When fluids are subject to external perturbations or put in certain geometries, they always show arguably the most fascinating structures : Starting from the eddies and vortices in pipe flows at large Reynold numbers, vortices of flows past an obstacle, or the vortices in fluids confined between two rotation cylenders. Yet, a fluid at rest is often thought of just as a completely structureless phase : Apart from very small-scale features such as the arrangement of nearest neighbors or solvation structures in liquids, it is just a random phase and the trajectory of a tagged particle from the fluid simply obeys the Brownian law, without any other features. But what I somehow intuitively expect is that most of the above mentioned "non-equilibrium" structures appearing under well defined conditions are present in the resting "equilibrium" fluid. This means if were to look at the trajectory of some fluid particle I expect to see imprints of the coherent non-equilibrium structures in the equilibrium trajectory, and not just see a Brownian trajectory. Can you somehow relate to this ?


Lets imagine you have a universe where the average matter density is extremely close the the boundary between one that will expand forever and one that will eventually re-collapse. Now lets imagine that there are very large scale fluctuations in density at the Gpc/Tpc scale and above. Could this result in differences in the rate of expansion between some parts and others, causing parts of the universe to collapse and parts to expand forver? If yes what would this look like near the boundary? What if the region was non simply connected. (E.G. a torus, or hollow ball.) How about a universe with bee marginally above the critics density with some very large under dense regions, perhaps ones large enough for the centre to become causally disconnected form the edges.


There is a good discussion here (Referring to past times with "hence") that concludes that hence can't be used like "ago" to refer to a past event, not even to describe how much time has passed "since" that event. However, what if using the past perfect and then referring to time forward from that point, Could "hence" be used there? The specific construction I'm wondering about goes like this: They had left me there to die. Even now, one-and-a-half moons hence, it would be a simple matter to follow their trail. In that case, "hence" is a time forward from a point in the past as made clear by use of the past perfect ("had left me"). Is that appropriate usage, or can "hence" only refer to the actual future, not a future relative to a point in the past?


There is a closed continuous loop, and you have a stick of a random fixed positive length,so that You can slide the stick on the loop. Take the stick's midpoint. It's track will form another closed continuous loop. Take another stick of another random fixed positive length. Slide it on the new loop again. It's midpoint will form another loop. Do this again and again. Will the shape of the loops tend towards a circle(well, the area of the loop may decrease, but you can magnify the area to not let it become a point)? Or could it become another shape? The thing I found was that circles and ellipses will tend towards a circle, and I think the question is intuitive, because this operation of sliding sticks round's spikes and makes concave shapes convex.


Imagine a ball with diameter "a" at rest and another observer at rest at a distance of 'l' w.r.t to the ball in this rest frame.Say the ball suddenly starts moving at velocity v towards the observer .(Assume it accelerates instantaneously).What does the observer measure the distance between them?What does the observer measure the diameter of the ball to be?If the diameter is contracted by gamma factor then why is not 'l' contracted by the same?Because acc to me by time dilation,as the time interval is zero wrt to the observer between the events (ball moving and ball at rest),so the distance moved is zero acc to observer and hence observed distance remains 'l'.I asked my professor,he also said that diameter would contract but distance of ball wrt observer wont , but I didnt understand why this partial treatment of two lengths ?.They are identical in properties with regard to lorentz transformations so if one contracts wrt observer the other should also.


Maybe this belongs on "Linguistics" but since it's about an English word, I suppose they'd only send me here. Over time, some words gather more and more meanings to themselves. Ironically, these words are often small like, "by". Other words seem to shed meanings, or rather, they devolve meanings to new words. In the New Testament, Jesus tells people to "arise" from their bed, elsewhere "the resurrection" appears. But this is actually just the noun form, i.e. "the arising." Now, no one except by poetic license would refer to their "resurrection" from bed; it has become a specialized form of "standing up" (from death.) Is there a pair of words to describe on the one hand the accreation and on the other the fragmentation of words and their meanings?


In my current understanding, I have been told that the spin of quantum particles is simply intrinsic to them. That, particles are simply right-handed or left-handed... just because they are. To me, that seems then that we should then treat handedness like it is just as distinguishing of a property as mass, or charge, or color charge. Almost like the standard model should be doubled, or mirrored. Like, right-handed, blue, up quarks should be considered as a wholly different particle from a left-handed blue up quark. But wouldn't something HAVE to catalyze the spin? Like a ball rolling down a hill hits a rock, and how it first strikes that rock determines whether it continues its rolling spinning to the right or to the left. Could spin be acquired, for instance, during the big bang, by the first interaction of a particle as it "hits" the higgs field? And some characteristic of the particle to that point determines which way it "glances" off of the higgs field?


At the end my document, I get a new page that contains, Temporary page! LATEX was unable to guess the total number of pages correctly. As there was some unprocessed data that should have been added to the final page this extra page has been added to receive it. If you rerun the document (without altering it) this surplus page will go away, because LATEX now knows how many pages to expect for this document. I know that by just recompiling the document, I can get rid of this error, but I was wondering why does this occur and what unprocessed data that LaTeX couldn't process. For citations, table of contents, table and figure references, LaTeX needs to be run twice, but I have never ever seen this "Temporary page!" page before.


As we know, in EM wave, with the oscilation charge (for simplicity, charge was moving at constant speed, then accelerated and then continued moving with constant speed again), kinks are produced. The funny thing is as the kink moves outward, its length actually increase(the joint vector between the new field line and old field line) due to the fact that new field lines move at a higher speed then old ones. I wonder what we exactly call a wavelength of wave then. Is it the kink's size ? but if it's kink's size, then it means with just one acceleration, kinks size increase and wavelength increases as well which seems to me wrong. one acceleration only should produce one wavelength. So what's the connection between kink and wavelength? If there's no correlation between kink and wavelength, then wondering what's the wavelength truly is in terms of EM wave?


I want to simulate typewriter style typesetted math articles. There are a few major requirements: All characters in all environments be tt style and of the same size and width (including super and sub scripts in math mode). In math mode, sub and super scripts should be one character higher than the base. All emphasizing should be replaced by underline. Spacing between all kinds of lines, including spacing between lists or figures and texts, should be always integral multiples of character height. The multiple can vary according to the height of inline equations (e.g., when too many levels of superscripts, should leave space for superscripts to avoid overlapping). All spaces being the same width as characters, no adjustments to fit line breaking. If a word cannot fit in one line, leave it to the next line. To conclude, all elements should be arranged as if the page is separated into "grids", each grid of the size of a tt character. All elements should be placed exactly fitting one or multiple grids in both vertical and horizontal directions, there is no "half" space or something like that. Is it possible to fulfill these requirements in LaTeX?


We know that when the wave model of light was being used in physics there was Thomson Scattering Model to explain the interaction between a free electron and a light. The Thomson Scattering was explaining the force that acts on an electron when a light hits it, with the force that the electrical field component of that light possesses. (Rather than billard ball explanation) However, I do not understand Thomson Scatter in relation to Doppler Effect and Low-Intensity waves. For the wave model of the Photoelectric effect, it was clear for me to understand where was the wave model wrong when explaining the necessity of the quanta of energies. But, How was Thomson wrong when talking about Doppler shift and low-intensity waves not being able to explain the movement of electrons? I know for a fact that low-intensity waves were insufficient to supply energy for an electron to have the speed and kinetic energy it is supposed to have. I am not sure how built-up energy by low-intensity waves have been thought to be a supply of energy for the acceleration of electron in the first place and how is this related to the Doppler shift by Thomson.


I am trying to have better understanding of localized wave functions. Apparently free particle de Broglie waves are NOT normalizable and act as delocalized functions which was the original rationale behind their use for explanation of electrons in double slit experiment. so far so good ! But then we use Fourier transform to make them localized that they can represent a particle and behave normalizable . Here is what I can not understand: if wave function gets localized ( say to the size of an electron - which actually is supposed to be point-like ) , then such wave can not cover the distance between both slits to be able to make interference patterns in the double slit experiment, .... and if wave function remains delocalized , then it will not be able to get normalized and consequently its wave function will not present accurate probability amplitude . The only way I can think of this is to have a localized wave function that is substantially larger than the slits distance but vanishes at the infinity, but it will be much much larger than a particle - let alone a theoretically point-like particle. My question is: when making a localized wave function, how big the particle size or wave area should be to make interference patterns and be normalizable at the same time? are there known limits for that Fourier section?


I want a tool to easily make geometry diagrams. For example, say I wish to overlay two circles, and then fill their overlap with the color red, whilst coloring other areas with other colors? Let's say I want to make a triangle with curved sides and a flat bottom side; that could be done in no time with the right tool. I could insert a square and then delete circles from its upper two quadrants. To be more clear, here are the list of features I'd like: A gallery of shapes that can be sized and colored as one wishes. The ability to fill any area enclosed by lines with a color. The ability to draw lines between highlighted points on shapes, like the centers of their sides, their corners, points between the quadrants of the circumference of a circle (allowing the drawing of diameters), etc. The ability to insert text into the diagrams. Is free to use. To give examples of close contenders: Draw.io has all features except number two. Microsoft paint and sketchpad has all features except number three. Does an app that has all features exist?


Much of physics education focuses on equations of known possible reactions, how to balance them, how to predict the result if certain elements happen to react with each other. What I don't see a lot of is an explanation of what chemical reactions are likely to happen in a complex system. Assume I was building a physical simulation where I know the position/velocity of any number of elements or compounds. What would the process be to determine which chemical reactions are most likely to occur? Given my assumption: Treat the collision of each atom as a sphere of varying size depending on the size of the type of atom's valence field (sphere collision is to reduce computational complexity). Calculate magnetic attraction/repulsion between local atoms and manipulate the velocities accordingly. If the valence fields of two atoms overlap (or are near each other?) due to attraction, electrons are exchanged, and other atoms that are the least magnetically attracted are either re-positioned or rejected from the compound depending on the magnetic forces and electron availability. I assume "heat" based reactions can be derived from velocity of a collision? Would this be a sufficient model to predict an arbitrary chemical reaction or non-reaction?


I am studying the Newton's Law's of Motion, and really confused about one thing. I know that bodies of very small mass get attracted to our earth due to a force it exerts, due to its huge mass. So that, according to the Newton gravitational equation, a given small mass on earth will be attract by a certain force and depends on the distance of separation from earth's core. Right now I am dropping a pen on a table and observing the sound. Higher the height from which I drop it more the sound it makes, showing that (I believe) more force is being exerted. Although from Gravitational equation of Newton, we can observe that the same force is being exerted on the pen because I am dropping the same pen from heights not so far away from each previous drop. My question is that from second law of Newton, change in momentum causes force. But from gravitation law of Newton, the same force pulls the pen down. But the change in momentum is happening due to gravity, but the gravitational force is constant so that, the pulling force is constant, thus, why every different drop do I hear a different sound. What is the right thing here? I am really confused...


This question is perhaps more on the use of certain parameters in BibTEX than on the use of TEX/LaTEX itself: nevertheless I think it could be of some help to other persons who, like myself, need to redact large bibliographies. In order to list information on the existence of a review in the ZBmath database on a bibliographic item, I am aware of the existence of the following three parameters: ... zbl = {}, zbmath = {}, zblnumber = {}, ... Until a few years ago, only the last one seems to have been used (possibly my opinion is biased by the heavy use of Jabref I do), therefore I'd like to ask why the other ones have been introduced, if there's one of them which is preferable to use. P. S. Finally, let me point out that the American Mathematical Society always codes the analogous information in the univocal parameter mrnumber = {}, which works perfectly in many bibliography styles.


We know from molecular spectroscopy that incoming light on a molecule can change a molecule's rotational, vibrational and electronic energy levels. If the incoming light is, on the far-infrared and microwave region the molecule gets rotational energy.(microwave spectra) on the near-infrared region the molecule gets both rotational and vibrational energy.(infrared spectra) on the ultraviolet & visible light region, the molecule gets all energy levels rotational, vibrational, and electronic energy (electronic band spectra) We also know that, Rayleigh Scatter does not excite electrons of atoms to another electronic level but only causes them to oscillate. Thomson Scatter happens when an electron is free or quasi-free and it also makes the electron oscillate and more, accelerate. Taking into account that both Rayleigh and Thomson most probably happen at low energies and vibrational and rotational energy changes of molecules happen at microwave and infrared spectra: QUESTION: It is clear that, Thomson Scattering can not cause rotational spectra or energy build-up, which requires the whole molecule to rotate since Thomson Scattering is about free and quasi-free electrons. But what about vibrational energy spectra, or energy build-up on such a free electron? Rayleigh Scattering happens on the atomic level so it begs the question, "can it cause both rotational and vibrational energy level changes in atoms or molecules?"


For a given n dimensional chaotic system such as a chaotic attractor or really a time dependent chaotic dynamic system does there exist a higher dimensional representation where the behavior is in fact deterministic? This question comes from a data science problem in multivariate time series forecasting. Most real world behavior is chaotic in nature, such as the stock market or Earth climate cycles (milankovitch cycles etc.). Lets say I could build a really big model that takes into account everything that might impact the system of interest, would there be some set where all the chaos would just go away? Take the TV show foundation based on the books by Isaac Asimov. In there they postulate the Abraxis conjecture which is solved by folding space and is critical to the success of psyco-history (which is really just data science). One of the steepest challenges with time series forecasting is the ease at which it is possible to leave conditions where your inital state no longer applies. If chaos could be eliminated then this problem could be pushed out. Effectively with infinite information we could forecast for infinite time (in the theoretical limit only).


While playing with GeoGebra a few days ago I came up with an initial property of my Cardioid curve, I heard this curve is well studied so I'm not sure on my guess if it's already discovered or new, adding references would be appreciated You have a Cardioid and its circle of perpendicular tangents The red point is an arbitrary point of the circle of perpendicular tangents The violet line is a right angle bisector The blue line is the hypotenuse of a right triangle The green line is the reflection of violet with respect to blue The brown line is the reflection of blue with respect to green The brown line is tangent For giving me Cardioid. I don't have progress on it yet, but I do have other cardioid guesses that I can include in a follow-up question on the site.


I tried lots of research, but still couldn't wrap my head around the difference between free and mobile electrons. Free electrons, from wikipediaLink(I mean free particle bullet point, because other bullet points are not really free electron - as an example valence electron is not free, but mobile) are electrons that don't experience any external force, hence it experiences zero "electric field" from others. as for the mobile electrons, they are loosely bound to the atom (valence electrons). Though, even if these electrons are ejected from the atom, they still experience some electric field from other atoms and could jump into their orbitals. If the above is correct, how can both of these cases exist? how can free electrons not experience any electric field, while mobile electrons do? I know that in sun's core, there're free electrons moving through material. Why don't they then experience the electric field of other atoms?


For some extended context, this is how Heads Up Displays (HUDs) and Reflector Sights work: However, as far as a I know, a lens will take collimated light and focus it to a very small point called the focal point. And since the reverse is also true, only the focal point can be collimated and "focused to infinity" to create the virtual image in a HUD. The reflector sight diagrams clearly show a source reticle that is larger than a focal point, and HUDs in general usually use an LCD display to create the initial image. Although modern HUDs use several optical elements to correct for aberrations and other distortions, most early HUDs and all reflector sights used a single large convex lens to collimate the image. Is this merely an issue of lens diameter? Would using a larger lens allow for the focal point to be larger in diameter too which would allow for a small display to be completely collimated?


If we bring a separable dipole (say a single ammonium chloride molecule) from infinity to the center of a capacitor, once the dipole stabilizes and the rotational energy is lost to electromagnetic radiation, the net work done is negative. If the capacitor plates be perforated and the field be strong enough to separate the dipole and the ions pass through without collisions, they then may be accelerated in the same direction by two other capacitors with perforations, in the same direction and then may recombine after exiting because of their electromagnetic attraction. Once they combine, they still have the velocity gained by the two additional capacitors but none of the three capacitors lose their charge. The dipole gains some kinetic energy so where does the energy come from? The capacitors are not connected to batteries and are pre-charged, the entire setup is in vacuum and gravity is non existent


Recently, I play a game of Turing completeness where I utilize various gate circuits such as NAND, AND, and NOT to construct a circuit that satisfies the given truth table. I didn't learn digital circuits or relevant content. When I play this game, I want to know whether there are any theory to deal it. For example, given a truth table, is there any theory to get the circuit with least basic gate (or only Nand gate). It seems to be a bit like graph theory (but the vertices are special). On the other hand, I know a little about retina from From Neuron to Brain. The retina responds to certain shapes of light, such as a ring of light. I've always wondered how build a neural network with neurons that respond to ring of light but not solid circle light. Of course, I'm not really asking for the details of these questions. I just feel they may are the thing not liking analysis and algebra. If I want to deal these questions, what book I should begin from ?


A parabola is supposed to be a conic section, obtained by slicing through a cone with a plane, like this from Wikipedia: But if you do this, what you get is just part of an elipse, since if the cone and the plane were bigger, eventually you'd end up with an elipse. And a section of an elipse is not a parabola - there is no possible focal point that satisfies the condition that all points on the curve are equidistant from the focal point and the directrix - at least not when I try to construct this in my CAD software Answered by JonathanZ, the key thing was the parabola needs the slicing plane parallel to the slope of the cone (not illustrated or explained clearly in either Wikipedia or Wolfram IMHO). Here's what it looks like in my construction:


I've attached an image that describes the light-rays refracted through a thin lens. P' represents the first real image that is created when light is refracted through the first convex mirror. But how does this translate to the second image point P'' through refraction? I understand the convex mirror formula and how if you plug in the numbers it just works out that Q'' is where it is, but intuitively, this does not make sense. If the image P' is what P'' "sees", as it is the image, how can the refracted image (P'') be on the same side as the object image (in this case P')? As far as I understand, P' acts as a real object for the second surface (the right side of the lens), and the rays that diverge from P' are what get refracted onto P''. But how is this possible, if they are both on the same side? As far as I understand, the model above depicts virtual object through refraction. How is this shown in our case?


The reason we can see the color of objects is light rays undergo scattering in the subsurface of a material, and in their walk in the material lose some wavelengths, and finally exit at a random direction (link) However, aside from this, you see white highlights on an object from the light source. These can be diffuse or specular depending on surface roughness (I'm assuming they're both from surface interactions) Look at the highlights on the shoulder here for what I mean. Not where the arrow is pointing. However, I just found that that aside from metal, most common surfaces, even when highly polished, specularly reflect only a few percent of the incident radiation. How are these highlights then so pronounced, especially for the diffuse highlight case where the light is distributed in all directions.


(Edit: This was closed because it was marked as "opinion based" and I was asked to edit the question so that it could be answered by facts and citations. @Fumblefingers gave citations, thank you. On re-reading several times, I think yours is the correct interpretation, rather than assuming the existence of some actual physical flag. It's like as if we referred to captains & sailors dying in the line of duty as a "badge of honour", there's no physical badge. In my opinion, expecting facts about poetic allusion is a bit unrealistic. Thank you @KillingTime, I hadn't known about the literature forum. I can ask such questions there, and keep this forum to linguistic questions. ) ORIGINAL QUESTION: Does any English literature expert know what kind of pennant Walt Whitman was referring to by "a pennant universal" in the poem "Today a rude brief recitative of ships" in "Leaves of Grass" ? Flaunt out O sea your separate flags of nations! Flaunt out visible as ever the various ship-signals! But do you reserve especially for yourself and for the soul of man one flag above all the rest, A spiritual woven signal for all nations, emblem of man elate above death, Token of all brave captains and all intrepid sailors and mates, And all that went down doing their duty, Reminiscent of them, twined from all intrepid captains young or old, A pennant universal, subtly waving all time, o'er all brave sailors, All seas, all ships.


I am aware that in the theory of classical infinite sums , one can not generally interchange the order of a double sum or do other infinite sum manipulations. However, these infinite sum manipulations can be valid if absolute convergence is available. I am working on a problem , where I would like to manipulate a double infinite sum but absolute convergence need be satisfied in my context. I am under the impression that these classically illegal infinite sum manipulations can hold under less strict conditions in the context of non classical summation methods. I would be glad if someone can write an answer that discusses when interchanging the order of a double sum is allowed in the non classical setting and direct me to a place where I should read about this. Thank you a lot.


After verifying that it was a real word, I just posted a chapter named "Processioning". I have begun to wonder what kind of thing that is. As far as I know there has never been a verb, "to procession", and "to process" is something else entirely. Yet, "they were processioning through town" makes perfect sense. I am thinking that there may be a class of such words, apparent verb forms made from nouns. Am I misguided, or simply ignorant of what the heck that is? It is also weird that processioning would not be used as a gerund, because the noun form is procession. I guess it could be an adjective though, "the processioning group". edit: talented musicians an unconvincing argument an enterprising foreign policy a fun-loving guy are sound examples with no corresponding verbs [EA].


Suppose there is a ridiculously large bridge, fixed at either end (light seconds long at least). The bridge is constantly under the influence of gravity. If the ends are severed simultaneously, the whole bridge will fall. I assume simultaneity isn't a problem as the bridge is all in one reference frame? Will an observer in the middle register the change of inertia, but be unable to see or detect any damage to the ends of the bridge until the light from each end or the signal from electronic sensors has caught up at light speed? If so, does this mean that they can infer from the fall that the ends must be severed I.e. have they recieved information about the state of the ends faster than lightspeed? Or will they not begin to fall until the information about the severed ends has caught up?


Usually in a Cartesian form of derivation in math and physics, I have seen that a particular formula is derived for simplicity by taking concerned points say in the first quadrant. Few examples are the derivation of mirror formula, lens formula, lens maker formula, distance between two points. Later, they extend the formula to all cases and say : "The formula is valid for all cases independent of quadrant and position because we have taken proper care of signs". For example, see the case of mirror formula derivation in the book (especially the pointed last paragraph): Similarly here for distance formula in SL loney. I do not understand deeply that why do this happen ? I know that it can be proved in the above cases by actually deriving for all cases and observing that it is confirming, but how can we say it generally? So the question is : Why do the sign convention in coordinate geometry work the way it does ?


I am an undergraduated student from a non-native English speaking country. In my country, when doing math, which in many contexts that we will be proving something like a theorem, a proposition. What matter is I feel like we don't have a variety of words while doing math. Everything we used in our assignment are just around "we have, we get, hence, thus, therefore, we have this, we have that, we have and we have ...", which is kind of boring to me. I love math and I love solving math problems, but I would love to bring my product with an abundant style of words, just like it's not simply a proof of math, but more like a nice essay of math, to make my products more lively, agile, elegant, refined. So I really appreciate if you guys could enlight me with your knowledge. My very thank you.


We are extracting transformation matrices for different joints from ARKit. These are relative to the parent joint. Since the positions are not very accurate, we have developed a machine learning model that corrects the positions based on ground truth (Vicon data). We only use position data in the model, so now the rotation part of the transformation matrix is no longer valid. My question is: Given that we have the initial transformation matrix (I can have the rotation, scaling, translation separately) is it possible to get an updated transformation matrix that uses the positions of the model? My end goal is if possible to calculate the Euler angles of the updated position for the initial coordinate system. I have checked the Rodrigues' rotation formula but I am not sure if I can use it somehow.


Have an excel financial model, that pretty much is a time series of assets and liabilities. I would like to know what the optimal liability structure / mix is, subject to multiple balance sheet constraints. My objective function is to minimise funding cost. I tried to optimise using Excel Data Solver, but encountered the following issues: Excel may not arrive at a solution (can't converge at times even though there is a solution) Solution is not consistent, eg same starting point (ie initial guess) but different end points (not all are optimum) Solution may not be global minimum Am open to using other softwares like Matlab or Mathematica, with my own "constraints" I won't be able to simplify this down to an analytical equation so will need to be in the format of a financial time series model Ideally arriving at global minimum Scalable Are there any guide you can point me to? Open to software suggestions, algorithms, approach, etc... Thanks


In Meissner effect we say that below critical temperature there is an expulsion of external field lines or field lines do not penetrate inside the specimen. But here is what I am not able to understand, if the sample creates a mystical force which do not allow field line to pass then it can be imagined as if there is an field due to specimen which cancels its effect. I know experimentally field lines are expelled, but why they are expelled I want to know the reason like who are the one that causes no fields to penetrate inside it. Somewhere I have read that a persistent current rises in such a way that it creates a field which cancels the external field. Now how is that possible that a superconductor inside in it already has a field?


I've been told numerous times in high school that "first and second person pronouns in academic writing" should be avoided. This supposed wisdom is echoed in various style manuals as well. However, if you look at any authoritative text, like The Stanford Encyclopedia of Philosophy A simple search of "we" will yield countless articles using the first pronoun all throughout. Given this situation, is it realistic to teach anything stylistically related to using first pronouns? Also, based on experience, trying to stick to this kind of rule damages the entire paper in many ways It takes time to conjure up passive tense sentences which usually is all but required to avoid you/we completely Some concepts more effectively expressed through these words may fall between the cracks and never see it to fruition, which is truly a shame Grades may even suffer trying to stick to the rule -- an irony given that adherence to rules are always motivated by desiring praise and getting good grades, etc. ad nauseum...


Background: ZFC in classical first-order logic. To make a definition, we first need to proof something does exist. So, this question is actually about proving something exists based on a recursion. I know that when you want to prove the existence of a certain function with the domain of ordinal numbers, you can use the transfinite recursion theorem. But our things are not always a function of this sort. For example, when we define the truth value function on the set of formulas in predicate logic, we usually define it recursively based on the number of the logical symbols in a formula(or complexity). But for a certain complexity of the formulae, there're many cases to deal with. I can't figure out how to prove such a definition does make sense, aka the function does exist.


Good day, everybody. I program a simple computer game and there is a "dumbbel" inside of it, which should using some kind of rocket engine. The engine is fixed on one side of the dumbbel and creates a constant force F, which is ALWAYS orthogonal to the dumbbel. Since the dumbbel is ideal, the both sides have only mass m and zero size. The junction is absolut hard, has the length l and doesn't have mass. S. also this figure: There are no other forces than F (especially no gravity) and there are no other bodies than the dumbbel. Therefore no collisions or bumping. The motion happens flat. How can I calculate the trajectory of the "dumbbel"? Is there an analytical or numerical approach I can use in my game program? Thank you very much!


For all the words I know, I can't seem to find the right word for this strange emotion. Some words I have tried come up short in one way or another. Confusion lacks the underlying hatred, annoyance ignores the triggering bafflement, skepticism is to forward to properly describe the abhorrent serenity caused by this mixture of emotions. I tried looking on Google and several other sites for some synonyms for each of them, but my results kept coming up short. Even my 'Better-words-for-authors' guides proved insufficient. I'm starting to think there might not even be a single word in the English lexicon for this emotional mixture that is hatred and anger caused by disbelief. Example of how it could be used: He couldn't help but feel [blank], had the youtuber really sunken below rock bottom. - or - [blank] began to well up inside of her. The others had resorted to Orwellian new speak to justify their problematic behavior.


The normal proof goes as follows: if H(f) is a program that takes the source code of any program f and return whether it halts, we define G to be if(H(G)){ loop forever } else { halt } Which is a contradiction if you run H on G. My problem is the unjustified assertion that a program like G even exists, as presumably it is able to print its own source code and run H on it (or something analogous to that) and halt or not halt based on that. My intuition tells me this violates some information theoretic result I am not smart enough to know about. Furthermore, since we have no idea what H even looks like, this proof only works if you can prove that for every program P, there exists a program G which can print its own source code, run P on it, and then do some stuff based on that. Since the mere existence of Quines, which still conserve information, is already a deep and complicated result (from what I can tell), the truth of Gs existance is sketchy to say the least.


Suppose I have some process that I believe to be the result of a differential equation. It would be reasonable to regard this as the path of a point particle in some notional space. Suppose further that I know the exact shape thaat the path follows, and it is one with a simple description, such as a conic section or a polynomial in two dimensions of low order. Is there a determinate series of steps that will give me a differential equation which has my specified shape as a solution for some initial value, or produce increasingly good approximations of that shape on iteration? If there is an exact solution tell me how to find it. If there is a way to find increasingly good approximations to a numerical solution, describe that way. If the answer or answers to the questions above are reasonably clear and straightforward, but hard to describe compactly, I will accept a citation to a work that contains such answers. However if it is a citation to a book-length work, I want a page range or small number of chapters that contains the answer.


In ceveral textbooks it is said that an ideal LC circuit does not lose its energy, it just bounces from the capacitor to the inductor coil back and forth. On the other hand, any system where charges are going to be accelerated radiate energy (that's should always happen when involved charges are going to be proper accelerated, which is a consequence of Maxwell equations). But then due to this, even an ideal LC circuit should lose after any cycle a certain amount of its energy as radiation, since the charges there are accelerated at all? Therefore I'm confused at that point: Does Bremsstrahlung not occur in an ideal LC circuit (why? That does not make any sense, the charges there are going to be accelerated, so this should give by Maxwell theory radiation losses)? But if this accelerations of charges takes place there, then the energy loss cannot be ignored even in this ideal case, or can it? Does anybody see how to resolve my confusion?


Everywhere it is written that electrons flow through through wire of an eletric circuit, and the reason given is that the battery maintains high potential at one end, and low at another. But, if you think about it from first principle electricity& magnetism, then one would realize by intuition that the provided reason only would imply that the electron would move from the high potential region to the low potential region, not that it takes the specific route of the wire. To find a specific route, it would involve solving of the differential equations. In regards to solution, many places I've seen have written that the eletric field caused by the battery only exists inside the wire, but how do we know that/justify this approximation? I have found this related question, but I felt the answers quite unsatisfactory in regards to the above problem. Also, I Understand that to study circuits, we need more than Maxwell's equations, that is a theory of the matter (charges involved and such) as discussed in this question, but my question is, what is the basis for the justifying that the eletric field is confined to the wire?


Central idea of Michelson and Morley experiment was that if single speed of light predicted is speed of light relative to ether, then as the earth moves through the ether, light travelling along the direction of motion must travel at a different speed relative to the earth and light moving at right angles to the motion. If I get the argument then something like saying say velocity of ball is fixed in air. Then someone watching the ball from a moving train will see the velocity of ball different in the direction of motion. But in the experiment, the measuring apparatus ( equivalent of person measuring speed of ball in train) and the what is being measured i.e. light are in the same frame (frame of earth). Why should the velocity come different in that experiment as they thought? The argument could have worked if someone to look at this experiment outside earth frame? I know I am missing something just not able to pin point.


I don't know if this site is a suitable place to look for examples, but I have a lot of ideas that I would like to find a good example of, and I try to organize these things in a later book. I hope I get to a place where I can discuss my thoughts so that I find people taking these questions seriously, and I think they are thoughts of many people. I am searching for an interesting example/example of a theorem, issue or case in mathematics, in which there is a specific variable that affects the result, while the intuition tells that the result is independent of this variable. In other words, I would like a case where intuition fails to estimate the effect of a factor on a particular problem and one thinks it is ineffective, while reality is otherwise. Are you welcome to find examples of searching for examples on this site?


Urban Dictionary includes one explanation of "play" : Anything to do with sexual relations: fooling around, making out, oral sex or having intercourse. However, most of the formal-language dictionaries don't include this explanation, only interpreting the word as "to tease" which can describe something childlike such as flirtation, except the phrase "play ground" meaning: Having sexual relations with other people who is not your partner. Merriam-Webster Dictionary also suggests "play (n.)" as "flirtation", which can be converted into "flirt" with "play (v.)" Is this a difference between the formal language and slang? This question also leads to an answer which regards "play" as a synonym of "flirt". word for a person that flirts with everyone? Does the slang term have a much broader meanings than the flirty activities implied in the formal language? Does the term imply being unfaithful?


Any introduction of relative simultaneity usually is like this : Alice on a train car shines two beams of light towards both ends where two clock are. She sees both clocks see the light at the same time. Bob on the station looks at the same event and says the trailing clock was hit before the leading clock. I have no doubt about what Bob is measuring, but if speed of the light stays independent of the source, then it should not matter if the frame of reference is moving with the targets -i.e clocks- or not , both Alice and Bob should see clocks are being hit non-simultaneously . Is there any actual experiment that proves Alice is seeing simultaneous clock hit ? I mean has anyone placed clocks on very fast moving objects to prove light hits both leading and trailing sides at the same time ? Or do radio signals arrive at different times to equidistant Eastern and Western stations due to rotation of the Earth?


This question has come up with me in two different Final Fantasy games, where you want or need to collect every memory or skill through random selection where not all memories/skills are available until late in the (main) game (not in the random selection "game"). Collecting a memory/skill does not remove it from the set (you can randomly land on it again) My question is: In terms of getting every entity in the set, does it help (in terms of the expected number of plays to pick every entity) to play this sub-game before all entities have been added to the set (assuming even probability distribution over all entities)? If what I am asking doesn't make sense to you, please ask let me know in a comment and I will try to clarify the question.


I'm trying create a new single-item selection system that has either identical or close to the identical probability of an existing multi-item selection system. The current system has multiple "tables", each containing multiple items. Both a table and each item inside of that table have a selection probability. That is to say we determine if a table is used based off of its probability, and then we go through each item in that table and determine if that item is selected based on the specific items probability. Using the new system, I want there to be only one table containing one item. The one table should have the combined probability of ALL the old systems table probabilities and the one item should have the combined probability of ALL the old systems item probabilities. Would this mean that I would simply find the average of all the tables and the average of all the items and that is the new probability for the table and item? I'm also curious if there is way to shift the old systems table probability on to the item? That is to say, the table would ALWAYS be selected, and the probability of the item in that table would adjust based on the tables previous probability. Is this possible?


In the book To Have or to Be Erich Fromm claims using "have" in English increased due to the rise of the market economy and Protestantism. Where one is alone in the market, with their personal relationship to God, rather than being part of a feudal economy and church where they are taken care of. How well is another conversation, but the point is they didn't talk in terms of how much they personally have so much, unlike in the market economy, where what they have is all they can depend on, with the rise of private property. He wrote that this is the cause of "have" being used in more cases such as "I want to have a party", instead of "I want to make a party", as one may see in other languages. England was ahead of others in the assimilation of a market economy (besides the Dutch). I'm not sure how English compared to others concerning amendability to change though. I did not see similar explanations in the etymology of "have". So the question is, how accurate is his explanation? This question can be answered with examples of people changed lifestyles of people along with changes in language. Things happen for reasons, the answer would give the reason, whether agreeing with the question or not.


When I try to solve a hard problem, I have no idea on what to do before I write anything on the paper: apparently, only after I spent hours on a particular problem I find the key idea by just a coincidence. This makes me feel that training myself by solving tons of hard problems on integration or geometry etc... is a waste of time since for the most of it I randomly run into the solution by coincidence and that is what I mean by randomness when solving a math problem. So my question is: how to overcome that? Should I try to solve each problem on my head without writing down any thing? Or what do you advise me to do? And is this a thing that I can solve or I am just not smart enough?


The main statement is: the EM coupling constant is energy dependent, and the fine structure constant (FSS) is the low energy limit. This means that the flow of the coupling constant vanishes when the energy does not suffice any more to create the lightest charged particles, electrons and positrons (there only remain the classical tree diagrams). Below the electron rest mass energy virtual electrons and positrons are frozen out. Consequently, the FTS is a function of the electron mass, and if the electron mass comes from the Higgs coupling, of the Higgs coupling. All QFT literature is about the scaling limit and high energy (where the electron is effectively massless), and and no clear statements about low energy can be found anywhere. But in statistical mechanics terms the above statement appears to be the only reasonable.


The problem arose of choosing a direction for studying and working within the framework of this topic: neural networks as universal approximators for solutions of partial differential equations. At the moment, as a rule, neural networks are used to obtain more accurate numerical approximations, stable, compared to classical difference schemes. Today, there are many different libraries for training models and predicting solutions. My problem is mainly that I want to try, based on already selected numerical approximations, using neural networks, to obtain a number of analytical solutions. The question is: is it possible in principle to obtain a similar result, and are there any practical problems described in scientific articles in precisely this context (not refining numerical solutions using neural networks, but approximating already obtained solutions and obtaining analytical expression). That is, maybe there is some small problem that has important practical significance, the solution of which can be presented in this form? In fact, my question is quite targeted, since I am not asking for an overview of the available methods for solving partial differential equations using neural networks, but rather solving the inverse problem. In this case, I am more interested in the mathematical formulation of the problem and the mathematical aspect of its solution. If possible, please provide an answer, links to similar topics, and possibly literature. Thank you very much in advance!


I have proven this question but somehow I feel like maybe I have done something wrong or I have been reckless somewhere would you please help me to verify my answer If S is a set with least upper bound and greatest lower bound property assume X and Y are nonempty subset of S such that every element of X <= every element of Y so prove Sup X <= Inf Y I have proven this in the following way from the hypotheses we can conclude everything in Y is an upper bound of X and vice versa. so Sup X and Inf Y exists in s and lets denote them by a and b now lets prove this by contradiction lets suppose a>b since b is infimum of Y everything greater than b is not lower bound and this is in contradiction of a not being a lower bound of Y because a is supremum of X and since everything in Y is an upper bound of X . is my proof true or I am missing something in somewhere?


It is said that if one would attach a load physically to the rotating disc from center to rim no current would flow because both in the load wires and the disc itself currents would be generated in the same directions, everything would cancel out, right? But I don't understand why this is a problem? there is a simple workaround: The wires can lead out of the field or extend far away from the magnetic field range, like so: The blue arrows symbolize induced force vectors when disc with shaft and wires rotates, current should flow from the rim to the center of the disc just like in Faraday brushed version, the upper part of the shaft and the wire does not cut the field lines. However I have build this and no EMF was detected, I don't understand why, the voltage induced in the shaft should not block the current coming from the rim to the center of the disc?


Like 'pretty ugly'. The most used meaning of 'pretty' is contradictory to 'ugly', and it could feasibly be used to mean something like (if it were an oxymoron), 'someone who may be considered ugly, but is actually kinda pretty'. But of course it actually means, considerably ugly. Is there a term to describe such pairings? Edit: I'll add more examples as I think of them: Don't trust casinos, they're fairly rigged. I'm an f'ing celibate. The doctor gave me a sick healthy report. I'm a bad good kid. -- Like they are morally good, but are bad (not morally bad) at actually making much out of it, as some role-models may do. Like they are comparing themselves to others. As opposed to a kid who is generally a good kid, but does still get into trouble some, so they may be teetering on the edge of not being considered a good kid. Which would actually be an oxymoron.


This is a question from an interested amateur. Math welcome (as I or another may investigate it some day) but don't expect me to understand it in your answer. Consider a star emitting light. In order to talk about it simply, think of a point light source, and ignore spatial perturbations (for the sake of phrasing the question). At at time t, the light emitted at a single instant at now-t forms a sphere at a distance from the source. What is the density of photons in the surface of that sphere? Perhaps the reply is that the energy is traveling as a wave, but I understand light energy is quantized - that is, the energy has no representation described by infinitesimals, but will be in minimum quantities. How much must the sphere expand so that there are gaps in the image when sampled at different locations on the 'instant-sphere' image? Or, is there an explanation as to why the energy is quantized, yet can manifest at any separation (due to distance) despite the energy density falling below the quantization when the sphere is of sufficient size? What if the source is a candle? Is the original source always sufficient to 'map' adequate photons to any sphere that may be created in the dimensions of our universe? Are there objects we cannot see because they are so far away (or dim) that we are 'in-between' the photons that form their expanding image? Of course, I may simply not understand, but the explanation would be welcome.


I play football. However, I dislike going and sometimes I just dont want to play it. But whenever someone tells me that i do not have to do it or that I can stop when I want, I put all the work into it. One reason I can assume is that my father played ball and I am the last son, he and my mother want me to play but I have "mixed" feelings to say the least. Whenever I constantly tell my parents that I dislike football, they obviously get very saddened and I dont care about that in particular. The strange thing is that when I get the choice alone and with nobody around, I always play ball because it makes me happy. But whenever I'm told that I have to, it feels like a chore. I can't explain this to my parents and I feel the need to get this off my chest as I am heading out soon.


As we know, In a circuit, simple or complex, electric fields created by surface charges move electrons which creates current which creates magnetic field which can be coupled to other lines and induce voltage if it is changing by time, which also creates current and this current also creates magnetic field and voltage drop along where it is coupled so on and so forth. Voltage and electric field are related just like currents and magnetic fields are. Changing electric fields create magnetic fields and the opposite is also true. I mean what is the independent variable here? Everything is like being affected by one another continuously. Is it power or energy or fields? What variable or thing I must I hang on to or take as starting point for better analysis of circuits?


Currently the proven theory is the quantum field theory. This theory defines fields in "all spacetime" and particles are disturbances in these fields. These particles are punctual and interact through virtual particles. But string theory is not clear to me. This defines the strings (bosonic or fermionic) as the smallest entities and these "strings are in a spacetime" but they are not defined unlike the fields in all spacetime. How is it possible? How is the zero point energy then described? So we must understand that at every point in space there is at least one section of string? Here the difference is analogous to the first quantization with the second quantization. Is a string field already confirmed in all string theories? Is there talk of virtual strings? If I take a field, for example the Dirac field or the EM field, these are defined in all time space. It may be that there are disturbances that we interpret as particles, but if there are not, the field exists the same and has a zero field energy value not equal to zero, with virtual particles being created and destroyed. Or for example, the higgs field directly has a non-zero energy in all space. Do we have a defined string in all space?


I am wondering how heat loss happens in sound waves through air. First of all, for wave energy to be transformed into heat, absorption should happen by a medium. But why wont absorption implicitly translate to faster vibrations which would state that even though wave energy was translated into heat vis absorption, that again must have transfered back into wave energy. My logic is that some particles that absorb wave energy will in turn increase their their own kinetic energy, and those that dont absorb wont. So some particles will vibrate more, some wont. Then due to this difference between particles, redistribution energy will occur and since particles that absorbed energy, they might not actually vibrate in a pattern but randomly start moving who knows where. I believe this random motion and not in direct vibration pattern is why we say energy of wave was lost due to heat conversion. Correct ? But I wonder why absorption(even though it increases KE of particle) does not implicitly give it more vibration in the same pattern as it had before the absorption ?


We call G the formula that in the formal axiomatic system employed by Godel to formalise arithmetic has the meaning of: "I am not provable in this system". We know that this formula G is true in that axiomatic system since we can exclude that it is false. But why can we say that it is also true in arithmetic itself? After all, within arithmetic such a formula is only a set of symbols, and even if well formed it can still be a false formula. How can we say that such a formula is also true within arithmetic and not only in the aforementioned formal system that formalises arithmetic? We come to this conclusion because the formula G also within mathematics retains the meaning of: "I am not provable in this system" or are there other reasons for considering it true within arithmetic?


In security, we speak about different strategies of protection. Mainly, there are three to consider. First kind, usually called shell protection corresponds to the act of preventing the intrusion before the access. For instance, a door, fence, password on a website or silver galvanization of the utensils. That's usually very powerful and easy to maintain. Occasionally, there's a breach and then, we have problems, unless there's also the other kind of security implemented. Here, we're talking about smoke dispensers, mazes, ambulating patrols, honey pots, fake repos etc. What would be a good term for that? (Extra great if it's formal'ish.) The one I've been using is impregnation protection but I'm not sure if it makes sense, let alone is correct linguistically, let aloner comprehensible to regular people. (The third is deterrent protection and not relevant to this question.)


I have seen videos that interview accent experts, like on why Americans do bad English accents (a reason given is a lack of exposure). But are there something like references that one could point to to rate accents? I would think at least one should survey native speakers to get a rating (I met a Kiwi who said he would get mistaken for British by Americans even). Maybe AI could do ratings nowadays if trained on enough. I figure one would need to do individual ratings for each case, even if a reference existed, like having to break down examples, showing where they messed up. If ratings were compiled somewhere that'd be easier though. Like something one could use to show how bad Americans doing accents in lotr were, except like wormtongue, which one may figure influenced having less in the hobbit.


As oppose to a cylinder in two pieces rotating against each other, as in O'Neill's cylinder, what if the cylinder were all one piece and an external thrust, like rockets, set the cylinder in motion, rotating it, would it rotate indefinitely? Would inhabitants of the cylinder experience artificial gravity indefinitely, or would their movement against the rotation of the cylinder slow it, or would movement with the rotation of the cylinder even it out? Or would neither effect the closed system because they'd be pressing against the cylinder from inside the cylinder? Would friction from the air inside slow the cylinder over time, or would the air turn with the cylinder and artificial gravity? If any of the above would slow the cylinder, how frequently would the rockets around it have to add more momentum to keep the craft spinning? Would it at least be a number of years?


What does it mean, "gravity is so strong that not even light can escape from a BH..."? This can physically interpreted IMHO that beyond the event horizon of the BH, light is phase transitioned (accelerated) to a FTL superluminal energy that breaks known physics. The inner of the BH therefore appears in our spacetime as "Nothing"! Exactly as the rest of vacuum space appears to us, basically as nothing. Someone could therefore infer that vacuum space we generally describe us nothing is actually something and considering the physics of BHs it points to the possibility that vacuum space could be a superluminal FTL (i.e. faster than light speed c) unknown type of energy. There is light, matter and vacuum in our universe. Seems to me that vacuum could be a different energy than matter and light, defying our known physics which we mastered over millennia and concern more about matter and light and their interactions. The vacuum is still a mystery today. Much to learn in the future. Maybe sci-fi Star Wars creators imagination was right after all when saying "...prepare for a jump to hyperspace!".


classical electromagnetism , from my understanding can be derived completely from coloumb's law , charge invariance, superposition principle and postulates of special relativity. Biot savart law and lorentz force law are all consequences of it . For derieving faraday's we need not do any experiments, flux rule : e=-d(phi)/dt (e is induced emf and phi is magnetic flux) can be proved from motional emf . From flux rule , it can be understood that emf is induced when area of conducting loop changes in constant magnetic field or magnetic feild is a function of time with constant area. As the flux rule rule is derived from no experimental information other than lorentz force law biot savart law ,we should be able to prove that changing magnetic feild induces emf and hence current is induced . It seems that magnetic feild cannot induce current because it acts on moving charges (which in our case is not moving).experiments shows that it is non conservative electric feild that induces current , but why do we need to do experiment because we already proved it in flux rule. My question is that this information of inducing emf was in flux rule , but it seems that we can not prove induced non conservative electric feild from basic priciples (coloumb's law , special relativity,charge invariance....)


The photoelectric effect is most probably seen when the incoming light has lower energy than the energy needed for both Compton scattering and pair production to happen. The probability of the photoelectric effect to occur also increases when the matter that light interacts with has a big atomic number and high atomic/electronic density such as Lead, Tungsten, and even better Uranium. Given the energy difference between Gamma and X-Rays, and the for high probability, photoelectric effect requiring low energetic photons to occur, how is it possible that X and Gamma rays are both prevented by the photoelectric effect that is observed in those materials? Is it that energy levels of photons that photoelectric effect has to occur on materials like Tungsten are on the high energy boundary of X-Rays and Low Energy boundary of Gamma Rays, and thus rest of the matter-photon interaction effects such as COMPTON and PAIR PRODUCTION are almost always on the GAMMA Spectrum then?


I was watching Susskind lectures in string theory. There he explains that open strings can both, split at any point, and also join at the ends when the ends touch at a single point. I have one question about each of these two processes. Is not the likelihood that the two ends of a string end up at the same spatial position of measure zero? Or the two ends do not need to really meet at the same point, but only be close and then they will be attracted to each other to make it closed? If and open string breaks at an arbitrary point, would not this create particles of arbitrary mass? as the rest mass is proportional to the length? but we know particle masses do not form a continuum. What am I thinking wrong?


If we shine a light that has less energy than the work function energy of the electron of an atom on a metal, the electron is not released but excited and electron gives off this excess energy as heat or collisions or light. We know that the intensity of incoming light does not affect whether the electron will be released from the metal or not, only the energy of the incoming photon determines it. So, Assume that we hit an electron with an energy lower than the work function energy so that the electron gets excited and builds up with energy. WHAT IF Before that excited electron releases its build -up energy as heat or light or collisions we hit it again with a light that again has an energy lower than work function energy, why isn't it released from the metal?


Current in a wire is defined as the amount of charge that passes through a cross-section of that wire in a single second. By this definition alone, it is clear that a current relies on the motion of some charged particle. I believe it is possible that electrons could transfer energy to each other in every direction and when the current starts flowing those energy transfers are more directional. This could lead to electrons always having the same velocity independently of the current. Is this the case? I came up with this wile trying to understand magnetic force around wire, if electrons are traveling in the same direction their spins would also be in the same direction, which could cause magnetic field similarly that would spinning ball in viscose liquid cause flow of that liquid.


In cubical homology you have to consider the group of degenerate cubics and use the group of cubes module degenerate cubes (Massey). If you not do that you get the wrong homology for one point space. In singular homology (simplices) every textbook explain you do not need to do that because you are going to get the same groups. It is easy to check that you get the correct homology groups for a point space without exclude degenerate simplices. But I would like to know a GENERAL PROOF of that. That is the equality of both groups in general order and spaces (excluding or not degenerate simplices). Because it seems that textbook writers consider that evident for I have seen the statement many times but no proof at all and I am not willing to take it by faith. I have read an old paper by Tucker available in internet by googling: Degenerate cycles bound. But I dont know how to apply that to singular homology Thank you very much


I have a set with unknown cardinality. The cardinality can be from a number a to a number b. How can I indicate the number a (or b)? The question in texts is: "What is the least value of n(A)" (A is a set with a few possible cardinalities) How can I properly express this in mathematical notation? I used the notation(If you call it a "notation") least of n(A) I have seen this question: Mathematical notation for the maximum of a set of function values The things that differ is that I need the notation to express the minimum (or max) number a function can give for a constant x, not the min (max) for a range of x Also, the set of all possible sets isn't given and is there to be found out by the question solver. (as it implies, each solver may generate a different one) n(A) is the cardinality of the set A I do not have a strong mathematical background. so please avoid very confusing notations. thanks in advance


Clearly particles individually pass through slits, be it a single or a double slit experiment. The fact that wave interference is evident in their trajectory may be due to their interaction upon entering the slits. If water particles, or sand particles can together form waves, one can assume they will act in such way that their individual trajectories will reflect the wave they form, and once that wave, if it were to pass through a couple of slits will alter their individual trajectories to satisfy the interference pattern we see on the plate in the double split experiment. In other words, the interference pattern in the double slit experiment (based perhaps on the my sorely naive point of view), are the result of particles, that together form a wave, and upon the entrance to their respective slits, their altered trajectories through the slits reflect their wave relationship. I need to know if any of this is absurd.


[Note I am asking up to, but not including, consciousness as this bleeds into philosophy and is a much messier question] Assuming that the laws of physics have remained constant across space & time since the big bang, has the way the universe evolved been entirely predetermined? While impossible to know the physical parameters & interactions across all space and time, these unknown states would have been acted upon by consistent forces since their onset. Thus, while not knowable, the universe grew along only a single possible path forward, meaning from the instant of the big bang to (cosmic) now, everything has been entirely determined? Phrased another way, was there any physical process that introduced true randomness that caused the universe to have multiple different potential paths? Note that I don't mean things that seem random - such as paths of specific gas molecules when dispersing - since these are ultimately entirely predictable given sufficient knowledge of states & environment. Or is my premise somehow wrong?


If the Hamiltonian manifold for the moving surface of the standing wave is smooth then it must be minimum surface of revolution. The frequency of the string is the capacity of the manifold that is a periodic solution to the Hamiltonian. The frequency cannot be a rational number frequency = velocity/wavelength because the solution to the Hamiltonian is a real given by the Dirac measure. There is only one Hamiltonian so I think string tension and length are determinative. All I want is the mean curvature of the manifold. You are being asked here to prove the manifold is not symplectic. Obviously, you cannot. I say you must conclude curvature is constant. You have no way out of this except to say I have not explained this to you well enough that the string makes coherent theory. Do you have a calculus for the string as a moving surface?


I. I thought he was swimming - I thought he swam for a while but found out it was a lie I thought he is swimming - I thought he was swimming now, but he's not. I thought he swam - I thought he once swam (don't know when). I thought he has swam - I thought he has already swum recently (a couple of minutes ago). I thought he had swam - I thought he had already swum recently (a couple of hours ago). I thought he ... but i changed my mind, and now i dont think in this way If I can't frame my thoughts in this way, how should I do it? Specify proposals, add new ones? But is it possible to designate meaning with the help of time?


If electric fields are created by an accelerated charged particle, such as an electron, and magnetic fields are generated by electric fields in motion, what are the individual fields that make up Electromagnetic Radiation oscillating between? What is the Y axis in their sinusoid function measuring, and how does that continue to oscilate in the absense of moving charge once the photo is emited? Is the electronic field aspect oscilating between a negative and positive charge relevant to overall energy levels, or is it confined to one side of that? I am asking as I have an incomplete understanding of how a photon, without a charge, continues to oscilate at the speec of light. I fully expect that I have an oversimplified understanding of both repsective fields, and I hope to learn what unknown forces are at play.


Hi there Wise people of the internet, I am trying to do analysis some data gathered from a gamma scintillator setup, its stored in root. So i have to do some coincidence measurements, and i found that in Krane, you normally use a TAC (Time to Amplitude Converter) to check for it. However, this gaussian peak expected there has count per channel in y axis. Should i just divide the counts that i recieve for each channel by channel number? I dont see how that will be useful.(this will disproportionately eat away any meaningful data stored in higher channels, which corresponds to higher energy,after calibration, breaking expected stable background of chance coincidence) I am kinda new to detectors. Any help is much appreciated and also any resources/references will also be. Regards


There is a group in which is embedded the set of rational functions considering the formal composition binary operation? More generally how (and when) can we extend a non-commutative monoid (or semi-group) into a group ? I was looking at the interesting case that the natural numbers can be extended to de integers by the matters of considering it as a monoid over the sum and constructing a group of the fractions and then extended again into the rational by applying the almost same method considering the multiplication operation this time. Thinking in this context i try to apply the same ideia to the set of rational function (with rational coefficients - quotients of two rational polynomials) but considering the composition binary operation as it is associative, has a neutral element, etc... But obviously isn't that simple and after searching for a answer i discover the concept of group completion of but not much more and specially not much for the non-commutative case. I'm thinking in a abstract way, thinking in a rational function as a pair of finite sequence of rational numbers, as was already noted in comments that not all rational polynomial are invertible, but. Considering that the case is analogous to the fact that not all integers are multiplicatively invertible


I am stuck writing this sentence: Relics such as Fortran, B, D and other programming languages continue to stay alive, stories of programming languages in the graveyard, and those that spurred magic have antecedent the continues quest of improving programming. EDIT: Clarity. Think of this as a battle cry among programming language developers. In this battle cry, stories of both success, and failures emerge overtime and historically, nevertheless, the journey is not even complete or done. Essentially, events, programming languages in the graveyard, and programming languages that made magic and allowed some companies to achieve technology never seen before, the combination of all of this has thus, "......" the continuous quest of improving programming. I am trying to make the reader, from also previous text and stories discussed in the text, to understand that, technologies that left a MARK in an industry or across industries as a result of a different programming language have caused: a curiosity and quest to build even better programming languages that can move industries forward. I HOPE THIS IS CLEAR ENOUGH.


I've read quite a few things about rarefaction waves in terms of giving weak and entropy solutions to certain PDE problems with fixed initial time data. I understand that the consideration of these solutions provide some sense of density that decreases with the passing of time, but I've not heard much else. My concerns are: Why are these kind of solutions specially noticed as one of the very first examples given in several books?. Maybe it's some sort of "unwritten" but written tradition to expose these solutions. Are they directly applicable in some specific physics topic or something? I'm pretty sure they are since every attempt I've done to try to research about the topic results in some content including the word "density". Any thought, suggestion or reference could be very helpful to me ;)


Papers often use one example throughout as their ongoing example. Similarly, it is possible to focus on one particular case and show a broader claim by using that case. I remember there was a really nice verb for using x at a test case, or perhaps x serves as a use case. I tried a thesaurus searching for exemplify (which is close, but not exactly right) and some other searches ... but none of these helped much. An example sentence might be: Testing food in general, we verbify eggs. or Eggs verbify food in our context. More elaborate examples: Using the RR dataset as a test case, we investigate how different aspects of metric design affect the computation cost. We perform an analysis study on efficiency, using the popular RR dataset as a test case. As a test case for investigating dataset efficiency, we analyze the results of RR.


Could somebody untangle following statement I found here: the integer cohomology groups correspond to the quantization of the electric charge. I know from pure mathematical side the meaning of cohomology groups, but not understand the translative part between physics and pure math at this point: Could somebody borrow some time to sketch how this identification/ correspondence is in detail established? ie how can the (presumably geometric) quantization procedure be "interpreted" / encoded purely in terms of certain cohomology group? And furthermore, what is here the precise meaning of the notion of "a charge" from viewpoint of pure mathematical terminology? Up to now I thought that a charge in mathematical physics can be recognized as a quantity comming from the existence of global symmetry of the given system, so in simple terms certain intergral/ number witnessing the existence of such global symmetry. Do they endow in the quoted sentence above certain "bulked" meaning to the term " charge" in order to making it to becomes the object subjected to quantization procedure?


I am trying to find some sort of motivation as to why we integrate manifold over differential form and why especially does it in some form corresponds to integrating the surface of the area. I have already passed my university courses that have covered these topics, including Stokes's theorem with proof (My uni has this approach of definition - theorem - proof structure in all courses with very little motivation). The issue is that sadly I've never intuitively seen that it could have anything to do with the real area if the manifold would be an earth-like object or any other intuitive geometric structure. Is there any free material/youtube class that illustrates that it truly corresponds well? I don't need to see proof as I quite understand the technical side of things. My issue also is that I cannot see differential form in any other way than only technical mathematical extremely abstract definition. Please keep in mind that my knowledge is limited to European standards of a bachelor's degree...


Let say we have some water in the sink and open the closure.. The water starts to move towards it in a whirpool like manner.. If we have a table tennis ball and leave it near the hole of the sink it will rotate like a planet instead move right towards the hole... If we suddenly open up more the hole the water will start to move faster towards the hole but even the water moves towards the hole the orbiting ball will still be able to detect the change of the water speed due to the hole of the sink being openned more... Can this be compared with gravitational waves where even the spacetime tissue moves towards the BH binary star the orbiting planet is still in position to detect this sudden perturbance moving in opposite direction regarding spacetime tissue simmilarely as the table tennis ball get the perturbance moving in oposite direction than the water moving in direcion of the sink hole?


Hi everyone, lets say we have a circle shaped space station and there was an accident where we got a hole (hole A on the picture) on a station and now most of the station is vacuum except for one enclosed part that is still filled with air. Now if we make holes B and C at the same time in the enclosed part, what will then happen with the air? Will it: because air moves in random directions, half of the air will go through hole B and then through hole A and other half will go through hole C, go through entire station and then through hole A because of difference in pressure all air will go through hole B and then through hole A (I am not sure if this is true or I am misremembering, but I read somewhere that air takes the shortest path to equalize pressure so that is the reason that all air will go through hole B? If this is not the true is there some way to influence all of the air to go through hole B, change the size of the holes, make a longer distance between holes C and A?) this and next scenarios are highly unlikely but I have left them just to cover all possibilities , all of air will go through hole C and then through hole A air will stay where it is and it will not leave the room


First, there are only three type of leaves: enter image description here If we assume lamination is compact ,all leaves is only can be a simple closed geodesic. I want to ask about how could union of uncountably many leaves be a minimal compact lamination,actually all leaves is complete simple geodesic,but all leaf in a compact minimal lamination in a punctured torus can only be a simple closed geodesic cause leaf of compact lamination can not goes up to cusp ,so it can only be closed simple geodesic ,so we take closure of any leaf ,it can only be itself, that contrict to minimal. cause i dont know that question above, i can not understand why we cut a punctured torus along a compact minimal lamination which is not a simple closed geodesic ,we can gent a ideal bigons which is patch two side of two ideal triangels.


If I were to measure some quantitavie metric of a sample population and record its mean, and then I were to split by random selection all members of the population into two groups of equal size and record the means of each group on the same metric. Is their a statistical term/measure of the difference of those means from each other and/or the mean for the whole population. This comes as I'm trying to analyze an experiment in which both the control and sample group where told information regarding the experiment during a pretest where I fell like the information could have been left out due to the large size of the population. My rational behind that is the difference in the split means would decrease as the overall population increases and one could assume that both sample groups are evenly distributed.


The court's ruling is claimed to be (...) because it has only counted specific actions to set up some (...) simple structure of justice that only encompasses the last hour neglecting the history of events. (...) = ? = "has only lasted for a short amount of time prior to being mentioned" Referring to current attitudes that have arisen in the general span of an hour or day or other fairly newly developed engagement. "Of the moment" "According to recent neoteric dispositions" The structure is simple because it is founded on only one or two just-happened pieces of evidence. Adjacent to "transient" "transitory" "ephemeral" but in reference to a newly grown/spoken structure of logic. "recent" and "latest" seem too general and encompassing an indeterminate amount of time, whereas the word I'm looking for refers only to the nature of the just-referenced events, which is (...) Transient = lasting for a short amount of time (...) = has only lasted for a short amount of time


Simulate a magnet sticking to a metal door. The side that is against the door, means that side is the pulling side, and must be the inwards movement of a toroidal magnetic field, pulling the object inwards resulting in getting stuck to the object! Does this mean, the side that is against the object is the side where the field goes in through the magnet and will always be the actual South pole of any magnet? When the magnet is flipped onto the object, the field is in the opposite direction pushing against the object, which will be the North pole of that magnet. Does this mean that when holding a magnet and you can diffirentiate between the actual physical pushing and pulling side of a magnet, them to know that the pushing side must be North and the pulling side must be South? This has Nothing to do with: Stringing the magnet and it will find magnetic North or South and zero to do with earths magnetic field direction, but everything to do with the toroidal magnetic field of the magnet itself and which direction the magnets toroidal field flows?


In school, I learned the mechanism of high and low pressure areas, which roughly goes like this: In the tropics, the sun warms up the air during the day. Water evaporates, so that the air gets warm and moist. Warm air is lighter than cold air, so it rises up, leading to a low-pressure area because the warm air is now "missing", so we need air to horizontally flow in. The rising moist air cools down, so it starts to rain, that's why there are rainforests in the tropics. Some thousand kilometers to the north or south, the air is sinking to compensate for the rising air in the tropics. Sinking air warms up and gets dried, so it is very dry (Azores High), leading to the big deserts in the subtropics, e.g. the Sahara. I hope this is more or less correct. There is one thing I do not understand: Why is the air pressure lower in the tropics, and higher in the subtropics (Azores High)? My intuitive physical reasoning tells me the opposite, for various reasons: The reason warm air rises up is not that because "it is light", but that it is lighter than cold air, so the cold air can replace it by flowing under it and lifting it up. So the phrase "warm air rises up, leading to a low-pressure area because the warm air is now missing" is very misleading: it is the inflowing cold air that lifts the warm air up. So there is never a "moment" when we have a lower pressure - the warm air rises because it gets replaced by cold air, and there isn't a reason why the pressure should drop. The air pressure is more or less the combined weight of the air above us. The heavier the atmosphere above us, the higher the ambient pressure. Now, when the air warms up and gets lighter, it gets replaced by the cooler air flowing in. So after the warm air has risen up, we should have more air above us: the warm air that has risen up, plus the cold air that has flowed in horizontally. So the atmosphere above us is heavier in total, which again means we should have a high-pressure area. Another view could be that since the air in the tropics is warming up, it expands. This large-scale expansion should lead to a high-pressure area. After all, if you expand a large amount of gas, its pressure increases until it has had time to expand properly. I am very much aware that all these reasons must be wrong in some way. I don't doubt the existence of the Azores High or the rainforest. I just miss an intuitive understanding. How can the tropics Low and the subtropics High be explained in intuitive physical terms?


My understanding is that in commercial nuclear reactor operations, fuel rods are not used up to the point where they're fully depleted and unable to support fission, but are replaced while they still contain an appreciable amount of fissionable isotopes, to ensure that the reactor stays in a stable operating regime at all times. However, would it be possible, for any reason, to run a nuclear reactor without replenishing the fuel as long as any useful energy at all could be produced, that is, actually "running the fuel rods dry" (or, more formally, until criticality is irrecoverably lost and decay heat is the only remaining output left, at which point the reactor is essentially just a fancy spent fuel pool)? Could such an end state be safely achieved, with power just gradually fading away while the control rods extend out more and more to compensate for the declining reactivity? Or would unstable and potentially dangerous operation ensue, as in Chernobyl where operator response to a poisoned core started a catastrophic chain of events? The question is motivated by the realization that one of the safety-enhancing measures proposed for the RBMK reactor design after the Chernobyl disaster was to raise the enrichment grade of the fuel, apparently to reduce the type's susceptibility for core poisoning and the resulting power fluctuations (at least that's how I understood it). I'm interested if the converse is also generally true, that is, whether it would be hazardous to let the fuel deplete too much in modern PWR/BWR reactors that don't share the design flaws of the RBMK.


In biology, the scientific name of a species (known as the "binomial name" or just the "binomial" or sometimes even just the "binomen") is written as a pair of words in italics (or underlined, which is the equivalent of italics in handwriting). For example, modern humans belong to the genus Homo and, within this genus, to the species Homo sapiens. The first word specifies the genus (meaning a group of related species) and has an initial capital, while the second word specifies the species, which doesn't have an initial capital, even if it is derived from the name of a person, e.g. the scientist who discovered the species. https://en.wikipedia.org/wiki/Binomial_nomenclature For example, that Wikipedia article says, "[...] the binomial name of the annual phlox (named after botanist Thomas Drummond) is now written as Phlox drummondii." Note that in the phrase "annual phlox", which is not in italics, "phlox" has a lower case initial. So in this case it seems that Wikipedia thinks it is acceptable to write "phlox" in all lower case. And yet I've often read that a genus name must always start with a capital letter. For example https://en.wiktionary.org/wiki/genus_name says, "genus name (plural genus names) (taxonomy) The scientific name of a genus, which is always capitalized; the generic name or generic epithet. Usage notes The scientific name (binomen) of a species is a two-part name and is typeset in italics, the genus name (the first name) has an initial uppercase (capital) letter and the species epithet (or specific epithet) is written with a lowercase (small) letter; for example, the scientific name of the wild Rock Dove is Columba livia.". First, notice how the dictionary quoted above wrongly has "wild Rock dove". It should be "wild rock dove" of course. Ha ha. Now, if you will, imagine people somewhere had started to refer to one or more locally found species of the genus Columba as "Columbas"/"columbas". This could happen because calling them "rocks" would be confusing. After all, rocks do sometimes fly, land, and so on. In writing they write them without italics or underlining and sometimes without an initial capital. So my question is: Would the latter be acceptable in formal writing among nonscientists? To me it seems a bit anomalous for something that is a type of animal to get a capital letter. It's as if biological jargon has spilled into ordinary English, and where it conflicts with the rules of the latter, it just brushes them aside. The italics/underlining rule is rarely followed by nonbiologists, so do we really have to always follow the capitalize all genus names rule? Can't we follow the English rule that says types of animal are not capitalized?


The following question was discussed in my Discrete Math class, but we couldn't reach a consensus. Think of a set as a collection of bins. Each bin contains exactly one object distinct from all others. While we are allowed to move the objects around between bins, we cannot remove any object, nor can we place two objects into the same bin. Someone hands us a new object. Can we place that object into a bin, after moving around some of the objects already sitting in bins? A. Yes, for some sets this works just fine. B. Sets are not composed of bins and this question makes no sense whatsoever. C. No, there is no room for another object, all bins are already filled. D. It depends on the new object. I'll try to summarize the general arguments for why each answer is correct or incorrect: A: A few of us think this is right, because we can insert the new object into the set iff it's not already in it. However, this is similar to answer choice D, which was given to be wrong (see below). B: Many of us shy away from the answer because it "sounds" incorrect, but I personally believe this is the right answer. This is because I disagree with the representation of a set as a collection of bins (each of which contains exactly one object), for the simple reason that the model does not seem to account for adding new bins (and hence does not account for adding new elements). C: Some thought this was correct, if we assume that the number of bins cannot increase (i.e. if we cannot add elements to the set). Some thought this was incorrect for the same reason; because it would mean we can't add any elements to the set at all. This answer was given to be wrong. D: Most thought this was correct - if the new object is different from all the objects already in the set, then we can add it, and otherwise we can't. However, this answer was given to be wrong. And the fact that this is wrong seems to also imply A is wrong (i.e. the intended answer isn't talking about whether or not the element is already in the set). Could someone give a justified answer to this question using concepts from Discrete Math and Set Theory?


I have seen different explanations to understand why there are no local gauge invariant observables in gravity. Some of them explain that diffeomorphisms are a gauge symmetry of the theory and thus any observable evaluated at a spacetime point will be gauge dependent and therefore not an observable. This line of reasoning then argues for Wilson loops, or asymptotic charges as good (non-local) observables in gravity. This explanation, in my opinion, is purely classical, it doesn't rely on the uncertainty principle, or commutation relations, etc. However, other explanations give the argument that any device trying to measure a local observable will have a finite size, and therefore a finite accuracy. If the device wants to probe local physics, it should be smaller. However, the uncertainty principle forces the device to collapse into a black hole before allowing the experiment to give us local information. Alternatively, it is also explained that the commutator of two (quantum) operators has to vanish for spacelike separations but that a metric that is dynamical and fluctuates will mess with the causality of the theory, therefore making the operators not observable. These arguments seem absolutely quantum mechanical. I have a couple of very similar questions: Is the statement "no local gauge invariant observables in gravity" true in classical GR, in quantum gravity or both? If it is true in both, why do people treat the statement "no local gauge invariant observables in quantum gravity" as something special? Do statements about observables in classical and quantum gravity mean different things?The arguments given to explain each one are pretty different and seem to involve different physics. The first one relies heavily on diffeomorhism invariance while the second one relies on holographic-flavoured arguments about how much information you can concentrate in a given volume before you form a black hole.