I am looking to study more about electron impact ionization. I noticed that this topic or anything collision-related, is not present in Chen in detail. Is there another resource that I can use to understand this better?


suppose we have a circle to which we make a target and rotate it without breaking the basic definition of a tangent how many tangents are drawn as rectangles strips i have tried with integrals but I really want to know the answer to this question


I searched on the Internet for the answer and found none. How can one find it? Also, I write "SLITEX" because methinks every letter of the name is, even if small, raised or lowered, the capital one but feel free to groundedly correct.


If still charges in a wire loop do not respond to a(or have their own) magnetic field, then how is current is generated by changing a magnetic field? And why only a changing magnetic field? What actually happens at the atomic level?


Though both of these terms come from other languages, they are both said in English, depending on where one is. One (ay wey as a more English form) can mean holy crap!, and the other can mean something similar. Maybe it can be due to a related background?


I have not got any solution after a few try. In latex, how we can comment out a single word from a line. I mean for example Hello %this is so on it comments out this and the rest of the line as well. Is there any way to comment out only this part, not the rest of the line ?


When an ideal gas is compressed isobarically the given energy by the external source is let out as heat alongwith some more heat that comes from the decreement of internal energy. My question is why doesnt the energy given by the external source increases the internal energy and some lets out some heat?? Maybe I am missing something conceptually about so please help.


While proving associative property of vector addition graphically, everything else appears symmetric except the fact that vector a do not have a derived line as part of the final graph. Is it due to a being considered base?


Let the Zhegalkin polynomial for function consist of those and only those monomials that are not included in the polynomial for function . How different can the functions and be? (Write the maximum and minimum number of sets on which they different for each ).


i used to substitute a value into the actual book prescribed answer and the into the answer i got to see if the equations yield the same answer. i now want to confirm if the method i am using is legitimate. this is my solution to the question this is the book solution


This question looks like me value theorem by canceling of a way to apply it in the problem quite yet. An image I messed around with the given equation and tried to set it equal to the second derivative but it I came up with nothing. I'm suspecting that my method to solve this problem is incorrect.


I asked a question a while a go what is a bound state and the question was closed because there is a similar question. Now since best description we have to describe nature in quantum field theory How we define bound states in quantum field theory?


have you ever encountered works on set partition in which the principle of inclusion exclusion is used in the case of a single parameter and reused in the generalized case (with several parameters) ?


Does there exist such a notation? It looks similar to old factorial notation, but its in upper right corner rather than lower left. I need it as math environment or with some simple code so as to achieve: These are made in ms word.


If particles are excitations of fields: how does the field record the spin of a particular particle? There is no space to record the spin states of two particles, of opposite spin, in the field.


I what taught that my last name which starts the Mc that the c should be above the symbol =, does anyone know what this means? I can not really type it out on a key board but if you write it out with a pencil you can see the M "lower case c with = under it".


I have the following in my notes: I dont understand the statement 'simple model fails' Is the graph shown supposed to be theoretical model or is that the experimental data. This are equation derived before the graph:


I feel that the reason to define thing like current density is that we want to use the directional nature of the current like a vector quantity and since electric current is not a vector so we define something which has same direction as current but a vector? is is true?


my sir told me that he will get a shock, but why not bird then he replied bird has legs acting as insulators i am in confusion what is the right explanation to this


I make reference at this question: Package pgfkeys: I do not know the key '/tcb/O' and I am going to ignore it and to accepted answer. If the text of question and or of the answer span on the next page, like here: the left arrow dont span on the next page (with both tcoloboxes breckables). Can we solve this problem?


I was at a complexity optimization class and they said all constant sheaves are a twist of a perverse sheaf; is it specifically in complexity optimization or is it true for all constant sheaves?!


I'm using the Overleaf documentation to produce a bibliography with in-text citations, using bibLaTeX (compiler is XeLaTeX, document class is article). I've tried all of the citation styles on the overleaf webpage, and none produce superscript in-text citations (i.e. numbers as superscripts). The closest I can get is numbers in square parentheses. Is there no citation style that has superscript numbers? Or does this indicate that something else is going wrong?


I have seen that the notion of affine scheme is a generalization of the notion of affine varieties where the coordinate ring is replaced by any commutative unit ring, and the variety with the Zariski topology is replaced by any topological space. Does this mean that every affine variety is an affine scheme? I think that if we have an affine variety X and we construct its structure sheaf then the affine variety is a locally ringed space but how we conclude that it is isomorphic to a spectrum of a ring? Or there is another proof of this statement?


Is there any expression in English that refers to the situation when someone who is just an acquaintance (not a close friend) happens to be present around you at a social gathering (be it at a house party, a club ...), and you are debating whether to show them that you have seen them out of being shy or to avoid an awkward greeting? I'm seeking anything that fits the bill, whether it's a single word, or a phrase.


A directed graph G=(V,E) is given and I am asked to prove if there is a tree of shortest path of this graph that cannot be returned using BFS. Meaning to say if it is true or not that for every tree of shortest path a BFS can return it. I thought that the answer is true, for every tree of shortest path there is a way of arranging the adjacency lists of G such that a BFS can return this tree, but I cannot find a mathematical way of proving it. Or maybe it is false, meaning there is a counter example of a tree of shortest path that cannot be returned by BFS but I can't find this example either. I would appreciate some help, thanks!


Airplanes traveling at supersonic speeds produce sonic booms because the pressure waves they create converge to produce an extra high intensity wave. But what about planes traveling at slower speeds? These create pressure waves too (although they do not converge). Are these pressure waves the sound you hear from an ordinary plane in the sky? I always thought the sound of an aircraft came from its engine noise.


Chess is an incredibly intricate game, offering an immense number of moves and combinations. Due to this complexity, determining the precise count of legal chess games poses a significant challenge. As of my knowledge, the total number of possible legal chess games remains unknown due to the vast array of potential moves and positions that can emerge during gameplay. However, I've been pondering an intriguing question: Can we establish whether the total number of legal chess games is odd or even? It's a captivating puzzle that has captured my attention. I've given it considerable thought, but I must admit that I'm still uncertain about the parity of the total number of games. Can anyone direct me how to approach this?


The Hills mechanism postulates that when a stellar binary system is perturbed by a supermassive black hole (SMBH), the tidal forces at play result in the capture of one star while simultaneously ejecting the other. My question is as follows: Why does one star become captured while the other is ejected? Is it not possible for the SMBH to capture both stars and maintain the binary system within its orbit, or alternatively, eject the entire binary system?


A graph is homogeneous when every isomorphism between two finite induced subgraphs can be extended to an automorphism of the whole graph. I was reading Diestel's graph theory where he describes the following graph and says its clearly homogeneous. It's not immediately apparent to me why. If I have an isomorphism between two finite induced subgraphs, and map every other vertex outside the subgraphs to itself, does it become an automorphism of the entire graph? But wouldn't that make almost anything homogeneous? What is special about this one?


Most eye floaters are caused by age-related changes that occur as the jelly-like substance (vitreous) inside your eyes liquifies and contracts. Scattered clumps of collagen fibers form within the vitreous and can cast tiny shadows on your retina. The shadows you see are called floaters. If you look in a mirror while getting floaters would you see an image of the floaters in the mirror? Since they are basically shadows on the retina, I think we should theoretically be able to see an image but practically not because the shadow image would be very small.


I have been trying to find the proof of the derangement theorem, and have checked out some materials on them (since its proof has not been taught to us). But I have not been able to understand much of it. So, I was thinking, is the inclusion-exclusion principle used? Since the expression seems somewhat familiar. If not, then could someone please explain the proof in a slightly simplified manner?


I am trying to understand what fugacity really is. I asked a related question on chemistry stack exchange but am not satisfied with the answer. In many books, such as Physical Chemistry by Engel and Reid, Physical chemistry bh Castellan, etc. fugacity is called as effective pressure. My questions are a)What does the word 'effective' is trying to say? and b) If we already have various equation of states that have already accounted for intermolecular forces, then why is there a need for fugacity? I'm sorry if this is a stupid question but english is not my first language and maybe I am not interpreting fugacity correctly. So any help is appreciated!


Objects appear in different colours because they absorb some colours (wavelengths) and reflect or transmit other colours. The colours we see are the wavelengths that are reflected or transmitted. As far as I understand, when an atom absorbs a photon, one of its electrons gets excited (and unstable). So it drops back to ground state, emitting that energy in the form of photons of a specific colour. This means that we should find objects to be in certain colours, but, for example, hydrogen gas is colourless. How is this possible? Shouldn't Hydrogen gas have a specific colour related to its emission spectra? I was reading an article on this topic today (https://www.khanacademy.org/science/chemistry/electronic-structure-of-atoms/bohr-model-hydrogen/a/spectroscopy-interaction-of-light-and-matter), and this came into my mind. Can anyone please help in clearing my understanding?


What I mean is how the shape of an object affects it's physical properties. I've been studying Marine Biology and there's constant talk of how a fish's shape determines how well it can swim but no explanation of why, say, "Stream lining" reduces water resistance outside of it "just making sense" and I'd like something more in-depth. Similar phenomena would be: How a blade's shape determines how it cuts, Why certain shapes are more structurally sound, What determines a shape's aerodynamics. Is there a field of research dealing with this? It's been driving me mad.


An excerpt from my grammar lesson: In a formal context, 'due' should always be treated as an adjective, and 'due to' must therefore follow or refer back to a noun, as in: his success was due to hard work. In this case, how is 'hard work' considered a noun? In a prior lesson, a noun was defined as a word that names a person, place or thing. by this definition, isn't 'hard work' more appropriate to be a verb? Am I missing something?


I recently engaged in a lively debate with a friend about whether the term "hot dog" qualifies as a misnomer. My argument stemmed from the fact that a hot dog doesn't actually contain dog meat. Instead, it's typically made from beef, pork, or a combination of meats. My friend argues that the term "hot dog" supposedly originates from a playful association with the dachshund sausage, given its resemblance to the dog breed and therefore, it shouldn't qualify as a misnomer. While this may be true due to the term's roots in historical references, it still inaccurately portrays the contents of this food today with the lack of any canine involvement.


The connection between Markov processes and Potential Theory is well known, as is conformal invariance of Brownian motion which allows probabilistic proofs of statements in Complex Analysis, like Picard's theorem. What are some results in these areas (Potential Theory, Complex Analysis, maybe Complex or Differential Geometry) that admit probabilistic proofs, and possibly for which no other proof is known? To what extent are probabilistic methods in these areas actually crucial in proving new theory and to what extent are they more of an interpretation of facts that are already known? It would be great if there were some survey about this. Thanks in advance for your answers.


My friend is working on a snow gun, which involves using compressed air to blow apart water coming out of a pressure washer. I read that you can cool air by compressing it, which makes it hot, then letting the compressed air lose energy by cooling to room temperature. I was going to use the ideal gas law to find the change in temperature when compressed/decompressed, and Newton's law of cooling to find the time it'd take to cool down. However, I don't think the ideal gas law can be used in this case. What equations are appropriate to find the temperature of a gas before and after it is compressed?


Suppose we have two entangled particles, A and B. If we measure particle A and then subsequently entangle particle B with a third particle, C, would the state of particle A influence the combined states of particles B and C? Specifically, would there be any correlation between the combined spins of B and C and the measured spin of A? Have there been any experiments conducted to explore or validate this scenario?


"We understand that a set is a collection of objects, but not every collection qualifies as a set, as exemplified by the collection of all sets (consider Russell's Paradox). To define the collection of all sets, we employ the term 'proper class.' Now, I would like to confirm if my understanding here is accurate: A class is a collection of objects, and these objects can be sets. Classes are categorized into two groups, namely small classes, known as sets, and large classes, referred to as proper classes. Is this correct?"


I am studying a rotating cylinder model in general relativity. The interior metric is given, but I am trying to express the exterior metric. It has to be practical and describe spacetime around a rotating cylinder. So far I have considered the Kerr metric, of course, the Janice-Newman-Winicour metric, and the Tomimatsu-Sato metric. For the Kerr metric, I am struggling to express it in cylindrical coordinates. Doing so will make it easier to match it to the interior metric. Any suggestions or inputs would be appreciated, thanks.


As a high school student, I'm curious whether everything emits thermal radiation, including monatomic gases and substances composed of a single type of atom and no molecules. Can these objects emit radiation in all wavelengths, and if so, how is it possible in the absence of molecules?Is it necessary for an object to be a blackbody in order to emit thermal radiation across all wavelengths?


I was watching this video where the English gentleman asks the Scottish MP to speak in "Antipodean English". From what I know, 'Antipodes' refers to the southern hemisphere and I am unable to understand why the gentleman would want the other person to speak a dialect from that broad region in particular. Because this seems unlikely, I want to know what the person meant by the term "Antipodean English".


Assuming the earth is either at the perihelion or at the aphelion, it is easy to see the Runge-Lenz (RL) vector is directed along the line joining the perihelion and aphelion. Since the RL vector is a constant vector, its direction (and also magnitude) must remain the same at all points of the orbit. So if we want to draw this vector at other points of the orbits, it will be drawn as a vector parallel to the line joining the perihelion and aphelion. Is this right?


I'm looking for where best to start on the following problem. I have a graph of N nodes and each node as a weight. I need to group all nodes into X groups such that the sum of weights in each group are approximately even and that each group is a connected component of the original graph. After some simple searching I came across the lukes partitioning algorithm implemented in python. https://networkx.org/documentation/stable/reference/algorithms/generated/networkx.algorithms.community.lukes.lukes_partitioning.html#networkx.algorithms.community.lukes.lukes_partitioning This is approximately what I am looking for but performance feels slow for the size of the graph I am working with. Do you have any potentially better ideas that worked for you?


It's a common myth (especially in popsci) that a black hole contains a singularity. However, I cannot find an explanation for what we think a black hole actually does contain. The best I've seen is "in order to answer this question, we have to unify GR and quantum mechanics." So there are no attempts to answer this question despite our incomplete picture of physics? Let's say Schwarzschild black hole to start with.


I am writing a research paper where I want to say The rest of the paper is organized as follows. in the introduction of the paper. However, the word "rest" doesn't sound good as it is perceived as if I am explaining about an unimportant part of the paper. I checked some synonyms on Grammarly, but nothing really fits well. I thought about the following: The remaining paper It also sounds odd as it sounds like "leftover" to me. I can be wrong as well. The remnant of the paper This is used mostly for historical events as far as I know. Any advice is appreciated.


I'm not very well-versed in topology, but I know the basic concept of topological equivalence can be approximated by "counting the holes" in objects, in the sense that a sphere is different from a torus, but a torus is the same as a coffee mug because they have the same number of holes. I know this roughly approximates being able to smoothly deform one object into another. Can this be applied to interior "holes" as well, such as a hollowed-out sphere? Does the hollow center count as a "hole"? Is there a way to smoothly deform a hollow sphere into a torus or a sphere? If not, is a hollow sphere in a different topological equivalence class from both a sphere and a torus?


I have read that projectile motion is parabolic in nature. I know that the equation of projectile motion represents a parabola. But when do we get a parabolic curve? Does it happen when there is downward acceleration and horizontal velocity? What condition results in a parabolic shape? Is there any possibility of elliptical motion or any other curve? What would be the determining factors in that case?


It's clear to me that when an object is in free fall in a uniform gravitational field ("Einstein's Elevator") it feels no tug of gravity. However if it is in free fall in a nonuniform gravitational field does the object (or more specifically an accelerometer) feel a tug as gravity pulls in one direction and then another? Asked another way, is an object in free fall in a nonuniform gravitational field considered to be in an inertial reference system?


I was recently studying a paragraph in which I came across a sentence, which was below: The British king, Charles I, had inherited a very difficult financial situation from his father. However, I don't understand why 'had' is being used here. My understanding is that had is used only in past perfect tense, where there are two events in past and we use had to signify that event that finished before the second event started. So I modified the above sentence like below: The British king, Charles I, inherited a very difficult financial situation from his father. Is the modified sentence correct? Please help me to understand.


My uncle is a retired math professor. He has had some serious health issues in the past year, but he is still mentally sharp. For both fun and a distraction, he has been working on a textbook. However, I believe it is hard for him since he has trouble sitting at a desk and typing. Does anyone have any recommendations for good voice-to-text software that he could use? Latex and math formatting would be very helpful, obviously, although I could possibly help here.


I have a set theoretic question. I don't understand why the collection of open balls of a metric space is a well-defined set. From my understanding of sets, a set is well-defined in Naive set theory if we can check whether an element is in it or not. But how can we check whether a subset is either an open ball or not? It seems very difficult to check. Maybe my view of a set is too rigid. Is there a better view of whether a set is well-defined? Thank you.


I came across the following sentence while reading "A Clash of Kings" book by George R. R. Martin: Whitetree was the fourth village they had passed, and it had been the same in all of them. The people were gone, vanished with their scant possessions and whatever animals they may have had. None of the villages showed any signs of having been attacked Why is the "may" word used here instead of "might"? I thought that "may" can be used only in the present tense and have never seen it used in the past context


I recently saw a video from a passenger of Concorde after it goes supersonic. Inside the aircraft, there is still a loud roar similar to what we hear in aircraft today. However, since the aircraft is supersonic, shouldn't the noise be left behind? Why is the roar present? I was expecting the aircraft to be nearly silent due to being supersonic, and it appears that is far from the reality.


So I know that the statement "If two random variables are independent, then the two random variables are uncorrelated" is true. Does that mean its contraposition "If two random variables are correlated, then the two random variables are dependent." also true? In terms of the theory of logic, the contrapositive statement should also be true, but then again, I learned "correlation does not imply causation". So I'm a bit confused. Can someone clarify this? Is the contrapositive statement really true? If so, what is the difference between that statement and the statement "correlation does not imply causation"??


Suppose a person is using an electrical appliance and suddenly there is a power outage- would it be correct if that person immediately said to their family members that 'the power is out'? Or would it be better to say 'power has gone out'? In what way are they different? Are both idiomatic? (And similarly, is there a distinction between 'the power has come back' and 'the power is back on'?)


I was reading about how reflection and refraction work at a quantum scale. What I understood was that the photon "can" take multiple paths to reach the end point and when we consider all these possibilities, most of them cancel out as they interfere destructively, while there's constructive interference around a certain direction. That is what we observe is the the sum of all the probabilities of the possibile paths the photon could take. Please correct me if I'm wrong. My question is how do we calculate that probability of each path? Or do we only calculate the sum of the probabilities directly?


When @anna v explained why the planetary model of the atom does not make sense in this post, she said the electron in an orbit is accelerating continuously and would thus radiate away its energy and fall into the nucleus. However, when the Earth orbits the Sun, isn't it accelerating continuously toward the Sun? If this is the case, should the Earth radiate away its energy via gravitons and fall into the Sun eventually? I also checked questions about Why doesn't the Moon fall onto the Earth?, but the answers didn't talk about radiating away gravitons.


From Beauville's book on complex algebraic surface, in Chapter IV, we know each Hirzebruch surface contains a unique irreducible curve with negative self-intersection. If we remove this curve from the Hirzebruch surface, different from contraction, it will result in a non-compact surface. I'm wondering what we can say about this non-compact surface. For example, what kind of symmetry does it have, will it still be toric? More generally, what if we remove an irreducible curve different from the one with negative self-intersection instead, how will it affect the situation above? I'd appreciate any relevant discussions or comments, thank you in advance!


I'm trying to represent a company as a feature vector. The vector has multiple stats describing the company such as earnings growth, capital expenditures, gross margin, etc. The goal is to represent companies with these vectors and find the most similar companies based on some similarity measure (ie euclidean or cosine). My questions are: Would this be an accurate way to represent companies? Is there some normalization I have to do to the vectors before finding similarity? And what is the best measure of similarity for these vectors?


According to Introduction to Solid State Physics by C. Kittel, An ideal crystal is constructed by the infinite repetition of identical groups of atoms. A group is called the basis. The set of mathematical points to which the basis is attached is called the lattice. The crystal structure of copper is face-centered cubic (fcc). So we can say the lattice is fcc, and the basis is composed of one single copper atom. I was wondering if it is wrong to say, instead, that the lattice is simple cubic (sc), and the basis is composed of four copper atoms. Both give the same arrangement of atoms in space, but in the second there are fewer lattice points.


Examples of solutions to linear differential equation are (from Wikipedia): exponential function, logarithm, sine, cosine, inverse trigonometric functions, error function, Bessel functions and hypergeometric functions. So in this sense these are linear functions (for some generalized notion of "linear"). But I wonder if there are examples of functions that are "strictly nonlinear", in the sense that they are the solution of some nonlinear differential equation but not of any linear differential equation.


I have an application I'm trying to model that involves using superconducting coils to produce a steerable dipole. To some degree, the problem is easier if I can modulate the current through the coil, but I have read in a few places that there can be practical challenges related to the superconducting coils being "perfect inductors." Can someone help me get a better understanding of how to mathematically model a superconducting coil in both a DC and AC circuit? Specifically, I'd be curious if trying to vary the current quickly produces operational challenges like excess heat which makes it more difficult to keep the coil in a superconducting state. Thanks!


Bicycle's brakes generally fall into two broad categories: rim brakes where the braking pads pinch against the wheel's rim, and disk brakes where the pads press against a disk (rotor) that is rigidly attached to the wheel's hub. General wisdom is that disk brakes are higher quality and offer better stopping power. But disk brake pads are much closer to the center of the wheel, which means that they have to exert more force to produce the same amount of torque. Wouldn't this be less stopping power?


I noticed that dew only falls on horizontal surfaces, while vertical surfaces remain dry. But dew is the condensation of water vapor contained in the air. And for the condensation process, the direction of the surface does not matter. If you bring a smooth cold object into a warm room, it will be covered with condensate evenly on all sides. Why do vertical surfaces stay dry?


I have a idea I'd like to explore: If a black hole can bend light due to its immense gravitational attraction, could we imagine an 'inverted black hole'? In this hypothetical scenario, this object would behave similarly to a black hole but with a force of repulsion instead of attraction. This repulsion would cause light to curve away from it, making it impossible for light to reach the white hole. I'm also curious about whether there are software or codes available that could help me visualize this concept, similar to how software like 'Space Engine' visualizes regular black holes. Right now, to keep things simple, I wish only to explore how this hypotetical object would affect light and images.


Using a triangular glass prism to disperse white light into a rainbow works because different wavelengths/colours of light have different refractive indices in glass, doesn't it? The different refractive indices apply regardless of the shape of the glass. Therefore, wouldn't a simple rectangular slab of glass also disperse white light? The different wavelengths would be differently refracted and emerge separated from the other end. Why is a triangular prism always used to demonstrate dispersion? Is there something I'm missing?


Mine is a very basic question yet I haven't got a satisfactory answer so I turn to you. Special Relativity is based on two assumptions, one of which is that the speed of light is constant for all inertial frames. While I know the disprove of universal reference frame, Aether, I want to know how we came to the conclusion that the speed of light REALLY is constant for all inertial frames (light traveling in vacuum).


The gravitational binding energy or self-energy of a system is the minimum energy which must be added to it in order for the system to cease being in a gravitationally bound state. Equivalently, the gravitational self-energy of a body is defined as the work done by an external agent in assembling the body from the infinitesimal elements that are initially at an infinite distance apart. A gravitationally bound system has a lower (i.e., more negative) gravitational potential energy than the sum of the energies of its parts when these are completely separated. Is this type of energy a source for gravity? In order words, does a star have a smaller gravitational field due to its negative gravitational self-energy?


I'm wondering which of the following sentences is correct: They realized that if the threat WAS not contained, it could endanger the nation." They realized that if the threat WERE not contained, it could endanger the nation." My understanding is that the subjunctive "If X were" is used for a counterfactual to a present situation. But here, the threat not being contained is a real possibility being considered in the past. If the main clause were written in the present tense, "to be" would be present indicative: "If the threat IS not contained." So I'd think it should be past indicative when the main clause is past tense. Am I right?


I just read an article describing the project Scorpius. The aim is to use explosives to compress and heat microscopic quantities of Plutonium in order to study the states of the matter in extreme conditions. I wondered why they needed to recreate those conditions with explosions. Couldn't they achieve the same states with a Diamond anvil cell? With such apparatus they could compress a microscopic quantity to more than a million atmospheres and by passing some laser pulses trough the diamond anvils they could heat the matter. Am I missing something about the project purpose?


During the Electroweak era, the Higgs mechanism had not yet distinguished particles such as electrons and muons by mass. Is there a conceivable experiment, or a hypothetical cosmological observation we could make today, that could have identified electrons and muons as distinct particles during that era? Alternatively, might there be features in the Cosmic Microwave Background or the distribution of matter in the early Universe that could reflect the distinction between the electron and muon fields prior to the onset of the Higgs mechanism?


Why do charges placed inside a metallic cavity have no effect on outer charges and vice versa? My teacher explained it by electrostatic shielding. He said that the conductor does not allow the electric field to pass through it. I agree that inside a conductor the electric field is zero, because free electrons of the conductor move opposite to the external electric field to create an equal and opposite electric field. But what about inside the cavity? There is nothing like free electrons in the cavity, so an external electric field should reach there.


I am wondering if the physical shape of a current carrying conductor will have any effect on the shape of its magnetic field or is it that the magnetic field surrounding any current carrying conductor will always be perfectly round. For example, let's say that current is passed down a long L-shaped metal bracket, such as the one shown in the drawing below. Will the magnetic field follow the path drawn around the L-bracket, or instead will the magnetic field be perfectly round and will encompass the entire L-shaped bracket? Does the physical shape of a current carrying conductor affect the shape of its magnetic field?


I was fairly convinced that usual QM formalism didn't necessitate the use of complex numbers and that ultimately they're just a matter of convenience and utility rather than anything fundamental. This paper claims that isn't the case. But their demonstration doesn't satisfy me because it's muddled up in this Bell-like experiment. Can anyone explain what's going on in a simplified manner? Their argument does not make it clear to me why splitting the wave function into its real and imaginary portions and treating them as two quantities which are coupled via Schrodinger evolution isn't always possible---even in the case of multipartite states.


I've seen several sources stating that Einstein was partially inspired by the Ehrenfest paradox when thinking about general relativity. However in the Ehrenfest paradox the non-Euclidean nature of the space in the rotating frame of reference is linked to Lorentz contraction rather than gravitational forces like in GR. Is the link simply that the rotating disk question got Einstein thinking about non-Euclidean geometry as a way to treat non-inertial reference frames?


Forward and reverse synchronization between the TeX source and the PDF output using SyncTeX is an immensely useful tool when writing larger documents. Sometimes, especially when the synchronization is not working as expected, I would like to see the synchronization anchors in the PDF file. I know that the information is stored in .synctex.gz file in some form, but this is hardly transparent for a human reader. Some graphical indication over the PDF output would be great. Does this exist? I am working on Windows. Solutions for other systems are of course welcome, though they won't help me.


Two generalizations of graphs have been described in a number of articles: multigraphs and hypergraphs. A multigraph is a graph in which multiple or parallel edges between nodes are allowed. These edges have the same end nodes. A hypergraph is a graph in which an edge can join any number of vertices. An edge can therefore connect sets of multiple nodes to one another. Both concepts allow one to model relations among objects with more detail and precision than with 'ordinairy' graphs. However, I have yet to find an account in which a combination of both structures is described: the hyper-multigraph. Such a graph would allow for parallel edges between the sets of nodes. Question: are there any references on such hyper-multigraphs?


Yesterday in my English class, I tried to say 'I hadn't got that book' and my teacher told me my sentence was incorrect, she said I should say 'I didn't have that book'. I bought Grammarly, and it shows me my sentence is correct. Or maybe there is a difference between 'I didn't have that book' and 'I hadn't got that book'. I would like to know the opinion of a British person.


Fraternal twins is what you call a set of twins who look different, but the word fraternal also has the meaning 'of or relating to brothers', and comes from a Latin root meaning 'brother'. The question is, how do I refer to a set of twins who look different but are both girls? May I use the phrase sororal twins to describe them? Or can fraternal be used here, too?


I'm trying to prove that the covering space X of a manifold Y is a manifold of the same dimension. I was stuck on proving X is second countable, more specifically, on proving the fiber of any element in Y is at most countable. I find a proof at Is a covering space of a manifold second countable?, but that proof is based on the countability of the fundamental group of a manifold. Could there be a more elementary approach without using this fact?


I imagine the following thought experiment: Let us prepare two identical neutral fermions with identical spin and antisymmetric spatial wave function in one spatial dimension. Let the initial state be two Gaussian wave packets which start at some large distance and have identical speed, but opposite direction. Since the particles are neutral, the wave packages should move through each other undisturbed. But at the time, where the particles are at the same place, the antisymmetrized wave function adds up to zero, so you cannot normalize the state. The exchange interaction should be zero because the particles are not charged. For the same reason there should be no magnetic moment. Where is the error in the argument?


In my notes regarding entanglement and CHSH inequality it says: Further, this is the case even for incompatible observables: measurements on one system allow us to predict with certainty the outcomes of measurement of incompatible observables on the other system. However, the uncertainty principle tells us that we cannot predict with arbitrary accuracy the results of measurement of incompatible observables. How is the first sentence allowed. My understanding was that if you measure one set of observable, then the other incompatible observables are still in superposition.


Given the following Bayesian belief network, what is the probability that a non-smoking patient with a smoking parent will develop lung tumor? In this question, isn't the answer already provided within the question itself? As I understand it, I required to calculate the probability of developing a lung tumor given that the patient does not smoke, but their parent does. However, this information is already provided in the table at the bottom right. Am I missing something?


Suppose I have a "moving" frame of reference where I send a light beam from point A to two equidistant points B and C (equidistant in the moving frame of reference). From a "stationary" frame of reference, A to B and A to C are different distances because they have different diagonal components. But the stationary observer still sees light travel from A to B and A to C at the same time. How is it possible given constant speed of light? Just started learning about time dilation. Any help would be appreciated!


I am an engineering student who is slowly branching into applied mathematics. While I can find plenty of good textbooks on the rigorous treatment of the Fourier Transform itself (including functional analysis and the Schwartz space and the result of this for the Fourier Transform) but I am having difficulty in finding an equivalent for the Discrete Time Fourier Transform (DTFT). Does anyone know of books that rigorously handle the DTFT, particularly on the convergence of functions that are not absolutely summable and its DTFT?


I tried to prove that the centers of two tangent circles are colinear with the point of tangency. If I just assume they have a common tangent at their point of tangency then it's pretty easy. However, how do we prove that two circles do indeed have a common tangent at their point of tangency? (without using the collinearity of the centers and the point of tangency to avoid circular logic)


In the Spanish language there is a word that is used to describe a person that has been tagged as a deadbeat; it is Largo. Largo in Spanish means Long as in the measurement of the distance between two points or places, etc. If you loan any money to this person, it will be a long time before you get paid back, if you get paid back at all, or if you receive any kind of payment. They can also be known as a Largo if they are sly, tricky, or conniving with bad intentions.


I'm looking for a formal language that has the following properties: Contains finitely many words (and you can prove it). Decidable/recursive (there's a Turing machine that always halts, that can determine whether any given word belongs to the language). It's impossible to compute the exact number of words it contains. Meaning, even if you've already found all the words in the language, you can never know if there's another word that you just didn't try yet. Such a language feels "illegal", but I didn't manage to disprove its existence.


A Chinese teacher of English asked me about the following, taken from an English test for Chinese people. It's quite tricky I think. I would like to know three things: Which answer or answers do you consider to be correct? If you consider an answer to be wrong, why do you think it is wrong? How would you reword these answers to make them acceptable? Nanjing is a popular city with a long history. Could you tell me ______ . A. where can visitors enjoy delicious Nanjing snacks? B. when is the best time to visit the Confucious Temple? C. how will people get to the Zhongshan Hill Scenic Area? D. why was the city also called Jianye in ancient times?


I am pretty proficient till Calculus II, but I want to learn about more advanced methods of computing integrals like Gamma function, Elliptic integrals and Leibniz integral rule. I also want to improve my mathematical problem solving ability with this. Should I read Advanced Calculus by Frederick S. Woods or Inside Interesting Integrals by Paul J. Nahi? Or would it be better to first learn multivariable calculus? (any other book suggestions are welcome)


A group action on a set is said to be transitive if it has only one orbit. Now, a group action on the same set is said to be regular if all the stabilizers of the elements of the set are trivial. My doubt is, if there is only one orbit, then, by orbit-stabilizer, all stabilizers must be trivial right? What am I missing here? Any hints? Thanks beforehand.


From my understanding, the frequency and wavelength of a mechanical wave can be observed by the motion of the particles (i.e., as the particles oscillate from their state of equilibrium, the wave propagates through the medium). But, with electromagnetic waves, how do we measure its frequency and wavelength? If an electromagnetic wave travels across a vacuum, isn't it pretty much invisible to the naked eye?


I am tasked to simplify this circuit to one source and one resistor, which is equivalent to the original but obviously easier to solve. My problem is the fact, that I don't know how the nodes A and B affect the process. I'm more used to the circuits where the A and B nodes are on one end of the circuit, but these are in the middle. Not sure how this is supposed to be interpreted.


I am wondering if this statement about parallel circuits and conservative fields is true?: The voltage drop over resistors in parallel is the same because the electric field is conservative, so the work done on a charge only depends on the start and end point. The field is conservative because the force it generates consists of the electrostatic force (Coloumb) and a magnetic force. We can show that the electrostatic force is conservative, and the magnetic force on moving charges is perpendicular to the velocity vector, so it does not do any work on the charge. So the only force that does work on the charge is the electrostatic force. Is this statement correct?


I'm playing around with modeling an archway of a building using a solid modeler, and have come to a problem that exceeds my mathematical ability. Given an ellipse that has a known major radius, and is tangential to a line of a known length, angle, and offset on the Y-axis, can I find the ellipse's minor axis? It's been decades since I took a math class. I can handle basic trig, but I can't figure out how to get started on this one. Edit: Thanks everyone! You can see your solution in the flying buttresses of this (very preliminary) render.


The company "Fluke" makes a DMM that they claim can measure AC voltage by simply clamping the connector around a live wire, and using the E-field to induce a voltage/current into the capacitor. I can understand the concept if we assume the person holding the DMM is grounded and the measuring loop is sufficiently sensitive. But what if the person holding the DMM is not grounded, or is in a vacuum, etc. Some people on the internet claim it works even if the DMM is hanging from the wire in open air. I have my doubts... So, I am wondering if someone can explain to me if current can flow in a capacitor that has no path to ground. It seems like an obvious "no."


I am trying to understand the Von Neumann decomposition, according to which every Von Neumann Algebra can be uniquely decomposed as integral (or direct sum) of factors. More specifically, I am trying to connect (if possible) the Von Neumann decomposition to the GNS theorem. Can someone please help me answer the following questions? Given a normal state, is it true that its GNS representation is always a factor? The answer would be different if the dimension of the algebra is either finite or infinite? Is it possible that the algebra obtained by applying GNS theorem on a irreducible state is not a factor? From the quantum mechanics point of view, why is the Von Neumann decomposition important?


There's a clip going around on the internet of the driver Ross Chastain, 'wall riding' in a race that allowed him to pass many opponents on one turn. Here is the clip. What I do not understand is how did this work? The only explanation I can think of is that the normal force of the wall acted like a centripetal force, which meant he didn't have to brake on the turn, but this doesn't make too much sense to me since the frictional force of the wall on the car would slow him down quite a lot.


I'm working on a project involving non-standard dice where each face can have a different size, but the overall shape and weight distribution of the die are uniform. I'm interested in finding out how to calculate the probability of the die landing on a particular face based on the size of its faces. Assumptions: The die maintains a uniform shape and weight distribution. The die has 'n' number of faces, each potentially with a different area. We ignore the complexities of rolling dynamics and assume the die lands flat on its faces. I am looking for an equation or a model that can accurately calculate the probability of landing on a face, given the area/dimensions of each face.


Does anyone know where I could find a copy of Ron Graham's PhD dissertation, titled "On Finite Sums of Rational Numbers"? I found this paper, with a similar title, but I could not find Professor Graham's dissertation. I need to double-check some detail(s) in Professor Graham's dissertation which may not be in the published version. This is in consideration of my (current) approach to odd perfect numbers via Egyptian fractions.


I am not a pure mathematician but an engineering professor. Right now I am going through the first chapter of Topology without Tears by Morrison. One of the questions is as follows. A set T consists of a set of real numbers, the null set and the closed interval [-r,r] where r is irrational. The question is whether (T,R) form a topology where R is the set of real numbers. The answer seems to be an obvious Yes irrespective of whether r is rational or irrational since both intervals can be infinitely large and can contain arbitrarily large numbers. But I am suspicious as it seems to be too obvious and I think the actual answer is No.


I am new to linguistics and am currently learning how to figure out phrase markers for sentences. I am unsure about this sentence: She went off in search for her hedgehog. I know that "she" is a noun phrase, but I am unsure about the VP. Is the VP "went off in search for her hedgehog"? And does that split again into the three phrases: went off, in search and for her hedgehog? Does it look like this?


I am trying to find the directional vector to hit a moving target. For example, as seen in my drawing, Lets say a space ship at position Sxy needs to fire a projectile to hit the moving asteroid (currently at position Axy). The asteroid is moving with a linear velocity vector Av. The projectile has a constant speed (magnitude) of Ps. How can I solve for the projectile directional vector Pv? Both the asteroid and the projectile have a radius.


I am studying simplicial homology and I am a bit confused about the definition of an orientation of a (finite) geometric simplex. The definition says that two orderings of a simplex are equivalent if they differ by an even permutation, and so we get two equivalence classes - two orientations. My question is, what is an ordering of a simplex? Is it just a permutation of the vertices or does it have to satisfy some other rules? If it's defined to be a permutation of vertices, then we draw paths between vertices indicated by the permutation of vertices (I am thinking geometrically), then my lines might cross each other. Is it that this will not happen because vertices of a simplex are in general position? Thank you!


I've been studying the Lax equivalence theorem . I've learned that it allows you to establish the convergence of a scheme through proof of stability. The Wikipedia article says that the method it's important because "it is ordinarily difficult to establish [convergence] because the numerical method is defined by a recurrence relation while the differential equation involves a differentiable function". I used the equivalence theorem to establish the convergence of the Crank-Nicholson method in an assignment. I was wondering what a direct proof would look like (I'm assuming it depends on the scheme). Can you please provide any references for this? Thank you in advance!


With some formal systems, it's possible to enumerate all the theorems. This is the case for instance in propositional calculus or in some first-order theories with a recursively enumerable set of axioms (e.g. group theory, field theory, Peano arithmetic, etc.). I'm looking for a software that would implement such an enumeration. Or more basically, a software that would produce random theorems (which may or may not be of interest). A software doing that in any of the formal systems listed previously would be acceptable.


My teacher taught us that a square matrix is called a diagonal matrix only if all the entries besides the ones on the principal diagonal are zero. He further cautioned that there is no restriction on the entries of the principal diagonal and that they may be zero or non-zero, BUT there must be atleast one entry on the principal diagonal that is non-zero. We were specifically told that null matrix is not a diagonal matrix as there simply is no entry along the diagonal. However, most if not all sources on the internet say that null matrix is also a diagonal matrix as all the non-principal diagonal entries are zero. Which one of the above definitions is correct?


Imagine the following conditional: If washing machines are humans, washing machines are quadrupeds. It seems to me that the truth value of the conditional as a whole is ambiguous. Since its antecedent is false, logic tells us that the conditional is (vacuously) true. But in fact, the conditional as a whole does seem false: if we grant that washing machines are humans, then washing machines are clearly bipeds. But how can the same conditional be both vacuously true and (at least intuitively) false?


I just read this answer to "What exactly is a Photon?" which has me a bit confused. It seems to be arguing that "photon" is just a catch-all term for any sort of interaction with the EM field and the implication is that it's not even a particularly useful concept, in contrast to the fundamental particles of other fields. But how is that any different from other elementary particles, when all particles are just excitations in quantized fields? I also found several threads talking about whether photons have wavefunctions or not, such as this one: Wave function of a photon? and I'm seeing conflicting information. If photons don't have wavefunctions, are they even particles at all, in the sense of being localized disturbances in a field?


I am doing some interactive plots about cosmological horizons and in my research I stumbled upon the sound horizon, the baryonic acoustic oscillations and how it had an impact on the formation of the first structures in the universe. Some people have calculated the radius of the sound horizon at the time of recombination using the same equation used in the particle horizon but with the speed of "sound", which itself is a function of the density of matter and radiation. But I've never seen anyone actually plot the radius into the future like some have done with other cosmological horizons. Is there a reason for that? Does it not play an important role in the universe anymore? Was it only important at the time of recombination?


The nlab claims that the maximal ideal theorem in Heyting Algebras (i.e. for every proper ideal in a Heyting Algebra, there is a maximal ideal that contains it) implies the axiom of choice. Sadly, it does not cite any proofs and I've been unable to prove this, I'd appreciate any proofs/references. I find this particularly interesting as it's known that the maximal ideal theorem for Boolean Algebras is strictly weaker than the axiom of choice. Yet, the same theorem applied to the apparently slightly bigger class of Heyting Algebras suffices to prove choice.


I have difficulty understanding when we use "an analogy", "a metaphor" and "a figure of speech". I feel like in a casual conversation, people just use any of these words to say something that is different from its normal use. Does a native English speaker distinguish the difference between them in a casual conversation?. Is it correct to say: "I am like a bee in a bottle is an analogy" or "I am like a bee in a bottle is a metaphor" or "I am like a bee in a bottle is a figure of speech"


In the study of the geomagnetic field, an expansion in spherical harmonics is used to represent the scalar magnetic potential: the first terms give the dipole approximation, then the quadrupole, etc. For instance, we can find here (International Geomagnetic Reference Field) reference data for the magnetic field, given as coefficients of a spherical harmonics expansion. My question is: how are those coefficients calculated? I was thinking of the Monte Carlo method, but I couldn't find any information on that. Note that I would also accept answers related to the gravitational field as the question also applies there.


I'm an English learner, and sometimes I cannot understand why some words are used to express a meaning. For example, the word "Down". What's the difference between "I walked on the street" and "I walked down the street" or "Alan drove down the street" and "Alan drove on the street" ?? As far as I know, these verbs are not phrasal verbs, so how can I know when to add this "adverb particle-like" with a verb ?


I saw this in a textbook and I want to confirm if it's true If a body of mass m rolls down an inclined plane of height h and length l (hypotenuse), the work done is mgh. It is the vertical height of the inclined plane that is used in the calculation, not the length (hypotenuse) of the inclined plane. Please how true is this because I believe we should be using a little bit of trigonometry since the object also moved a horizontal distance. Please if true, how?


I'm trying to understand how car wheels work. In a perfect world the bottom of the wheel has no velocity. The wheels are in a state of pure rolling, which means there is no static or kinetic friction because the bottom contact point is not in motion. This would mean that in a world without air resistance a driver wouldn't have to push on the gas pedal, as there aren't any forces opposing the cars forward velocity. Is this right?


I have a question about the EPR phenomenon of the quantum-mechanics. When you observe the EPR phenomenon, you separate two spins in the singlet state. Then, you and your fellow independently observe the direction of each spin, and you compare the directions. If one spin is in New York, and another is, for example, in Tokyo, how do you assign the directions? Because the earth is round, I do not know which direction is opposite to another. In reality, it is difficult to make such a large separation without losing correlation. However, in principle, it is irrelevant how large the distance is. Thank you for your attention.


I found this problem online somewhere (cannot remember the exact source, sorry) and initially I thought it looked fairly simple and fun to solve. I've now sat for several hours and even asked some friends to help, but none of us can actually solve it (without complex algebraic equations). I am by no means an expert on geometry or trigonometric (or mathematics for that case), so I would assume there are some identities or theorems that would make this problem easier to solve that I just don't know. Can anyone help please. The problem is to solve for x in the below diagram (excuse the bad drawing in Paint):


Generally speaking, the universe is electrically neutral and the universe abhors an unbalanced charge. Wherever there is a positively charged object, you can bet there is a negatively charged object not far away. However, no laws of physics prevent the accumulation of positive charge, say, so long as the negative charge is sent somewhere else. This must be true on a cosmic scale as well. Hence my question: What is the most electrically charged celestial body in the universe, and what is its charge, i.e. positive or negative? Dipoles don't count.


From many papers I've read in condensed matter physics, it seems "breaking time reversal symmetry" is often equivalent to "applying an external B field", while "breaking inversion symmetry" can be done (but not limited to) by "applying an electric field". Is this true in general? A side question: if a perturbation breaks the inversion symmetry of the original Hamiltonian, does that imply it mixes states of opposite parity? Similarly, if a perturbation breaks time-reversal symmetry, does that mixes time-reversal pairs? Looking forward to any counter-examples. Thanks!


At first glance, it seems like the "many" in "There are as many A's as B's" is an adjective. However, we can't substitute an arbitrary adjective in this sentence. Compare: "There are as many A's as B's." (correct) "The A's are as many as the B's." (incorrect) "There are as numerous A's as B's." (incorrect) "The A's are as numerous as the B's." (correct) I'm curious about how to describe the reason why two of these feel 'correct' (to me, as a native speaker) and two of them do not. Is 'many' here a different part of speech than an adjective? Or is "as many as" just its own grammatical construction with its own rules?


David Cameron, the previous Prime Minister that has been away from politics for many years, has just come back as the Foreign Secretary. The BBC refers to him as the "old new face". Politics aside, just as an English grammar question, I wonder why it's "old new face" instead of "new old face": is there some convention/rule to determine the order of the two adjectives? Or is it just slang? I did some searching but was overwhelmed by face recognition AI links...


I have an anodizing rack that has hundreds of prongs. These prongs are made of titanium and need to be squeezed together in order to place the parts on the rack. I am wondering how I would calculate the force needed to make the prongs touch so that parts can be added? I have seen some people saying to use F=kx treating the prongs as a spring, would this be the best way? Any help is appreciated.


I was at a jewelry workshop today and saw somebody place a gold ring under an intense flame. Weirdly, the ring looked normal when underneath the flame, but glowed "red-hot" as soon as the flame was removed. The latter behavior is what I expect of a hot object emitting blackbody radiation. It makes no sense to me why the ring would not look like a blackbody object in the presence of the flame (where it could only be hotter). Maybe ions in the flame are affecting the physics at the surface and making it effectively opaque? I didn't capture the effect on camera, but this video sort of shows the same thing happening, though not nearly as clearly as what I saw in person.


I'm trying to construct a table that looks like the following: I know I need to use multirow package in the tabular environment, but I can't figure out how since I couldn't find examples that are similar to my need. It seems like I need to use a multicolumn environment inside multirow for the 'Experiment' cells. Can someone guide me in the right direction? Thank you!


As I read on Wikipedia, a phosphor coating of a CRT screen lights up because an electron in one of the phosphor atoms emits light when it goes down from its excited state. When a single electron hits a phosphor coating, is it the case that exactly one atom of the screen will glow? If not, what is the shape of the spot on the screen for a single electron?


If we create a pair of rotary wings like those on a helicopter, and spin them in a vacuum, would it be possible for a virtual particle and anti particle pair to spawn, be split by an approaching wing, and take opposing paths around the wing, before re-combining and annihilating? Would this create a pressure differential as it does on macro scales in fluid dynamics? Does the particle and anti particle pair even need to be split or is it sufficient for multiple such pairs to spawn and move around the wing before annihilation? If we rotate the wings at relativistic speeds, does this increase the rate at which this occurs and the pressure difference generated?


Given that our best models so far suggest that the universe has been expanding a lot ever since the time the CMB was emitted, I'm wondering: How large was the (visible) universe back at the time it became opaque and emitted the CMB? Specifically, I'm curious as to the ("original") circumference of the spherical shell we "see" out there, in comparison to how far the light from it has traveled to reach us.


I was trying to figure out some details about a specific decay channel for a generic particle accelerator experiment. For instance if I have a proton proton collider with fixed energy, I would like to figure out what is the average decay length of a B meson, what would be the energy of its decay products (for a chosen channel) and similar things. I know there is a lot of MC software used in high energy physics but I was just wondering if someone had a recommendation for one which is free and is not hard to use and configure when doing such simple things.


In his classic poem which pretty much everyone knows by heart, Edgar Allan Poe writes as follows: Ah, distinctly I remember It was in the bleak December, And each separate dying ember Wrought its ghost upon the floor. Why? "... a bleak December" would have sounded (or would it?) more logical: not all Decembers are bleak. Or: "This occurred in bleak December" would have scanned just as well. What gives?


I have recently moved to an institution run by the British government where staff use the term "indent" as a noun to refer to an inventory exercise done to work out what they need to buy. This is consistent with several dictionaries, such as Merriam Webster: chiefly British a : an official requisition b : a purchase order for goods especially when sent from a foreign country Does anybody know where this usage comes from?


I was reading a book, the author described the scenery of a certain place, and then said: "You're hooked for life." The search engine told me it means, you have been fascinated by this scenery all your life, "for life" represents the duration of the action of fascination. My questions are: Can this sentence only be used after the previous sentence describes the charm of something? If I want to express people's obsession with life itself, can I use "hooked on life"? Your reply will be greatly appreciated!


I've recently read about a subject field called "Boolean differential calculus", which discusses changes of Boolean variables and functions. This subject defines the derivative of a Boolean function, its minimum/maximum, etc. While the field sounds very interesting, I couldn't find much about its applications. I found a book by Steinbach and Posthoff called "Boolean Differential Calculus" which describes applications, but it's under a paywall. For any of you familiar with the subject, what are some interesting applications that can be done with it? I would also be interested in sources that are open-access. Thank you in advance!


I don't have any physics background aside from intro physics so apologies if my question sounds very shallow. If lens power is based on focal length, but our eyes adjust its focal length based on the distance of the object we are observing, how do we know that our lens prescription is correct for all distances? Most eye tests I have done involve focusing on objects at a far distance. If our lens prescription is based on focal length when we focus on objects at a far distance, is it incorrect to be using this prescription when doing near distance work?


I'm curious, if based on what we know with Newton's law, can we determine if a random planet, knowing it's mass and gravitational pull, can hold a moon in it's orbit. Or to phrase it another way, is there a gravitational force "range" in universe where if a planet has a force less than that range, a moon won't stay in orbit and just escapes and if the planet has a force more than that range, the moon is pulled into the planet.


Let's say we have an undirected graph G, where V = number of nodes and E = number of edges. In this graph, there are vertices with odd degrees as well as even degrees, but I have to transform the graph to one with only even degree nodes and I can do that by removing some edges. How do we find the number of ways to remove the edges so that there are no nodes with odd degrees left (this could also mean removing all edges altogether)? Thanks a lot!


The elongation of light waves occurs not because galaxies are moving away from each other, but because light waves are stretching along with the fabric of expanding space (cosmological redshift). Although the Milky Way is held together by gravity, space expansion occurs everywhere, including within the Milky Way. Since space expansion occurs in the Milky Way and light stretches due to space expansion, why don't we observe cosmological redshift within the Milky Way?


In the derivation of potential energy of two-point charge system, we define the potential energy of the system as the work done in giving both charges to their present configuration, and while doing the derivation, we assume the work done in bringing the first charge to its desired configuration to be zero, but I think it shouldn't be like that. Why is the force = qE acting on the first charge due to the second charge being ignored while calculating the work done in bringing the first charge to the desired configuration?


How can I learn General Relativity? Am a undergrad student, who happens to be interested in general relativity. I want to learn it by heart. What are the prerequisites for it? As a fresher who just got into undergrad, what resources are best for me? Feel free to suggest any YouTube playlist which can help me understand the math of relativity and any book that teaches me from scratch presuming I know nothing.


I'm doing 'longitudinal' qualitative research, and to avoid continually being accused of assuming that I will find change, I need a term that covers both change and stasis - so I can say that I'm looking for instances of [change-or-stasis] in particular areas of experience over a particular period of time. One researcher gets round this by saying that stasis is a type of change, but I'm not sure that works. Any thoughts gratefully received!


I have been curious about intuitionistic logic for some time and I want to know about it and I have a question, the law of the excluded middle and double negation elimination seem completely logical to me and I think they are always correct. But intuitionistic logic does not accept these two laws. What is the cause of this? Are there really propositions in mathematics for which these two principles do not hold? If so, please give examples of propositions that are examined in intuitionistic logic and the law of the excluded middle and double negation elimination does not work in them.


I have been trying to get my head around this proof, but i just don't manage to do so. I am ok with the first two lines, it is just the definition. But on the next line, how is Cauchy-Schwartz inequality exaclty applied? And on the next line, I also don't understand how is Taylor's Theorem used. Update: I just managed to figure out the usage of Cauchy-Schwartz. But I still don't understand how Taylor's Theorem was used.


The Nyquist limit for interpolation by trig functions states that one must give at least two data points per wavelength, because data just above and just below this "folding frequency" cannot be distinguished. I am currently playing with Hermite interpolation, providing the first few derivatives as well as the function values, and find that the Nyquist limit no longer applies. This cannot be a new observation, but I don't find anything about it online. Maybe I have not got the appropriate vocabulary?


I've been reading a lot of various classic literature, and at times there is the sort of casual misogyny or racism that was commonplace and (within certain cultures) the social norm at that time. Such things are no longer so commonplace and generally aren't tolerated. I wish to have a concise adjective or noun to describe those things when discussing them that isn't so oblique and timid as "outmoded" or "archaic", but I cannot recall one nor find it with my (perhaps feeble) attempts to search for it online. Has anyone come across such a word? If not, I'm open to a suggested synthesized word or a potential loanword that might one day make it into a dictionary if we use it enough.


I wonder how can I constuct a sphere inscribed in a tetrahedron using GeoGebra? I've thought I can construct it using the intersection of three planes bisecting angles between some pairs of faces in a tetrahedron, but when I contructed it, it didn't seem to work. Is this construction valid and I made some mistake while constructing the planes, or is it not a proper construction? If it is the latter, what is the proper way?


If amber and cloth are rubbed together, it is known that some electrons are transfered from cloth to amber. In this case, cloth acquires a positive charge and amber a negative charge. If these two have opposite charge, shouldn't there be electrostatic force between these two ? So, the next time you rub these two items, shouldn't the electrons be transferred from amber to cloth?


A solid sphere rolls without slipping up and then down an incline. Why does the static friction force point up the incline both when the object is rolling up and also when it's rolling down? I know it's necessary for the force to point up when the object is rolling up, because then it's the only source of torque on the object, and for the object to roll back down the incline, there must be a torque which acts in the opposite direction of the initial angular velocity. However, I don't understand why this is the case. Why does the force need to point up the incline to create the necessary torque, particularly when the object is rolling back down the ramp?


"Mill" is usually used to describe the process of making flour (grinding), to describe a wind or water mill, or a factory. Since the process of milling or grinding consists of making small pieces or powder out of objects, I was wondering if the word "mill" can be used figuratively as in kill, destroy, obliterate or tear to pieces? For instance, in the context of a competitive computer game, would it make sense to say something like "I was milled" as in "I was obliterated" when losing a game as a kind of way to add a bit of color to the expression?


According to the S-B Law, an area of mass will radiate out heat energy equal to the fourth power of its temperature in degrees Kelvin. If energy cannot be created nor destroyed, how can a body give off more heat than it receives? I understand how a magician seems to make a pigeon appear from nowhere, but I can't understand how so much more heat is being radiated out than was absorbed by the mass. How is this possible?


I'm going to take an introductory class on dynamical systems at the level of Guckenheimer & Holmes, or Arnold's ODEs book, but I don't have any topology background. I know analysis at the level of Stein's Fourier and Complex analysis, but have not had exposure to full-blown measure theory or manifolds. Is there a book that covers basic point-set topology in an accessible manner but perhaps without unneeded generality? Thanks!


I'm reading through the paper "Lagrangian Intersection and the Cauchy Problem" by Melrose and Uhlmann, and I'm having trouble with the definition of intersecting pair of Lagrangian manifolds. I am aware of the definition of conic Lagrangian manifold, but what is a conic Lagrangian manifold with boundary? Do we require the boundary to be isotropic or is it that if a conic Lagrangian submanifold has boundary then the boundary is always isotropic, or do we just define it to be such that the interior is a conic Lagrangian submanifold?


I have a circle and a line with the length of its circumference. One end of the line is attached to a point on the circle circumference, forming the shape of the lowercase 'b'. Now, the line begins to wrap the circle clockwise, until it completes one revolution around the circle, where there's no more "line". What is the path created by the free end of the line?


I've always thought of a transformation as "the movement of a mathematical object through space", but I found this definition that seems to be better: "An operation that changes the status of a (mathematical) object to another state." Does anyone have a better definition that works for both dilations and isometries and also with more advanced math like linear transformations? Or, what would be the best way to combine these definitions? Is the "space" part important?


The proof is quite obvious but how do you write it down nicely? I have this problem a lot with graph theory proofs where I don't know how to write them down so that they are rigorous enough. The proof itself would go something like this: Let C be a cycle in G.If we remove the edge e = {u, v} from C then C becomes a path between u and v. C is still connected so G is also still connected.


I am looking for a word that describes advice that is too obvious to be helpful. Imagine asking for advice, and someone responds with simply saying to "do what is desirable" or "avoid what is undesirable" in the context of the advice. For example, if you asked someone for exam prep advice, and they simply said "don't fail", or if you asked for advice in marathon preparation and someone said "don't get tired". One might use the adjective as follows: "Please spare your ____ advice"


I'm not a physicist so I'm asking you guys. We have been told that a device ("medical"...) uses a laser to irradiate a biological body (human, animal,...). The laser should apparently penetrates through metal and concrete (without destroying it). Can a laser penetrate through metal, concrete, ... without destroying the material? I don't believe that. After all, laser is light and light does not penetrate through steel or concrete. Except it's super thin, so the density would not be enough. Am I right with that? Thanks!


I've only started my own studies on grammar recently and as I was reading CoGEL by Randolph Quirk, I came across this category called 'Catenative verb constructions'. According to the analysis used in the book, catenative verb constructions cannot be interpreted as verbs followed by a nonfinite clause acting as a direct or prepositional object. However, 'manage to' is also included yet when I look up the dictionary, 'manage' can be transitive even when used with the meaning 'to succeed in doing something, especially something difficult' like in these examples: In spite of his disappointment, he managed a weak smile. I don't know exactly how we'll manage it, but we will, somehow. Shouldn't 'manage to' be listed elsewhere then ? Or am I misinterpreting something ?


I am trying to use the Poisson process for BLE(Blueetooth Low Energy) neighbor discovery. In this context, there are two devices called Advertiser and Scanner. The advertiser and scanner emit and listen to beacons in a duty-cycled fashion. This leads to longer discovery latency sometimes and I want to use the poisson process for the scanning and advertising patterns and prove the discovery is going to happen. Basically I want to merge two poisson processes and make sure there are two events within a small interval delta.


I recently used a shear interferometer for aligning two lenses in an optical setup (a Galilean beam expander to be precise). I was able to align the lenses such that the vertical stripes were perfectly vertical (i.e. the beam is collimated). But I noticed that the image showed slight fanning: With fanning I mean that the stripes seem to be expanding when going to the right. Does anyone know which type of aberration causes this? I'm using this shear interferometer:


Is it possible to find the highest "noise" value along a line segment defined by two endpoints within a Simplex (or Perlin) function by using an equation? The reason I ask is to avoid expensive ray marching. It's fine if there are multiple points along the segment with the maximum value. It would be preferred if it could work with multiple octaves of Fractional Brownian Motion as well. I'm fairly certain it's possible but it might be too difficult to derive. Corresponding code for the simplex (snoise) function can be found here. I've added a rudimentary image to show what I'm trying to do here :P


I have a background in acoustics and do more and more of quantum mechanics. I wonder why don't we use more often the bra-ket notation in acoustics and vibrations like in quantum mechanics? In modal analysis, we deal everyday with Hilbert spaces, inner products and linear algebra so I think that it would make sens to use the notation in the same manner. Is it because the notion of state vector is less important in those contexts?


Reading through wikipedia's page on international law, I noticed that there is never a definite article "the" before the phrase. It is a set of rules, norms, and standards generally recognised as binding between states, so it should be quite a definitive thing, right? So why is there no definite article? Compare: the constitution, the second amendment, the Bill of Rights, the Defense of Marriage Act, the Non-Proliferation Treaty.


Is there more than one definition to "polarization" in the context of waves? In some cases, I've seen the term "direction of polarization", which I presume it refers to the direction of oscillation. In other context, I see this word use as a process of turning unpolarized waves into polarized waves. Also, there are times whereby people say that "polarization only occurs in transverse waves", which implies that polarization is a property/phenomenon of transverse wave. So is polarization a property, a process, an adjective to describe the direction of oscillation, or is it all of the above? Is there any other definition of polarization I'm missing here in the context of waves?


I am studying about fibre bundles. If a fibre bundle is locally trivial then I understand the fibres are locally isomorphic. Also I understand that any compact Hausdorff base space cannot have a disjoint finite open cover for if I have a disjoint finite open cover then base space is disconnected. My question is, if my base space is connected, then are all the fibres isomorphic to each other?


I have been watching videos of WIMP's and have a simple question.. The larger the particles, the shorter their lifetime. A top quark is so massive that it cannot form a "stable" bond with any other particle to form a meson or baryon. If WIMPS are too massive to make in a particle accelerator, how can they still be around today to affect gravity of galaxies? Wouldn't they have decayed by now into things we could see or detect?


I have an enclosure mounted to the chassis of a vehicle. There are multiple points where the enclosure is fastened to the chassis as well as areas where it rests on the chassis without being fastened. Is there any point in putting vibration pads on the areas where the enclosure rests on the chassis? Logically I feel like anything fastened rigidly to the chassis would vibrate the same as the chassis.


Could anyone help me to understand the difference between the usage for and to in the below sentences. They will be more interested in the struggles of the downtrodden, in those who are fighting to obtain the title deeds for respectable humanity. They will be more interested in the struggles of the downtrodden, in those who are fighting to obtain the title deeds to respectable humanity. Ideally, both the sentences are correct, however the preposition for is mostly followed by a noun whereas the preposition To is mostly followed by verbs. So, why is "To" correct in the second sentence in comparison with the first sentence?


On plotting the angle of deviation vs the angle of incidence for a prism, we find that the graph dips only at one point. It is this result that is later used to prove that for minimum deviation to occur in a prism, the angle of incidence must be equal to the angle of emergence. But why exactly does the graph have only one minima? Are there any mathematical or physical reasons behind this phenomenon, or is it a purely empirical result?


In his famous hit Working Class Man, Jimmy Barnes sings: He believes in God and Elvis He gets out when he can He did his time in Vietnam Still mad at Uncle Sam I can't make sense of the second line. What could it mean? What does he get out of? I thought that the line in question might be some idiom or slang known among workers, so I looked in dictionaries, but found no idiom or slang that would fit the context. Maybe it's something obvious to native speakers, but I am not one. I humbly hope that users of this SE can shed light on this mystery.


I am a beginner to Lie groups and Lie algebra. Recently, I learnt that the dimension of the Lie algebra generated by a Lie group is the same as that of the Lie group. I have known for a long time that the space of vector fields on a manifold (even when not considered a Lie group), forms a Lie algebra. Is there any conclusion that can be made about the dimension of this Lie algebra? It seems to me that the Lie algebra generated a Lie Group (space of left-invariant vector fields) is the subalgebra of the Lie algebra generated by the underlying manifold (space of all vector fields), so their dimensions should not be the same. Any enlightenment is greatly appreciated!


While scrolling through an online document, I came across the statement, "If charges carried by the black hole are large, the curvature at the horizon is small." How is this so? I tried to think of an answer to this. My thinking goes this way: I know curvature is related to mass. If an object is more massive, it bends the spacetime fabric more, producing large curvature. So curvature small means mass small. So how does a large charge mean a small mass for a black hole? What is the relationship between the charge and mass of a black hole?


Is there, or has there ever been, an unambiguous word for "the leftmost extreme" or "the rightmost extreme"? Such words exist for the other two dimensions. Imagine you are floating in a cube in space. You can go up until you hit the top. You can go down until you hit the bottom. You can go forward until you hit the front. You can go backward until you hit the (back/rear). You can go left until you hit the _______? You can go right until you hit the _______?


In an acute-angled triangle ABC, let us denote H as its orthocenter and I as its incenter. Let D be the perpendicular projection of I on the line BC, and E be the image of point A in symmetry with center I. Furthermore, F is the perpendicular projection of the point H on the line ED. Prove that the points B, H, F, C lie on a circle. I think the idea here is the circumcircle BHC is the reflection of circumcircle ABC respect to BC. But I don't know how to proceed, can anybody help?


So, the uncertainty principle states that one can not measure momentum and position with accuracy simultaneously. However, we know from relativity that simultaneously is something frame dependent in nature, so how can we reconcile these two in relativistic quantum mechanics or even QFT? I know that in QFT we define observables in different points in spacetime such that their Lie bracket is zero, but I'm not really sure how to go from this to an answer.


I noticed that whenever I bunch up aluminum foil (into a ball), it becomes extremely hard to compress. If I use another piece of the same amount of aluminum foil, and keep folding it in, I arrive at a much smaller volume. It is full of air and much lighter than the same shape of solid aluminum, though, it is still extremely hard to compress. Question: Why does bunched up aluminum foil become so extremely hard to compress?


I am a graduate student in mathematics studying algebraic logic. For quite some time now, I have been meaning to study physics -- it turns out that some ideas in foundations of mathematics have found promising applications in theoretical physics (topos quantum theory and quantum logic come to mind). I want to get a somewhat rigorous picture of classical and quantum mechanics, so that I can contextualize some of the things I will be reading later on. Is there a good resource that I can use for this? I would prefer to read something that has a careful development of ideas starting with classical mechanics, eventually culminating in a broad formulation of quantum mechanics. All suggestions are welcome!


I came across some interesting dialogue in a tense scene in a novel, Salvation Lost by Peter F Hamilton: "We'll know exactly what the other [people] are seeing and doing." "Doing and seeing," "What?" "Better grammar." "Jesus, man; keep it relevant" Though this is intended as comic relief, it has me wondering if there is any reason why "doing and seeing" is preferable to the reverse. Is there any actual guideline/grammar or does it just sound better to have the more active verb first?


We recently attended a concert in a protestant church in England that was advertised as "free admission with retiring donations". The concert was indeed free with a voluntary (optional) donation. What is the meaning of retiring in this context? Googling without quotes gives results for people retiring from work, which is obviously not meant here, and googling with quotes returned similar events being advertised as such.


A friend has pointed out that in an Oxford comma separated list, either every clause needs a verb, or each clause must be the object of the first verb. This seems to make sense to me, but would this mean that this sentence is a comma splice? "Next week, I will go to the gym, beach, and get a haircut." The above sentence might be ambiguous, but are there any strict grammatical rules that this is violating? Does each clause have to have a verb, like so: "Next week, I will go to the gym, go to the beach, and get a haircut?"


Everyone knows this experiment: You mix salt and pepper and use a charged balloon to separate the pepper from the salt. I never really understood how this works. In school (long time ago) we learned that unlike charges attract each other while like charges distract each other. In the experiment the negatively charged balloon attracts the pepper. Does this mean the pepper has a positive charge? Why is pepper charged? How can I predict for given particle if the balloon will attract, distract or not interact with it?


I have often read that our universe was once small enough to be subjected to quantum mechanical effects, potentially altering how our universe turned out. This is a large theme in Laura Mersini-Houghton's Before the Big Bang. Michio Kaku also mentions this in his book Parallel Worlds. What exactly are they referring to? The initial singularity? The observable universe shortly after the big bang? The inflaton particle that inflated space giving rise to our universe? It is my understanding that the universe might have always been spatially infinite after emerging from the initial singularity so I'm not quite understanding what aspect of the universe was subject to QM.


I have started to study formal languages, especially finite automata and regular languages and I encountered some difficulties, i.e. Is this true: Automata will be called isomorphic if, by changing the names of the states, one moves from one automaton to the second. Let be a regular language. How can I Prove that all minimal automata for the language are isomorphic? Thank you for any hints and answers!


I am dealing with a PDE and I am trying to characterize the solutions: Galois differential theory might be of help, but I can only find books about this theory applied to ODE. Are there resources, books or papers, on the application of Galois differential theory to partial differential equations? I know the basis of the Galois differential theory applied to ODE as explained in Magid's Lectures on differential Galois Theory and Galois theory of linear differential equations by van der Put and Singer, and I am looking for something along the same lines, but maybe with more examples and at least at the graduate level.


If I have two identical magnets of equal strengths producing equal repulsive forces between them, and if I replace one of the magnets with a stronger magnet (everything else remaining the same), how would the forces on both of them change, i.e. will they exert equal repulsive force on each other or not? Why? Also, does the size or strength of the magnet affect this conclusion? In other words, replacing one of the magnets with a significantly larger or stronger magnet will produce uneven/unequal forces on both the magnets?


Lately, I observed a rainbow-like ring forming in my window during nighttime when the light enters from a street lamp. Interestingly, the sequence of colors in the rainbow seems to be reversed from what I anticipated. Normally, I would expect the inner part to be red and the outer part to be blue due to the greater refraction of blue light (like a sun halo). What could be the reason behind this color reversal phenomenon? Edit: To clarify, the window has a decorative film on it which causes the rainbow. The core question is still the same: Why would blue be on the inside and red on the outside?


I am trying to understand the logic of negative exponents and I stumbled upon the video "Negative Exponents" by YouTuber JLC. He says negative exponents are "defined". But why is it defined the way it is? Is there some kind of international agreement or some kind of international organization that maintains this definition of negative exponents? What if I make my own definition of negative exponents? Please enlighten me, many thanks.


I am trying to figure out what this phrase deconstructs to. Does it imply a logical OR, like: "attention-deficit disorder" OR "hyperactivity disorder"? For additional context, there are different types mentioned here: There are three main types of ADHD: Predominantly inattentive presentation. Predominantly hyperactive/impulsive presentation. Combined presentation. ...which seems to indicate that it is always both, but a little more of one than the other. If this is the case, does the grammar check out? Also when it is spoken as: "attention deficit hyperactivity disorder", or abbreviated as "ADHD", the slash is dropped, which loses meaning and adds confusion. Would there be a better way to phrase it? Would there be a better way to say it?


Does anyone know a simple and accurate way to measure the size of an iron core using a magnetic sensor? Imagine having a collection of identical iron plates. I expect the strength of the magnetic field around the core to increase with its size. Would it be possible to determine the number of stacked plates by measuring the magnetic field strength using e.g. a Hall effect sensor placed on top of the staple? Are there better sensors/methods?


This sentence is an excerpt from The Economist website. The Kremlin denied the advance, saying that Russian troops had repelled Ukrainian attacks in the region. What is the grammatical rule that's governing the "saying that Russian troops had repelled Ukrainian attacks in the region" in the sentence above? I've been looking into the types of clauses to find any applications that match this, but I think I'm lost.


I am wondering if anyone happens to know the computational methods - and hence the computational complexity - of simulating the distribution of an electric charge (how electrons distribute) over a surface. In particular, I am curious about the computational difficulty in simulating the distribution of electrons over a highly asymmetric and non-uniform surface (like the surface of a rough rock). How does the simulation complexity scale with the complexity of the surface and the precision/accuracy of measurement?


I am trying to write a proof regarding the following question and am stumped on this particular version of the problem. I believe that I have written a correct proof when asked if the columns of A are linearly dependent. That is solved by showing that the non-zero vector y can be used as a nontrivial solution of linear dependence. But I don't really know how this information translates when analyzing the rows of A instead. I tried to look for a counterexample that IS linearly independent but cant find one.


I've recently read that what most people learned to think of as the 'speed of light' is actually the 'speed of causality', and that light just happens to travel at that speed (through free-space.) I'm also aware that light travels measurably slower through transparent materials such as water or glass. This leads me to ask: is light traveling slower through such materials, because the speed of causality is reduced in such materials? And if so, what does this tell us about how the speed of causality is determined by permittivity and permeability of various materials?


I'm trying to reconcile two seemingly conflicting sources of information about Lorenz force. Let's consider a solenoid in proximity to a permanent magnet with the field pointing in the same direction as the axis of the solenoid. From the right hand rule I conclude that the wire should either be pushed towards the center of the solenoid or outside. However, the solenoid also acts as a magnet with poles on each sides and will attract/repel other magnets. How does one reconcile the two explanations ? Is the solenoid both contracting on itself and moving towards the other magnet at the same time ? If yes which force/equations describe the movement of the electromagnet towards a permanent magnet. Thank you!


I am aware there are similar questions already asked, however, I find none of the answers satisfactory, they either do not contain any mathematics at all, or mathematics of a level I am not capable of comprehending. My doubt is that, if the change in kinetic energy is dependent on frame, then how is the law of conservation of energy valid in all frames? One possible explanation I have thought of is that the change of potential energy is also frame dependent, but I cannot think of a suitable example to justify the same. Please answer only using newtonian mechanics I do not know statistical or lagrangian mechanics.


I have recently started to study integrals and faced some difficulties of understanding why we need antiderivatives to calculate area under curve. I am trying to visualize it using graphs but it just doesn't make sense to me. If anyone can explain me if it's even possible to understand them the way I try. here is the image: enter image description here Thank you :)


I was curious if anyone knows the font used by the Desmos graphing calculator. All the formulas are formatted as LaTeX, so I wanted to know if there was a way to use this font in my own documents. It seems at first glance like standard Computer Modern, but upon closer inspection as you can see the "w" and "v" characters are rendered in a style that makes them pointier so that they are much easier to distinguish from omega and u respectively (and imo they just look a bit more pleasant). Here is a side-by-side comparison for context. Computer Modern: Desmos Font: (This is my first time making a post on the site, so apologies if I have made any mistakes!)