darklight03/StackMIAsub huggingface dataset:
I have been brushing up on real analysis recently and, knowing nothing about topology but having seen some very general YT videos on the subject, I have developed a vague feeling that it somehow is very closely related to real analysis. For instance they both seem very concerned with continuity and continuous maps. Could someone please elaborate on the contact points between these branches of math?
I was taught in my image processing class that when a mean filter is applied infinite times on a given image, the intensity of each pixel reaches the same value. I understood this that time entirely on intuition. But looking back, I tried to come up with the formal proof of this result but couldn't make any progress. So is there a way to actually proof this notion. Any help would be appreciated.
My question is on the nature of oscillating electric field. From Maxwells equation, we know that the solution of the electric and magnetic field for an empty space are always oscillating fields. I learned that oscillating fields are connected to oscillating charges. However, what about the case when the the charge is not oscillating but rather, lets say, just linearly accelerating over a given distance. The field gets disturbed once the particle accelerate suddenly but what I don't understand is why this would still cause an oscillating field rather than just a locally distribution of the field. Hope my question is clear.
I would like to know whether bounded linear integral operators (defined on a separable Hilbert space of functions) always have a countable point-spectrum. Or if not, what would be a practical counterexample. The following situations for instance already lead to a countable point spectrum: self-adjoint (or symmetrizable) operator kernel integral operator with bounded kernel But what about bounded integral operators in general? I couldn't find any helpful theorem.
I'm trying to ad some probabilities on my model and I'm wondering which conditions are needed on my initial space. I'm starting with a metric space X and consider the set of operators from X to itself equipped with the topology of compact convergence. Let's consider the Borel sigma algebra of this space. Does there always exists a Probability on this measurable space ? Do I need some stronger assumptions on X ? EDIT : I forgot to specify that I'm asking about the existence of a probability with a "large support", equal to the space or infinite Thanks !
I'm trying to find models that allow me to calculate dropped objects, specifically, a model where I can consider lateral displacement due to wind force and vary the weight and resistance force of the object. This is crucial because I'm trying to calculate with utmost precision the point of impact of a tool falling from a height at a construction site, to determine the safety zone that must be established below to prevent injury to anyone. I appreciate any information in advance.
It is my understanding that sets are said to contain their elements but also a set is said to contain its subsets. However, I feel like this might lead to a confusion since while a set contains its subsets, its subsets are not necessarily elements of it. It seems to me the word "contain" must be restricted to only one of these meanings and there must be another word for the other. Am I missing something here or is this supposed to be understood from the given context?
As far as I understand a material is a better thermal conductor if its thermal diffusivity is higher. The thermal diffusivity is inversely proportional to density, thus denser materials have higher thermal diffusivity all else being equal. Intuitively I would have expected the opposite to be true! Atoms close together interact more often with each other, transporting thermal energy more easily. In my mind that would explain why gases (and 'zero density' vacuum) are good insulators. Why exactly low density improve thermal diffusivity?
I believe anyone who has driven a car without power steering has noticed this effect. I am not able to find a justifiable argument online. My thought process suggested it must have to do something with the static friction in the case of a moving car which provides it a centripetal acceleration, but I am not sure. A reasonable and complete argument will be much appreciated.
It is easily shown that in proving compactness, we can assume the arbitrary open cover is given by a basic open cover. However, the same proof does not work when proving countable compactness. The reason is that not every open set can be written as a countable union of basic open sets. So my question is: in proving countable compactness, is there a counterexample showing that we cannot assume the cover is by basic elements? Note that such a space cannot be second countable. The trivial example of a discrete uncountable space is not a counterexample.
In A_Treatise_of_Human_Nature, of David Hume, it is written: For if truth be at all within the reach of human capacity, it is certain it must lie very deep and abstruse: and to hope we shall arrive at it without pains, while the greatest geniuses have failed with the utmost pains, must certainly be esteemed sufficiently vain and presumptuous. I gathered that the usage seems to be to lie deep instead of to lie deeply. Why does an adjective follow the verb lie instead of an adverb? Is lie here a linking verb? Would the phrase It is certain it must lie very deeply and abstrusely. be incorrect?
There is a book that is called embeddings in manifolds that studies topological embeddings and how they relate to each other (by homeomorphisms). I was wondering if there is a study of isometrically embedded polytopes in manifolds. "Isometric" here would probably mean conservation of the metric (as in metric spaces) rather than the Riemannian structure. So I would like to ask if there is anything I can find on the topic, if there is such topic, preferably a book but papers will do as well?
I understand that the steady-state Kalman filter is more computationally efficient because the noise in a measurement equation and the shock in a state equation have a constant variance over the time period. However, what model should I use if I want to study the effect of a noise shock, which refers to a large shock that was not imaginable? Should I use a time-varying Kalman filter that allows for the variance of noise to vary? I am faced with this question because I cannot exactly tell the differences between these models and their respective advantages and disadvantages. Please help me out.
I need to know the grammar and syntactical explanation of a phrase in the following sentence: The protest was reportedly linked to members of the group objecting to the replacement of the group leader. Is this phrase "members of the group objecting to the replacement of the group leader" an absolute phrase? If so, can absolute phrases act as the objects (noun clauses) of verbs?
The back side of a coin is called reverse, the left or back side of a bound manuscript is the verso, but what is the back of a piece of paper such as a letter (not the envelope) called? Dictionaries give a wide selection of options (rear, back, reverse, backside, flipside, back page, etc.), and I don't understand which of these a native speaker would most likely use here. I would like to direct the reader of a formal letter (which in this case is a legal document) to further information on its back, e.g. "bla bla (see the back of this letter)".
I found this https://en.wikipedia.org/wiki/Harmonic_oscillator in my search results when I search for "electrical oscillator Hamiltonian" and some other things too. But none of them answer my question about the form of the kinetic and potential energies of an electrical oscillator. Searching for the kinetic energy and potential energy does not show up fruitful either such as here. I am specifically looking for an electrical oscillator Hamiltonian. Any help would be appreciated.
We are learning the rational root theorem right now. It is pretty clear how to prove a number is irrational when we know how to construct a polynomial that only includes integer and has the irrational number as a root (rational root theorem). However, if it comes to sum of two number with different power, it is pretty hard to construct the polynomial. Is there a general way to construct polynomial? Or is there any alternative ways to prove this question?
If we have a smooth and frictionless pulley with two different masses attached at the ends of the rope (meaning they will accelerate) and the pulley is attached to the roof by a string, why cant we consider everything as one closed system and calculate the wieght of the that system to find the tension on the string attaching the pulley to the roof. After thinking for some time I feel that somehow allowing the masses to accelerate in one direction is relieving some of the force on the pulley. But why? Why can't we consider it a single system and find its wieght?
I think it is computing the fine-grained entropy. However, I am confused by the case that when there is a black hole in the bulk. The Ryu-Takayanagi surface may include the horizon of the black hole in this case. However, the Bekenstein-Hawking entropy is a coarsed-grain entropy. I have also heard that the RT formula is computing the coarse-grained entropy of the complementary region. What does this mean?
I am working on a model of a Fe square planar complex with nitrogen and oxygen given below is a monomer of that polymer. While constructing the hamiltonian matrix of this monomer I'm confused as to if I should take the orbitals of Nitrogen and Oxygen as normal atomic orbitals or hybridised orbitals? I tried to search up about it but couldn't find any method that used hybridized orbitals
While solving this problem, we arrived at the solution in a form of discontinuous function of electric field. Intuitively it doesn't seem right. What is a physics explanation behind this discontinuity? Or is the mathematical approach wrong? Or is it just so idealised problem that it doesn't exist in reality anyway, so discontinuity doesn't have any physics meaning? Tried solution (electric field as sum of electric fields of all thin rings):
I am a mathematician and a while ago I started studying quantum physics. I know the concept of superposition mathematically, but my question is regarding a famous experiment that Aaron O'Connell performed. Link to the TED talk that describes the experiment: https://youtu.be/dvYYYlgVAao He says in this video, "The small piece of metal was vibrating and not vibrating at the same time." I did not understand this phrase because it seems illogical. Can this be explained in the experiment and is his statement correct or is it just to stir up the audience?
I a m a high school student who has spent a lot of time learning about limits but still do not understand how they work completely. What I have understood is that sometimes a function does not have a y-value/s for one/(or more?) x-value/s however, since we know the y-values of the x-values close to the x-value with an undefined y-value we can approximate where the y-value for that x-value will be. To express this we use a concept called limits. My question is, can we use limits for functions where there is a y-value for every X-value? If yes,why would we do that? To show the behaviour of the points of a function as they get closer to a certain point? Thank you in advance!
(I know this is going to sound odd just please consider it) What if you were to throw an indestructible ring straight into a black hole with a hole larger than it's singularity? Would its trajectory be straight through the black hole missing the singularity? If this is the case then wouldn't it reach escape velocity and as result break the speed of light? Think of it like a ball rolling down a hill and being able to speed up enough to roll back up a hill of the same height. I understand that a black hole achieves it's event horizon through distortion of space time so what I am truly pondering is if it would have a straight and un-impeded trajectory through the black hole.
I found this problem on Facebook from a Mathematics group. Apparently it had appeared as one of the problems on a college entrance exam. The person who shared it on Facebook did not disclose the correct answer. I'm going to post my own approach below, please let me know if my answer is correct or if there are any other methods possible. (I even encourage solutions that may typically go beyond pre-college level mathematics.)
Every well-ordered set is linearly ordered. However, is there a notion of well-orders that generalizes to partial orders? Maybe, the generalized definition will be that every non-empty subset has a minimal element, not necessarily a least element. I would be very interested to hear about such a definition, and more importantly, I would like to know about research on this topic about these "generalized" well-orders.
I use VS code with the Latex workshop on Mac and it works perfectly fine when writing a paper. But I don't understand "pdflatex", "mactex", "texlive" or something other people are saying What are the essential components to compile a paper using Mac? How can I check what is working behind my VS code? (I don't remember what I installed so far to make my environment possible )
I was reading in a patent that a rotating spherical body in space will experience a curvilinear translation, due to the conservation of angular momentum. This confuses me, as per my understanding if there is no friction, or any other force applied, conservation of angular momentum will keep the same rotating speed but I do not see why it will make it move/translate from its location. On the other hand, I have seen a video of a spinning object in the ISS changing direction alternatively, and I also do not know how it happens, in that case there is no translation that can be observed, but there is still friction with the air. Anyone can give some insights? Thanks!
I am just wondering how do we transfer IMU readings to Body frame? Especially, linear speed and acceleration. In following diagram, the IMU is mounted on one of the surfaces of the body (you can imaging it to be a bot, vehicle, drone, or etc.). The center of the vehicle is at Body (green) point in this diagram, and it is on a different surface of the body. Both IMU and Body point are on a rigid body which moving in an inertial frame. Any hint is highly appreciated. I'd really appreciate if there are some references. Thanks!
I am wondering about the correct verb form to use here: This article limits itself to consider... or This article limits itself to considering... For this example both sound dubious to me (possibly because of the presence of 'to'), and those I have asked are also unsure, including a native English speaker. And what about the passive counterpart: This article is limited to consider... This article is limited to considering...
Earlier I was using texstudio with MikTex. In MikTex, I am facing some issue with its updates. So, I downloaded MacTex. How to configure texstudio with MacTex? When I am running texstudio, it is taking MikTex compiler. The error is shown below. Sorry, but /usr/local/bin/pdflatex did not succeed. The log file hopefully contains the information to get MiKTeX going again: /Users/bp/Library/Application Support/MiKTeX/texmfs/data/miktex/log/pdflatex.log pdflatex: major issue: So far, you have not checked for MiKTeX updates.
I am confused by the definitions of a gauge transformation, a coordinate transformation and a diffeomorphism. In particular, should observables in GR be fundamentally invariant under gauge transformations, coordinate transformations or diffeomorphisms? For example, in these notes by Bertschinger, observables should be gauge invariant, and he defines a gauge transformation as an infinitesimal diffeomorphism. This appears standard and in line with e.g. Sean Carroll's GR textbook. On the other hand, there is this paper by Yoo and Durrer, which claims that cosmological observables are invariant under diffeomorphisms but not gauge transformations in general. Their definition of a gauge transformation appears to refer to simply "an equivalent" coordinate transformation - the sort which Carroll disagrees with being strictly correct in his textbook.
Is it correct to say "None can stop me?" I feel like "No one can stop me" is the one I've seen used most often, but I'm not sure if "none" works too? Does there have to be a specific context where there are several people: e.g. "There are many who will try, but none can stop me." And in other contexts (without specifically mentioning other people) "none" doesn't work?
I'm having trouble reading documentation from packages. It seems like there's something wrong with the way I'm reading them. How should I read them, in order to learn how to use the packages correctly and efficiently? Also is there a place where you can learn typography related topics in LaTeX like glue and boxes? I don't find it talk all that much, and if they do they don't teach me why that's necessary. I haven't learn the details of typography other than fiddling with MS Word. And I would like to learn how to make is so that the fonts and font size aren't too long for the average characters per line.
On the average Semi-trailer truck, assuming an average load, can the engine overcome the brakes and continue spinning the wheels despite the operator applying the brakes fully? This would be assuming the brakes are in good operating condition, and are not overheated or failing in some way. Additionally, would this also be the case if the tractor unit had no trailer on it, and no weight? How about fully loaded?
I am having a conceptual difficulty understanding the following issue regarding space-time: it is clear to me why a full description of coordinates requires three spatial dimensions plus time. However, Linear Algebra also teaches that dimensions needs to be independent from one other, such that my movement along one dimension has no bearing on my movement along another dimension. This does not seem to hold in space-time, given the effects of time dilation and space contraction, which is essentially a rescaling transformation of one dimension based on the object's velocity along another. This seems to preclude such independence between said dimensions. Can somebody clear up my confusion?
In the case of big freeze, space expansion will be accelerating and there appears to be a lot of different phenomena occurring. However, in the case of big rip, the expansion is super-accelerating so that light speed will be reached, ripping apart every structure. How come this speed is not reached in the case of big freeze. If the speed of expansion is accelerating, it should reach light speed at some moment. How does the evolution of scale factor in this case look like?
So far I have not needed to insert jpg images in my LaTex files, but now I want to write a text that will include some pictures. I would like a format where the picture doesn't take the full width of the page but be rather set to the right or the left and the remaining space on its side filled with text. I have tried to use the figure environment from the graphicx package together with a minipage environment, but failed. What would be the best way to proceed? Is it safe (or even possible) to change from a one column format to a two columns format in the middle of a page?
It's on the tip of my tongue. The great thing about this word is that there's a connotation that a person it describes is not just smart or curious, but that they find a childlike joy in learning. This word is paired with the word "intellectually", as in "intellectually curious." But when it's on its own, it's more likely to be describing someone who's cheeky or sassy, particularly a child. But in that usage, there's the connotation that their cheekiness or sassiness belies gifted intelligence. I am pretty sure it starts with a p, or even maybe 'pro,' but I could be wrong about that.
In indirect speech some modal verbs usually change. can -> could He said "I can ride a bike" = He said that he could ride a bike may -> might/could He asked "May I use the bathroom?" = He asked whether he could use the bathroom But what about dare? It can be a modal verb. "No one dare go there" he whispered = He whispered ...
I'm from South America and I've been fluent (at least in my opinion) in English for several years as of now. With this in mind it has really called my attention lately that I've heard Afro-American people say, for example, "I don't like no cats" instead of "I don't like cats". With double negation. And the other one is, for example, "Look at them kids" instead of "Look at the kids". With them instead of the. It is not really confusing tbh but I'm just curious about why this happens?
Would it be feasible to extend the range of an EV using a generator? I would guess that F=ma and perhaps a myriad of other factors would make a generator placed on an EV not work very well. I think of things like a wind generator to capture the wind power but the drag would slow the car down thus canceling the added energy. Or how about a generator in the rear wheels of the car to add back some of the loss? I could imagine something like that working in a downhill scenario but i can't see it adding much if any in the whole driving scheme. Seems Elon would have figured this out by now if any such thing would be feasible. What you guys think?
This question is not similar to what I have read so far in this site. I am just trying to understand what electricity actually is. I have come to know that when charges move or vibrate in a wire by the supplied voltage, the electrons drift velocity is very slow. And the bulb glows instantly because energy travels through EM field, which is perpendicular to electric and magnetic field. Now, comes the question, suppose if I create a circuit, and connect it to the battery. So, the bulb will glow, but what if I put another unconnected wire near the circuit connected with another bulb. Will the bulb glow? Because electricity travels through EM field. Shouldn't the bulb glow? Which actually doesn't happen. I am just confused.
Let's assume a galaxy has super massive black hole in it's center. We can't see what's inside the black hole because light cannot escape from it. In simple terms, black hole is a hole in the space and also assume that this black hole goes down all the way somewhere, like a straw/pipe that we cannot see. So now imagine a a spaceship is traveling exactly beneath the supermassive blackhole (but far away from event horizon where the blackhole can't pull it inside). And now what happens if the spaceship hits the blackhole straw/pipe? Just trying to understand is there any science that we can use to see where this blackhole straw/pipe goes to from outside. Or what happens if a spaceship or a planet hits this pipe?
The s-procedure or s-lemma tries to solve a system of quadratic inequalities via a linear matrix inequality (LMI) relaxation. There are many different enunciations and if you are not sure what i mean you can check here. I have seen it be used in papers where the inequalities are not quadratic, for example here, but I cannot find a form of the theorem that seems to apply. Can anyone help me? Thanks in advance.
I remember hearing at some point that pertubation QFT using Feynman diagrams can be thought of as certain limits of lattice QFT. Is there a precise statement of this fact? Or is it just a heuristic with no mathematical formulation/justification? Alternatively, is there a way to do calculation on lattices, that gives the same result as calculating Feynman diagrams, taking into account things like Pauli Villars/dimensional regularizations and renormalizations?
I have read so many posts about the implication, but I am still confused. The statement "if A, then B" is always true when A is false. But I think that when A is false, nothing can be concluded. Moreover, I do not understand the example, "If today is Saturday, then tomorrow is Monday." As today is Wednesday, both sentences are false so the implication should be true, but it seems in daily life that this implication is false.