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What is the current consensus about the age variation of the existing galaxies in our observable universe? Not to be confused with the age of very distant galaxies as observed today by our telescopes which is actually observing these distant galaxies as they were many billions of years ago in the early formation but instead I am asking about their absolute age today assuming they still exist? I mean, a hypothetical observer located in these very distant galaxies and looking at our home galaxy would also see our galaxy not at its current state but how it looked like billions of years ago. I am not asking this but if all the galaxies were formed more or less at the same time according to the Big Bang theory? In general how you estimate the age of a galaxy? | 0 |
Let's assume that a cylindrical magnet is clamped by an horizontal arm extending out of the wall. The top of the magnet is its north pole and the smooth face facing down is its south pole. If I bring another cylindrical magnet and position it perfectly in line below the fixed magnet, with its top being its north pole and the bottom being the south pole, would it be possible that the weight of the magnet going down and the magnetic force attempting to pull the magnet upwards balance out so that the magnet can levitate in mid air? Conceptually, it seems to make sense as in this case, the magnet is not prone to flipping since it poles facing each other are not repelling. However, I'm not sure if this violates Earnshaw's theorem. It it's not possible, then why isn't it possible? Did I miss something in my assumptions? And if it is possible, Is it also possible for the levitating magnet to rotate about its central vertical axis and still remain suspended in space? | 0 |
I'm looking for an accurate method to measure the tension in a guitar string, without using a sonometer setup nor by measuring the frequency. the current method that I have in mind is to measure the force perpendicular to the string and the string's displacement, and then try to resolve the vectors (the method is explained in more detail here), but I don't think that is an accurate method for a couple of reasons: we have to consider the uncertainty of the newton meter and the ruler the propagation of uncertainty when resolving vectors makes the final uncertainty much larger even after resolving the vectors, we are calculating the tension in the bent string, not the equilibrium position. are there any other methods I can use to measure the tension more accurately? | 0 |
Clicking through the OEIS I found this sequence that seems curious - the number of "labeled" groups. It cross references the usual sequence of the number of groups up to isomorphism (the very first sequence in the OEIS) as well as saying it is "a sequence related to groups". So I've tried to look up this concept, but Google with "labeled group", "labeled group maths", "labeled group theory", and so on, to no avail at finding anything. Groupprops (essentially a wiki for group theory) doesn't seem to have anything either. I've clicked through the links on that OEIS entry and cannot find anything of help. So as a last resort I'm asking here (I know this isn't necessarily the kind of place for definition questions though, sorry). So, what is a labeled group? | 0 |
Neither my wife nor I have English as our mother tongue, but we use English to communicate to each other, which sometimes causes confusion. My wife often uses the expression "until now" to mean "so far" or "yet", meaning that the action is not yet finished. I didn't receive any answer from the landlord until now. To me "until now" means that for some time she didn't get answer but now (recently), she did. So I'll usually reply something like: Nice, so what did he say? And she looks at me weirdly, as if I would have asked something silly and we both laugh. My question is. Does "until now" always imply that the action you are talking is now finished, or can sometimes be used with the same meaning as yet? | 0 |
If i place some charge on a conductor then it will distribute itself in such a way that electric field everywhere inside is zero. My text book says that only one kind of such charge distribution is possible. Or if i place charges outside the conductor then charges will be induced on the surface of a conductor to make field inside zero and again such charge distribution is unique. Intuitively it all seems correct but i am trying to figure out right arguments for this. One such argument is that suppose i place some charge on conductor then i can solve poisson's equation to find potential at the surface(at boundary). Charge distribution outside the conductor is known. Then uniqueness theorem says that only one such potential function is possible which satisfies the given/known boundary conditions and poisson's equation outside. Since a particular potential function corresponds to a particular charge distribution the charge distribution must also be unique. This argument seems correct to me but at the same time something seems missing. Am i correct? | 0 |
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. | 0 |
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? | 0 |
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. | 0 |
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? | 0 |
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. | 0 |
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). | 0 |
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. | 0 |
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. | 0 |
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. | 0 |
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? | 0 |
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. | 0 |
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? | 0 |
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. | 0 |
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)? | 0 |
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 | 0 |
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. | 0 |
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. | 0 |
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? | 0 |
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 ? | 0 |
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. | 0 |
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"? | 0 |
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? | 0 |
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? | 0 |
This is a follow up to a previous question. I use pandoc to create PDFs from markdown files, which uses pdflatex behind the scenes. After a bunch of system upgrades, the type in my printed PDFs started to look off.The weight was heavier, and the the kerning was bunched up. At first I thought this might be a problem with LaTeX or my MacTex install, but now I believe it's a problem with MacOS. When I print on MacOS via Preview or Acrobat, I get the font that looks janky.If I switch to a windows machine and print there (just via the Edge web browser) the type matches more closely to what I see on the screen and used to see when printing from my Mac. I now think the "bad" font I see is just good old Times New Roman, being substituted as a default. Does anyone here know why PDF viewiers/printers on my mac might not see fonts embded in PDFs generated by pdflatex? | 0 |
I work in quantum field theory in curved spacetime. Within QFTCS, we have a bunch of phenomena showing that the notion of "particle" is quite subtle. For example, the Unruh effect let's us know that the notion of particle is actually observer-dependent. As a consequence, I tend to be quite skeptic about fundamental theories that rely too much on the notion of particle. While I am far from being an expert in string theory, the things I've heard about it make me feel as if that is precisely what string theory does: to take the notion of particle seriously and change it into a quantum string, which would then be able to solve numerous problem within quantum field theory and quantum gravity. I have also heard one or two words about "string field theory", which appears to be a quantum field theory with infinitely many fields. The name, of course, also suggests it is somehow related to string theory. With this in mind, my question is: QFT is a theory about fields, in which particles sometimes appear as cute interpretations. Does the same thing happen in string theory? How exactly (to the best possible presentation for someone who knows nearly nothing about string theory)? Has string theory (or string field theory, or any other similar idea) successfully derived the Unruh effect or is it only "likely to be there" based on being a generalization of QFT? | 0 |
I have a GRIN lens with the refractive index varying linearly with y, and supposedly this lens tilts the wavefront. Since the rays are travelling normally in both refractive index materials, they stay normal to the plane of the lens at all times. However the wavefront travelling with them tilts because of different speed of light for different indices, and gives a wavefront like this. Is this correct? Or do the rays themselves tilt? I don't think this is a possibility since the rays are always perpendicular to the plane. If my reasoning is correct, does this satisfy Huygens' Law? It states that the wavefront must be perpendicular to the path of the waves at all times, but this doesn't seem to be the case here, which leads me to thinking its the waves themselves that are tilting. Diagrams like this can be found on the internet, but I do not understand how are the waves tilting when they enter normally, What am I missing? I found this situation while solving a problem in a book (Pathfinder for Olympiad Physics) but upon thinking a bit more was fell into this confusing situation. | 0 |
When one studies representations of (bosonic) Lie groups in physics, whether dealing with spacetime symmetries or gauge symmetries, it is often left implicit whether the representations are over real or complex vector spaces, without acknowledging of course that the representations of an algebra or group can be different over different base fields. A simple example is that the adjoint representation of the Lorentz group is irreducible as a representation over the reals, but reducible when take as a representation over complex vector spaces. When it comes to spinors, often people wish to treat components of spinors as Grassmann variables - this being the 'classical limit' of the fermion anti-commutation relations. My question is whether or not there has been a rigorous treatment of the representations of semi-simple (bosonic) Lie groups over Grassmann-valued vector spaces. EDIT: I suppose the direct analogy I anticipated isn't there. When we allow complex fields, we also allow complex changes of basis. This changes the reducibility of representations, because real Lie algebra elements now can have imaginary components even though the boost/rotation parameter remains real. For Grassmann numbers, it appears that we dont extend the reals as in the complex case, but supercede them. So we dont gain access to a new change of basis like before, where the algebra elements would have Grassmann-valued components while keeping the boost parameters real. Unless I am mistaken and people do work with spinors with "real parts" and "Grassmann parts". | 0 |
What's the term for a line of dialog that is repeated later in a different context? Christopher Nolan seems to be fond of these lines. In Batman Begins, Ra's first says: Always mind your surroundings And then in the climax Batman says You never learned to mind your surroundings. In The Dark Knight Rises, Bane first says: Then, you have my permission to die as he's breaking Batman, and later Batman claps back with Then, you have my permission to die. Same line, different contexts, often as witty comebacks. I've thought of a couple of possibilities I've heard of, none of which works quite well: Callbacks seem to be a reference to a previous gag, rather than an exact quote. Comebacks and clapbacks can be any kinds of clever retorts, not necessarily the use of your opponent's words against themselves | 0 |
The general bin-packing problem is NP complete. I have read several papers and other source but I am still not clear about whether a bin-packing problem with a fixed number of bins is NP-hard. Wikipedia says: "Computationally, the problem is NP-hard, and the corresponding decision problem, deciding if items can fit into a specified number of bins, is NP-complete." and later: "On the other hand, bin packing is solvable in pseudo-polynomial time for any fixed number of bins K, and solvable in polynomial time for any fixed bin capacity B." In another paper "Bin packing with fixed number of bins revisited" I have read that for a fixed number of if the sizes of the items are polynomially bounded integers, then the problem can be solved in time n^(O(k)). My specific problem is the following. I have a fixed number of bins m with fixed capacity. The items arrive online. I one case the set of items is known in the other case it is not. Given on what I have read I assume this problem is not NP hard as it could be solved by exhaustive search. | 0 |
I'm learning about functional programming, In functional programming, functions are treated as first-class citizens, meaning that they can be bound to names (including local identifiers), passed as arguments, and returned from other functions, just as any other data type can. This allows programs to be written in a declarative and composable style, where small functions are combined in a modular manner. -Wiki And.. I started to wonder, how it would be if we tried to adopt this method as foundations to mathematics. Would it be meaningful to consider mathematics with functions being taken as the fundamental object rather than, say, that of the set or non function types? If it is, what would be the difference between the produced mathematics of that procedure, and that we have when we consider sets as the fundamental objects? What would be the advantages/ disadvantages to this approach? | 0 |
I am seeing many people claiming that cumulus clouds that sometimes form periodic wavy patterns (see images for "altocumulus undulatus" or "Radiatus" for instance) have no explanation aside from being chemtrails, and I'd like to be able to respond with a sound scientific explanation. I'd like to understand the phenomenon and my guess is that it's about the cloud blanket being forced by winds with the result of a periodic pattern appearing, much like sand waves form in shallow water at the beach. But is it really the case? Searching for a more detailed explanation I ran into Tollmien-Schlichting waves that are a path to turbulence, but I admit I did not understand much, so here is my question(s): What is the physics behind such cloud formations? And Given an estimate of the spatial period of these cloud waves and the cloud height, can one infer the windspeed at that height? | 0 |
If we are moving a block on a rough surface extremely slowly (quasi-statically) with the help of an external horizontal force, then is it the case that no heat will be produced, but the work done by friction force will still be equal to the work done by the external force for any given displacement? The reason this question comes to my mind is, in thermodynamics, if we say that a process is reversible when it is slow, then there should not be any heat production despite having frictional/viscous forces in the system when the process is slow. So, this implies that the existence of a reversible process rests on the idea that friction should not produce any heat. So, it is correct to assume that friction can do work without producing heat when the process is extremely slow? Because if in all the examples we see around, we see that work by friction is always accompanied by heat production. Kindly help. | 0 |
Suppose we have a system of electrons a very tightly confined space (like a tiny magnetic trap). Let's say we continually increase the degree of confinement such that the electrons are confined into a smaller and smaller space, to the point where the degeneracy pressure reaches a scale comparable to the masses of the other charged leptons. Does there come a point where it's favourable for an electron to "decay" into a heavier lepton due to degeneracy pressure (producing neutrinos and antineutrinos in the process)? My thought is that this process would be somewhat analogous to the formation of a neutron star (or hypothetical quark star). What would happen if we kept confining the particles into a smaller and smaller space to increase the potential further? Would even heavier particles be produced? | 0 |
Consider a simple experiment, such as boiling water in a pot in your kitchen, is it possible to estimate the time needed for the water to boil based on elementary properties of water ? In the physics and chemistry literatures we find many computational and theoretical works attempting to predict the boiling temperature of water, from elementary properties, such as the molecular bonding strength or angles. But what about the boiling time ? The basic parameters for the boiling time should the heating power, the heat conductivity of the pot, the heat capacity of water, and possibly other properties that characterize the transfer of energy from the environment to the water. My question is about the time needed from the moment where the water temperature is close to boiling point, to the moment where water has left its initial liquid state. It is mentioned in many lectures and textbooks that nucleation is a limiting step in the formation of a new phase, in particular for the boiling of water. But I was never able to find a critical comparison of such statements. So I would appreciate any references or indications. Which kind of computations do you expect to be necessary ? What kind of empirical parameters enter in the calculation ? Was this done in a reference? | 0 |
Suppose we place a current carrying loop of wire in a magnetic field. Of, course its easy to see that for a uniform field the wire will experience uniform force in all directions and so it'll get stretched out into the shape of a circle. But I wanted to ask if we could make any comments about the shape of the wire, in any random magnetic field without knowing the exact function describing the field. At places where the field is stronger, the wire will tend to get stretched out more so I suppose the local radius of curvature there would be less as compared to a point where the magnetic field strength is lower. But thats just a qualitative rough way of looking at things. Is there some exact way to find out the answer to this | 0 |
I just started reading about the conduction mechanism in polymer. From what i read, polarons are used as method of charge transportation in non-degenerate polymer. While for degenerate polymer, both charged soliton and polaron will do the job. As i go a bit deeper, polaron can be defined as: A polaron can be thought of as a bound state of a charged soliton and a neutral soliton of which the midgap energy states hybridize to form bonding and antibonding levels. A Quasiparticle that is created through the interaction of charged particles ( electron or holes ) with Phonon. If i understand it correctly, the definition of the polaron contradicts with its existance in non-degenerate polymer as non-degenerate polymer doesnt have soliton due to its topological reason. Also, as i am quite new to all these terms (polaron, bipolaron, soliton, charged soliton and phonon), i tend to get confuse and couldnt differentiate each of them precisely in layman terms as most of the literature are mostly explained in what could say too "academic" for a starter like me. Would someone be able to help me clear my confusion? Any help, advice and guidance are really appreciated. Thanks in advance | 0 |
Basically, as the title says. Maybe this is trivially true or false, but I have not enough intuitions about topological surfaces or aperiodic tilings. To make it a bit more precise - I mean the kind of aperiodic tilings such as Penrose tilings or the tiling generated by the recently found hat tile. By "gluing" the patch into a torus (or some other surface) I mean essentially just like you fold a net into some geometric shape by connecting edges. And the edges should be glued in such a way that all matching rules of the tiling are satisfied. If this is possible - would the resulting "tiling of the surface" also be aperiodic, or could it be made to be? Here I would consider "aperiodic" to mean that there are no non-trivial symmetries between any two tiles on the surface. | 0 |
I saw a question asking "If the Moon was moved to the surface of the Earth, from how far away on the Earth's surface would it be visible?". (The question says to ignore any other factors like gravity making it smash into the Earth's surface and effectively destroying both bodies). Basically, it's this, I think: We know AB, which is the radius of the Earth, and we know CD, which is the radius of the Moon. AC is the sum of these two. Even though it doesn't really look like it in my sketch, I think that the lines CD and AB are parallel, and that ABE and CDE are both right-angle triangles. We need to work out the angle at A, which we can use as a ratio of the circumference of the Earth (which obviously is another known quantity), to work out the distance around the circle to B. Are my assumptions correct? Is it possible, from those givens, to calculate the angle at A? | 0 |
EDIT: Completely rewritten because of the 'needs clarity' tag and some useful related questions appearing in the side-bar. I hope this is clear now This answer gives a long list of properties of particles whose value differs by a minus sign when comparing a particle to its antiparticle. We know that anti-particles exist, so apparently for every particle there is a particle where the value of all the properties in this list are 'flipped': i.e. the same magnitude but of opposite sign. My question is: given a particle, say an electron, does there exist a different particle where some of the properties in the list in the linked answer are flipped and some are not? If the answer is no, why is this not possible? If the answer is yes, what is an example of such a pair of particles? | 0 |
I am beginning to learn some very basic electronics. I was learning how and why lightbulbs light up. It turns out it happens because they have a very thin filament which makes the passage very narrow for electrons so they lose a lot of energy colliding with the molecules making up the filament instead of using it to drive the passage of the current. This causes three things. Voltage drop Decrease in intensity Transfer of the electrons' energy to the atoms making up the filament so they vibrate causing it to heat up, and the energy is also used to excite electrons to a higher energy level momentarily where they release the energy in the form of photons where the lower the wavelength, the brighter the light color and the greater the energy. So the filament also glows and lights up. I am interested in the latter and I have a couple of questions. Why don't the electrons remain in the higher energy levels? Why do they choose to release the energy as photons? | 0 |
I've been struggling with proposed answers to the twin paradox. I know that an object traveling at relativistic speed ages slower than a stationary object. This must be caused by some interaction with spacetime that leads to a slowing of time. But how does this work when we are looking at relative time dilation between two or more individuals. Imagine a huge crowd of relativistic rockets moving inertially and crossing each other in all directions at different velocities. How can one pilot claim to be younger than another, while the other is younger than a third who is younger than the first? Then it came to me that this all might just be the same as relativistic length contraction where it is only occurs while they are in motion, and all effects stop once they stop moving. Once they stop the only age effects are measured against an earth clock. We know that for kinetic length contraction, the man on the train station platform will see the relativistic train length contracted while the main on the train will see the train station platform length contracted. But the instant the train stops moving, all measurements return to normal. Is it the same with the twin paradox? That any age difference as seen by one pilot over another disappears once the motion stops? And the only real aging is when measured against an earth clock? | 0 |
Exactly what the title asks. "Diode" comes with it the ideas of depletion layers and forward/reverse biasing and electron-hole recombination, but SPAD physics doesn't seem to be dependent on any of that. You just put a giant electric field on, and as soon as a photon comes and promotes a charge to the conduction band, it gets so much energy that it can knock other electrons out of their atoms, creating the avalanche. In other words the fact that it's "reverse-biased" relative to the doping of the semiconductors seems irrelevant, as does the fact that the reverse-bias is above the breakdown voltage. Could I take a piece of undoped silicon and use it as a SPAD if I put a sufficiently large voltage on it? Or else why is it important that it's a diode? | 0 |
I have figures which I use in several places, some of them in dark context and some of them light (for example, a my obsidian vault is dark but a PDF of my thesis is light). As such, some of the figures are colored to work in a dark environment, and some in light. But sometimes I want to take a figure from a dark context and use it in a light context, or vice versa. So far I have created a copy of the figure for the light/dark mode by simply inverting the colors. I use color schemes in which this looks mostly fine. But manually inverting the colors and saving a copy is exhausting, and inefficient. I want to do this automatically, and possibly without saving a copy of the original figure. Is there a way to invert the colors of a figure while compiling the output of a latex document? Thanks a lot! Edit: I saw this question, but the answers were basically "you can't". Since the answers are unsatisfactory and the question is VERY old, I thought it would be best to open a new one. But if this is not the custom here, please let me know and I'll ask there instead. | 0 |
Noether's theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. Energy, however, is not conserved in an expanding spacetime because there is no symmetry on the time axis. This enables dark energy to exist and is the reason that general relativity does not conserve energy (except in symmetric situations). My question is about how this translates to theories of quantum gravity that predict the existence of spacetime that becomes spatiotemporally isolated near the center of a black hole. Examples are white holes, fecund universes, and collapsing wormholes. I know that black holes conserve energy, at least for an outside observer. So can energy be created in a disconnected region of spacetime at the other side of a black hole center for certain theories of quantum gravity? Or does Noether's theorem prevent this? (Bing insists that the theorem applies here, but I don't trust the answer.) | 0 |
I was taught that there are four types of nouns: singular countable: journey, sheep, child plural countable: journeys, sheep, children singular uncountable: travel, water, fruit plural uncountable: groceries, customs, thanks Some words are only used with: countable nouns: one, two, three, many, number, few uncountable nouns: much, little, good/great deal, quantity, amount Am I right? What about the word 'cattle'? You can say neither 'three cattle' nor 'much cattle'. (According to Practical English Usage) What type of noun is it? If it is an uncountable noun why I can say 'many cattle' but not 'much cattle'? Isn't 'many' used only with countable nouns and 'much' only with uncountable nouns? Also, you might consider 'staff' or 'jeans'. You can say for example 'four staff' but not 'a staff'. (According to Practical English Usage) | 0 |
While trying to understand General Relativity, I'm struggeling with disentangling coordinates and curvature. The metric tensor contains information on both: coordinates as well as curvature. Curvature is a physical characteristic of spacetime, while the coordinates can be chosen completely arbitrary. Is there a method to disentagle coordinates and curvature? (Stephan Hawking must have had one as he managed to show that the singularity at the event horizon of a black hole is only a coordinate singularity, not a physical one...) I fistly thought that one could simply use cartesian coordinates - and then would be left with only the curvature (in this question: Determinant of metric tensor in Cartesian Coordinates constant in vacuum) However, I was taught there, that cartesian coordinates can only be used in flat spacetime. Now, I'm totally clueless again how to disentangle coordinates and curvature. I'm looking for a tensor/measure/metric/field, where only the physical curvature is in - and no coordinate curvature. Where the coordinates are flat like in a cartesian grid. Where the metric tensor is simply that of a flat spacetime in cartesian coordinates, if the spacetime is flat (Minkowski spacetime). | 0 |
We know that a complex manifold with a hermitian metric h is Kahler if and only if the corresponding Kahler form (which can be obtained from Hermitian metric) is closed.The complex manifold has an underlying real smooth manifold whose Riemannian metric can be found from the Hermitian metric of the corresponding complex manifold (Re(h)). We also know that a Riemanian manifold is Kahler if and only if there exists a smooth almost complex structure on it which is metric compatible and whose covariant derivative is zero. Now, the question is: if a complex manifold is Kahler then does this imply that the underlying real smooth Riemannian manifold will be also Kahler? I think that it should be Kahler as if the manifold is already Kahler in complex case so there should be no obstruction to the existence of compatible almost complex structure with vanishing covariant derivative. | 0 |
I have been using lists inside multicols for a "condensed" typesetting of lists of short items. However, I do not like what happens if a page break occurs inside, and I couldn't find a way to automatically force a page break before multicols if it cannot fit onto the current page. So I have been thinking about using some kind of a table environment to typeset a single-row table with lists in cells. However, I would like to make the lists in cells spread vertically so that their tops and bottoms be aligned (except when this wouldn't make sense), like columns in multicols. So far I do not see how to make a table whose height is determined by the maximal natural height of its cells, and where I could make the contents of any cell spread vertically to align top and bottom baselines. Is it possible? How? This might look like not a very natural thing to do, but I am used to the behaviour of lists in multicols, and I would like to avoid situations where, despite having the same number of items, the bottoms are not aligned because of accumulation of minor height differences. | 0 |
In quantum field theory (e.g. lattice QED), perturbation theory can "break down" when interactions become too strong. Can something like that happen in classical non-linear optics? Can there be any combination of material (absurdly non-linear) and light (attosecond pulses, petawatt lasers, etc.) which can (somehow) be described by classical electrodynamics but the usual "expansion of polarization density inside the material in powers of electric field" (maybe later adding magnetic field as well...) breaks down and doesn't work? I.e., where no finite expansion order gives a reasonable description? Alternatively, if the "describable clasically" assumption is weird in this context, is there a regime where some very strong coupling leads to the perturbative non-linear optics approach breaking down for a stable bulk material that can be produced on earth? I mostly care about "shooting light at something and describing the light that comes out". I've read claims about high harmonic generation (HHG) being such a phenomenon but it's not really clear to me how/why the perturbation theory (at any finite order!) does not apply. You can do it in bulk crystals but I guess the "classical system" restriction would not apply there, as it is usually described using some semi-classical models? For example, Wikipedia on "nonlinear optics" writes about some series "not converging" but it's not quite clear what is meant there. Some articles on HHG (e.g. this) use the phrase "non-perturbative" but what exactly do they mean by that? | 0 |
Plasma and fusion physics experts, help me with this one: Suppose we have a D-T plasma with net positive electrical charge inside a positively charged metal sphere. As the containment sphere's net positive charge increases, my ignorant intuition would expect the plasma to be compressed and its internal pressure to increase. Sputtering would likely occur as electrons from the plasma escape and are drawn towards the containment sphere, releasing atoms from the containment metal into the plasma and cooling it - but ignore that for a sec. Suppose we could both arbitrarily increase the temperature of the internal plasma and the charge of the containment sphere. Would this be a viable means for nuclear fusion containment? Help me shoot down this idea. What are some practical limitations that would prevent this from working? For starters, at arbitrarily high positive charge, the containment medium itself would disintegrate due to the electrostatic repulsion of atoms in the containment vessel, correct? Is there any existing means of calculating charge distribution in a plasma or solid at very high positive charge? Or is there a means of calculating the theoretical spatial gap between the plasma and containment vessel in such an arrangement? | 0 |
Consider a horizontal long rod that is undergoing free fall. Consider the torque about an axis through the rod (perpendicular to the rod and to the direction of gravitational force), that is a little to the left of the rod's centre of mass. Clearly, there is a net torque about this axis, but we know that the horizontal long rod will only translate downwards. I suspect it arises from somehow this reasoning being inapplicable for some cases of translating axes about which torque is taken, but in the case of a cylinder rolling down a rough inclined plane, the same paradox is not seen by taking torque about the axis passing through the centre of mass of the cylinder, even though this axis is translating downslope. What's going on? | 0 |
Is there a distinction that we make between what we call "elements" and what we call "sets" in Von Neumann type set theories? Two textbooks I have used for an introductory study of NBG both made distinctions between sets and proper classes (the prinipal primitive undefined notion being that of "class"). The distinction was ofcourse made to avoid various logical paradoxes, in particular, Russell's Paradox. However, what wasn't clearly stated in either textbook is whether the concepts of "set" and "element" are identical. Both did state that a "class" is considered an element if and only if it belongs to another class. Would it be safe for me to identify the notions of "set" and "element" completely interchangeably? I don't see an elementary distinction between the two. Then the only two distinctions we should make are of "sets" (elements) and of "proper classes" with the general notion of "classes" referring to either one of the two distinctions. Thank you in advance. | 0 |
When studying supersymmetric QFT's, it is very common to compute the moduli space of the theory by solving all F-term equation (derivatives of the superpotential). More precisely, one should also quotient by the complexified gauge group. Here are three fact that I believe to be true about moduli space and IR flow: Let us consider a supersymmetric QFT, dumbed theory A. Let us now take flow towards the IR and obtain an affective theory, called theory B. The moduli spaces of theory A and theory B are not necessarily the same. More generally, the moduli space of a theory changes with the flow. Now, on the contrary, a moduli space can be seen as the space of theories, i.e. the set of all possible vev's configurations for all the scalars. Choosing a point in the moduli space therefore introduces a scale, and implies that we are at a certain point in the flow of the theory. Certain regions of the moduli space can then correspond to strongly coupled or weakly coupled regimes. The moduli space of a theory, i.e. its space of vacua is the most low-energy thing you can do, no fields can be excited so they all take their minimum values, i.e. their moduli. So the moduli space could be seen as the bottom of the IR flow. Those three pictures are clearly in contradiction. What is the problem with this way of thinking ? I would be very interested by references about the link between RG flow, low energy-limits, and the moduli space. | 0 |
I had a guess from three years ago that I couldn't prove It is impossible to find a polyhedron whose perimeter (represented by the sum of the lengths of its edges) is equal to its area (represented by the sum of the areas of its faces) and numerically equal to its volume It is easy to find a similar shape with any shape such that two of these are equal to each other, but the question is whether all three are equal to each other. All I have been able to prove is that this shape cannot be a Rectangular cuboid, I also proved that it is impossible for it to be a Platonic solid but I do not know if this would hold in the general case. Here is a proof of the condition of the Rectangular cuboid by arabic language | 0 |
The infinite square well (and variations) are some of the best-studied systems in quantum mechanics and are often used as the starting point for any quantum mechanical education, as the Schrodinger equation is easy to solve for such a system. I'm looking for specific research papers that describe experimental results that affirm the particle-in-a-box theory because, while the theory is beautiful, it is nearly always presented to students without any discussion of experimental confirmations. Obviously, any real system is going to be more like the non-infinite square well. I'm well acquainted with the other experimental results that affirm other introductory aspects of quantum mechanics, such as the double-slit for affirming particle-wave duality, the stern-gerlach experiment for spin, and various interferometers for superposition, but have yet to come across any for particle-in-a-box theory. The best that I am aware of is Bose-Einstein condensate experiments, but those are typically focused on affirming that bosons can indeed occupy the same state and are not so much what I am looking for. The simplest practical system that comes to mind to demonstrate what I'm looking for would be something along the lines of a Jaynes-Cummings model single-atom cavity experiment where the atom is repeatedly found in the most likely predicted locations. Is this question misguided? Are there actually no such experiments and we only know the theory to be true because more sophisticated developments of the theory align with experiments and extrapolate that the basics of the theory must therefore also be true? | 0 |
I am aware that this series is incomplete, but it has a large body of existing content, and I am also aware that it is written to be "accessible to non-specialists", but that is obviously quite vague. All that really tells me is that I don't have to be a CFSG scholar, and on the other end that I can assume it's probably not, yknow, an undergraduate text. But what is the actual background needed to follow it? Group theory is a pretty vast field, and it is unclear to me whether GLS is written for people on the level of "grad students who took abstract algebra courses", "PhD candidates in group theory", or what. Specifically the kind of answer I want is describing what the curriculum might look like for a hypothetical course whose purpose is GLS reading prep. | 0 |
When a tiny spherical charged particle (having a conductive surface) moves at a relativistic speed, the Lorentz transformation for EM-fields predicts that its electric field increases at the top and bottom of the charge whether or not it has been made of something conductive, whereas the field decreases at the left- and right-hand sides along the motion direction. On the other hand, we know that as the tiny spherical charge moves at a considerable portion of the speed of light, its geometrical shape shifts to a spheroid. Taking account of the fact that the charge density is no longer uniform and it increases at the top and bottom of the spheroid, I want to know whether it is the new arrangement of the charged particles that implies the electric field to be like that shown in the figure for the relativistically moving charge, or the Lorentz transformation predicts the EM-field independent of the charge distribution. Remember that the electric field of a charged conductive spheroid is very similar to that predicted by the Lorentz transformation, however, I do not know how far they match. | 0 |
My only familiarity with topology is the basic theory of mereotopology, so apologies is this sounds strange. An open region is a region without a definite boundary--that is, there is no way to draw a boundary including only points in the region such that every point in the region is inside that boundary. I think of it as the interior of, for example, a polygon without the edges that define the polygon. A closed region includes a definite boundary. The complement of a region is all of the points that are not in the region, so in a normal topological space the complement of a closed region is an open region and vice versa. This strikes me as related to continuity in the real numbers. If you imagine the real number line as a topological space where a range of numbers is viewed as a region, then an open range is an open region and a closed range is a closed region. In this case, the statement that the complement of an open region is a closed region seems equivalent to the statement that every range has a least upper bound and a greatest lower bound--which is equivalent to continuity of the real numbers. Does this make sense, and am I right about it? | 0 |
I have combed through various sources on the internet and I don't have a definitive answer for the above question: The best that I can come think is the following: Because when I remove synthetic clothes, the body gets charged one way, the clothes the other way. Then as I continue the process of removing the piece of clothing off my body, the charge build up is significant, owing to which the potential difference between the cloth and my body is greater than the breakdown potential of the air. Thus, electrons flow between my body and the piece of clothing I am removing. The air heats up, gets ionized, energy is dissipated in the form of heat, light and sound; in other words, as sparks. Also one more follow up question: Do the sparks flow "inwards" between my body and the piece of clothing? Or just dissipate "outwards" into the air, towards nothing in particular? Is the above reasoning sound? Also, does the same reason hold when I comb my hair? However, if this were true, why do I NOT SEE any sparks when I comb my hair but only hear a crackling sound? | 0 |
Consider two free-falling observers in the Earth's gravitational field, A and B, who meet at point C, where A orbits the Earth at a constant radius from the Earth's center, and B falls towards the Earth's center along a constant radial line: Can either detect the other accelerating locally at C? As an attempt to answer this myself: it's worth noting that I can detect the acceleration of an object dropped onto the surface of the Earth where I am, and that the Earth is free falling along a geodesic within the solar system. However, I don't think the Earth's surface is a geodesic because I can drill a small hole from the surface towards the Earth's center so that a small mass will accelerate towards the center where, at the center, it will experience a zero acceleration. Hence an observer at the Earth's center can't measure the acceleration of another moving mass there, and on another geodesic. So I'm inclined to believe that neither A and B can detect one another accelerating at C locally when they meet, and that more generally: observers on different geodesics can't detect one another accelerating wrt one another locally when they meet. | 0 |
I am using the word 'avatar' in my work in the sense that an avatar is the physical representation of some higher entity. We could say that a particular person, animal or object is the avatar of a deity, or that a symbol or picture is the avatar of a person, or that a player-character in a computer game is the avatar of the player. For example, my Stack Exchange avatar is a red 'W' on a blue and green field. I am looking for a single word, or failing that, a short descriptor that is a generic term for the person or entity that has one or more avatars. To use an example from mythology with the correct semantic context for my question, we have the sentence: Rama says, 'I am an avatar of Vishnu'. I'm looking for a word (or short phrase) to fit in the paraphrased sentence from the example above: Rama says 'Vishnu is my _____'. Where _____ is the relational antonym of avatar. Edit: 'opposite' changed to 'relational antonym'. | 0 |
Nondeterministic refers to a system or process that does not have a single predictable outcome. In other words, when a system is nondeterministic, it means that multiple outcomes are possible for a given set of inputs or conditions. Given that it can be associated with randomness. For physics nondeterministic is the lack of predictability (am I right?). The unpredictability in classical regime (say the motion of classical atoms in a box) is given because of existence of many variables that is hard to keep track of and also the ignorance of the human with regard to the system. The quantum mechanics has nondeterministic behavior only in the measurement process. My question is that other than the measurement which is nondeterministic (in the sense that the outcome would be randomly distributed over a probability distribution), is there any other process that has such property (in any other regimes such as relativistic)? If yes what and if no, why quantum measurement seems to be special in this case? | 0 |
I've seen the notion of the models in the title a lot in the context of automorphic forms and representations, but I wonder if there's any nice reference for the definition of them for general reductive groups, with some motivations for their namings. I'm pretty sure that the name Whittaker model comes from the Whittaker functions that give Fourier-like expansion of Maass wave forms, and somehow generalize the notion of Fourier coefficients for the general automorphic forms. But I have no idea with the rest of them, although I've seen the definition of them for some classical groups in many papers, especially related to Gan-Gross-Prasad conjectures. My naive guess is that Bessel model should be motivated from the classical Bessel functions but don't know how. Also the name Jacobi in Fourier-Jacobi may come from theta functions, since all the Fourier-Jacobin periods I've seen are defined in terms of Weil representations and theta functions. | 0 |
I want a term for someone who generally doesn't care about their individual popularity or how other's view them and as such is willing to break social expectations about behaviors and do their own thing, even hen doing so may affect how other's view them. So something close to the phrase "Marches to their own drum", but I'd prefer something more concise and possible less colloquial. I don't want a term that implies one is incapable of understanding societal expectations, I prefer the implication be a choice or lack of desire to try to fit expectations rather then an inability to do so. I also do not want to imply a willingness to harm other's as a result of ignoring social expectations, via causing someone embarrassment, hurting feelings, or general rudeness for example. The term should focus on situations where ignoring of expectations would mostly only affect the person ignoring them, not when doing so is likely to harm others. I'm okay with, and may even prefer, if the term comes with a implication that the person is less popular, less understood, or even mildly ostracized as a result of their ignoring social expectations. | 0 |
I have some idea of what the main concepts of calculus are about but I have never actually taken a calculus class or studied myself. My general understanding is that before the concept of limits were formally defined by later mathematicians, the first people that actually invented calculus explained the idea of derivatives and integrals using the hyperreal numbers, something for which they were actually criticised by some for the concept being more intuitive rather than mathematically provable/valid. I also believe that modern, standard calculus does not deal with hyperreals at all but rely solely on real numbers in explaining all of its concepts. So I was thinking whether it would be a good idea to get the gist of what hyperreals are and how they relate to limits as a foundation before actually taking on calculus, or would it be not worth it. I'd appreciate any comment/suggestion you guys would make. | 0 |
Could you simulate a Turing machine by a sequence of Turing machines each with strictly fewer states than the simulated machine? By a sequence of Turing machines I mean this: the first machine is given the input tape, then executes until it accepts, then the contents of the tape are given to the next machine in the sequence as input, and so on until the final machine accepts. If any machine in the sequence rejects, the simulated machine would reject, and if any in the sequence would fail to halt then the simulated machine would fail to halt. If this is possible, the more interesting question would be whether there is a bound on the number of machines needed to simulate a machine with more states. The motivation behind this question is that Busy beaver numbers are known for very small machines, so I was curious about whether machines with many states could be (reasonably) reduced to machines with fewer states. It seems to me that there must be some error in reasoning or understanding if the number of machines needed to simulate another could be bounded, because that would imply any machine could have a bound on steps needed before halting, but I am unsure where that disconnect is. | 0 |
I recently came across some discussion on the fact that "no bueno" is not gramatically correct Spanish, and generally not a phrase Spanish-speakers use, unless they find it funny. Of course, "no bueno" is just a literal translation of the common English phrase "no good". Sometimes, the phrase appears with the implied subject/verb attached: "it is no good" or "this is no good" (accompanied by a sad shake of the head, or a downcast expression). One can of course say "It/This is not good" or for short "not good", which is unimpeachably fine grammar, but I'm not so sure on the "official grammatical status" of just "no good". Note: there is the hyphenation "no-good", which can be used as an adjective, but that is not the usage I'm talking about in this post. | 0 |
sorry if the title seems vague. English is not my first language and I dont know what to search to find the answer to my problem. I will preface this by explaining the purpose of this question. I have a robot that I know the position of (x,y) through odomotery. and I also have an IMU that returns the robot's yaw angle with respect to the positive x axis of the map. In this map are objects, that I need to rotate the robot to face them. I will attach an image of how It would look like. Some sample cases My problem is I cannot find a generalized way to cover all cases of robot's position and yaw with respect to the object and map. If someone can point me in the right direction or tell me how I could solve this I would be very thankful | 0 |
I have a data set of values each with a different associated error. If I take the mean, I can use standard error propagation to calculate a much smaller error. This will therefore incorporate the individual uncertainties of each data point. However, is there a method to propagate the individual uncertainties when taking the median? If I take the standard deviation this only provides a measure of the data spread and not the individual instrumental errors. I have tried adding the standard deviation and the uncertainty on the actual median value in quadrature. However, as this only takes in to account one data points' uncertainty this is high and so my total error does not get reduced. Is there a way to propagate these errors when taking the median? | 0 |
What would happen to metals or ceramics (non-porous) when subjected to extremely high pressures (dozens of GPa) ? I feel like the sample will deform elastically only. So a ceramic object (think carbides) wouldn't crack. Typically, we assign a maximal compressive strength to metals and ceramics. After which, they undergo plastic deformation or fracture but these compressive strength values derive from experiments involving the pressing of a sample with an hydraulic press. So in these experiments, there is "space" without stress on the sides where the material could expand. But that isn't the case in an hydrostatic context where stress/pressure is applied evenly on all surfaces. Also, in the case of ceramics, I feel like fracturing would create more volume which seems contradictory to me as the sample is being compressed.(in the sense that work should be done against the pressure if the sample gets more voluminous and that wouldn't minimise the energy) If I am correct, you could have solid matter even at gigantic pressures. Any idea? | 0 |
I'm interested in writing a document following not only the Chicago Manual of Style in principle but also very literally the style used by the University of Chicago Press in their publishing. In other words, I'm not just trying to follow the style guidelines outlined in the manual, but also directly emulating the full style used in their publications. Here is a sample page: Obviously their style has a large number of rules (many more than used on that particular page) and I may not need to use all of them, but even a small subset would be painful to implement manually. So what I'm wondering is, is there an existing (la)tex template that has already implemented most or all of their style? When I search, I only find results for templates for their bibliography style, and that's more about following their guidelines rather than directly reproducing their formatting. | 0 |
Suppose we have an arbitrarily well-isolated empty region which we initialize with a cold, diffuse, homogenous photon gas, such that the only energy density gradients are random thermodynamic fluctuations and the only massive particles are those that are created after initialization. Thor pokes a tiny hole in the arbitrarily isolating boundary, puts his magic omni-detector on the hole, and measures an interaction event every time a particle - typically a photon, but sometimes perhaps a massive particle or antiparticle - reaches the omni-detector. Magic omni-detectors always interact with every kind of particle. However, Loki has secretly tampered with Thor's omni-detector so that it records everything as if time was proceeding many times faster - not just the time stamp on each measurement, but also the values measured, as if the SI Second had been re-defined to be some multiple of its current value with all the downstream consequences on other units and constants. Is there any way Thor can tell, by looking at a subset of his data (i.e. not just comparing the total run time of his experiment to a clock that hasn't been tampered with), that he was looking at a large, diffuse, cold region, and not a small, dense, hot region? | 0 |
I have been revising some maths and in particular I have been focusing on vector calculus. Today I have just finished looking at surface integrals. I do understand how we parametrize a surface integral starting from a region R, but I am having troubles actually understanding how these integrals are put in context when having a scalar function or a vector function. For instance, a popular example is that you have a surface (for instance a metal sheet) and then you have a function which gives the density of each point of this metal sheet (if I were to plot this function, would it have the same shape of the surface?) Further, I dont understand why calculating the surface integral of the density function gives me the mass (isn't the definition of density mass/volume and hence I would need some kind of volume?) Thank you for whoever can explain this in simple terms | 0 |
Something got messed up in my lyx settings and now the editor window previews the text in a very ugly and informal looking font. The pdf output is still the regular latex font. How can I reset the editor window font? This would probably be fixed by a complete uninstall and reinstall of lyx, but I can't figure out how to do that either. Here are some screenshots of what the editor window and pdf output look like For context, this happened after I tried to add a lyx -> markdown converter (https://wiki.lyx.org/Tips/ConvertMarkdown). I couldn't get it to work, and quickly gave up. But not my lyx font is hideous! Things I've tried: Uninstalling and reinstalling, but all my previous settings are maintained, including this font issue. I tried looking for any saved files in the indicated directory here, but don't see any files. How can I completely uninstall lyx? Tweaking the screen fonts options, but not exhaustively Running "tools/reconfigure", which produces an error | 0 |
My question is both naive and subtle. Naive because I don't know much more than the layman about physics and in particular quantum physics. Subtle because physics is an attempt to model the world, and as a computer scientist, with a strong interest in machine learning but also formal logic and models, a model is just that, a model. Not necessarily reality. It is not because a model fits reality that the model is the truth about reality. From my understanding, we know that: the quantum physics model has not been contradicted on the notion of superposition that it introduces there are experiments that can be explained using the quantum model where the classical model fails Am I correct to say that this only proves that the classical model, is just that, a model, and therefore incomplete? And that for those cases quantum mechanics has a better predictability power. Now the question: Has it been somehow proven that a physical entity can a some point in time and space have a dual state (independently of the model)? Or is it only that quantum mechanics is the only known model that allows us to explain things we otherwise couldn't? I would like to know if objects of our world can be in two states at the same time or that it is just more practical for predictability purposes to model things this way. | 0 |
I live in an arabic speaking country, and while I was sitting in a food court today, I noticed that the sound of the crowd was the same as the one I see in the stereotypical sound of the crowd. I have seen many threads discussing this on different forms, eg: Quora , reddit, Metafilter, and on this site, there is this tangentially related question. The problem with all of these is that, none of them is concrete, in sense of not having any calculative evidence. Going down the reddit rabbithole and reading the metafilter thread gives me some idea on how a calculation should look like, but I don't feel so confident in my abilities to do it, could someone provide a fermi type calculation of this? Else, link to a source which does? | 0 |
Recently I was on an airplane on a sunny day. The sun was shining on the other side of the plane and noticed a bright patch on the ground following beside us. Eventually I noticed a dark centre to this bright patch, the plane's shadow, which became more distant as the plane descended. When the plane flew over a city road signs in this bright patch lit up brightly because of their retro-reflective paint. My question is, what was acting as a retro-reflector to produce this bright patch when it wasn't passing over a road sign? When flying over water the bright patch disappeared, or was very faint. I could see it over forests, more clearly over cut grain fields and, at least faintly over a wide range of terrain. | 0 |
For a general linear map, there are infinitely many matrix representations. You choose a basis of the vector space, and this gives you a certain matrix for the linear map. You choose a different basis, and this gives you another matrix different from the first. For a linear map that does nothing however, we only get the identity matrix no matter what basis we choose. It isn't hard to see that the identity matrix will always do what we want from this linear map mechanically. But I feel there is some deeper intuition for why the matrix for this linear map shouldn't change with the basis, without thinking mechanically about the matrix. On a side note, is there any other linear map where the choice of basis doesn't change the matrix representation or is this the only one? | 0 |
In quantum mechanics, unbound states will tend to spread out in space over time according to the Shrodinger equation. So it seems to me there is a degree of freedom for "wild" particles (i.e. those have not been prepared in the lab) as to how much they are spread in space and momentum. So my question is: do we have any knowledge of how disperse "wild" particles (i.e. those which have travelled a long way such as photons from the Sun or cosmic rays) are? Is there any way to tell, or are we completely unable to extract this knowledge? Fundamentally we can't for an individual particle, but I mean statistically. I know that the spread must satisfy the Heisenberg uncertainty---my question is what do we know beyond that? | 0 |
Why can't we see light from beyond the observable universe? I've done a lot of research on this and all I've found is unsatisfactory answers and straight up nonsense. Some claim that the universe "expands faster than the speed of light" beyond the observable universe. Such a claim doesn't even make sense because the units of speed are m/s and for expansion are Hz. That's like saying "the area of this square is larger than the volume of this cube". All that the expansion can do to light (as far as I know) is redshift it. And light doesn't have a minimum possible frequency or energy value. So even if the expansion of the universe is very rapid, why does the light of distant objects "never reach us". Surely it still would, just extremely redshifted. In this case it does still reach us, and yet the claim is that it cannot. We often detect redshifted light, and that light has not been slowed down. When we detect it, it still goes at c, even though (in fact a better word than "even though" here is "because") it is redshifted. Light is always propagating at c no matter the reference frame. More precisely: does the light really never reach us, or can we just not detect it? If it never reaches us - why? If we cannot detect it although it does reach us - why? | 0 |
Let's suppose the earth is perfect sphere and let's ignore its rotation and movement. What would happen if i would be in the center of the earth? Would the gravity be zero in any direction so i wouldn't feel any gravity force? Or would there be 'pulling' forces (of half of the gravity force on the surface) that would attract me in every direction. Or in other words - If i would be in the center of the sun, would i be in a weightlessness state or would i be torn to a pieces (except of having a serios sunburn of course)? I know there are similar questions here: Effect of gravity at center of Earth Would you be weightless at the center of the Earth? but they are ot answering whether there would be no force or there would be forces to all direction. Some answers implies that the forces would be the same but in every direction so they would 'cancel' each other. But i'm not sure what does it mean. If I'm dragging a paper to different directions the forces are not zeroed, the paper is torn apart. | 0 |
If I'm trying to advertise that you can scroll through this webpage to find additions that go in/on your home, would it be... "Find new additions for your home." "Find new additions to your home." Does "find for your home" technically mean that your home itself is literally looking for new additions and you're looking on its behalf? And therefore "to" is correct? The problem is, what if it were something like advertising gift-giving, and said... "Find shoes for your friend." In that case, it could be read as finding shoes on behalf of your friend, but also finding shoes that suit your friend. Certainly, "find shoes to your friend" is absurd. So, why does it work in the home example (assuming "to" is correct in the home one)? Thank you :) | 0 |
I was once asked the question: What French word is commonly used in English for which an English word is commonly used in French? The answer was respectively rendezvous and date, which I found very unsatisfying. So, does such a situation exist, in which a French loanword is used in English to mean something for which an English loanword is used in French? Criteria: Both loanwords should be fairly common in the language that borrow them. By that, I mean that it is expected of the general population to know this word. A more precise definition of this criterion would be the absence of the "specialised" tag in the Cambridge dictionary. Both words should ideally be actual loanwords, meaning that their spelling hasn't changed, except for any punctuation. ("Gastronomy" or "beef" wouldn't count.) Research done: I've looked at various lists of French words used in English and did not find a match, as well as this related question. ChatGPT was no help. | 0 |
I am working with optically active nanomaterials (quantum dots, perovskites), that have pretty large exciton binding energies and can form multiexcitonic complexes, e.g. biexcitons, relatively easily. It has been well established that they also exhibit biexcitonic lasing/ASE under above threshold excitation, so here I have a question. To have lasing behavior/ASE, stimulated emission process should occur which requires population inversion between upper (biexciton) and lower (exciton) states. However, in the current literature when this effect is reported for any new material, population inversion requirement is barely ever mentioned and I wonder why. Is it for some reason biexcitonic recombination is a "special" transition, and therefore this requirement is lifted, or population inversion indeed takes place? If any of these options is correct, could you, please, elaborate why that happens? What is the mechanism lasing or inversion build-up? Thank you! | 0 |
Massive compact halo objects ("MACHOs") include a wide variety of hardly detectable bodies such as brown / white / black dwarfs and black holes, to name a few. If we take into account the inevitable end of all stars into either a white dwarf, a neutron star or a black hole and we compare the average lifespan of a star with the time star formation has been active since the Universe started, then we can tell galaxies already contain many dead stars which still have mass and can therefore interact gravitationally with the rest of the bodies in said galaxy. Furthermore, stars tend to accumulate in the galaxy's bulk (primarily) and in the galactic disc as well, so if dark matter happened to be MACHOs, which are distributed in said proportions, then this would account for the fact that models such as the dark matter halo need to have a density such that it decreases the further away we get from the centre of the galaxy. Finally, MACHOs are bodies we know to exist for a fact, whilst particle dark matter (e.g. WIMPs) has not been found. Is there a reason for which MACHOs are not the most likely candidate for dark matter? | 0 |
I have recently watched a couple of lectures in order to revise some of the notions that I will have to tackle in the coming months, and as I did so I have stumbled over something which baffled me a tad. Now, within these lectures the lecturer described a relation as being a function solely if it maps one element from the domain to one sole, unique element from the codomain (or range assuming it is a surjective function we are dealing with); I found this a tad confusing since, ever since I can remember we considered functions to map a value from the domain to one sole value from the codomain/range, and if no value from the domain is mapped multiple times to the same value within the codomain/range then it is injective, hence abiding to the lecture's definition of a function. Have I been missing something, do people automatically discard non-bijective, or at least non-injective functions for whatever reason? | 0 |
According to Archimedes' principle, the buoyancy force on a submerged object is given by the product of the mass of the displace fluid and the gravitational acceleration. Effectively, it is determined by the net pressure difference of the fluid between the bottom and the top of the object, or in other words the weight difference of the corresponding columns of fluid. However, if we have a weight placed on a scale for instance, this weight will change if we accelerate the scale upwards or downwards. As scale in freefall for instance will register zero weight. Now as the buoyancy force will accelerate an object initially at rest, should it not be reduced as a result of this (the buoyant object can be considered the scale here and the fluid column the weight being measured)? | 0 |
I want to start learning some optimization theory (mainly to get into ML in the future but I'm also just very interested in mathematics that intersects heavily with engineering) and I would like to get some recommendations to start tackling the subject. I found two books: Bierlaire's Optimization book and Convex Optimization by Boyd and Vandenberghe but the former seems to lack convex optimization which I've heard is quite foundational and the latter seems to target more practitioners of optimization algorithm where I want to get into theory to understand exactly why the algorithms work and both mention that they target engineers in their preface and I'm not sure if that's a good or a bad thing. So are there any other introductory references to the subject that target mathematicians which I'm missing? | 0 |
Very roughly, dark energy tends to cause space to expand and mass tends to cause space to contract. If nothing gets in the way, the math on that contraction breaks down when it forms a black hole singularity. So the question is, can enough energy halt the contraction before it forms a singularity, but after it forms a black hole (i.e. an event horizon)? Are there any theories that have tried to use this to avoid the the math breaking down? Edit: after some digging, I have found that the Penrose-Hawking singularity theorems gives conditions where a singularity must form, but I've yet to find a description of what happens (or could happen) if those conditions are not met. Namely, it seems like "interesting" thing might happen if negative energy became part of the conditions inside a black hole. | 0 |
For each segment of a piecewise Lyapunov function that exhibits asymptotic stability, we can utilize the LaSalle-Yoshizawa theorem and solve it using a differential inclusion. This allows us to merge all the piecewise Lyapunov functions and demonstrate that the entire system is asymptotically stable. Now, if each segment of my piecewise Lyapunov function achieves finite-time convergence, specifically in the form of square root, how can I utilize the differential inclusion? Can you explain whether the overall system is finite-time stable? Upon searching through some resources, I found that these studies generally assume each segment to be asymptotically stable and then prove that when the function obtained through the differential inclusion exhibits square root convergence, the overall system is finite-time stable. However, if my current equations do not exhibit asymptotic stability for each segment but instead demonstrate finite-time stability, how can I prove it? | 0 |
I'm not a physicist in any way, but I'm curious enough to watch and attend some pop-science lectures. Let's imagine the following situation - there is a free-standing unbound electron. It has its wave-function describes probabilities to find it in particular position. Also there is incoming photon with a proper wave-length to be absorbed. How the electron "decides" to absorb it, from what distance? If it's probabilistic too then electrons can absorb photons from centimeters away? And we shoud be able to verify it statistically by observations of many photons and electrons. Since the photons are stretched out by cosmological expansion, then I assume that they have the length in physical space. Is the absorption instant or does it take time to absorb the "whole" photon? If the absorption is not instant then it's possible to make a trick: make the electron annihilate with positron - after it started to absorb photon, but right before it finished. What will happen to "unabsorbed part" of photon? What is a correct theoretical answer to my questions? | 0 |
Was all the energy that is the universe today, present at that femtosecond theory has pushed back to or was there an influx of energy for a period of time? My question was: is the big bang like a pin prick in an infinite balloon of water? Initially, you have very high pressure, then as the hole expands the pressure drops but the flow doesn't stop. In the case of the universe is the big bang still pouring gluons and muons in everywhere at about the same rate as it always has, it's just that over this much larger volume it is less dramatic than the first few centuries. Does our universe create a back pressure at its various stages that would explain the changes in rate of expansion over time? It is equally likely I am fundamentally misunderstanding what it means to be in an expanding universe. I suspect it's fairly obvious to those that might answer that I've watched more science documentaries than is healthy but fail to comprehend most of the wikipedia pages describing the physics and maths that support them. I expect you see lots of these stupid questions and I suspect this has been asked before, but I couldn't find it. | 0 |
I was interested in electromagnetic induction in loops, since ground loops and such tend to be a problem in electronic system engineering. To me, it seems like induction in loops may tend to cancel out. For example, if the plane of a simple wire loop (that is just a ring of wire and nothing else) is perpendicular to an electromagnetic wave, using the normal rules for induction in wires, the E-field of the electromagnetic wave would be parallel to the top and bottom of the loop, and induce a current in the same direction in both the top and bottom. The sides, would be perpendicular to the E-field, and have no electromotive force on them. I asked about this on Quora, and was told that a "counterclockwise circulation" would be induced per Lenz's law. I know inserting a north pole into a wire loop does that, but, that was not what I asked. I found this question here on Physics Stack Exchange, EM waves from AC current in a loop?, and it seems to suggest cancellation of forces for a radiating wire loop. A receiving wire loop seems like it might have similar behaviour. So my question is, what current is induced in a wire loop whose plane is perpendicular to an electromagnetic wave, when the electromagnetic wave passes over it. | 0 |
We, as humans, given our height and size, view the world from the same general perspective. An ant, on the other hand, will understand the same world in a completely different way, given how limited their knowledge and experiences can be. When we spend a couple seconds taking a step out onto our front yard, it might have taken a microbe years to travel that same distance (assuming it can survive and all that). To the ants living in your front yard, your back yard might just be the "edge of the Universe" for them, let alone the other side of the city or continent. For an organism smaller than even Planck's length, assuming such thing exists, then Planck's length might just be the limit of their world, and who knows what's going on down there. Let's reverse that thinking and apply it to something that's bigger than us. Now, we are playing as the "ants" or the "organisms smaller than Planck's length". A bear, for example, can travel farther than a human given the same amount of time, and what we view as "further apart" might seem "closer" to the bear. To something as big as a planet, the Earth might be its "front yard" and the moon its "back yard". For something much much bigger, the size of a galaxy, lets say, what would be its "other side of the city"? By "taking a step", that thing would travel a distance worth of many light years in just a few seconds, right? | 0 |
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