darklight03/StackMIAsub huggingface dataset:
I'm looking for a word similar to "free" or "without cost", but that makes it explicit only that no money is exchanged, while still allowing (or implying, or explicitly specifying) that some other exchange of some value has taken place. For example, if I rent a room to a friend in exchange for language tuition, I have a form of non-monetized exchange. I would like to say that the room is rented for free, except that it isn't for free here, because tuition is given in return. I still have to pay taxes and bills in money, and to describe this I want to say that the room is rented for zero-money (that way pointing out that I can't use the exchanged tuition service to pay the bills, and that therefore the exchange is somehow of less value than its cash-value equivalent). Saying it's "free" is not correct, because something of value is given in exchange. "Costless" I think has similar issues. "Barter" is a related meaning, but here I'm dealing in an on-going exchange-of-services relationship, for which "barter" doesn't seem right. "Non-monetized exchange" or "on a non-monetized basis" is the best I have so far. I can't think of term that is succinct and gets this meaning clearly. Any ideas? A technical accounting term would do, but ideally I'd have something that is clearly understood by a lay person.
Imagine a train moving relativistically according to an observer at rest wrt to the tracks it's moving on. The train seems contracted in the direction of motion. At the points where the wheels make contact with the tracks, the wheels have zero velocity, so the distance between these points is not Lorentz contracted (like the caterpillars of a tank are at rest). Now what will happen when the train suddenly decelerates, simultaneously at all parts, as seen from the observer at rest? Obviously, the wagon will expand, but what will happen to the wheels? Will they follow the expanding motion, which leads to friction between the wheels and the track? On the other hand, when the train starts to accelerate from zero velocity, the distance between the wheels will not Lorentz contract, so the wheels seem to have to slide with friction, dragging behind the contracting wagon, so to speak. Will the reverse happen when decelerating?
I had a recent conversation with a professional mathematician about the status of relations, functions and predicates. I was arguing that it seems intuitive (to me at least) to classify them in this hierarchy (as to which is more primitive): All predicates are functions. All functions are relations. The obvious problem here is that it seems intuitive that unary or even nullary <predicates/relations/functions> are more primitive than their n-ary variants. Is there a way to compose functions as at least unary relations or vice versa (relations as unary functions)? If not, is it possible to order them in such a hierarchy given the binary restriction. Finally, if there is something more primitive that has a formal definition, then that would do as well. A resource or explanation pointing to how at least functions, relations and predicates are composed using this notion would be helpful.
It's well known the effect of Rayleigh scattering on the color of the sun, and it's explained several times on this website. Here's one of them. The summary of these explanations is, that when light travels through a colored medium, that color is being "used up" to make the medium the color it is, and only the other colors will go through. Make very much sense. But the problem is that our experience in everyday life is just the opposite. When we shine a light through a colored medium, the light becomes the same color as the medium. Although it's hard for me to understand why (this was asked here without an impressive answer). And that's also happening with sunlight that goes through colored glass. So why does the atmosphere behave different than anything else? And is there anything I can experiment with that would have the same behavior as the atmosphere?
I did some online search and found the explanation using the following two diagrams. It's not perfectly convincing to me. Or at least it is not clear to me in the following details of the process: For the secondary rainbow to have the inverted order of bands, do the light rays have to exactly reflect twice and cross each other twice inside the water drop? If so, what are the necessary and sufficient condition for that to happen? It seems to rely on the relative direction of the incident light and the surface of the water drop and the position of incidence? The diagram only shows one particular position and angle of incidence, so it seems kind of accidental to me that it inflects twice and cross each other twice and got to the inverted order.
I was reading a physics problem related to astronomy, and upon re-reading it, I realized that it could be really indicated to extrapolate some really interesting physics-related information. One of these is: How could we measure the ratio of a planet's radius to a star, for example using transits? The only idea I have is to compare them when the planet passes exactly in front of the star (i.e. they are aligned with our view), but this only makes sense if the distance between the two is much smaller than the distance between us and that star system (which I think is true enough for every system except the Solar system) and if it is possible to obtain such high resolutions (and I already had my doubts about the distance between the two, which should be much greater than their radii anyway).
Suppose I have a nonlinear ordinary differential equation, in several variables, with a stated initial condition. How would I go about finding a nonlocal linear approximation? What is known about such approximations? By nonlocal approximation I just mean that the would-be approximation minimizes the maximum distance between the solution of the approximation and the solution of the original, nonlinear initial value problem, or meets some other criterion such as minimizing the integral of the square of the distance. At least in the case of minimizing the integral of squared distance, this problem amounts to a nonlinear least squares problem, and the criterion is a differentiable function of the parameters of the linear approximation, so maybe a Gauss-Newton type algorithm or even simple gradient descent (at this point I'm not worried about efficiency) could be applicable. For the problem I have at hand, I constructed a linear approximation at the initial point, via the Jacobian of the original system of equations, but the solution of the local approximation is very different from the solution of the original nonlinear IVP at later times, which inspires me to look for a nonlocal approximation.
While reading the Wikipedia article on Drag Crisis, I found: The drag crisis is associated with a transition from laminar to turbulent boundary layer flow adjacent to the object. While, the Wikipedia article on Turbulence states: In general terms, in turbulent flow, unsteady vortices of many sizes appear which interact with each other; consequently, drag due to friction effects increases. If drag (due to friction effects) increases due to turbulence, then why does drag crisis occur when flow shifts from laminar to turbulent? Shouldn't drag be less for laminar as compared to turbulent flow? Why else would surfaces smoothen due to drag over time? The Wikipedia article on Drag Crisis also goes on to say: For cylindrical structures, this transition is associated with a transition from well-organized vortex shedding to randomized shedding behavior for super-critical Reynolds numbers, eventually returning to well-organized shedding at a higher Reynolds number with a return to elevated drag force coefficients. I am having trouble understanding this. A related question, Drag Crisis and Terminal Velocity?, examines the relationship between drag coefficient and terminal velocity.
I'll put pictures from the book (Introduction to the Structure of Matter: A Course in Modern Physics by John J. Brehm and William J. Mullins) as I think they are relevant to understand my problem: I have trouble understanding the case where the observer watches the source in a direction perpendicular to the magnetic field. The electron will rotate around B axis, so the observer will only see a linear oscillation of the electron hence linearly polarized light. But how can the Lorentz force explain the splitting of spectral lines (i.e. the change of the frequency of the electron)? The book suggests to view the linear oscillation as a combination of two counter-rotating motions like this: But if this is the case, the Lorentz force would act in a plane perpendicular to the image so it won't explain the change of the frequency of the circular motion of the electron (and so the Zeeman splitting, classically). Instead the situation is clear when we observe along the direction of B, as in that case Lorentz force would act radially.
The standard explanation in textbooks goes that in the presence of electric field (e.g. external electric field) the free electrons inside the conductor will keep moving until electrostatic equilibrium is reached and that they will create a field which will cancel out external field (e.g. electrons move to one side leaving positive charge on another side, this creating electric field that cancels out the external field). But what if the external field is so strong that there is just not enough free electrons to cancel out external field? What if there is nothing left to be moved? E.g. I have electric field lines going from left to right, all free electrons move to the left surface, there is nothing left to move, but it's still not enough to cancel out external field. Will there still remain an electric field inside the conductor then? If not, what will happen?
I'm looking for a phrase that describes a problem whose complexity starts to increase exponentially, either because the problem is recursive, the definitions/conditions of the problem interlink with themselves, or it turns out that it's connected to a great many other issues, and changing one changes all the others in turn. This phrase would be useful when, say, discussing how to solve a simple, harmless, trivial little bug in some software...that now has eight people standing around debating business logic and corollaries to corollaries and exceptions to exceptions. Phrases that are close, but not what I'm looking for: "Tip of the iceberg" could be used later on when describing the original difficulty, since it would become much worse later, but the rest of the situation wasn't originally an issue or complicated, and only by investigating the original, simple issue did the full complication arise. "Can of worms": the rest of the difficulty isn't inherently a problem; the original, minor issue is the only part that's an actual problem, and the rest of the complication is created by trying to solve it. "Cure is worse than the disease": it's not that the solution is bad or undesirable, but that the original problem was far more complicated than it first appeared. "Gordian knot" describes a problem that is already complicated known to be difficult; what I'm looking for is a situation that seems simple or even easy until you try to solve it.
A congruence is a useful notion in general relativity, relating mathematcal definition and physical interpretation: "A congruence (more properly, a congruence of curves) is the set of integral curves of a (nowhere vanishing) vector field in a four-dimensional Lorentzian manifold [...]" Following up some intuition (mostly concerning timelike congruences) I find the following description further down on the Wikipedia page especially relevant: "The integral curves of the vector field are a family of non-intersecting parameterized curves which fill up the spacetime. The congruence consists of the curves themselves, without reference to a particular parameterization." The emphasis on non-intersecting is placed in the Wikipedia article already; however without any further supporting explanation or link. To me, this raises an ambiguity: The phrase "non-intersecting curves" might be understood as referring to each of the (unparametrized) curves as a set (consisting of events), meaning that the intersection of any two of these sets is empty. In other words: (pairwise) disjoint sets. Or: The phrase "non-intersecting curves" might be understood (in a geometric sense) as allowing the possibility of at least some instances of being tangent to each other; corresponding to the general geometric description of such mutually tangent curves as "touching but not intersecting"; thus allowing at least some pairs of curves to have at least one point (event) in common, "in which they are tangent to each other". Accordingly (and without hinting at my intuition) my question: Which is it?
In my physics textbook, the foundation for work is derived using newton's third law, where F_surr = - F_gas, where surrounding represents a piston-cylinder device and gas is pushing against the inner surface of the piston towards the right. My questions are: Given this information, it is obvious that W_surr = - W_gas. So, shouldn't the terms cancel each other and result in zero acceleration of the piston, i.e., the piston remains stationary? I understand that newton's third law is applicable for forces acting on different objects in contrast to second law which is used for analysing forces acting on a single object. So using second law, if i were to dissect the system and piston into two bodies and assuming no friction, we will have F_surr acting towards the left on the gas and F_gas acting on the piston towards the right. So, I am confused here, shouldn't F_surr being the only force on the gas cause it to compress and F_gas on the piston cause the piston to move towards the right? Finally, during a quasi-equilibrium process, will there be any accelerations of the piston at all at each equilibrium states or is the equilibrium state like a point at which the system and surrounding are in complete mechanical equilibrium and are stationary? Essentially, I am having a hard time applying newton's second and third law to derive work equations.
Given two probabilistic distributions (red and blue) it is well known that a linear interpolation between them is well defined (see this). For example, by the Wasserstein metric we have the following interpolation: When I first saw this approach, it crossed my mind that the Bayes 's posterior could be explained by a similar geometric argument. Perhaps, the mean between these distributions by the Wassertein metric. However, when I plot the Bayes's posterior (in green) we see that it does not belong to this linear interpolation. My question is: is there a geometrical reasoning that we can use to interpret the Bayes 's posterior? If there isn't a geometrical reason , can there be a variational one (the Bayes's posterior is the minimum of some energy in this space)?
I am curious about magnets and their interaction with the electromagnetic field. So I tried to find this answer and stumbled upon (Magnetism and Photons), this is an interesting description that I quite like, but it doesn't really explain how a magnet can keep a piece of metal on a string hovering in mid-air indefinitely, the magnet is not moving, yet there is an interaction there, as I understand it via photon exchange. I did read in a magazine (New Scientist particle physics special from memory) that electron spins create a non homogeneity that interacts with the electromagnetic field that then "tends" to allow virtual elements to appear through particle/antiparticle annihilation, with a resulting photon being generated. I don't love this explanation, perhaps because I find it hard to get my head around it. Any advice or references you can point to explain the electromagnetic field (this is confusing in itself, I thought long ago we gave up on the idea of an "ether", it seems it's back) and then the interaction of electron spins (magnetism) on this field. Thanks in advance, Tibor
I am looking at a tutorial create for the FENICs finite element method package. The tutorial shows a system of advection-diffusion-reaction equations occurring in a solution that is moving according to the incompressible navier-stokes equations. In the tutorial, the authors simply present the weak form, without explaining the derivation. I was hoping someone might be able to explain how to derive the weak form, or even identify some good references to look at to find these derivations. The system of equations is below. [NOTE: I included the equations as images because I could not get the latex formatting to work for an aligned environment. I can put in the latex, but might need some direction on how to format the equations.] The provided weak form is as below. I understand the steps in computing a weak form, especially the integration by parts piece. However, I am not clear on all of the steps in the derivation below, such as why a vector PDE is represented as a single equation. I imagine that the different species are tracked in a vector, and hence the sum makes sense. BUT, in the equation below, it is a bit difficult to see those nuanced elements. Hence I was hoping someone could explain this derivation or provide some links to resources that provide a more detailed derivation of this weak form.
I am really confused about the part where the photon shares a part of its energy with the electron. It is said that the photon loses some of its energy to the electron causing the change in its wavelength. In my understanding photon isn't something that contains energy, it's the energy itself. A single photon has a fixed amount of energy (E = hv) ;the smallest possible value of energy of the photon at that particular frequency. How can it break into something smaller, doesn't that voilate the postulates of Plank and Einstein about the quantum nature of light? Or does a photon break into many, in that case, is the quantized nature of light even meaningful anymore? Moreover if the x-ray photon has surplus energy after knocking the electron out of the atom, why doesn't it contribute to the electron's kinetic energy? Like it does in the case of photoelectric effect. And it leads to another confusion I have now. If photons can share part of their energy then can't we simply increase the number of photons hitting the photoelectrode such that the free electrons can slowly store enough energy to break the required energy thershold.
A person might say on one day: It is hot outside - let's go out for a picnic! It is healthy. Another person might say on the same day in the same place: It is hot outside - stay inside where it is safe from extreme weather! It is unhealthy. The weather and health together is our internal model of what is happening at a physical and biochemical level. It is not actually what happens - simply an analogy we can use to communicate an idea. Let me give another example: One teacher might say This group of students are so quiet. It's unhealthy. We need to break up the group. Another teacher might say for the same students on the same day: This group of students are so quiet. It's so healthy. We need to encourage more students to be like this. In this example health is a metaphor for something that is assumed - but not defined. It is used both ways. The analogy of 'healthy group behaviour' is ambiguous in this conversational context. One possible answer to this might be metaphysical conceit - but that seems like a poetic term, and possibly limited to spiritual poetry. My question is: What is a word for when an analogy could be used for both sides of an argument?
I have two lists, and six elements: V V V M M M I would like to compute every possible combination of those elements into two lists (left and right): So, five possible solutions could be: ([V, V, V], [M, M, M]) ([], [V, V, V, M, M, M]) ([V, V, V, M, M, M], []) ([V, M, V], [V, M, M]) ([V], [V, M, V, M, M]) My first approach was reducing the problem to that which had less elements (two). So, all possible solutions in that case are: ([], [V, M]) ([], [M, V]) ([V, M], []) ([M, V], []) ([V], [M]) ([M], [V]) I can't seem to fit this problem into a basic standard combinatorics one, but perhaps there is something I am missing. How can I approach this problem ?
I need clarity on some definitions and mathematical "skepticism". In a recent video by Matt Parker, he says "(...) although the existence of the sign function, which says if a value is positive or negative, upsets some people (...)". What does he mean by this, and can this "upsetting" be extended to functions like the absolute value or the indicator function of a set? Any suggestions for further reading would be highly appreciated! Edit: Before this question is closed, I would like to clarify that by no means is this question intended to incentivize discord or negative emotions regarding certain groups of people. I do not wish to offend, nor suggest I am taking part in any ideology regarding the nature of mathematics and its definitions. I am merely curious about what Matt meant, and what is the deeper motivation behind his brief comment, hopefully generating a healthy discussion. Thank you.
Consider the following sentence: "This paper introduces a new alternative for generating synthetic data based on images." What I want to say is that "the new alternative" is "based on images". Thus, the noveltiy of the new alternative is in the fact that it is based on images. What I would like to avoid is that one think that I mean "image-based synthetic data generation already exists and I propose a new alternative to it." So, to avoid the ambiguity I add ", which" to the above sentence, so it becomes: "This paper introduces a new alternative for generating synthetic data, which is based on images." However, after looking for the usage of ", which", I am still not sure that the sentence means what I mean. More precsisely does "which is based on images" refer to "generating synthetic data" or to "new alternative"?
I'm a native English speaker, but I'm drawing a blank on how to describe this complicated exchange of situational emotions in words. Maybe someone can help. I'll just illustrate the situation with a narrative scene. Two doctors examine a patient. "Have you been eating any protein-rich foods lately?" one of them asks. "No," answers the patient. "No protein at all." Their suspicion confirmed, the doctors looked at each other, nodding _____ly / nodding with _______. What emotion am I looking to describe? I can picture the facial expression in my head, where the doctors aren't personally involved in the consequences of what's going on, feel somewhat sad about what the other person is going through, and yet understand that the consequences are justified based on the person's actions, all of which is shared between the two people who both come to this conclusion. In this case, they both know the patient should've been eating more protein so they wouldn't have this affliction, are somewhat apologetic that the patient didn't know this, but have some sympathy in explaining afterward that in order to stay healthy, one must consume protein. Anyone know what words I'm trying to express?
In the nuclear physics book by Krane, I was reading about how mathematically and numerically Q value of the reaction are measured. So he decided to prove it by solving it in Lab frame, And he goes saying this, "Let's apply them (the formulas he wrote to measure Q value) first to the laboratory reference frame, in which the target nuclei are considered to be at rest (room-temperature thermal energies are negligible on the MeV scale of nuclear reactions). If we define a reaction plane by the direction of the incident beam and one of the outgoing particles, then conserving the component of momentum perpendicular to that plane shows immediately that the motion of the second outgoing particle must lie in the plane as well." So my first doubt is, what is a reaction plane, and why the nuclear reaction always occur in a plane only. I read some more stuffs online but was unable to understand this reason as why nuclear reaction occur in a plane. Why is so that the momentum perpendicular to plane is to be conserved and is of interest?