Why not explain the apparent attraction of masses by a repulsion coming from all directions in space (perhaps the dark force)? I.e. there is no gravitational force, just a repulsive force. A point in midspace would be receiving repulsive forces equally from all directions and appear to not accelerate. Mass near the outer reaches of our universe would have more repulsive forces coming from behind, and therefore accelerate outwards. An object near a large mass would be receiving less repulsion from the direction of the large mass (because that mass has absorbed some of the force) and therefore be pushed towards the large mass.


This question was inspired by the very first exercise in Thurston's Three Dimensional Geometry and Topology, where he gives a picture of a very tangled up loop and asks what manifold it depicts. I understand there are several knot invariants and algorithms that can be used to detect the unknot. But where are the practical implementations? Are there programs into which one can feed an actual picture of a knot (as opposed to more directly computational representations) and have it at least recognize unknots? One test would be whether it would accept the above picture as input and give the correct answer.


I have noticed when I view a hose pouring water the color of water slightly changes when looking from above compared to when looking from the front. Also, the same thing applies when you pour water from your mouth. If you look at it from above there's a color I can't describe very good (but I think there's a yellowish in it). But when you look at yourself in a mirror pouring water from your mouth, the water is colorless. To see the phenomenon better do it immediately after you've finished brushing your teeth. Just put a little water in your mouth and pour it down (looking from above). Why do I perceive this color change?


What is the proof for this (assuming that we draw infinite field lines). I understand why flux through some area is proportional to the number of field lines through that area only in the case of an isolated singular point charge. However, I dont understand why this would be true in other cases, like two unequal charges. To make it clear, when an area has twice the amount of field lines going through it than another area, why is the flux through it exactly twice the other area. Why not thrice or half? Also, why do field lines that originate isotropically from one charge always end isotropically at another charge.


I want to strap down a big heavy cylinder on a flatbed truck. The strap is attached to the truck bed as shown in the picture and also behind. Will the strap slip off as in the next picture? PS. This is a purely geometrical question about the length of the strap and the shape of the cylinder. Please consider that the block cannot move sideways and is blocked in place by some metal "foot" at the bottom.


I've read that the equatorial radius of the ergosphere of a rotating black hole is the Schwarzschild radius. But in this animation made in natural units by Yukterez it doesn't apply. I then used Yukterez equations in Geogebra and I observed the same thing. No surface scales accordingly. In the extremal case, the event horizon at the equator should be half the Schwarzschild radius, but it isn't. Is this a consecuence of using natural units? How could I "fix" this?


An FTIR spectrum was measured for me (device from Alignment and MCT Detector) and since I'm not very familiar with FTIR I'm looking for some advice. The absorbance spectrum was obtained by measuring the reflectance of a reference sample and the actual sample. The reference sample was stainless steel and the actual sample was probably stainless steel coated with dopamine and a peptide. The spectrum shows some kind of oscillations. It was suggested that the oscillation is due to interference caused by the dopamine coating. The coating could act like a thin film. Unfortunately, I haven't found anything paper/ literature supporting the suggestion. Have you seen something similar?


I have a question. I have a problem that when I learn science, I like to think 'how do they know this is right?'. When we learn physics or science from a textbook, we read and understand it, and then we have exam. But how do you know what is given in a textbook or said in research is correct? Because we didn't do this experiment, or we don't know what the process is to get this scientific result by ourselves? So, should I trust it and accept it? If not, what should I do?


Suppose a body is kept in space and is not fixed about any point (i.e, not hinged). If an external force is applied, the body rotates about the centre of mass. I saw one answer pointing out that moment of inertia is minimum about the centre of mass, and thus motion around any other axis is unstable. Is this view correct? And if true, how do we go about proving that it indeed is the most stable?


Both nematic transition and Jahn-Teller distortion distorts the lattice and reduces the symmetry of a crystal; both of the structure transition can take place at higher temperature than the magnetic transition. Why in iron-based superconductors it is called nematic transition, while in molecular system it is called Jahn-Teller effect? (Particularly in the papers talk about iron-based superconductor they never talk about Jahn-Teller effect.) Is there a deterministic experimental fact to distinguish the two? Thanks.


I'm writing a report. In the report we refer to program participants as either dedicated or auxiliary. Each has a unique situation in which they enter the program. Since they are a particular type of participant, I have been capitalizing them to avoid confusion, but I'm not sure if this is correct. For example: When Dedicated participants submit an application they must... Is this correct? I understand it as a proper noun. I'm confused about the rule as this comes up many times in reports I need to write.


What is the synonym/counterpart for the word "Default" in NON-IT WORLD? We use the word in computer programs/apps, meaning that there is a pre-specified/pre-defined setting or situation. The word has also different meanings in Law and Banking. But I want an alternative to use in our daily life: for instance, a physician may choose a "default" treatment for a common disease, or one can have a default position/view on a particular topic .... I want a counterpart meaning of the same word for everyday uses (non-IT world).


Could the gravitational effects of exotic matter be replicated by using something like a torus of mass around a point? Because spacetime is relative (I'm pretty sure), for observers on the torus, or just any celestial object i guess, would they not see the other points in spacetime around them having a negative energy, and repelling objects instead of attracting them? I know I've made some sort of misconception about this, but I can't figure out what it would be, except maybe spacetime not being relative. Edit: I drew an amazing picture to demonstrate what i was thinking


This is a snap of a textbook while describing Uniform Circular Motion. For all the diagrams, and by the vector law, the directions of the del(v) vector is perfectly fine. But I am confused why this del(v) vector will always be pointing towards the center. As it is stated here, just because v is perpendicular to r and v' is perpendicular to r', therefore del(v) is perpendicular to del(r) does not sound satisfactory. Any other logical reason why del(v) vector will point towards center only when del(t) is long enough as in diagram a and b.


Just wanted to question how Hilbert space dimensions work. From what I've seen, they're used to generalize the mathematics of finite-dimensional Euclidean spaces into infinite-dimensional vectors. I've seen them used in quantum mechanics, and in some cases stated to "inherit finite dimensional space-time" despite the Hilbert space having infinite dimensions. So I'm basically asking how Hilbert spaces correlate to the physical spatial dimensions they represent. Are there Hilbert spaces that represent topological spaces with infinite spatial dimensions? same question for any function spaces with infinite dimensions.


I'm not sure if this is the right community I could ask to, but I'm looking for a reference in business calculus that has challenging problems. The ones that I found so far have easy problems and can be solved within a day. I'm looking for problems that will probably take time to solve and will use a lot of thinking (and not just copy the methods/insert some formulas to solve). I would appreciate it if someone can recommend to me a reference that I'm looking for.


I'm trying to modelise the ping pong and water cup experiment. They were already questions on stackexchange about this: Why does a ping pong ball bounce higher when it is dropped together with a cup of water? but I would try an approach with Archimede Force and superficial tension. I want to apply Newton's law on the ping pong ball, but the water and the cup are moving too and I don't know how it works in that case or how to modelise it... What do you think about it?


I have some powder in a container with one side open. If I then compress it from that open side, does the powder evenly compress, or will the side closer to the opening (or furthest from the opening) compress more, and end up with a gradient of compression? For example, when tamping ground coffee in an espresso portafilter - is the coffee density even distributed through the portafilter, or will it be denser at the top or bottom? And what factors (if any) affect this? Regarding individual particles - shape? size? texture? uniformity? compressibility (of each particle)?


I recently delved into the topic about the reality of the Dirac string. Although several authors show how this object should not be real, this article "Field momentum and the reality of the Dirac String" proves they're wrong. I think the thesis of the article is very strong and I would like your opinion. I am writing my bachelor thesis on this study, I am learning both the Lagrangian and Hamiltonian formulation of the system to understand if I can find a solution, however the authors of the article seem to me to use only classical arguments. Any ideas?


I read about the 'Method of undetermined' in the wikipedia and it said as following: In order to find the particular integral, we need to 'guess' its form, with some coefficients left as variables to be solved for. This takes the form of the first derivative of the complementary function. Below is a table of some typical functions and the solution to guess for them. I can't understand the sentence "This takes the form of the first derivative of the complementary function" what does it mean? Is there any relation between particular integral's form and derivative of the complementary function?


I see the statement that: Similar to how a car needs fuel to run, a computer needs electricity to power it. In this case, I comprehend its meaning. However, I don't really get why "similar" is used at the top of the sentence here. If "similar to how" is an adveribial clause, my searches don't find any results or any envidence. Likewise, I can't figure out clearly any type of reduced relative clause.


I have read that if helicopter's engine fails, you set the pitch to minimum but not reverse pitch. The rotor would have to be tilted back to provide an angle of attack on the blades so it glides like a fixed wing aircraft but with the rotor spinning to keep the blades from buckling under the weight of the craft. I think it would have to go into a slight reverse pitch to spin the same way. That would also provide more angle of attack on the receding blades where it is needed because of the lower air speed over those blades. How does it keep rotating in the same direction with the air going up through it if it can't go into a slight reverse pitch?


I know that we have to use 'the' when it comes to groups of people, for example: The French are good at cooking. When nouns and adjectives are the same, we can omit 'the' Americans are good at sport. However, I do not know what to do with a word "Brit". I'd like to know Brits more OR I'd like to know THE Brits more? Can I omit the definite article?


In an electric circuit, electron flows from the negative terminal of a voltage source to the positive terminal, which in turn, gives us the the conventional direction of electric current. So, the first question that arise in my mind is: On moving to the positive terminal, does the electron gain stability? If it does, then we know that an electron moves from a higher energy level to a lower energy level it releases a photon of a frequency dependent upon the energy gap, to gain stability. The second question is: Why does an electron move towards a region of lower energy but in a circuit, towards higher energy (positive potential is obviously a higher energy level)


Suppose I have a rhombus based penrose tiling like the following: Is there some coordinate system that allows me to easily uniquely refer to a specific tile, and also from the coordinates efficiently calculate the coordinates of the tiles that border it? I also need to be able to easily tell what tile is on the "opposite side" edge of a tile with respect to a different tile that also borders it. Basically my end goal is, if I have some instruction like "Straight, Straight, Left, Straight, Right" and follow them on the grid to calculate if I ever cross my own path and when.


I am attempting to apply the Grubler formula (which can be found here: https://learnmech.com/how-to-calculate-degree-of-freedom-of/) to determine the number of degree of freedom, but it does not seem to yield the expected in some cases. Is it correct that there are some cases where the Grubler formula should be avoided? If yes, how can I recognize these cases and what method could I use instead? EDIT: I've edited it to only ask about the underlying concept as required. Now, one might say that it lacks an example where it doesn't work ... tell me if that is the case, I'm just trying to comply.


Interference is explained by showing a diagram of two parallel waves of light that are in phase or out of phase and showing the result of the fields constructively or destructively interfering. Implicitly it seems to be the case that they are polarised in the same plane as well. What I wanted to know is what this picture looks like when we're talking about unpolarised light. I'm reading about XRD and Bragg's, and can't seem to find any indication that polarised X-rays are used. And if that isn't the case, then even if two waves are in phase, the net E fields are polarised in random directions to each other and wouldn't undergo constructive interference. Where am I going wrong here?


The wavelength or frequency of light determines its color. Photons seen as particles are said to have a frequency, determined by its energy, so I assume that 'is' the same color. But being quantum particles, photons also have a probability wave, determining how likely it is to detect it in a certain state (like, a position). How do these waves relate to each other, if at all ? Iow, would the double slit experiment have a different result when performed with 'red photons' instead of 'green photons', and if so, how would it differ ? Sorry if I get things wrong, I'm just grappling with the nature of these things.


How strong of a gravitational field do you need for a projectile to make a full loop? By full loop I mean it curves once around the by dot, and then it ends up on the same trajectory as it was one before it approached. How many times the Schwarzwald radius would the projectile have to get for this to be possible? I know the limit in Newtonian gravity is half a loop.


I had the feeling that physics has moved on from the idea that spacially bounded objects located in spacetime (such as particles) can be fundamental. Instead, QFT describes everything by quantum fields which are better considered as an everywhere present property of spacetime rather than embedded in it. From this perspective, it always felt to me like string theory is a step back, talking again about bounded objects located in spacetime (i.e. "free floating strings", as this answer states). Question: Is there a flaw in my understanding? How can free floating bounded strings give rise to spacially extended quantum fields? I am a mathematician and fine with advanced mathematics, but am a layperson when it comes to physics (you might consider me as an undergrad in this respect).


I want to know the mean first passage time (MFPT) on a unit interval for two boundary conditions (please see attached figures a and b for your reference). This is in the context of the hydrodynamic dispersion of a particle. In case a) the b.c switch from reflection to absorbing whereas in case b) the b.c is reflective. Can anyone help me to figure out the mean first passage time in this problem?


I am currently looking for a cyrillic font (for russian typewritting). It must be an opentype font usable with lualatex and fontspec, and must be complete : roman, italic, bold, small caps, superiors letters and figures, in serif and sans serif. For all I know, Libertinus is very close, but it lacks cyrillic small caps and superior cyrillic letters. Does a more complete font exist?


I have a knot about the "secondary" colours and its spectra. So If I look the spectra of white light, I can see that yellow, which is between R and G, has some wavelength. If I do an addition of light of Red and Green, I get yellow. Ok. Now If I do an addition of light of Red and Blue, I get Magenta. But how come that Magenta is only a perception and has no wavelength, as Yellow for example ? I understand that some colours physically do not exist, but I dont understand why yellow exists as a real physical wavelength but not magenta. Can someone help please


According to Stephen Boyd's textbook on ADMM, the global variable consensus ADMM updates are given as the following, which is pretty straightforward to understand: However, I've recently seen another variation of the consensus-ADMM updates from this work I tried very hard to derive the primal updates shown above but I just cannot arrive at the same equation. I wonder if this is the same as global variable consensus ADMM at all?


So, I'm trying to make a basic shooter game in desmos, and I have a list of points that act as bullets. The (current) enemies right now are only some triangular jets that fly in a linear manner. My current goal is: Make it so when the point hits the triangle, it deletes itself from the list, and thus itself from the screen as well. The Triangle, upon getting hit, moves downwards in a curved path and stops when it hits the ground (a line). https://www.desmos.com/calculator/sjezpkhhxl Here is the graph itself. The lines for the bullets themselves are in the folder titled bullet and the lines for the triangle are labelled as enemy: fighter jets. If any more details are needed, please do let me know.


My wife mentioned today that she would love to know more about 'how the world is made up'. She stopped learning science at a young age and finds most stuff meaningless or incomprehensible. She is not a child - so the challenge is, how to introduce the history and development of atomic physics with minimal specialist vocabulary, in a manner that reliably explains how physics got to the standard model. Pictures are good, waffle is bad, mathematics and equations is a no, hand-waving or similar patronising (eg '...for idiots!') statements are a no. Her wish is aspirational - so the presentation needs to be reasonably effortless... Any thoughts or suggestions?


I remembered this really corny phrase today when I was on a walk but I forgot later before I could write it down. It's kind of like "You are what you eat" but with different words. It's not: You reap what you sow. You're the sum of your actions. Easier said than done. (It is a different one but they're both about contiguity. I don't remember what it is.)


Why does she try and pull the stunt? This is probably an elliptic sentence - Why does she try the stunt and why does she pull the stunt (off)? - but it still seems like very bad style to me. Why does she (even) try to pull the stunt (off)? I am aware that it is not clear that the attempt was successful, however the to-infinitive comes much more naturally to me, for whatever reason. From context, it is clear that it was successful. So, would you consider it weak style to use an ellipse like this?


I am trying to determine a statistical test to run. My dependent variables is employee retention. The independent variables are exam score and interview score. Unfortunately because employees who failed the the exam did not moved onto the interview, I do not have interview scores for all cases. Do I run a logistic regression? Is there any other test I can run to be able to use exam and interview scores to predict employee retention?


To my understanding, the curvature of spacetime is determined by the stress-energy tensor. I was wondering if we could calculate some of those components using radiation. Is it possible that objects that possess immense electromagnetic radiation, for example, will follow with a strong gravitational pull? Does radiation occupy physical space? Just had a thought experiment where I visualize spacetime like water that engulfs any solid object and doesn't go through it. But in my visualization, the actual spacetime didn't touch the solid object since the solid object occupies more space than its physicality. It engulfs the radiation and interacts with it. Just thoughts :).


So in my course of quantum computation i came across this question that "What guarantees a quantum gate to be unitary?" i was specially curious about photonic quantum gate. At first i thought that maybe unitary is not something that is guaranteed by any special phenomena but its inherited by the math behind quantum physics and that we use to describe any physical operation with an unitary operator but is it really the answer?


We need to solve this problem without using trigonometry. KLMN square is inside ABCD square. Prove that midpoints of the segments AK, BL, CM, and DN are vertices of a square. KLMN can be situated anywhere inside ABCD. I think that we are going to use congruent triangles. I tried to draw so that the intersection point of diagonals of ABCD and KLMN coincide, I also tried the opposite to consider the general case, because KLMN can take an arbitrary position. So, I would like to see approaches, how to prove.


I wonder if above some energies (or diagram level?) the virtual particles would rather behave (or become) like a micro black hole. Does that make sense? Could this somehow introduce a cut off above which Feynman diagrams would be unnecessary (the micro black holes would always evaporate in perhaps a few different ways), and in doing so avoid renormalization? This are speculations from a freshman, so please don't be hard on me.


An oxymoron is a figure of speech in which contradictory terms appear in conjunction with one another. Can there be a figure of speech in which similar compatible terms appear in conjunction with one another? For example, how would I describe the following sentence? Nice should be great. Nice being the location, but also a compliment. Play on words? Comedic effect? Feel like there isn't one and idiom or a proverb doesn't quite fit.


As my title says, is a "change in scenery" as correct as a "change of scenery"? I am self-conscious of how a "change in" might sound odd or off or be even absolutely incorrect. Are both forms of this sort of expression truly correct? I know the latter one is (a "change of scenery" makes full sense), but what about the former version of this idea with "in" as its preposition instead of "of"?


So there's this debate ongoing on one post that this phrase is incorrect. People are getting confused and wondering how can Adele take a divorce from her own son. The sentence is: Adele says new album will explain her divorce to her young son. Is the above sentence correct? I always thought a divorce was taken from someone and not to someone. For context, the sentence is being used a headline caption for an image.


If all bodies emit thermal radiation, does it mean that it's impossible to build something that can retain all of its energy - and last indefinitely? i.e. is everything in our world, including a dynamic, living system that can self-repair, eventually gonna collapse? OR - there is something in Thermodynamics similar to Superconductivity in Electromagnetism that would help prevent this? OR if there is no such effect, is there, even theoretically, a way to mitigate the effects? I can only think of creating new energy within a system faster than the system can lose it due to thermal radiation, but doesn't this violate conservation law(s)?


I'm looking at an example system which has a very large number of microstates (effectively infinite - not possible to enumerate exhaustively, but possible to sample from). The energies of the microstates however are drawn from a normal distribution with some average energy and standard deviation of the energy. What is the partition function for a system like this expressed in terms of the average energy and standard deviation of the energy? What is the entropy?


If the vacuum energy of the universe is not at its minimum, that is, the universe is at a false/metastable vacuum state, then it could decay into the true vacuum state. However, if our universe is already at the true vacuum state, then, this decay would not take place. However, in the far future, if we would wait enough time, could the true vacuum transition into a false vacuum level somehow? Or is this utterly impossible?


I was thinking of converting a disused chimney into a static periscope, using long focal length lenses at top end of the chimney. A static design would be easier, so I was wondering how one would design the mirror shape such that it converted a cylindrical view onto the lens, which could then be processed at the other end of the chimney to generate a cylindrical view once more. Is it even possible?


Assuming right to be positive, in this question, is the work done by the boy positive or negative? I would think it is negative because the force that the boy exerts is towards the left while the movement of the boy is towards the right. I would think that work done ON the boy is postive because the force that the string exerts on the boat is towards the right which is the same direction as the motion of the boy. Am I correct? I don't quite understand how to think about the idea of work done BY the boy in this context. What does it mean for work done BY the boy to be negative?


I'm looking for open books on abstract algebra, other than Judson's AATA. I want one with a good number of exercises (and solutions, preferably), this one covers the rest. Does anyone know of one? This and this are similar questions. The first one has many answers and the second is markedly different, but answers specific to abstract algebra, with brief descriptions of each book (too much to expect in the first) might help someone someday.


So I was wondering if it was possible to prolong the "entropy" of the bottle by making sure the heat of the contents of the inside of the bottle doesn't escape by first having a secondary bottle on the inside of the existing container. Then there would be a vacuum. So would the heat stay trapped if there were mirrors on the inside of the outer container bouncing infared radiation as it gets radiated from the inner container. Is this a means of preventing entropy increase in the universe, so could be used with dyson spheres to trap heat preventing it from escaping in the distant universe to preserve more energy? Are there any issues with this approach?


According to numerous questions (e.g Is it recommended to use "we" in research papers?), one should use "we" instead of "I" while writing a scientific paper. However, it's unclear to me if, for example, "In the following image you can see..." should be replaced with "In the following image we can see..."? Personally I think, that the latter sounds better and also includes the reader together with the authors in the discussion at hand, as already mentioned in the above question. But it's still unknown for me, if it could or should be mentioned while reviewing another paper.


Superdeterminism is one wild conjecture which is an alternative to the standard quantum mechanical interpretation and preserves local realism Superdeterminism seems to be too much of a stretch. If experimental results are pre decided it simply renders the experiments useless. Is there any alternative in the academia where something like minimal determinism is discussed? Like just the minimal amount of determinism needed to retain local realism. For eg: In the classic Bell's test, if the entangled electron pair generator knows the direction of measurement of spin beforehand for even one of the electrons while generating the entangled pair of electrons, hidden variable theory can violate Bell inequality. Instead of the entire system being superdeterministic, only one pair of entangled electron pair generator and electron spin detectors are determined.


I'm looking for a single word that means "to bring within reach"/"make possible" for a technology or promising development. For example Mobile phones _____ photography for the common person. "Facilitate"/"make accessible" isn't quite right - I want to convey a step change, a paradigm shift. In particular, I'm naming a business. I work in robotics. Robotics has long been touted as a way to give people mobility, automate the mundane, and facilitate humans focusing on what matters to them. My goal is to build tools for other companies to accelerate them and _____ the promises of robotics for humanity.


I work in STEM and a problem I often face with is the use of "by" or "through". Let me explain it with an example. I have a quantity named "variance" that is derived after a certain number of tests (called Monte Carlo simulations). So, I write it like the following: The performance of the estimator is evaluated by comparing its numerical variance with other estimators obtained by/through Monte Carlo simulations.


this was my first question here and I asked it before I knew anything about this site like the dup questions I want resources (books) for studying Euclidean geometry beyond the high school level. I have been searching for good books on this topic but haven't found any. Could you please provide me with recommendations or suggest good books for this subject to study it in more depth than high school geometry?


So, back in my school, our teacher was teaching structure of an atom. I had some doubts, like why do hot objects emit radiations and so many. I noticed that every question I had asked had an answer. If I added a why to it, thats another question. It mostly came to a conclusion that all atoms want to attain stability. My question is why atoms want to attain stability and can I relate it to entropy?


How is this construction correct? The drawing-room began to look empty: the baccarat was discontinued for lack of a banker; more than one person said goodnight of his own accord, and was suffered to depart without expostulation; and in the meanwhile Mr. Morris redoubled in agreeable attentions to those who stayed behind. (Robert Louis Stevenson) In my humble opinion, the correct sentence should be: ... banker; more than one person said goodnight of his own accord and was suffered to depart without expostulation; .... We are not supposed to use commas when a second verb (was) has the same subject as an earlier one (more than one person). Of course, my problem lies only in that part of the brief.


I am looking for a word for an environment where people feel choked. For example, the environment in medieval Europe (or in communist era) was far from free. One felt choked/strangulated. Is there is word to describe such an environment? For eg, for happiness, one could say An era/regime of happiness. What would that be for choked? Update: I was trying to say that the age of Enlightenment was a response to a 'choking kind of environment' where the church had too much control over people's lives.


Whenever there is contact between objects, there exists a normal force. If we apply force on an object, our hands are in contact with the block, so there will be a normal force on my hand and normal force on the block, so the net force acting on the block will be the applied force plus the normal force. Why don't we include the normal force acting between our hand and object?


I've seen a lot of proofs of the convergence properties for the Fourier series of continuous periodic functions that use the assumption that the function's derivative and second derivatives exist and are also continuous. I was wondering is the stipulation that the derivatives exist considered not that restrictive of an assumption because we can consider the Stone-Wierstrass polynomial approximation of the function instead which converges uniformly to any continuous function on an interval and does have continuous derivatives?


According to Transformation of the energy-momentum tensor under conformal transformations The schwartzian term in the transformation properties arises due to the stress tensor being defined as the normal ordered classical expression, where the normal ordering happens with respect to the given coordinate frame. However, the infinitesimal transformation properties of the stress tensor can also be derived in terms of conformal Ward identities. I think it is kind of magical, that apparently the derivation in terms of the path integral "knows" about the appropriate change in normal ordering under coordinate transformations and I wanted to know, why the path integral implements the right normal ordering procedure, without having somehow incorporated it explicitly?


I'm writing a book using the book document class and, for whatever reason, one of my chapters has its contents start on the next page rather than the current page. I have no idea why this is; I looked for manual page breaks or anything that might be causing an issue to no avail. Each chapter is broken up into a .tex file that inputs other sub-chapter .tex files. So, I'm not really sure what to post as far as a MWE since this issue plagues exactly one chapter. Is there some arbitrary setting or command that I accidentally enabled that does this? "Broken" chapter: Example of a "correct" chapter:


In The Symmetric Group the author, Bruce Sagan, uses the terms degree and dimension of a matrix representation synonymously to mean the number of columns of the matrices. However, I have also seen elsewhere that the degree has been defined as the number of columns in the matrices, whereas dimension refers to the dimension of the vector space on which the group acts. I guess the author can define the terms how he wants, but is there a convention for what these terms mean more precisely?


For an animation with TikZ and Asymptote (that I am somehow familiar with), its size is accumulative with the number of steps. I am mimicking animation for From Random Polygon to Ellipse, take a polygon with random vertices, find the mid-points of its sides, and use these to create a new polygon, replacing the original. Do this repeatedly and an ellipse will eventually form. See here for more explanation. In this situation, the number of steps/frames is large or extremely large, and only last step/frame is needed. Can we do it with TikZ or Asymptote? Do you have any other recommendation? PS: I can add a MWE with TikZ/Asymptote, but as said above, I think it is useless.


I am creating fill-in the blank notes for my class using the Censor package, a solution I found on this forum here. What I can't figure out is how to make the blanks for my students to fill in longer -- ie, I don't want text to be ____ but more like ____________ since most people don't have tiny, perfect handwriting. Is there a way to add a multiplier to the length of the blank or replace a character with, say, two or three underscores instead of one so that normal people can fit their handwriting in one of the blanks? If someone has a different, cleaner solution that would be great too; the post explains the gist of what I need to do.


I am searching for the title and author of a book once lent to me. I cannot recall much about it other than the color scheme used on the cover. The cover was black with orange lettering. Perhaps there was a gray strip at the very top. It looks somewhat like this cover. However, as I recall, the book had some depiction of a matrix on the front cover. The title had Matrix or Matrices in it. Perhaps Theory or Applications. If anyone knows what book I may be thinking of, a citation would be very appreciated.


Consider the above diagram. I am currently learning about standing waves in an open and closed tube. To me, it is trivial why at a closed end, there must be a node, as particles are not free to move around back and forth. However, to me an open end is not so trivial. Why must it be an antinode rather than some other part in the wave?


Could someone help me constructing such segment AB in Geogebra that would satisfy such properties: by moving the point A the segment moves as a solid body but does not rotate by moving the point B the segment changes its orientation and size the segment should be able to move like a solid body even if one of its edge points is attached to a line I have already tried to use vectors but got the result that don't satisfy those properties above.


I am wondering if an incident gamma could compton scatter from the delocalised pi-electrons in an aromatic compound (e.g. liquid scintillator). I understand that compton scattering is valid for free electrons, and is usually described in terms of the outer most atomic energy levels where incident photon energy >> binding energy. However, how about compton scattering from molecules, and particularly delocalised electrons in aromatics? Furthermore, if it is possible, would the scintillator be excited and scintillate after a pi electron is removed? Thanks!


The following are the formatting guidelines for citation call-outs and formatted bibliographic entries provided by the journal International Economic Review: Unfortunately, the journal does not provide a BibTeX bibliography style file that would implement these guidelines, and no template is avaibable in https://www.overleaf.com/learn/latex/Bibtex_bibliography_styles. I was wondering whether there is an encyclopedia of bibliography styles that I could choose from. Or, how shall I customize it by myself?


basically the problem is the same as in a question i previously asked: Draw shaded region using tikz and pgfplots Now the problem is that the region i want to draw is more complicated, mainly because of the functions involved: This is what i want to obtain, but i don't know how to change the code in the answer in order to draw the graph showed in the image, also because tikz does not really get along with sqrt function. Does anyone know how to solve my problem?


There are many methods for diagonalizing matrices; probably the most widely used is the combination of household transformations and the QR algorithm. Is there any superior method for diagonalizing large, non-sparse real symmetric matrices? Superiority can be a bit muddy, so I define it as fast, numerically stable, does not require large amounts of extra memory, and lends itself to parallelization and vectorization. Crosspost: SciComp.SE


If a stationary charge is kept in front of current carrying wire , from charge's perspective the protons are at rest with it and electrons are moving , so shouldnt there be a lenght contraction in electrons and there density increase and stationary charge should feel a force , but it is seen that it doesnt?? please answer the question in stationary charge's frame of reference when electrons are flowing in the wire


It doesn't have to be a MacBook exactly, other switched power supply powered metal case devices are good enough, but MacBook is the most common. It must be AC powered; the effect disappears when unplugged. Your other hand (which doesn't slide on the surface) must not touch the surface; the effect disappears when touching the surface with the other hand. You must move your finger; the effect disappears when you stop. The same effect can be felt by sliding on someone else who touches such a device. It's definitely not a potential difference. I suppose it's mechanical resonance, but can't explain.


True or false question! Conduction is the slowest mode of heat transfer - true (given in internet) Now my question is, let's say you want to boil some water, in this case should not be the conduction mode be faster than convection? I mean within seconds, we can feel the heat energy if we touch that beaker, where we need some time to feel the same amount of heat energy from water. Kindly elaborate on this.


Let there be some air contained in a vessel whose walls are elastic. The vessel is closed and we impart some velocity to it. Please note that the gas inside the container is stationary w.r.t. the container. But the container, itself, is moving with some acceleration. The gas particles should press against the walls in a direction opposite to the direction of motion. My question is can we say whether the internal pressure created by the gas particles increase or decrease with increase in the acceleration of the vessel?


The electrons in the hydrogen molecule experience the same potential and are thus in the same state, so the Pauli exclusion forces them to have opposite spins. Since the protons are identical by symmetry, are these in the same state too? If they were, then the spins would be forced opposite and orthohydrogen would not exist. So in what way am I thinking about this wrong? Is it that the ground state for the protons is degenerate?


Aperiodic tilings for the plane are quite popular. I can't find any papers on aperiodic tilings on infinite surfaces like a cylinder. One reason might be that these are no longer called tiles but something else (a curved finite surface). Do references for this exist? Periodic tilings of cylinder can easily be accomplished with a machine: https://en.wikipedia.org/wiki/Knurling A lathe machine has parts that rotate and easily create a periodic pattern that has one tile type. Not sure if there exists a machine that would do it with multiple tile types. My end goal is to find a similar method to aperiodically tile a cylinder. At least one part of rotation would have to exhibit aperiodic behavior, but I leave that outside of the scope of this question.


If we have a set of sentences S in first order logic. We know that we can create a first order theory Th(S) from S, which is the "set S" union "the sentences which we can prove them from S" . Also as first order theory, this theory follow the rules (axioms and rules of inferences) of first order logic. Does it has sense to define the Th(S) as above but with an extra union with "axioms of first order logic" ? Also can i consider the Th(S) as formal system which has "intersection" with the formal system First Order Logic?


In Bell's Spaceship Paradox, are there any direct observations that the stationary observer can make that would justify the breaking of the string without taking into consideration as to what's taking place in the frames of the ships? In other words, can it observe any changes in the string's properties that would justify the break without having to switch or calculate in other frames? If the string is replaced by a bar that can stretch a bit before it breaks, in the rocket's frame it would appear that the stripes become broader before the bar breaks. What would the stationary observer observe?


Can the longest length of a regular icosahedron be equal to the longest width and the longest height? Since the icosahedron is regular, this would mean its faces would still have to remain as equilateral triangles. If this is not possible, are there any ratios between length, width, and height? As a follow up question, can we use edge lengths to determine the length, width, or height? Are there any possible equations?


Example: She ate one or more apples, and each apple of the one-or-more apples was either red or green. In the example, if "one or more apples" is the antecedent, should the reference back (i.e., "the one-or-more apples") be hyphenated? My thinking is that "one-or-more" in "the one-or-more apples" works as a compound adjective or "compound quantifier" when referring back to the antecedent of the example, and hyphenation of the compound adjective or quantifier improves readability. Readability is not an issue in the example, but repeated references back to such an antecedent with the hyphenation seem to be easier to read in longer forms of writing.


I understand "what (a) nerve!" means "how rude". I somehow always thought "what a nerve" is more common and was surprised at the exchange made in this question. Apparently American English speakers feel "what a nerve" is strange. My questions: Is it correct to think that the difference is regional? More specifically US/UK? If there are native speakers from other countries, how do you feel about "what a nerve"?


A vertex is defined as a 'meeting point of two lines that form an angle'. When I increment the circumference of a circle into infinitesimal small increments I get something like shown below in the picture. This infinitesimal small increment of the circumference can then be divided into left and right which is shown by the black arrows in the picture. My question is, if the part of the circumference is infinitesimal small, the left and right part of it would not be curved and therefor the above mentioned description of a vertex would be true.


I have heard of definitions along the line of saying that a transformation T of something with respect to a certain property p is symmetric if, after T is applied, p remains unchanged. Is there a way of refining this definition? How could you use this definition to show that fractals are symmetric? What are T and p? If something satisfies a recurrence relationship, is that a type of symmetry? ################################################## From the comments, I am getting the impression that there is no specific definition of mathematical symmetry. Fractals are frequently described as being highly symmetric. The idea is that you can uniformly contract or expand a fractal and lay it on top of itself.


Given a stochastic differential equation SDE the mathematical definitions are clear to me, i.e. we want to find a filtered probability space, a BM and a process which satisfy the SDE. However in the real world we are given a probability space and a BM on it (e.g. the uncertainty on the stock market). Why constructing these weak solutions which are purely mathematical objects would help?


When two black holes are merging, one of them can be ejected if it has less mass than the other black hole, so the gravitational waves emitted by both of them is unbalanced, and the more strong ones would "kick" the smaller black hole out of the merger, sometimes even kicking it out of the galaxy. This is known as gravitational recoil. Can this also happen with other objects like stars or planets so that they can get ejected to very high speeds? If not by the same mechanism, maybe through a similar one?


Newb here. When I look up how to create two columns, I get a bunch of tutorials that assume one text will run from the bottom of column A to the top of column B, like a newspaper article that spans two columns. That's not what I want to do. I want to put one text in the left column, a different text in the right column. Like two separate newspaper columns. When the text reaches the bottom of column A, instead of picking up at the top of column B, it should start at the top of column A on the next page.


This is not a hint-to-solve-exercise question, but just mathematical curiosity involving one my favorites branches of this beautiful science: group theory. Having done some research in databases like this one, I came across with some different ways to construct groups of a given order. Since I don't know about extensions, my knowledge is more limited but I have a question: how many "different" group/subgroup products are there? In the database I wrote, I found things like "central product", "wreath product" (apart from the typical direct or semidirect ones), etc, but, are these products the only constructions? Am I missing other types of products?


As I wrote in the question, what is thermodynamic equilibrium? From what I understand is a state where the thermal, chemical properties don't change with respect of the time and there is a mechanical equilibrium as well. But can this be true for example, in a body with a temperature field with gradients? I was reviewing this concept because thermodynamic equilibrium is critical in defining a reversible process and I want to understand it well.


I have a presentation with one of the slides titled Results: limited-data in-the-wild semantic segmentation What would be the correct way to capitalise this title? https://capitalizemytitle.com/style/APA/ recommends either Results: Limited-Data In-The-Wild Semantic Segmentation or Results: Limited-Data In-the-Wild Semantic Segmentation with the latter of the two being the one I originally used. However, this seems to be in conflict with the recommendations from this question, which say words like "in" or "the" should never be capitalised.


Imagine if a box tied on string is swirl around a rough surface. The centripetal force might be contributed by both tension and friction. Since the surface of the object is flat instead of spherical surface, it might not rotate along its motion (rolling like a tyre). So, what type of friction should we consider that contribute to centripetal force?Is it static friction or kinetic friction


I know that: At this time tomorrow, I will be playing cricket. is used since the so-called future continuous is used to talk about a continuous activity at a specific time in the future. Even present continuous is used to talk about a planned activity in the future. So my question is can we say this? At this time tomorrow, I am playing cricket. I am eagerly waiting for the scholars' or experts' comments on this.


I was looking at the modeling of ttW production (but the question is general) and came upon an off-shell corrections paper. As I am no expert on MC generators, I am not sure if these off-shell corrections can lead to different amounts of radiation compared to an on-shell sample. My intuition says that maybe a Sudakov factor contribution might shift slightly and because of the Pt shift of jets the PS might differ but I do not expect large differences in number of jets as I can't think of an off-shell channel diagram that would differ significantly in extra radiation. Is this the case or are there examples where one does see noticeable differences (only in the modeling/in data as well)?


The narrator in this video has a very clear native English accent. What caught my attention is near the end of the clip when he said: They spend so much time in the tree and they sleep while hanging upside down like a bat. Though I know that it's grammatically correct to compare a plural noun with a singular noun, such as: They are much slower than him. Or: They ran as slow as a turtle. However, I am still wondering how often (or rare) for native speakers to compare many "lemurs" with plural "bats" instead of a single "bat" in the first sentence, or at least isn't it abnormal to do so?


In an attempt to understands some basic concepts about manifolds and differential geometry I encountered a concept of a parallel transport. This perhaps may be very naive question, but could one consider an analogy to the parallel transport (as defined in this Wikipedia article) being a ball rolling on a flat table surface? If so, is then such an example also an example of holonomy or not?


Recently, I was attempting to describe the experience of traveling with strangers on one of those pedal trolleys which serve beer - how producing physical effort with others towards a shared goal inspires fraternity or camaraderie. I did some googling for inspire synonyms and a few searches along the lines of inspiring fraternity, but haven't been able to find the word I'm looking for. It would be a synonym of inspire, with this added connotation of specifically inspiring fraternity. Example sentence: Digging a ditch with another man ____ between the two of you. Any clever linguists know a single word to describe this?


I understand that a taylor series is used to approximate a given function at a point to any required degree of accuracy, I have not however understood why when we take the taylor series of a function at point, expanding to infinite terms it perfectly represents the function over the entire domain of the function. Is there a mathematical proof for why this results from approximating a function and its infinite derivatives at a single point


If you have a point mass moving with some velocity 'v' and then a force is applied perpendicular to 'v', physics/mathematics claims no change in the magnitude of 'v' , just a change in direction. But that doesn't make intuitive sense to me because wouldn't a force cause an instantaneous change in the point mass velocity perpendicular to 'v'? Therefore, wouldn't there be instantaneous velocity vectors perpendicular to each other and a resultant vector with a greater magnitude than 'v'?


My professor was discussing this question. After finding out that CR equations are satisified at origin for the function he used the definition of derivative to check for differentiabilty. Also, he mentioned that we can't compute partial derivatives of u(x,y) and v(x,y), and for example one of them turns out to be discontinious at origin then say that the function is bot differentiable. I couldn't understand his reasoning. Isn't that the sufficient condition for differentiabilty?Thoughts? Click here for question


I'm trying to write a book (documentclass "memoir") and I would like each chapter to begin alternatively with a greek or arabic letter. I have no problem with greek thanks to the package "babel", by contrast whatever attempt to write a-single-arabic-letter creates big conflicts in my template, such as compiling errors, and other problems (such as, the letter is automatically written on the right hand side: and it makes sense, of course, but I just need it as a symbol). I've looked through many other questions, which usually concern writing sentences, and I wonder if there is a simpler method that fits my case.


The way I have it is: the Observable Universe looks as follows. In some ball, all the galaxy clusters exist, then in a bigger concentric ball the dark ages exist (no galaxies), then on the surface of this larger ball at the edge of the observable Universe the Cosmic Microwave Background exists. This leaves no room for an infinite or curled back on itself Universe, so it must be a wrong picture. How do you picture it? It would be advantageous to get a correct picture.


When submitting a memorandum request for approval of something and providing options of for the signer... what is the best way to word the denial option ? Disapproved? Denied? Unapproved? This is intended to allow the approver to circle their response. Approved / Denied Approved / Disapproved Approved / Unapproved I feel disapproved is not the correct word... since they may disapprove of the request in theory but that doesn't mean they're saying no to the action. Denied seems to be more definitive. Unapproved works but seems lackluster in some way and may indicate the action was taken without prior approval or something similar to that.


I am writing my Doctorate thesis in Latex. I am using book class, which by default is twoside and openright. But for the preliminary pages, specifically for page number ii, certificate, it needs to be printed on a new page, but the result is coming out as a larger right-side margin page for being an even-numbered page. The same problem arises for the list of figures and tables also, they are also starting on an even page and the output is a bigger right margin. Please help to overcome the situation. Thanks in advance.


I'm doing undergraduate research on the history of Abstract Algebra (specifically permutation groups) and the notion of symmetric groups in indigenous artwork has come up several times. Is anyone aware of evidence of this 'intrinsic' human desire to construct groups I could point to? Or maybe some natural phenomena that conform to the definition of group? I'd like to have specific examples I can explain (and possibly map some ideas of formalized group theory onto). Thanks!


So I had taken a course on BEC and Cold Atoms. I have read about the properties of non-interacting Bose gas and I was a little concerned about what we mean by two wave functions (of bosons) being the same. Does it mean that they are physically located at the same point in space? I understand that they have the same spatial probability distribution. But it is more than that. I basically want to know given two wave functions how do I say they are the same? A more physical meaning of two wave functions being the "same". P.S. This is my first time posting. So please be kind.


Regarding an acceleration in an empty universe, from the special relativity we feel an acceleration in an empty universe since there is still the presence of an space-time with respect to which an acceleration is measured and the equivalence principle says that locally the effect of gravity and acceleration is indistinguishable i.e we feel the gravity whenever we are acceleration and conversely we feel acceleration when there is gravity that means according to equivalence principle we can't feel an acceleration in an empty universe since there is no gravity but special relativity says that we can feel an acceleration in an empty universe ,for me there seems a contradiction between special relativity and equivalence principle. Does anybody make clear this doubt?


I imagine a swinging pendulum being held outside a black hole supported by the normal foce of a jet propelled rocket. The rocket's velocity is approaching light speed, therefore hovering the swinging pendulum right above the event horizon in the strong gravitational potential outside of the black hole. My first question is: what would the frequency be of this pendulum and would this be the natural limit to swinging frequencies? How could the frequency possibly increase more in potential in nature (besides by shortening length)? And following the consequences of this: I believe photons and pendulums behave identically under the influence of a gravitational potential; does this then imply photons also have a natural cut-off limit for their frequency?


If I have an action that is not invariant under a change of coordinates. Does this effect the renormalizability of the theory? The procedure of renormalizing a field theory essentially boils down to carefully keeping track of observables while changing the length scale of the theory. If I have an action that is not diffeomorphism invariant changing the scaling of the action has physical meaning and also has consequences on the parameterization of the theory. How does the renormalization process differ between actions with and without coordinate invariance?


My electromagnetic waves professor said any time an electromagnetic signal is received by an antenna, a transmission also occurs. I have not independently verified this, but I believe it is likely true. Does someone know if this is true, and can the basic physics be explained, please? Note: This question was inspired by my own comment on Antennas and relativity. I'm thinking of this more from a Maxwell's equation POV then relativity, however. ANOTHER NOTE: Per comments below, transmission is usually called "scattering" when unintentional.


As an example, why does glass tend to break by means of intergranular fracture? My analysis so far led me to believe that the tendency can be related to the intermolecular bond strength in a given crystal grain and the intergranular bond strength. A stronger bond means that better load bearing. If the intermolecular bond is stronger than the intergranular bond, then a material is more likely to experience intergranular fracture. But is this correct?


So, I know the proof for Halting Problem is not recursive using diagonalization. We prove it using proof by contradiction. First we assume HP is recursive which implies there is a Total Turing Machine. And prove it is not possible by contradiction. But the proof just says Halting Problem is not recursive. It doesn't say Halting Problem is recursive enumerable. So how to prove Halting Problem is recursive enumerable


So in math, analysis in particular, divergent series are not particularly interesting since most sequences are divergent. However convergent series are interesting and they are useful. However, mathematicians have realized they had to be careful and define two types of convergence. Most of the theorems rely on absolute convergence. Therefore my question is, if your series turns out to be conditionally convergent, are you stuck? That is, can you not conclude anything further or can you still work with a conditionally convergent series?


Consider how the wizarding world is hidden from the muggle world in the Harry Potter/ Wizarding World universe. Or in Hell Boy II. The troll market is hidden under a bridge. Red can see it but it's veiled from the human eye. Shadow Hunters. Clary can see the runes outside of the nightclub but Simon can't see them. I have been referring to it as a troll market concept but I feel like there is a better word(s) for it.


So , I will be completing S.L Loney Coordinate geometry book very soon and I feel like I need more practice in Conics especially in Ellipse and Hyperbolae section , Can anyone suggest me a problem solving book in Conics (Not theory, Because S.L loney has a very vast theory and sufficient for my preparation) and the level of the book must up to a competitive level exam or Olympiad. Thank you:)


From Boston Legal (or your everyday conversation): "He and I will be going on a little vacation." "Where?" "Probably best you don't know." Being short for "It would probably be best if you didn't know," the spoken version changes the past tense to the present in the second half of the sentence. Is this something a grammarian would find fault with? Or dismiss it as colloquial?


From how I've learned it in school the magnetic vector potential is used as a mathematical tool to simplify problems with current-carrying wires in classical electromagnetism, but is never treated as bearing any physical meaning. After doing some research on it myself, I found Feynman's lecture on the subject, which outlines how this vector potential actually takes center-stage in QED, not only featuring in its primary equations for concepts like phase change but also as capable of producing physical changes such the Aharonov-Bohm effect that can't be explained by the magnetic field alone. Are there any comparable physical effects the vector potential has in classical electromagnetism?


I understand that thunder clouds are a congregation of millions of water droplets. As water evaporates from the ground, these water molecules rise up and collide with water molecules of the cloud, sometimes ripping off electrons. This creates a seperation between positively charged water droplets moving up inside the cloud, leaving electrons behind. Why do the electrons stay at the bottom of the cloud and not move up due to attraction to the positively charged water droplets?


I'm new to fractals and especially the Mandelbrot Set. I've noticed these never ending and self-similar spirals all around the Mandelbrot set, just like the one below: At lower max-iterations, there is a black point in the middle of the spiral, which suggests me that there could be a minibrot there. I've read that minibrots are centered on c values with super-attractive orbits. However, since the spiral never ends, I cannot precisely get the middle point and calculate its orbit. Is there really a minibrot infinitely small at the end of the spiral?


I've been learning about set theory and have heard of something called a "group". I didn't understand what it was, even after researching it, but I did find out one piece of information: All groups are sets but not all sets are groups My question is: what is an example of a set that is not a group? That would help me better understand it.


More precisely, "to" is banned in the active form: The headmaster makes us honor our teachers but required in the passive (with no agent stated): We are made to honor our teachers I can just say it's something -- for lack of a better term -- mechanical: "to" is required to mark the infinitive in the passive voice. But then, why does "mechanics" matter in the passive voice but not in the active voice? More puzzling is that this phenomenon seems to apply to causative make, but not to causative force: The storm forced me to seek shelter I was forced (by the storm) to seek shelter Unlike make, "to" is required in the active voice of force. Thank you.


Imagine a stationary transmitter which vibrates the surface of the balloon and a stationary receiver half way around the balloon that can pick up these waves. Let the balloon expand. Will the frequency of the received waves change due to a Doppler effect? As the balloon expands, the tension in the surface will increase which could alter the velocity of the waves. This is not a Doppler effect and could be allowed for. I ask the question because the motion of both the source and the receiver due to the expansion is perpendicular to the direction the waves are travelling in, though there is also a velocity of recession, which is different to the velocity of expansion.


In the same context as of "Trendy names", why is "Brandy names" not used? Brandy comes naturally/intuitively as an adjective to brand in the minds of a new English speaker, instead of the commonly used words like brandworthy, brandable, branded. I want to understand the reason behind such usage. If we assume the word brandy does not denote a specific alcoholic beverage, is it grammatically correct to use "brandy" as an adjective of brand?


What are the different methods using contour integration that can be used to count the non-trivial zeros of the zeta function? I know of Backlund's exact formula which is similar to Riemann-von Mangoldt formula. What are the other ways? If there aren't many other different methods, then where can I find the exact derivation, like from scratch to end of Riemann-von Mangoldt formula? I have managed half of it till the point I can derive Backlund's exact formula. Also an evaluation of the limitations would be nice. I know Titchmarsh's works are useful but what other resources can help me?


In the sentence: "There were people everywhere. They thronged the streets, choking the alleys. They had come in their thousands to see the light show" I would group the infinitive of purpose and past perfect together, so that the sentence refers to a plan formed prior to the events of the story. At the same time, the arrival itself takes place before the crowds assemble and begin 'thronging'. Is my analysis off? EDIT: I apologise. I had to cut this short before clarifying. The sentence seems to be referring both to the arrival and a preformed plan.


I'm trying to prove the above statement, and I had a look at this site. This ends the proof with the following statement the rank is the number of non-zero eigenvalues But, this is what makes me confusing, because it's not all the case that rank of a matrix equals the number of non zero eigenvalues as far as I know. This shows a counter example of this. Any comment about the rank of idempotent matrix would be grateful.


I understand the general notion of twist like that of a mobius band(like a cylinder cut open and the glue the edges together with a "twist") , and the torsion in the context of algebraic structures like groups, in which a modulo-n torsion indicate when we multiply some element with n, we get the identity(please correct me if there is an error in my understanding). So in the context of topological groups, can these ideas be related, like in the case of a lie group, where we can talk about both twist and torsion and whether these ideas turn out to be related?


I want to add a citation inside a pdf figure, and would like the citation to be a link to the bibliography. I know embedding a pdf with includegraphics normally mangles the link, but I have found a solution (in pdflatex), using the pax package. But am open to any other solution. pax preserves the hyperlink, and I tested it is working with a www address. But for the internal link to bibliography, I am not sure what link address should I use. At the moment, I insert this link in the pdf figure "cite.citation_key" But the result is that the hyperlink points to /home/user/cite.citation_key instead of the bibliography. Does anyone know how to achieve this.


We were happy before. There was a market next to the school before. I saw a whale before. These are from a grammar book for kids that I'm proofreading, and this part is about "past simple tense". I would like to ask if these sentences are grammatically correct regarding the use of "before." It seems that "before" should be used with perfect tense? Should I make the following changes? Thank you! We were happy then. There was a market next to the school (sometime ago). I saw a whale (sometime ago).


We have the adjective "facial" for the noun "face", and I am wondering what the counterpart for the noun "head" is. I could not find it on Google, and ChatGPT told me "cranial" which seems a bit different from what I expected. Thank @Stuart F for the comment! My original thought was to make a joke on myself who can have some kind of "head expressions" (corresponding to "facial expressions") because I am a bit bald. And I am also happy to see that the vocabulary can be different in different contexts!


Newton's globe experiment: two globes that share all their features are connected with a rope in an otherwise empty universe. Newton introduced this experiment to show that even though the cases where the globes don't move and where they rotate are not distinguishable relationally (i.e. considering their distance to each other), we can nevertheless distinguish them due to inertial forces that only occur in the rotation case (tension on the rope between the two globes). My question: could Newton also have chosen an example using linear accelerated motion instead of angular accelerated motion? Inertial forces should occur there as well. If not, why does he need circular motion to make his point?


I know that "soliton" waves can consist of a crest without a trough. One would expect the reverse to be true as well. However, this Wikipedia excerpt says, So for this nonlinear gravity wave it is the total water depth under the wave crest that determines the speed, with higher waves traveling faster than lower waves. Note that solitary wave solutions only exist for positive values of H, solitary gravity waves of depression do not exist. Is this true, and if so, why?


I'm still introducing myself in derivation, antiderivation and integrals, because I haven't studied it in class yet (I will study it next year and derivation in some months), but well, about my question I have seen that integrals and antiderivatives, at least, looks like the same, so, is that true? and also I didn't understand at all why the integration gives you the area, I mean, realize that derivation gives you the value of the slope is easy, but why the antiderivation gives you the area is a bit harder for me to understand, thanks!


Alright so I am confused somewhat about Neutron Heavy Nuclei not decaying, or more specifically why it is that Neutron Heavy Nuclei sometimes decay by emitting Neutrons. Someone has already answered why neutron-heavy nuclei decay in general, weak force beta decay basically happens until it can't, but I do not understand why neutron heavy nuclei can, sometimes, emit Neutrons. I am talking about the light elements that have neutron upon neutron upon neutron. Hydrogen Isotopes beyond Tritium decay by Neutron Emission, why? Why doesn't it decay by beta decay? Why do other neutron heavy light elements perform neutron emission?


We know that flowing water remains liquid even in the below-zero temperatures (a good example is rivers in arctic regions). Of course water doesn't remain liquid forever if temperature goes down constantly. Also, I know water freezes from surface. But that's for stationary waters like lakes. Not for flowing waters like a dam output which is so fast. I think it depends on a sort of things, like the speed of molecules, the viscosity of water, etc. Can we formulate it to find the answer? Perhaps with such a formula we can find the freezing temperature of other flowing liquids as well. And a minor question: How does the density of flowing water behave in below-zero temperatures?


Consider a one dimensional chain of N classical point masses interacting with harmonic neighbor forces (with periodic boundaries for specificity). If the positions and velocities are prepared in a configuration corresponding to a specific oscillation mode, can the energy be transferred to other modes ? Or does the chain oscillate in that prepared mode forever ? Intuitively I do not see a reason why the energy should be transferred to other modes, but I am not sure. However supposing that is the case, does the chain return to that mode if small random disturbance is added at some time ? How is the stability of this mode to perturbations ?


I was planning on using whipping boy but it seems that means more like a scapegoat (ie someone who is innocent but made to take the blame for others). Moreover, I want something that means that everyone seems to point to one particular person as the bad example and turn a blind eye to the same behavior in others. The issue is more with reputation or perception from an outside group than guilt per se, but I don't want to suggest that the example person is innocent. The context is history.


I would like to calculate the repulsive force between two permanent magnets in a different variations of distance between the two polarity faces. Can the different distances associated to different repulsive forces be related to the magnetic field? Can the magnetic field have a variation when the distance between the magnets change? Can a tesla meter measure the magnetic field in the space between the two magnets? Can the repulsive magnetic flux between two permanent magnets be measured with a tesla meter? Thank you!


I was writing a Statement of Purpose letter and I am stuck with this dilemma. Here is what I wrote: I pray that this letter attains your approval. I first used "gains" but my mind instantly switched it to "attains", as the word seems more formal...and then I thought of "obtains", which only got me more confused. All I know is that these words have the same meaning of getting something but I'm still struggling with the context/condition it should be used on. Which one should I use?


I found that the definition of coherent phonon: A femtosecond laser pulse can initiate collective, in-phase atomic motions in solids called coherent phonons. My question is: what is the difference between coherent phonons and acoustic phonons? Acoustic phonons also mean the displacement of atoms has the same amplitude, direction and phase. Does it mean atoms move in phase and collectively for acoustic phonons? Thanks in advance.


Is there a word that symbolize a status of either unstarted, started or completed ? For example a football match can be either unstarted, started or completed. I would like another word than "status" to indicate those lifecycle adjective, as we have a lot of other "status" in this context. Something like "completionStatus" : but this is not really it, as this is just about completion. I thought about lifecycle but that's too biology oriented.


Throwing a small stone on the surface of a calm pond of water creates concentric ripples on the water surface starting from the point of impact and travelling outwards like transverse waves. At the same time the momentum of the sinking stone creates longitudinal underwater pressure waves that propagate outwards with the speed of sound in water. I am interested for an equation that describes the propagation speed relation of these two different types of waves created. Will the surface transverse waves have the same propagation speed with the underwater longitudinal pressure waves and if not how can the propagation speed of each type of wave calculated for water?


How do you find the area of the highlighted subsection of a circle? Linked Below. The circle has a known radius from A-O and a trapezoid that lays inside it. The trapezoids longest edges contact the circle circumference at points D and E. All edge lengths of the trapezoid are known. The shortest distance from the center of the circle to the inner most edge of the trapezoid is also known, O-F. I'm trying to design a lid with these dimensions and need to know the area to get the lids weight. The trapezoid portion with the highlighted area will float on top of a fluid. Unfortunately, I don't remember much from my High School geometry class. Thanks for the help in advance.


I'm curious about pendulums and their behavior during a solar eclipse. I also wonder about gyroscopes and if their behavior during stellar events has been observed. I wonder about the both of them, and of their relationships to each other. I wonder about Mobius loops and gyroscopes; what if a gyroscope was a Mobius loop in rotation, would it precess in a traditional manner? What if you lined the outer rim of a gyroscope with a plurality of strong magnets, all spinning the same way outwards? What would happen next? I wonder endlessly.


Suppose, I have a sample from an unknown distribution. I want to prove/disprove (mathematically!) the following statement: The maximum likelihood (ML) estimator for the (unknown) population mean will always be equal to the sample mean irrespective of the likelihood function. We suppose that, the likelihood function is twice differentiable with respect to its parameters and the mean and variance is finite under the inverse likelihood function (pdf). I believe for symmetric distributions it is straightforward. To maximize the likelihood the center of the mass should be placed at the mean. So, sample mean should be the best estimator to maximize the likelihood of the observed data. But I can not relate to it mathematically.


I have a question about contact force and normal force. whenever two bodies are in contact and one body tries to push another body there will be a normal force acting between them. like when we push or pull a body, when we hit a ball by a bat, when we lift a body from ground, like pushing a wall at horizontally or at an angle etc... these all are contact forces does that mean that all contact forces are normal force?


Basically something that starts with the substitution principles, definition of an equation, of subtraction and division, axiom of opposites, your usual commutative, associative, distributive axioms, and covers all the way up to the end of high school trigonometry. Dolciani's two volume "Algebra: Structure and Method" was good but I need another similar work. Serge Lang's "Basic Mathematics" wasn't good enough. Reading the first few chapter of abstract algebra books wasn't that helpful as they were too broad in scope. Any other suggestions?


Lately I've been approaching problems with conservation of momentum of systems, I was wondering, if I draw the free body diagram, then write the equations and then add them togheter; if momentum is conserved along an axis I will always get that the sum of the forces must be equal to zero in the equations of the dynamic along the axis where momentum is conserved, is that right?


I see that the Oxford dictionary has a plural dice for "a small cube with a different number of spots on each of its sides, used in games of chance". However, there is no plural listed for "a block with a special shape that is used for shaping pieces of metal" (both definitions summarized). Researchgate has a photograph of "Two dies arranged side-by-side". Is dies the correct plural of this meaning of die? Why do the two meanings have different plurals? Which was in use first?


For a new book I am writing with two friends, I like to include three biographies (mine as well as the other two friends) after the preface. Ideally, we like to keep three photos (A.jpg, B.jpg and C.jpg) on the left and our biographies on the right.. The book is in single column format, and I don't know how this biographies page can be done in double column format. Also, the biographies need to be aligned with our photos. We are OK with any format or style in a professional way as long as we can post our biographies along with the photos. Your suggestions are most welcome. Thank you.


I have a pretty strict requirement for a document (I'm currently using memoir), which must: use biber/biblatex in the usual way based on a primary bib file (e.g., main.bib), and sort the resulting list of refs by author then year. at some other point in the text, I want to print out a list of all the refs of type article from a secondary bib file (e.g., extra.bib), sorted by youngest to oldest. Is there some biblatex voodoo that would make this easy-ish?


This particular example comes from a peer-reviewed publication with authors who seem to be native speakers: This trend is both popular and has presented a variety of challenges I wonder if this is grammatical? If this sentence didn't have "both", we would parse it as follows: This trend ((is popular) and (has presented a variety of challenges)) But with "both", what do we have? This trend is both ((popular) and (has presented a variety of challenges)) but this gives us "is has presented", so this parsing is incorrect. Alternatively, we could try This trend ((is both popular) and (has presented a variety of challenges)). but then "both" goes inside the first coordinate. This is different from, say, You and me both where it's at the end.


How or where can I find the Absorption cross-section of Gaseous Ions? I have looked over the internet and can only find it for neutral atoms. Does it increase or decrease for ions? I have found formulas but they include the imaginary part of the refractive index of the specific medium and I can't find how to calculate or get a relative order of magnitude for it even for plasmas. Scattering cross section will do too Please help.


Let ABCD be a convex quadrilateral with AD = BC. Show that AD and BC determine congruent angles with the line passing through the midpoints of sides AB and CD... MY IDEAS MY DRAWING As you can see i noted some points. Okey, so, i thought of similar or congruent triangles. But i don't know where to start. Hope one of you cn help me! Thank you!


There is literature on the lie symmetries of quantum harmonic oscillator differential equation. The generators satisfy certain lie algebra. On the other hand, we have ladder operator method. The relevant operators satisfy some lie algebra. I was interested in knowing if there might be some connection with the two approaches. Is there some relation between the two lie algebras. Cause if there is, it might help me explain ladder operator method better.


In the Einstein field equations, the only tensor that shows up is the Ricci tensor and the metric tensor, together with the Ricci scalar. The Weyl tensor though is a tensor that is a part of the Riemann curvature tensor as well (the off-diagonal part). It represents tidal forces, which are non-local. Why doesn"t the Weyl tensor show itself in the Einstein field equations? Is it because gravity is based on the equivalence principle, which is a first-order equivalence?


I had recently come across "Russian for the Mathematician" by S.H. Gould, and saw it as a wonderfully useful resource. I have very little experience in Russian (only skill is ability to read Cyrillic), but I saw it as a useful introduction to the language, in the sense that I would be able to decipher Russian language math literature without the need to be entirely fluent or understand every word. Reading through a preview, I like how it breaks apart the words piece by piece to I was wondering if there were any resources for other languages in this similar vein, e.g. learning French or German with the main focus of reading scientific or mathematical literature. Downloadable here


I hear this kind of false implications pretty often, e.g.: If you're smart you'd agree with me People who understand the situation would all agree that ... Anyone who says something else must be trying to confused us These often come up in social deduction board games, where the good team is trying to gain information and the evil team is trying to spread misinformation. If someone says one of these lines and I disagree, they can say "see, he disagreed, so he must be on the evil team". Is there a term for this kind of faulty logic trap?


Is it possible to use magnets to lessen the force of a pull that is associated with Cervical Dystonia (spasmodic torticollis) which is causing an involuntary pull of the head over one shoulder. Having the magnets attached to clothing on one side such as a shoulder pad and more on a collar. Would the strength of magnets keep the head from pulling to other side ? As a person with Dystonia I'm trying to find a way of creating some sort of "crutch" to allow the head to look straight :)


Imagine a tank that accelerates towards a relativistic speed. As the caterpillars are at rest with respect to the ground, the caterpillars will show no length contraction there. But the tank body above it will. The upper part of the caterpillars, on the other hand, will show more contraction than the tank body does. What will happen? Will the lower parts of the caterpillars slide along the tank body to "deliver" an extra piece of caterpillar for the upper part which contracts more than the tank body? Will the caterpillars snap? I'm not sure.


This question may have been asked a bunch of times, but I have not found exactly what I was looking for. I say calculus, but mathematical/real analysis works as well as long as it does not have calculus as a prequisite. I have been looking for books that take an inquiry based approach, where the theory and everything else is build by solving problems with minimal explanations from the Author aside from introducing some definitions (Bogart's "Combinatorics Through Guided Discovery" is exactly what I am looking for, but for calculus/analysis). I am aware of jiblm, but I was wondering if there are published books on calculus/analysis using this approach that may be more suitable for independent learners?


According to MathWorld, the probability distribution of the length of lines between randomly selected pairs of points inside a cube has a very complicated form. In particular it is not uniform. This distribution relies on the fact that the pairs are chosen uniformly and independently. My question is - is there a way to do a reverse engineering and enforce uniform line lengths density? Namely, say I want the probability distribution of the length of lines between randomly selected pairs of points inside a cube to be uniform, from what probability distribution should I sample my pairs? In particular, I want this sampling to still be independent, but not uniform.


If I use a transformer the get a high voltage, the power of the transformer remains the same, as the current is reduced proportionally. However, according to ohms law, the current through a resistor connected to that secondary side of the transformer will be the voltage divided by the impedance of the resistor. But that is impossible, because this way both the voltage and the current felt by the resistor will increase, meaning an increase of power, while the power of the transformer remained the same, compared to the first side of it. Please sort out my confusion over it.


Yesterday I heard this: suppose you have a rubber hose, in which you inject helium, which replaces the air which was there before. Now, because there is no helium in air, so the partial pressure is zero there, the helium escapes through the rubber into the surrounding air and creates vacuum in the hose (insofar it is possible in a rubber hose, effectively you'll get substantially lower pressure than before). Is this correct? If so, se can create vacuum using a solid membrane, which permits the gas (that we used to replace the air inside) to escape to the outside world, where its partial pressure was zero/low?


I have found a sentence that goes like this: "Serenity may be the name of the company, but it doesn't describe the company." Someone translated this sentence to my language as if "may be" meant there was a possibility that Serenity was the name of the company (and a possibility that it wasn't), but I feel like the sentence really means "While Serenity is the name of the company, it doesn't describe the company". Is there a specific name (akin to "non-continuous verb" or "inclusive or") for this usage of "may be"? I want to be able to say "This sentence is using a [...], so the correct translation would be...", chere [...] might be something like "comparative may" or "comparative auxiliary".


[A partial groupoid (half-magma) is a set S equipped with a (single-valued) partial binary operation, as in Bruck's Survey of Binary Systems.] This question may be nonsensical, given that the duality between Cayley graphs and groups is deeply related to the mutual complete-ness of each structure. If that is the case, perhaps what I am really asking for is a common diagrammatic representation of partial groupoids. Here is a crappy mock-up of what I'd expect from diagrams of half-magmas in the latter case. If Cayley graphs encode the abstract structure of group, is there a similar type of graphs that encode the abstract structure of a partial groupoid? If there is not, is there some other commonly used diagrammatic representation used for partial groupoids? Thanks!


There is a medical specialty called obstetrics and gynecology. In the UK I have only heard this truncated to something like the first syllables separated by a truncated "and". This could perhaps be written as "obs 'n gy-nee". I have multiple times recently heard this specialty referred to by Americans by spelling out prominent early letters, an initialism. This could perhaps be written as "oh bee gee why en" (as in one spelling out OB GYN). Is this a UK/US difference? Is there a "right" way to abbreviate this specialty? I am particularly interested in how it is said, as opposed to how it is written.


While solving inverse function problems, I got confused in a part, like for any Inverse function to be defined, it must be one-one and onto, then in many questions why the codomain is given more than the Range as if we know that the codomain must needs to be equivalent to the Range for the Inverse function to be valid or defined, then why does they gave us the codomain different from the range? Kindly help me solving this doubt.


So I understand that matter emits EM waves when hot. And that the higher the temperature, the shorter the wavelength, so cooler flames start off orange and the hotter flames reach light blue and white. What I'm wondering is if, hypothetically, you could take that blue-white flame and could make it considerably hotter, would the emitted wavelength reach UV thus making it invisible to the human eye? If so what temperature would need to be reached?


I am interested in modeling the populations of low-lying, long-lived "dark" electronic states after excitation of an atom in the ground state with a laser. Using just a regular rate equation solver, I get nonsensical results. Provided there are no interactions between the low-lying states and the branching ratios to these states are known, is something like an Optical Bloch Equation solver needed to determine their population as the laser irradiates the atom?


I downloaded a nice latex class, but I'm having trouble with one aspect of it - every page's header gets marked with either the chapter name or section name. As you can see, it makes the page look really cluttered. The code for the class is here. I've tried for a really long time to solve this, but I can't. I searched online and found that some commands containing mark have that info, and I've tried commenting out lines containing such commands. Still no luck and could use some help on this - I just want the page numbers to show in the header.


If a block of mass m is attached to one free end of a spring and is performing simple harmonic motion of some amplitude A . Now if a wall is placed at some distance x (x < A ) from its mean position . It collides elastically, how is its amplitude going to change ? [ Will it be different in the side with the wall or uniform throughout? ]


We know that sound travels through a medium through vibrations in the form of longitudinal waves. An example of it is here: We also know that particles of any medium vibrate when we give them more energy in the form of heat which gives them more kinetic energy. From this a logical inference can be drawn: heating the medium might disrupt the propagation of longitudinal sound waves. Consequently, should the disruption occur, would sound production be inhibited altogether?


I am taking the second law of thermodynamics in the 'useful energy minimized' sense, although I am not sure if that is necessarily the most correct. Take the example of two electrons: electron A and electron B. Electron B has a wavefunction. Can we say that electron B's wavefunction will not have high probabilities close to electron A because that would lead to there being more useful energy than otherwise? Otherwise, is there a similar type of 'wasted energy maximized' or 'entropy maximized' for singular wavefunctions rather than groups of particles? Instead of treating entropy as an outcome of different particles, perhaps the wavefunction's different possible positions of particles can do a similar job?


I know that the work done by gravity in going from A to C, and from A to B to C is the same. I also know that the work done by an applied force in going from A to C is the same as negative of the work done by gravity if there is no change in kinetic energy. But what is the work done by an applied force in going from A to B to C if there is no change in kinetic energy?


It is assumed that a valid QFT, must have states/observables in a Hilbert space, transform according to the projective unitary representations of the Poincare group. I can understand why the observables must transform according to unitary representations of the Poincare group, but don't see any compelling reason for that symmetries must be projective as well. Some have explained it, as it is required for dealing with the phase factor redundancy in the wavefunction,but I guess this redundancy is dealt by the unitary representation itself, as the phase factors cancels out. I don't see why a phase factor redundancy demands the symmetry be projective.


Exactly the question above. I've recently posted a question, and the more I look at it, the more it seems like a bug to me. The upstream code base on GitHub seems pretty inactive too (for about two years), having many open issues. I'm missing some experts who could help me or even the maintainer. Do you know whether there are good alternatives for PGFPlots which I can integrate easily into LaTeX and are quick to learn?


Imagine spreading double-sticky tape all over the surface of a car or a plane. Would there more significantly more aerodynamic drag as a result of the adhesive 'sticking' to air molecules and slowing down? This would certainly cause more resistance in solid mediums but would this cause more resistance in air or water? If an adhesive surface can drastically increase friction with solid surfaces, could one also increase friction with the air?


In finding this article, I questioned my understanding of the word 'gadget'. My current understanding is the word is (and please let me know in your answer if I'm right or wrong on this one) synonymous with 'accessory'. The only difference really being that accessory somewhat precedes the technological age. Hence, 'gadget' carries a sort of electronic/technological connotation (but not necessarily?). The article enumerates a 'backpack' as a gadget. Is that correct?


A lever and a fulcrum is able to to multiply the input force to the output force and trade moving distance for increased force. The same happens in automobile transmissions, taking advantage of high motor RPMs. What exactly is the physical mechanism behind this? I am very aware you can mathematically prove it, for instance via an energy approach where energy in roughly equals energy out. Still, is it possible to explain the phenomena down to atomic level? That is, how is it in qualitative terms, that the force on the other side of the lever is amplified?


I will have to teach a first course in differential equations. A good motivator might be to promulgate modelling with differential equations but I have seen some teachers have made polemic against modelling. Are there any really good resources on modelling with differential equations? I want something which will have an impact and motivate the learner. I am not really looking for artificial engineering examples but some bona fide real applications. Also, would like to bring in some history of differential equations to motivate the learner.


What are local laws? I was reading this line in a book... Newtons second law is a local law. This means that it applies to a particle at a particular instant without taking into consideration any history of the particle or it's motion. What does this mean in simpler terms? Are their non local laws as well? Please explain with an example. I'm a high school student.


It might well be that universal constants, say the speed of light, are only constant as far as we can tell in our chunk of the universe - in the same way that the Earth looks flat in the area you live. Are there any ways to tell that the speed of light is actually the same not just in our "neighbourhood" but in all the observable universe?