darklight03/StackMIAsub huggingface dataset:
I am reading Middlemarch by George Eliot. I encountered this long sentence and only understood the gist of it. I can't figure out the grammar style and I am looking for the verb of the subject "he". Here is the sentence: Thus, the mind of Frisk was exactly of the sort for Mr. Solomon Featherstone to work upon, he having more plenteous ideas of the same order with a suspicion of heaven and earth which was better fed and more entirely at leisure. Why does the sentence have "he?"
There are projective sets in descriptive set theory. For them, the axiom of determinacy is not contradicting the axiom of choice. Given the axiom of choice, every set is a projective object. But in a world without AC, is any projective set a projective object in the sense of category theory? Or did those terms happen to use the same word and there is any counterexample?
Assuming the object positioned in front of the mirror at some point on the principal axis before the focus, each point of that object emanates light in all directions, so what happens to the rays of light that do not fall parallel to the principal axis and those that pass through the focus, do they not converge at some point in space? Do they diverge, not forming a sharp image? Why is a real image formed in a concave mirror considered only the two rays of light, the one that falls parallel to the principal axis and the one that passes through the focus?
During the rotation of the Earth, a centrifugal force is formed, according to the actions of which ocean water should be collected at the Equator, forming a "hump" hundreds of meters high. However, there is no water hump on the Equator. The depth of the ocean does not depend on latitude, although there should be such a dependence. at the same time, the centrifugal force is sufficient to deform the shape of the earth from a regular sphere to a geoid!
For a given game, is there any relationship between the computational complexity of weakly solving the game and strongly solving it? I ask because a weakly, but not strongly solved game with simple strategies to guarantee at least a draw could still be interesting to play if the players place little value on a draw, and do not know how to win if their opponent is also trying to achieve more than just a draw. However, if simple strategies weakly solving a game guaranteed simple strategies strongly solving it, then this point would be moot. So, if for a given game, as some parameter scales up we have a weak solution of a particular complexity class, can we bound the complexity class of a strong solution?
When I saw factorials, I immediately thought about Wilson's theorem. However, I didn't succeed at all. I also thought about cyclic groups, but with such amount of information I haven't find the use of them. How can I prove it, where to start at least? Source of the problem: practice for exam made by my teacher in the university. Motivation is simple - I want to see more abstract algebra ideas since I don't even know how to start correctly.
As a non-native speaker I currently have a doubt about something. I am writing a technical note about units of measurements (kilogram, meter, second...) and physical quantities (mass, length, time) for software (and I am talking about abstractions in software architecture) and I am note entirely sure whether I should write : Unit and quantity abstractions Units and quantities abstractions when talking about "the software abstractions of units and quantities". Is only one of them grammatically correct? Or if both of them can be correct, what is the difference and which one should I pick?
Is there an adjective or shortish expression for an item that is "not known to be needed". Context: at the beginning of a process, all items are "unknown as to whether they will be needed or not", so all items are "X". As I progress through my process, at any point in time I may realize that some item is now or will be needed for sure. At that point, the item moves from "X" to "needed". Chat gives "discretionary" :) , good but not quite...
Magnetic field lines are loops, so at the surface of a magnet pole they look like this? there is a dead zone at the center where there is no lines/magnetic field present? Or lines converge in the middle to a single line or point? like so? Which picture is correct? also on some googled pictures I can see a single straight line coming from the center of the pole that goes nowhere... I know that lines are really an mathematical artifact, I just want to know how the field is shaped near the surface at middle of the magnetic pole.
I believe that Star Trek has advisors so that their science has at least a tenuous theoretical basis. The latest episode is based on interacting with an "improbability field". Is this just gibberish or is there some conceptual basis in quantum field theory? The only reference I can find is in this book that says which entails the process that allows for the development of Multidimensional Geometry through its use of Vibrations, the actual nature of Numbers and Math as a whole, the formation of Particles and Fields, and how Quantum Entanglement is instrumental to that
So I came across this game called "Getting over it". It is a game where you propell yourself upwards with a hammer, sticking it into various places to keep it in place. My question is, if you were to do the same thing in real life, how would the force applied by you onto the wall above you lift you up? You can check the game if you want to understand what I mean.
I am writing lecture notes for a course I am teaching. A Module on Matrix Computations involves a special mathematical expression and I have no clue how to typeset in LaTeX. I have enclosed an image of this calculation and I request your kind help how to typeset it. There is a special type of arrow in this image, and I have never seen it listed in the standard list of arrow commands in LaTeX. I thank you for the kind help!
Is Point O called the center of the half disk (informally, semicircle)? It's not the centroid since it's not at the center of mass. I'm not sure if it should be called the center if it lies on an edge of the shape and it's only the center of the full disk. I've never seen an official term for it for just a half disk. Does anyone have an answer for this?
I have a game in which pieces normally get tiled in an arrangement like this (which can be shortened or extended arbitrarily): But in special circumstances a piece may be permitted to be placed in between and on top of the other pieces, like this (red circle shows the one piece placed on the corner of four other pieces): I'd like to briefly describe this placement something like this: A red piece may be placed _____ in relation to normal pieces. What the best, short, clear way to describe this placement?
A pipe with a pump and two open ends is placed under water. When the pump is initially turned on, will the pressure at P_out and P_in initially rise/fall, but then stabilize as a flow of water V_flux develops (and the suction pressure is equal to the pushing pressure at all points)? The assumption is that the pressure from the pump first travels as a compression and rarefaction wave, and then once those meet and a flow forms, they cancel out to some extent.
Can anyone tell me what are the relevant pieces of information that are required to find the amplitude of a speaker membrane for a particular input ac signal frequency and voltage amplitude? Could you also show the method by which I can achieve that? Do I need to do an energy balance? Does that mean I would require to know the elastic modulus of the material of the membrane? Or is there any other predefined parameter of the speaker usually provided that I can use to get around that kind of calculation approach?
As Photons do not experience time or space, then according to my thought experiment, all photons must occupy some kind of singularity as well as what WE observe from earth. I was also thinking that the faster you travel the smaller the universe becomes and although you experience time, that time is dependent on the observer. I'm trying to get my head round this (I have no science background) It appears to me that the universe is at the big bang all the way to what we observe at the same instance
The text defines the torque of a particle system as being connected with the axis of rotation. "Taken two particles of masses m connected to a thin rod of negligible mass etc", but later when the text explains the torque due to gravity defines the torque of a body as the point of origin (I assume that the axis of rotation is orthogonal to that point, right?), but should I guess this body is connected to the axis of rotation or not? However, in the case of a torque due to gravity, how should I imagine point O? Because in the end the axis of rotation passes through the point O and belongs to the body, right?
It is clear that infinitesimal strains are obtained from finite strains (e.g. Green-Langragian or Eulerian-Almansi tensor) by removing nonlinear terms which smallness order is greater than that of linear terms. But it is not clear for me whether such a procedure is justified, if we bear in mind that finite strains are subject to differentiation when plugging their into the equilibrium equation. After all, the contribution of nonlinear terms in equilibrium equation can be significant, as the derivative of function can be much greater than function itself, even if that function is considered to be sufficiently small.
I found a metal rod lying on the floor and I found out that it sticks when brought closer to a steel sphere like a magnet, but, neither the sphere nor the rod is a magnet. More astonishing is that the sphere sticks to the rod to only a single side of the rod and not the other side as if it was a uni-directional magnet, but none of those two objects interact with any other metal objects and the rod and the sphere are not repelled by a permanent magnet either. Please can anybody explain this anomaly.
I recently came across this document which I am trying to recreate, and this is what I'm dealing with: I have no clue where to start in trying to replicate a header and footer like that, I new to LaTeX classes so I'm not really sure where to start. Youtube is pretty plain on this topic and Overleaf learn section doesn't help me much either. I really want to create a .cls file in order to adapt it to other related documentation... If anyone could point me in any direction it would be great, or if you already have some experience and easily recreate it yourself, that would be awesome! Thanks in advance!
I understand numbers to be defined as objects defined to have certain convenient properties in relation to certain operations. It is very surprising that the exact same group objects should be applicable to the modelling of such seemingly disparate concepts as length, area, degree, cardinality, time, wavefunction, etc. I also find it surprising that the qualities of numbers, which appear quite contrived, should correspond to anything real in the first place. How can I make the relationship between these concepts and the concept of number more intuitive and expected?
A signal in additive noise is sampled by a receiver with an unstable clock. Maybe the clock has absolute bounds on its frequency drift and sample time aperiodicity, or maybe the clock has some statistical characterization (maybe an Allan variance). Maybe the signal being sampled was also produced by such a system. Can anyone recommend any brief treatments on this situation's signal model, its spectral analysis, and efficient (or typical) signal processing approaches in this domain? I notice there is a brief Wikipedia article for "Coherence theory (optics)" which is closely related, if not more general.
For example, suppose a spaceship explodes. The sound of the explosion will travel in the expanding cloud of escaping gas. I gather that it should rapidly weaken and the pitch should drop, but I'd like to get more specific than that. Are there any models that suitably describe precisely this sort of situation? Additionally, is it feasible for there to be a relatively safe distance away from such an exploding spaceship where the expanding gas cloud isn't yet extremely rarified and an audible sound might actually be heard through your space helmet?
There are two facts about elementary row operations and column/row spaces: Column space is affected by elementary row operation, whereas row space is not. (the issue of finding a spanning set of column/row space) Linear dependence relations between columns are preserved over elementary row operations, whereas the same between rows is not. (the issue of finding linear independence between columns/rows) In finding the column space or row space, we lose and gain something in either case. I am not sure why this is possible intuitively: the issue of finding a span is linked to having linear independence between the vectors. Therefore, my question is a more conceptual one: why are the two above statements intuitively consistent (at least to an amateur like me, they seem opposing)?
What is the exact difference between a moment and a couple? In some YouTube channels and books, they say the moment of a force produces a translational as well as rotational motion whereas the concept of couple produces a pure rotation. How and why is it so? I am not convinced about this yet. Once you fix your point of axis or rotation axis. How come when you apply a force the body would make a translational motion?
My objective at this point is not to learn all and advanced mathematical proof concepts/techniques. At this point my objective is to get basic understanding of what is mathematical proof and learn only basic concepts/techniques behind them. Also, I need it to have as minimal prerequisites as possible. Best variant will be some article or tutorial or a small part of a big book. Can you please provide such material?
My goal is to implement a mathematically correct multivariate normal distribution using Sobol QRNG sequence as a source of randomness. The implementation should NOT produce the whole set of a given size at once, but instead it should reveal next values one by one, also depending on getting new values out of the Sobol sequence. So the algorithm should be streaming the data with minimum memory requirements. This guy claims he implemented MVN out of Sobol, which is officially endorsed on the MATLAB website. However, he has another call to rand() in the middle of his code, so it feels he does not fully understand what he is doing. That's why I hope for your help with a strictly mathematically and statistically valid approach here on Math.
For example, if you say New Orleans like "New Orleens" it's considered wrong, instead of "New Orlins" (Orleans is pronounced different in France though), or Iran like "I ran" is wrong, instead of "Ear on". But then people have names for other places all over, like Germany or Allemagne, instead of Deutschland, which would have their own pronunciation. An even closer example would be France, pronounced like "Fronce" for the locals, and closer to "Frants" for Americans. Maybe this is protested in France, I'm not sure. Or Greenwich, UK vs. USA (I wonder if each says the other should be pronounced like their own). But when does pronunciation of somewhere else become wrong instead of their own way of saying it?
I read that the standard depiction of covalent bonding is misleading. It gives the impression that the valence electron exists at that bond site. But really it exists as a wave throughout the crystal. So is it a standing wavefunction that has a peak at the bond site? From this can we infer that conduction electrons have propagating wavefunctions? Furthermore, how is it that an electron can mantain its wave characterictics in a crystal? We know that oberservations (more accurately interactions) collapse the wave function. Certainly the electron is interacting with many potentials and forces in a crystal.
Let us assume that the classical spacetime describes relative positions of mass particles, and the relative positions are in fact described by quantum entanglement. That is, the stronger the entanglement, the shorter the classical distance. If a group of mass particles are being pulled very close together by classical gravity, then can we argue that interactions behind entanglement become fuzzy and that infact weakens the entanglement, hence the classical distance can't be arbitrary small and gravitational singularities don't exist?
I study the magnetic trapping recently. I met a problem. I consider the original quadruple magnetic trap. In the original quadruple magnetic trap, the magnetic vanishes at the center. The Majorana flips happens at the center. If we want atoms move in the trap adiabatically, the Lamor frequency must be larger than the trapping frequency of the trap. What does the 'adiabatically' mean and how come is the condition? I saw this statement in my lecture note and other prof.'s lecture note but I have no idea why it is.
I have read the related discussion and some text and get some basic ideas. Geometrical interpretations of SVD Visualization of Singular Value decomposition of a Symmetric Matrix What I don't understand is why are we rotating twice? I imagine we only need to rotate once, and scale once? Is there any significance in the angle we rotate? One answer did say "the value of this is not too important as long as you pick a sensible number as the rotation matrices are not unique" which makes it more confusing to me ...
I am confused about the curve carried by the train track. I am using two references: Mosher's note (via arcs) and Johnson's blog (via bands). By Mosher's definition, I wonder if or not, in the following picture, the blue arc is carried by the train track determined by the red and green arc (we can make the blue arc arbitrarily close to the train track). Similarly, by Johnson's definition, I wonder if or not the blue arc is carried by the train track determined by the bands from the red and green arc.
We know that displacement is change in an object's position (here position means 'position vector'). Then velocity will be change in position of the object with respect to time, simply displacement/time taken. But I was taught that velocity is change in displacement with respect to time, then it means velocity will be equal to change of (change in object position =displacement) with respect to time. So, which one definition is correct?
I want to understand something: when someone measure the pressure, temperature and specific volume of a material, I think the pressure is applied in all directions of the material. This can be understandable for liquids and gases because the pressure in a closed container and for instance, you can used a piston to increase the pressure in this system. What about in a solid? the pressure or normal stress applied in this solid doesn't mean that the stress in other directions exists, so what kind of equipment is usually used to measure and control this thermodynamic property (oressure) ? Or is my assumption wrong and is not necessary that all direction of the body to be applied the same pressure?
I am not sure if I understand the energy band diagram correctly. In a pn junction situation below, before the p type material come into contact with the n type material as in figure a), an electron at the conduction band minimum in the p material has the same(or roughly the same) energy level as an electron at the conduction band minimum in the n material. Is it still true after they are fused together? If so, how come it is showing on the band diagram there is an energy difference as in figure b)?
Many textbooks on tensors start with a sentence like "An equation written in tensor form is valid in any coordinate system." The second sentence is often "Closely associated with tensor calculus is the index notation." According to me, this seems contradictory. The first sentence seems to state that coordinate systems should be avoided while the second one emphasizes the need of a coordinate system. How far can we go in tensor (calculus) without relying on index notation? Are there books specifically following this approach?
Consider no air resistance. If we throw a ball vertically upward from my point of view it goes vertically up and down and moving along straight line. But from point of view observer in outer space it is parabolic because of rotation of the earth. We know vertical acceleration/deceleration is due to earth gravity. But where component of velocity parallel to the Earth's surface come from to make parabolic path?
In the book Then I'm trying to understand the reasoning behind constructing the matrix, I get the unique scalars P_ij part, representing the one basis's vectors with another basis vector's linear combination. But then he uses same matrix for getting another basis's vectors. I'm confused about this kind of reasoning and it appears in other proofs too. How these scalars i's and j's satisfy these equations? At this part, he uses same matrix but using j's and i's at the same time. Sorry this question is not clear but in general i am confused about constructing matrix P, can someone explain the reasoning?
I thought about the question in the title for a ring that is not a Dedekind domain (here, the unique decomposition is a well-known theorem and also characterizes Dedekind domains!) but only a one-dimensional noetherian integral domain. The following is the motivation: In a Dedekind domain, every fractional ideal is invertible. Therefore, my idea is that actually the invertibility of such a fractional ideal makes a unique prime decomposition possible. Is this true? If yes, how could one proof this?
I'm going to study plasma characteristics by using Particle-In-Cell (PIC) method and I know that plasma has fluid behaviour and can also consider as particle behaviour also. To understand a basic of PIC simulation, I would like to understand an origin of the PIC that why it used differential equation to describe plasma properties from Eulerian method? and what is the difference between Lagrangian and Eulerian method for plasma physics?
When one says light comes on immediately when you turn on the switch of a flashlight cause the electric field is set up in the wire with a speed approaching the speed of light, what does exactly this sentence mean "electric field is set up in the wire with a speed approaching the speed of light" does the electric field travels from the source? If it doesn't how can it has a speed to begin with? Is there any sort of electromagnetic wave that originates upon turning the switch on that signals the farthest electron to move?
How to cut the square which tessellates to octagon using straightedge and compass? What are the exact measures of colored sides? What is the angle marked with red color? Edit (I added vertices): Edit. Acknowledgement after answers. Thanks to Jean Marie, Blue and Daniel Mathias for tackling the problem. I accepted Blue's answer for its breakthrough, but I also weighed Daniel Mathias' solution for its geometric simplicity. Initially, I didn't expect the problem to be so laborious. I invite you to my puzzle in a similar flavor of dissections, which after many years still has no answer nor proof that the solution does not exist. Variation of Haberdasher problem of Henry Dudeney - dissection of equilateral triangle into square with flipping pieces
What is the difference between the two sentences below? And what does the first sentence express exactly? Also, could you provide a grammatical analysis of the structure of the first sentence naming each of the parts of the sentence (which one is the subject/object, etc) and parts of speech (noun, verb, etc.)? We were here for a search of what the first sentence was in comparison with. We were here for a search of what we were comparing the first sentence to. P.S. We had an argument with a friend and we tried to present as objectively both sides as possible and refrain from showing any sides.