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.


Is the word multiperspectively (sometimes polyperspectively) only used to describe literature, as in narrative perspectives? Could I use it to describe a personal trait? For example: In debates people despise that I multiperspectively argue.... I did see a post asking what the word was for "seeing from multiple points of view" and this word didn't come up. Also, I understand this is a serious English site, my question is not for any scholarly reason. I am trying to write to a person and I got stuck while trying to find an adverb that means this.


I have a Kalman filter that uses location as part of its internal state. This location is expressed as x, y coordinates in meters north and west from a fixed point (the home location). Part of my measurements for the filter are GPS coordinates. Can I translate the measurements back to the state space using the H matrix? The range of movement for the tracked object is small enough that I can use a fixed multiplier to scale the longitude to meters. But how do I add the transpose to the matrix, so that the GPS coordinates use the home location as the origin?


I am self studying algebraic topology, the fundamental groups and covering spaces and homologous groups. I have read the topological properties of classical matrix groups like connectedness, path connectedness and compactness. Now I want to study the algebraic topological properties of these groups. In particular what are the fundmental groups of patnconnected components of the matrix groups? Does there exists any relation among the matrix groups via covering spaces? If anyone provide some referances regarding these questions that will be very much helpful. Thanks in advance.


When an electromagnetic wave interacts with a free electron the electron starts to oscillate in the direction perpendicular to the propagation of the wave meanwhile when a photon interacts with a free electron what will the electron do? Will it oscillate in the direction perpendicular to the momentum of the photon? or will it travel in the direction of the momentum of photon? How can someone describe this interaction in photons form so that the electrons oscillates?


In present tense or past tense, subject and verb can be omitted after conjunction. For example People tend to laugh when (they are) imagining pleasant things When (I was) going home, I met my father Then could it be possible to omit subject and verb in case of the past perfect tense as well? For example, As it had been written in a hurry, the report showed many errors. As written in a hurry, the report showed many errors. Or, because this case is of the past perfect tense is changing this way only correct? (Having been) written in a hurry (without conjunction 'as'), the report showed many errors.


I live in this region and is studying to retake the national test and I need a big amount of tough exercises to practice but I can't find good documents/ textbooks over here. Any appropriate textbooks help me greatly. The topics are: calculus, algebra (especially logarithm), complex numbers, Oxyz algebra(?) (I really don't know what to call this one. It includes the equations of lines, planes, spheres, etc. And vector formulas. No matrices.). Thanks for your help.


From the perspective of an electron in the double slit experiment, does the "rest of the universe" that it is disconnected from behave in the same manner as the electron does in our view? As in, is the world around it, from the electrons point of view, as a closed, separate system to be interpreted the way we see the electron? Is the effect relative? From the electrons perspective, could it be in a "collapsed"-state and consider only the universe around it to be in a superposition? I hope the question makes any sense, because I am really curious and a little confused about the whole thing. Thanks in advance!


I was reading Lazarsfeld's Positivity in Algebraic Geometry, and I realize that Cartier divisor can have torsion elements, but I can't find such an example. I know for the Noetherian scheme that is locally factorial, the Cartier divisor is isomorphic to Weil Divisor and we know Weil Divisor group is free therefore it's torsion-free. So this can only happen when the Weil Divisor do not isomorphic to Cartier divisor is there any example that the Cartier divisor has a torsion element.


Okay, apart from the American slang term, or the reference to a male chicken, what is the meaning of the syllable "cock"? For example, shuttlecock ... in badminton. Or, as I'm doing housework and learning about plumbing, a "ballcock" valve in a toilet, or a "sillcock" valve in pipes? Does it refer to a specific shape, or a mechanism or something? Which might make sense for the valves, but then what about the shuttlecock?


I can't decide what I prefer - having empty space at the end of a page or a split box. For example: Should I be aiming to make sure the page is full with no need to break boxes? Like, I can add more text I probably don't need further up the page so that the last box starts on a new page, is this something people do for aesthetics? Or should I be happy enough with a break like this? Or the empty space?


Consider the small weight is attached to a big diameter wheel and the big weight is attached to a smaller diameter cilinder. I've read that if you ignore friction and all the forces are balanced perfectly, you could consider the possibility of this machine lowering the lighter weight through the distance of one wheel circumference at uniform speed while the heavier weight is raised by the smaller distance of one circumference of the cylinder. In this situation, what would be the forces that balance out the differences in the gravitational forces that act on each mass?


I know the restatable environment from the packages thm-restate combined with thmtools, which allows to formulate theorems that can be restated later in the document. Now I am wondering, if there is a similar solution for equations like equation, align, etc.. I have already tried to just reuse that restatable environment for equations, but as expected I got an error. I have already found "solutions" that suggestion to copy the content and tag the reference number. However, this seems to be a stopgap.


I would really like to create these pictures to help with fractions. I know how to begin using tikz package. I am not sure how to go about setting it up so that I do not have to draw it separately each time. I would love to show something that I have started with but have no idea. And what if I wanted to make the rectangle horizontal and not vertical? Thank you so much for any responses!


Suppose that I have two electrons in a box that were originally spread by a infinite potential wall. The two electrons are in their ground state wave function (sine function). Now I removed the barrier for the two electrons to interact. How is the Pauli antisymmetrical pair formed? Do I have to impose them to become the antisymmetrical pair in the beginning, or can I derive the antisymmetrical form from the original sine functions?


If I were to have asked this question about local systems, it would be extremely easy to give examples from physics/biology, etc. I can give example if people want, but this should be easy to Google. Question: What examples of perverse sheaves naturally show up in nature? More specifically, I mean examples which are not local systems, and the example only applies to perverse sheaves (or constructible complexes of sheaves) and not sheaves of vector spaces in general.


I am not sure why the following configuration can not be a configuration of a electromagnetic wave. I mean, i am sure it can not be the configuration of the fields in a plane perpendicular to a wave, but i can't explain the reason in a satisfactory way. Look, it seems there are souces propagating with it, since these configurations are the configurations of a point electri ccharge and a infinite line with current. But i would like i better way to explain why we can't have it.


At a weight m, a plane is acceleration upwards at rate of a. We also remember the value of g. From my understanding, we have two opposite forces that we care about. The force due the gravitational acceleration, which points in the negative y-direction, The force of the upwards acceleration, which points in the positive y-direction. The first force must be equal to m times g, while the second would be equal to m times a. However, that is not the case with the upwards acceleration. My physics book gives this reasoning for calculating the upwards force: I don't understand why we would add g, since it points in the opposite direction?


The question is simple. Could analog computers (for example, electrical analog computers) in effect avoid traditional numerical instability problems that come from solving a set of PDEs over a discretized domain? Progress on significant physical problems (the computational aspects at least) are often substantially delayed because of time spent figuring out how to avoiding numerical instabilities, and often the solution ultimately utilized is just to hit the problem with lots of artificial diffusion, which seems unrigorous at best. Also, since many physical theories are often significant approximations themselves, maybe some accuracy could be sacrificed (in moving to analog computers) for the greater benefit of avoiding numerical instabilities.


To my understanding, normal reaction is a force that pushes in the opposite direction of gravity, in order to keep the object from falling, or phasing through the surface. Though, what happens when we enter a circular rail? In this case, the guide book is saying that the normal reaction is pushing alongside gravity. Then, shouldn't it fall down? I thought the normal reaction was supposed to be pushing upwards in this case. If not, then is the normal reaction always bound to the center of the circle? I can provide the exercise with the answers, in case there's not much details in there.


A grad school problem has been haunting me for years and I have never been able to find it on the google machine nor was the professor who assigned the problem able to describe how to actually solve the problem... How do you find the surface area of an ellipse projected onto a sphere? The real world application is what is the footprint area of a satellite transmission signal so you know how long a ground station will be in contact with the satellite?


Say a spaceship is traveling at a certain velocity v (>>c) and it emits light from the nose of my spaceship in the direction of travel. The speed of light is finite and hence there should be a change of wavelength of light. How does the speed affect the momentum transfer does my light have lower momentum thanks to the direction in which it has been emitted?


My math professor told us that the upper and lower bounds theorem of polynomials (pictured) applies not only to rational roots, but also to the modulus of complex zeroes. They were unable to come up with an explanation for why this works, and I do not grasp this concept at all. Why/how does the bounds theorem apply just as well to to complex zeroes as it does to rational zeroes? How does the modulus come into play at all for that matter? Bounds Theorem


I heard that if Universal Expansion continues at some point galaxies will be so separated that a future civilization would have no way to know there are other galaxies, for them the Galaxy they live in is alone in the universe. (I'm planning to write a sci-fi story about it). Thus more or less in what time frame would that expansion would cause that effect? Thank you in advance.


I am new to it, so please help. As I know, AC current is back and forth motion of electrons, which radiates electromagnetic waves, so there is energy radiation from power line to our home. But in case of DC, if the magnetic field and electric field are static then how does energy radiate. I have searched that drift velocity of electron is slow, so DC flows at the same speed as of AC which is C. So, how does this happen if there is no radiation of DC? Please answer.


I have a body on a horizontal surface with friction. My thermodynamics system is only the body and I want to describe this with the first law of Thermodynamics. Suppose the body has a start kinetic energy and for the friction in the final state it has no more kinetic energy. For this problem there isn't a heat exchange but only a work exchange caused from the friction. How can i write the fist law with this system?


I have Two parallel lines, I want to create from one line to another using a cubic Bezier curve (S shaped basically), I'm free to pick any start point and end point. But the constraint is this path should be optimally shortest, also another constraint is I don't want the curves with too much curvature, I just want to switch to the other parallel line smoothly without creating too much curvy path. My question is how to pick start and end point? for having such a curve? I'm considering Bezier curve would be best for this with my very little research Thanks :)


I was thinking about the definition of reversibility but there is something that I cannot understand. In real life the surrounding of a system (anything else in the universe) is always changing making impossible a reversible process even if the process is slow enough to guaranteed the lack of friction heat loss and this surrounding are not in thermodynamic equilibrium so I cannot understand how people calculated entropy under such conditions, am I wrong in the assumption of choosing the surrounding as everything outside the system or it is enough to choose the immediate surrounding that is in thermodynamic equilibrium?


My question is whether there are some books on chess mathematics with theorems and their proofs concerning winning strategies for classes of initial positions. Despite there being a lot of books on chess theory and surveys on the use of AI in chess, it seems there is no purely mathematic chess theory that may use the results of game or group theory. If this is the case, then someone has to start it (despite obvious combinatorial difficulties).


What is the best way (as in most efficient) way of writing pseudocode like below? In particular I like that it is well-structured but gives enough freedom in writing (in reality it it is just standard inline text aligned in a specific way). If a library was used I would be grateful if anyone can spot if. In case there isn't such library, how can I achieve the result?


The following graph is a sensor measurement. Scatter plot The data points are also available in this excel file. I want to do some offset correction/scaling to make the measured points match the actual points. What is the simplest algorithm to do computationally by programming language like Python or MATLAB. These data are acquired in MATLAB continuously point by point, and I have to apply these offset to each point. Will something like finding a linear fit for all possible points and multiplying it with a polynomial work? or if there are any better methods, please suggest.


I am wondering if a vertex can lie on an edge in a planar graph- I am not sure if an edge of this vertex is regarded as crossing the edge on which the vertex lies. I have two questions here: Is the actual edge from the vertex tangent to the edge on which the vertex lies? Even if it only is tangent, does it still fit the definition of a planar graph that edges don't cross- as crossing would mean it would have to go through the edge and out of the edge, rather than just touching it at a single point? Is a vertex simply a point and an edge a line segment in the sense in which these terms are defined in plane geometry?


I'm not scientifically adept, so bear with me. Would it be feasible to retain enough fuel on new satellites/space stations, etc., that we could dispose of them when obsolete by sending them to the sun where they would likely incinerate before they came close to the "surface." How would we ensure burn up rather orbiting the sun? Already in earth orbit, how much speed would be needed to "hit" the sun? Impractical because of fuel needs? We could use the space station as an example. It would seem to me that eventually the gravity of the Sun would pull it lessening the need for fuel, it's getting way from the rest of solar system's gravities that would require fuel.


according to my book the perpendicular component of reaction force is called normal force when there is contact between two bodies. I can not understand that how for example when we jump we pushes the ground downward(action) and the ground pushes us upward(reaction) how this reaction is normal force? or for example if someone jumps to the shore from a boat, the boat moves in the opposite direction we push boat in backward direction and boat pushes us in forward direction(reaction) is this also normal force or friction force plz someone explain i dont understand.


Countable atomic structures are homogeneous, i.e., all finite tuple of the same first-order type are conjugate under automorphisms. While this is classical, I can't think of an example of a countable structure whose homogeneity is best proved via atomicity. Competing proof strategies include construction of that structure as a Fraisse limit possibly in an expanded language. Is there a countable structure whose homogeneity is proved via atomicity in a manner at least as good (whatever that means) as via Fraisse theory?


here is a vague question: I recognized that adjoints of operators and commutants of non-degenerate self-adjoint algebras share similar features, e.g. the double adjoint of a closable, densely defined operator is the closure of this operator. The double commutant of an non-degenerate self-adjoint algebra is the strong/weak closure of it (double commutant theorem). It seems to me, that this property of double(something) implies closure is a recurring feature and I was wondering whether they are somehow related in a maybe category-theory kind of sense? Thanks


When is a category of sets and spans just a category of sets and kleisli morphisms? There is a link between spans and kleisli morphisms, but I feel like they are the same things. I guess I am assuming that the base category for the monad is Set. Is this only true for the multiset monad? To see how we can have a category of sets and spans, just see the nlab article here:


We know that a pair of scissors is hard to use at maximum opening angle, because the force from the blades pushes the object away from the scissors and the actual cutting force is reduced. If we would design a pair of (simple, two-piece, with one point of rotation) scissors whose blades have a limited maximum incident angle, or even better, a constant incident angle, then what shape does it look like? Does it have a mathematical form? image source amazon.com


If the large space voids between galaxies had uniformly-distributed "negative" gravitational lensing, would its presence be obvious from photos in the same way that the presence of Einstein crosses and Einstein rings make clear the existence of positive gravitational lensing? Or would it only cause subtle distortions, such as reducing the angular diameters of distant galaxies as a function of their distance from us or by distorting how light transitions into and out of a positive gravitational lens? If it would only cause the subtle effects (not obvious ones involving the inversion of images), is it possible for this effect to exist given what Hubble, Planck, etc. has received?


I have a question about article usage. I have two sentences and I'm struggling with the articles in the second sentence: There may be invoices on each table. When (?) invoices lying on (?) table disappear, it's a problem. I would guess the invoices and a table. My reasoning is that I'm not talking about specific table so it should be indefinite article. But definite article for invoices because we talk about specific invoices on that table.


If we get magnetic field lines on a piece of cardboard, shouldn't it form a gradient of metal pieces across the board (thick layer of metal pieces near the magnet and thin layer of metal away from magnet) rather than forming lines of metal pieces leaving crevices between them. Like in the red areas it is somewhat evenly distributed on board. But in the blue area they forms pattern of lines. Why do they forms lines? Why don't they form a gradient of metal pieces? According to me there are infinite field lines, and only the magnitude of magnetic field decreases as we go away from the magnet.


Lately I've encountered a few situations in discussions where I feel like there may be a word that is either more succinct and/or perhaps more wry than just 'ethically convenient'. An example sentence would be: You can only say you wont be involved in unethical thing because it so happens that you cant be because reasons unrelated to your ethics. You are being/You are _______. Basically something just fell on your lap with no personal effort, but you then take advantage of that to further your moral standing. Often it is done willingly, but it could be done with less awareness, perhaps through a form of delusion. 'Casuistic' from the answer here is interesting, but not quite right.


A dude has the power to split a nucleus. But this nucleus doesn't hit anything, it just explodes, and that's that. He explodes the atom. He can let the nucleus explode, or he can absorb it. I was making this character thinking he could just split an nucleus, and it would make a nuke-level explosion, and then he could absorb the power of a nuke. But is ONE SINGULAR nucleus splitting, with NO help whatsoever, would that ONE singular nucleus explode with the force of a nuke?


What bugs me the most is that since there is net charge, i dont know two things, how would that charge distribute across the surface (even though my intuition says that one of the sides will have more charge than the other) and how would one calculate the dipole moment since the distribution wouldn't be symmetrical about the X-axis (in this problem the equator of the sphere), i've found the potential around it via legendre polynomials which i guess could help, but i just... dont really have a good idea of what to do neither the confidence


To clarify with a made-up example: When you address the nation state ... Above, 'state' could be taken as either a part of an open compound noun ('nation state', see footnote), or a verb ('to state'). As a non-native speaker I notice that such ambiguous interpretations do briefly hinder my reading flow, as I have to retract a bit. Also, similar ambiguous cases don't seem that rare. Hence, must, or at least may, one use a comma to disambiguate such cases? In the running example, is this allowed (or even required)? When you address the nation, state ....


In piezoelectricity, the piezoelectric tensor is of rank three and its a reason why piezoelectricity is not present in all materials whereas for electrostriction, it is present for all crystal symmetries because the electrostriction tensor is of rank four. So, I think the presence of the electromechanical phenomenon is related to the rank of the related tensor because the rank is related to the crystal structure ?


I've always curious about one interesting fact. I will explain it with example. Currently, I'm watching an example which shows the following picture and says that box at the left is the speaker and it produces a sound wave. I wonder why it's being drawn like the above. since it's a longitudinal wave, which actually doesn't have the shape like that. Since particles move in the same direction as wave, their motion is right and left back-and-forth. On the graph though, I get why they end up drawing sin/cos functions, but in the real motion, they don't have the shape as on the above picture. Transverse waves do though which I get why. Thoughts?


I am learning multivariate regression, with matrix equations and heavy use of: Transposes Inverses Identity, idempotent matrices Can anyone refer some books where I can learn the properties of matrix equations. Practice problems would be a great help. (I looked up some standard books on Linear Algebra - for example, Anton - but the topics are different from what I need. For example, Anton has Vector spaces, subspaces, eigenvectors, which is not really what I need).


I have a question about the potential and wave functions of discrete energy levels of quantum point contacts. Assuming one has a rectengular potential well: Can the potential of the QPC be assumed to be infinite or is it finite and the charge carriers also have a probability of being outside the potential well? Is it also necessary for the wave functions to drop to zero at the edges so that the quantized conductance can be observed?


On looking upon the process of penetration of radiation. I came upon this: "If the particle has enough energy, it can deposit its energy to break the molecular bonds displacing the electrons, hence it can penetrate the object. So, if the same thing is done in a large scale with higher concentration of radioactive particle hitting an object per unit area, would it be possible to break nearly every single molecular bonds, and resulting into the object breaking apart in dust?


I'd just like to enquire on how this sort of chapter style is done. I am aware it is some permutation of the memoir class with hansen style chapter headings, however the number seems to be in sans-serif, the text in CMU serif, and the document to be in a different font to what I typically find on hansen templates. If anyone could replicate this for me, it would be greatly appreciated.


Is the wire in the figure below homotopic to a wire between the two points that just go straight above the pipe? My knowledge of topology is very basic, and it seems to me that the pipe can be seen as some kind of hole, so the wire seems to really be impossible to untie. However, there is a youtube video that seems to untie it. Is it a trick or is my intuition wrong, and if so is it easy and evident to tell why they are homotopic without having to watch the video? Or something else that I am misunderstanding? enter image description here


In the Wikipedia entry associated with the Kahn-Kalai conjecture, there is the following assertion This conjecture concerns the general problem of estimating when phase transitions occur in systems. As this entry notes, the conjecture (technically "theorem", as is recently proved) is in the field of graph theory and statistical mechanics I am particularly interested in the consequences of this conjecture in view of statistical mechanics and (even if possible) its implications and applications to formation of complex networks in engineering fields. The main article, as far as I can see, heavily focuses on graph-theoretic facets of the conjecture. So, is there any applied exposition of this conjecture?


For example: Should the free market then only capture that which is beyond necessity? I'm not sure if "then" is accomplishing anything, or if it is even correct in this sentence. Perhaps it should be written more like Should the free market only capture that which is beyond necessity? But this sentence doesn't sound like it is referencing any contextual information. Perhaps this Then should the free market only capture that which is beyond necessity?


So some authors consider (sigma additive) signed measures on algebras (that are closed only under finite unions) instead of on sigma algebras. This happens for instance when considering the subset of finitely additive signed measures that are sigma additive. So my questions is what can I do with these a sigma additive measure on an algebra? I.e., do the usual convergence theorems hold? What kind of restrictions arise?


When firing single photons and letting and interference pattern occur. If you were to draw a straight line from one of the slits to a particular interference gap on the screen, and continued this line through the screen, when you moved the screen to different distances from the slits, would an interference gap be present along the lines trajectory at all screen distances? Or would the interference alternative between gap/no gap along the lines trajectory at different screen distances?


I'm using the FrontiersinHavard Latex class provided by the author guidelines of the Frontiers Media. Strangely enough, even though their guidelines specifically state table captions should be placed above the table, the class forces the captions under the tables. There is also little to no documentation on that document class itself. Are there any options to disable this behavior when declaring the document class, or does one need some other workaround to override the document class?


When water molecule is cooled, it is well known that it expands: That's the reason the rocks in the desert explode in the cold of the night: the rocks have water inside that expands with cold and breaks the rock. Other molecules compress when cooled. Could this effect (expansion/contraction when heated) be used to moved an electron as a POC of doing work? For example, water is a polar molecule, so it's mere expansion/contraction seems to point that it can move electrons. Given there are random temperature fluctuations everywhere, it seems possible to use these micro contractions/expansions. Does this break any known rules? If we manage to use a diode equivalent, it would move electrons in only one direction.


We consider an electron "Bob" trying to escape from a metal surface as shown (blue circles represent electrons, red circles represent ions). I know that Bob can't leave the metal surface is because the attraction due to the ions is greater than the repulsion due to the electrons below. My question is: why F(attraction, by ions near the surface) is greater than F(repulse, by electrons)? Thank you!


Suppose I have N unique objects, where N is known, but I have a subsample of only M of them in a bag, where M is unknown. I draw X from the bag with replacement (where X is known) and observe A unique objects (where A is known). If I next draw Y from the bag with replacement (where Y is known) what is the distribution for B, where B is the number of unique objects I will observe?


In one of our class discussions about the origins of the word Bank (Money), a guy guessed that maybe it comes from the Bank (the land alongside a river) since the sand gets deposited there, as an equivalent to Bank being a money depositing place. It felt right, except Bank comes from Banque, a Latin word for table, as that's how the financial transactions used to happen, over a table. Now, what I want to know is could it be that Bank and Bank are related? One coming from the other? If so, which one came first?


Is there any difference between an axiom and a postulate? I would say that while an axiom is any proposition fixed as true without premises in a theory, a postulate is (also according with its etymology postulatum = something that is requested) something that is necessary to assume to make a theory fit (also in a consequential way) with some kind of experimental measurement. Is there any convention about it?


Although inertial forces are just our mathematical creation to help apply Newton's law even in non-inertial frames, can we assume or is it (mathematical proof) that inertial forces also create its field. I think it does so, because when we say that we cannot distinguish between gravity and accelerated motion, we assume them to be same. Is it just as assumption or a correct assumption?


Im currently a math masters student. Im doing good and i obtain good grades. Im interested to pursue PHD. At a masters level course i have never been able to solve all exercises which a lecturer has given. This really bothers me and i feel like i need to solve every single exercise problem easily if i want to pursue PHD since these problems in most cases have known solution, hence how i should be able to do research in mathematics if im not able to solve these known problem? My way of thinking might be too black and white in this sense. So my question: people who have completed a phd or a fellow masters student; were you able to solve every single exercise at classes?


I know that if a wire is carrying current, the force is F= iBl sinc But what if the wire does not have current, and rather the magnetic field around it varies. Will there still be a force? If so, what will be it's magnitude. You may consider magnetic field direction as required to get a value of force if possible. Note: This is not repeated question. Other similar questions are regarding wires carrying current. Mine is for wires with no current.


Why are terrestrial planets like the Earth so much smaller in size than the gas giants? Why can't they be the larger ones? Why are there no stars made up of something solid(/rocky)?(To generate energy, it could maybe use fission in place of fusion)But anyways, Is it because that would violate some laws of physics or mathematics or is it just the way the universe formed? If it is due to the former then what fundamental laws does it break?


Is there a general book of mathematics which starts from first order logic, treats set theory (in ZF setting) in which first order logic is used, then abstract algebra (in which set theory is heavily used), topology (at least point set topology) and finally mathematical analysis (in which all the previous concepts are used). In all these parts most important results should be covered. In this way one can study mathematical analysis having solid foundations of mathematics? Am I dreaming or a book of this kind exists?


I know from calculus that Riemann Integrals are Riemann sums. We use such integrals to calculate areas under curves. But why does adding an infinite amount of small terms give a finite result for non divergent curves? I know the obvious answer would be to just look at the graph of the function and we can see there is a finite area. But I keep thinking of the Harmonic series and how adding an infinite amount of smaller and smaller terms has as a result infinity. How can we be sure the small terms if the Riemann sum also don't add up to infinity?


I have read yesterday the following statement: An initial object when it exists is the colimit of the empty diagram. A terminal object when it exists is the limit of the empty diagram. And I found some justification of the second statement here: What means an empty diagram? but still the justification is not very clear to me, I am wondering if someone can explain this to me in a simpler way please?


Using a magnifying lens with the sun as the source, one can focus the light in a single point. Is it possible to use just lenses, or any number of transparent materials of any particular shapes, to focus sunlight into a line and not just a point, and turning it into what I think is called a "focused beam of light"? Is it possible to do with just transparent materials?


In a recent chat conversation, I asked someone quite high up in my global organization but not belonging to my regional org setup - "Shall I send an email copying my supervisor", and he replied in the chat window with a "I am good". What does it mean? I wanted an opportunity to introduce my supervisor to him through email, but what should I make of his response? Is he asking me to send the email (OR) is he asking me not to send the email?


According to several dictionaries I have, "validate" is a transitive verb. But both Grammarly and ChatGPT judge the following (imperative) sentence as correct: Ensure feature quality by validating with the UI team before release. I would use: Ensure feature quality by validating it with the UI team before release. Is the sentence without it correct? If yes, is it because the verb can be intransitive or because of something else? Edit: corrected them to it.


It is well known that the electric field by a ring on a point charge can be easily calculated by integrating the each infinitesimal electric field by each infinitesimal charge. However if one were to attempt to calculate the contrary, that is, electric field exerted by a point charge on a ring, how can one integrate each infinitesimal electric field by the point charge on the ring, since unlike the former, the electric field in the latter case is not generated by an infinitesimal charge, rather by a well defined, let's consider, Q charge?


After learning about progressive waves, I struggle to understand what physically causes particles to move up and down in such a wave. What mechanism is actually transferring energy as these particles move up and down? Only longitudinal waves seem intuitive to me. Furthermore, consider water waves. Do we still consider this transverse, even though the particles move in circles rather than up and down. What causes this movement? Is it true water can only be transversal due to the attraction and interactions between water molecules?


In all the textbooks that I've seen complex numbers were presented as a given (from history - Cardano's formula etc.). Fundamental theorem of algebra, Frobenius theorem and others use complex numbers as some answer. But is there a way to deduce complex numbers constructively from some (algebraic) reasons without any knowledge about its existence? I'm looking for questions where the only best answer would be complex numbers. Something like "what is algebraic closure of the field of real numbers?", but without any a priory reference to complex numbers. Maybe someone can tell me where I can read about this?


I'm currently learning Latex and using Overleaf plus documentation. Sometimes I want to quickly see what other options a package has to offer. Since Overleaf shows a PDF preview, I would like to know if there is a way to quickly list that package's options. Basically what I'm looking for is a page like TeX_primitives_listed_by_TeX_engine that shows each primitive but with a link that explains watch each one does. Like the docs for any programming language, for example.


Imagine if you have a molecule with a large size, e.g. proteins. Anything that rotates experiences pseudoforces if my understanding is correct (e.g. coriolis, centrifugal etc). Since every molecule rotates, does this have any effect on these molecules that we could measure? Or properties that arise because of these pseudoforces? My guess would be more smeared out electron clouds, or weaker covalent bonds or something. Or a clustering of more heavy atoms in the centre point since farther from the centre would mean weaker bonds. My guess could also be that the effect is so small that it is negligble. Probably only really big molecules that spin very fast could have an effect (if any).


I hope I can ask this question here. I have downloaded Asymptote from windows-binaries and want to run an asy-program that contains the row (it has worked a long time ago) import gsl; But that seems not to be included in the distribution. The manual says something about If asymptote is compiled with... you have access to the gsl-library. Obviously the version I downloaded from the webpage is not compiled with gsl. How do I find a version that is, och how do I add it to the version I have? -anders


I'd like to achieve the following using Latex: I'm open to still-better ways of formatting such block matrices. But the above is what I'm thinking towards, right now. I'll be doing a lot of these and I'd like to beautify the stuff I've been posting before. Especially since directed graphs, column and row spaces, the left and right nullspaces, modified network analysis, and the Schur complement method can figure prominently in analysis. I'm not very familiar with using tikz packages. But I'll learn more, if needed. The following is an example of the above that I do not want to produce: It's sad-looking and I'm hoping for something significantly better, more in the direction of the first image above.


We know that inside solenoid the magnetic field are same at all point's and are parallel to each other My question is why they are parallel as my teacher explained it as solenoid is set of multiple circular loops and one face coil form North pole while other will form South pole and magnetic field lines from both poles will join in parallel I think this explanation is not correct Any response would be appreciated


A long rectangular cardboard thrown in air starts spinning automatically on its way down. Few observations in this experiment: Long rectangular cardboard pieces spin well Doesn't work well with square / circular pieces Long ellipsoid pieces seems to spins the most Also note the spin axis on the way down. Even if you give a little torque to it spin it vertically upwards, eventually on way down the spin stabilizes so the body starts spinning around the horizontal axis This doesn't work with paper / foam as they start floating. Higher density materials like iron / glass of the same size/shape also don't spin A small demonstration can be seen in this video. Can someone explain the exact physics/aerodynamics behind this?


In a system, the total energy is composed of Electronic >> Vibrational > Rotational >> Translational. As far as I understand, temperature is only related to the average kinetic energy, which includes Vibrational, Rotational, and Translational components and not electronic energy (excitement states)?? If theoretically, so much energy is added to the system that initially, the temperature increases due to these first three degrees of freedom, but then more and more electronic excited states are populated because the molecules kinetic energy provides enough that electronic excitation occurs, which, of course, takes up that kinetic energy, wouldn't this lead to brief drops in temperature with energy input?


I wonder about the Hmiltonian form fo the Potts model and would like to know about its matrix form using the generalized Pauli matrix. I believe it might be possible in a smiliar way applied to the Ising model, as you can see in the website below: How can I explicitly express the Ising Hamiltonian in matrix form? I'm not sure if it's just changing the Pauli matrix into generalized form from the Ising model.


I try to apply the replica trick in a two-dimensional gravitational model. I interesting for the case where the semiclassical geometry is an evaporating black hole. These geometries have no time-reflection symmetry. In many related articles they say that the geometries must have time-reflection symmetry, but I can't find why. It seems to me that the method can generalize to geometries with no time-reflection symmetry.


I am organizing an algorithmic lecture at my university, which is more theoretical. Regarding Landau-notation and runtime analysis, using algorithms that implement mathematical formulas is great. In my opinion, it helps the students see the connection. Of course, I also want some difficult exercises, and I am searching for mathematical functions that have a very algorithimal touch, if you know what I mean. Because it is hard to describe, here are two examples of such functions: nth prime number and nth Dedekind number. Can you provide me with similar formulas?


Force acting on person when jumping from boat is denoted by red colour and force acting on boat is denoted by yellow colour. I have not included the buoyant force and boat weight force on the boat. Coming back to my question:- The force applied by the person on the boat(F)can I Say it friction force which moves the boat in backwards direction and in reaction force acts on person feet can also be called friction force?? Or it some other kind of force like muscular force which moves the boat in backwards and person in forward??


The force repulsing two protons never goes away, meaning that there is a constant force pushing the protons apart forever. Firstly, where are these protons getting the energy to constant apply this force, and where is the nucleus getting the energy to continue to hold them together? Is it the binding energy? If so does it eventually run out of energy to hold them together?


I'm writing an academic article title, and, for matter of capitalizations, I would like to know: if "-based" is appended to a word, is "-based" treated as another word and should be written with capitalized "B", or is "-based" treated like a suffix that became one with the word, and thus is written with a non-capitalized "b"? Title example: "Computer-Based Strategies For Asking About Title Capitalization On Stack Exchange"; "Computer-based Strategies For Asking About Title Capitalization On Stack Exchange".


I need a reference for this elementary result which I assume to be true and known. Adding a citation in the paper I'm writing (even to a problem in a textbook) would be faster and more elegant than adding the proof itself (which may elude my decaying brain): Given a finite set S of real closed intervals, if each pair of intervals has an overlap (intersection) of length at least k, then the intersection of all the intervals in S has length at least k. Thanks.


Which could exhume a response like, "Lul, nice term for this type of pun subgenre." A pun is the humorous use of a word or phrase so as to emphasize or suggest its different meanings or applications, or the use of words that are alike or nearly alike in sound but different in meaning; a play on words. But I'm not sure of this type of subgenre. Edit: If it's not a pun (the jury seems out on whether it counts, I didn't think it was myself until I saw it included any play on words), an answer on what type of joke it is would then suffice.


How do we define radius in quantum mechanics. A particle has wave-particle duality, so do we define the radius as Compton wavelength or Bohr radius or some sort of superposition of both? I don't think that either of it works, because of the Heisenberg uncertainity principle, the Bohr radius would vary because the electron shell is probabilistic and requires a wave function. Also we have defined classical electron radius, but as the name says it is classical rather than quantum mechanical. I was curious about this because as I was estimating the Kugilblitz Swarzchild Radius that is required to produce a Micro Kugilblitz with LHC at it's maximum energy, I got the result but I was unable to find the particle at the same or below size Thanks!


I've been having a little bit of a debate with one of my friends about the correct use of a word. I hope the native English speakers can help me put an end to this :) I know the word "bold" is mostly used as in brave and courageous, but is it right to use it as prominent, too? This aspect of this issue seems bolder than the rest. Can it be used like this or is it meaningless?


I have seen in this link that there is a way to calculate the rolled material length and the number of turns based on: the material thickness inner diameter of the roll external diameter of the roll Is there a way to calculate the length of a material roll (in my case fabric) without knowing its thickness but with knowing its number of turns? So calculate the length by knowing: the number of roll turns inner diameter of the roll external diameter of the roll


"Norton's Dome is a thought experiment that exhibits a non-deterministic system within the bounds of Newtonian mechanics. " A ball rolled to the top can reach it in finite time with zero energy at the end, which means if you time reverse it the ball appears to star moving for no reason. By don't use just use an inverted cone with a point at the top? Whouldn't this be much simpler?


We see only stars whose light is not older than the age of universe. This is understood. If a star is too far away that currently we don't see it then possibly we will see it later, then the age of the universe will grow. What I don't understand is how a star that we see now can move away out of visible universe in future.


I have created some otf files of fonts for symbols and I'd like to write a package from them and then upload it to CTAN. But I have no experience in writing a package. So is there any reference to write a package from otf files? Please give some advice on the references which should be very basic. Specifically, with otf files of some fonts for symbols in hand, how to write a package like MnSymbol?


I'm a beginner in the subject & my question can be meaningless, so I'm sorry from start if that's the case. I just don't understand why all of the image f(L) can be a sublattice of L' when f is a homomorphism from L to L'. I learned that all subsets of a lattice don't have to be a sublattice at all. If I don't understand wrong, it's not about the elements of the lattices actually but about the operator differences between them. So why there is not a possibility of a non-inherited operator of f(L), or a non-sublattice lattice of the L' as f(L)?


Like the multiple pronunciations of "ough" or different spellings for the same sounds in English I've read come from mixing different dialects into one language. Whereas with French, they imposed on centralized set of rules. Or in Spanish, there are multiple languages/dialects in Spain (though I think at least Franco did try to impose Castellano for all). And other countries have their own histories of language homo/heterogeneity routes taken. What made English turn out the way it did, as opposed to others? In other words, did England ever try to impose one set of rules on how to say written letter combinations (e.g. 'ough'), like other languages have done, one way or another?


Suppose a square table. You are shooting from a corner. A ball ends its path upon landing in a corner. At what angles can you shoot a ball from your corner such that the ball will never come back to your corner? Edit: seemingly my question would be how can you prove that if the trajectory is periodic, how can you prove that it will encounter another "corner" point before it returns to itself?


I have a matrix, if I apply inverse function from existing softwares, I get an inverse matrix for every non singular matrix. I would like to know if there is a measure that tells that how close the approximation to check weather to consider the inverse matrix not. In my application, there is a possibility to discard an inverse matrix that's not well approximated. I am looking for few hints, keywords to look for or an advice in calculating the measure. I really appreciate any kind of help.


I know that particles don't go from one end to the other, they just vibrate and transfer sound energy to next particle. How do they return to equilibrium again? Do they return just after collision with next particle in the medium or do they return as a consequence of reflecting off objects like a slinky? And if they return after reaching objects and reflecting back, is that how we hear our own sound. What will happen if you levitate in the atmosphere arbitrarily at one place (assumption), will you be able to hear your sound provided there are no objects around?


Is it important to understand the concept of a universal 'now', to explain that time is a local phenomenon, and the rate of time varies throughout the universe, always dependent on the local distribution of mass, and within passing particles dependent on their relative velocity? I currently think that there is a universal now, but each point in space 'thinks' that it is a different age. I understand that photons don't age (relative to us) because of their relative speed, and they see their entire path since they were formed, (perhaps in a primordial star) as all squished into a single instant, including my retina as I see it.


Couldn't we just assume that waves have mass and momentum and can become localized? Dirac Deltas can be given a rigorous mathematical foundation but physicist do not use the Gelfand triple. Why not just assume that electrons, photons, etc., have the same dimension as space? I know string theory does this but are there reformulations of QFT that use this idea that together with the insight that renormalization has a connection with scale invariance gets rid of all the problems that plague QFT? It just seems to me that the concept of a particle is a very unphysical idea.


This is the answer. My question is why the integral of E is not zero if there is positive charge on the inner surface when the field lines end on the shell. And why the integral of E is zero when all charges is negative on the inner surface, the field lines also ends on the shell. What is the difference between these two situations?


Molecules in solids get polarized and the resulting dipoles align with the external bias field. Can this be done with lightweight spheres that are oppositely charge on their surfaces? Like surfaces that are contact electrified. If a powder made of such oppositely charged fused spheres is placed in a bias field, will each individual particle (two oppositely charged fused spheres) align themselves? If clumping becomes an issue, I guess we could vibrate them lightly while in the bias field.


How do they calculate the line of sight of a galaxy or binary star system from an observer's point of view. the velocity of a star in a binary star system depends on the line of sight so how do they measure that crucial parameter with precise values? but in case of unresolvable binary star system like spectrometric and astrometric binaries how do they take the (sin i) parameter


I know this may sound very weird, but I'm curious on what makes the water be like water? Like why, when we get into a pool, we are like in a "lower gravity" pull? All our movements are slowed down in water in comparison to air? I very much know it is a strange question, but I've really never thought of it this way, I've always assumed water is like that, but never actually started to think why we move like that underwater. I'm guessing, it has to do with the properties of the water, but if someone can shed some more light on this, I'd appreciate it?


Apparently, the special relativity theory is based on analyzing the motion of a point particle with relative speed near the speed of light. However, if I apply and combine this theory with a collection of particles with Newton's third law acting, is it correct to analyze the mass as a whole, whose dynamics equation is still modeled by F(net)=dmv/dt, where F(net) is the sum of all forces experienced by sub-particles within the system, and that m is modeled by Is this correct? If so, please provide reasonable reasons.


Suppose a bunch of micron sized particles are charged positively on one half of their surface, and negative on the other. If they are arranged randomly, I understand they will have high entropy, as the system is in disorder. If under the influence of an electric field, we shake these particles such that they align themselves (they will, right?), the dipoles are now in the state of least energy and the system is more ordered, so I understand the entropy must have decreased? Yes, not the total entropy, just the particles themselves. Is my understanding correct, and if yes, will this bring a decrease in temperature of the particles? Before: After applying electric field and shaking:


The white line is a flexible conducting foil suspended on the metal wire. I think the external DC circuit will induce a charge on the flexible part and the other charges move to ground. The charged flexible part will incline towards the oppositely charged electrode from the AC circuit and once the voltage reaches peak, the air breaks down and the charge is transferred. When the current switches, the flexible part moves towards the other electrode and a spark transfers the charge in the same direction as before. Am I hideously mistaken?


In Arabic, this idiom is used to refer to the fact that when your survival, basic, or primitive needs are not met, you can't think of arts, music, your ambitions, or luxury. e.g. A: "Why is he not planning with us for the vacation? It will be a once in a life time tour around the world!" - "He has a trial next month and could lose half of his savings to his wife, a hungry man can't enjoy the sunset" In the example, the primitive need for the feeling of safety blocks the way of planning for luxarious vacation.


I am doing competetive programming task called "Ptaszki" Its easy dfs with some observations. But i came across the following problem: I have two sequances of size n and m respectively and i want to count the number of ways to insert x different elements into this sequances You can choose for every element to which sequance it will go. You can also insert every element to the start or the end of the sequance. I dont have any idea how to solve it. Could you please help me?


I see some textbooks showing how Hubbard model with Mean field approximation can explain ferromagnetism of band electrons(stoner theory), but intuitively I can't understand why an on-site Hubbard repulsion term (rather than say Heisenberg term which correlate neighboring sites) can lead to a long-range ferromagnetic correlation. Although hubbard U seems to prefer non-zero spin at a certain site, whether one site's spin is up or down will not have any influence on another site's spin direction, so by what means are different sites correlated?


Now (hypothesis) -- And the Physics of Time In this page the author talks about (find in page) (in the context of LIGO Observatories-->) "In their most powerful event, reported just this past February, the combination of two massive black holes should, according to general relativity theory, have resulted in a massive amount of new space in the vicinity of the final star" Please explain my question. I don't get this!!!


I came across this sentence: But I craved it so hard at the time, didn't I just? The author appears to be a native speaker. The part that confuses me is the "just" at the end of the question. Is it meant to lessen the guilt of "craving it"? If so, can question tags be ended with other words than "just" to slightly alter the meaning of the question?


Dark matter is believed to be a substance of unknown origin with mass that is distributed in space. Can the same observed effects be explained by an intrinsic curvature of regions of space without presence of any mass energy in them? Say during big ban the space got randomly curved such that there are areas with curvature that is just the property of space, like valleys and hills in a landscape. If so, it might be that there are no dark matter particles or any baryonic matter to explain visible effects.


What is the correct way to describe people inheriting a castle? (American and British English) Castle heirs or Heirs of the castle or Heirs to the castle It's just a phrase I've heard, but I've seen it written in various ways and I want to know which is the correct way. "heirs to the kingdom, heirs to the castle, heirs to blah blah". Sometimes they put, example, "heirs of the crown" or "crown heirs".


I have three doubts regarding the magnetic field due to the charge:- Is a magnetic field generated when a charged particle moves with constant velocity or it only generated when the charge particle is accelerated or both? Why and how does movement of charge result in the creation of a magnetic field around the particle? We also know that motion is relative so if we observe in frame of moving charged particle then its magnetic field should not be there, then what we do in this case? Note (related to third question): as we use pseudo force in non-inertial frame then what do we use in this case?


I was studying various cryptographic encryption schemes and why modular arithmetic is being used so much while encrypting messages. Why don't we use simple multiplication with a large known prime with the message, as factorizing large numbers takes exponential time complexity, so it would be hard for an intruder to decrypt the message? So, simply using an encryption scheme that multiplies a large prime on the message to be encrypted is also hard to decrypt for an intruder without knowledge of the prime number multiplied. Why was there a need felt for modular arithmetic-based encryption schemes? wanted just a motivation why we should study modular arithmetic...


Can modular exponentiation be used to shuffle a list? I have noticed that modular exponentiation sometimes repeats every number only once, and therefore works like a shuffle mechanism for a list. I do not exactly know within what constraints it does this. I am interested in modular exponentiation for shuffling lists to reduce the computation needed to know the new position of any given object (that each object can know its new position, without knowing the position of every other object or the list as a whole).


A transgender colleague requires being referred to as "they" and not "he /she" In what number do I use verbs with this pronoun when it refers to a single person: they is/ are(?) a transgender ? Anther example: When a person is elected the chair of the committee, they give(s?) an inaugural speech. I know I could say "he or she" gives, but this sounds to me too formal or too long. With a transgender, this option does not work; So, back to the transgender example: Do(es ?) they require being referred to as "they" ?


Here contains a template for a thesis. In style_sheets there's a template.tex file. The template works by editing that file and compiling. When one does so there is an appendix A and appendix B at the end of the pdf. I have used this write my thesis, but still am not able to remove appendix A and B. Erasing the appendix code in template.tex makes it where the file won't compile. How can I remove the appendix chapters?


The above is one of the four two-dimensional lattice systems. See this article for some info. The parallelogram is the primitive unit cell, because it can be translated in order to recreate the entire lattice. So, what is the hexagon then? The conventional unit cell, whatever that means? Is it shown because the symmetry operations that can be performed spans a greater part of the lattice than that which is provided by the primitive unit cell?


In very long wires, such as hundreds of kilometers, when we flip a switch and turn on the electricity, does the magnetic field (that iron filings organize around) start to form in one direction and then move through the conductor to the other end? Such that iron filings around the whole wire, would start to react to the magnetic field at different times, with delay the further from the start we get? If yes, does the magnetic field forming propagate in the opposite direction to conventional current, the direction that the electrons travel in?


Assuming that light travels as photons that can interfere with themselves while also only being detected as specific localized points, the results of the Elitzur-Vaidman quantum bomb tester experiment demonstrate that the interference of a photon can be affected without the photon itself being absorbed/affected by the object which affects the photon's interference. Does this mean that events that could have happened but did not ("counterfactual"), e.g. the photon being absorbed by the "bomb", are what determine what actually is observed, e.g. the photon being detected at Ddark? If so, how can this occur -- is the explanation for this, for now, strictly dependent on interpretation?


In the above question , there will be induction of charge on inner sphere , hence on its surface there must be non zero value of electric potential . But in the solution , the electric potential on the surface of inner sphere has been taken as zero. Inner sphere is connected to earth does it mean that electric potential on its surface is zero despite of charge on the sphere ? Does it mean that any conductor even if it has charge is connected to earth will have zero electric potential? I can't digest this , pls provide some useful insights


Suppose event H is a subset of G, where everything in the event H is contained in G but not vise-versa, the probability that H occurs is r and the probability G occurs is s, what's the probability that either G or H will occur, but not both occurring? The probability that G will occur but H will not occur is (s-r). The probability that H will occur while G not occurring is zero.


I am curious to know if the relatively newly developed melanin-based semiconductor materials made with polydopachrome have been able to create films or crystals which visually reflect blue, purple or green hues (translated in to nm). If so, was the form of synthesized polydopachrome utilized in creating those outcomes more resembling that found in in the plant/fungi or more represententative of those found in the animal/insect kingdoms? If blues, greens and purple light reflecting melanin based semiconductors exist, are the structures associated with concentrated or unusual copper bonds? Zinc? Manganese?


Considering the probabilistic relationship between the momentum and position of an electron, are the pictures that show the interaction between electrons and positrons an accurate description of particle motion? We often see visuals that represent the probabilistic nature of particles as probability clouds. How can this converge to well-defined paths? (I never studied this subject, this is just knowledge from general physics books and videos.)


By this I mean say you had and entire galaxy to work with, and you could measure the exact wavelengths of the waves you send out and record it. You would shoot this light beam across the galaxy and measure the exact wavelength of the light you are transmitting, and receiving. Could you compare the difference in the wavelengths between the same light and calculate the one way speed of light from it?


Since value of electric field decreases as we go from surface to the centre for a solid insulating uniformly charged sphere then why does the potential increases as we go from surface to the centre?According to me we will need to do less work against weak electric field therefore the potential will be less as we go inside the sphere from the surface since electric field is decreasing as we go from the surface to the centre.


i think the title speaks for itself, since the defining trait for NP class is that - they are the set of decision problems verifiable in polynomial time by a deterministic Turing machine. thus it shouldn't matter if the algorithm computing the function belongs to any particular class - exp or factorial, even non-PR , as long as they are computable, but i am unable to find any specific example. if there aren't any known such algorithms or problems, can it be proven that they atleast should exist ?


Just as underwhelm/overwhelm exist without any usage of 'whelmed' (generally speaking) I'm wondering if there's any adverb ending in -ly without an adjectival counterpart (or that has dropped out of usage). For example where 'he walked XYZly' is recognisable, correct, or common for this or some other time period; whilst 'his walk was XYZ' is not. NB: I've found no other answers for this question elsewhere, and search engines usually lead to explanations of 'flat adverbs' not ending in -ly which is not what I'm asking.


I can't make out clearly what could be defined as a "topic element". In "A Communicative Grammar of English", Geoffrey Leech has written: Fronting is often accompanied by inversion; that is, not only the topic element, but the verb phrase, or part of it, is moved before the subject. Accompanied by this diagram: Now, "x" in the above diagram is stated to be "the topic element". However, from what I've learned, the topic is usually either "given info" or used to link statements. Moreover, they are usually subjects. Yet here "down" and "never" are considered to be "the topic element"? I'm truly desperate for an answer here.


Reflective is obvious, but what about the other properties? It's weird when we check the symmetric property of the identity relation. The symmetric property happens when (in a matrix representation) the upper triangle and the lower triangle that exists in a matrix that's divided by the matrix's diagonal get mirrored. Also, you can make a graph representation of the relation and see that it's just loops. So is it symmetric or not? (also, check out if it's antisymmetric or not because I think the matrix representation shows that it's antisymmetric)


"Underpinned by left-wing perspective,the defence gains in significance,based as it is on the development of critical art of inheritance,of combing past and present and establishing links to shape a better world." In this sentence how the word "based" is used as i have read based on but here is nothing such.please reply correct should have been "Underpinned by a progressive left-wing perspective, the defense gains in significance, as it is based on the development of critical art of inheritance, combing past and present and establishing links to shape a better world.". so how it is used?


How rare or how common are the analytic (differentiable infinitly many times) functions in real applied math? Is it like a "this almost never happens" or is it "most of the time you can approximate a physics experiment with analitic functions (and if you failed, you are bad at math)"? I ask because there are so many nice properties of analytic functions in complex analysis. But are they relevant? Seems like "differentiable forever" is too strong of a limitation.


So - I know that when we observe galaxies, we are observing their past. And that if the same past-them were to be looking at us, they would see our past. But when we look at their past - that is a moment in time that would, to them, be the present. So, what I cannot find an answer to (possibly because I don't know how best to search for it) is: does it then make sense to say that it is possible that a galaxy that is future to us, could be observing our present - but as past to them? (if that is so then that opens up a whole can of worms that all time is occurring at all times, yes?)


Hypothetically speaking if you had a satellite going near the speed of light in a straight line towards an exoplanet light years away and that satellite had a telescope pointed at the surface of an exoplanet. What could a year long journey look like, while traveling near the speed of light, through the lens of that telescope that what light years away pointed directly at the surface of the exoplanet? would the telescope capture the planet in fast forward as the light information would be entering the telescope faster? Would it be slower as time dilates for exoplanet because it not moving the sleed of light?


I have to prove that if the dimension of vector spaces V and W is the same then T: V->W is injective if it is surjective. My idea is as such: know that all elements in W have corresponding element in V (surjective) Know that all elements in V correspond to some element in W (map) I want to use that I know that V and W must be over the same field (by homogeneity) and generalize that since the dimension is the same then the number of elements must be the same, thus proving injection. Is this valid?


one day I kept my phone in my pant pocket, I haven't opened it for a long time. My phone is covered with a plastic case. So why the phone is getting wet when I observed it after some time even though plastics are non conductive towards heat. When I kept my phone in pant pocket without the case then no wet has been observed by me on the phone Whether it is due to heat of the phone or the sweat of mine due to the heat released by the phone or the heat is mainly due to heat from the screen of the phone (as some background apps were running at the time) which is open and it is not close by any non-conductive material


The normal force may be unevenly distributed on the surface of contact between two bodies. I had a discussion with my friend but was unable to prove whether this assertion is true or false under two RIGID bodies. I know it is a unreal question. Therefore, I am seeking a mathematical solution to this question. (If this question is independent from classical mechanics, how to prove?)


In India, most educated People say that " my exams have been preponed" As an English teacher I know that Americans say moved up and Britishers say that advanced in the context. One of the best English professors I know in India says that prepone is the right word in the context.Even some native speakers said that the word prepone could be accepted. I would like to know the opinion of the experts in the group.


When we work with rational numbers, our continued fraction will have a finite number of elements. Are there ways to estimate the number of elements of a continued fraction when expanding a rational number? For example, in the following form: "For the chosen rational number N, continued fraction will contain at least/at most/strictly M elements" The obvious way is to implement the continued fraction decomposition algorithm and simply count the number of elements each time the loop passes. But this method can be quite time-consuming if there are a large number of elements of the continued fraction. I tried to find the answer and the only thing I found was this thread: Nth number of continued fraction But it ultimately came down to what I said above.


Like how Taylor Swift fans say that she needs to take private jets because fans would make her taking public transport untenable. So it's the fans themselves that are the reason for her behaviour, which they are making excuses for. Is there a term for such people, or for what they are doing? An example of how it could be used: Fans are (creating the exceptional need for an excuse they are making) Taylor Swift's use of private jets' or something like that.


I need to write an equation to determine how rotating a digital balls path with the ball as the center of rotation effects the amount it moves on the x&y axis each frame. I made a table of points on the equation to help me write it, but I'm unable to convert the equation into points. I linked an image of the table here. Does anyone know the answer? I think it's a helix. It's fine if you need to switch which axis each column corresponds to in the equation, as long as you mention which ones you switched. Click here to see the table


I'm looking for an interesting application of Sequences and Series (theory from mathematical analysis) in the field of Operational Research, in particular involving combinatorics or linear optimization problems. I have tried only chat GPT for now and all I could get is an application of Dynamic programming to solve the knapsack problem where series and sequences are defined for recurrence. I was wondering if there is another one or maybe an interesting application not from the Operation Research fields. Thanks in advance.