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Suppose I have a hose pipe connected to a tap. I turn on the tap and water flows out of the other end of the pipe. Now imagine I install two valves at different points along the pipe. Water will flow normally if both valves are open, but not if either one of them is shut. No doubt you can now see where this is going. If the spacing of the valves along the pipe is sufficiently long, and the duration for which the valves are open is sufficiently short, we can always imagine a second reference frame in which at any given time at least one of the valves remains shut, so how can we account for the fact that water flows in that frame even when the valves are not both open? I imagine the resolution is somehow to do with the fact that the flow requires a pressure wave to propagate along the pipe when a valve is opened, but I haven't been able to find a convincing resolution. Has anyone come across an analogous problem before? | 0 |
I'm implementing physics for a computer game, and came accross something that looks unintuitive to me. Consider two bodies at rest: Let's say we momentarily apply the same amount of force to them, but in different locations: at the center of mass for one body, and off-center for the other. The second body will gain angular velocity, while the first one won't. Both will gain rightward velocity, in the same direction. But will the second body gain less velocity than the first? Or the same velocity? Naively I would expect less velocity, since some of the force was "expended" on giving it angular velocity, but I googled around and experimented in Algodoo, and it seems the velocity ends up the same. Is that correct? Is there an intuitive explanation for that? | 0 |
I've got a few months to go before I enter in graduate school, but I have to say that my math background is not as good as I would like. Thus, I would like to take the opportunity to strengthen it and work on undergraduate analysis. I had in mind to work on the Rudin book I've heard so much about, the book is very concise and covers a wide spectrum of mathematical analysis (and a bit of topology) seen in undergraduate so I thought it was suitable for what I want to do with it. But I'd like to hear your opinions and recommendations if you've ever been in this situation and/or know of any other cool books (I'am also interested in functional analysis and especially Hilbert spaces so...) for the self-taught to get through the summer. Thank you a lot ! | 0 |
Let's assume we have a standard double-slit experiment, just for illustration's sake. When an electron collides with the detector, is its momentum parallel to the emitter's output, or is it something else? I know that the uncertainty principle forbids us from knowing too much about the momentum when we know the position, so let's assume we have a magic detector that will only give us the momenta and not the positions. This magic detector can detect the momentum of a single electron. If we shot one electron through this apparatus at a time, what would we see (on average)? I say on average, because there's a non-zero chance that the electron lines up with the emitter by chance. We should still be able to see the standard deviation of the momenta; if it's tight, then the momenta line up with the emitter, otherwise, it's relative to something else. I'll admit this question may be nonsense, but to my knowledge, the only limitations to this are engineering ones (due to the magic detector), not physical limitations. If this experiment is nonsense due to a physical limitation that I've misunderstood, I'd like to know why. | 0 |
I've been reading some category theory texts for my undergraduate monography, and I've found that one can talk about small and large categories using small/large sets, or do the same using sets/classes. Most introductory texts just shrug formalisms off, mention one of these formalizations and goes on. I've noticed that most category-focused authors and texts (such as Categories for the working mathematician and Sheaves in Geometry and Logic by Mac Lane) tend to prefer the small/large set approach, while texts that don't focus that much on categories (Topology and Groupoids by R. Brown) tend to prefer the class/set one. My question: Is there a categorical/set-theoretical reason to prefer the small sets approach? I'd assume that there's a benefit in having a set of objects even in large categories, but fail to see an actual difference. I found the paper Set Theory for Category Theory by Michael Shulman, but it was way out of the scope of what I can understand. | 0 |
I'm looking for a real analysis book that covers measure theory and I've been recommended Folland's or Royden's books, but their price is quite high and I was wondering if any of the books below would serve as a cheaper substitute. Theory of Functions of Real Variables by Lawrence M. Graves; Real Variables with Basic Metric Space Topology by Robert B Ash; Theory of Functions of a Real Variable by I.P. Natanson; Real Analysis by Gabriel Klambauer; Nonstandard Analysis by Alain M. Robert; Foundations of Analysis by David F. Belding and Kevin J. Mitchell; Foundations of Mathematical Analysis by Richard Johnsonbaugh and W. E. Pfaffenberger; Mathematical Analysis by Tom M. Apostol; Elementary Real and Complex Analysis by Georgi E. Shilov; Foundations of Modern Analysis by Avner Friedman; Real Analysis by Norman B. Haaser; Introductory Real Analysis by A. N. Kolmogorov; Introduction to Analysis by Maxwell Rosenlicht. | 0 |
I'm having a little bit of a hard time understanding some concepts to do with self-inductance in class. I understand mutual inductance. That's when a loop has some time-dependent current going through it, and because the current is time-dependent, the magnetic field the loop produces is also time-dependent, and the magnetic flux in a nearby loop is time-dependent. With self-inductance, though, is the idea that a single loop carrying a time-dependent current gives rise to a time-dependent magnetic field, and that very same time-dependent magnetic field gives rise to a change in flux in the single loop itself, which then generates an emf? If I'm correct in this reasoning, then that should mean that the new emf gives way to a current, which opposes the previously existing current. Why doesn't this just result in an infinite loop creating an opposing current, and then one that opposes that, and then one that opposes THAT...etc.? | 0 |
Given a mapping f: R->R, what properties must f have in order for its range to be countably infinite? I'm personally not sure, especially in terms of formalization, but ideas of non-monotonicity or some construction of countably infinite equivalence classes come to mind. For context, in quantum mechanics it's said that a system's energy can only take discrete values, but this confuses me since energy is defined as the sum of a function of position and a function of momentum, which can both take any real value. The indexing of the sets doesn't make much sense to me, especially since the kinetic energy, the function of momentum, is very clearly uncountable since it's strictly monotonic. So even if the potential energy has a countable range, it doesn't make sense that the total energy would be since the union of a countable set and an uncountable set is uncountable. Like I said, I'm more concerned with the question I asked above rather than this confusion that created it, as I'm sure an answer will either lead me in the right direction, or at least give me the means to reasonably formulate a new question- so don't worry about addressing this. Any answers or information are greatly appreciated. Thank you! | 0 |
I was solving this question in my coursebook and was not able to understand it conceptually. Though I did understand the mathematical approach to it. Now I understand that when the rod is given an initial velocity towards right, due to its motion an EMF is induced across it. Current starts flowing until it reaches a steady state. However, once the current becomes constant, no potential difference exists and the current slowly starts dying out at which point the rod starts moving back to its mean position. Then the velocity is in the opposite direction and the 'polarity' of the induced EMF reverses and current starts flowing in the opposite direction. My questions are- Doesn't it take an infinite time for the inductor to reach its steady state (where it behaves as a connecting wire). So, wouldn't the rod move in the direction of its initial velocity forever? What will happen once a steady current is achieved in the circuit? Will it be correct to say that since the potential difference across the ends of the rod is zero, the current wants to decrease to zero immediately, but the inductor only supports a slow decay of current? Can we also say that the reason the circuit doesn't heat up due to short circuiting is because there is no actual battery in the circuit? I tried asking my teacher but I couldn't get a satisfactory answer. | 0 |
Consider a circular horizontal plane (like a round tabletop) rotating around its center. Consider a body A resting on this tabletop. Since the tabletop is rotating around its center, the body is moving along with it in a circular motion around the center of the circle. I understand that in circular motion, there's always a centripetal force, pushing the object towards the center of the circle. And in this case, this force is the static friction force, operating on the radial axis towards the center. I have two questions regarding this: I was under the impression that a static friction force, or any friction force, always operates in the opposite direction of some other force which operates on the same body. Is this correct? If so, what is the additional force (opposite to the friction force) that I'm missing in the above scenario? Is the static friction force operating only on the radial axis towards the center? Or is there also static friction along another axis? | 0 |
I'm wearing glasses with a sunglass clip-on. This means I have my regular glasses and, on top of them, I have a second pair of lenses that work as sunglasses and attach to my regular glasses using tiny magnets. I'm also in a car, and I'm looking through a window layered with Insulfilm. Curiously, when I look through the Insulfilm with the sunglasses on, I see a rainbow pattern. If I keep my glasses on, but remove the sunglass clip-on, the rainbow is gone. If I look through a window without Insulfilm, the rainbow also seems to be gone. If I look through the Insulfilm with the sunglasses clip-on, but without my regular glasses, the rainbow is there. I believe my sunglasses are polarized, and the rainbow fades away when I tilt my head a little. Why exactly is this phenomenon happening? I guess it has something to do with polarization due to the rainbow fading when I tilt my head, but I can't figure out why it is a rainbow and why I can't see it without the sunglasses. | 0 |
This popular question got me wondering if there is a simpler way to reduce radon in homes. Our house has a fairly standard radon mitigation system. It has a lot of parts. There is a hole cut in the basement floor. There is a thick pipe which goes up through all the floors with lots of bends and turns, like the old Microsoft "pipes" screensaver. There is a powerful, always-running fan in the attic pulling the air up and out through a roof vent. There are also other components, such as a manometer and other sensors. It all has to be serviced by technicians. Like any system with a lot of components and a lot of puncture points through surfaces, there are many points of potential failure. Why not have a standalone unit in the basement with a fan that blows the air through an electrostatically-charged baffle of gills or fins or whatever, collecting the radon until it decays? At least then it won't end up in the lungs, where it causes the damage. We could put lead shielding around the unit, if necessary. | 0 |
I found a few questions that are similar to mine, but I do not think that either of them answers exactly what I want: Is there a Birkhoff-like theorem for stationary axisymmetric metrics? or Gravitational Collapse: Kerr solution is a vacuum solution but not for any rotating body? My question is this: When we look at simple models of gravitational collapse for spherically symmetric matter distribution (such as Oppenheimer-Snyder) we always exploit the fact that the solution to the Einstein field equations just outside any spherically symmetric collapsing matter is Schwarzschild and that this is the only possible solution. This is thanks to Birkhoff's theorem. An analogous theorem exists also for charged matter. Unfortunately, no such theorem exists for rotating matter, as the spherical symmetry is broken. This means that we can't approach the gravitational collapse of rotating matter the same way we would for spherically symmetric charged or uncharged matter collapse. So how do theorists actually study rotating black holes? Do we need some sort of computational or numerical methods? | 0 |
My friend was sitting in a wheelchair, with a cup of coffee in one hand. That made me wonder, how would the wheelchair move if he turned only one rear wheel? He tried it, using his right hand to turn the right rear wheel. The wheelchair starting moving in a circle. No surprise. But I was interested in the radius of the circle. The circular track of the right rear wheel on the ground, had a radius that was slightly larger than the distance between the rear wheels (i.e. the distance between the two points of contact between the rear wheels and the ground). What determines the radius of the circular track of the left rear wheel (the unturned rear wheel)? Possibilities that come to my mind are: Location of all four wheels The size and shape of the contact patch between the wheels and the ground The coefficient of friction between the wheels and the ground, and the weight of the system The diameter of the wheels Angular speed of the right rear wheel How the person is sitting within the chair (centred, or leaning to the left, leaning to the right) (Assume the wheels are in parallel planes.) | 0 |
I apologize if this question has been asked before, but I haven't been able to find a post which covers this exact question. Let's say that we have a driving car which is moving forward. At some point, one of the rotating tires hit a stone on the road. Which way does the stone move, in relationship with the angle in which the tire hits the stone? Can the stone ever be thrown backward onto another car? Does the general direction (forward, backward, sideways) depend furthermore on the shape of the stone? Whether it is for example perfectly spherical or asymmetric? I am kind of struggling how I would answer such a basic question, since there seems to be many factors at play here. Can you even answer this question without fully knowing all the nuances and details? | 0 |
Suppose we have an isolated n-type semiconductor having varying concentrations of carriers over x. After some finite time, the concentration becomes uniform for a given constant temperature. We always say that this is because of diffusion which allows the movement of charge carriers to move from higher to lower concentrations. But shouldn't this diffusion occur because of the repulsive forces between electrons? why do we say it's because of the random motion? (the random motion allowing for the carriers to rearrange should be because of the electric field produced by them due to the gradient of charges). If we see some excess charges added to an isolated conductor, then the electric field produced by these excess charges allows the charges in the conductor to redistribute to settle on the outer surface where the net force experienced by the excess charges is zero. Shouldn't the same principle apply to semiconductors? What exactly is happening inside? I hope to get a clarified answer. Diagrams will be highly appreciated. | 0 |
I'm trying to deepen my understanding of Von Neumann entropy (out of interest in the quantum Bekenstein bound) by learning more about the reduced density matrix (aka local density matrix), which is given by the modular Hamiltonians (aka entanglement Hamiltonian). As I understand it, modular Hamiltonians are only known to be local (i.e. an integral of a local operator) under three specific situations. When the state is the vacuum state in a half-plane (the only case where the modular Hamiltonian is given by the traditional Hamiltonian) When the state is the vacuum state and is limited to a ball and the field theory is a CFT When the state is the vacuum state and is limited to an arbitrary piece of a null plane I could not, however, find any explanation for how or why these are the three known cases where it is local (and additionally why only the first one is given by the traditional Hamiltonian). I'd appreciate any insight you could provide on this matter. | 0 |
Adjectives and adverbs can be formed by adding "y" or "ly" to e.g. a noun, such as: heart -> hearty, heartily However, sometimes these words are not in use, or make no logical sense, such as: chair -> chairy, chairly Although you could come up with a situation where you'd use "chairy", e.g. "this block of wood looks quite chairy", you'd probably be more likely to use "chairish" or "chair-like". And if you read all the books in the world, you'd probably never come across the word "chairly" or "chairily". What is the status of words like these? Do they exist because they are technically correct and their meaning is obvious? Do they not exist until someone (notable) uses them in a (notable) text? Or, to make the question more specific: if I were to create a spell checker, should I allow or disallow these words? | 0 |
I am teaching students at a high school about bending of a beam (the beam is clamped at one end and then an applied force acts downwards close to the end of the beam to bend it). I thought that it did not require further explanation that the deflection of the beam due to the force at the end is inversely proportional to the stiffness of the beam, proportional to the size of the applied force, and that it is dependent on the distance from the fixed point where the beam is clamped. However, I have been asked to provide justification for these statements and not to appeal to physical intuition which is not obvious to students who are not seasoned physicists. What type of justification would be enough to use these statements at an elementary level? How could I explain that these statements are true? | 0 |
Compressive strength is how much inward force a given area of material can withstand before failure. The force tries to compress the atoms closer together. Mohs hardness is the difficulty of a material to be scratched. Abrasion resistance. However, when an object scratches another, there first is a point of contact where the harder material is forced into the surface of the other, and then lateral forces shear off a grove from the softer material. But at this point of contact, the forces are compressive, trying to force the atoms together, until one of the materials give way. As such, I would expect materials with a high Mohs hardness to have a high compressive strength and vice versa. This holds for ceramics such as silicon carbide, aluminum oxide, and others which have both a high compressive strength and a high Mohs hardness. Does this trend hold for most/all materials and to be expected for the above reasons? Are there any counterexamples? If so, what could explain a high Mohs hardness and low compressive strength or vice versa? | 0 |
I really don't like the phrase "excited for" which seems to have become very common in recent years, as in "I'm excited for the weekend...". My sarcastic reply would be "I don't think the weekend cares or appreciates you being excited on its behalf". Anyway, I'm aware I have to accept that language changes, even though here I think it's just bad grammar. The question is ... are there any studies looking at how (the wrong use of) this phrase came about? I have a theory based on absolutely nothing, that perhaps it came from Spanish where 'por' can be used in this sense, e.g. Google translates "excited about this weekend" to "emocionad(o/a) por este fin de semana". Is it possible that this usage came via speakers of Spanish, perhaps in the US? | 0 |
"Longingly" is the adverb form of "longing". Depending on where you look, "longing" is described as an adjective, a verb, or a noun: Oxford defines it as either a noun or an adjective. Cambridge, Collins, and Merriam-Webster seem to define it exclusively as a noun. Wiktionary says it's either a noun or the present participle verb form of "long". Despite the disparate definitions, I can see justification for all three forms. This raises a question: which distinct form of "longing" is "longingly" derived from? Or is the adverb formed indistinctly? I know that adverbs ending -ly are typically formed from adjectives, but English is full of exceptions. I checked a few etymology sources but they either only had information on "longing" itself, or only described longingly as being formed from longing as a general word. | 0 |
My question is closely related to the answer of this question: Why is general relativity background independent and electromagnetism is background dependent? General Relativity is often stated to be "background independent", because it calculates how spacetime is curved. That's in contrast to other theories like classical electrodynamics which act on the manifold without interacting with it. I understand that GR interacts with spacetime (and that that's a great advancement in comparison to the theories before) - however, I do not understand, why it's called "background independent". Matter curves spacetime. That means to me that there has to be a spacetime first which can be curved by matter. If the universe is empty (in GR), Minkowski spacetime is still there. If spacetime were produced by matter like the electromagnetic field is produced by its sources I would understand the term "background independence". But to me it's only interacting with the background, not background independent. (Like an artist who is forming any possible object out of clay is not independent of clay... Just acting on it in every possible way) How shall I understand the term "background independence"? Is the term not precise? Did I get something wrong? | 0 |
Quite naturally, the observable Universe is the only bit of the Universe we can extract information from, as light from farther away has not reached us yet, and there are zones from which we'll never even be able to extract information, as they are causally disconnected from us due to their high speed. Therefore, how do we know the cosmological principle holds at larger scales if we can't observe them? It just seems to me a quite bold assumption to make that the entire Universe is homogeneous and isotropic. Perhaps it presents irregularities which we'll never know of, and which would be necessary for us to not mess up with our cosmological models. Frankly, I don't quite see the point of building cosmological models if we only know what seems to be a small portion of the Universe, just like it makes no sense to try to see the picture of a jigsaw puzzle just by examining one random piece of it. | 0 |
Recently I came across Normal matrices and their properties, one of which states that their eigenvectors are the same as their adjoint and are orthogonal. I've gone across some proofs and I understand it but when I tried to prove the same using the inner product, It somehow states that the above is true for any arbitrary matrix. Can someone possibly help me point out where I'm going wrong? Let's say A is a matrix and B is its adjoint. If x is an eigenvector of A with eigenvalue k, then, < x | A | x > = < x | kx > = k< x | x > also < x | A | x > = < Bx | x > hence < Bx | x > = k< x | x > = < kx | x > So, x is also an eigenvector of adjoint of A. | 0 |
In the English language there are many words where the letter "a" is pronounced as a short (continental European) "e". Or at least very close to it. However dictionaries point out that in these cases the correct pronunciation is more an "ae" sound. Several websites on English pronunciation take the same view. So I decided to test this assumption. Over a period of weeks, whenever I watched an English or American television program, I payed attention how these "a"-words where actually pronounced by native English speakers, and made notes of my observations. I found out that in fact there are many words that have the pure "e" sound. A few examples: back, cash, cat, fact, relax, track, crash. So what is going on here? Are this words pronounced incorrectly, or are the dictionaries and websites promoting viewpoints about pronunciation that are outdated? Or has it to do with regional variations (dialects)? | 0 |
In Photoshop (and many other computer graphics applications) there are are various ways that images/layers can be composited with their underlying background. These are sometimes called blending modes. One blending mode in particular is multiply, which has the convenient feature that it eliminates white backgrounds from black line illustrations while properly dealing with anti-aliased edges (see the example below). Is there a way to do this in LaTeX? I have a large number of images with a white background that need including in two documents with different coloured backgrounds. I'm looking for a quick way to remove the white background from the images without having to manually add an alpha mask to every image in Photoshop. Normal blending mode: Multiply blending mode: Close up of anti-aliased edge, properly feathered: | 0 |
I found this MC question that asks about a soft iron bar entering a solenoid: At first, my answer was (B) because the iron bar should increase the magnetic field induced by coil M and, according to Lenz's law, this magnetic field should oppose the original current, making the lamp dimmer. Also, the magnetic field created by M should then induce a current in N in the opposite direction to the current in M (also Lenz's law), so since the current in M is right to left, then it should be left to right in N. I wasn't too confident in my answer though. Then, I noticed that they had drawn a DC power source in M. This would indicate no changing magnetic field, thus no current induced into either M or N. This would mean that the lamp doesn't change brightness and there is no direction of current in N. Right? Is this just a mistake in the question, or am I missing something? Thanks to anyone who can help. | 0 |
Given a path, how do the magnitude of the velocity and acceleration vector along the path correlate? I am confused due to the fact that the acceleration is the change of velocity over time and in general if you plot the velocity and acceleration over time, you can see a correlation (e.g. constant velocity means zero acceleration). So I expected the same relationship for the magnitude of the velocity and acceleration vector. Moreover, I got confused because the magnitude of a vector is always positive and in the general plots of the velocity and acceleration over time it is not the case (e.g. linear decrease in velocity means constant negative acceleration). Therefore, can someone clarify if there is a clear relationship between the magnitudes of the velocity and acceleration vector and how this is defined? And if not, is there a relationship between the velocity and acceleration vector, so you can verify if your velocity and acceleration vector are correctly calculated? | 0 |
Relativistically, the electric field of a moving charge is not purely radially directed, but is instead concentrated perpendicular to the line of motion. So, a current loop consisting of electrons as charge carriers should generate a "charge separation" effect, where the negative electric field is concentrated perpendicular to the loop, and is not perfectly cancelled by the stationary positive ions in the (electrically neutral) conductor. Conversely, in the plane of the loop, the positive electric field should dominate. This "charge separation" effect is separate from the magnetic field produced by the current loop, as it should have the effect of accelerating stationary charges in the surrounding space, which are unaffected by a magnetic field, because their velocity is zero. I have never heard of such an effect, which should be small but measurable, hence I am asking the question. | 0 |
A term in a Lagrangian is gauge invariant if one makes sure to use quantities which transform in proper representations of the group of gauge transformations. This means that one cannot write terms in the Lagrangian using a 'bare' gauge field, but one can use the associated field strength or act on a gauge field with a covariant derivative. On the other hand, if one writes a term using a 'bare' gauge field (for example, a combination of the gauge field and the associated field strength tensor), although it will not be gauge invariant, will this term still be invariant under diffeomorphisms of the spacetime on which the field theory is defined? Since diffeomorphism invariance is sometimes taken informally to be ''background independence of the field theory'', I guess that this term should be invariant under diffeomorphisms? Or does diffeomorphism invariance fail because it is a specific example of a gauge symmetry? | 0 |
This may be an amateur question, however here goes. I understand the double slit experiment creates a situation where detecting which slit the particle/wave goes through causes a collapse of the wave function and for the particle to act like a particle and not cause an interference pattern. What I want to know is this: If say for example, the detector emits a signal to a speaker if it detects the particle and the speaker emits an audible sound, my understanding is that the wave function would collapse - we measured the outcome. However, what if we left the detector doing its thing, but turned off the speaker so that we didn't know? Would this still be considered "measurement" or does measurement require recording/knowledge (future included) of the outcome? | 0 |
I'm considering to choose between "Convex Optimization" by Stephen Boyd and Lieven Vandenberghe, and Yurii Nesterov's "Lectures on Convex Optimization" to supplement for my university course. I have had Calculus and Linear Algebra, my course covers constrained and unconstrained convex optimization, linear programming and constrain programming so i want to know which one of these two books will be good for me. The lecture i'm having mostly just focus on the implementation of the method but not the mathematical proof behind it so i want to learn more about the math behind optimization algorithms. I've read a few section from each book and personally i think Yurii Nesterov's book is more rigorous and easy to understand, but i still want to ask because i don't have enough time to study both and i want a strong foundation on this subject for my AI major. P/S: I'm also really enjoy rigorous books like Apostol's Calculus I,II so i want to know which of the two books above have similar style. | 0 |
I often see "functional" used as an adjective in situations where I think that "function" would actually be the better choice. Specifically, I am referring to translations of German compound nouns. Here is an example: Funktionsarchitektur. This is the architecture of the function/functions. To be specific, we are talking about a technical function in a machine or vehicle. This is the architecture of the devices, control units, etc. which perform the specific function. ABS would be one example. In this case I would use "function architecture", translating it as a compound noun in English as well. For me, "functional architecture" is an architecture which functions and not an architecture of/for functions. I see a lot of cases in which the adjective "functional" is used instead of expanding the compound noun. Any thoughts on this? | 0 |
while working on problems related to fluid mechanics i came across a problem in which we were asked to find the kinetic energy of a sphere which is under pure rolling and has non viscous fluid filled inside it. here you can see the figure : so my initial thoughts were to write the moment of inertia of the whole body about center of mass and then proceed as normal. But to my surprise this was not correct so i thought more about this and made a conclusion that if the fluid is non viscous then maybe we don't have to consider its rolling motion. This assumption made me do this question but i am not sure whether it's correct or not. I want to know what you guys think about this assumption. | 0 |
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. | 0 |
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. | 0 |
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. | 0 |
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? | 0 |
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. | 0 |
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? | 0 |
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 | 0 |
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. | 0 |
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. | 0 |
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? | 0 |
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? | 0 |
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) | 0 |
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? | 0 |
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? | 0 |
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? | 0 |
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! | 0 |
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. | 0 |
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. | 0 |
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. | 0 |
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. | 0 |
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. | 0 |
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? | 0 |
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? | 0 |
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? | 0 |
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 | 0 |
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? | 0 |
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 | 0 |
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"?)? | 0 |
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? | 0 |
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. | 0 |
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? | 0 |
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? | 0 |
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. | 0 |
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? | 0 |
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? | 0 |
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 | 0 |
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? | 0 |
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...) | 0 |
"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? | 0 |
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? | 0 |
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? | 0 |
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 ? | 0 |
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? | 0 |
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? | 0 |
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? | 0 |
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. | 0 |
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. | 0 |
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! | 0 |
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! | 0 |
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? | 0 |
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" | 0 |
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? | 0 |
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 | 0 |
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. | 0 |
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! | 0 |
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?) | 0 |
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? | 0 |
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. | 0 |
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. | 0 |
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. | 0 |
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. | 0 |
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? | 0 |
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)? | 0 |
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? | 0 |
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. | 0 |
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