I tried finding a decent book to study nonstandard analysis from and found Goldblatt's Lectures on the Hyperreals. However, I was very disappointed to find out that the text is not rigorous at all --- Goldblatt doesn't even prove the transfer principle. Are there any completely rigorous treatments of the hyperreals? Ideally, I would like them to cover the same content that Goldblatt's book does: construction, nonstandard analysis and miscellaneous applications. However, I'd be more than satisfied with just construction and nonstandard analysis. I would also like the book to be more-or-less self-contained in this regard. That is, if the book simply states the transfer principle is a consequence of some stronger model-theoretic theorem, I don't gain anything from that. Of course, some knowledge of analysis, algebra and logic must be assumed, and I'm fine with all of those on a basic level. To clarify, I am also not interested in axiomatic approaches to these structures. So, for example, a construction of the hyperreals would be much preferred over an axiomatic introduction.


A popular misconception in the layman public is that the Big Bang was some sort "explosion" at a single point of space, where originally all matter was concentrated and then it "exploded" outwards. This is of course different from the modern general-relativity understanding of reality, which is that it is space itself which expands - not the content of the space moving, and the Big Bang did not start at a single point, but everywhere. My question is - what experimental evidence do we have that can convince people that the explosion model can't be right. Note that I'm not asking why GR guarantees that the space-expanding is the correct model, not the "explosion". I know that. I also know that GR has a lot of experimental evidence for its correctness at least in smaller scales. Rather I'm asking which evidence we have from astronomy, CMB measurements, or whatever, showing directly that the "explosion" model simply cannot be a valid explanation of the universe's history.


You know how when you talk to someone about a bad time you're going through, and they feel the urge to one-up you to achieve some sort of imaginary victory point? E.g. School is challenging because I have to catch a bus early in the morning. Pfft, that's nothing, I used to have to WALK to school. I did the dishes and the laundry this morning so I'm a little tired. You're tired from that? That's nothing, I swept and mopped my entire house AND my parents' house, I'M tired. Are there words/phrases that describe this behaviour? Either as verbs to describe the act itself, or as nouns to describe the competition. I've seen "struggle Olympics" used before and that's pretty accurate but I'm pretty sure that's just internet slang.


A sufficiently strong electromagnetic pulse can/will destroy smartphones and computers. I know somebody who went into MRI machine and forgot a Visa credit card in his pocket. The card was toast and he had to get a new one. A mobile phone in an MRI probably wouldn't fare better. But a big part of the human body itself is based on electric signals. The brain and nervous system, including the heart, works on electric signals. And those signals have to go to very precise places. There is an area of brain processing vision, another is responsible for speech, etc. Also the heart function depends on precisely timed signals traveling very specific routes. So it would seem that a trip to an MRI scan should totally fry anyone's possessing brain and heart. Except it doesn't. An MRI scan is harmless (if you are not allergic to those injections they give). Why? And then there are those electromagnetic pulse devices they show in Hollywood movies. While totally trashing electronics of bad guys, fellow humans are always shown unharmed. Again, why should the brain be different?


Suppose there are two arbitrary side lengths of a right angled triangle that are known to us. There are two possible cases here that I can see: Either one of the side lengths given is the length of the hypotenuse. Both the sides lengths given are the lengths of the legs of the right angled triangle Now, additionally one arbitrary acute angle measure is also known. There are again two possible cases that I can see: The angle lies between the two sides that are known to us (in which case it will then be confirmed that one of the sides given is the hypotenuse). The angle does not lie between the two sides that are known to us (in which case it will then be confirmed that the sides lengths given are the lengths of the legs of the right angled triangle). Now, I want to know if there is any possibility of determining the third side of the right angled triangle using the conditions given above (two arbitrary sides and one arbitrary acute angle) and without any additional conditions for eliminating the cases given above. If possible, please tell me the formulae/procedure to follow. Thanks in advance.


While the gravitational path integral is not a well-understood concept mathematically, a number of works (particularly in recent research connected to AdS/CFT) emphasize the importance of integrating over metrics on surfaces with nontrivial topologies as well as surfaces diffeomorphic to a flat spacetime. These metrics are obviously not continuously deformable to a flat spacetime metric. A superficially similar concept occurs in Yang-Mills with the notion of a large gauge transformation, which relates gauge configurations that are equivalent but not homotopically equivalent. In Yang-Mills one can also introduce a "topological term" which has no effect on the equations of motion, but computes the homotopy class of the gauge configuration. There is a somewhat analogous object in gravity, the Gauss-Bonnet term, which has no effect on the EOMs and which computes the Euler characteristic of a spacetime. My question is: is there a sense in which integrating over nontrivial topologies in a "gravitational path integral" approach to quantum gravity can be understood as including large gauge transformations in the gauge group of the theory?


I am new to algebraic geometry. I can't understand why an etale space is considered locally homeomorphic to its base space. The analogy I have heard is that etale space can be visualized as "stacked over" the base space, or that it is like a "puff pastry". I understand that we can (locally) map this stacked/pastry space to the base space, but I don't understand how an inverse map could exist. It seems to me like the inverse map would be multi-valued as it takes a point in the base space to several points in the etale space. As an example, consider the base space to be a circle and an etale space to be a circle winding around itself twice. The map from the etale space takes two points to one point of the circle. Doesn't this mean that the inverse map is multi-valued? Michael


I know that white light, upon entering another medium from air/ vacuum, disperses into its constituent colours. Essentially when travelling in the air, all of the constituent colours have the same speed but when entering another medium, they travel with different speeds and hence they are refracted by different amounts - red the least and violet the most among the seven colours. So we are able to see the constituent colours (especially in the case of a prism since they are diverging.) But I came across the following situation where green light (mono-chromatic) changes into red, as perceived by the human eye as well as the camera. Attaching an image of what happens: Why does this happen? I am a high school student and I haven't read anything so far that could explain my observation. Also as far as I know the properties of light are dependent on its frequency and frequency does not change from medium to medium. I saw the other posts but it would be nice if someone explained it in a much simpler language.


We commonly anthropomorphise or personify non-living things by giving them human characteristics: "the angry storm". We zoomorphise things by giving them animal characteristics: "the storm roared". But what word would describe giving general (not necessarily animal) attributes of living things to non-living things: "the storm died"? A more specific example would be treating geomorphological landforms as if they have a life-cycle, but in the context of drawing parallels with the life-cycles of trees and forests. The additional human or animal connotations of "anthropomorphise" and "zoomorphise" do not apply, so it would be good if there were a more general word for this practice. This is very similar to the question "Equivalent for "personify" that's not human-specific". However the accepted answer in that case was "zoomorphise" since the OP was talking about a self-driving car being "hunted" [like an animal], whereas I am looking for the right word for a more general case.


Semantic consistency of a theory T is defined by there being a model M in which all theorems of T are true. Syntactic consistency of a theory T is defined by there being no formulas A such that both A and it's negation ~A are provable. So let's assume semantic consistency and then assume syntactic inconsistency. Since T is inconsistent, both A and its negation ~A are provable. But since it's also semantically consistent, there must be a model where all of its theorems are true. So there must be a model M where both A and ~A is true, which is not possible. So semantic consistency implies syntactic consistency. Is this conclusion right or am I severely misunderstanding something? I thought that semantic consistency only implied syntactic consistency under the additional assumption of soundness, but soundness was not assumed here. Is this conclusion a consequence of defining semantic consistency too broadly? Should it be limited to there being a model where the axioms are true?


In classical mechanics , the Laplace-Runge-Lenz (LRL) vector is a characteristic feature of the Kepler problem. This enables a very simple discussion of the properties of the orbit for the problem. It is an extra conserved quantity besides the total energy and angular momentum for a particle moving under the influence of an inverse-square force field. More importantly, this is associated with some symmetry of the problem. However, if we consider motion in the gravitational field of a black hole, the analysis of the orbits of test particles become more complicated and we require to use general relativity. So, is there any analog of the LRL vector in general relativity? The description of black hole geometry by pseudo-Newtonian potential (e.g., the Paczynski-Wiita potential for non-rotating black holes) is a well-known concept. Is the LRL vector applicable in such systems for motion around non-rotating black holes having spherical symmetry?


So I've recently started taking maths seriously in hopes of doing physics at university and have gotten pretty good at algebra and learned some very basic calculus but I'm having trouble finding a good introductory geometry textbook. I know this question has already been asked at least twice before and I looked at the at some of the recommendations, but I feel they're geared toward more undergraduate study as I'm assuming the guys who asked the questions already did high school geometry which isn't what I'm looking for. I'm looking for comprehensive high school equivalent introductory geometry textbook that's fairly challenging and would set me up for learning more in depth topics that crop up in physics later. (I say fairly challenging because, like I said, I feel like I'm pretty good at algebra). Consider me somebody who has never done geometry before because that's basically what I am. Thanks. (PS. I've never asked for book recommendations before so forgive me if I'm being too vague, I'll gladly amend my request if anyone needs clarification)


I am struggling to see the motivation behind Radon measures. In many places I see that they are described as measures that "interact nicely with the underlying topology", but without further elaboration. On Wikipedia it states A common problem is to find a good notion of a measure on a topological space that is compatible with the topology in some sense. One way to do this is to define a measure on the Borel sets of the topological space. In general there are several problems with this: for example, such a measure may not have a well defined support. Yet they go on do define a Radon measure as being defined on Borel sets so how does that solve the problem they mentioned? What is the usefulness of Radon measures and why are they defined as they are?


Background: My understanding is that model-theoretic semantics (MTS) and proof-theoretic semantics (PTS) differ in the following ways. In MTS, you first define the notion of truth in models and then having this notion you develop a "good" (sound, complete, decidable) proof system. In PTS, you just define a proof system and take this system itself to be the the definition of "truth", without even talking about models. So both MTS and PTS have proof systems, but these proof systems have different roles. I understand why proof systems are needed in PTS: because that's the only thing you have in PTS -- there are no models. But what is the motivation behind studying proof systems in MTS, given that in MTS there is already a notion of truth (with respect to a model) defined? Why do we need an extra apparatus of proof system specifically in MTS?


I'm confused by how thermometer works based on The Zeroth Law of Thermodynamics. The Zeroth Law of Thermodynamics said that "If a body, A, be in thermal equilibrium with two other bodies, B and C, then B and C are in thermal equilibrium with one another." Then, about thermal equilibrium, it said that "If when two bodies are placed in thermal communication, one of the two bodies loses heat, and the other gains heat, that body which gives out heat is said to have a higher temperature than that which receives heat from it." Then if we use a thermometer to measure our body temperature, if the temperature of our body is higher than the temperature of the liquid inside the thermometer, will the heat flow from our body to the liquid inside the thermometer to reach the thermal equilibrium? If yes, then will our body lose heat and temperature of our body decrease? How does the Zeroth Law of Thermodynamics on Thermometer actually works?


Greetings fellow physicists. I have some questions about the ability of different electromagnetic waves to pass through materials that I hope you can clarify. It seems that microwaves can go through concrete, wood, etc. since we can listen to the radio inside houses. However, it doesn't make sense to put a TV antenna indoors because the pictures might flake; so it is apparent that radio frequency doesn't have trouble passing through walls and so on, but the TV frequencies have. I'm guessing as the frequency increases, it gets harder for waves to get through non-conductive stuff. That would explain why we can't see inside boxes - visible light reflects instead of going through. Things get more confusing with X-rays and gamma rays. They seem to move through matter easily, like how X-rays image our skeletons and are used for security screening. Also, radiotherapy uses gamma rays to reach tumours inside patients. So I'm not sure if microwaves really penetrate concrete and other materials or not. But visible light, x-rays and gamma rays acting so differently is puzzling. Can anyone explain the physics behind why different waves can or can't pass through various materials? Some insight would be greatly appreciated!


In Australia, we use the expression Tall poppy syndrome for a "social phenomenon that occurs when someone's success causes them to be envied, resented, criticized or discredited." https://en.wikipedia.org/wiki/Tall_poppy_syndrome Conversely, is there a term or expression that's used for a social phenomenon where successful and popular figures are respected and blindly defended only because they are, well, popular and successful. I notice this behaviour in the YouTube world where smaller channels (or lesser known Youtubers) get extreme flak for criticizing popular channels, no matter how right they can be. Example: Critic: I cannot stand this person. His opinions have no basis. I can't believe he's so popular. Fan: Oh my god, you're such a pathetic jealous hater. Get a life! Critic: Wow, you sure suffer from [insert expression or word]. People like you are hilarious. I am fine with an idiom, a single word or an expression. Just whatever fits.


I would like to know of any techniques that can be used to measure the electric field strength precisely and accurately in both time and space. I know that there will be physical/ practical limitations to this and that it is not possible to measure at an infinitesimal position, but I would still be interested to know what techniques are out there. Assume the electric field I am interested in varies significantly in all spatial directions, and in time (I believe this means the use of a typical antenna is not possible as it sort of depends on a plane wave i.e. non-varying field in the z and x directions at a point in time, if the wave propagation is in the y direction). Assume that any electric fields are oscillating at frequencies at or below microwave frequencies. Thank you in advance for any insight.


I've asked this question in Quora and the answers I got were: First answer: Using "more" and "less" helps maintain clarity and consistency in comparative forms. It provides a straightforward and predictable way to form comparatives and superlatives without relying on irregular or unpredictable suffixes. Second answer: Longer adjectives often have complex or multisyllabic structures, making it more challenging to add suffixes like "-er" and "-est" without affecting the pronunciation or flow of the word. Using "more" and "most" allows for a smoother and more natural-sounding comparative form. Third answer: Using "more" and "most" for more-than-one-syllable adjectives maintains consistency with two-syllable adjectives that also use "more" and "most" for comparison. This avoids creating separate rules or patterns for different types of adjectives Is there an identifiable reason for tending to restrict the forms -er, -est to single-syllable adjectives? It may be one of these I suppose; or something else.


I am actually having an introductory course in Special Relativity in which I was looking at the Michelson Morley experiment. And I have this silly confusion. The setup for the Michelson Morley experiment looks something like this in every textbook (assuming the observer is in the comoving frame) When this same setup is being observed by the observer in the hypothetical frame of "ether" (which this experiment was designed to detect) , then the vertical beam of the light is no more vertical but along the hypotenuse(the dashed red line) and this should be true so that the two observers agree on the results of the experiment. But I really don't understand why the light should not go vertically as the shown path of the light beam seems to violate the laws of reflection as the angle of incidence is no more equal to the angle of reflection. So how can then the two inertial frames be equivalent ? PS :- Nobody in the class bothered about this, not even the instructor of the course.


I know that the tangents from a point to a conic section subtend equal angles on the focus. However, I have mostly studied conic sections from the perspective of coordinate geometry, so even when there are properties common to all conic, I have to prove them separately for parabola, ellipse, hyperbola, and circle. The only common denominator between these figures I know of is that they are the locus of points which have a constant ratio of distance from a fixed line and a fixed point. However, I haven't been able to use that much to my aid, having hardly any experience in dealing with these figures in such a way. Observing the apparent simplicity of the result, is there a simple proof for the theorem? P.S.: I would be thankful if someone could suggest resources that deal with such results about conic sections, especially if they use synthetic geometry.


The Leibniz formula for determinants starts with multi-linearity and the alternating property and builds from there. I asked a question about why we should start with multi-linearity: What's so special about multi-linearity?. And the response (which had an excellent alternate motivation for the determinant which I wasn't aware of) seemed to be that there is indeed no good reason. The thing that keeps bothering me is that we are talking about linear maps here. And the determinant is really a property of the linear map. And given the definitions of linearity in the context of linear maps and the multi-linearity of determinants look exactly the same, I would assume a statement like this exists - "because the determinant is a property of linear maps, it clearly must satisfy the property of multi-linearity because _____". Is there no way to fill in the blank?


I just stumbled across this older video of a girl trying to "talk" to Alexa (the voice assistant). She says "I am trying to talk to you, hen". Now, I am not a native English speaker, but I am familiar with the endearing term "honey", also sometimes written "hunny" or even "hun" or "hon" depending on the pronunciation. However, the pronunciation "hen" made me think of the word for a female chicken. I've heard the word "chicks" to refer to females (but I think this is generally not an appreciated term? - again, not a native speaker), so there is a connection to use terms for poultry to refer to people. So my question is, in Scottish, what is the root/etymology of "hen"? Is it related to "honey" or to "hen" (chicken), or maybe something completely unrelated?


Consider a metric space with a path between any two points, so a real line segment of some length between them, and the length of this line is the same as the distance between the two points in the metric. This is called a geodesic metric space. Any two points are connected with a shortest path. There could also be multiple such paths as with the antipodes of a sphere. What I mean by homogenous is that any point can be mapped to another while preserving the structure of the space. Another way of looking at it is that the space is the same no matter what point is taken as the origin. Geometries like Euclidean and non-Euclidean geometry would be examples. However, I am curious about examples which are not locally Euclidean. So, they would behave kind of like Euclidean space in that there is a measurable shortest path between two points and everywhere in space is the same. So you could "move around" in a given "direction" in this space, but it does not locally resemble Euclidean space. Is this even possible? If so what characteristics would these spaces have and could they be classified? What about the underlying topologies of these spaces?


Say that I have two long cylinders of material as shown that are in a region of oscillating magnetic field where the field direction is aligned with the long axis of the cylinders. I want to calculate the inductive heating of each cylinder (let's pretend for a moment that none of the parameters like resistivity are temperature independent). I know how to calculate the skin depth and therefore the heating when only one is present, but don't know how to deal with two cylinders. Does the inner cylinder "steal" some of the magnetic flux by reducing the flux from its own induced current? If so, how much? I would think there would be time lag between the flux and the induced current. Any thoughts or references explaining how this can be attacked? Thank you


I hope that this is the correct site to ask this question, but since I was unable to find one specifically for modeling, I decided to ask here (and please feel free to correct me, if any of my statements is bogus): There are tools which allow one to do parametric modeling like Rhino's Grasshopper or Blender's Geometry Nodes. The modeling happens in a bottom-up manner, e.g., twisting and bending some input shape according to some parametric functions. And then there are implicit modeling tools like nTop (formerly "nTopology") which allow modeling with implicit functions, and therefore enable a top-down workflow, e.g., you have some shape and describe with combinations of implicit functions that is shall have rounded corners and be filled with a fancy pattern. But the implicit modeling tools are, I think, relatively new on the market and generally only smaller and light-weight tools. I specifically wonder about the reasons for that. And my question would be: Is there something that can easily be done with parametric modeling which is hard to be done with implicit modeling? And if so, what and why?


I have been interested in hyperbolic (negatively curved) space and I have been reading enough about it to feel that I understand it relatively well intuitively (e.g. the Poincare disc). But the problem is when I flip the curvature sign in my head to positive curvature and elliptic space nothing seems to make sense. For instance: In elliptic space, locally "parallel" lines will converge and intersect given a great enough distance, but if there is really a positive curvature, then it seems like the lines will eventually converge after the intersection to intersect again. This leads to the result that the lines will intersect an infinite number of times. Is this correct? I cannot find the right search terms to find this answer. Is elliptic space really a sphere? Does the positive curvature mean that if you travel far enough, you will end up back at the starting point? If so, then it seems to imply that elliptic space is finite, where Euclidean and hyperbolic space are infinite, which seems wrong. I likely have some misunderstanding of elliptic space, so If you can correct me where I may have gone wrong, please do. Thanks


I'm looking for the best BE substitute for the AmE word "ornery" in the phrase "an ornery bunch". Complicating the task for this second-language speaker of English is that according to the Oxford Dictionary of English, "ornery" means "bad-tempered and combative" in AmE whereas it means "bad-tempered or difficult to deal with" in BE. Does this mean that BE speakers tend to understand the word "ornery" slightly differently than Americans generally do (that BE speakers may perceive ornery characters as not necessarily being bad-tempered?)? Also, are there any synonyms that BE speakers usually prefer as substitutes for "ornery", or do BE speakers rather use several words to cover the AmE meaning of "ornery" (in the latter case, for my task: instead of "an ornery bunch, perhaps "a coarse and combative bunch"?)?


When I compile my document with pdflatex with natbib and either the plainnat or the dinat style on my computer, the author list is translated to German. E.g., I get "und" between authors instead of "and" and instead of "et al" I get "u. a.". I am very puzzled, because I don't know why. In my bibliography file all references are in English and I also write "and" between each author. So the original is in English and there obviously happens an unwanted translation somewhere. When I upload the very same document (i.e. all .tex files and also the .cls file, exactly as they are on my computer) to Overleaf, everything is fine and the compiled document has all references in English. I use TexStudio to write and I also compile from TexStudio. Could it be because of TexStudio? What could I change to make sure that nothing is translated?


Consider two samples of visit lengths to two different Emergency Departments. We want to test whether or not the means of the samples are different. A basic requirement is independence between and within samples. However, while we know that there are no patients that had visits in both samples, there are patients who had multiple visits within a given sample. Therefore, we cannot consider the visit lengths within a given sample to be independent. How do we deal with this? The visit lengths are labeled with patient identifiers. If, for every set of visits associated with a patient (most of which will contain a single visit) we randomly sample one visit, can we use the resulting sub-sample for hypothesis testing, or will this process inject bias? If so, how could we control for the bias? I've done a fair amount of searching on this topic, but there's a lot of obfuscation. Maybe I just need someone on here to tell me what terms I should be searching for.


I started running a tabletop RPG campaign set in the distant future where mankind has degraded to a primitive hunter-gatherer society, and I'm looking for flavorful terms for NPCs to describe directions. As research, I started watching the show "See" which has a similar premise, and they use the term "sun grave" to refer to west, because that's where the sun sets. And I love it! But what about the other directions? (I'm only midway through the first season of "See" and I haven't noticed them refer to another direction.) East could be "away from the sun grave," but what about north and south? "Right of the sun grave?" That feels awkward to me. Is there another more primitive way of describing a perpendicularity like that, or perhaps another point of reference besides the sun?


I saw this post which says: The way you do it is just a simple logic. Imagine a tennis ball and imagine you can't really see it, just like you can't see an electron. So the only way you can see where the tennis ball is is to hit it with another tennis ball or with an enormous amount of light that it will actually displace the tennis ball position, move it that is. At that moment the light brings you the information into your eyes and you see the tennis ball. Does the author mean the reflected light? Because if the light isn't reflected (i.e. a new photon emitted), how would we see the electron or tennis ball? It seems like, unless light is reflected/a new photon is emitted, we won't be able to see the electron. If my reasoning is correct, then we can't detect electrons in the photoelectric effect this way, because no photons are emitted in that case, but we can in the Compton effect. Is my reasoning correct or did I make a mistake?


I am doing some research into social interactions and the data is represented in a series of square matrices for analysis. I am interested if the grouping of matrix elements above/below off-diagonal, as highlighted by blue and green in the following diagram, are known by a specific name? These groupings of matrix elements differs from the upper/lower triangular matrix as they don't include the main diagonal and all the elements can be greater than zero. I have searched for a name but have not found anything yet. Also does the sum of these groupings have a specific name? (Akin to the sum of the main diagonal being called the trace). If there are no conventional names, I am okay to defining a name for each grouping and their respective sums as part of the research.


I am trying to understand the basics of quantum mechanics and I am having some issues in understanding the main mathematical properties of quantum operators. In particular, it's clear to me that: Quantum operators must be Hermitian to guarantee: real eigenvalues, that can be measured an orthogonal basis, so that after each measure the state of the system collapses onto a precise base-state Unitary operators are needed because they preserve norms: if we apply a non unitary operator to a system, its total probability would not be preserved. It's clear to me that the Hamiltonian is both a hermitian and unitary operator. In any book I have read, I have only found statements such as "operators in quantum mechanics must be hermitian" or "time evolution is unitary", but I've never found a statement such as "operators must be both hermitian and unitary": why is that? Shouldn't all operators be both, to guarantee measurability and conservation of norm?


Although neutron stars are mostly made of neutronium, the pressure at the surface is not very high which allows regular atomic matters to exist. Emission spectrum can reveal the chemical composition of distant stars. However, neutron stars are surrounded with extremely strong magnetic field which is enough to distort the atomic structures. Atomic nuclei should be more resistant against the magnetic field because they are much more compact and tightly bound, but I am not sure if atomic nuclei emit characteristic spectrum like the electron clouds. If we can determine the composition of the neutron star crust, will there be any variations? Neutron stars formed due to iron core collapse should have an iron crust. Some neutron stars are formed due to the electron capture of the O-Ne-Mg core. Would these neutron stars have a different chemical composition in the crust? Many neutron stars are accreting hydrogen and helium gas from companion stars. Will the accreted matters show up on the surface?


this is my first post, please excuse my non-technical language. I would like to simulate the return of a stock, that is correlated with other stocks, that meets the mean and variance of a given empirical time series. The usual way to do this is to set up a geometric brownian motion and to use a cholesky factorization on the correlation matrix of the returns of a set of stocks. As far as i know the moments of this GBM are functions of time but the log of the GBM doesn't contain any skewness. My question is: In what way can i extend the GBM to introduce a target level of skewness while keeping the other moments as in the standard GBM? Would you mind providing the explicit form of the SDE? Thank you very much in advance! Thomas


Ok, this is my first question on this site. But it's one I've been thinking about for a while. Say through whatever means, we place a device capable of generating thrust/ kinetic energy on the surface of the Earth. The goal being to remove Earth from its orbit and exit the solar system. By "perfect conditions" I mean that hypothetically when this does happen, no other planets or space debris will be in its path. I'm thinking that first it would be easier to stop the rotation of the Earth, and then use our device, but I don't quite know how the Sun's gravitational field will affect all this. Also, there can be two versions of this: one where the device generates continuous thrust and the other where it's more of an impulse. How much energy would the device need and, if any, what other conditions are needed in order to exit the solar system?


I'm a native English speaker, and I noticed that I sometimes use accusative pronouns (him, her, me) to replace actors in certain clauses. I have a feeling this is prescriptively considered incorrect usage, but I want to be able to describe it. "What does that have to do with me not coming to class?" "I want to hear about them starting a new game." "Him winning that contest has nothing to do with his family." Maybe the prescriptively correct form is the possessive, and this is confusion based on "her" which is the same in both? Or maybe it is because when the clause comes second, the pronoun makes sense as an object of the larger sentence? But that doesn't apply for the third sentence as much unless you flip it around: "His family has nothing to do with him winning that contest." How do you describe what is happening here? Why does it happen? Is it considered "incorrect", and if so, how widely? (I have the intuitive sense not to write it in an essay, but it comes so easily out of my mouth...)


"Such" has many meanings, one of them being to refer to something of a particular kind/type (see "of this or that kind"). However, in many legal documents and laws in Malta, "such" is used instead of "this" or "that", or instead of "the said thing". For example, a legal document might read as follows: The following are the terms and conditions regulating the relationship between A and B. Such terms and conditions are.... Here, "such" is not used to mean "of this or that kind", but is used instead of "these" (the terms and conditions previously mentioned). Is this use of "such" limited to the legal English used in Malta or is this a common feature of the legal English used in the UK, USA and countries where English is an official language?


Not really sure if physics question or engineering question. If we can apply energy to make an object in space move faster, the reverse should be possible - we should be able to extract energy while causing the object to slow down. So in theory, if we use rail gun to launch a projectile in space, we should be able to extract energy out of it if we "catch" it with another rail gun. Considering that, since we can use gravity slingshot to accelerate objects, we should be able to extract MORE energy out of the projectile when we "catch" it. The extra energy would come from the gravity of the celestial body used for the slingshot, which is practically huge. We could use part of this extra energy to redirect the object to the original course, thus creating a stable loop that would allow us to extract vast amount of energy from the gravity of other planets. Why wouldn't this work?


I ordered a book online, unseen, and the invoice told me the book, or at least its pages, were 'foxed'. I had never come across the expression, did not know the word could be a verb and discovered : Foxing is the age related browning, or brown-yellowish spots, that can occur to book paper over time. When this aging process happens to the paper in a book it is referred to as "foxed". The term may come from the rust brown color of the paper aging process or from a chemical used to coat paper called ferric oxide. Foxing may also be caused by fungal growth on the paper, chemical reactions, or high humidity. Biblio.co.uk Is there any more information anywhere to clarify the idea of ferric oxide being involved, which sounds a little far-fetched to me, or to demonstrate the history of the word in that particular context?


In condensed matter and materials physics it is often assumed that the response of a condensed phase to some perturbation is determined by the fluctuations of the system at equilibrium (without perturbations). For example, the electric resistance of a liquid (the response of the current to an applied field) is determined by the mean squared displacement of the charge carriers (the fluctuations) at equilibrium. Pictorially, this makes sense because if the charge carriers move slowly at equilibrium it will be probably not be easy to make them move, so for small fluctuations the current response would probably be small. Another way to say this, the difference between motion at equilibrium (without field) and motion out of equilibrium (with applied field) is that there is a small bias in the direction of the field, but the motion is otherwise essentially the same. Formally, this relation is proven by the fluctuation-dissipation theorem for a statistical mechanical (micro-canonical or canonical) system subject to small perturbation (small term added to Hamiltonian). My concern is that I think that the validity of this FDT relation can be formulated for systems more general than statistical mechanical (micro-canonical and canonical) systems. Pictorially, I think whenever the system can be described as a stochastic variable fluctuating in some potential landscape between some potential minima, then the response should always be related to the barrier heights of those minima, which also determine the magnitude of the fluctuations at equilibrium (without bias). So is it possible to derive FDT also for this setting ?


In order to obtain phonon spectrum, we usually do Born-Oppenheimer approximation and assume that the electrons are always at the ground state when the atoms move, and by calculating the force on each atom (usually using DFT) we are able to assemble the dynamic matrix of the atoms and obtain the phonon modes. After this we can calculate the electron-phonon vertex and then do Feynman diagrammatic calculations to get things like polaron. So the whole formalism is based on the BO approximation. The question then is how we can write down a set of Feynman rules about both electrons and atoms without the BO approximation. Can we still define phonons (hence phonon propagator, etc.) in this case? If we already have a theory under the BO approximation, is it possible to, say, add more interaction vertices and/or adjust the coupling strength to go beyond BO?


In An Argument Against Abolishing Christianity it is written: Another advantage proposed by the abolishing of Christianity is the clear gain of one day in seven, which is now entirely lost, and consequently the kingdom one seventh less considerable in trade, business, and pleasure; besides the loss to the public of so many stately structures now in the hands of the clergy, which might be converted into play-houses, exchanges, market-houses, common dormitories, and other public edifices. I hope I shall be forgiven a hard word if I call this a perfect cavil. I readily own there hath been an old custom, time out of mind, for people to assemble in the churches every Sunday, and that shops are still frequently shut, in order, as it is conceived, to preserve the memory of that ancient practice; but how this can prove a hindrance to business or pleasure is hard to imagine. What if the men of pleasure are forced, one day in the week, to game at home instead of the chocolate-house? Are not the taverns and coffee-houses open? I looked up chocolate in Wiktionary and chocolate house in Wikipedia. Of what chocolate house does Swift write here?


In a power grid, the grid itself has a certain amount of inertia from all the spinning loads and generators. If at a given moment in time the production and consumption of active power does not match, the power mismatch will be provided/absorbed by the spinning energy stored in the grid, and the frequency of the whole grid will raise or lower in lockstep to compensate. Eventually the generators' governors and the load/frequency relationship will arrest the frequency drop, and it will arrive at a new steady-state value, but it's not something that happens immediately. Does it work similarly for reactive power? If a capacitor bank supplying reactive power suddenly trips, does the voltage immediately drop to its new steady-state value? Or is there a period of voltage decline as the missing reactive power is siphoned from somewhere?


Suppose there is a plank on a smooth surface and a man is standing on one end of it. The surface of plank is rough. Now the person starts to move towards the other end with some acceleration and the plank also starts to shift to keep the COM stationary as the only forces acting here are all internal. Now my question is,as the direction of the friction force on the man is in the direction of the displacement of man and same is the case with plank. Then in this case isn't work done by friction on the whole system coming out to be positive which should not be happening as if it is the case of static friction then the net work done should be zero and if it is the case of kinetic friction then the net work done should have been negative? Also please tell me whether the friction present here is kinetic or static.


Note: this question is not a duplicate of the following questions: How to remove vertices from a graph that are not coverable by cliques? How to remove vertices from a graph while preserving clique coverage of specific vertices? This question is with respect to removing vertices that are not reachable from a set of vertices, regardless of clique membership. How can I remove vertices from a graph that are not reachable from a given set of vertices? Here is a toy example and solution: In the graph below, I want to remove the vertices that are not reachable from any vertex with an "x". The orange vertices are the vertices to be removed. toy example graph image The current approach is doing depth-first search from each of the vertices marked with an "x", and keeping track of which vertices are visited. All vertices from the graph that are not in that set of visited vertices are removed from the graph. However, this could take a while if the graph is large, so I was wondering if there is a faster way of doing this.


Let's say we entangle four electrons based on their spin. Now, let's say we measure the spin of electron A along the z-axis (up/down), and we observe a spin of 'up'. My understanding is that this means that the remaining three electrons would need to have some combination of spin which adds up to 'down'. Next, we measure the spin of a second electron, but along the x-axis (left/right), and say we observe 'right'. My question is, what does this mean for the up/down spin of the entangled system? I'm thinking since it can't be that the remaining two electrons add up to a 'down' spin, it must be that the second electron we measured also still has a state for up/down spin that has not been observed yet. But that would also imply that it's still entangled with the other electrons; this is where I'm confused, cause it was my understanding that a measurement would break entanglement. Is that correct, or no? What actually breaks the entanglement in this case? Hope the question makes sense, and thanks in advance!


I need a word that describes someone who advocates for harmful laws or policies; it would describe someone who writes policy without listening to the people it affects or someone who doesn't pay attention to actual effects of that policy, kind of like politically or socially tone-deaf. They might continue to stand behind that policy even after it was proven to be ineffective or harmful. It would be like ignorant or deluded, but specifically regarding knowledge of people's situations or societal problems and solutions. A word to describe the actions of that person would be helpful as well. As an example: It was very __ of the senator to vote for the harmful bill that would require every homeless person to own a car. The word wouldn't refer to someone doing this knowingly and maliciously, but I would like it to have a negative connotation. Thank you!


I want to start learning model theory for my master's thesis, but I can't find the right book for me. For some context, last year as an undergrad I had a class on logic where we learned the following: So I already had contact with formal languages, structures, models, and both the semantic and syntactic aspects of first-order logic and now I want to follow from this and start learning some real Model theory. I started using David Marker's "Model Theory. An Introduction" but I'm personally not a big fan of that book: It's full of typos and I don't like how informal some of his proofs and expositions are, for example, this definition: Do you have some recommendations for a rigorous book/resources to learn Model theory, ideal for someone with a little background on the subject?


Whenever I get into a disagreement with an individual I notice that this phenomenon always occurs. (I either notice it during the argument or afterward.) Allow me to provide you with an instance to help describe exactly what I am talking about: I am having a disagreement with an individual regarding bottled water VS tap water. I am supporting the consumption of tap water and simultaneously attacking bottled water. At one point, it seems like I have won the argument as my points outweigh his. However, he then starts bringing up additional points and he now feels that he has won that argument. I tell him, "Any argument could be supported with the English language, due to its vast vocabulary, but my points are logical and outweigh yours". Therefore, here is a summary of what I am asking for help with: What is this phenomenon called when someone uses the vast vocabulary of the English language and all associated knowledge to formulate points to support their argument? What is a good word for this phrase/How could this phrase be reworded to a simpler form without out sounding cruel? - "any argument could be supported with the English language, due to its vast vocabulary, but that my points are logical and outweigh yours"


I'm trying to understand how Newton's third law works with springs. If we hang a block on an ideal spring mounted to the ceiling, in equilibrium, the block is affected by gravity downwards and the spring force upwards. So according to Newton's third law the spring itself is affected by the spring force downwards. This makes sense in my brain, but what would be the force upwards, since the spring is also in equilibrium, right? Is this the reaction force to the gravitational force on the block? That wouldn't work if the block was accelerating e.g. downwards, since then the spring would be accelerating upwards which is not what's happening? There also has to be some force from the spring on the ceiling, does the reaction force to that just always cancel out the reaction force to the spring force so the spring stays in rest? But shouldn't that force depend on the weight of the block somehow?


My question may be simple but I'm curious, let's say I start bouncing a ball like a footballer with my foot on an elevator, and it starts moving upwards (with acceleration) and then it stabilises itself moving upwards with no acceleration at a constant velocity. My question is if the ball will or not start bouncing less due to the upward motion of the elevator. My thoughts are that it depends, if the elevator is accelerating upwards then the ball will get a force pointing downwards so that it will start reaching less height than if we would be bouncing it on a stationary case. And if the elevator is not accelerating but moving upwards with constant velocity, it doesn't feel any difference with the stationary case since there is no acceleration, but I don't know


I am wondering how this sentence is to be paraphrased: At weekends they prefer to stay home and visit some friends. I am not sure which ones are close to the original: They like to remain in their house at weekends and they also like to go and see their friends. They like to remain in their house at weekends and to go and see their friends. They like to remain in their house for a while at weekends and then go and see their friends. They go and see their friends on weekdays but would rather remain in their house at weekends. They generally go and see their friends but would rather remain in their house at weekends. They like to remain in their house at weekends or else to go and see their friends.


The task is: The farmer has several sheep, each responding to one or more names, and several sheep may respond to the same name. Each sheep answers to at least one of the names, the farmer knows which sheep respond to which names. The farmer has two paddocks for food and a haircut. A farmer can say a name, and then all the sheep who answer that name go to another paddock. Prove that farmer can say names several times, so that at least half of all the sheep will be in the haircut paddock. I really want to solve this problem using probabilistic method - I need to show that probability of this event is more than zero, then the task will be solved. But I can't come up with a strict solution how to do it. (I am new to probabilistic method) Please, help me out!


I was talking to a colleague professor the other day and he said something that got me curious. The way I remember it, he said basically that in experiments a Bose-Einstein condensation is usually trapped by some external potential, which I imagine to be an electric or magnetic field. Then, the temperature is considerably lowered, the trap is turned off and one studies the evolution of this state. Can someone further elaborate this kind of experiment? For instance, it was not clear to me what exactly one tries to measure with it; as far as I understood, the temperature is cooled down so to form a condensate in the first place and the release of the external field is intended to study its evolution. What one wants to understand with the evolution of such a condensate? Moreover, what material is used in such experiments? (I am thinking about liquid Hellium?)


If all matrices can represent a linear transformation, can I refer to some general notion of the transformation associated with the matrix to make conclusions about the matrix? For example, I want to prove a simple true/false statement, "If A and B are square matrices such that AB is invertible, then both A and B must be invertible." I believe that the statement is true for the following reason: Assuming AB is well-defined and A and B are square, A and B have the same dimension. The product, AB is square as well then. AB is invertible and square, so the transformation AB represents is an isomorphism, thus the transformation associated with AB is a bijection. The transformation associated with AB is a bijection, so the transformation B represents is surjective and the transformation A represents is injective. Since both A and B have square matrices representing their linear transformations, injective and surjective respectively, the transformations must also be isomorphisms. Hence the transformation associated with A and B has an inverse, and thus A and B (the matrices) are invertible. This feels convoluted and likely isn't the simplest solution. I am just wondering if it is OK to bring in the notion of the transformations the matrices could represent like this? Am I missing some subtlety?


I do not understand at all why, if an object is sitting on a spinning platform, the friction force is towards the center. I understand the need for a centripetal force during circular motion, but friction is only in opposition to a force being applied to an object / system, why does it act as centripetal? I understand for a car, for example, the wheels providing the forces needed, but, for example, a penny sitting on a rotating disc, why would the friction be towards the center, would it not be in opposition to the impending motion of the tangential velocity? I have seen a lot of explanations on why there would obviously be a centripetal force, the need for one because there is a change in velocity, etc, but none of these explain why friction acts as this force, or how.


A flat spiral in immersed in a homogeneous magnetic field. An electric current is flowing in the spiral. The directions of the B field, the spiral and the current can be seen in the picture. As the result of the electric current and the B field, Lorentz Force appears acting on the electrons, aka the Hall Effect. The electrons are flowing from the edge to the center of the spiral, the Lorentz Force F also points to the center. Seemingly the Lorentz Force is helping the electrons to move to the center of the spiral. If that reasoning is correct, can we say that e.m.f. (electromotive force) is created because of the Lorentz Force? The bigger is the current, the bigger the emf gets, as opposed to the Ohm Law. Voltage is created out of nothing. How would you justify the energy conservation? If you think I am wrong, please point out my mistake concretely.


It is stated that an object in motion acquires "kinetic energy" while an object under the influence of gravity when raised to a height acquires "potential energy" but I have a doubt that what leads to the object acquiring the energy in actual sense? What special is happening that an object in motion or an object at a height acquires the ability to do some work(i.e. energy)? If I start thinking about it more energy is just an "abstract thing" so is it just that we assign this value to an object under a specific condition and we say that it "gains" this energy? Also, isn't this energy just a "numerical value" with no real meaning so what is it even representing under a specific condition of motion or change in height? P.S. I tried searching for similar questions on the internet and Physics S.E but couldn't find a satisfactory answer yet.


I am doing an experiment with the overall research question of: To what extent does the amount of fluid within a hollow cylindrical can affect its dynamics while rolling down an inclined plane I was able to derive an equation for acceleration for the case of a fully solid cylindrical shell and then used law of conservation of energy to determine equations for the velocity of a solid and hollow cylinder and I understand that the moment of inertia's of the aforementioned cans can also be found easily using the radius. However, I am finding difficulty in finding a method to determine the moment of inertia for cans that are partially filled other than the parallel axis theorem which I am not sure can be applied to this scenario. Furthermore, I am also unsure about the actual experiment that I am doing to answer the initial research question (I have access to a motion sensor and photogate sensor) and was wondering if I should simply measure the final velocities and relate it to varying moment of inertias if it is possible to calculate but would appreciate any alternatives to this experiment. Finally, I am not sure if I should treat the water as an inviscid fluid as that would mean there would be no effects on the moment of inertias but the mass would still change so that would affect it in that regard, right? For now I have been treating it as an inviscid fluid for the purpose of the derivations that I have made.


Also, how do different sets of principles affect the results we can get in our meta-theory? The more concrete questions that lead me to ask the above two questions are stated below. If we are studying intuitionistic logic, should we also drop the excluded middle rule in our meta-theory? Besides, in classic logic, I seem to notice that a proof of the compactness theorem must utilise the axiom of choice, be it directly or indirectly. The proofs I ever saw so far all implicitly assumed the axiom of choice in its meta-theory. (Actually, so are the proofs of the completeness theorem.) Does that mean, if we are to drop the AC, we can't even have the compactness theorem? What if we are studying a set theory without the AC? Should we assume the AC in our meta-theory when studying a set theory with the axiom of determinacy instead of AC? Or maybe we should adopt the AD instead of AC? Are we supposed to use the same rules in meta-theory as those in our formal theory?


I've been exploring the concept of gravitational wave (G-wave) emission from symmetrically accelerating systems and have encountered a puzzling question. Standard sources typically state that symmetrical systems, such as a perfectly rotating sphere, do not emit G waves. As they require a changing quadrupole moment, which such systems do not exhibit. However, this leads me to draw parallels with concepts like standing waves and the Fourier transform of a static field, where individual waves can cancel each other out, resulting in what appears to be a static field. My question, therefore: Is it that each part of a symmetrically accelerating system never actually emits any gravitational waves, or is it more accurate to say that any potential gravitational waves are effectively cancelled out due to the system's symmetry? To illustrate this, consider a hypothetical scenario: If a rotating spherical mass were to suddenly appear in an ideal space-time, would an observer detect non-zero gravitational waves during the initial moments when the gravitational effects from different parts of the sphere begin to propagate at the speed of light (c)?


Is the essential argument that these systems are microscopically chaotic enough that we can approximate their evolution as random (vastly simplifying calculations) and still make accurate experimental predictions? Or are there deeper physical/mathematical foundations? A related question: how does one interpret the distributions we end up using (e.g. over particle velocities)? E.g. when we calculate expectation values are we (A) implicitly averaging over an ensemble of universes with different microscopic system configurations? If so, why is it "okay" to make predictions about measurements in one universe from an average over universes? Does one have to invoke some kind of further self-averaging argument? Or (B) are we taking a more Bayesian perspective, where there is only one universe, with the distribution representing our uncertainty over the system state? If this is the case what other assumptions are made? Should we think of the distributions we manipulate as posteriors, given some prior? If so, what is that prior? Or are (A) and (B) equivalent in some sense? Or am I thinking about this the wrong way entirely?


So, I have been studying thermodynamics, in it I read that state functions are those functions which depend only on one state of the system and are independent of the path taken, which is pretty easy to understand. And this clears why properties like enthalpy, pressure, volume etc are state functions. Then there was the statement, Change in a state function is not a state function. which took me some time to understand, but I got it once I referred to the original definition of a state function. But now when I read about Hess law, which basically states that change in a state function is independent of the path taken, I'm confused as doesn't this imply that change in state function is infact a state function. Can someone please clarify this.


I often come across materials discussing convergence spaces and their relevance in various contexts. It's commonly mentioned that the existence of a natural convergence on the space of continuous functions (turning them into exponential objects) makes the category of these spaces a suitable environment for studying homotopy. However, I've found only a limited number of resources that actually delve into this idea (mainly this and this). As someone who doesn't engage with algebraic topology on a daily basis, this leaves me with a few questions. Are there any works that effectively highlight the significance of these spaces for the typical algebraic topologist? If not, could the issue possibly be attributed to an "excessive" use of filter-related terminology? I would greatly appreciate any insights or references that could shed light on this matter.


Say we're drawing marbles from a box. The marbles can be labeled X, Y, or Z and can be either black, brown, or white. The probability of drawing a marble with each letter label is unknown but fixed and marbles are replaced after each draw. A marble labeled X is half as likely to be brown as the other colors. A marble labeled Y is half as likely to be black as the other colors. A marble labeled Z is half as likely to be white as the other colors. Say on our twentieth draw from the box you see a black Y for the first time. Question I came up with: What is the best estimate for the probability of drawing a marble labeled Y from the box?


In the "talk" tab for Wikipedia Heat Engine article, someone is questioning whether an internal combustion engine can be modeled as a heat engine - and therefore is limited or is not limited by Carnot efficiency. The arguments are that the input is chemical energy, not heat. And also that IC engines dont operate on a closed cycle. Fresh air enters and is expelled each cycle. There are some answers there as well, but I'd like to get input from you guys to settle this matter once and for all! Is it ok to model IC engines as heat engines? And in a wider sence; can you model a solar cell as a heat engine and calculate carnot efficiency with T_c being the air temperature, and T_h being temperature of the sun. (I saw this in a textbook once, I believe.) https://en.wikipedia.org/wiki/Talk:Heat_engine (section headline: "Internal combustion engines can't be considered heat engines")


If I were sorting, for example, audio recordings based upon the performer, then "Vince Guaraldi" and "Bob Seger" would be sorted as "Guaraldi, Vince" and "Seger, Bob" On the other hand it's not immediately clear to me how I should sort a recording by "Vince Guaraldi Trio" or "Bob Seger & The Silver Bullet Band." My instinct is that this is now an entire proper name for a group and shouldn't be split up (just like I wouldn't sort things under "Zeppelin, Led"), but in both cases the group name is created to highlight the most famous member. Both of my print style-guides are silent on this; the MusicBrainz style guide, is clear that e.g. "Guaraldi, Vince, Trio" is correct, but I'm not sure if that's just a quirk of a single site, or if it is something more universally done.


I am thinking about the following question: when I look at the atomic emission spectrum of a specific element and I am only interested in the visible section of this spectrum and the method of retrieving it is via putting the element inside a flame, does the type of the flame in any significant way impact the observed spectrum? For example, does it matter wheter I excite lithium with a standard propane flame or instead a hydrogen or acetylene flame? I am trying to find ressources on this online but so far I haven't found any spectrum of any element that was measured using anything else than "the flame method" which I suspect refers to using propane gas (or other falmes with less known composition) as the exciting medium. I was tasked to find such spectra measured using different types of flames but since I couldn't find anything on this I am wondering if this question makes sense in the first place.


Suppose we have a loop of copper wire moving perpendicularly through a constant finite rectangular magnetic field directed into the screen . When the loop enters the field, the induced current would be counter-clockwise and when it leaves the field the induced current would be clockwise according to Lenz's law if we view the loop moving through the field from above. But the question here is that when the loop is moving through the middle of the field where it is neither entering nor leaving the field, can we say that the magnetic flux is changing and thus producing an induced current in the loop? The field is constant throughout, so will the change in flux be zero? But if not, how can we tell in which direction the current will be flowing? How could Lenz's law be applied to such a case?


I'm studying about electric field and referring to an article about electric field in wikipedia And in here, there are some doubtful sentences: The electric field is defined as a vector field that associates to each point in space the electrostatic (Coulomb) force per unit of charge exerted on an infinitesimal positive test charge at rest at that point First, in this sentence, i'm doubtful about 'electrostatic force'. As i know, moving charges also make electric field. This implies there are two kinds of electric fields: electrostatic fields and fields arising from time-varying magnetic fields. And i think, in here, there is electric field which is not belonging to 'electrostatic field and field time-varying magnetic fields'. For example, steady current, not in wire, makes electric field but don't make time varying magnetic field.. Is there something I'm thinking wrong? I think above sentences are uncomplete


In the vacuum state, particles and antiparticles can spontaneously emerge as virtual pairs due to the uncertainty principle. These virtual particles have a brief existence before they annihilate each other. This process is a manifestation of vacuum fluctuations. Similarly, in particle interactions, intermediate states involve virtual particles that can exist for a short time before contributing to the overall interaction process. These virtual particles mediate the exchange of momentum and energy between the initial and final states of the particles involved. If the intermediate state is represented by virtual particles, should we distinguish the actual physical process of the in-between, intermediate state from the vacuum state? Are intermediate states just mathematical constructs only useful for calculating; or they have a physical significance within the framework of quantum field theory? We cannot think of the intermediate state as the vacuum state despite having the same description involving virtual particles that cannot be observed due to their brief existence?


I am what most of the world considers a "non-native" speaker of English. I, however, consider myself a "native" speaker of English because: I grew up speaking English since birth, English is the language that I am most fluent at, I have never formally studied English, I just picked it up as a child as the first language that I acquired. Recently, a "non-native" friend of mine asked me whether it is "a helicopter" or , "an helicopter", and I realised that I would personally say "a helicopter" when the stress in the sentence was on the noun helicopter, but when the stress was elsewhere, I would say "an helicopter". Is this correct in the main standard dialects of English or is this some bad habit I have picked up?


In all of the treatments of elementary Euclidean geometry which I've seen so far, the section about triangle congruences introduces S.A.S. criterion as the basic postulate from which A.S.A. and S.S.S. criteria are deduced. I remember reading somewhere that one could choose any one of these three as "the congruence postulate" and deduce others from it. I am able to produce proofs for S.A.S. by taking A.S.A. as an axiom and vice versa, but S.S.S. seems to be the "odd" one since I cannot reach either S.A.S. or A.S.A. by taking it as the axiom. I was unable to find anything online that shows such a proof so my question is whether the premise that any one of these three criteria can be picked as the axiom is true or not. If it is, how can we prove, for example, S.A.S. through S.S.S.?


How would little, native English speaking kids say "Tell me a story about the bear!"? In my language, they might say something like "Tell me the bear!". Does "Tell me the bear!" sounds like a valid "little English" question from a little kid who is catching up the language? Context: Each language has different logic, and the kids build up the logic differently. I am very much curious about how young kids in native English environment build up the logic of the English language. For example, in my language, they might say something like "Tell me the bear!" instead of "Tell me the story about the bear!" Do English-speaking kids make similar mistakes, like saying "Tell me the bear"? What are the common mistakes they make in this context?


I hope this question hasn't already been asked, but I looked and couldn't find a question with a similar title. It is my understanding that Maxwell's equations and the Lorentz force law form the foundation of classical electromagnetism. These are fundamental because (in the classical limit) they are always obeyed, in contrast to Ohm's law, which is still a valid empirical law, but is restricted to a specific scope. As I understand them (correct me if I have made a mistake), these describe basic assumptions used to create a model of electromagnetism: the net outflow of electric field lines through a closed surface is proportional to the amount of and polarity of the charge inside the net outflow of magnetic field lines through a closed surface is zero a magnetic field that changes with time will create a circulating electric field an electric field that changes with time will create a circulating magnetic field, taking into account the polarization current the electromagnetic force on a charged particle depends both on the electric field and the magnetic field I've encountered some equations pertaining to phenomena described by QED, but I was wondering if there is a set of fundamental equations like Maxwell's equations and the Lorentz force law that describe all of the core requirements for accurate predictions in its model. In other words, can I summarize QED the same way as I have done above for classical electromagnetism?


I am currently reading "No-Nonsense Quantum Field Theory" by Jakob Schwichtenberg and in several derivations in the book it'll pull a partial derivative operator out of an integral, or change the order of full and partial derivatives on a function, etc. without explanation. So my question is: Are the differential and integral operators commutative? Is this always true? And if not, what are the conditions for it? Furthermore, in the more general sense of abstract algebra, if/when are unary operators commutative? A proof to accompany a given answer would be much appreciated, or at least the citation of one which is free and public. It's certainly intuitive that this is true, at least in the special case of the calculus and physics in this book, but I'm not sure why it is in a more fundamental, mathematical sense and I'm also curious about the general case of all unary operators. This is the kind of thing I would love to try to solve and research on my own but I simply don't have the time right now and I'm having trouble finding a straightforward answer online, so any help would be greatly appreciated. Thank you!


I would like to learn Mathematics for understanding GR, Differential Geometry, Riemannian Geometry and related research papers rigorously. I would like to carve out a clear path to understand these topics by listing out all the necessary prerequisites. I have undergrad Math under my belt such as: Real Analysis, Algebra, Topology and ODEs. I am missing intro to PDEs at this point. I have also created a diagram of the prerequisites in which each bubble represents a subject along with textbooks written in blue. Please take a look at the attached image/file. UG means "Undergrad" in the diagram. Specifically, I need help with the following questions: Is my goal (the center bubble) well defined? I know it may not be specific enough yet, but I have tried to list down some topics I am interested in RED color. Have I listed all the subjects? Am I missing any subject? Is Lie Groups worthy of mention here? Or, would it just fit under Algebra? Is Hyperbolic Geometry worthy of mention here? Is it relevant? How do I learn it? Any textbooks for it? Would anyone please help me break down the following subjects into specific topics that are necessary for my goal: Manifolds, Riemannian Geometry, Real Analysis (grad version), PDEs (grad version), Algebra (grad version).


I was explaining the tidal locking phenomenon to a friend. First I started with the formation of solar system and how at the beginning the planets were actually like balls of magma-esque rocks. And then how the force of gravity would eventually affect their shape and mass distribution and consequently, moment of inertia. But I myself wasn't satisfied with this abstract argument and tried to come up with a more intuitive one. So I said: imagine a fresh egg. Its yolk is at its center (almost) and the mass is evenly distributed. Then after a few days, the earth's gravity pulls the yolk down and moves it closer to the shell. This changes its moment of inertia. But I didn't know how to continue and pull a tidal locking explanation out of this. So I said never mind!! There are large-scale forces and objects involved in tidal locking that I cannot scale them down to something more accessible and easier to demonstrate. Now this has gotten on my nerves. I thought a lot about it and also searched [fruitlessly] quite a lot to find an intuitive, accessible way to understand this phenomenon. We may not be able to demonstrate the effect of gravity on a small scale, but can we use another force that behaves like gravity (e.g. static electricity) to design an experiment for tidal locking?


I've recently started using Texmaker to edit on my LaTeX documents (up to now, I mostly used Overleaf or TeXworks), so I am still new to this program's shortcuts. I have been looking for a shortcut in specific that I can't seem to find, and I would like to know whether this is something this specific LaTeX editor has. I am working on a lengthy file that has several chapters saved as separate .tex documents, which are all called by a single main.tex document. As such, any time I want to edit a specific part of the compiled .pdf file, I need to know in which line of which of my subfiles that specific part of the document was encoded - which is always a bit annoying to find. In Overleaf, this is made simple, since double-clicking on the corresponding line of the .pdf I wanted to edit would link me to the corresponding code line in the document. Does Texmaker also support a similar shortcut? And if so, what is it?


This is a question that a high school student asked me and I couldn't give him a satisfactory answer. He started by saying that An object appears red because the energy corresponding to a "red" (don't take it literally) photon isn't equal to that of the energy gap of the elctronic energy levels of the object. And now if I take the same object in complete dark and shine a red laser from above why don't I see a red beam coming out from the bottom surface the object ? My first reponse was that 'since the object is opaque it reflects most of the red light falling on it' but he wasn't convinced and I think that he is basically trying to ask what causes an object to be more reflective than refractive from the atomic point of view . Can someone help me with this ?


I would like to find the distance function for a curve comprised of all points within a lune that are of equal perpendicular distance to both circular arcs. In other words, what's the minimum distance by which a given point inside of the lune would need to move in order to become equidistant to both arcs? My use case would require both arcs to be of equal radius, if this helps. Constructing a signed distance function for the lune itself is fairly straightforward, but this particular problem seems considerably harder. I've been mulling over it for a few days without any real idea of how to approach it. Can anyone point me in the right direction? Edit For clarity, the curve marked in red is what I would like to find the distance function of:


I am searching for a book on group theory that follows in the style of textbooks written for math students (since I was one), but that covers all (or at least most) topics that would be needed in physics, such as in QFT for example (Think of the topics in Zee's Group theory in a nutshell). My problem with Zee is that it is too verbose and not as mathematically rigorous as I'd prefer (I can understand the concepts better when they are presented in a formal mathematical way). And my problem with other standard group theory books for math is that they usually do not cover all the topics needed for physics (Lie Algebras, Poincare groups...) A single or many-book-combination recommendation that would fill in these criteria is welcome.


Please correct me if I am wrong. Electron is fundamental and is zero dimensional (probably made up of strings). Electron is bound to the nucleus. Electron in an atom has velocity and position which are statistically distributed. Electron mainly interacts electromagnetically. Temperature is the measure of statistical kinetic energy of particles. In the nucleus of an atom, proton and neutron have temperature because they are a bag bound by color forces , valence quarks and a sea of quarks and anti-quarks. Quarks interact electromagnetically. Saying that protons and neutrons have temperature means that there is a randomness in the kinetic energy of the constituents of proton and neutrons. Electrons are bound to the constituents of protons and neutrons through electromagnetic force. Therefore if there is randomness in the kinetic energy of the constituents of protons and neutrons then the electrons must be experiencing wobblyness in its orbit, that is , it must be experiencing some kind of little randomness in its motion around nucleus. And therefore the electron must have a little temperature in the atom. My question is : Does the electron have a temperature in the atom?


I am a veteran of the wars in Iraq and Afghanistan. When talking about the people opposing us, I have always referred to them as the "enemy." Now, as I get a little older, and a little more aware (specifically, as I became a determinist), I have lost any hostility I feel towards them, and I feel that calling them the "enemy" demonizes them in a way I no longer wish to do. So, I was hoping there was some neutral term applied to enemy combatants, even if it's historical or in another language. I think opponent comes close, but it's not quite there. I was hoping there was a word like "interlocutor" meant for combatants of war. I have checked the OED (paywall, I think, I get it through my library), and the words I think that come closest are two obsolete words "contrary" and "adverse." (obviously those words are still in use today, just not in the noun sense of a contrary or adverse being someone who opposes you). Does anyone else have any better matches? One note, some may suggest I call them by the name they go by, such as the Taliban. The problem is that the Taliban is one of dozens of different groups of fighters and we rarely knew who belonged to which. So, I was hoping for a neutral term I could apply to all of them.


I recently received a newsletter from an entity I previously thought to be credible which is embarking on a brand/campaign around the concept of ambition which makes me wonder if my understanding of the word is "off". Can the following be examples of Ambition? Starting a new hobby. Getting out for a walk on the beach. I would have thought that while the above are goals/things a person may strive for - they are not ambitions because they are not substantive enough to qualify as an ambition. Similarly I see an ambition is a goal you strive to reach, while the "Ambition" campaign in question is about the journey (and indeed with no expectation of completing the journey) In other words I think "Planting a new garden" would qualify, but "Planting some plants in my garden" would not, or "Completing a marathon" would qualify, but "getting out for a walk on the beach wouldn't". Do I misunderstand the meaning of the word?


I got a bit confused about independent clauses,so I decided to ask ChatGpt, which has given me three different answers for the same sentence I think he is getting too old, suffering from Alzheimer's disease. The sentence is: I bring a natural positivity and a can-do attitude, allowing me to effectively solve challenging problems even under pressure. Here are my questions about that : First of all "allowing me to effectively solve challenging problems even under" is that a gerund clause functions as an adjective for the main clause describe it Now regarding the clause I have two possible analyses The first one is that: "allowing me to effectively" is a gerund phrase that acts as a subject. "solve" is the main verb. "challenging problems" is the direct object. "even under pressure" is a prepositional phrase that acts as an adverb The second one would be: . "allowing me" is a gerund phrase to start the caluse "to effectively solve" is an infinitive phrase "challenging problems" is the direct object to solve "even under pressure" is a prepositional phrase that acts as an adverb Does anyone have a clue which analysis is right and why?


I was reading a paper on black-hole information loss and it mentioned backreaction. I had never heard the word before so I googled it and was surprised to find no cohesive definition that I could understand. The best I found was this wikipedia article which gives a definition akin to "a backreaction is a field that is often used to calculate forces in quantum systems." However, it gave no strict definition or real physical justification. I was wondering if someone who understood the topic could explain them to me better, because right now they seem to me to be more like quantum epicycles than anything. P.S. I also found this article talking about back-action in quantum measurement which used the term back-reaction once. I could't quite figure out precisely how they were related though.


I am currently studying Linear Algebra Done Right (LADR) by Sheldon Axler and also How to prove it by Daniel Velleman. Currently I am in the chapter of Bases in LADR and I already read the chapter on equivalence relations in How to prove it. I tend to be somebody who looks for analogous things and something that I wanted to verify was whether there is a resemblance of Vector Subspaces and Equivalence Relations: Can one think of subspaces being equivalence classes? What I would suggest would be to say that vectors in a subspace form an equivalence class if they can be written as a linear combination of the basis vectors of that subspace. Moreover, there is a theorem which says: Every subspace of V is part of a direct sum equal to V. Wouldn't then the vector subspaces be analogous to a partition? Am I thinking along the right lines? Doesn't that mean that there is a sort of universal structure of mathematical topics?


A common construct for comparison (making parallels) is "the X of A is similar to that of B". Quoting an example here ... the animals' situation is similar to that of the plants. which is equivalent to the situation of the animals is similar to that (i.e. the situation) of the plants. I often wondered whether this construct can be extended to make parallels in more complex scenarios, where one needs to compare or contrast two relationships (between two pairs of objects). A contrived example is as follows: "The difference between werewolfs and vampires is similar to that (i.e. the difference) between wolfs and bats." Here, what I am trying to compare are two relations: the one between werewolfs and vampires and the one between wolfs and bats. As the two relations are each between a pair of objects, should one say "... is similar to that between X and Y"? If not, what is the proper way to express such comparisons/parallels? (I thought about using such patterns from time to time, but haven't seen it used. Hence the question here.)


I have the following diffraction pattern produced by a fabry-perot etalon The red represents the center of that pattern. Taking the mean intensity of the pattern as a function of the radius from the red dot and applying a cut-off radius results in This clearly shows a superposition of at least two diffraction patterns. Is there any way to automatically decompose the patterns? I originally tried Fourier analysis but that resulted in a complete mess as the peaks appear to be Gaussian, not sine functions. I can fit Gaussian to all the individual peaks. But then my question is if there is an algorithm or method that would allow me to determine which gaussian belongs to which diffraction pattern. I have quite a few more diffraction patterns some of which are more complicated and I wanted to see if there is a way to automate this?


Two up quarks in a proton lead to an imbalance, which results in the proton having the ability to attract electrons. Two down quarks in a neutron lead to balance in the electromagnetic force, leading to no interactions happening with electrons. The strong force and electromagnetic force appear to be directly related. Unless I am misunderstanding something, it appears that a particular arrangement of quarks leads to a specific kind of baryon that has a specific kind of electromagnetic behavior - meaning the strong force is indirectly leading to the behavior of the electromagnetic force. I don't understand why the electromagnetic force is considered to be a separate force than the strong force for the following reasons: Quarks clump together to form baryons due to the strong force, not the electromagnetic force. I understand that quarks have electromagnetic charges, but I don't understand whether the cause for this is due to it "fitting into the math" or it actually being a fundamental property of the quarks. I understand that the strong force overpowers the electromagnetic force, but how do we know for certain that quarks even have charges? Any insights or references would be greatly appreciated! Thanks for reading. So, my question is: why is the electromagnetic force considered separate than the strong force?


I am trying to simulate the reflection of a sound ray, that goes from a sound source, bounces off a wall, and is received by a microphone. The wall has a an absorption coefficient, and a specular reflection coefficient, both of which vary by frequency. Thus, the sound reflected by the wall specularly can be characterized by a certain frequency response curve. LTI filters are characterized by a frequency response and a phase response. Thus, we can treat the contribution of the wall's specular reflection as a LTI filter (applied to the source signal) if we know the correct phase response. The reflection path (shown in red above) corresponds to a time delay proportional to the length of the path. If we assume a constant time delay across the frequency spectrum we get a linear-phase filter, that is symmetric about the time-delay corresponding to the reflection path length. However, this filter clearly has "anti-causal" components: the filter is nonzero before the red line. Thus, the filter begins to have an effect on the signal before the length of the reflection path would suggest that it should. It seems that either the assumption of constant time delay across the frequency spectrum must be wrong then? If so, I wonder what the correct phase response of the filter is.


I have a couple questions on the mechanism for how solar projection through a telescope works: I recently took a small telescope focused at infinity, aimed it at the sun, and held a sheet of paper behind the eyepiece. It projected a focused image of the sun onto the paper. When I changed the focus away from infinity in either direction, the picture became less focused. I'm confused as to why this works since I always thought a focused telescope produces a virtual image that can never be projected onto a screen (and your eye turns it back into a real image for your retina). How can one focus setting for the telescope make an image that your eye can see and that can also be projected onto a paper? If the image coming out of a telescope can be projected onto a paper, I assume it is a real image. However, I find that no matter how far or close I move the paper, the image remains in focus. I always thought that a real image is projected onto a single focal plane at a fixed distance from the lens, so how does this work? Any help would be greatly appreciated.


Suppose we have two bodies at different temperatures, and we let them interact thermally in such a way that the process is not quasistatic (e.g. two different metal spheres touching). Do we arrive at the same final temperature as if the same spheres had equilibrated via a quasistatic process? If so then, rigorously, why? Of course I expect the answer to be yes but I can't convince myself of why in the context of thermodynamic theory. Edit: I think I see that since I can establish that in a final equilibrium, they have equal temperatures, then I can imagine a quasistatic process taking them to this final state? It seems thus that in the case of non-quasistatic heat transfer, ceteris paribus, we have the same entropy change as for the corresponding quasistatic heat transfer?


Clarification: the word "important" is ambiguous. Here, I use "important" to mean that we want to know the value of a quantity accurately. For many optimisation problems, it looks as if we are more interested in finding the minimum value, rather than finding the argmin. For instance, in machine learning models, we care about how well the model can predict, and we do not care about the internal argmin inside the black box that makes this possible. For a linear programming problem, say, for an airport, we do not really care about the exact way to schedule flights to get the maximum efficiency. We just want to come up with a plan which can be completely different from the optimal one, but have similar efficiency in terms of saving time. This is due to the existence of many equally good solutions. Note: one might argue that we do need to know a solution. Of course we do. But we don't care which solution we get, as long as it is a good solution. So it is unimportant to know the solution itself accurately; only the objective value is important. Nevertheless, argmin is discussed a lot in literature (like parameter estimation in statistics). However, I have not seen anywhere why this is useful. Could anyone give an example application where the argmin is more important than the min?


Some English nouns are identical to their verbs (and their adjectives) both in spelling and pronunciation, for example: "This is fake"; "to fake"; "this is a fake" "To tear"; "a tear" "To parody"; "a parody" "A misfire"; "to misfire" There are some examples where the (disyllabic) noun is spelt exactly like its verb but the noun is pronounced with the opposite stress (at least, in the standard accents which I hear day-to-day in the UK), for example: "An escort"; "to escort" "My recall of the event is faulty"; "I cannot recall what happened" "To repeat"; "a repeat" (this last one is a bit weak since people use either stress pattern in different contexts) This phenomenon is observed with some verb-adjective pairs too: "To perfect"; "this is perfect" In all three cases, I would be interested in: The corresponding formal linguistic term (if there is one) Examples! Although I know there are plenty more than I've shown here, they elude me. What's worse is that there's a fair few I've thought of but can no longer recall. Bonus: are there any examples of noun-adjective pairs that are pronounced with opposite stress?


Imagine a situation where I'm moving at the same velocity as the electrons in a conducting wire. In this scenario, from my frame of reference, the electrons appear to be stationary and thus there is no current from my POV. Now, for a brief moment, these "stationary" electrons come into contact with (or pass right next to) a light bulb in the vacuum of space and closes the circuit for a brief moment, causing it to illuminate, and then we all continue moving together. (water this down as much as possible to reach "spherical cows in a vaccuum" state for the purposes of this problem) Question: From my perspective, the light bulb lights up even though the electrons (which I observe as stationary) aren't "flowing" in the traditional sense. How is this phenomenon explained? Would I effectively be seeing a light bulb illuminating without any observable current thus creating something out of thin air (obviously no). Or rather, what do i have to "give up" for the laws of physics to be consistent.


I am trying to check if my understanding is correct. This is not really a question but a request for validation. I hope this is allowed. When painting with a heat reflective paint there are two options A- The room does not contain a significant heat source. This will always lead to a reduction in ambient temperature as heat from the outside is not allowed in. Heat by convection and conduction will still penetrate but overall the energy transfer will decrease. B- The room contains a heat source emitting IR rays. The rays will keep bouncing over the painted walls until it is absorbed by the air or the body of the heat source itself. Depending on the ratio between the external and internal IR sources, a IR reflective paint can lead to either increase or decrease of ambient temperature


In algebraic number theory, we would like to study rings of algebraic integers but sometimes they are not PIDs and thus they don't possess good properties. Because of this, we have introduced the notion of Dedekind domains, in which rings of algebraic integer are included and one can study the nice properties of Dedekind domains to understand algebraic integers. While doing algebraic geometry and commutative algebra, I have encountered polynomial rings in several variables, whose properties are quite different from polynomial rings in one variable, as the former ones are just UFDs while the latter one is a PID. So this "degeneracy" of polynomial rings from PID to UFDs reminds me of the similar case of rings of algebraic integers but I don't think they are related since the dimension of polynomial rings depends on the number of variables , which does not fit the definition of Dedekind domains.But I wonder if there is a way to classify polynomial rings of several variables(especially those over an algebraically closed field) into a specific type of rings, such that this type of rings has nice properties just like Dedekind domains and from which one can classify the prime ideals as well as determine the irreducibility of polynomials easily?


I came across a question a while back. It stated that a oil droplet was suspended vertically within an electric field. The man who suspended it had left to eat a very LONG lunch, and came back to see that it was splattered on the top plate (nothing with the setup had changed), and the question asked us to state a physical reason for such an event. I thought it was because of a transfer of momentum from electrons moving from the negative plate on the bottom to the positive plate on top, which just so happened to collide with the oil droplet. Someone else answered saying that the droplet evaporated slightly, which caused it's mass to decrease and so the force acting upon it managed to accelerate it more than that of gravity, causing it to move upwards slowly. That was the correct solution, but I'm just a bit confused, since wouldn't it lose some charge when it evaporates as well? So overall the q/m ratio would still be the same? They said it was unlikely that it evaporated some of its charge, so I'm just wondering how the charge within an object is distributed. Is it distributed evenly throughout, or is it more concentrated in some areas?


I have learned some basic measure theory as covered in the first chapter of Durrett's Probability: Theory and Examples, which includes the construction of Lebesgue integral (from simple functions to bounded to positive to integrable ones), as well as some basic properties such as Fatou's Lemma, monotone convergence theorem, dominated convergence theorem, the Fubini theorem, etc. I will be officially taking Real Analysis (measure theory), next year. However, as sometimes functional analysis turns out to be sometimes illuminating for my undergraduate research in mathematical optimization, I want to have a basic pass through the most fundamental concepts of functional analysis. I have read some posts here stating that it's possible to learn functional analysis before measure theory, but the other order is preferred due to the need for integration. Since I don't have much time nor want to stumble upon something that requires measure theory, does anyone has some recommendation on a list of topics or theorems that might be needed for a basic read on functional analysis? Thanks in advance! Edit: I have some basic background in general topology.