darklight03/StackMIAsub huggingface dataset:
A thin charged wire conductor, with a semidiameter of a is placed into a vacuum at height h (h >> a) above a big grounded conducting plate. The wire is placed parallel to the x axis, and the area of the plate lies in Oxy-plane. The potential of the conductor compared to the plate is V. Find the equation for the areal charge density in the coordinate system (O). I tried solving this problem, I could find the equation for V using method of images but I don't know how to find the areal density. Can you help?
Is the Heisenberg's microscope gedanken experiment valid when considering spacetime kinematics? That is, if we consider a small region of space and try to measure its curvature, then we may use photons and see how the trajectory changes. On the other hand, the photon's energy changes the very curvature that we are about to measure. Here I assume that a high energy photon "maps" curvature more accurately. The ultimate goal would be to find an uncertainty principle that connects curvature and the rate at which the curvature changes.
I've gone to Wikipedia and watched a few videos regarding Linear Codes, and from what I see, the Parity Check Matrix is always given, and because of that the Generator Matrix (consisting of Identity Matrix, Parity Matrix) can be made. But what if the Parity Matrix isn't given, and instead, only k and n? I understand the general layout, but I do not see how to figure out the actual contents (or identity, I believe its called) of the Parity Matrix. I should add that the specific scenario I am asking about is in binary, so maybe its only doable there?
I've recently started learning about the physics of fluids and I've found that euler's equations exist and that I can use them to compute the flow of fluids with zero viscosity. But I have a problem that I don't see any easy way of adding an obstacle to a compessible flow as shown in this gif How should I do it? Is something similar possible also with navier-stokes equations? (I am aware that this questions is probably a duplicate of some older question, because this sounds as a really basic problem, but I've search for the anwer for some time and I even asked on the physics discord server and I found nothing)
Is there a way in TeX (preferably plain XeTeX, but other engines are fine) to automatically choose between different options (e.g. between "and" and "&") so that the badness of a paragraph is minimal? For example, for a small hsize, the following is fine | I read a book by Smith & John. | But if we increase the value of hsize, we have to add space: | I read a book by Smith & John. | So we prefer to use the other variant: | I read a book by Smith and John. | If I remember correctly, the first printers used this kind of trick to avoid line breaks.
I've heard about solitons in dense mediums (water), sparse mediums (acoustic) and optical fiber. But I can't find a good overview work on solitons in radio spectrum. Something like generating EM solitons with a peculiar antenna setup. It seems like an obvious thing for people to be working on, but looks like I just lack proper terminology to google it. Is there something that can be recommended, preferably with an experimental setup described?
Davisson and Germer's experiment confirmed the wave-particle duality of matter. In this experiment a slowed down electron beam is reflected by the surface of a nickel crystal, which is able to split the electron beam into several parts, each of which is characterized by a reflection angle, rising a diffraction pattern quite similar to that of the X-rays. Is what I'm about to write right? This phenomenon is possible because the surface of the nickel crystal is not regular, but is formed by microscopic grooves capable of rising to the phenomenon of diffraction.
I understand it is possible to find the shortest distance from a point to a straight line, but how do we extend this to any function. For a straight line it is simply the equation of the perpendicular which intersects the point of interest. If we have a cubic or a quartic function how would I model the shortest (straight line) distance to that function from somewhere on the plane? I anxiously wait a response.
Is there a way to make meaning of it? I was thinking that maybe because the covariance is divided by the percentile rank of std dev of A, the magnitude of A is to a degree bounded. But how does it change the covariance, as covariance is about the co-directions of A and B rather than the magnitude in the first place. Essentially, what does "covariance(A,B) divided by percentile rank of std dev(A)" reveal? Any thoughts would be greatly appreciated!
I have learnt that a planar graph is a graph which can be drawn on the plane without any of its edges crossing. What exactly is "the plane"? In plane geometry, we define a plane to be a flat surface of two dimensions; however here the use of "the" suggests we have a particular instance of a plane in mind, not just any plane. Is there a certain specific type of plane which is meant here, other than the general plane from plane geometry?
I am reading the paper "REPRESENTATION AND INDEX THEORY FOR TOEPLITZ OPERATORS" by Gerald Murphy and came across this passage: I was wondering why we care about the strict topology or why is it important for multiplier algebra. So I know how these are defined but I've never understood why we should use it. Am I supposed to assume we are talking about the strict topology every time I see the multiplier algebra? Now for this specific example given below for the Toeplitz matrix, why is it better to identify the operators on Hardy space with the multiplier algebra when talking about convergence?
Based on my intuition, once light is monochromatic it will stay that way unless you actively try to change its frequency. My intuition stems from the intricate optical laser setups in my lab that sometimes (often?) work with monochromatic light. Sound, on the other hand, seems to change its frequency easily when it interacts with the environment. Again, this is based on intuition so I'm not sure if its true. So my questions are Does light stay monochromatic once it is created? Does sound stay monochromatic once it is created? Why? Under what conditions do waves stay monochromatic?
Considering no air resistance, the acceleration time graph for a free falling object would be a horizontal line as the acceleration remains unchanged. When we do take into account air resistance, the graph becomes a curve with negative gradient. However, to me, this is unclear. I understand why this must be a negative relationship, but why can it not be a negative sloping line. When we graph acceleration against air resistance, we see this is a negative sloping line. Why must this be the case? I am looking for both an intuitive and mathematical answer.
We have a sentence like this: Transients are individuals just passing through, not contributing to the local population and its demographic rates, and they are thus very unlikely to be recaptured. Is it possible to remove "they are" behind the comma? So it would sound like this: Transients are individuals just passing through, not contributing to the local population and its demographic rates, and thus very unlikely to be recaptured. It would be much more elegant. But the catch also is that the "thus" doesn't refer to the most proximate sentence "not contributing to the local ....", but to the one at the beginning - "just passing through". Is that a problem somehow?
In this quote (from this essay), Russell talks about a theorem showing that "three or more lines meet in a point". "In geometry, instead of the tedious apparatus of fallacious proofs for obvious truisms which constitutes the beginning of Euclid, the learner should be allowed at first to assume the truth of everything obvious, and should be instructed in the demonstrations of theorems which are at once startling and easily verifiable by actual drawing, such as those in which it is shown that three or more lines meet in a point. " What theorem is he talking about, and what is its proof?
I'm new to manifolds and in my Computer Graphics class, we briefly explored the topic superficially through visual examples. From my intuitive understanding, we should be able to place a Euclidean grid on a manifold's surface or it should appear flat locally. However, I'm puzzled by how a pyramid with sharp edges and corners can still be a manifold. When we zoom in on the edges or corners, we only see flat areas along the faces of the pyramid, in any other direction it doesn't look flat. What is the difference between this case and two pyramids sharing a vertex (as an hourglass shape), which we classified as non-manifold? I'd appreciate if someone can explain what I'm missing in my intuitive understanding.
I've been looking into profinite groups, their topological subgroup lattices, etc. I asked the question does every profinite group admit maximal subgroups? I can't find an example of a profinite group which does not admit maximal subgroups, though I also cannot prove that each must have one. I think that if this is true, then it is related to profinite groups being residually finite, something about being able to tell apart elements using a projection onto a finite group, which must admit maximal subgroups. Can someone answer this question with the affirmative, or give a counterexample of a profinite group which has no maximal subgroups?
I know how to perform the double slit experiment in high school: a dark room, a source of light like a consumer electronics laser, a black board, and a card board with double slits. Is there way to extend this experiment to the one where we observe from which slit the photons pass (and causes them to not create the interference pattern anymore) without sophisticated measurement devices? Something that we can perform at home or at a typical high school? I have searched a bit but haven't found the answer. Does observing which slit the photons pass through require sophisticated measurement devices?
I recently found that a rope suspended from two points forms the shape of a catenary curve, which brought me to another question: What is the shape made by a rod of uniform density when only one of its ends is held horizontally? Obviously the material and width would affect the strength of the rod and thus the shape it makes, however I assume they must all resemble the same curve. (similar to how all shapes and sizes of ropes will resemble a catenary curve when held at two points) I would love any help with this question!
Suppose light from a laser passes through a single vertical slit. The width of the slit is certainly a factor how the light scatters after it goes through the slit; the thinner the slit, the wider the scattering. But my question is about the material surrounding the slit. How does the depth of the material influence the scattering pattern? And how does the composition of the material itself influence the scattering pattern? Does the nature of material around the slit significantly affect the scattering pattern?
I know that slope is the tangent of angle of inclination. But are they just two ways of defining the same thing? Or do they have any significant differences regarding what purpose they serve in mathematics? The way angles are defined as arc divided by radius, while slope the height divided by the base, doesn't show much difference. I only have a basic understanding of calculus.
So on the german wikipedia page of topological vector spaces it is written that, if a topological vector space is a Hausdorff space, then translation by a vector and dilation by a scalar are homeomorphisms. However, I have looked at several lecture notes and it is always stated (and proofed) that these two mappings are homeomorphism for general topological vector spaces (even those that are not Hausdorff). So I wanted to confirm that it is true that for any - possibly non Hausdorff - topological vector space the translation and dilation are homeomorhpisms. Also, can anybody think of a reason why on the german wikipedia they have restricted that statement to topological vector spaces with the Hausdroff property?
I have heard that light is a transverse wave which is necessary for effects like polarisation, but I am unsure of what exactly this means. Is this saying that light's electric and magnetic components oscillate in perpendicular directions of three dimensional space? i.e could the electric (or magnetic) component literally oscillate say 'up' and 'down' making the electric field strength of points in space above or below the wave stronger or weaker according to the light's phase? If not, then what could it mean for light to be a transverse wave, and how does polarisation work in this context? Thanks in advance for any help with understanding this.
Where do the electrons in a lightning bolt come from? There are problems with the explanations given up to at least two years ago. The explanation that the electrons come from inside the thunder cloud or are derived from cosmic rays are inadequate. There are a lot more electrons in a lightning bolt than those that come from these sources. Is there research which shows were the electrons actually come from ? Thanks.
So I know that the main trig functions are sine, cosine and tangent, while cosecant, secant and cotangent are simply known as the "reciporicals", but wouldn't it make more sense for sine, SECANT and tangent be the main trig functions while COsine, COsecant and COtan be the secondary ones? After all, secant is just the reciprocal of cosine, so everything you can do with cosine you can do with secant.
I was reading geometric optics when I came across the concept of virtual objects. I found the concept counter-intuitive, as far as reflection goes. (I perfectly understand the formation of such an object when the light ray is refracted through a glass slab.) Attaching the definition provided in my book: My confusion: A mirror forms an image when the rays originate from a certain object. So, how can an image of a virtual object form if the rays do not diverge from a particular point in the first place? What are some real-life examples of such a situation, as depicted by the figure below: I have checked answers to similar questions but my doubt is not cleared.
I don't know exactly how boomerangs work, but I imagine that the fact that they are less symmetric than a sphere is part of the reason they can be thrown in such a way that they return to whomever threw them. Eggs also appear to lack some symmetry, because one side is usually pointier than the other. Of course eggs aren't ass aerodynamic as boomerangs and here it is claimed that aerodynamics plays a vital role for how boomerangs work, so probably the answer to my question is no: Will boomerang work in vacuum chamber on Earth? But I am still curious as to what would happen if you did throw an egg (shaped object) really fast like a boomerang. Does anybody know?
Please correct me if I am wrong. Almost all materials are made up of atoms with electrons. If we rub two different materials, like plastic scale and oily hair on the head, then static charge develops on the surface of scale. This charge can attract tiny bits of paper. Also during lightening tiny droplets of water and ice crystals rub against each other to create static electricity. However if I rub my hands on the wall then no static charge develops on the hand. I checked it, as my hand did not attract tiny bits of paper , which means no charge developed on my hand due to rubbing. My question is : On which atomic factors of the materials does the phenomena of static charge depends ?
In mathematics, scalars are defined as elements of a field. Are scalars in mathematics invariant under coordinate transformation? In physics, the way they are defined, I know they are. Since in physics, they are defined to be quantities having only magnitude and no direction, they are invariant under rotations or translation. Some sources say that in mathematics, scalars are, by definition, invariant under coordinate transformations. Can anyone please explain how defining scalars as elements of a field would intrinsically assume invariance under coordinate transformation? If not, can you give an example of a field whose elements may not be invariant under coordinate transformation?
I am currently studying Analysis on manifolds for my thesis. I am looking for some references about the McKean-Singer conjecture and Patodi's proof. I read with much attention to detail his paper: Curvature and the eigenforms of the laplace operator. However I am having difficulty over some of the arguments. I would like to kindly request some references about this specific paper, for example some extra details or maybe if this is contained in a book. Thanks in advance.
I am a chemist currently trying to dive in to details of random motions. I have been studying Einstein-Smoluchowski treatment, and want to learn further (Fokker-Plank's equation and Langevin treatment etc). Although I am comfortable with mathematics some of the textbooks I came across include lot of symbols I am not aware of, and hence it gets overwhelming. Could someone please recommend me textbooks which provide detailed conceptual and mathematical treatments, assuming I am not aware of all the symbols or denotations?
Most objects in the universe cycle around their centre of mass: earth, sun, galaxies. Even galaxy clusters. However, the whole universe doesn't possess an angular momentum. It can't: If it had one, the centre would be a special point (which isn't allowed) and the velocities of the outer regions would easily become infinite. How does the angular momentum vanish with increasing scale? Shouldn't the laws of nature that lead to cycling for smaller objects also hold for larger objects?
The Doves Press was one of the major players in the early British private press movement, hand-printing books in Hammersmith that were known for the absence of any illustrations and a focus on the aesthetics of beautiful type. Below is the famous first page from the Doves Press Bible. Can anyone provide some guidance on how to replicate this "I" that extends downwards to the height of the text? I find this quite interesting because there are several letters that could elegantly be extended vertically by stretching the vertical strokes of the letterform while leaving the proportions of other parts unstretched (I, J, H, E, P, L, F, T, Y, U, M).
I read the paper Example of indeterminacy in classical dynamics. I understand the paper: Because the differential equation does not satisfy the Lipschitz condition, its solution is not unique, and thus there is indeterminacy. But is it really possible in physics, as in the example described in the paper, for the ball to rest at the origin to suddenly deviate to the left or right from the origin, which violates the law of causality? Or does the physical world not follow the law of causality?
Are electric and magnetic multipole transitions are classified based on the nature of change in angular momentum of an electron during a transition between two states? In electric multipole transitions, the electron changes its orbital angular momentum, leading to transitions between different electron shells or orbitals within the same shell. In magnetic multipole transitions, the electron changes its spin angular momentum, resulting in transitions between different spin states.
I'm looking for examples of dramatic typos - where a minor edit can dramatically change the meaning of a phrase, and would like to know if there's a name for this phenomenon. Paronyms are a similar concept, but seems to incorporate pronunciation - "love" and "loathe" might be paronyms but they're not especially similar in spelling. For example, "symptomatic" and "asymptomatic" are one letter apart but have opposite meaning, "heroin" and "heroine" have very different meanings. By edit distance, I mean the similarity of the two words - the number of single-letter additions or substitutions to turn one into the other.
I did find the proof of pseudo torque about the center of mass of a body in combined rotation and translation being zero, and thus I'm convinced that in this frame of reference, we can conserve angular momentum. However, although the net torque about the point of contact in rolling is zero, I don't understand how we can conserve angular momentum about the point of contact in rolling motion, even though this axis itself is not fixed and constantly translating. Any help is appreciated.
I have a question which asks me to compute the mapping matrix, but I have no idea how to do this. I understand the concept, but I am unable to understand how I actually solve problems. Would appreciate if someone gave me a step by step explanation of how I would tackle problems like these. Would it be the same method for finding a change of basis matrix as well? Here is the image of the question: Mapping matrix question
If the axis of rotation is not passing through the centre of mass but some other point of a rigid body then how do we define the direction of angular parameters that is angular velocity, angular acceleration and angular displacement. For example : This is a sphere with a void in between rolling on a flat surface.C is the centre of mass and O is the axis of rotation. So basically my question is, will the angular velocity of the point A be perpendicular to OA or CA? Also will the angular acceleration of A be parallel or antiparallel to the angular velocity and correspondingly the centripetal acceleration would be perpendicular to the angular velocity right? I am confused about this.
In the article about Coulomb's law There is following sentence: Coulomb's law is an experimental law of physics that calculates the amount of force between two electrically charged particles at rest. However in the Griffith's Introduction to electrodynamics Introduction before explaining Coulomb's Law, there is following sentence: To begin with, we shall consider the special case of electrostatics in which all the source charges are stationary (though the test charge may be moving). What i'm confusing is: Wikipedia applies to two charges at rest but Griffith's book applies to stationary source charge. They collide with each other. Which is correct one?
I tried to implement Kruskal's minimal spanning tree algorithm and I found that the output is essentially a set of the edges... The output is not a tree at all. In my opinion, a tree can be represented by a parents array, and each node associates a key which is its parent. For a tree we need to know which node is its root at least I think. But the output of Kruskal's algorithm are just edges. How to fill this gap?
All ferromagnets have domains in which the magnetic atomic dipoles are alligned. However, soft ferromagnets like iron respond quickly to the external magnetic fields while the hard ferromagnets(permanent ferromagnets) don't. In atomic view, what causes this difference? My fist guess is that in the hard ferromagnets the atomic dipoles are stronger and their density is higher so that their greater interaction keeps them more alligned compared to the soft ferromagnets. So,the hard ferromagnets are not affected by external magnetic field as easily as soft ferromagnets? I would appreciate if anyone please help me with that.
Given two positive semidefinite matrices X, and Y. My question is, can we use von Neumann or the log-Determinant divergences to measure the distance between the two matrices. Most Manifold measure distance require two positive definite (PD) matrices. In this paper, the authors used the log-Determinant divergence with two positive semidefinite matrices (PSD). However, from what I know, the two divergences are for measures involving positive definite matrices (PD). Any insight please.
Consider a mass rotating in a plane around a fixed point O at constant speed. In such case the angular momentum with respect to O is constant and its time variation is zero according to the fact the torque is zero.But if you change the point to which you evaluate torque and angular momentum, the time derivative of angular momentum is not necessarily zero. So?