the peak electric field. However, can this current be increased to arbitrarily high values is the charged air sheathing the conductor is removed quickly through some other mechanism? In essence, what I am asking is if the current limitation of corona discharge is due to the sheathing effect of the charged air or something intrinsic to the conductor-air system such as the probability of a free electron jumping into the valance band of the conductor being low. Where can I read about the microscopic nature of corona discharge? Any good books would be appreciated. I would like to learn more about the statistical mechanics behind the free electron and valance band interaction of metals.


Perhaps you might want to take this question as some sort of challenge. I'm really just looking for the "most efficient way" of making a table similar to that in the image directly below (in LaTeX, of course): I've attempted several different things, and ended up with a huge mess and a thousand packages in my preamble. (Here "thousand" is a hyperbole for "too much for my brain to keep track of".) Because I provided an image of what I want, I don't feel like any MWEs are necessary here. Whatever looks most like the image I above, and gives the least headache, I'll mark that as the solution. Notice the days of the week are aligned to the horizontal centre of the cells, unlike the rest of the content. Note. The table I provided as a reference was made using LibreOffice writer, and the immortal CMU Serif font. If you have a good eye, you'll have noticed that the content of the list elements are not aligned perfectly, as the second and following lines of all the list text blocks are pushed ever so slightly forward. This is neither intentional nor wanted.


If I want to calculate the length of a string that is wrapped around a cylinder what mathematical equations can I use? The thread is tightly wrapped around the cylinder, creating a spiral that has no gaps between each consecutive loop. Moreover each complete spiral creates one layer. and as a single layer finishes at the end another start to form up to the top creating a second layer, and so on. The catch is, is that each consecutive layer would require a larger length of string than the one before it because the circumference enlarges. Since the yarn ball is manufactured by machine, it leaves no room for error because the machine is in the same constant motion, the gaps between each string, as well as the gaps between each layer, are all the same, as is the tightness of each loop around the centre cylinder that holds the yarn together. So, how would you suggest I calculate the length using only external measurements and observations?


When a thermodynamic system, like an ideal gas within a piston immersed in a heat bath, is subject to changes, such as compression or extension of the piston, then the work that can be extracted from this process is maximal if the process is carried out quasi-statically, that is, at each step of the expansion the gas is allowed to relax to equilibrium. Why is the relaxation to equilibrium at each step related to more efficient energy extraction (or reversibility) ? Here is another way to ask. If I make a large change for the piston, I imagine there will be an abrupt relaxation to equilibrium and this will be related to large dissipation. But If I make many quasi-static changes, then at each change the dissipation should be small, such that at the end more energy is extracted. But the problem is that I do not see any physical picture or equation why the sum of the very small amounts of dissipated energies during the sequence of quasi-static changes should be smaller than the amount of dissipated energy during the one abrupt change. The difference between my question and similar available questions such this one, Is there a quasistatic process that is not reversible? , is that I insist on pictorial or physical answers of the questions.


Its kinda confusing but I will explain. I saw a term a while ago that explained how people can disagree with another. Instead of disagreeing because they do not like their opinion, the person will disagree because they dont like someone or something that also likes the opinion. Example: Person A thinks that driving on the road should require a license. Person B disagrees because a politician Person B doesn't like agrees with Person A, so Person B decides to disagree. You see that the argument Person A gave is reasonable, and Person B would completely agree, but the problem is that because someone Person B doesn't like agrees with Person A, they decide to disagree. This example is something very popular in politics (ex; only voting for a single political party, instead of voting depending on the politician's opinion). Another thing its similar to is contrarianism. You are disagreeing based not on the logic, but for another reason.


How is the background noise of gravitational waves modeled? Is it a thermal model, giving a stochastic distribution of the curvature tensor (field-strength tensor) in ambient space? That is, every binary star, every orbiting planet, every orbiting black hole or neutron star -- anything that accelerates -- is emitting gravitational radiation. The grand-total of all of these sums up to what looks like noise. Is there a "well-known" distribution for this noise? Some power law? Can an argument be made that this noise has a thermal profile? Are there specific equations describing this noise, and what are they? How do they scale? Should one suppose that this varies from galaxy to galaxy, and depends on the local environment? Or can one argue for some generic form that is "typical"? Say, for binary clusters? Somewhat related: what is the order of magnitude strength of this noise, compared to the instrument noise in current gravitational wave detectors? Yes, of course, its frequency dependent, so its a graph, but is this "natural noise" strong enough to be detectable? (Ignoring Earth-bound sources.)


A non-spontaneous change occurs when an external effort is being done to it. Since the external effort is also a natural source , does this mean that there is no truly any non-spontaneous process in nature (universe) ? Example to illustrate my point : An electrochemical process (like in a galvanic cell) can be reversed by providing an external voltage in opposite direction whose magnitude is higher than the cell potential. This external voltage is given by humans. Humans are basically converting some form of energy like mechanical energy of rivers in a dam into electrical energy. Also, humans are doing so because they eat food and they get their energy to do through spontaneous metabolic processes in their body. Also, the conversion of energy in above process is spontaneous because humans are making use of a spontaneous process ( a river flowing down from a height) . So, is there a truly non-spontaneous process ?


I asked my coworker to fix something in a program. When he fixed it, he replied with, "I already fixed it." -- this wasn't intentionally misleading, but was an incorrect translation of "ya". But for a moment I was thinking... "no, you fixed it after I asked". It was a bit jarring. Having learned a decent bit of Spanish when I was younger, I paused and realized there was some confusion when translating between "ya" and "already". BUT...I really struggled to find a clear/cogent explanation to help my coworker--from my English perspective. I finally found an article in Spanish, but honestly it was crazy how many meanings of "ya" there were, many of which I hadn't realized, and it was slow-going to read through it and make sense of it. My Spanish skills are rusty. So nobody else whose primary language was English was likely to realize what happened, had they seen this mistake So, here's my question: How can I help both the ESL coworker, other coworkers, and others in general to be aware of this pitfall, and what is a kind, cogent, and English-centric way to explain it? Thanks. Secondary question: And are there blogs/references or sites where I can learn more patterns of this sort of mis-translation across many languages? As a privileged white guy from a homogenous suburbia in the USA, I probably have a number of ways I might misunderstand cultural differences, and likely need to improve my awareness bit by bit.


I want to use Matex for the figures in my plots which I eventually would like to use in overleaf. I create a plot and use Matex in it and right click on it to save as pdf. I then use this pdf in overleaf. So, when I compile the latex file and download the pdf, the Matex font looks rather shabby compared to other text on the plot (such as numbers on axes for which I didn't use mathematica) when I do not zoom in. . The font on latex can be seen on the description of the figure. The Matex font looks spotty with kinda white gaps in it. The problem with this is that when I print the file onto paper, the font doesn't look good. I don't think the problem is resolution since when I zoom into the pdf file, I get a very sharp Matex font. Does anyone know what's going on?


Consider the sentence: So how can a computer think if it knows nothing of what it means to be a human being. Initially I thought that because "of" in this sentence basically means "about" the latter part of the sentence is not a relative clause as it is not adding any additional information, therefore is the subject of the sentence and cannot use which. Furthermore, as the contents of "to be a human being" is not known, "what" is used opposed to something like "that". However, upon further thinking I'm not sure if this is a relative clause or not as technically it is adding information to the sentence by specifying what the "nothing" the computer knows is. However, none of the resources I looked at concerning relative clauses used "what" in any example sentences. In other words, I'd like to know why this sentence uses "what" and what type of clause/grammar this sentence is.


This is a broad question but it's well documented that GR and QM are very well tested in their own domains but they conflict around black holes. Picture a neutron star slowly accreting matter until it's mass is sufficient to bring about an event horizon. It resists gravity owing to the Pauli exclusion principle and it must surely be comprised of the same 'stuff' as the event horizon forms. Why do we then rely on GR and assume everything collapses to a singularity which seems illogical in nature when the most sensible ('we don't know yet') answer surely should be that there exists a 'black star' under the event horizon? It seems that QM is overshadowed by GR in this instance when GR seems to give more illogical answers. As a thought experiment, if we had a very heavy neutron star and fired one photon at a time at it; I would imagine the surface begins to redshift more and more as time goes on. There would surely reach a point where the miniscule deviations mean that a 'ravine' can no longer emit anything to an observer but a 'mountain' could. It would seem to be on a tipping point of being both a black hole and a neutron star but the mountain is still supported from below from seemingly below the horizon that is forming.


Is there an easy way to draw complex images in LaTeX? By this I mean: Have a look at the following question. Some answers provide incredible pictures, very elaborated. In the (rudimentary) way I can create figures with tikz or pstricks, it would be impossible for me to come up with the correspondings codes. This made me think: Perhaps there are some tools that help write down the code. I am aware of things like tikzcd-editor or LaTeXDraw. I also know about InKscape, although I have never used it myself. Howerver, none of them seem to be appropriate to draw tori or spheres. Yes, with LaTeXDraw one can draw ellipses and create things that resemble those surfaces, but the quality of these draws is as good as the quality of Paint. I am thinking more of some type of image recognition software that identifies the geometric object (as well as some text the figure might have) the user has drawn and produce the corresponding code. Does anything like that exist?


I am seeking a term for what can collectively be referred to as "the leader and/or the most important and powerful roles" in the hierarchy of a society. I have an example from the YA series Warrior Cats by Erin Hunter. Warriors are the generic members of society. Meanwhile Leader, Deputy, and Medicine Cat together are the important and powerful positions. A term, for them together, as a collective, is what I seek so I can quickly an efficiently refer to them instead of some nickname like "the Big Three" every single time. I would like for a term generally applicable to most of societies, as it can be used in many contexts. I've been unable to think of any possibilities, hence I ask. Example Sentence: "The X are foundational to how our clan runs."


Poi are tethered weights used for dancing, which often have battery powered lights in them. Clearly work is being done on the poi, first to accelerate them, then to keep them going at constant velocity despite air resistance. I'm curious whether it would be possible to power the lights via the work done by the dancer rather than needing to replace/recharge the batteries externally. Obviously the amount of energy the dancer needs to exert would go up as well, regardless of the method. One idea that clearly wouldn't be sufficient would be to use current induced by the Earth's magnetic field. Are there any methods that might produce more power? It'd be cleanest if these methods would work while the poi were spun at constant velocity, but methods relying on variations in velocity or tension in the tether would also be interesting.


When you ask search engines or dictionaries, they don't seem to recognise the word 'orthodontry' and all point to 'orthodontics' and 'orthodontia'. I suspect 'orthodontry' is a mash-up of either of those and 'dentistry'. However, there are many businesses that use the phrase in their advertising and online communications. It seems to specifically refer to the practice of orthodontics. Is this an example where the dictionaries are just behind the times and is 'orthodontry' just an example of a neologism on the rise - a new 'aluminum'? Or is there some good reason the word should really be avoided? Examples of sites using the phrase: https://orthodonticsinlondon.co.uk/blogs/benefits-of-orthodontry.html https://www.columbiaasia.com/indonesia/specialties/orthodontry There's more like it - none of them any kind of language authority on the subject, but I started looking at it because two colleagues (I live in Australia) mentioned it independently in chat, and upon asking, one shared a mail from their orthodontist that had the word in there.


you are a player in a game where each contestant has to choose between three boxes to win a prize. The prize is distributed uniformly between the three boxes. Each player makes a choice independent of one another. many or none of the players could choose the right box which is revealed at the end of each round. However, you have info that the other players don't- you know that the prize will be behind a different box from the previous round. Find the probability that you will have chosen the right box more than the other players by the end of the game. The problem doesn't give me how many rounds per game or how many players there are which made me first think that there was an issue with the problem, but i'm not sure. Would appreciate if someone could please confirm or help me out if its possible. Thanks


I wonder what are the advantages of using the MPS-MFS (method of particular solution combined with method of fundamental solutions) for nonhomogeneous PDEs? In order to implement MPS with radial basis functions (multiquadrics or compact supprt, doesn't matter) you still need to mesh the entire domain (like in FD) but, unlike FD, you eventually end up with either a fully-populated matrix (when using MQ) or something anyway denser than in FD (when using functions with compact support). So, if anything, the MPS-MFS must be more expensive than FD, not less. The only advantage compared to FD or FEM seems to be that MPS-MFS is a meshless method in a sense that you don't need to build the connectivity matrix (which takes anyway just a fraction of the total time in a typical simulation job). Am I missing something?


I want to self study Real Analysis, and I want to choose between these books. This is my background I am a teacher of mathematics for highschool, which means (at least) in my country, that I had seen abstract algebra untill ring theory and calculus untill integral in several variables, but not things like Stokes theorem (because standards for teachers in my country solicites some very basic ODE and, so I had seen ODE instead Stokes). This calculus class was a mix level between engineering and pure mathematics Could you tell me about the pros and cons of each of the following Real Analysis books? (Suitable or not for self-study, content quality, difficulty of the exercises, etc.) Real Mathematical Analysis, by Charles Pugh Mathematical Analysis, by Tom Apostol Principles of Mathematical Analysis, by Walter Rudin


If we look at the majority of useful or industrial materials surrounding us, like metallic alloys, glasses, ceramics, or plastics, it is often the case that these materials went through really hard times or difficult stages of their life during synthesis and processing. For example this could be the heating of a metal to an extreme temperature and abruptly putting it in a cold environment to quench it, thus typically resulting in good mechanical resistance. Perhaps the question may be not so precise, but is there some abstract reason why good material properties (mechanical, thermal, electrical, and so on...) are typically obtained through procedures that really drive the materials out of their native solid stable states? Can this somehow be related to the difference in phase flow topology between very stable states (liquid, solid, gas) and the "non-equilibrium" states obtained through extreme conditions (like heat-treatment and so on)?


I use Beamer presentations in workshops where I set questions, have students work on the answers, and then display some sample answers. I give the students a version of the presentation without the answers, so they can read back through the presentation as needed. It seems the usual way of doing such conditional processing in LaTeX would be to define a command to mark up an answer and then provide two definitions of the command, one that includes the answers and another that omits the answers. The usual way of switching between these definitions is to use distinct top-level source files. I would prefer to avoid creating an extra top-level source file for each presentation and instead control the formatting behaviour using command line arguments, an environment variable, or such like. Then I could write a script to format the presentation twice, once with the answers and once without answers. Looking through the arguments of pdflatex, I don't see a way of passing a flag that could be used in the LaTeX source to choose between the definitions. Related questions and solutions: Passing parameters to a document has answers that solve my problem, but the question does not express the context and requirements as clearly as the question here. Conditional text lines latex document - expresses a similar requirement. The exam package - seems to require multiple top-level source files.


I know the heading may be a bit misleading one. But I can't find a better one. Anyone can suggest a good one. By writing Weird function I mean those functions that are integrable but not easy to do so. I have come to know that uniform convergence of a series of integrable functions on some set confirms the fact that we can interchange the order of sum and integral. In other words, this leads to perform term by term integration for infinite series of functions. I am very curious about: Can we write any weird function as the limiting function of uniformly convergent series of simple integrable functions? If the answer is yes, then I think we can integrate those functions easily. But then after integrating we again have a series, that must be again uniformly convergent I think. In that case it may not be possible to write it as closed form. Is my observation is correct?


I'm trying to replicate a specific script 'X' that I'm seeing in a textbook: This lines up pretty well with the computer modern script X: But I'm using the unicode-math package, and the script characters in the default font (latin modern) look nothing like the script characters in computer modern (even though it's supposed to just be a modernized version). The unicode-math documentation includes a symbol list of a few included fonts, of which the concrete math font gets the closest: But I really don't like the uniform stroke thickness - it really doesn't match with the rest of the characters on the page (since I'm otherwise using computer modern / latin modern). Does anyone know of an OpenType math font that has a similar script X that I'm looking for?


Maybe the questions is too stupid to be asked or I do not know the technical words, but I could not find any answer to this question. Here is how I started to think the title: First I thought of if we look far enough if we could see Big Bang theoretically. The answer was "No", due to reasons such as opaqueness of the early Big Bang and expansion. This is okay. However, then there was a second question arised: Even if we could, which direction to look at? Then I thought if it is the furthest and the oldest, it should not matter. Because any direction we looked at furthest and oldest we should see the Big Bang if we could. Then, I know I made too many assumption, but does this conclude that we are surrounded by Big Bang and the universe is expanding not outwards but inwards? As I said maybe question was way too stupid, but still I want to hear some ideas/facts. Thank you. By the way this is the first time I post a question, I have no experience, so sorry about any mistakes I made.


So I'm writing a book, and I've used straight quotes (") in my latex files throughout the entire project. Now whenever I compile my document in Overleaf, these, to no ones surprise, come out as no starting quote, and a straight quote at the end of what's quoted. What I wish is for all straight quotes in my tex files is to actually appear as straight quotes in the generated pdf, regardless of whether they are situated at the front or end of a word. Here's an example: % In my .tex file This is a "nice quote". I want this to appear exactly like this in my document: This is a "nice quote". Is there anything I can do to override the default behaviour? I really do not want to go through my entire project consisting of many .tex files finding and replacing characters, all I want is simply the straight quote symbol to appear as an actual straight quote in my text regardless of whether it's a start-quote or an end-quote.


Consider a person pedaling a bicycle, if we consider the system consisting of the rider and cycle as a total and apply work energy conservation we can see that whatever force the rider applies on the pedal will also have an equal and opposite reaction and as both the pedal and the foot of the rider will have equal velocity the net power cancel outs? I don't know where the mechanical energy is generated in this process. I have attached a picture of a question from my engineering textbook (Engineering Mechanics by P.C Dumir) from which I got this question. I have attached a portion of the solution as well, I am not sure why It has considered the reaction force to be acting at point B and not the foot of the rider, due to which it has calculated the power by taking the difference of velocities of the subsequent parts.


My understanding is that without acceleration the "movement" of a body is a relative concept, i.e. we can choose an inertial frame of reference where the body is at rest and there is no property or experiment that can tell us that the body is in movement, because it's a meaningless question. In the same way, can I say that the movement of earth is arbitrary, just choosing a non-inertial frame of reference? The fact that I need to include fictitious forces to explain for example the movement of a Foucault pendulum, means that the earth rotation is "absolute"? The law of physics should stay the same if I choose the earth as frame of reference, but does that mean that there is nothing absolute about the movement of the earth?


Traditionally, mathematical work is presented in a linear fashion. Books, papers and articles are single streams of text meant to be read sequentially, from beginning to end. However, mathematical content often has a not-so-linear underlying structure. Sometimes it can be imagined to be tree-like, with nodes being results and directed edges being dependencies. Question Is there a format for presenting maths that is faithful to some underlying logical structure of the work? The 'logical structure' could be defined by the author. Using digital devices, we are obviously not restricted to linear text anymore. Have you seen such an 'untraditional' format being used? Prototype Imagine a PDF-viewer that can collapse and expand certain blocks of texts, as defined by the author and with the possibility of nesting. In proofs there are often steps which are very unclear to some readers and trivial to others. These steps could get elaborated on in an expandable block -- providing the necessary details for the people who want it while maintaining reading flow and brevity for the others. Using layers in LaTeX something similar can be achieved as described in this question


I'm writing documentation for a piece of software I worked on and I came across an odd sentence format that puzzled me. I was wondering if there might be a conclusive answer on the matter: In the event that a complex compound sentence contains two independent clauses and one dependent adjective clause, but each independent clause would require different articles to tie with the dependent adjective clause, would the best article to use be the lattermost article? Example: "create a" vs. "update your" Everyone needs an updated google account to access the document. Please create, or update, your account before the meeting. I do not have a background in English, please excuse any incorrectly used terminology! I did some googling beforehand in order to better articulate my question so it's possible I misunderstood something in my haste.


In order to define my question, I will demonstrate what I consider an appropriate answer. My question is as follows: How do I develop fast "shortcuts" in math? What I consider a shortcut is a means to solving a problem in a fast and creative way. My own answer would be to look for patterns and model a formula to solve the statement in an appropriate and timely manner. Maybe one could also introduce something brand new to the problem, like the perfect square technique. As one can see, I am not asking how to approach the problem, but what to do to find new and fast ways to solve it. This is obviously getting back to my question. How do I develop fast "shortcuts" in math? I value the community's intelligent feedback on this question. Thank you for your time and patience with me.


It is my understanding that there are two types of random number generation used in computer science. True random number generators that use principles of a physical property to determine their generation and pseudo random generators that use algorithms based upon mathematics such as chaos theory. True random number generators use a proposed indeterminate properties of the physical world because of the principle of uncertainty and pseudo random number generators employ chaos theory which are non linear deterministic equations whose outcomes are sensitive to initial conditions however they are deterministic and with the same initial conditions the outcome will be the same. My question is the principle in true random numbers generators and the principle of using chaos theory in pseudo random number generators the sane, as both principles use a deterministic factor the schrodinger equation in true random number generators and deterministic equations of chaos in pseudo random number generator,so what's the difference? Is randomness actually certainty? or vice versa?