About a year ago, I came across a really cool property of the envelope curve of a parabola that I couldn't prove. I'm posting it now for help: If we have a straight line and a circle that belong to one plane, then the enveloping curve of the parabola whose focus is a moving point on the circumference of the circle and whose guide is that straight line forms two parabolas that can be drawn with the four information: the perpendicular from the center of the circle on the line is the axis of symmetry of the two parabolas, the distance between the guides of the two parabolas is equal to the length of The diameter of the circle, the center of the circle is a common focus of the two parabolas, the straight line is midway between the two guides. If you start from a circle that does not share the line at any point, the segment will leave traces of two parabolas inside each other, as shown in the picture But if you start with a circle cutting the straight line, leave the traces of two intersecting parabolas at two points on this straight line. But if a circle starts touching the straight line, then the traces of one parabola touching the straight line will be left I was hoping to prove it myself, but unfortunately my level of proof does not allow me. Please do a complete proof that discusses the three cases, please, and thank you. Also, is this feature previously discovered or is it new, please attach a reference if it was previously discovered


My question probably sits more on the applied mathematics side. There is a spring mass system, for the case where the mass moves into the spring (vertically downwards) the spring experiences compression and the second order equation for a spring mass system will provide a solution. However, when the mass moves vertically upwards the lets say that the spring cannot experience tension and thus the mass moves freely without the effect of the spring. It is pretty easy to develop the differential equations for each of the two scenarios individually. However would the following procedure presented a decent manner of modelling the system as a whole: Lets say we apply a displacement as an initial condition which causes compression in the spring. Thus initially the differential equation of the spring governs the system. At each iteration the results are reviewed to check if the spring experiences tension. if the spring has tension then the initial conditions for the last iteration of the spring mass system is used as initial conditions for the differential equation which presents free motion. The above mentioned points will work in both directions, thus if the free motion differential equation is in use and the system begins to move vertically downwards the initial conditions from the last iteration will be used as initial conditions for the spring mass system. I am not looking for a coded solution for this since it can be done easily on Python and Matlab using the ode solvers. Would the presented method provide relatively accurate results for the system?


As a rule of thumb, vapor condensation usually happens at the interface between the system and the heat reservoir. Now, according to my analysis below, it is the only way for vapor to condense, which implies the near impossibility of condensation in the bulk of vapor. : As the temperature drops, within the vapor system, whose thermodynamic behavior can be described by a canonical ensemble, energy is favored over entropy. That is, the energy should escape from the system to the heat reservoir to create a large entropy overall. This means that the system has the tendency to evolve into the low energy configuration when the temperature drops. Thus, for such a phase transition to take place, the passage of energy from the system to the reservoir is crucial. The case for vapor is unique in that its only access to the heat reservoir is on the surface, while for other systems, such as the Ising model, it has access to heat reservoir (the phonons, for instance) within the bulk so that we can see "bubbles" of broken symmetry, domain walls, forming inside the bulk. At the interface between vapor system and the heat reservoir, the mechanism for such a transfer of energy is the conversion of kinetic energy of vapor into the phonons of the reservoir. In the bulk of vapor, however, there is no direct way for the energy to be transferred. Classically, the energy released by forming a droplet can only be carried to reservoir by vapor. This process is extremely inefficient, and its explicit mechanism is obscure. Quantum mechanically, it should be possible that the release of energy is in the form of radiation, where the "radiation background" plays the role of a heat reservoir. Clearly, these two passages of transferring energy to the reservoir should be negligible compared to the mechanism taking place at the interface. So, in any realistic cases, condensation of droplets never happens in the bulk without impurity. It remains to be checked whether such an analysis is physically well-founded and consistent with the experiment.


I am using ZFC as a tool to demonstrate my problematic logic. In zfc we construct a proof system for zfc in zfc (a simulation of a proof is what I mean); we will call it inner proof system. We establish that if there is a proof in this inner proof system that concludes A then A holds. Turing machines can be formalized in zfc with the notions of halting and not halting. The whole question relies upon the argument that the zfc can prove that for every halting turing machine on a certain input there exists a proof of this in the inner system (which I am not entirely sure it is provable in zfc without further assumptions). A turing machine that takes an input (and treats it like an incoding of a turing machine and also as an input) and goes through all possible proofs (in the inner proof system) that conclude that the input halts on itself or does not halt. If such a valid proof was found our machine does the opposite of the proof's conclusion (meaning that if the proof demonstrated halting the machine would enter an infinite loop and if the proof demonstrated not haulting the machine would halt). Under the assumption that zfc is consistent it holds that this turing machine does not halt on itself. This is a construction used to demonstrate the first and second incompleteness theorems in zfc for computer scientists. Assuming the consistency of zfc, the statement that the aforementioned turing machine does not halt on itself cannot be proven by zfc and thus cannot be proven by the inner proof system. Assuming the consistency of zfc, according to the completeness theorem there exists a model of zfc where the statement: "the aforementioned turing machine does not halt on itself" is true and another with this statement false. My problem is with the model in which it holds that this statement is false. This means that the relevant turing machine halts. It follows that there is a proof of that in the inner proof system. It follows that the inner proof system is inconsistent. I know there is something wrong with this logic but I cannot pinpoint it (obiously because I did not formalize this argument sufficiently). Where in this sketch of a proof does the argument fail (for example because a statement is not directly provable from zfc without certain assumptions)? Note: my backgroung is in computer science


"For classical (non-quantum) systems, the action is an extremum that can never be a maximum; that leaves us with a minimum or a saddle point, and both are possible." The above statement is an excerpt from the "Introduction" (preface) of the book "THE PRINCIPLE OF LEAST ACTION - History and Physics" by ALBERTO ROJO & ANTHONY BLOCH. I want to know whether the "For classical (non-quantum) systems, the action is an extremum that can never be a maximum" aspect of that statement is true because it looks pretty definitive. (definitive in the sense certain or assertive) Note: Now I know there are a lot of related questions that look like this, but not any of them looks for a direct and definitive answer for this direct and definitive question, most are descriptive questions for descriptive scenarios and most answer's given are describing particular scenarios, incomplete ones or ones that asserting irrelevance of such question's for actual path determination as we only seek stationary action not whether that is minimum, maximum or inflexion point. (This is intended as to why this should not be labelled as a duplicate, not as a judgement on other questions or their answers as they serve their intended purpose. It is however important to differentiate between the scope of this question and other similar questions. I hope the Phys.SE community will respect the original poster's judgement on the relevance and uniqueness of their own questions unless there is overwhelming evidence to say otherwise.). I have already browsed similar questions as indicated by the system and have not found any definitive question or definitive answer. This definitive question clearly expects a definitive answer, so I hope it will remain a question, not a duplicate.


I am an A-level physics student, and I've been taught that temperature is the average kinetic energy of a particle. So when gas particles are heated, they move faster. This makes sense as an airplane traveling faster does make the nearby air warmer when measured from the plane. Say I release a box of room temperature atmospheric pressured gas in the vacuum of space. Assuming there is no gravity, all the gas particles will be traveling in a straight line as they won't be bumping into other particles. And since space is only a few degrees above absolute zero, the gas particles will cool down after they transferred a lot of their "thermal" energy via radiation, and thus should slow down (lower temperature = lower speed). Looking at a single gas particle this breaks the conservation of momentum, as it is quite literally slowing down to nothing. So what's going on. Now imagine there is indeed gravity in space, the gas particles will eventually start traveling and be accelerated toward a source of gravity. So its kinetic energy increases and so does its temperature (higher speed = higher temperature). First of all it is heated up by nothing without any sort of heat transfer taking place, and would there be any distinction between temperature and speed? Why does this only seem to apply to gas and not solids - a fast moving car wouldn't look hotter would it (ignore friction with air). So can anyone point out where the chain of logic breaks down because it is not making any sense.


Many electromagnetic interactions are modeled as exchanges of a real photons: e.g. an excited electron can relax and emit a photon. Somewhere else, a photon and an electron can interact, "consuming" the photon and leaving the electron in a more excited state. Electromagnetic radiation is modeled as a flow of real photons. The flux of energy radiated through a surface over a time is the sum of all photons' energy that pass through that surface. Other electromagnetic interactions are modeled as exchanges of virtual photons: e.g. the electrostatic force between two electrons is mediated by virtual photons exchanged between them. I understand that these virtual particles are not "real" in the sense that they can't be measured directly, and are just a representation of whatever actual physics our models approximate. They "exist" transiently, in infinitude. In a vacuum field, virtual photons pop in an out of existence, with their total energy and momentum summing to zero. Can electro/magnetostatic fields be modeled as flows of virtual photons to/from on their sources? In the absence of a field, the "positive" and "negative" virtual photons cancel out in energy and momentum. In the presence of a charge, are positive virtual photons flowing one way, and negative virtual photons in the other, relative to the charge? What would the net energy and momentum flux through some surface around e.g. an electron be? Part of why I ask: radiation pressure is quantized in the sense that each individual photon imparts momentum on an object, instantaneously, yet it's difficult for me to imagine the situation for static charges where they accelerate continuously.


When calculating area of a hole in an irregular surface for water flow calculations, what defines that area? I need to calculate the amount of water passing through holes in surfaces in a fixed time, knowing the pressure on either side. This is mostly a straight-forward process (discounting the discharge coefficient, but that is a different matter), but the formulas depend on the area of the hole. What I know about the hole is its boundary. If the surface is planar, this is obvious enough and can be calculated via Green's theorem. But when the hole is in a non-planar surface, it becomes much more problematic. What does the area of the hole even mean? The area of the surface? The hole is exactly where the surface does not exist. Outside the hole there is some surface, which I could presumably find information about (though at a serious increase in complexity), but where the hole is, there is nothing. I cannot even be sure how the surface where the hole was cut was shaped. That information was lost with the hole. "Yeah, there was a kilometer long capped tube there! Good thing that was where the cut-out needed to be." Besides, the water doesn't know how this mythical surface was shaped either, so its behavior will not be influenced by that. Whatever definition of area is appropriate here, it cannot be dependent on the exact shape of some surface. The areas of all possible surfaces filling the hole is clearly bounded below, so there is some minimal area. Presumably this would be the best choice. That is a mathematical rather than physical question, but I want to be sure I am properly handling the physics instead just jumping on an idea.


What is electric current? An electric current is a flow of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is defined as the net rate of flow of electric charge through a surface. How do we generate electric currents? We know charge is the property of a particle due to which it can interact with electric fields and experience electric forces. We use this property of charge to have electric currents; we connect a conductor with a potential difference (battery) so there an electric field sets up in the conductor causing the free-charged particles (electrons) to drift opposite to the direction of the electric field. Hence we get an electric current. But electrons have another property too i.e mass, which is the property of a particle by which it can experience Gravitational forces. I need help in designing a setup to produce electric current using the property of gravitational mass of electrons. So if we have a long conductor wire and a Gravitational field is switched on such that it's direction is along the length of the wire then electrons (which are free) will experience a Gravitational force and we should have a flow of charges, the electric current. But there can be some problems in this: The gravitational forces are way more weaker than the electric forces. As free electrons move away from one end of the wire, they leave this end positive, so a mean electric attractive force acts on the free electrons tending to prevent their further movement. Possible solutions and crux Electrons in a conductor are loosely bound to the nucleus and at any temperature they will have enough thermal energy so that they can be free and do continuous random motion like gas molecules in a container which is also called brownian motion. And our sir has told us that brownian motion ceases at absolute zero temperature. If we connect the wire from top to the earth then the charge vacancy can be fulfilled by charge flow from the earth. So above thought experiment suggests generating current using gravity instead of electric potential difference. Some study materials I don't thought that I will need to do it but now I felt the need to do this after an answer asking for "how one would switch on a gravitational field" and there can be more like that comments in future maybe. So here is a Wikipedia page about thought experiment


The way I work on homework questions, especially for analysis and topology, might be a little different (or maybe not). I would remember the questions and think of them when I run, take a shower, and during other periods when I don't need to use my brain; later I will write out the solution without much thinking. This typically works well even on harder questions that my professor expects more than two hours to solve. However, sometimes I forgot about what I thought before, and my notes for insights, which are random diagrams or phrases on a scratch paper hidden in a stack of papers, are normally either not found or incomprehensible, so I have to rethink the questions. When I actually write out the proof, it is very readable, but as an undergraduate, I have many other time commitments, so I don't have enough time to completely elaborate or type out what I'm thinking. In that case, how can I record my thoughts quickly so I will be able to reproduce them? Also, sometimes I would try different approaches, and typically most of them do not work, so how can I record these attempts and not go into loops? What is frustrating is sometimes I go back again and again on a "branch" that is seemingly close to the answer but is very unrelated and other times I spent a too short amount of time on something that is close to the solution. Is there a way to prevent this? For mathematical research, I would imagine the complexity dramatically increases. For my research project (as an undergraduate working in mathematical optimization), my professor typically introduces to me some very nice lemmas, which help me to prove the optimal goal. However, if I'm working on complicated questions by myself, how can I keep track of all the methods I tried and not worry about forgetting about them the next day? Of course, typing everything out is a solution, but how exactly can we type something that is just vague intuition in our brain instead of proofs for lemmas and theorems?


In machine learning sometimes we build models using hundreds of variables/features that we don't know (at least at first) if they might have a relation with the target. Usually we find that some of them do and others don't. Some of them even have a true relation that we couldn't think of at the begining. Once we build a first model, sometimes we have an idea to include a new variable that we know it has a natural relation with the target and that we didn't think of at first. Sometimes this new variable is sparse, though, what means that it's constant or null for the major part of the data. The problem then is that to use the information that the new variable has, we need to find in some node a cut point of that variable that reduces the loss_function more than all the other cut points of the other variables. However, a sparse variable usually doesn't reduce the loss_function a lot because the major part of the data ends on the same side and only a very few part of the data goes to the other side. Also, when we have that amount of variables, statistically we find cut points in other variables which are not related to the target that reduce more the loss_function for those data points in that node. Not because there is a true relation, but just because of statistics. This ends in overfit and also not being able to use the prediction capacity of our new variable. In this circumstances, what can we do to extract the value of our new variable?


In Einstein's original thought experiment involving "a (very long) train running along a [straight] railway embankment", of essential importance appears the prescription that "[E]very event which takes place along the [railway track] line also takes place at a particular point of the train." The constituents of the train are surely distinct from the constituents of the railway embankment and track (especially since they are supposed to have moved wrt. each other; each constituent of the train and each constituent of the track segment under consideration have separately met each other in passing). Is it consequently correct to say that each separate event involving train and embankment/track has one particular constituent of the train and one particular constituent of the track as participants in this event, such that this particular pair uniquely identifies the event ?, and that each such event has two distinct parts, namely one distinctive part attributable to the participating train constituent (which is characterized by the train constituent indicating being met, in passing, by the track constituent; perhaps with additional characterizations), and another distinctive part attributable to the participating embankment/track constituent (being vice versa foremost characterized by the track constituent indicating being met, in passing, by the train cconstituent) ? Note that the event-parts in question are not presumed to be separate from each other, or resolvable, in a geometric (spatial) sense. My question is not whether and how certain sets of (finely-resolved) distinct events may be considered and addressed as one (coarsely-resolved) event; nor whether any one actual particle (or even several) may be considered fully contained in any spacetime region of finite spatial extent. My question is rather conceptually: Whether (at all) and (if so) How to reconcile speaking of an event as "having distinguishable parts", as described above, while also speaking of an event as "a point in spacetime", or "a point of spacetime" with the understanding that: "A point has no part(s)." ?


I'm currently taking this college calculus course, and this exercise has stumped me. It is in German, but hopefully what it's asking is fairly clear. To summarize, the problem first asks me to prove that the preimage of an intersection of a family of sets is equal to the intersection of a preimage of a family of sets, where f: A -> B and Ui is a family of subsets of B. To do this, I just gave the general example of x being an arbitrary element of A, which means that x is an element of the preimage of the intersection, which means for all values of i f(x) is an element of Ui, which means x is an element of the intersection of the preimage of Ui. Image of my answer cause I can't figure out MathJax: Then the problem asks to show that for a family Vi, a subset of A, an image of the intersection of a family of sets is not in general equal to the intersection of an image of a family of sets. Yet, for every example formula and sets I can conjure up, I always seem to prove that the two sides are in fact equal. No amount of coaxing and complaining on Bing or ChatGPT causes them to bring an example up either - they just keep giving examples where the two sides are equal. So, is this a typo from my professor and this expression actually is generally correct, or am I missing something? What family of sets and functions would make these not equal? Thank you!


How is quantum entanglement different from a controlled experiment where a pineapple is smashed at a high speed to a perfectly symmetric object and measuring one of its piece in air for spin, speed, direction etc. and finding correlations with other pieces of pinapple in mid air? In quantum mechanics information cannot travel from piece A to piece B at faster than light to explain the correlation so the result is seen as unexpected (at least to people outside the science community like me) and at any time a piece can have any property but once the properties of a piece is measured the wave function will collapse and the properties of other piece or pieces becomes clear. But isn't the correlation simply due to how the piece or particle was generated? No information needs to travel because the particle is simply continuing to react to the same common generation event? If we can call it information isn't it already with all the particles or pieces of pineapples in air already? I (am a beginner, used to be interested in physics but nobody gave me answers so lost interest eventually and took a career in finance. this one was one of my questions) also view quantum mechanics as deterministic rather than probablistic but think the probablistic approach is necessary because we cannot measure all the variables with our very limited measuring and processing capabilities. For example, a roll of die by a machine is deterministic if we measure and process everything from the light fluctuation in power, air movement in real time, mass of the die, initial position, curvature, area of contact, etc. but because that is too complicated we use it as random with a probability. I feel in quantum physics we are doing the same thing as it is simply more practical, but everybody I talk to truly believe I am wrong. Mentioning this as it is related to the question, that is, the properties can be calculated without measuring in quantum mechanics if we have virtually unfathomable knowledge, measurement capability and processing power. We don't have that so in my view too probablistic quantum mechanicsm is the way science can progress, so no question about that, but is it established that quantum mechanics cannot be deterministric in reality? Just want to know where and why I'm wrong. Thank you in advance.


It occurred to me that the limits of possibility to the nature of the universe is it is either deterministic ie we are all at the will of natural laws that determine the outcome of events from the moment of inception and we are philosophically dust in the wind. Or is the world random and our future is uncertain and indeterminate we have freewill and there is nothing governing our future but our free choice or is it both of these scenarios happening concurrently. I am no expert, but from what I gather from quantum mechanics is that the cause of events at the quantum level is indefinite and the outcome of these uncertain and probalistic. There is the principle of uncertainty in which determining the property of a particle results in an uncertainty in determining the nature of another property of this particle is it this principle that makes it difficult to get precise measurement that is a factor in us being able to predict the outcome of events in a linear deterministic equation that gives us the unpredictable nature of quantum mechanics ie there is an equation governing the outcome of events at the quantum mechanical scale and classical scale but we are unable to feed that equation with precise data to enable us to accurately predict their outcome ,kind of like how dynamically chaotic systems are mathematically determined by there initial condition but a small perturbation or difference in initial conditions results in a different outcome. I know its that old chestnut I am not a mathematician or physicist but there can only be three possible scenarios any response will be welcome even if its disparaging.


The proofs presented in lectures, textbooks ect. are usually cleaned up versions that show just the necessary steps for logically proving the theorem, not the thought process that went into the proof. To give a concrete example, I'm working through an (overall quite good) MIT Open Course Ware class on real analysis. I just paused a lecture where the professor said: Now, when you write a proof, as you'll see, it's going to be magic that somehow this h does something magical. That's not exactly how you come up with proofs. How it comes up is you take an inequality that you want to mess with, you fiddle around with it, and you see that if h is given by something, then it breaks the inequality or it satisfies the inequality, which whichever one you're trying to do. And proceeded to just write the finalized proof. Because that's not the part I care about. The main thing I want to learn is the part were you do the fiddling around to come up with the proof to begin with. Verifying proofs other people made is (relatively) straightforward, and the property being proven (Q doesn't have the least upper bound property) is important but I'd be willing to take it as an assertion if I was just trying to learn about Q rather than how to do analytical proofs. I would really like to see examples of someone who is good at proofs showing their work in creating a new one including the dead ends, fiddling around ect. I have tried to teach myself this step by just going out and proving things, and have made a little bit of progress but thing I would greatly benefit from more examples of deriving proofs.


In gel electrophoresis, there needs to be an electric field created in the gel. I realized that I have some gaps related to my understanding of electromagnetism because I cannot fully understand in what conditions a field gets created. Are the electrode plates on either side in contact with the gel or insulated, and if so, how does this affect the field? My understanding is two insulated plates with different voltages would still create a field between them. Is the issue that the surface charges in the gel would align themselves to cancel out this field on the gel inside? Thus, do you need to actually pass a current through the gel in order to create an electric field inside? I think this is related to the nature of the electric field in a current carrying wire (where I treat the gel as a really thick rectangular wire), which I also realize I don't understand. Namely, do the electrodes need to span the full width of the gel rectangle to create a uniform field across the width? More generally, what are the possible ways to create an electric field on the interior of a conductive object such as a gel? Do you have to pass a current through it from one side to the other? My other thought is that a changing magnetic flux would also create an electric field, although it would be of a odd shape. But would this induced electric field exist on the interior or also be subject to some cancelling effect as is the electrostatic case?


Recently I saw a conversation between Sean Carroll and Slavoj Zizek concerning the MWI. One of the questions that drove Slavoj concerned with this question of Ontology vs. Epistemology, as the way Sean had described the current understanding of MWI is that we merely 'don't know' where we are on the wavefunction which seems to keep the openness of reality at the level of epistemology rather than ontology. The question I have concerns entanglement with new information that an object encounters as it is coming in from outside the object's light cone. In my understanding when entanglement-measurement occurs, some pair (or more) objects in the wavefunction take on a correlated value, but then the price of this is that other values become randomized (so electrons take on opposite spins, but then the spin of each electron is not determined). What I am wondering is as we pass from before the moment of entanglement with new information coming in from outside the lightcone to after that moment, some of the information will be consistent (or not consistent) with our current history - that information gets zipped with (or excluded from) the entanglement event, but all the sort of 'random' information that is really new... is it the case that the wavefunction of our object has a different space (or location on the wavefunction) for each of these random timelines before the encounter with new information - or is that space created at the moment of entanglement with the new information? That is... is it the case that our original object is actually already always split into all of its future possible timelines in a way such that various points on its original wavefunction can be assigned to various final 'futures' - or is that split not possible in principle or in the mathematics from the beginning of the calculation (so that in reality many futures pass through the same original point on the wave function)?


I would like to forecast demand count time series of taxi fleets at different locations on the map at different points in time. I.e. multivariate demand Time series forecasting. Given hierarchinal demand time series. where the region level time series is the sum of district level time series at each point in time and district level time is the sum of city level time series at each point. sum of all the city level time series at each point in time should equal region level time series. There are also covariate time series (both numerical and categorical) associated with each city's demand time series. e.g. rainfall level, temperature, traffic level, taxi supply amount, date information like weekday or weekend, holiday information and Time invariant features like type of city. (e.g. industrial, CBD, residential etc. ) Note: The city level time series are quite sparse ( have many zeros). How do I go about using a bayesian framework/statistics to model and predict the mutivariate (city-wide) demand time series? The reason for choosing the bayesian framework is to estimate uncertainty in the prediction. Build a strong prior sequentially over time which can capture the fundamental data distribution/patterns without being affected by noise in instantaneous data. incorporate the various pieces of covariate information together to enhance prediction accuracy. Here are my thoughts: Use summary statistics of covariate time series and response time series like ACF to compare similiarities between and try to explain causation. Try to improve dataset quality by (probably using bayesian a hierarchical model using partial pooling) to combine sparse and low quality (lowly correlated time series to response time series) covariates to model a better covariate. Use bayesian sequential learning to sequentially accumulate learned patterns over time into the prior distribution. Could someone give me your thoughts and suggestions on how to go about doing this? Thanks.


When a mathematician says that two categories are the same thing, they may mean there is an equivalence or an isomorphism between them. I am wondering if there is a precise way we can say that two categories that are isomorphic are different categories. I have no idea what I mean, but perhaps it has something to do with models of categories in Set. I just feel there may be justification sometimes when two mathematics feel that two isomorphic categories should be considered independently. Is this possible? Does it ever come up in discussion? For example, consider this language on modules "The concept of a Z-module agrees with the notion of an abelian groups." This language "agrees" really means there is an isomorphism of categories. It seems like the theory of categories simply deletes any way of seeing them differently, but is there another way of seeing them where one might say"No, we consider them differently!" I think this might have something to do with the way computer scientists talk about monads in a different way than mathematicians. If you've ever looked at these two literatures, they are nearly incompatible at times. This may have something to do with the notion of "up to isomorphism" in category theory. When, in category theory, do we say, "no, I won't forget details that are deleted when we work 'up to isomorphism'"? Is there category theory that doesn't work "up to isomorphism"? Which category theory concepts and constructions only work up to isomorphism and which, if any, go beyond this?


Where from Hawking radiation actually arise? I would like to connect the answer with the technical derivation along the lines of the original calculation by Hawking (a modern account of which is given in Harvy Reall's notes). I have seen this where the three answers apparently don't agree with each other entirely. i. The first answer points out to tunneling process (whose math I would like to see) or particle-antiparticle production near event horizon none of which isn't directly connected to the original derivation. Well, the particle-antiparticle picture is claimed to be somewhat connected because we can find their mention in many "formal" places (also) including Hawking's original paper. (Somewhere else it is claimed to be wrong/heuristic). In the technical derivation in which step exactly do we utilize this "vacuum turning to particle-antiparticle pair" thing implicitly? I am actually partially convinced with Ben Crowell's answer that the radiation comes not exactly from the horizon but from a region away from it. But then a) How do we justify the use of geometrical approximation far from the horizon in Hawking's derivation? b) In an alternate derivation (for Schwarzschild black hole---not a collapsing spacetime like in Hawking's derivation) detailed in Carroll's book, the author uses Unruh temperature for a static observer and redshift it to infinity to find Hawking temperature; there also the static observer is necessarily near the horizon---otherwise we can't apply the Unruh temperature formula correctly because only the near-horizon geometry of Schwarzschild spacetime is Rindler. These two observation seems to point out the fact that the radiation is necessarily coming from near the Horizon---although the second point seems invalid because accelerated observers do Unruh-radiate in Schwarzschild geometry too. And how do we know that the particles are not coming from a time when the black hole didn't form? If we look at this diagram (adapted from Harvey's notes) used for backtracing things become more confusing (to me) Reading this would be like (according to an observer at infinity)---A wavepacket started from past null infinity away from the horizon at a retarded time after which if it had started it would go inside the hole. It went near the horizon before it's formation at very late times and stayed near the horizon for the rest of it's life while coming to us---the observer. It was joined by another wavepacket coming from infinity which started at a later retarded time, hence heading for the event horizon but got scattered towards future null infinity instead. Much of the results of the derivation follows from the change in frequency of the Schwarzschild modes due to extreme blueshifting near the collapsed matter . So apparently the collapsing spacetime(partially Maimon's POV in the linked answer)/some stuff at past null infinity created the Hawking radiation. So how do we prove that these two are not responsible and the event horizon is the culprit of creating Hawking radiation, from this backtracing framework? As mentioned above, there are reasons to believe in event horizon creating the Hawking quanta but I can't understand how that fits within Hawking's original derivation. This might seem like a duplicate of the linked question but since most of the answerers there are not active members of this community anymore and that many years have passed since the Q/A and (possibly) we have a better picture, I took the risk of reposting essentially the same question but nonetheless with new issues raised.


Consider a loop gap resonator for electron paramagnetic resonance which has static (but sweepable) magnetic field in one direction, "x", and a GHz RF magnetic field in a direction "z" that is perpendicular to the static field." I had assumed that I had built the loop-gap such that the H_z field was constant over the sample volume. We get a reasonable EPR signature from it, can identify some hyperfine components, etc. I finally got around to doing a finite element for the field and to my surprise, the calculation appears to show that I am generating positive and largely constant Hz on much of the "left" side of the resonator and exactly the opposite Hz on the "right" hand side of the resonator (i.e. negative Hz, with zero field in the middle). I am now completely confused. I would not have thought such an RF field could give any EPR signature because the spin contribution on the left should cancel that from right. If so then my FEM model is wrong and I need to look there. But maybe not... in which case I need to look into fixing whatever is driving the wrong eigenmode because obviously I will get more signal if all the Hz is uniform. So that's my question can you get an EPR signal from an Hz which is flat over two areas of the resonator but such that the average Hz is zero? Or maybe in math terms: Given a computation of the RF field in a resonator and a theoretical spectrum of a species, how can I determine the corresponding EPR spectrum of the sample? E.g. does it depend on |H_z| or H_z ?


In everyday talk we say things like: we must use the fundamental theorem of calculus to calculate this integral; we must use the results of analysis to proof the fundamental theorem of algebra; we must have a function that maps well formed formulas to numbers and that is bijective (Godel beta function); we must have a ring with the fundamental theorem of algebra to be able to proof this result; etc... Sometimes it seems that, when we say this, we are saying that the theorem in question must be made an axiom, then we will try to investigate what results hold with it, or without it, like in the case of the fundamental theorem of algebra. Sometimes it seems that we are saying that to proof some theorem we need to use a central theorem that mobilizes a particular set of axioms, this seems to be the case with the fundamental theorem of calculus, or the case of a theorem that say that some function or property exist and we must use it. In this last case, which is the case I am interested, it seems that the construction we are trying to make needs another construction, that in turn needs a set of axioms. But it's not the case that, if we simply fix it as an axiom, we could see what thing depends or not on it, for fixing it as axiom would turn the axiom system redundant (meaning: one or more axioms could be eliminated). So, my question is: what means to say that a theorem depends on another theorem? Is there some logical analysis of what means to say this?


Constant motion can not be detected by neither particles (because of inertia) nor mechanical waves ( because they need a medium ). However when you consider light for example and assume it does not need a medium to travel through you could detect constant motion. For example, when someone is traveling at the speed of light in the x direction, and fire some photons in the x direction the person would see light as standing still while someone who is not moving will see the light passing at c. So why would we consider constant velocity to always be relative and so on undetectable. Clarify: Its obvious that the discovery was made, however for Einstein to actually get anywhere he needed to assume c to be constant. Which is exactly what I am wondering about, how did he come up with this. So that is why I am asking about constant relative motion and trying to come up with a thought experiment which proves the distinction between being stationary and constant velocity reference frames. However every single object with or without inertia which includes particles and mechanical waves are shown to give no proof of one or the other the one being in constant velocity. Which brings me to a light wave traveling through no medium and then would suddenly give us a way to detect constant motion, because now the so called ether wind is not causing the wave of light to go slower because the "wind" is moving in the opposite direction, no, it is now possible to say that light moves at a different speed for one of the two experiments on board the two and then proof that the one where c is slower, is the one actually moving and the one other one stationary. So from knowing this to be forbidden there is only one logical conclusion c must be constant for all observers, its the part where knowing this to be forbidden to c is constant for all observers is where I am stuck. Ps: please don't just give me a mathematical proof, this doesn't really help me, Much appreciated!


A 'blunt' statement is when someone says things to the point and factual. But wouldn't 'sharp' (or some other word that implies frankness or sharpness) be a better word than 'blunt'? As blunt has the meaning that it is not sharp. When someone says something that is not direct or something soft (and if I am to make phrases or rules in English) then I am more likely to use 'blunt' for such a thing. All I am saying is the real meaning of the word is not appropriate(actually opposing) for the phrase. Being direct is never 'soft' and can hurt and blunt real meaning is more close to 'soft' and cannot hurt you (compared to a sharp object). Blunt: (of a cutting implement) not having a sharp edge or point. "a blunt knife" (of a person or remark) uncompromisingly forthright. "a blunt statement of fact" Etymology here points to some interesting meanings and usages but none quite explain the usage in phrases . Edit: what I am trying to say explained better in comments by other users "why blunt means what it does, when it doesn't sound like a piercing effect" The most common, physical meaning of 'blunt' is 'dull', 'rounded', or lacking edges - the antonym of this context is, as you say, 'sharp'. But when 'blunt' is used more metaphorically to mean 'direct' or 'unnuanced', the antonyms are very different. Also "Blunt knife hurts more" is creepy thinking and language usages do not take shape on those lines. It is usually simple popular opinions that turn to usages/phrases.


According, to definition, provided, as I understand by Newton, there are frame of reference, where all constantly moving bodies keep their velocity constant, untill the force is not applied to such bodies. Some sources, in particular Russian Wikipedia inertial frame of reference article, state, that there is no inertial frame of reference in the world (translated from Russian): Absolute interial frames of reference are mathematical model, and do not exist in real world Or, for example this answer from this forum: When you ask for a "perfect" or "true" inertial reference frame you are asking for something that cannot be answered in physics. But each time sources either do not explain why, probably considering it obvious, or the explanation is not satisfying for me. I do not know, do I understand the reason correctly. Is the condition of frame of reference to be inertial applied for any time? Do constantly moving bodies in such frame should always move constantly if the force is not applied, so if the force even only once is applied to the body, which is linked with inertial frame of reference, that frame will never be intertial? Or frame can variate: for some period it can considered absolutely inertial, even in real world, for some period not? Is there no inertial frame of reference because there is no constantly moving bodies, i.e. every body have at least tiny, yet acceleration? If yes, why all bodies have acceleration? Due to mass, and the fact, that gravity force goes to infinity? Okay, I understand, that massless particle will move at speed of light, but what if there is a particle, that doesn't not affected by gravity or eletric field, i.e. will be "fixed" at space? Will it have intertial frame? Also what about particle with speed of light? Since their speed is constant, don't they have interial frame of reference? (Extra) Also, since the motion, even non-constant (i.e. with acceleration), as I understand is relative, then for some frame of reference, attached to the accelerated body, there are "constantly" moving bodies, that "actually" move with acceleration, but relatively to such frame - constantly. I do not understand why this frame is not inertial? Because, by definition, bodies should move constantly and without a force being applied to them? But how do we know is the force applied? No, if someone punches a ball, okay, but what about fields of something like this?


I am in advanced UG (in Elec Engg). I was consolidating a mental model to help in (most of) engineering applications. I read Griffith recently. I learned that there exists no magnetic charge, (although, as Griffith said - how cute the scenario would have been, if they did.) Intuitively so; as Magnetic fields are engendered by time-variance of electric phenomenon (in field or charge (as current)), I never expected their existence. But, why the world needs a concept that is "charge"? The Gauss law, firmly relates the charge concept to the field. Intuitively, the field fluxes out of charge. It may seem that charge is engendering the field. But, let's believe in absence of such concept; the field is inexplicably (concentrated) sourced at "some" places - say S. Here is an argument - It is ambiguous whether the charge carries the energy or field. In the absence of charge concept; the field carries the energy; then the source to this energy are S. Energy is a transient phenomenon; Charge concept could represent the source to these new perturbations in the field. Coming to the title of the question. There is no magnetic monopole. Why is there an electric mono-charge? Clearly, charge represents transfer of energy from other domain of energy into electrical domain. Transfer of energy is relative. Can't we say the other half of the charge exists in the other domain of energy? (imaginary pole). I have an argument to address the polar nature of charge as well. But, I have already revulsed enough physicists here. (Excuse me, I was just thinking.)


In Einstein's equations the curvature of spacetime and energy-momentum-pressure density are correlated. Is it clear when changes in matter energy density affect causally to curvature and when changes in curvature affect to energy density? Or can these kind of changes be non-local just as gravity correlated with matter in general? Is orbiting-like repetition needed for causality considered? I'm studying the principle of locality in physics. But the curvature of spacetime means that the fly-by has momentum of spreaded curvature of meeting bodies and those keep colliding into each other early. Where is the source of the cause? The fly-by anomalies has been mentioned. Are there any fly-by research projects that take the principle of locality into account? I appreciate answers that focus on localizing the physics so that time travel or any exotic negative energy etc. is left out. My problem is identifying on what basis something is cause or effect with gravity or when it is a non-local correlation of the curvature of spacetime. For example, gravitational waves act locally, but gravity acts non-locally in general. In which situations do changes in the gravitational field transmit causal information? Especially in fly-by passing: where is the origin of the causal information? Another case. Point-like fermion particles obey the Pauli exclusion principle. Although they appear to be free, they are subject to field forces and together other particles can "feel" tidal forces - if I interpreted correctly what is meant by "feeling gravity". Thus, the local limiting flat tangent space must be considered relative to the surrounding other tangent spaces (even if it is also point-like) and real (e.g. em) forces occur. The farther the particles are from each other, the more significant are the changing field effects compared to Decoherent interaction. How much has been loaded into the finished space-time curvature into the future without the exchange of causal local information and energy? Does it depend on the internal structure of the particles or the size, the mass? On the relationship to the dominant curvature?...


The conservation of energy says that it can neither be created nor be destroyed, it can only be transferred from one form to another. i had this doubt in my mind for a long time that why do we need to save energy if the total energy is conserved. Energy can never be destroyed so why do we need to save energy?


Is there something we can tell about the function, if we know that its Hessian matrix is the zero matrix? Is it concave, convex? What about quasi-concave and quasi-convex? As I am a beginner on this topic, I would be interested, if there's maybe an intuitive answer to this. Thanks!


Why, when calculating contour integrals, we need to avoid the singularities inside the region delimited by the closed path we're integrating over? Wouldn't we need to care only about singularities laying on the actual closed path?


I am looking for exercises/examples of "simple enough" morphisms between varieties being flat/non-flat, with solutions. I have only ever found a handful of examples, however very rarely with solutions to check my work. Any help is appreciated.


The English word "pepper" comes from Latin word "piper". But why is there a double 'p' in the English word, when there is no double 'p' in the Latin word? Where does the English spelling of the word "pepper" come from?


I came across this sentence: The dogs bounded past their leader. What does "bounded past" mean? I know the past tense of bind is bound. But here the verb "bounded" is used. Is the root of "bounded" different from the verb "bind"?


Is there a point of balance where the gravitational pull of a sphere of electrons is equal to their electromagnetic repulsion? That is to say, could it be possible to create stars that are made purely with electrons and that are stable and don't fly apart?


I initially tried to convert the whole thing into a Conditional via Material Implication but just ended up in a loop of assuming the antecedent and then using DeMorgan's. If anyone has any insight I would appreciate it.


Does the word "alumni"/"alumnus" always pertain to a particular university or school, or can you say e.g. "history alumni" meaning all the people who graduated in history at any university (e.g. in some particular geographical region)? Would "history graduates in France" be more correct? Thank you


In an integrable quantum system (say XXZ model), where there is an extensive number of conserved charges, does the set of local conserved charges obtained from expanding the log of the transfer matrix form a complete commuting set? In other words, does specifying all conserved charges obtained from the transfer matrix specify a unique state of the system?


I have recently started to study quantum information theory on Nielsen and Chuang book, but in order to understand the theory better I need to solve more problems, there are in this book. Can you recommend a book or a book of exercises, where there will be problems on the topics from this book?


By reading the first sentence in this article I interpret that, for every projective space, every isomorphism of its underlying vector space gives rise to an isomorphism of projective spaces. Is this also true for affine spaces (and their underlying vector space)? Or, at the very least, is every isomorphism of affine spaces induced by an isomorphism of the underlying vector space?


I've been browsing through a bunch of questions here about how diagonalisation fails for Q. What I am finding hard to wrap my head around is how every bijection from N to Q, if used with the diagonalisation argument, will surely yield a rational. I can't really see how we can go about this. I'd appreciate any help.


I can't think of any reason as to why one would have a directional derivative using a vector that's not of unit length. It would always "mess up" the derivative by scaling it by the magnitude of the vector, would it not? I tried searching online but can't seem to find any particular purpose.


In TexStudio, one can open two or more tex files (or .bib, .aux, etc.) in the same window. But with my little experience with TeXShop so far, if I open a tex file and then a bib file, they open in separate widows. Is it possible to make all files to appear in the same window in TexShop as well?


Is the following sentence grammatically correct? The word "if" is not used in this sentence, which I'm not sure is a mistake or not. And if anyone has a link to a reference on conditional clauses within relative clauses, that would be great!


Specularity is a value that defines how glossy or matt a surface is. For example, a mirror is highly specular, but a brick wall is not. Is the specularity of a material a fraction of its total reflectance?


I am relatively proficient in most material found in high school math courses but one of my weaknesses is in parametric equations and expressions; I really struggle to understand and apply them. Does anyone know of a good book or online resource written by an expert that really introduces and explains them step-by-step from the very basics? Thanks in advance.


To estimate the parameters of a truncated distribution (lognormal for example), we can use the Maximum Likelihood Estimation or Method of Moments. For the Method of Moments Estimation, one needs to write down the mathematical expression of the expected value of the truncated lognormal distribution. Is it possible to do so? Or can we use a numerical method?


In the textbook I am reading for class it is talking about the interactions of photons with atoms and how when an atom emits a photon it must have the same and opposite momentum of the emitted photon. I am confused on what the book means by "the atom must also have a 'recoil' kinetic energy".


Orchestra seats are mentioned in this answer but why are seats closest to the orchestra (or stage) called "orchestra seats"? They are certainly not in the orchestra. What is the history/origin of such an expression?


As a speaker of English as a second language, I've long been curious to know why English speakers would choose to say "Easier said than done" over "Easier to say than do". Why should it be said in the passive voice, not in the active one? Or is it simply a matter of being idiomatic?


Forgive my naive usage of the term 'classical physicists' but what I mean is physicists who were either unaware of probabilistic quantum mechanics or refused it entirely. I am interested in whether they thought there was an infinite amount of matter in the universe, and whether space was infinite.


The parameters is getting updated when you change the project rate. The parameters is being updated when you change the project rate. Is there a difference between these two statements? Which statement is more correct from a grammatical point of view?


When the IV drip for a patient is completed, the patient's blood will flow back. This is apparently due to the pressure difference. However, IV needles are inserted into veins in the direction of blood flow. Hence, the blood is flowing against the direction of blood flow. Is the pressure difference so high? Can someone explain this? A non mathematical approach is fine


I read on this page that an endomorphism of degree one of a smooth projective algebraic variety must be an automorphism. The proof uses Zariski's main theorem. My question is this: are there examples of nonsmooth projective algebraic varieties having endomorphisms of degree one that are not automorphisms? Note that for a morphism, having degree one is equivalent to being birational.


What would you call a place or thing that's well-known or renowned but only among a locality, family, or small group? I've looked into it and have failed to find anything that fits.


Could you please provide suggestions on how to include a list of my publications in my PhD thesis? I would like it to appear at the end of the thesis, following the "References" section. Additionally, I would like the title "List of Publications" to be listed in the table of contents, with dotted lines connecting it to the corresponding page number.


I'm currently wondering how packages like tikz can possibly work. I've read the TeX Book and found no evidence of TeX primitives capable of drawing things. LaTeX is just a collection of macros so the use of LaTeX cannot provide this capability. Is the pdfTeX engine providing the primitives necessary for producing drawings? If yes, which are those primitives?


Let's assume there are two observers 'A' and 'B'. B is at ground and A is moving with an uniform acceleration ('a') with respect to B. Will A be an non inertial observer with respect to B? (As we know that non inertial observers are those which are moving with an acceleration with respect to an observer).


The ring of representations of the symmetric group is isomorphic to the ring of symmetric functions. The Schur-Weyl duality relates the irreducible representations of the symmetric group and that of the general linear group. Using the Schur-Weyl duality, is there way to find the ring of representations of the general linear group in terms of the symmetric functions ?


I would like to know if there exist concepts of ergodicity and mixing properties for projective representations. If they do, do these properties exhibit similar characteristics to those observed in the context of unitary representations? Thank you in advance.


HAMTREFOIL = {(G,s,t,u,v) | there exist paths s->t, s->u, s->v such that every vertex(except s) belongs to one of the paths} I want to prove that HAMTREFIOL is NP-complete by reducing Hamiltonian path to it but I am stuck in the construction part. Any tips or solutions would be great. Thank you.


I want to make LaTeX Workshop's PDF preview fit my Visual Studio Code's theme, and I managed to change the background color of the previewed file following this guide, however if I make the background a darker color, the black text becomes very hard to read, and I'd like to modify it to a lighter color, but I'm not sure how.


I am studying formalism of QM from the book by Griffiths. He illustrates via two examples that momentum and position operators have no eigenfunctions in Hilbert space. In that case, how can we expand a general state vector (in Hilbert space) in position/momentum eigenbasis, as the eigenvectors constituting the basis do not lie inside the space?


I am an art student and I think my work could benefit from broadening my horizons on mathematics. I am mostly interested in Topology, but starting from high school mathematics I don't know where to begin learning. Could you give me an order of subjects to learn before starting topology? Thanks in advance.


In UK English, the idiom "more fool me" means something like "and I'm a fool for doing so". But how might you try to understand the underlying syntax? Is "fool" an imperative here? A noun? Is the idiom an ellipsis of a longer phrase?


Given a convex quadrilateral, and a point inside it, I want to find all ellipses that are inscribed in the quadrilateral and passing through the given point. My attempt: is outlined in my solution below Your comments, hints, and answers are highly appreciated.


Which is considered (more) correct: wracked by [X] or wracked with [X]? Example sentences: He is wracked with grief. He is wracked by grief. The Cambridge definition of wrack contains examples of historical use of both forms. Thanks! (Apologies if this has already been asked; I searched first, but only saw the more common question of rack versus wrack.)


The sign of the Jacobian determinant of a two-dimensional transformation tells us if the transformation is locally orientation-preserving (if it's positive) or locally orientation-reversing (if it's negative). Is there a similar interpretation of the Jacobian determinant for a three-dimensional transformation?


Could anyone help me understand the use of 'that' in the below sentence? (I took this example from an old book) The morality of Buddhism is essentially that of the Upanishads except in the matter of forbidding sacrifice. What is the role of that in the above sentence?


Distinguishing between distinct unitary irreducible representations is important from the point of view of distinguishing between different sorts of particles; the eigenvalues of the Casimir operators provide a way of doing this. I have never properly understood why this is the case. Is there a good way of seeing why any two distinct irreps must, necessarily, have different eigenvalues for at least one Casimir?


Consider a location-scale family of random variables. Is the variance of a random variable within the family always monotone in its scale parameter? What if the random variable is also symmetric around zero? I would be interested in a proof or a counter example.


Is there a theorem that states that Integer Linear Problems with a Totally Unimodular constraint matrix are solvable in polynomial time? If the answer is positive, is it also valid for Mixed-Integer Linear Problems? And in which books can I find such theorem?


I understand that special relativity is what help creates a magnetic field in currents. But what I don't understand is how this plays into how changing magnetic fields lead to electric fields and how changing electric fields lead to magnetic fields or if special relativity even plays into this. If anyone can clear up my misunderstandings, that would be greatly appreciated.


Recently I started learning about matrices and what you can do with them. I learned how to add them, subtract them and multiply them, but I found out that apparently you can't divide them. Why is this?


Examples: "Two miles is too far to walk" vs "There are two miles" In the first situation, the noun is seen as singular, while the noun is seen as plural in the second situation. What is the reason that allows this noun to be both plural and singular.


I'm reading this paper "Billiards In Polygons" by Boldrighini et al. They say that polygonal billiards have zero measure-theoretic entropy, because a given element of the configuration space is almost surely determined by its forward hit-sequence(the sequence of sides it collides with). Why does being so determined imply zero entropy?


I know that when a cavitation bubble collapses, heat is given off and a shockwave is formed. What else happens? Is there increased water pressure in that region? Can the intensity of this implosion be measured using a hydrophone?


I'm looking for a word to describe a person that is: always trying to treat others nicely, kindly, afraid of saying or doing something that makes others upset, never say or think about what he/she really wants, only try to make everyone happy. This word is usually talking about a child or woman that has suffered from something terrible that made them become this.


There is a famous song called Baby I'm-a Want You. The name sounds wrong. Why is it not "Baby I want you"? Auxiliary verb "am" and article "a" look excessive in this phrase.


What is the intuition behind Koszul graded-commutative algebras and Lie algebras? Why is it an interesting property to study in commutative algebra? I know why it's interesting in homotopy theory but what about in algebraic geometry or commutative algebra?


Why does a changing magnetic flux induce an electric field? When a coil moves through a magnetic field, the induced emf is due to the Lorentz force. But why does a changing magnetic flux produce an induced electric field and emf even though the coil is stationary or there is no coil?


I'm a first year graduate student in math and I'm currently studying fiber bundles. The definition is clear and I understand how it generalize concepts as (co)tangent bundles or vector bundles. What for me is unclear is why we are interested in spaces which are locally product of spaces. What is in general its usefulness? Thanks.


I assume that the Milky Way has a dark matter halo just like any other. If that is the case, if we look at a huge part of our own galaxy, do we actually see the gravitational lensing effect? How intense is it?


When I tried to solve some motion questions, I got complex numbers for time, displacement, etc. And my teacher said my answer was correct. Is it possible to have a complex number solution in the equation of motion? What does the complex number mean in the equation of motion


In a, let's say rectangular container, the water surface always aligns itself perpendicular to the direction of net force acting on it. Why exactly does it happen? (For example when this container is accelerating towards left, the surface of water aligns it self in such a way that the right end of the liquid is at a higher level than left end.)


While researching rotation curves, I've noticed a variety of velocity behaviors in different galaxies. In some, the velocity decreases, in others, it remains relatively constant, and in some cases, it increases, particularly in the outer regions. Is there a mathematical equation or method to objectively determine these patterns rather than relying on visual inspection?


From the movie "No Time to Die": Nomi: The world's moved on since you retired, Commander Bond. Perhaps you didn't notice? Bond: No, can't say I had." Why did Bond use "had" and not "did"?


A goat is tethered to a rope of length L. Each end of the rope is attached to a post, the two posts being distance D apart. The rope passes through a ring attached to the goat's collar and can move freely. What is area that the goat can graze?


Is is correct to say that, elementary particles have different masses, because they have different coupling strengths to Higgs field? And if yes. Does it make sense to question, why they have different coupling strengths?


Newton merely stated the law of inertia as: An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. How does it prove the existence of inertial frames? I see no mention of it in the original law.


In mathematical logic, there is a notion of "predicative definitions". I understand what a predicative definition is, intuitively, but I am interested in a formalization of that notion. Has any mathematician ever formalized and made rigorous a definition of predicative? I would love to see such a definition.


As per the conservation of mass, matter cannot be created or destroyed. Doesn't this contradict the big bang theory? Like, it states that it all started from a single point. But seeing the massive size of the universe, isn't it weird that a single point held this much debris for the formation of planets, stars etc.?


I'm new to Tikz and I've already made a few diagrams. However, I'm having trouble making this one. Would it be possible to help me? It doesn't matter if the colours aren't right, I'm just trying to draw the central block and the two arrows marked "seuils...". Thank you for your help.


I am looking for texts that discuss the logic of the game of chess. I am sure there are a few such texts out there. Such a text might formalize chess in first-order logic. I would be very grateful if someone gave me a list of texts on the logic of chess.


As the title says I want to know if the result is true, ie, if a linear projection in a normed space is bounded iff its kernel and image are closed. I know the result is true if the space is complete but I cannot prove nor find a counterexample in the non-complete case. Any ideas?


I'm a final-year physics degree student and I want to know about QFT, but my knowledge about tensor mathematics and groups of symmetries is really low, as my university doesn't offer us advanced mathematical subjects. So, would you please suggest me some books or any other resource that make it clear and easy to understand these mathematical basics and that digs into QFT?


Consider the topological group(s) in https://en.wikipedia.org/wiki/Solenoid_(mathematics)#Pathological_properties This seems like an interesting and bizarre group. It is obviously not a Lie group and obviously not a manifold. However, I was wondering if it satisfies geometric properties like local contractibility? What about its homology groups? Is there a nice reference for the topological properties of this group? Could anyone tell me more about this group? Thanks!


I'm currently trying to figure out a function for a real coordinate space, which satisfies all the rules for an inner product, but with which the standard basis is not orthogonal anymore. Does such a function even exist? Or does someone here know an example of such a function?


Recently, I'm learning FFT and I want to read this original paper("Theoria interpolationis methodo nova tractata", by Carl Friedrich Gauss. ). But I can't get a valid url from Internet. Would someone give me a valid download url about this paper(or just send me that file via any viable method)?


There are many quantum Monte-Carlo methods. Many of them can be used to calculate or estimate the ground state energy. The problem is, is the estimated energy an upper bound of the true ground state energy? We know an estimate based on a variational wave function has this nice property, but is it the case for the QMC methods?


When we speak about Discrete random variables we have PMF. When speaking about Continuous random variable we have PDF. So my question is if we already have PDF or PMF with us then why do we use or calculate CDF, what's the intuition behind CDF


This is a question about the proper use of the word "thrust." Suppose a freewheeling prop is turning in the wind. Is the air coming out the back (downwind) side properly called thrust? I understand the basic physics. I just don't know if thrust is the proper term to use for the air coming out the back.


I was wondering, if I theoretically shake a magnet like a sine wave with a very high frequency (Let's say a frequency in visible spectrum), will I see a visible light coming out of it?


I am looking for a verb with the meaning of ceasing to like something/someone. To dislike is not fitting here, as it means the opposite of like, not going back from liking something to having a neutral opinion about it.


For a differentiable function f, we know that the gradient is orthogonal to the level sets of f. What if f is Lipschitz continuous? we know that its gradient exists almost everywhere. Is the gradient (when it exists) orthogonal to the level sets?


I am wondering when the Legendre transformation is irreversible. I know if Y is a linear function of x, then the Legendre transformation of Y is irreversible. However ,what if Y is a multivariable function? Could anyone give necessary and sufficient conditions when Legendre transformation is invertible? Thanks!!


I have to work with the radial velocity dispersion. Could someone help me to understand what this actually is and how I work with velocity dispersion? What happens if you multiply the radial velocity dispersion with the radius? Is this something that is often done? In which systems does the radial velocity dispersion change with the radius? How is it related to the velocity?


I know that the purpose of differentiation is to find the rate of change of a function. But i don't really understand how this is concept can be used to find the tangent of a specific point. It would be greatly appreciated if you can explain this to me.


You can use the Euler Lagrange equation to show how the shape of a suspended cable with no load is simply a catenary. However, if you suspend a much heavier load (i.e. a bridge) with the cable, the cable forms a parabolic shape. Does anyone know how to show this fact using the Euler-Lagrange equation?


Where did the slang term juice/sauce for steroids come from? I suppose that it may be a metaphor because of being liquid or maybe the similarity in colour. I cannot find any information on its origin, can you help me out?


I'm typesetting a multi-page text with a fancy frame on each page. As you can see on the attached picture, the inner corners of the frame are taking extra space and I would like to increase the margin for the first/last two lines (or for a certain distance from the top/bottom) on every page. Is there a simple way to do that?


I'm trying to find a Scottish colloquial idiom that means that people talk too much (especially in a gossipping, hot air, or inconsequential way). Perhaps this might be along lines of the English 'more babbling than a brook in flood'. I'm ideally after an idiom, not a single word.


I'm currently working on proving an iff statement and was wondering is it allowed for me to prove the two statements required to prove an iff statement using two contrapositive statements. (for example, I'd prove if A then B using if ~B then ~A) and similarly for the second statement Thank you!


I was wondering if there is any connection between number theory and operator theory. Especially the applications of Hardy spaces, de branges-Rovnyak spaces, Dirichlet spaces in number theory. For example to study modular forms, and etc.Something except Riemann Hypothesis. Does anyone know any books or papers about that?


Is it possible to separate fluids or at least alter their relative concentrations in the presence of a magnetic field if one fluid is diamagnetic and the other paramagnetic? If so by how much? Are there any studies or data on this? How could this effect be calculated?


If someone uses the name of a book/object with punctuation in it, does one need to capitalize the word after? Do I write: In the book Who's Afraid of Virginia Woolf? the main character... Or In the book Who's Afraid of Virginia Woolf? The main character...


I'm writing a speech about someone and I would like to describe them as intelligent using a similar phrase to the way that "she has fire in her belly" would describe her as being passionate and driven. Something along the lines of She has fire in her belly and {something} in her head.


I am writing a scientific article and I need to describe the two hatched areas in the Figure below. How do you call these? Upward hatching and downward hatching? Or what? E.g. The upward hatching corresponds to newly captured individuals, the downward hatching to the recaptured ones.


I am searching for introductory books that explains in pedagogical way how scattering experiments are done . I want to understand how the experimental apparatus works and how experiments are done. For example in electron electron scattering I want to know what are the initial condition of the electrons like impact parameter, momentum etc,and how the detectors are placed.


I would like to know which of the following sentences is correct: Ideally, we should leave tomorrow early OR Ideally, we should leave early tomorrow? I know an adverb of time comes after the verb or object and usually at the end of the sentence by I'm not quite sure in this sentence. I would really appreciate if someone would help me with this.


I took an English test where I had to complete sentences. After I finished the test I was told that my answer "If I didn't have..." was wrong and I should have written "Hadn't I had...". That sentence: "Hadn't I had the chance to make that acquaintance, I would never have been able to enjoy such a huge success."


Find all positive integers, if any, such that all of its divisors can be arranged in a rectangular grid where the sum of all rows and the sum of all columns are equal.


I am reading the Wikipedia article on mathematical structure, and it says "a structure is a set endowed with some additional features". Features can encompass a metric, a topology, etc... I'd like to know if there exists a rigorous definition of this meaning of structure, especially of this term "features". Could "features" mean just a signature along with an interpretation ? Thanks in advance.


Recently, I needed some generalized version of Gronwall's Lemma, which I couldn't find in a quick search. However, I discovered that MSE is full of questions differing only in details on this very topic. Hence, I was wondering if there is a general version of Gronwall's inequality which covers most (if not all) of the different cases.


I had a friend over recently and he expressed happiness for our family and home life. He was envious but with no negative connotations. It is something he wants for himself but is happy we have it. What would you call that?


And in general, what else can we say about the set of normal numbers? We know that a normal number must be irrational, and hence it is possible that the set of normal numbers is countable or uncountable, since the irrational numbers are uncountable. I have been unable to find much literature or research about this topic.


I was curious about solving the reflection and transmission amplitudes for the dirac delta potential barrier using fourier transform. I'm able to take the fourier transform but I'm unable to interpret the pole prescription that comes up. I didn't completely understand the following answer: Poles for a particle scattered in a delta potential


I want to create my own latex package with a .sty file in it, and install it via tlmgr command. I've looked it up in Google, and found only this page https://tug.org/texlive/pkgcontrib.html, where I found no detailed procedure to create a new package. Is there any document about this kind of things?


Are there any simple obstructions for the existence of a simplicial map between two given simplicial complexes? A natural candidate would be some simple to compute function that is non-increasing under simplicial maps. In general I'm interested in finding introductory references that explicitly discuss simplicial maps and their properties.


If a website is blocked, and I wish for it to be unblocked, which of the following is the best way to put it? "The blockade will be lifted tomorrow" "The block will be removed tomorrow" "Lockdown will be lifted tomorrow" Or something else?


Is there a reasonably simple (pen and paper) yet good approximation for the refractive index of an ideal gas as a function of temperature, pressure, and density. How would one derive this? Thanks for any help.


I know the average potential over a spherical surface equals the potential at the center of sphere. But is it also true for average over spherical volume ? In both cases the sphere is empty of charge.


I am searching for literature explaining the Spin/Helicity-Density Matrix formalism in order to compute "spin-dependent" matrix elements and cross-sections in particle physics. I would prefer if the source would contain minimal examples for elementary processes within QED or even the Standard Model.


In the general treatment of geodesics in the Schwarzschild metric symmetry is used to constrain the geodesics to a plane which is usually chosen to be equatorial. Is there a way to generalize this, without restricting ourselves to the equatorial plane?


I understand that a proton is composed of three valence quarks and a very large number of sea quarks that briefly come into existence and then annihilate as quark-antiquark pairs. Is it possible for the antiquark to instead interact with and annihilate one of the valence quarks and for the remaining quark to replace the valence quark?


Consider a nonlinear dynamical system that evolves with a continuously differentiable vector field that is there exists a compact set within which the solution trajectories exist and are unique. Is it possible that all the stationary points of the dynamical system are stable? (Apart from the trivial scenario of having only one steady state.) Thanks in advance.


I have txt file with table format, I did it in python but now I have to export this table to Latex but I don't know how. I'm not good at all in latex so I'll appreciate if you help me. Thank you!


Which preposition would be grammatically correct? I am confident that I have the character and integrity to be an asset to your organization. I am confident that I have the character and integrity to be an asset of your organization. I am confident that I have the character and integrity to be an asset for your organization.


It is a well known fact that there is a linear algorithm to determine if two given trees are isomorphic. But my question is: is there a (simple) way to expand that result to forests (a disjoint collection of trees)? Thanks in advance!


The following line occurs in Keats's poem 'Ode to a Nightingale': And haply the Queen-Moon is on her throne It seems to me that 'haply' means either, as Merriam-Webster says, 'by chance', or, possibly, the same as 'happily'. So, which is it?


I tried to use Torricelli's law to find the velocity of the fluid but I have to know the pressure inside the bottle, I can not calculate the pressure difference because the surface pressure is unknown


I have seen articles online discussing the roundness of electrons. This goes a bit against my understanding of electrons as elementary point like particles. How can a point have a shape? What does roundness mean in this context?


I saved a .tex file as a pdf with TexStudio. It has made a weird conversion and I was hoping to recover my original .tex file by saving it back as a pdf. However I now have this when I open it in TexStudio. Is there a way to recover my original file ? Thanks


I am looking for an introductory linear algebra textbook for undergrads. The basics: matrices, eigenvalues, change of basis, exponents, pseudo-inversion, SVD and QR factorization. But what I really need is a good set of problems, preferably with solutions. And it would be great if it is online or an open book. Do you know such a book?


How are vector spaces fundamental to higher mathematics (functional analysis, number theory).I'm familiar with vector spaces, basis vectors, inner products, eigenvalues,groups, rings , fields, modular arithmetic etc. I've read Axler's Linear Algebra Done Right and a bit of group theory.


I think I know what the displacement current is. But I don't know why they use the word "displacement" exactly. What is the literal meaning of "displacement" of the displacement current? Why don't they call it "electric flux variation current"?


If hawking radiation is the result of a pair of virtual particles popping into existence just outside the event horizon, and then one falling into the black hole through the event horizon, and the other escaping, how can this process reduce the mass of the Black Hole?


Scientists are always saying that they cannot fully explain how lightning works, especially when it comes to the fact that there doesn't seem to be anywhere near enough potential difference between the cloud(s) and the ground to generate it. How large is the difference between the voltage theoretically required for lightning to strike and that which is actually observed?


I find it hard to choose the most appropriate one among these three sentences: she has been busy recently, she has prepared for an exam. she has been busy recently, she has been preparing for an exam. she has been busy recently, she is preparing for an exam.


What is the path followed by free electrons in a conductor under applied electric field? Is it a zig-zag path (due to collisions with atoms of the metal) or a curved path? Why (in each case)?


I am trying to study, through the Riesz representation theorem applied to a space of compactly supported continuous functions, what hypotheses must be met to establish that the measure given by the theorem (which is unique, regular and Borel) is a probability measure. I have tried to find literature on this but I cannot find it. Any help? Thanks.


I have almost completed my mock test syllabus, but complex numbers are pretty complicated for me. Can anyone give me some tips to approach this kind of problem? There are tons of questions like this in the previous year's papers. Please don't downvote me, I just want a tip to improve in these kind of problems


I was reading about the proof of the Mordell-Weil Theorem in "Rational points on elliptic curves" from Silverman and Tate and in Chapter III is the hight function described as an indicator on how "complicated" a rational number is. Does this mean that if the hight function of a rational number has greater value that it requires more numbers to write it?


I have been trying to find if a real experiment has ever been done to prove and calculate the magnetic field of a rotating charged disk, whether it's a conducting or a non-conducting charged disk. All I found on Google and YouTube are just thought experiments, so please if anyone had come across an actual experiment let me know.


I thought a lot about it but couldn't come to an answer. It's obvious that there must be an equation telling the answer. If one of the electrons were fixed, by Integral calculus, I would have found an answer, but when both the electrons are free to move, I am struggling to find a solution.


A cyclist and a car, which is twice as fast as the cyclist, are approaching an intersection of two different streets perpendicular to each other. At noon they were at an equal distance from the intersection. Twenty minutes after noon the car was at the intersection. How many seconds from then on do they need to be equidistant from the intersection again?


When I select text in Texstudio it remains highlighted even if deselected. The last selected section of text always has a light-blue highlighting. This is visually very irritating and I have no idea how to get rid of it. I might have turned it on by accident, hitting some shortcut that I am unaware of. Does anybody know how to turn this behaviour off?


I have learned from a knot theorist that quandles were developed in order to more sensitively distinguish between distinct knots. However, they seem to be an interesting algebraic object in themselves. Are there any applications of quandles to quantum theory?


what is the definition of maximal connected subgraph? Is every maximal connected graph of the form K_n (that is n nodes and between any pair of nodes there is an undirected edge)? Thanks in advance


From what I read meteors compress the air in front of them, increasing the air's temperature massively; this in turn is what heats the meteor. My question, which I can't seem to find a clear answer to is: does the resulting extremely hot air primarily heat the meteor through radiation or convection, or some combination of the two?


I was wondering if this sentence is grammatically correct: I am not glad. Such is the sentiment of fools. Am I using such correctly here? I spent a good hour researching its use and couldn't find what I was looking for. I thought that perhaps someone on here could provide me with the answer I seek.


I was studying Quantum Hall Effect and there I came out with a question that why the plateaus in the plot of Hall Resistivity are robust ? I know by solving Schrodinger equation and using Landau Gauge Condition we can arrive at the conclusion of landau levels, but I am not satisfied that why the plateaus are robust ?


Injective objects in the category of abelian groups are precisely the divisible groups. However, despite using injective resolutions of sheaves everywhere, I realized that I did not know a single example of an injective sheaf. I have searched for such examples but unfortunately found none on the internet. What are some easy examples of injective abelian sheaves (if there exist any)?


I'm trying to use Georgian language with vimtex. I found out that I have to compile with XelaTex or LuaTex but i don't know how to make vimtex to compile with them. I would like to keep default compiler and add Xelatex as optional, is it possible? I have tried to read official documentation but I can't understand much.


I'm writing a research paper and for that I'm using TeX Live and TeXmaker. Sometimes I need help (correction, improvement) from my professor. So we want a software or something, in which, if somebody edits, we can see it in real time and all edits will be updated and stored. I don't want Overleaf as it requires a permanent internet connection to edit.


I am studying Game Theory and I have some confusions. Is actually Carrot and Stick strategy is the same as Grim Trigger? Is the Carrot and Stick only for Multistage game? Can you please give me example of case using carrot and stick strategy? I know that it's very basic questions but I am struggling to understand this topic. Thank you!


I am trying to prove the snake lemma. I want to show that the sequence of kernels form an exact sequence. Must I show that the functions that map one kernel to the other are R-module homomorphisms? Thanks in advance.


In Wikipedia's article "Complete Boolean algebra, the following example is given: The Boolean algebra of all Baire sets modulo meager sets in a topological space with a countable base is complete... This suggests that the Boolean algebra of all Baire sets modulo meager sets in a topological space without a countable base may not be complete. Is this true?


I want to model a pendulum consisting of two magnetic balls with the same 'sign' of the charge. How should I write the Lagrangian of the system? Will the Lorentz force appear in it? Also, what should I change in the Lagrangian in case of hitting one of balls with another?


I want to combine multiple departments of the same university in the institute section. Please refer to the image provided. This will save a lot of space and be much more clear as well. Please let me know what's the best way to go about doing this, as I currently have no idea. I'm using the LNCS template.


I'm not finding the answer to this on the internet after searching. When writing dialogue, do you use capital letters to spell out words? Jamie said, "I said P-L-E-A-S-E please, and don't you forget it!" Or should it be lower-case? Jamie said, "I said p-l-e-a-s-e please, and don't you forget it!"


I am a beginner with set theory. What is meant by " analytic set " ? I see this term in the context of set theory, calculus and real analysis but I have no idea what it means. See here : https://en.wikipedia.org/wiki/Borel_set where it occurs in the non-Borel set part.


I know that the particles that constitute air move freely about. There must be a significant amount of empty space between the bouncing particles. So why is it so hard to compress air without any machines/devices?


Why do vortices in a superconductor (i.e. magnetic flux-tubes) form triangular lattices? In one of the articles I found, I read that a square lattice would cause repulsion but a triangular lattice will cause attraction. Can anyone explain it?


Is there some way to argue that the Higgs VEV is unnatural, without even bringing up the Higgs boson mass? For example, can one show that the Higgs vev receives large corrections from new physics? The literature on renormalizing the Higgs vev doesn't seem to talk about this (or maybe I missed that part).


Some years ago, a weather satellite manufacturer in the SFO bay area was buying small quantities of piezoelectric acoustic guitar pickups from one of the big music supply houses, which greatly puzzled their sales rep. What were they doing with pickups intended to be glued to the underside of an acoustic guitar bridge?


I was trying to use the answer from here to ensure my tables fit to the width of my page. I'd like to redefine the tabular environment so that this change is applied automatically if the table's width exceeds the page width. Is there any way to do this?


I'm writing a report with the sentence: The vaccine has been shown to reduce diarrhea in an in vivo model. Colleagues use a sentence starting with The vaccine has shown to reduce... In my opinion, the vaccine obviously didn't show anything itself, so it makes sense to use passive voice. What is your expert opinion?


Is it true that the cardinality of a hypothesis class with finite VC dimension is less than the cardinality of real numbers? My intuition is that the number of functions in a hypothesis class with finite VC dimension cannot grow arbitrarily and there is a limit on that.


Is there a more accurate way to determine speed? I am asking because this has to do with an accident reconstruction where my son was killed. The investigators are trying to conclude his speed prior to impact and I want to know if this is the only way to determine speed.


I want to create a table of content like box on the left side of every slide with the name of every chapter/section and with the current chapter/section name highlighted while the other are slightly faded. Like in the following picture.


I am reading about the Douglas-Rachford Splitting algorithm and in that algorithm, the two operators resolvent and reflected resolvent are used extensively. I wanted to know if these operators have any graphical interpretations. e.g. What is the reflected resolvent of an operator? What does it reflect and etc.?


I want to know, are there any texts out there that take conditional probability as more primitive and define probability in terms of conditional probability? Perhaps they give some axioms for conditional probability. Anyway, I would be very interested in such texts.


Everyone floats in Dead Sea because it is so salty. Is it true in Great Salt Lake? How about Red Sea and Mediterranean Sea? Is it impossible for anyone to sink in these seas without added weight?


I'm wondering about lithium batteries in electric cars. Could time and technology improvements continue to improve charging speed, to the point where you could "refill" an electric car as fast as you can refill a gas tank? Or is there something potentially impossible to surmount there, like a law of thermodynamics?


I can't carry a heavy bag and a heavy heart, either. Is this sentence grammatically correct? Can I say this instead? I can't carry a heavy bag and a heavy heart at the same time.


I'm new to using chemfig. I'm trying to create chemistry handouts and I want to draw the tropylium cation: However I am unable to get the charge to come out of the ring. Any help would be appreciated.


I'm working on a project for which I originally put the title as: Methods for controlling the quality of PET bottle manufacturing process. However, now that I re-read it, I feel like its way more natural to say "of the PET bottle manufacturing process". I would normally go with the second variant, but is the original version grammatically correct too?


I'm a mathematician currently working on a problem involving splitting a square into two triangles, either by a line connecting top-left and bottom-right, or top-right and bottom left. I'm trying to think of natural names for these, but the only thing I could think of was acute and grave (after the directions of accents, originally Greek I believe). Are there other options?


I am currently studying X-ray-emitting plasma pervading clusters of galaxies, and I see many studies employ an ideal equation of state to infer the pressure of the plasma. Is this the correct thing to do? Apart from this, it seems like an ideal equation of state is thrown around in astrophysics many times without proper justification. What do you think?


A "stable" is the compartment where an individual horse is housed. A "stables" is the building which contains multiple stables/horses. So the stables is a singular building, but the word "stables" is clearly plural, not singular... Example sentence: "Where else would you buy a horse other than a stables?" she said. Is this sentence grammatically correct? If not, how would you correct it?