Google dictionary (based on Oxford dictionary) has an entry for the verb "reveal" as follows: make (something) known to humans by divine or supernatural means. And it includes one example: "the truth revealed at the Incarnation" Based on the definition, I find it hard to understand the role of "the truth" in this example whether it's an object (as denoted by "something" in the definition) or a subject (that is supposed to "make(something) in the def.)? My best guess is that it's an object because after checking other dictionaries such as Collins, Merriam Webster, Cambridge, etc., I found that they all define "reveal" to be a transitive verb. So I guess that the "to be" may be omitted in this case and the full sentence should be: "the truth IS revealed at the Incarnation". However, no resources I found have mentioned such type of ellipsis like this one. So my question is, is my guess correct. If it's indeed correct could you give me some resource covering such type of ellipsis?


If I have a stick that has a spinning disc on the end of it, and I try to rotate the stick, I will feel more inertia the faster the disc is spinning because I have to transform its rotational energy into a different axis. It seems intuitive to me that this would apply to electromagnetic fields as well, since you have to move all the energy in the field to another axis. So if you had an MRI machine on an airplane and the plane tried to rotate, would it rotate more slowly when the machine was powered vs unpowered? If so, does this depend on whether the axis of the field is aligned with the axis of rotation? If they were aligned so that the dipoles are parallel to the body of the plane, could it roll without extra inertia? I may be describing the Einstein-de Haas effect or gyromagnetism, but I am confused whether they are describing the same thing as me in different words or just a related concept.


Often we define a quantum phase of matter as a set of ground states of a gapped family of Hamiltonians, where the gap does not close anywhere in the family. Equivalently, we can define it as a set of states which are related by constant depth quantum circuits with local gates (equivalence of these two definitions is shown by the adiabatic theorem). Now consider a family of states which are ground states of a gapless family of Hamiltonians, and assume they have similar correlations and can be considered a "phase of matter" in the sense that all their observables vary analytically with the parameters of the Hamiltonian. Can we use the circuit definition of a phase of matter in the same way as the gapped case? That is, is it reasonable to define a "gapless phase of matter" in terms of states which are related to each other by a constant depth-quantum circuit? Maybe there are counter-examples to this which demonstrate this is not true. In particular, it seems reasonable to define gapless phases of matter in this way as (a) local unitaries do not change long-ranged entanglement properties, which intuitively define phase (b) unlike the Hamiltonian case, we do not need to worry about excitations to higher energy levels. Is there any literature on this at all?


This question might be very silly, but I am really confused about it from several days. Look transverse waves on a string propagate along the string due to the electromagnetic (EM) forces between adjacent particles of the string. So when one of the particles is displaced, it will drag nearby particles and thus the disturbance travels. In case of electromagnetic waves, the electric field values oscillate with time. But is there anything physical that connects the electric field at a point to that at its nearby points (just like we had for strings)? If not, then why does this information travel? And if yes, then what is that "thing"? Also, I would like to add that for strings the direction of displacement at nearby points was completely determined by the source point, but for EM waves the direction of electric field at one point does not necessarily imply that the electric field at nearby points will be in the same direction (take the case of unpolarised light). Why is that?


I am working on automated templates, and I want to use latexmake with luaLaTeX, as suggested in a comment on this question, where I attempted a similar setup with XeLaTeX. However, I am facing challenges with the configuration. My goal is to automate the process of typesetting my documents with luaLaTeX using latexmake, and I believe this will save me a significant amount of time. I have tried variations of latexmake main.tex -lualatex -outdir=output -auxdir=auxil in the terminal but it tells me that lualatex is not recognized as an argument For the code examples, I have a template that works and needs lualatex but needs several compilations. Hence the need for a latex maker. I also have tried to read the documentation for latexmake but I didn't find any solution and there isn't much content about this library online. If someone could provide guidance on how to configure latexmake to work with luaLaTeX, it would be greatly appreciated. Thank you!


For context, I watched PBS Spacetime's video on virtual particles (link goes to relevant timestamp) where they say that virtual particles aren't mathematically necessary, because the lattice version of nonperturbative QFT doesn't use them, and yet still makes all the same predictions as perturbative QFT. I was satisfied with that, until I had a brief exchange with someone in the comments of this answer where he says that, in most cases, it's impossible to actually do computations in nonperturbative QFT, and, when I asked if this was just due to not having sufficiently efficient algorithms, he said Note that in particular that even establishing the existence of a non-perturbative Yang-Mills QFT (which is what the standard model / QCD / QFD are) is a millennium problem. which implies that we don't currently even have a nonperturbative version of the standard model and that it's unclear whether one exists. However, PBS Spacetime is, in my experience, typically a reliable source for high-level explanations, so I wouldn't have expected them to mention nonperturbative QFT as the reason not to think virtual particles are physical if that theory wasn't actually useful for nontrivial calculation. Is it that most physicists think there probably is a nonperturbative version of the standard model and it just hasn't been discovered/created yet?


Let S be a compact orientable surface and U an open connected subset of S with finitely many ideal boundary points (or ends). U has a prime ends compactification which is a surface with boundary (following Mather). Let b be one of these ideal boundary points and Z(b) its impression in S. If Z(b) has more than one point, then there is a circle C(b) of prime ends associated to b. Each prime end e of C(b) has its impression Y(e) (another impression, the intersection of the closures of the sets of a chain that define e). This seem to be known by experts, but I could not find it written anywhere. Does anybody knows how to prove it? Knows a reference? I think I can prove it using scaffolds, a powerful tool created by Mather to prove various properties of prime ends. But it is a lot of work. Any good hint? Thanks.


In the situation above, we have a power line which uses an alternating current. This alternating current causes a change in magnetic flux through the loop below the power line, which induces a current according to Lenz's law/Faraday's law, which eventually goes to the farmer's equipment. This is classed as power theft. However, I was wondering, does this power tapping by the farmer in some way reduce the power delivered by the power line or cause an energy loss from the power line? If so how? If it doesn't, then how is this situation exactly consistent with conservation of energy? Does the induced magnetic field in the farmer's loop somehow cause energy to get lost in the power line? (Where does the energy from changing magnetic flux come from in the first place?)


I have a hard time understanding time dilation and special relativity; each explanation seems to contradict the other, and don't explain the apparent paradoxes they cause. Say clock A orbits clock B at a very high speed. According to the explanations I've heard, B would perceive A as ticking slower than B, because of time dilation. But since A has equal right to claim to be stationary, it should observe B as orbiting A, thus ticking slower than A. If at any moment the clocks' time would be measured, would their respective elapsed time be different? In other words, is one clock actually slower than the other? This seems unreasonable, since the respective situations of the clocks are identical. If the answer is no, what exactly happens if the clocks suddenly stopped orbiting each other and become stationary relative to each other? If each clock has perceived the other as ticking slower for a while, would the clocks instantly jump to the same time? This also seems unreasonable, since orbiting each other for a longer time then would imply a different outcome when stopping, even if the events of stopping are identical. How can this be resolved?


Usually when we introduce postulates it is so they are of some use to us. I do not see the reason for these two. The Line Separation Postulate: Each point on a given line divides the line into three disjoint sets: the set containing the point, and the two half lines on either side of the point. The Plane Separation Postulate: Each line within any plane divides the plane into three disjoint sets: The line and the two half-planes on either side of the line. I understand the postulates and what they are saying, but I do not see how they can be used. And I have not yet found a proof that uses them in the book. I guess I am not convinced of their fruitfulness so if anyone can help convince me these are useful/necessary postulates that would be of great help.


Suppose that, as a manager, I am creating a campign to evaluate employees based on the NPS (Net Promoter Score) of their sales - the objective is to reward high perfoming employees. However, due to the different nature of each role, some employees have a lot more sales than others. What is the best way to create a ranking system? The data available is: NPS for each sale for each employee (the value can be null, if the survey has not been answered by the customer). I thought about using a bayesian ranking system, but I think that it would be unfair to low seeling employees which sell little beacuse of their specific role. In a more abstract sense, how can I compare samples of varying sizes without skewing the comparison towards bigger or smaller samples?


I've noticed while playing Minesweeper that when I have too few bombs, I get very easy to play games. In other words, I get games that can be solved with very simple algorithms. When I play games with too many bombs, I get unsolvable games or games that can't be solved without guessing. There seems to be a region in the middle(not too few, not too many), where some games are solvable but require increasingly complex algorithms. I've also read that Minesweeper is NP-complete, which I take to mean that there is no upper bound to how complex these algorithms can be as n, in an n-by-n game of Minesweeper, is allowed to grow. This seems relevant to information processing(halting problem?) but not exactly sure how. First, is my phrasing correct? And second, where can I search for clarification in my thinking?


In an office, there are typically various different kinds of environments in which each employee carries out their work. Some have their own rooms, some have cubicles, some only have a desk in an open landscape etc. I'm looking for a word for the place where an individual employee carries out their work, regardless of if it's in a room, at a desk etc. Things I considered: Workstation - probably the best I've got so far, but it sounds kind of techy to me (associating with a workstation computer perhaps) - does this work also for someone not using a computer such as writing by hand (!), manning a reception etc? Workplace - to me this sounds like a common word for the entire office/building? Am I wrong? Furniture - kind of encompasses all kinds of surfaces you can work at, but it sounds too broad (and plain silly!) Equipment - to me this sounds more like advanced/specialized lab gear (labs, fume hoods, ...)? Does anyone have a good suggestion? Imagine e.g. this sentence: "All employees in the office are guaranteed their own [ ... ] where they can carry out their work". And I don't want to go too broad with abstract terms such as "location", "spot", etc.


There are Two different defintions for the event horizon of a black hole. The Absolute horizon and the Apparent horizon. An apparent horizon is a surface that is the boundary between light rays that are directed outwards and moving outwards and those directed outward but moving inward. While and Absolute Horiozn is a boundary defined with respect to the external universe, inside which events cannot affect an external observer at infinity. Normally these horizons coincide but they can differ during event such ad black hole mergers. Technically, knowing exactly where the Absolute horizon is requires knowing the entire future history of spacetime. So it seems to me therefore that in a universe destined to even in a big crunch, black holes wouldn't technically have Absolute horizons, only Apparent horizons.


I interpret the expression 'If P then Q' as asserting that if P is true Q is automatically true. So, we would say 'If P then Q' is true only when it indeed is the case that P being true implies Q is true. However, in logic, the truth of 'If P then Q' is determined solely on the basis of the truth values of P and Q individually and not by verifying whether Q follows from P, or is implied by P. So I just don't get how we can decide the truth of if-then statements by just looking at the truth values of P and Q. For it to be true don't we need to prove somehow that the truth of Q follows from the truth of P? In one of the logic books that I read, they explained conditional statements in this manner: 'If P then Q' asserts that it is not the case that P is true and Q is false. I liked this. It makes me understand the truth table of conditional statements well. However, by this explanation, I'm not able to see why would one use the words 'If then' then. How the idea that it is not the case that P is true and Q is false follows from the meaning of words 'if then' (or 'implies' for that matter). Should I completely forget about the meaning of if-then sentences as used in ordinary language and assign them a new meaning?


One possible formulation of the second law of thermodynamics is that the work extracted during the change of a thermodynamic system between two thermodynamic states is at most equal to the free energy difference between those two states. One possible derivation of this empirical result from statistical mechanics is the derivation of Jarzinski relation in the canonical ensemble. Also, the Jarzinski relation was also derived in a stochastic setting by Crooks. For the sake of further simplifying the statement of the Jarzinski relation, I consider a Brownian particle fluctuating within a double-well potential in one dimension, i.e subject a Langevin equation on the real axis with known potential and noise. What I can say for this system is that the particle will be undergoing transitions between the two wells with a rate that is exponential in the barrier height of those two wells. Also, the equilibrium distribution of the particle will be a bi-modal having a peak at each potential minimum, and the height of each peak is given by the barrier height. Now which additional useful information does the Jarzinsky relation tell me about this simple system ?


I'd like to prove: Given two points A and B and a straight line connecting them AB, any smooth curve passing through A and B will, at some point, have a derivative that exceeds the slope of AB. My proof uses the Mean Value Theorem. So, any curve passing through A and B has, at some point, a derivative equal to slope of AB, call this point C. Consider lines AC and BC. Together with AB, they form a triangle. Noting the angles a and b, we know that the slope of either AC or BC exceeds the slope of AB, and, by the Mean Value Theorem, there exists a derivative on any curve passing through either A and C or through B and C that equals the slopes of the lines AC and BC. Therefore, any smooth curve passing through A and B will, at some point, have a derivative that exceeds the slope of AB. Perhaps this is trivial, but for me it is important.


I'm trying to understand where heat energy goes in a substance. I've seen that mainly it's translation at lower temperatures, rotation at mid range, and vibration at high range, but I'm not sure of that. Someone made the claim that an atom can expand as it is heated, which I don't think is right, but in a way, this is a good description of black body radiation? Does anyone know some approximation of percentages for how much heat energy goes towards motion (translation, rotation, vibration) and how much is radiated as blackbody? And, does this change between the different phases: solid, liquid, gas, plasma? I'm just looking for a general concept here, so no constraints. If there's some special constraints to note, then please list them and the energy distribution within that particular system for comparison.


Recently I saw a video talking about the Yoneda lemma from category theory being used in neuroscience. It was my first introduction to category theory. In category theory we have objects and maps between the same objects called morphisms. For example we have one type of object called set and its morphism would be the function. A function is a map from one set to another. Another example would be the object vector field with its morphisms being linear transformations. This just looks like we are renaming the same stuff. Is a vector space not just a type of set we construct to obey our desired axioms? If so then introducing linear transforms would be truly just renaming stuff. And if they are not sets, why not allow non linear transforms?


My first name is "Jean-Baptiste". "Baptiste" is not a second or middle name, however I noticed that it's not unusual for native English speakers to address me just as "Jean". I don't mind it at all, I'm not offended and find it actually charming, but it would be quite unusual for a speaker of my native language (France's French) to call me like that (either native speakers address me with my full first name, or they use various nicknames - and the nickname is never "Jean"). So I'm curious as to why many native English speakers seem to spontaneously call me "Jean". The two hypotheses I have is that either my full first name might be a bit difficult to pronounce (so just saying "Jean" is easier), or because they mistakenly think that "Baptiste" is my middle name - as compound first names are unusual in English (correct me if I'm wrong). I'd tend to think it's the second option, as I noticed that it also happens in writing, but I'd be interested in having the insight of people who know English language and English-speaking cultures better than I do. Maybe I'm completely on the wrong track! When it happens, I usually don't have the opportunity to ask people why they are calling me like that (e.g. in a professional context we usually don't have the time for that, plus in this kind of situation I don't want to make people feel like they offended me), that's why I ask this question here. Thank you,


An event horizon appears in the Schwarzschild metric when considering a positive point mass in General Relativity. But for a negative point mass in the negative mass Schwarzschild metric, which repulses test particles, no matter how big the negative mass is, no repulsive event horizon forms. Intuitively, a big negative point mass should also create a repulsive event horizon, i.e., a region of spacetime that cannot be traversed from outside, due to respulsive gravity. If I make the negative mass big enough, I should create a region in which masses are so much gravitationally repelled, that they cannot enter (a repulsive event horizon). Why can't a repulsive event horizon form in General Relavity with negative mass? Is there any other metric which contains a repulsive event horizon with negative mass?


I am interested in calculating the power received by an object near a black body radiator. Say, for example, I had a piece of paper perpendicular to the earth's surface normal. If I make assumptions about the earth's temperature, I can calculate its black body radiation. If I had a range of wavelengths I was interested in, I could calculate the power in that spectral range. Now if I assume my paper is itself a blackbody in the spectral range of interest, then I believe I could say that if it were placed very, very close to the earth, then the power it absorbed would be the power emitted per unit area by the earth multiplied by the area of my paper. This is based on the image below. My question is, what if I start to move my paper up? Is there anything I can say about the power it receives at different ranges from the surface of the earth? (Assume we are not so far from the earth that it can be treated as a point source). What I would like to be able to say is that the paper receives that maximum possible power when it is very, very close to the earth's surface. I am unsure how to prove this or if it is even true. Any kind of bounding limit on the received power I could derive would be helpful


I would like to ask correspondence between Euclidean and non-Euclidean geometries. In the science and hypothesis, Poincare says that non-Euclidean geometry can be translated into Euclidean geometry with the following correspondence. ref)https://mathshistory.st-andrews.ac.uk/Extras/Poincare_non-Euclidean/ Space -> The portion of space situated above the fundamental plane. Plane -> Sphere cutting orthogonally the fundamental plane. Line -> Circle cutting orthogonally the fundamental plane. Sphere -> Sphere. Circle -> Circle. Angle -> Angle. Distance between two points -> Logarithm of the anharmonic ratio of these two points and of the intersection of the fundamental plane with the circle passing through these two points and cutting it orthogonally. Etc. Etc. Does he talk about Poincare's disk here? I cannot understand the meaning "Sphere cutting orthogonally the fundamental plane". If possible, can you please illustrate this? In addition, he says an theorem of Lobatschewsky's geometry: "The sum of the angles of a triangle is less than two right angles," can be translated thus: "If a curvilinear triangle has for its sides arcs of circles which if produced would cut orthogonally the fundamental plane, the sum of the angles of this curvilinear triangle will be less than two right angles." I would appreciate if you explain the above meaning more easily. Thank you for your help.


When it comes to topologies, my understanding is that we designate elements of that topology as 'open.' In the context of a metric space, a topology is formed by selecting open sets, as defined with respect to a given metric. Moreover, a normed vector space is equipped with an induced metric derived from its norm. This metric allows us to associate it with a metric space and, consequently, with a topology. In the realm of topology, a closed set is one for which the complement is open. In the context of metric spaces, it refers to the set containing all points to which sequences converge. Now, in regard to normed vector spaces, when discussing Banach spaces, it's often mentioned, as seen in Riesz's lemma, that a certain subspace is 'closed.' I wasn't familiar with a concrete definition of what a 'closed' vector space entails. To the best of my knowledge, this term has been used to describe the property of addition and scalar multiplication remaining closed operations. Furthermore, are the concepts of 'open' and 'closed' equivalent in these structures? Does dimension play a role? In the case of finite dimensions, all norms are equivalent, and open sets are the same regardless of the induced metrics.


My wife, a native Spanish speaker, today asked me about why a youtuber would call themselves 'craftypants'. I explained that -pants was added to something as synecdoche, so for example an intelligent person might be called a 'smarty-pants', a poor-humored person a 'grumpy-pants', etc. However, I then realized that while this is common in the US, in the UK 'pants' by itself usually means what this Yank would call 'underwear', while what I call 'pants' would appear over there as 'trousers'. How do such expressions as 'fancy-pants' sound to speakers in the UK? Are they understandable in meaning but recognizable as being North American? Does putting a -pants ending on a word remove the association with intimate clothing? Does it just sound bizarre? Do parallel expressions like grumpy-trousers exist? Are some combinations acceptable in polite speech, but others (e.g., pissy pants) not? Edit: There is much discussion in comments, and some in answers, about whether -pants is synecdoche. That's fair, and I am happy to have the point challenged. However, that's not my question. Even if I am wrong, my question is how "-pants" sounds to UK speakers, and whether it is clear or not that it is different from "pants" (underwear).


I've studied in my physics class, absorption and emission spectra for gases or more "spread out" molecules like what is done in atomic absorption spectroscopy (learnt that one in chemistry). As in what discrete wavelengths of light a singular atom absorbs and why it only absorbs those specific wavelengths because of what energy levels its electrons are available to jump to. I was wondering if this worked generally the same for solids and liquids? Like, when you have something painted black, is the black paint designed to be composed of molecules that absorb enough wavelengths of light to make it look black (it wont be entirely black of course), or is it more that when light hits the surface of the paint, it travels as far as it can through it before using up all its energy to cause electron (and to an extent proton) vibrations? Is it both? Sorry if this comes off as confusing, I've tried to convey my question(s) as best I can.


I have been reading into X-ray and gamma spectroscopy. I have found that they can both be done with scintillation detectors and work off similar principles. That is to say that when a sample is bombarded with high amounts of radiation, electrons will be excited, and electrons from other shells will fill the spot of the excited electrons. The ejected photons from this event then hit a scintillator, turning it into light (usually a NaI crystal), which is then turned into an electrical signal via a photomultiplier tube. It is then possible to ascertain the identity of the element using modern analysis tools. My question is: Is it possible for a scintillation probe meant for gamma spectrometry to be used to perform X-ray spectroscopy and identify the chemical composition of samples What would be different and have to be modified for such a setup to work with X-rays instead of radioactive isotopes?


Now this might be a very dumb question but this has been bothering me from some days. Imagine I want to create the real number line and for that I start with the rational numbers. So I start to put the numbers on the line (like toppings on a pizza). Now I know that between any two rational numbers I can add another rational number. So I keep on adding infinitely many rationals between any two rational numbers by adding more and more digits in the decimal expansion. I can keep on doing this forever but I don't think I will ever reach a point where I can't put a rational and only an irrational number can go there i.e. I don't think there will be any gap between two rationals which can be only fit in by an irrational (or is it ?) So where do irrationals actually fit in on the line ? I know this entire argument can be done the other way starting with irrationals too.


In accordance with special relativity, if we take into account length contraction then ideally the Planck length which is the smallest possible length possible should be frame dependent. Well we know that the number of particles are invariant in different frames. So, a frame travelling with some relative velocity will see the same number of particles but their length contracted by a factor. According to this logic, the Planck length should change with different frames. Yet it is defined by using the fundamental constants G,c and h which obviously do not change with frame. This could only imply two things :- One of the fundamental constants G,c,h are frame dependent (which I do not believe could be since they are fundamental constants) The shortest possible length cannot be written only in terms of fundamental constants The second conclusion seems more apt but is it right or is there something more to this?


I am a Canadian, but I study in Edinburgh, Scotland. I have discovered a peculiar feature of my speach that seems to surprise most people from here. When ill befalls others, I use the phrase "that's too bad" to express my honest heartfelt sympathy with the victim. I have been surprised to find that most people here interpret the phrase "that's too bad" more negatively, closer to "well- sucks to sucks" and such. I have posed this question to a few folks from both Canada and the UK, and the Canadians seem to "agree" with my usage, and the British folks seem to take it negatively. Maybe I just have a small sample size. My question is: is this a feature that others know about/have evidence for? Or is my sample size just small, and it's pure chance.


What is the simple relationship expressed verbally between flux, circulation, div, and curl, as captured by Green's, Stokes', and Gauss' Theorems? Below is what I've been able to assemble: Can you confirm or improve it? Given a vector field, we can measure its instantaneous rate of change in multiple ways, the two most important being div (a scalar) and curl (a vector). Each of these can be accumulated in a region. These theorems shows that this accumulation equals a property of the region's boundary: Flux (through a boundary) equals the accumulation of divergence (in the bounded region). Circulation (around a boundary) equals the accumulation of curl (in the bounded region). More fundamentally, neither flux nor circulation is limited to boundaries: they, or their equivalents, can be measured in non-closed curves as well. Only in that case, there is no relationship to any accumulation. That is: Flux can be measured across a simple curve or surface that is not closed. But this has no relationship to divergence. Likewise, a quantity equivalent to circulation can be measured through a simple curve or surface that is not closed, but if the curve isn't closed, it's called a line integral or work integral (or, for a surface, a surface integral). Neither has any relationship to curl.


I have been frequently stuck on this question since first realizing it in the shower. While watching the hot water drop from the above shower head, and then watching cold water drop, I realize they both appear to be moving at the same speed. This puzzles me, because I know that when an object is hot, its atoms are moving fast, or otherwise have energy that is propelling them to "move fast". Correspondingly, when an object is cold, its atoms are moving slowly, since they do not have as much energy to propel them to "move fast". I was wondering, if cold water appears to be moving at the exact same speed as hot water, then where does my misunderstanding come in? What law of physics and thermodynamics am I misunderstanding, or otherwise unaware of? Thanks for any answers!


The title might be a bit misleading, but basically I've been self studying differential equations so I could apply them in electronic circuit design. I was wondering what books could be recommended on differential equations, the Fourier, Laplace, and Z transforms, PDEs up to solving the wave equation, convolutions, filtering, and both discrete and continuous approaches to signal processing. One book that's caught my eye was Differential Equations: A Modern Approach with Wavelets by Steven Krantz, but I'm not sure how relevant wavelets are in modern signal processing. But regardless, if there are any books you've enjoyed a lot, please let me know! Also, if possible, I was wondering what a good book on information and coding theory would be, specifically with a focus on communications? I just want some physical reference books on these topics I'm learning by other means, since holding physical books is different from using an ebook. Thank you for your time!


I'm looking for an existing word that roughly means "the process or idea of replacing bad/negative elements/deeds with good/positive elements/deeds, such as when a negative thing is eliminated at the same time that a positive thing is added". In terms of words that would not fit this goal are words that could (to some people) mean only the elimination of bad or the addition of good without the other half. Here is an example sentence showing the usage: I live my life with _____ so that I'm not just removing bad habits, but I'm filling the hole created by removing the bad habits with the addition of good habits, which both maintains my ability to fight returning to the bad habit while also creating a positive element at the same time.


I am trying to do ray tracing for an endoscope, but I don't see how it could form a sharp image that could be viewed at the eyepiece. In the diagram the box in the middle is the "fiber", but the light from the object (the dot on the left) comes out of the fiber as if they originate from two locations (top and bottom), and if viewed through the eyepiece at the end, wouldn't two images form (the two dots on the right)? Also, if just looking at the image end of the fiber by eye directly without eyepiece, one should still see the image of the object, right? I don't see how it would work with ray tracing in this case either. And should the image in this case be inverted or upright, or left/right flipped, or does it depend?


I've been struggling with this one... I'm trying to figure out whether it's okay to use the expression "call of the blood" to describe the phenomenon of doing something naturally (or coming to like it naturally) because it's in your blood, because your ancestors did it/were skillful at it, etc. For example, my ancestors were always good at horse riding, and I should also be good at that, at least like that because it's in my blood- in English, could I call this the "call of the blood"? We do have this expression in Russian. The words themselves seem pretty self-explanatory, but since there are so many expressions involving the word "blood", I thought to ask the native speakers and English language specialists here. (What I found during my research online were the similar-sounding expressions "call out for blood", or "blood for blood", but theseare not what I need in terms of the meaning.) So, can I say "call of the blood"/"the blood call"?


It is said that most of what we call "mass" of nucleons are in fact from the kinetic and binding energies of quarks, and that the rest mass of quarks, from the higgs mechanism is much smaller compared to the nucleon. There are other examples, like the binding energy of a nucleus, that show how energy contributes to the mass of things in the subatomic world. My problem arises from the fact that it is also said that the gravitational effects are not really affected by how fast the object goes. Thus, a object won't get infinitely attracted to a planet even it goes at the speed near light. This seemed different from the previous examples, since in those cases the energy did affect the mass(in the classical way), which is basically what builds us up. What would be a appropriate way of understanding this discrepancy? Thank you for your time in advance.


I've been playing around with the concept of entropy and how it can be manipulated when I came onto the complex workings of quantum waves. To my understanding, everything we know as a particle is a wave on a quantum field. Those waves are the result of energy being introduced to, and then propagated through, the field---ei. The waves carry energy. Further, the wave's entire existence is defined by the energy it carries, no energy, no wave. This caused me to ask the question, is there anything in our reality that is fundamentally real? Is there anything that exists that is not just the extrapolation of the process of moving energy from a to b? To be clear, I think that, to a certain extent, the fabric of space-time counts, but it's not what I'm really looking for. Space-time is what our reality is built upon, but it does not constitute it. It is the physical stuff that is built upon and held within space-time that makes reality, and it is from that that I ask if there is anything that is fundamentally real.


I just watched a rather unreliable (or so I think) Physics YouTuber who asserted that absolute acceleration is impossible to measure, because it is impossible to ever calibrate an accelerometer accordingly, and I'm rather stumped by this argument. My understanding for all these years has been that acceleration can be absolute, and it can be measured easily with an accelerometer, but the YouTuber asserts that all accelerometers that are calibrated are done so in a frame that may or may not be accelerating, and to "correct" for this acceleration means having knowledge of an "absolute" inertial frame, which is defined using the accelerometer in the first place, thereby making the logic circular and invalid. Is he correct? Or is there a hole in his explanation? Or is this just an engineering problem, with a fairly easy fix in pure physics? What am I missing?


In a recent conversation with my professor, he explained to me a misconception I had, in that given how physicists says that the Higgs "gives" the mass of all other particles, I was under the impression that a particle's interaction with the Higgs field is what gives it the property of mass. However, my professor explained that the Higgs boson was simply the first term in a perturbative series where each term can be recursively defined as the one before it. So knowing the first term allows us to calculate all the other terms. The other terms were the masses of all other particles and the first term was the mass of the Higgs boson, so discovering the Higgs's mass allowed us to calculate ("gave us") the masses of all other particles. This topic came up because I am doing research in QFT with him and in the book we're using I recently got to the part where it describes self-interactions and Yukawa interactions of fields (not yet quantized) using perturbative analysis. So, my professor used the Higgs as an example of this which I may have heard of and I had to explain my misconception to him. My question is: Where exactly does this perturbative series come from, what was it attempting to describe, and what does it and its derivation look like mathematically? it's difficult to read the literature on this and, to be frank, I don't quite have the time. Any help is greatly appreciated. Thank you!


We know that a static charge only produces an electric field and a charge in motion (be it constant velocity or accelerated) creates both magnetic and electric field. But recently I came across that "changing electric field changing magnetic field, and a charge moving with uniform velocity doesn't create a changing magnetic field, and hence electromagnetic waves aren't emitted". Now I am confused since I can't find a clear explanation of accelerated vs uniform velocity in case of charge anywhere. Let's assume a source charge in space and the corresponding electric field associated with all points. Now if the charge starts moving (irrespective of uniform velocity or accelerated), since the position of the source charge is changing, the electric field associated with all the points in space are experiencing a change in both direction and magnitude and hence electric field is changing. So according to "changing electric field creates changing magnetic field theory" shouldn't electromagnetic waves be generated in this case? I really am finding it hard to understand the concepts. It seems that the electromagnetic waves are generated, only if the field lines are disturbed according to the following picture, but I am finding it hard to reconcile between the two concepts.


The diffusion coefficient is known in the traditional physical literature as an empirical parameter of Fick's law. Here the observation is that spatial gradients of densities are suppressed by a rate proportional to the gradient of the density, and the diffusion constant is identified as the constant of proportionality. An other empirical definition is the identification of the constant of proportionality between the mean squared displacement and the square root of time of colloids. One insight of the paper on Brownian motion by Einstein was that these two definitions are consistent with each other. However in the last decades we additionally learned from the computational physical literature that if we take N classical particles interacting via Lennard-Jones pair forces, we consistently observe the linear MSD vs square root of time dependence. However I never came across an attempt to derive this result analytically, that is, for a set of known LJ parameters (sigma, epsilon, T, V, N,..), is it possible to analytically show or derive this linear correlation ?


According to my environmental science textbook and various sources on the web, the thermosphere has the second highest density of any part of the atmosphere. It falls only behind the troposphere which can be explained as the most dense because the force of gravity holds gas particles there. But here is where I get lost, The thermosphere is located at a higher altitude than the stratosphere or mesosphere and has a higher temperature. It collects most of the thermal energy released in radiation from the sun. The thermosphere is more dense than the two layers directly below it (stratosphere and mesosphere.) But why is that? Pressure and temperature would be higher, but those shouldn't effect density. If density is mass/volume, then shouldn't the density be lower in the hottest part of the atmosphere due to Charles' Law? I can't really find any reason why it would be more dense other than this theory from Einstein that adding thermal energy to a gas particle would increase its mass. (can someone explain why this would be, I get increasing its kinetic energy but thats cause of speed increase, not mass.) Attached is a graph detailing temperature, density, and pressure in the different altitudes, I've been turning about this for a few hours and any input is greatly appreciated!! :)


Consider a simple chemical reaction, such as the association of two hydrogen atoms within the gas phase to form one hydrogen molecule. It is known that this reaction is related with energy release in the form of heat because the hydrogen molecule is more stable than the two hydrogen atoms. Within a gas containing hydrogen atoms and hydrogen molecules, kinetic energy can be stored in several modes, including translation, vibration and rotation. If the energy of the reaction is to be released as heat, then directly after the reaction the temperature should slightly increase locally near the reaction. This local increase of temperature results in a transfer of energy to the "bath", and this transfer of energy is the heat. Which one of the above mentioned modes of kinetic energy do you think is first activated directly after the reaction ? If I imagine two hydrogen atoms that decide to make a covalent bond at some "t", then directly after time "t" they feel a strong bonded radial force, and directly before they feel a weak "dipole force". So they should be pulled toward one another, possibly overshooting the equilibrium distance of the molecule, then relaxing back. So I guess the energy is dissipated in the vibration mode. For my understanding, I would first like to focus on the case without radiation.


I have been reading this article about the quantum vacuum state, and in the section that I linked to, there is a video showing an experiment that shows visibly that quantum fluctuations are actually happening (see the video in the link). The quantum fluctuations are visible because they amplified them using Spontaneous parametric down-conversion Now from what I learned here, quantum fluctuations (or virtual particles) do not exist (and links therein). They are just tools of perturbation theory in QFT when taking into account interactions to make the mathematics work. Another reason I learned is that the fluctuation-dissipation theorem states that when a system is in fluctuation, the energy dissipates as heat or other forms of energy. So then, the vacuum cannot be fluctuating. I seem to be confused now. Is the video fake or is what the video/wiki claiming not true and is something else entirely? This question is sort of a follow-up to my previous question.


I'm trying to figure out if a hexagonal grid can embed rectangular coordinates in whole numbers of "Y-steps". In the image below, one "Y-step" is the spacing between red hexagon centers in the Y dimension. For some arbitrary hexagonal grid size, how many hexagons do I need to produce some whole-valued number of "Y-steps" in the X dimension? Another way to ask this might be: Select four hexagons whose centers create the corners of a square. In the hexagon grid orientation shown below, how many horizontal hexagons are needed to create such a square, and then how many vertical "steps" are needed in the Y dimension? Both X and Y values need to be whole integers. In case it helps, this site provides great info in hexagonal coordinates, but I've not figured out how to pin down a way to solve this. We are using the "pointy top" orientation.


I would like to perform something that resembles a curve fitting optimization but for which i could not find much info. Lets say i have a function that yields a time series. What i would like to do is tune the parameters of this function so that the time series it yields has a specific shape. For instance i want it to output a step, or two steps back to back. What i don't know (or care about for that matter) is the height of the steps. For a single step, a naive solution would be to simply take the average of the output and compute the RMSE of the output to it's average and minimizing it. However this gets more and more complicated for more complex shapes (the two steps example is already a pain with this method) Would there be a mathematical tool that would facilitate this?


Classical mechanical systems observable on a dynamical scale are subject to Newton's laws. In this case, knowledge of the Hamiltonian allows us to minimize energy taking into account inertia. This allows us to calculate trajectories, find equilibria and derive many properties. Classical mechanical systems consisting of large sets of particles are subject to Boltzmann's law. In this case, knowledge of the Hamiltonian allows us to minimize energy and maximize entropy. This allows us to calculate phase space weights and derive many properties. Is it possible to proof that the two approaches are equivalent in some cases ? That is, proofing all assumptions made in micro-canonical and canonical formalisms based on Newtonian mechanics ? Or that the average properties generated by Newtonian trajectories are equivalent to Boltzmann average properties ? Or is it maybe possible to observe some kind of transition between the two cases if they are not equivalent ?


I have observed that when an insect is electrocuted in a bug zapper, there are typically a few sparks or flashes of light. Sometimes the insect will catch on flame in a tiny fire, and the body will burn for a few seconds before stopping burning. The lights and flames stop after a few seconds, but the insect body often, instead of falling, remains in position still connecting two wires. Given that the body is still in position connecting two wires, why is it that after a few seconds, sparks or flashes of light stop appearing, and any fire is extinguished? Does that mean that electricity is no longer flowing through the insect's body, and if so, why does the electricity stop flowing if the body is still connecting the two wires?


Is it okay (in terms of usage) to use present participle clause for an action that follows another action as a result? For instance, the following sentence seems correct to me: The bomb will explode, sending shrapnel everywhere. Yet the following sentence does not correct at all to me: I will run one mile to reach my target, grabbing him. I don't know how I feel about this sentence, I think it feels fine: The cat leaped into the air, landing in the river. This sentence feels pretty wrong: I noticed my target, shooting until I hit him. But this sentence feels fine: The man took off from one end of the street, reaching the other side just in time. And then obviously this sentence feels not great: I will live a good life, dying. But then this sentence feels fine: I will live a good life, dying when it reaches its conclusion. What's the differentiator here? Are my feelings correct?


The following problem has bugged me for a while, ever since I noticed it. On the Visible Spectrum Wikipedia, the following is the visible spectrum: Now, in Photoshop, or really any colour picker, the hue slider looks something like this: Or sometimes this: I noticed that in both of these, the colour loops back to red. Why is this? I believe that this doesn't happen on the visible spectrum. The visible spectrum goes from a violet-ish colour to a maroon-ish colour, with a whole range in between. But where does the magenta colour from the hue slider fit in? I take it that it is possible to have a purely yellow object, or a purely teal one, as it is on the colour spectrum, yet are magenta things and pink flowers inherently reflecting multiple wavelengths of light, from opposite ends of our particular viewing spectrum? All of this seems awfully odd to me, so I was hoping someone might be able to clear it up.


I understand why the temperature of the hot reservoir has to be minimally higher than the temperature of the hot working fluid during the isothermal expansion phase of the Carnot cycle (to limit new entropy being produced in the working fluid that we have to get rid of). But during the isothermal compression phase why do we need the cold reservoir to be only minimally cooler than the cold working fluid? The working fluid looses the same entropy independent of the temperature of the cold reservoir, so why do we need to minimize the new entropy created in the cold reservoir? What is wrong with just letting the cold reservoir get more and more entropy as long as the working fluid returns to the same state as before the cycle?


So, as I was browsing a bunch of the tachyon questions throughout the years in this forum, and an oddity of these hypothetical faster-than-light particles came to mind. Ordinary particles with mass always have an inertial frame of reference where they are at rest, thus their rest mass can be measured. Lightspeed particles, while never at rest, have an invariant velocity, so different frames of reference can agree on their energy. Not so for tachyons. Tachyons not only are never at rest (like lightspeed particles), but their measured velocity is frame-dependent (like ordinary particles with mass). So, my question: Question: Is there anything analogous to a rest mass that can be used as a common point of reference when working out the mass of tachyons, or is it completely frame-dependent? And, no, we cannot say the tachyon's frame of reference, because they cannot be treated as observers (same as with lightspeed particles, we cannot treat a photon as an observer).


Electricity is used for many things. One of the biggest uses is transporting energy, almost instantly, from a power plant to the machines in my home and many others'. I was wondering if a similar energy transport grid could be created using a fluid under pressure. The power plant would use its energy to pump a fluid into the main pipe, which extends all over the country, with many sub pipes coming out to each house. The high pressure liquid would act like high voltage electricity, ready to be tapped into by a user. At home, one would open a valve in the pipe, causing high pressure fluid to come out. This jet could be used to spin a rotor mechanically, in other words, energy was transmitted almost instantly from the power plant to my home. We would probably need to have the used fluid go back into another pipe which goes back to the power plant to be pumped again. In other words, something similar to an electric circuit but using fluid instead of electrons. Knowing about the analogy between electric current and hydrodynamics, I was wondering if my strange idea could be practical or at least physically valid.


I hope this isn't the completely wrong community for my question, please let me know. I did search for 'varifocal' and got some hits, it seems (still getting used to these glasses, so it is a bit hard to use my PC) So, it seems that when you get these glasses - we call them varifocal in UK, not sure if that is the international name? - they have these three zones with different strength, but for some reason, there are two blurry areas in the lower right and left corners. The top has the distance vision all across, but the computer- and reading strengths are only in the middle. I haven't been able to find an explanation online so far; does anybody here know of a good, technical reason for not simply providing three zones in the full width?


Even though a black hole has a Scwarzschild radius that indicates a finite small distance to the center of the hole, the distance traveled by an infalling particle seems a lot bigger than the Schwarzschild radius due to the extreme curvature of spacetime. An infinite curvature even seems to imply an infinite distance. Two particles falling into the hole, one after another, end up spatially separated. This holds true for all particles that fall into it. This seems to imply that the distance traveled is actually infinite. I mean, if particles end up spatially separated because of spaghettification, and they all fall the same distance to the singularity, only an infinite distance will do, so it seems. What (if) is the flaw in my reasoning? EDIT I'm not sure if the linked question is a duplicate as it asks about the time it takes to freely fall to the singularity. I'm asking about the distance traveled, and I read in a comment by @MichaelSeifert that there is a good explanation for why the concept of distance isn't well-defined in this case. I can't see why that is so. What if you, the freely falling indestructible observer, have an infinite rope, one end of which you attach somehow just above the horizon? Can't you see how much rope has rolled off from your device when you hit the singularity?


Short introduction to my understanding: As far as i understand, virtual particles are usually defined to be the internal lines in Feynman Diagrams. But we know that those are just useful tools to calculate amplitudes in interacting quantum field theories. In a free theory I have no interactions, hence no internal lines and no virtual particles. Virtual particles show up when, in interacting QFTs, we have the so called contact interactions showing up in the perturbative Dyson-Schwinger equations as deltas. My question: Can we regard virtual particles solely as mathematical tools needed to make predictions in perturbative interacting theories because we don't fully understand how interacting theories work? (I mean how they work in a non-perturbative setting) If the latter is true, and from my actual understanding I think it is true, what is our "hole" in the understanding of full interacting QFTs?


I was reading about effective field theories and wondering how much an EFT can tell you about the ''underlying'' theory which is then reflected in the EFT. Can one extrapolate back from what one sees in the EFT to derive features of the theory for the high energy degrees of freedom? As an example, general relativity is considered to be an effective field theory for quantum gravity, but that does not tell us much about what the correct quantum gravity theory would look like, with many people arguing that this underlying theory would be a string theory and not even a field theory. However, are there cases where one can say more or come to definite conclusions about the underlying theory? What is the systematic way for doing this? For example, in lecture notes of Pich on EFTs with Nambu-Goldstone modes, he mentions that one can ''uncover fingerprints of new physics scales'' from couplings in the EFT.


So, just to expand on the question in the title, I know group representation theory, and especially character theory, are quintessential tools for anyone hoping to study finite groups. That said, from what I've seen and read, some people study characters in and of themselves (say, density of zeroes, Brauer characters, etc), while another set of people uses them as tools for studying questions concerning the groups themselves (i.e., it's not clear from the get-go that characters will even get used; they are a part of a toolkit). My question is: what's the intersection between these two sets of people? Can a person spend some time working on Brauer characters and blocks and then move on to study, say, the subgroup structure of finite simple groups? Or are the areas far enough apart that character theorists simply don't have enough baggage to work in general problems on finite groups and vice-versa? Sorry for being a bit "vague", perhaps... I'm just very curious about the distinction (if it even exists at all!), since I really love both topics! Thanks so much in advance!


(Honestly I fear this is borderline off-topic... be gentle with the downvotes ;-)) Most of the packages I've written are for personal use but a couple of them are used by some other people. These packages change relatively often as I'm constantly implementing requested features. Right now I tend to ship out the most recent stable version by e-mail but then I get either answers of people that are not interested any more, or requests from other people who aren't in the mailing list yet. I do not want to upload the packages on CTAN for two reasons: There's maybe a dozen users. It's highly unlikely the packages will be useful to someone else. I find there is enough bad code on CTAN without me contributing to it. For now I've got an institutional home page but that may change any minute, so posting the code there for download (which I've done) might not be a long-term solution. Any advice? (My personal texmf tree is on Dropbox, if that may spurt a solution.)


It seems correct to write "A host of tools exists..." or "A range of tools exists...", i.e. the verb reflects the fact that you are referring to one collective noun. But, if I want to continue to talk about that collection of things, it often seems natural to say ". They aim to solve..." or ", which attempt to model...", i.e. now I am referring to the capabilities/intentions of the things within the collection. Which is strictly correct? And putting strict grammatical rules aside, which is an English speaker most likely to say? a) "A host of tools exist, which aim to solve..." b) "A host of tools exists, which aims to solve..." c) "A host of tools exists, which aim to solve..." My instinct is to avoid the construction and write "Several tools exist, with the aim of solving..."


It is currently understood that gravity is not actually a force, and a fact that is often used to show this is that an object in free fall doesn't "feel" that it is accelerating and is thus an inertial frame. However, it seems to me that Newtonian mechanics can already predict that this will be the case. Since all the parts of the object are being accelerated by the same amount simultaneously (at least approximately, like near the surface of the Earth) there won't be a tendency for this object to contract, and thus "feel" that it is accelerating. This isn't the case, for example, when I push the same object. In my understanding, the force needs to be communicated from the point I apply it to all the rest of the object, and this delay is the cause of a contraction/increase in internal forces or tension, allowing it to effectively "feel" that is accelerating. Now, suppose that there is a uniform field that accelerates any particles with constant acceleration. Like gravity, a free object in this field will not be able to detect that it is accelerating, since all of its particles are accelerating equally at the same time. My question is: Is this object inertial, or is it only inertial if this field is a gravitational one?


What is the current consensus about the age variation of the existing galaxies in our observable universe? Not to be confused with the age of very distant galaxies as observed today by our telescopes which is actually observing these distant galaxies as they were many billions of years ago in the early formation but instead I am asking about their absolute age today assuming they still exist? I mean, a hypothetical observer located in these very distant galaxies and looking at our home galaxy would also see our galaxy not at its current state but how it looked like billions of years ago. I am not asking this but if all the galaxies were formed more or less at the same time according to the Big Bang theory? In general how you estimate the age of a galaxy?


Let's assume that a cylindrical magnet is clamped by an horizontal arm extending out of the wall. The top of the magnet is its north pole and the smooth face facing down is its south pole. If I bring another cylindrical magnet and position it perfectly in line below the fixed magnet, with its top being its north pole and the bottom being the south pole, would it be possible that the weight of the magnet going down and the magnetic force attempting to pull the magnet upwards balance out so that the magnet can levitate in mid air? Conceptually, it seems to make sense as in this case, the magnet is not prone to flipping since it poles facing each other are not repelling. However, I'm not sure if this violates Earnshaw's theorem. It it's not possible, then why isn't it possible? Did I miss something in my assumptions? And if it is possible, Is it also possible for the levitating magnet to rotate about its central vertical axis and still remain suspended in space?


I'm looking for an accurate method to measure the tension in a guitar string, without using a sonometer setup nor by measuring the frequency. the current method that I have in mind is to measure the force perpendicular to the string and the string's displacement, and then try to resolve the vectors (the method is explained in more detail here), but I don't think that is an accurate method for a couple of reasons: we have to consider the uncertainty of the newton meter and the ruler the propagation of uncertainty when resolving vectors makes the final uncertainty much larger even after resolving the vectors, we are calculating the tension in the bent string, not the equilibrium position. are there any other methods I can use to measure the tension more accurately?


Clicking through the OEIS I found this sequence that seems curious - the number of "labeled" groups. It cross references the usual sequence of the number of groups up to isomorphism (the very first sequence in the OEIS) as well as saying it is "a sequence related to groups". So I've tried to look up this concept, but Google with "labeled group", "labeled group maths", "labeled group theory", and so on, to no avail at finding anything. Groupprops (essentially a wiki for group theory) doesn't seem to have anything either. I've clicked through the links on that OEIS entry and cannot find anything of help. So as a last resort I'm asking here (I know this isn't necessarily the kind of place for definition questions though, sorry). So, what is a labeled group?


Neither my wife nor I have English as our mother tongue, but we use English to communicate to each other, which sometimes causes confusion. My wife often uses the expression "until now" to mean "so far" or "yet", meaning that the action is not yet finished. I didn't receive any answer from the landlord until now. To me "until now" means that for some time she didn't get answer but now (recently), she did. So I'll usually reply something like: Nice, so what did he say? And she looks at me weirdly, as if I would have asked something silly and we both laugh. My question is. Does "until now" always imply that the action you are talking is now finished, or can sometimes be used with the same meaning as yet?


If i place some charge on a conductor then it will distribute itself in such a way that electric field everywhere inside is zero. My text book says that only one kind of such charge distribution is possible. Or if i place charges outside the conductor then charges will be induced on the surface of a conductor to make field inside zero and again such charge distribution is unique. Intuitively it all seems correct but i am trying to figure out right arguments for this. One such argument is that suppose i place some charge on conductor then i can solve poisson's equation to find potential at the surface(at boundary). Charge distribution outside the conductor is known. Then uniqueness theorem says that only one such potential function is possible which satisfies the given/known boundary conditions and poisson's equation outside. Since a particular potential function corresponds to a particular charge distribution the charge distribution must also be unique. This argument seems correct to me but at the same time something seems missing. Am i correct?


I'm looking for a word similar to "free" or "without cost", but that makes it explicit only that no money is exchanged, while still allowing (or implying, or explicitly specifying) that some other exchange of some value has taken place. For example, if I rent a room to a friend in exchange for language tuition, I have a form of non-monetized exchange. I would like to say that the room is rented for free, except that it isn't for free here, because tuition is given in return. I still have to pay taxes and bills in money, and to describe this I want to say that the room is rented for zero-money (that way pointing out that I can't use the exchanged tuition service to pay the bills, and that therefore the exchange is somehow of less value than its cash-value equivalent). Saying it's "free" is not correct, because something of value is given in exchange. "Costless" I think has similar issues. "Barter" is a related meaning, but here I'm dealing in an on-going exchange-of-services relationship, for which "barter" doesn't seem right. "Non-monetized exchange" or "on a non-monetized basis" is the best I have so far. I can't think of term that is succinct and gets this meaning clearly. Any ideas? A technical accounting term would do, but ideally I'd have something that is clearly understood by a lay person.


Imagine a train moving relativistically according to an observer at rest wrt to the tracks it's moving on. The train seems contracted in the direction of motion. At the points where the wheels make contact with the tracks, the wheels have zero velocity, so the distance between these points is not Lorentz contracted (like the caterpillars of a tank are at rest). Now what will happen when the train suddenly decelerates, simultaneously at all parts, as seen from the observer at rest? Obviously, the wagon will expand, but what will happen to the wheels? Will they follow the expanding motion, which leads to friction between the wheels and the track? On the other hand, when the train starts to accelerate from zero velocity, the distance between the wheels will not Lorentz contract, so the wheels seem to have to slide with friction, dragging behind the contracting wagon, so to speak. Will the reverse happen when decelerating?


I had a recent conversation with a professional mathematician about the status of relations, functions and predicates. I was arguing that it seems intuitive (to me at least) to classify them in this hierarchy (as to which is more primitive): All predicates are functions. All functions are relations. The obvious problem here is that it seems intuitive that unary or even nullary <predicates/relations/functions> are more primitive than their n-ary variants. Is there a way to compose functions as at least unary relations or vice versa (relations as unary functions)? If not, is it possible to order them in such a hierarchy given the binary restriction. Finally, if there is something more primitive that has a formal definition, then that would do as well. A resource or explanation pointing to how at least functions, relations and predicates are composed using this notion would be helpful.


It's well known the effect of Rayleigh scattering on the color of the sun, and it's explained several times on this website. Here's one of them. The summary of these explanations is, that when light travels through a colored medium, that color is being "used up" to make the medium the color it is, and only the other colors will go through. Make very much sense. But the problem is that our experience in everyday life is just the opposite. When we shine a light through a colored medium, the light becomes the same color as the medium. Although it's hard for me to understand why (this was asked here without an impressive answer). And that's also happening with sunlight that goes through colored glass. So why does the atmosphere behave different than anything else? And is there anything I can experiment with that would have the same behavior as the atmosphere?


I did some online search and found the explanation using the following two diagrams. It's not perfectly convincing to me. Or at least it is not clear to me in the following details of the process: For the secondary rainbow to have the inverted order of bands, do the light rays have to exactly reflect twice and cross each other twice inside the water drop? If so, what are the necessary and sufficient condition for that to happen? It seems to rely on the relative direction of the incident light and the surface of the water drop and the position of incidence? The diagram only shows one particular position and angle of incidence, so it seems kind of accidental to me that it inflects twice and cross each other twice and got to the inverted order.


I was reading a physics problem related to astronomy, and upon re-reading it, I realized that it could be really indicated to extrapolate some really interesting physics-related information. One of these is: How could we measure the ratio of a planet's radius to a star, for example using transits? The only idea I have is to compare them when the planet passes exactly in front of the star (i.e. they are aligned with our view), but this only makes sense if the distance between the two is much smaller than the distance between us and that star system (which I think is true enough for every system except the Solar system) and if it is possible to obtain such high resolutions (and I already had my doubts about the distance between the two, which should be much greater than their radii anyway).


Suppose I have a nonlinear ordinary differential equation, in several variables, with a stated initial condition. How would I go about finding a nonlocal linear approximation? What is known about such approximations? By nonlocal approximation I just mean that the would-be approximation minimizes the maximum distance between the solution of the approximation and the solution of the original, nonlinear initial value problem, or meets some other criterion such as minimizing the integral of the square of the distance. At least in the case of minimizing the integral of squared distance, this problem amounts to a nonlinear least squares problem, and the criterion is a differentiable function of the parameters of the linear approximation, so maybe a Gauss-Newton type algorithm or even simple gradient descent (at this point I'm not worried about efficiency) could be applicable. For the problem I have at hand, I constructed a linear approximation at the initial point, via the Jacobian of the original system of equations, but the solution of the local approximation is very different from the solution of the original nonlinear IVP at later times, which inspires me to look for a nonlocal approximation.


While reading the Wikipedia article on Drag Crisis, I found: The drag crisis is associated with a transition from laminar to turbulent boundary layer flow adjacent to the object. While, the Wikipedia article on Turbulence states: In general terms, in turbulent flow, unsteady vortices of many sizes appear which interact with each other; consequently, drag due to friction effects increases. If drag (due to friction effects) increases due to turbulence, then why does drag crisis occur when flow shifts from laminar to turbulent? Shouldn't drag be less for laminar as compared to turbulent flow? Why else would surfaces smoothen due to drag over time? The Wikipedia article on Drag Crisis also goes on to say: For cylindrical structures, this transition is associated with a transition from well-organized vortex shedding to randomized shedding behavior for super-critical Reynolds numbers, eventually returning to well-organized shedding at a higher Reynolds number with a return to elevated drag force coefficients. I am having trouble understanding this. A related question, Drag Crisis and Terminal Velocity?, examines the relationship between drag coefficient and terminal velocity.


I'll put pictures from the book (Introduction to the Structure of Matter: A Course in Modern Physics by John J. Brehm and William J. Mullins) as I think they are relevant to understand my problem: I have trouble understanding the case where the observer watches the source in a direction perpendicular to the magnetic field. The electron will rotate around B axis, so the observer will only see a linear oscillation of the electron hence linearly polarized light. But how can the Lorentz force explain the splitting of spectral lines (i.e. the change of the frequency of the electron)? The book suggests to view the linear oscillation as a combination of two counter-rotating motions like this: But if this is the case, the Lorentz force would act in a plane perpendicular to the image so it won't explain the change of the frequency of the circular motion of the electron (and so the Zeeman splitting, classically). Instead the situation is clear when we observe along the direction of B, as in that case Lorentz force would act radially.


The standard explanation in textbooks goes that in the presence of electric field (e.g. external electric field) the free electrons inside the conductor will keep moving until electrostatic equilibrium is reached and that they will create a field which will cancel out external field (e.g. electrons move to one side leaving positive charge on another side, this creating electric field that cancels out the external field). But what if the external field is so strong that there is just not enough free electrons to cancel out external field? What if there is nothing left to be moved? E.g. I have electric field lines going from left to right, all free electrons move to the left surface, there is nothing left to move, but it's still not enough to cancel out external field. Will there still remain an electric field inside the conductor then? If not, what will happen?


I'm looking for a phrase that describes a problem whose complexity starts to increase exponentially, either because the problem is recursive, the definitions/conditions of the problem interlink with themselves, or it turns out that it's connected to a great many other issues, and changing one changes all the others in turn. This phrase would be useful when, say, discussing how to solve a simple, harmless, trivial little bug in some software...that now has eight people standing around debating business logic and corollaries to corollaries and exceptions to exceptions. Phrases that are close, but not what I'm looking for: "Tip of the iceberg" could be used later on when describing the original difficulty, since it would become much worse later, but the rest of the situation wasn't originally an issue or complicated, and only by investigating the original, simple issue did the full complication arise. "Can of worms": the rest of the difficulty isn't inherently a problem; the original, minor issue is the only part that's an actual problem, and the rest of the complication is created by trying to solve it. "Cure is worse than the disease": it's not that the solution is bad or undesirable, but that the original problem was far more complicated than it first appeared. "Gordian knot" describes a problem that is already complicated known to be difficult; what I'm looking for is a situation that seems simple or even easy until you try to solve it.


A congruence is a useful notion in general relativity, relating mathematcal definition and physical interpretation: "A congruence (more properly, a congruence of curves) is the set of integral curves of a (nowhere vanishing) vector field in a four-dimensional Lorentzian manifold [...]" Following up some intuition (mostly concerning timelike congruences) I find the following description further down on the Wikipedia page especially relevant: "The integral curves of the vector field are a family of non-intersecting parameterized curves which fill up the spacetime. The congruence consists of the curves themselves, without reference to a particular parameterization." The emphasis on non-intersecting is placed in the Wikipedia article already; however without any further supporting explanation or link. To me, this raises an ambiguity: The phrase "non-intersecting curves" might be understood as referring to each of the (unparametrized) curves as a set (consisting of events), meaning that the intersection of any two of these sets is empty. In other words: (pairwise) disjoint sets. Or: The phrase "non-intersecting curves" might be understood (in a geometric sense) as allowing the possibility of at least some instances of being tangent to each other; corresponding to the general geometric description of such mutually tangent curves as "touching but not intersecting"; thus allowing at least some pairs of curves to have at least one point (event) in common, "in which they are tangent to each other". Accordingly (and without hinting at my intuition) my question: Which is it?


In my physics textbook, the foundation for work is derived using newton's third law, where F_surr = - F_gas, where surrounding represents a piston-cylinder device and gas is pushing against the inner surface of the piston towards the right. My questions are: Given this information, it is obvious that W_surr = - W_gas. So, shouldn't the terms cancel each other and result in zero acceleration of the piston, i.e., the piston remains stationary? I understand that newton's third law is applicable for forces acting on different objects in contrast to second law which is used for analysing forces acting on a single object. So using second law, if i were to dissect the system and piston into two bodies and assuming no friction, we will have F_surr acting towards the left on the gas and F_gas acting on the piston towards the right. So, I am confused here, shouldn't F_surr being the only force on the gas cause it to compress and F_gas on the piston cause the piston to move towards the right? Finally, during a quasi-equilibrium process, will there be any accelerations of the piston at all at each equilibrium states or is the equilibrium state like a point at which the system and surrounding are in complete mechanical equilibrium and are stationary? Essentially, I am having a hard time applying newton's second and third law to derive work equations.


Given two probabilistic distributions (red and blue) it is well known that a linear interpolation between them is well defined (see this). For example, by the Wasserstein metric we have the following interpolation: When I first saw this approach, it crossed my mind that the Bayes 's posterior could be explained by a similar geometric argument. Perhaps, the mean between these distributions by the Wassertein metric. However, when I plot the Bayes's posterior (in green) we see that it does not belong to this linear interpolation. My question is: is there a geometrical reasoning that we can use to interpret the Bayes 's posterior? If there isn't a geometrical reason , can there be a variational one (the Bayes's posterior is the minimum of some energy in this space)?


I am curious about magnets and their interaction with the electromagnetic field. So I tried to find this answer and stumbled upon (Magnetism and Photons), this is an interesting description that I quite like, but it doesn't really explain how a magnet can keep a piece of metal on a string hovering in mid-air indefinitely, the magnet is not moving, yet there is an interaction there, as I understand it via photon exchange. I did read in a magazine (New Scientist particle physics special from memory) that electron spins create a non homogeneity that interacts with the electromagnetic field that then "tends" to allow virtual elements to appear through particle/antiparticle annihilation, with a resulting photon being generated. I don't love this explanation, perhaps because I find it hard to get my head around it. Any advice or references you can point to explain the electromagnetic field (this is confusing in itself, I thought long ago we gave up on the idea of an "ether", it seems it's back) and then the interaction of electron spins (magnetism) on this field. Thanks in advance, Tibor


I am looking at a tutorial create for the FENICs finite element method package. The tutorial shows a system of advection-diffusion-reaction equations occurring in a solution that is moving according to the incompressible navier-stokes equations. In the tutorial, the authors simply present the weak form, without explaining the derivation. I was hoping someone might be able to explain how to derive the weak form, or even identify some good references to look at to find these derivations. The system of equations is below. [NOTE: I included the equations as images because I could not get the latex formatting to work for an aligned environment. I can put in the latex, but might need some direction on how to format the equations.] The provided weak form is as below. I understand the steps in computing a weak form, especially the integration by parts piece. However, I am not clear on all of the steps in the derivation below, such as why a vector PDE is represented as a single equation. I imagine that the different species are tracked in a vector, and hence the sum makes sense. BUT, in the equation below, it is a bit difficult to see those nuanced elements. Hence I was hoping someone could explain this derivation or provide some links to resources that provide a more detailed derivation of this weak form.


I am really confused about the part where the photon shares a part of its energy with the electron. It is said that the photon loses some of its energy to the electron causing the change in its wavelength. In my understanding photon isn't something that contains energy, it's the energy itself. A single photon has a fixed amount of energy (E = hv) ;the smallest possible value of energy of the photon at that particular frequency. How can it break into something smaller, doesn't that voilate the postulates of Plank and Einstein about the quantum nature of light? Or does a photon break into many, in that case, is the quantized nature of light even meaningful anymore? Moreover if the x-ray photon has surplus energy after knocking the electron out of the atom, why doesn't it contribute to the electron's kinetic energy? Like it does in the case of photoelectric effect. And it leads to another confusion I have now. If photons can share part of their energy then can't we simply increase the number of photons hitting the photoelectrode such that the free electrons can slowly store enough energy to break the required energy thershold.


A person might say on one day: It is hot outside - let's go out for a picnic! It is healthy. Another person might say on the same day in the same place: It is hot outside - stay inside where it is safe from extreme weather! It is unhealthy. The weather and health together is our internal model of what is happening at a physical and biochemical level. It is not actually what happens - simply an analogy we can use to communicate an idea. Let me give another example: One teacher might say This group of students are so quiet. It's unhealthy. We need to break up the group. Another teacher might say for the same students on the same day: This group of students are so quiet. It's so healthy. We need to encourage more students to be like this. In this example health is a metaphor for something that is assumed - but not defined. It is used both ways. The analogy of 'healthy group behaviour' is ambiguous in this conversational context. One possible answer to this might be metaphysical conceit - but that seems like a poetic term, and possibly limited to spiritual poetry. My question is: What is a word for when an analogy could be used for both sides of an argument?


I have two lists, and six elements: V V V M M M I would like to compute every possible combination of those elements into two lists (left and right): So, five possible solutions could be: ([V, V, V], [M, M, M]) ([], [V, V, V, M, M, M]) ([V, V, V, M, M, M], []) ([V, M, V], [V, M, M]) ([V], [V, M, V, M, M]) My first approach was reducing the problem to that which had less elements (two). So, all possible solutions in that case are: ([], [V, M]) ([], [M, V]) ([V, M], []) ([M, V], []) ([V], [M]) ([M], [V]) I can't seem to fit this problem into a basic standard combinatorics one, but perhaps there is something I am missing. How can I approach this problem ?


I need clarity on some definitions and mathematical "skepticism". In a recent video by Matt Parker, he says "(...) although the existence of the sign function, which says if a value is positive or negative, upsets some people (...)". What does he mean by this, and can this "upsetting" be extended to functions like the absolute value or the indicator function of a set? Any suggestions for further reading would be highly appreciated! Edit: Before this question is closed, I would like to clarify that by no means is this question intended to incentivize discord or negative emotions regarding certain groups of people. I do not wish to offend, nor suggest I am taking part in any ideology regarding the nature of mathematics and its definitions. I am merely curious about what Matt meant, and what is the deeper motivation behind his brief comment, hopefully generating a healthy discussion. Thank you.


Consider the following sentence: "This paper introduces a new alternative for generating synthetic data based on images." What I want to say is that "the new alternative" is "based on images". Thus, the noveltiy of the new alternative is in the fact that it is based on images. What I would like to avoid is that one think that I mean "image-based synthetic data generation already exists and I propose a new alternative to it." So, to avoid the ambiguity I add ", which" to the above sentence, so it becomes: "This paper introduces a new alternative for generating synthetic data, which is based on images." However, after looking for the usage of ", which", I am still not sure that the sentence means what I mean. More precsisely does "which is based on images" refer to "generating synthetic data" or to "new alternative"?


I'm a native English speaker, but I'm drawing a blank on how to describe this complicated exchange of situational emotions in words. Maybe someone can help. I'll just illustrate the situation with a narrative scene. Two doctors examine a patient. "Have you been eating any protein-rich foods lately?" one of them asks. "No," answers the patient. "No protein at all." Their suspicion confirmed, the doctors looked at each other, nodding _____ly / nodding with _______. What emotion am I looking to describe? I can picture the facial expression in my head, where the doctors aren't personally involved in the consequences of what's going on, feel somewhat sad about what the other person is going through, and yet understand that the consequences are justified based on the person's actions, all of which is shared between the two people who both come to this conclusion. In this case, they both know the patient should've been eating more protein so they wouldn't have this affliction, are somewhat apologetic that the patient didn't know this, but have some sympathy in explaining afterward that in order to stay healthy, one must consume protein. Anyone know what words I'm trying to express?


In the nuclear physics book by Krane, I was reading about how mathematically and numerically Q value of the reaction are measured. So he decided to prove it by solving it in Lab frame, And he goes saying this, "Let's apply them (the formulas he wrote to measure Q value) first to the laboratory reference frame, in which the target nuclei are considered to be at rest (room-temperature thermal energies are negligible on the MeV scale of nuclear reactions). If we define a reaction plane by the direction of the incident beam and one of the outgoing particles, then conserving the component of momentum perpendicular to that plane shows immediately that the motion of the second outgoing particle must lie in the plane as well." So my first doubt is, what is a reaction plane, and why the nuclear reaction always occur in a plane only. I read some more stuffs online but was unable to understand this reason as why nuclear reaction occur in a plane. Why is so that the momentum perpendicular to plane is to be conserved and is of interest?


This is a follow up to a previous question. I use pandoc to create PDFs from markdown files, which uses pdflatex behind the scenes. After a bunch of system upgrades, the type in my printed PDFs started to look off.The weight was heavier, and the the kerning was bunched up. At first I thought this might be a problem with LaTeX or my MacTex install, but now I believe it's a problem with MacOS. When I print on MacOS via Preview or Acrobat, I get the font that looks janky.If I switch to a windows machine and print there (just via the Edge web browser) the type matches more closely to what I see on the screen and used to see when printing from my Mac. I now think the "bad" font I see is just good old Times New Roman, being substituted as a default. Does anyone here know why PDF viewiers/printers on my mac might not see fonts embded in PDFs generated by pdflatex?


I work in quantum field theory in curved spacetime. Within QFTCS, we have a bunch of phenomena showing that the notion of "particle" is quite subtle. For example, the Unruh effect let's us know that the notion of particle is actually observer-dependent. As a consequence, I tend to be quite skeptic about fundamental theories that rely too much on the notion of particle. While I am far from being an expert in string theory, the things I've heard about it make me feel as if that is precisely what string theory does: to take the notion of particle seriously and change it into a quantum string, which would then be able to solve numerous problem within quantum field theory and quantum gravity. I have also heard one or two words about "string field theory", which appears to be a quantum field theory with infinitely many fields. The name, of course, also suggests it is somehow related to string theory. With this in mind, my question is: QFT is a theory about fields, in which particles sometimes appear as cute interpretations. Does the same thing happen in string theory? How exactly (to the best possible presentation for someone who knows nearly nothing about string theory)? Has string theory (or string field theory, or any other similar idea) successfully derived the Unruh effect or is it only "likely to be there" based on being a generalization of QFT?


I have a GRIN lens with the refractive index varying linearly with y, and supposedly this lens tilts the wavefront. Since the rays are travelling normally in both refractive index materials, they stay normal to the plane of the lens at all times. However the wavefront travelling with them tilts because of different speed of light for different indices, and gives a wavefront like this. Is this correct? Or do the rays themselves tilt? I don't think this is a possibility since the rays are always perpendicular to the plane. If my reasoning is correct, does this satisfy Huygens' Law? It states that the wavefront must be perpendicular to the path of the waves at all times, but this doesn't seem to be the case here, which leads me to thinking its the waves themselves that are tilting. Diagrams like this can be found on the internet, but I do not understand how are the waves tilting when they enter normally, What am I missing? I found this situation while solving a problem in a book (Pathfinder for Olympiad Physics) but upon thinking a bit more was fell into this confusing situation.


When one studies representations of (bosonic) Lie groups in physics, whether dealing with spacetime symmetries or gauge symmetries, it is often left implicit whether the representations are over real or complex vector spaces, without acknowledging of course that the representations of an algebra or group can be different over different base fields. A simple example is that the adjoint representation of the Lorentz group is irreducible as a representation over the reals, but reducible when take as a representation over complex vector spaces. When it comes to spinors, often people wish to treat components of spinors as Grassmann variables - this being the 'classical limit' of the fermion anti-commutation relations. My question is whether or not there has been a rigorous treatment of the representations of semi-simple (bosonic) Lie groups over Grassmann-valued vector spaces. EDIT: I suppose the direct analogy I anticipated isn't there. When we allow complex fields, we also allow complex changes of basis. This changes the reducibility of representations, because real Lie algebra elements now can have imaginary components even though the boost/rotation parameter remains real. For Grassmann numbers, it appears that we dont extend the reals as in the complex case, but supercede them. So we dont gain access to a new change of basis like before, where the algebra elements would have Grassmann-valued components while keeping the boost parameters real. Unless I am mistaken and people do work with spinors with "real parts" and "Grassmann parts".


What's the term for a line of dialog that is repeated later in a different context? Christopher Nolan seems to be fond of these lines. In Batman Begins, Ra's first says: Always mind your surroundings And then in the climax Batman says You never learned to mind your surroundings. In The Dark Knight Rises, Bane first says: Then, you have my permission to die as he's breaking Batman, and later Batman claps back with Then, you have my permission to die. Same line, different contexts, often as witty comebacks. I've thought of a couple of possibilities I've heard of, none of which works quite well: Callbacks seem to be a reference to a previous gag, rather than an exact quote. Comebacks and clapbacks can be any kinds of clever retorts, not necessarily the use of your opponent's words against themselves


The general bin-packing problem is NP complete. I have read several papers and other source but I am still not clear about whether a bin-packing problem with a fixed number of bins is NP-hard. Wikipedia says: "Computationally, the problem is NP-hard, and the corresponding decision problem, deciding if items can fit into a specified number of bins, is NP-complete." and later: "On the other hand, bin packing is solvable in pseudo-polynomial time for any fixed number of bins K, and solvable in polynomial time for any fixed bin capacity B." In another paper "Bin packing with fixed number of bins revisited" I have read that for a fixed number of if the sizes of the items are polynomially bounded integers, then the problem can be solved in time n^(O(k)). My specific problem is the following. I have a fixed number of bins m with fixed capacity. The items arrive online. I one case the set of items is known in the other case it is not. Given on what I have read I assume this problem is not NP hard as it could be solved by exhaustive search.


I'm learning about functional programming, In functional programming, functions are treated as first-class citizens, meaning that they can be bound to names (including local identifiers), passed as arguments, and returned from other functions, just as any other data type can. This allows programs to be written in a declarative and composable style, where small functions are combined in a modular manner. -Wiki And.. I started to wonder, how it would be if we tried to adopt this method as foundations to mathematics. Would it be meaningful to consider mathematics with functions being taken as the fundamental object rather than, say, that of the set or non function types? If it is, what would be the difference between the produced mathematics of that procedure, and that we have when we consider sets as the fundamental objects? What would be the advantages/ disadvantages to this approach?


I am seeing many people claiming that cumulus clouds that sometimes form periodic wavy patterns (see images for "altocumulus undulatus" or "Radiatus" for instance) have no explanation aside from being chemtrails, and I'd like to be able to respond with a sound scientific explanation. I'd like to understand the phenomenon and my guess is that it's about the cloud blanket being forced by winds with the result of a periodic pattern appearing, much like sand waves form in shallow water at the beach. But is it really the case? Searching for a more detailed explanation I ran into Tollmien-Schlichting waves that are a path to turbulence, but I admit I did not understand much, so here is my question(s): What is the physics behind such cloud formations? And Given an estimate of the spatial period of these cloud waves and the cloud height, can one infer the windspeed at that height?