Suppose I was standing in the sea, and touched an electric fence; I would receive an electric shock, because both my body and the sea are conductors, and create a path for the electricity to flow. The sea is so big that it has the capacity to draw and dissipate a great quantity of charge. However, if the sea was drained away, and I was standing on some rocks, would I still receive an electric shock, even though rock is not a conductor? And if I did receive a shock, where would this electricity even flow to? Would it just be dissipated amongst the rocks / soil / sand beneath my feet (even though they are insulators)? My question comes from me wondering why we receive electric shocks at all, when there is usually some form of insulator between you and the charged object, such as your shoes, the carpet, several layers of bricks, and indeed the entire composition of soil and rocks beneath the building that make up the "Earth".


I will be writing a report in which I analyse the performance of a piece of software. I would like to show my data using PGFPlot. My plan of attack is to schedule a script to run on the machine I am using, which will run the program a number of time with a variety of parameters (the main one being the number of processors I use) and dump all the data in to a .dat file. Can PGFPlot this? Can I ask it to average all the tests with the same parameters? I will be plotting a line graph with the average time, but I would also like to plot the standard deviation of the results. Is PGFPlot the tool for this? Or should I be using LibreOffice to generate the relevant data, and only use PGFPlot at the end? Thank you


I dont get why a forward gear allows sliding a car backwards. Any ideas? So this is a car with an automatic transmission. The transmission is in the drive mode (move forward). Car is at the slope facing the hill up. I am in the car pushing the gas pedal very gently. The car is supposed to move forward going up the hill. Instead it moves backwards down the hill, because I am not pushing the gas pedal hard enough. I haven't switched it to reverse. This is where I don't get it, because it turns out that the wheels are spinning in the opposite direction of what the transmission gears tell them to spin. How can this be possible? Or, to put it in a different way, how can a car move backwards when it is in "drive" mode (supposed to go forward)?


I am simultaneously taking courses in functional analysis and commutative algebra. In doing so, I found that there is, at least heuristically, some similarity between the notion of an algebraic variety (essentially, the zero locus of a family of polynomials) and the annihilator of a subspace of a Banach space (the collection of continuous functionals that "kill" the subspace). In doing some reading, I came upon the Banach-Stone theorem, which implies that the algebra of scalars is an analogue of the structure sheaf of a ring, and is evidently an essential ingredient in non-commutative geometry (thank you, Wikipedia). One finds a one-to-one correspondence between finitely generated nilpotent free k-algebras and affine varieties by first asking what algebraic structure (finitely generated nilpotent free k-algebra) corresponds to the geometric object (affine variety). Then, inspecting what sort of geometric object gives the correspondence with the more general commutative rings, we find schemes. This analogue of a structure sheaf makes me think that there might be analogous sort of correspondence in the Banach algebra setting. Is this the case? If so, can anybody provide some references, interesting papers, etc?


The following is a GRE sentence completion question. The most striking thing about the politician is how often his politics have been (i)_____rather than ideological, as he adapts his political positions at any particular moment to the political realities that constrain him. He does not, however, piously (ii)_____political principles only to betray them in practice. Rather, he attempts in subtle ways to balance his political self-interest with a (iii)_____, viewing himself as an instrument of some unchanging higher purpose. Options: a. quixotic, brandish, thoroughgoing pragmatism b. strategic, brandish, deeply felt moral code (correct answer) c. self-righteous, flout, deeply felt moral code d. self-righteous, brandish, profound cynicism e. strategic, follow, profound cynicism I am able to get how strategic and deeply felt moral code fits in the first and third blank but confused about the second blank. What is the intention behind choosing brandish? Why not follow or flout?


While studying Faraday rotation (linear magneto-optic rotation) I came across a fact that Faraday rotation can be enhanced. Verdet constant which depends on the wavelength can be enhanced as change in refractive index with change in wavelength is maximum near resonant light. Now if the incident light frequency is resonant with the atomic transition frequency then light will get absorbed in the medium and we'll get very less light at the output for observation. It is my understanding that optical pumping is done to make medium transparent to the resonant light, as when majority of electrons will be trapped in dark state we'll get considerable light at the output to measure change in the plane of polarization. Is my understanding accurate or is there anything that i have missed? Does optical pumping enhance faraday rotation in some other way also?


I had previously asked about how useful everyday solar physics data is to other astronomers ... But about a year from now, we will have another Venus transit, where Venus will pass between the sun and the earth ... I was just in a meeting with some EPO (education and public outreach) folks about trying to prepare something for the event, and we can do the basic 'make a pinhole camera' kids activities, and use it for awareness of our field ... But is there any novel science that we could use the event for? We're especially interested in trying to find something 'Citizen Science', where we could try to organize people to contribute some data from across the world that might help advance our knowledge of something. (it doesn't even have to be astronomy ... We had an idea of trying to get people to take pictures of the transit, and send the images, and we could try to use it to estimate how the quality of seeing varies, but we weren't sure if there were liability issues if someone managed to ruin a camera or their eyes in the process)


(Please note: I've seen some other questions asked like the one I myself am asking, I realize that I am indeed late to the party, but I am genuinely curious and none of the previous answers have satiated me as of yet. Note that this is a highly hypothetical question, asked by somebody who isn't even a college student studying the idea of physics at this point in time, so go easy on me here. Sorry if this isn't worded well, but I still haven't studied much on the subject, nor am I used to typing up thesis papers. Excuse me please.) Theoretically, it is stated that if you were to travel through a Kerr Black Hole- a Black Hole with a rotating ring singularity- it would allow you to go back to an earlier point in time. Another says you can travel through time using a Wormhole and exotic matter. What I'm curious about is whether or not time travel is hypothetically feasible were one of these true, as in whether or not the sheer force would crush a human by attempting to use one of these routes.


A BibTeX question (using BibDesk). I am having trouble working out how to make BibTeX handle the new Digital Library of Mathematical Functions (DLMF) web page. No authors or editors are listed. DLMF itself suggests (http://dlmf.nist.gov/help/cite) When referring to the web site from a print publication, it is recommended to use the following format. Digital Library of Mathematical Functions. Release date. National Institute of Standards and Technology from http://dlmf.nist.gov/. where the Release date can be found at the bottom of all DLMF pages. None of book, webpage, url, ... does it like this. Also, if I put NIST or DLMF in the author field, with or without braces, I get "NIST, " in the refernce list, i.e., it is expecting a first initial. This is going to be a very highly cited web page. It is the replacement for the venerable Abramowitz & Stegun and will be the standard reference for mathematical functions.


Sometimes I see something like "a mapping preserves the structures of its domain and of its codomain". From Wiki about morphisms in category theory: a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. I was wondering why the structure-preserving mappings between two topological/measurable spaces are defined by the "inverse" of the mapping, while the structure-preserving mappings between two groups/vector spaces are not? Why are the structure-preserving mappings between two topological spaces chosen to be continuous mappings instead of open mappings? I also see that "a mapping preserves some property of subsets, points or whatever". Such as Continuous linear mappings between topological vector spaces preserve boundedness. According to Brian's reply to my earlier question, this quote should be understood as "under a continuous linear mapping, the image of any bounded domain subset is also a bounded codomain subset", not as "under a continuous linear mapping, the inverse image of any bounded codomain subset is also a bounded domain subset". I wonder why? It seems at first to me like how continuous mappings preserve topologies, but it is actually in the same way as how group homomorphisms preserve group structures. Thanks and regards!


We tried my electrolyzer with a friend today, we filled a small bottle with HHO gas, and set it on fire. It was loud but not a big deal, like a firecracker. Then we added some soapy water and created HHO bubbles instead of filling the bottle with HHO underwater. Then we lit the bubbles and it exploded so loud that for a while all we heard was just ringing. It was way too louder than when before, same gas same bottle, we even tried it twice with same result. Im wondering why is it so unbelievably louder. I thought that maybe the sound just traveled in a different way, because the bottle was turned in the opposite way (towards the ceiling instead of towards the floor), but im not sure.


Sometimes in quantum cosmology, when we are thinking about 'wave functions of the universe' we have in mind some sort of formal path integral, where we include not just the variations in the dynamical fields (metric and so forth), but also possibly some sort of prescription for summing over all possible topologies. At least, that seems to be some sort of heurestic guess often encountered in the literature (Hawking et al). Now, in String theory, admittedly a different context but there is a rather well defined notion of how this works (basically as a generalization of Feynman graphs over Riemann surfaces). However I don't understand exactly how this is supposed to work in quantum cosmology exactly (say in the Wheeler-De Witt context). First of all there seems to be a massive amount of overcounting already at the metric level (where presumably one needs to mod out by all the diffeomorphisms), but how exactly does one deal with the topologies? Do they only include connected topologies? Is there even a mathematical formalism on how to even approach this problem? Any good papers that deal with this in a comprehensive way?


When I'm pushing a tall, cylindrical object stood on its end (like a Pringles can) I can push it near its bottom, and it will slide forwards, or push it near the top, and it will topple over. So, there must obviously be a point somewhere on the side of the cylinder where it stops moving forwards and begins falling forwards. My question is, how can I calculate where this point occurs? It might be something fairly obvious, like: "above the center of mass" or something like that, but I'm guessing that it's something more complicated, taking into account friction, etc. This knowledge will be very useful in my life, as the next time someone asks me to "slide along the beer" I can whip out my pocket calculator, my ruler and measuring instruments, and finally stop being such an embarrassment. I swear, it falls over every single time.


This image from Wikipedia shows how a black hole would look like: A black circle that acts as a gravitational lens for light rays coming from behind. How would a rotating black hole look like? How would it look like when viewed from side and when viewed from the top (along its axis of rotation)? If I understand it correctly, withing the ergosphere photons are dragged in the direction of the rotation, and just outside it photons would be still heavily influenced by the rotation. Also this video descibes in detail how would it be to descend into a black hole. What would be different when descending into a rotating one? In particular, what would we see inside the ergosphere? And how would it be different below the event horizon?


I've began learning LaTeX for the expressive purpose of eventually creating a resume template of my own. A business school friend of mine showed a very professionally-looking resume he created from a workshop, and I would like to turn it into a LaTeX template. I know there are already many existing resume templates, but I would like to create my own. I also know that there are many tutorials and guides for LaTeX, but I would like to know if there are any that can lead me towards creating templates, specifically something in resume/cv form. I know that it will be a long and arduous task to learn how to create templates, much less a template for a stylized resume, but I would like to make that my goal for learning the language. Would it be simpler to just modify or tweak a preexisting template?


I am currently a math major in college and my main problem is that it feels directionless. My college offers little in term of variety in undergraduate math so I moved on into taking graduate courses and I am actually loving it. But at this point I am just randomly taking courses that looks interesting and was wondering if there was specific directions to take within math. And even past that where will it lead. Thanks EDIT After reading the comments I decided to add extra info. On the courses that I like, I have taken Abstract Algebra and Linear Algebra and really enjoyed them and also Number theory. I have also taken the typical three semester Calculus courses, Differential Equations, Discrete Math, Probability, Numerical Analysis and Real Analysis. On these courses I like them though I am not as enthusiastic about them as I am for the other courses I have mentioned. Though I find their applications very interesting. And more specifically on the question it is on what paths are there to take in school and also what paths are there to take after graduation.


When creating a presentation, I sometimes create extra slides that contain additional information, a more thorough explanation, or an extra plot pertaining to certain parts of my talk. These extra slides are usually in a separate PDF and the document is usually only opened if an audience member asks a question or requests information and one of my extra slides supplements my response nicely. I am wondering if it's possible to insert these slides into my presentation with the two following options: The extra slides are skipped when progressing through the presentation unless... I click on a hyperlink placed somewhere on the slide. If clicked, we traverse to the extra slide. From this slide, continuing to the 'next' slide would send us back to the slide that got us here originally. The above can be accomplished with two (or more) PDFs (one with the presentation, one -- or more -- with the supplemental slides) via hyperref but ideally I would like to only have one document. I am not familiar enough with ifthen to know if it can be done with that package. Any help would be much appreciated.


Some time ago I learned the difference between a present participle and a gerund, so today I decided to pass any online test to make sure I understand it. I passed it having made only one mistake, which asked the difference between the two in this sentence: Nobody was surprised at John being absent. One needed to choose between present participle and gerund in reference to the word being. I chose present participle because the word being here plays a role of an adjective apart from a verb. I thought that if it had read John's being, then the word would have been a participle because it would be a noun in a form of a verb. I know, this question is a duplicate and I agree that it should be closed, but I just would like to find out whether it was me who made a mistake or whether it was the website that diddo.


Background: I was cooking eggs (very difficult) with a plastic spatula which was not very good, so when I set it on the border of the pan it began melting. In order to keep cooking, I reached for a wooden spoon which can touch the pan without melting. This led me to wonder about the behaviors of the materials. For example, if I had left the plastic spatula it would have melted, but if I had turned up the temperature enough it would have actually ignited and caught a flame. If I had done the same with the wooden spoon, it wouldn't melt as I turned up the temperature but it would ignite eventually. So why is it that certain materials (plastic in this case) will go through the process of melting, then igniting, but wood just seems to skip the melting and go straight into ignition? My only guess is that somehow wood has a melting point higher than its ignition point, but I am not even sure if that makes physical sense. A Google search led me to a strange-looking forum with disappointing answers. My understanding of thermodynamics is only as far as the math goes, but I am conceptually blind in this area. Maybe someone can shed some more light.


I think everyone should have seen a bowl of hot water moving by itself on a flat surface such as glass(seems like it is moving by itself but maybe there are some external force applied to it when it moves). What is the explanation of this phenomenon? Hypotheses : Since the bowl is heavy, for it to move the friction between the bowl and the surface must be very very low. I have heard that water may form a thin layer between the bowl and the surface, but l'm not sure whether it is the case or not. I think the bowl can move by itself because heat makes the air below the bowl expand. Edit : I will re-experiment those again and update the question in few days. Maybe I will take a video clip too. and I am not talking about a magic trick whatsoever.


For the absolute life of me, I cannot seem to wrap my head around the proofs given in my number theory and cryptography class. Maybe it's the teacher, or the textbook, both or neither, but this is causing me great concern for two reasons. First is obvious, since I hate solving math without a full comprehensive understanding of the concepts. Second, I have the feeling I'll be asked to prove concepts in my exams. So how do I go about this? Does anyone have any textbooks or something (or anything) that could help me grasp all this better? To clarify, the main concepts covered in class are GCDs, Euclidean algorithm, concepts of coprimes, Euler-Fermat theorem etc. Then these bleed into solving cryptosystems such as RSA, Chinese Remainder Theorem and others. Thanks!


"It is also my favorite quotation. " Does this sentence mean that this is my favorite quotation in addition to other quotations which are my favorites, or does it mean that I am a person who has this quotation as a favorite, in addition to the persons who have this quotation as a favorite? Does "also" indicate the presence of more (favorite) quotations, or more persons who have this quotation as a favorite? If it is the former, how what is the correct way express the later? I mean if a person says, "This is my favorite quotation", how should I express that I also have it in the list of my favorites? Should I say, "It is also my favorite."? I want it to be more formal than "Mine too". A side question which arose while asking the question above: Is the following sentence correct from the perspective of punctuation? I mean if a person says, "This is my favorite quotation", how should I express that I also have it in the list of my favorites?


I am trying to prepare a paper for submission to a journal that is not an AAS publication, but I would still like to use the AASTeX package for the tools it provides and stylistic reasons. The journal does not have its own LaTeX package, so I can use the AASTeX package as long as I make sure to conform to the few given guidelines. One of the guidelines I need to follow is using the in-text citation format of [n] where n is a positive integer. Each reference in the references section should be labeled with a corresponding number. If I were not using the AASTeX package, thebibliography environment would use this format. The modified version in AASTeX package forces you to use AAS format. How can I use LaTeX's default thebibliography with the AASTeX package? EDIT: In order to make this question more specific, I want to know what I can comment out of the class file (or add) in order for the default thebibliography to be used. Alternatively, what options can I add to a blank tex file to get a similar look and feel to AASTeX style without the AASTeX thebibliography? Alternative packages that are already made that are similar in style would otherwise be appreciated if I'm asking for too much.


This question actually came about from a discussion of another question posed here The neutron is known to be comprised of an electron and a proton, and there are observations that the neutron can be created by these particles or alternately decay into these particles. Furthermore the stability of the neutron is vastly improved if it remains bound within the nucleus. So presumably the nuclear force helps to maintain the neutron's stability. Once outside the nucleus the neutron has a much harder time keeping itself together, and once again becomes a proton and electron. And just to keep matters, and this post more simple and shorter, I'm neglecting the additional fact that the electron antineutrino is also produced in the decay. Now getting to my question, I'm wondering why we have to consider the neutron as a separate particle within itself rather than a proton-electron "system" of particles? Is there an experiment that refutes this way of looking at things? (note I'm asking for an experiment or set of experiments, not just theories)?


I have been performing an experiment at school in which I test the force on an iron surface from the magnetic field of an electromagnet. The electromagnet has a rectangular iron core. The theory predicts that the force increases linear with the surface area of the iron plate. This is because the volume between the plate and the magnet contains a certain amount of energy, which is equal to the force exerted on the plate times the distance between the plate and the magnet. I found that the force does not increase linear with the surface. This is because of the divergence of the magnetic field of a bar magnet, which is what the core of the electromagnet essentially is. Researching the magnetic field of a bar magnet, I discovered that there is a higher density of field lines at the edges of the poles, and thus a stronger force on the plate. There are some images on this website, such as the one below. I am curious as to why the field is stronger here. I know that the electric field is stronger at edges and corners because the electrons repel and end up at a higher concentration there, is it the same concept for magnetic fields?


I would like some help with what the word it in the paragraph below (with emphasis added; the word is not emphasized in this way in the original). Does it refer to the TV or the antenna outlet? The passage is taken from Haruki Murakami's TV People. Everything gets removed from the sideboard to make room for the television. The TV people plug it into a wall socket, then switch it on. Then there is a tinkling noise, and the screen lights up. A moment later, the picture floats into view. They change the channels by remote control. But all the channels are blank---probably, I think, because they haven't connected the set to an antenna. There has to be antenna outlet somewhere in the apartment. I seem to remember the superintendent telling us where it was when we moved into this condominium. All you had to do was connect it. But I can't remember where it is. We don't own a television, so I've completely forgotten.


I was finding the are the of hypocycloids. Then it struck me that apart from integration, there could be another method of finding the area of the hypocycloid with different curves. But the problem is I am not getting my answer right. So could somebody please help me tell if my logic is wrong altogether or if I am doing some other mistake. Here is the another method I am talking about:- Take case of a deltoid - we can make a deltoid by taking an equilateral triangle, and on all three of its vertices drawing circles whose radius is half the side of the triangle. Now the figure left in the middle is a deltoid. Similarly, we could use n number of sides in the regular polygon to draw hypocycloids with n cusps. (see the pictures below - the red coloured drawing in the middle is the hypocycloid) Is this idea wrong? Thank you so much :)


Materials in quantum information often emphasize that one and two bit classical reversible gates cannot achieve universality for the classical reversible computation, whereas universal quantum computing can be achieved only using one or two qubits gates. I want to understand why classical reversible computing cannot be achieved with only one and two bits classical reversible gates. I consulted with some of the materials, but all the materials I consulted with only 'illustrated' why it was difficult or seemed to be impossible to simulate some kinds of classical reversible gates only using one and two bits classical reversible gates, never giving a satisfactory clear mathematical proof as to that impossibility. Specifically, I wonder if there is any mathematically clear proof for the claim that Toffoli gate cannot be achieved only using one and two bits classical reversible gates. Thanks in advance.


If you make a straight cut through a square, one part can always be made to cover the other. (This is true by symmetry if the cut goes through the centre, and if it doesn't, you can shift it to the centre while taking from one part and giving to the other.) However, if you cut an equilateral triangle, it may or may not be the case that one part can be made to cover the other. In some cases it may depend on whether we're allowed to flip the parts; I'll leave that to you in case one or the other version has a more elegant solution. How can the cuts that allow one part to cover the other best be characterized? What is the probability that a random cut will allow one part to cover the other? Of course we need to specify a distribution for the cuts, and again I'll leave you to choose between two plausible distributions in case one yields a nicer result: Either Jaynes' solution to the Bertrand "paradox" (i.e. random straws thrown from afar, with uniformly distributed directions and uniformly distributed coordinates perpendicular to their direction), or a cut defined by two independently uniformly distributed points on two different sides of the triangle. Update: I've posted the case without flipping as a separate question.


I'm looking for an adjective that describes an object that, rather than is just the outside, is instead just the inside. Searching for antonyms of hollow, I have only found the word solid, which suggests both the inside and outside of an object. I really need a way to describe something as being just the inside of something. I can only think of words like core or heart to describe what I'm looking for, but those are nouns. When a tree has only its exterior, the tree is hollow. It is a hollow tree. When a tree has only its interior, the tree is _____. It is a _____ tree. Perhaps tree was a bad example, it was the only object I could think of at the time that made sense to be a hollow object. Any object where that description could apply is a reasonable substitute, it does not have to be specifically about a tree.


I've long wondered about the future of our species. Taking the long view, I find it very amusing to consider the challenges that humans will face and (hopefully) overcome on our journey to the end of time. The most vexing problem is the apparently insurmountable issue of Proton Decay and how this could put a premature halt to our species and in fact any complex baryonic structures. I know that Proton Decay has not been observed in practice, but the timescales involved for the effects to become appreciable have not and will not for a great long while elapse(d). The only possible solution I've been able to come up with is Anthropo-nucleosynthesis (this is the only term I could think of meaning "human caused matter generation" but maybe there already is a word/phrase for it that I am unaware of). That would only be effective if these newly forged protons/neutrons would have the same lifespan as those that were created from Big Bang nucleosynthesis. Anyways, I know this was a very long winded intro question/statement, but I'm curious as to your thoughts, solutions, refinements, or problems with anything I've laid out here. What do you think?


My aim is to produce an html-file with the same text as the pdf produced by LaTeX- The html shall represent the pagination and line-break structure of the pdf: When there is a linebreak in the pdf I want to produce a <br> in html, when there is a paragraph I want to produce a <p> in html, when there is a newpage in the pdf I want to produce a horizontal line in html. Handling of the paragraphs is easy since they are defined in the input file. But line-breaking and pagination depends on the font and on the width and height of the document (and maybe on some other things I cannot even imagine yet). Is there a way of getting LaTex to tell me where it broke the lines and where it started a new page?


I've been searching the internet all day for this particular case and can't find a single word on the subject - whether that's my poor searching or the subject is obscure, I do not know. I am also not particularly fluent with mathematical jargon, so forgive my fumbling explanation. Hopefully one of you may re-phrase my problem after I present it. This Tangent Chord Angle diagram is the closest visual representation I can find for describing my dilemma. I'll be referring to points from this diagram, although it doesn't demonstrate the problem itself, only the starting conditions of my problem. I have been trying to write a program that will rotate a vector TP from a point T tangent to circle O until that vector essentially describes a chord on the circle (possibly TA if the magnitude of A equals the magnitude of P). I know: T, the initial tangent point TP, the initial vector from T O, the circle on which T lies The radius of circle O I have no clue what to do to get that vector to "fit" inside the circle. I understand there are two possible chords depending on the direction of rotation - I'll be using whichever angle is closer to the vector's original angle. Again, forgive my inability. I'm afraid it would take considerably longer for me just to learn how to more appropriately describe the problem.


I have long found the Simulation Hypothesis to be better-able to answer questions that I have regarding the universe than the string theory and the quantum field theory. What I believe is that the notion of our universe being a simulation, especially in the last few years, has developed to the point where it should be seriously considered as a candidate for A Theory of Everything. Work on it, by the likes of Nick Bostrom (who proposed the Simulation Argument in his groundbreaking paper), has been cohesive and it has time and time again been proven to be just as plausible as any theory out there. Remember that a theory is just that, a theory. And as a theory, the Simulation Argument gives fitting answers to a lot of questions that physicists have found hard to answer for decades. Much to the dismay of pragmatists, the results yielded by the Double-Slit Experiment have been particularly insightful. The Simulation Hypothesis has much less irregularities than the other theories. It pays just as much heed to the laws of physics and mathematics, even being better able to address issues of epistemology and metaphysics. Then why is it that we don't even see it as a contender for A Theory of Everything?


An off-topic question posed at Mathoverflow by Andrew Stacey, but one which fits here: One thing that came out of Terry Tao's recent blog posts on this matter (first post and follow up) is that it's hard to get an overview of all the different ways of getting one's amazing mathematics onto the web. I thought it'd be useful to gather together a list of such. This meant to be a list of ways to do it, not examples of where it's already being done. Standard community wiki rules: one thing per answer and feel free to edit other's answers. Additional rules: it'd be useful to have a little more than just links. A brief description, pros and cons (be objective), platforms (does it only work on Linux, sort of thing) - things that might help someone decide which things to examine further.


To move charge from one point to another in an electric field, the force which we must apply is equal and opposite to the force due to the field. (Quoted from Engineering Electromagnetics by Hayt.) Here is my concern: To move a charge, shouldn't the force that we must apply be just a little higher than the force exrted on the charge by the field? I know that electric field intensity is the amount of 'kick' a test charge feels when placed in that electric field. Now if I apply the SAME amount of force against the force due to E field, the charge, according to my understanding, will just stay there and won't move. If I apply a force just a little higher than force due to E field, only then the charge should move. Why does the text say that to MOVE charge in E field, we should apply force EQUAL to that of the force experienced by charge because of E field.


I've come across a book that has this general questions about lines and planes. I can't agree with some of the answers it presents, for the reasons that I'll state below: True or False: Three distinct points form a plane - BOOK ANSWER: True - MY ANSWER: False, they cannot belong to the same line Two intersecting lines form a plane - BOOK ANSWER: True - MY ANSWER: False, they can be parallel and coincident lines. Two lines that don't belong to a same plane are skew - BOOK ANSWER: True - MY ANSWER: True If three lines are parallel, there is a plane that contains them - BOOK ANSWER: True - MY ANSWER: False, they can be parallel and coincident lines. If three distinct lines are intersecting two by two, then they form only one plane - For this last one there's no answer and I'm not sure about the conclusion. If you could help me, I appreciate it. Thank you.


This question concerns the residual heat (if any) contained within the Earth's moon. At the time of the Apollo moon landings, it was widely reported that the boot marks left by the astronauts would last "forever", as there is no atmosphere to disturb them and because the moon was geologically "dead." Ignoring cosmic dust, moon quakes (which have since been confirmed) and the blast created by the Eagle lander top section as it left the moon, is this idea of extremely long term preservation of the boot prints really valid? Do we know if the moon is capable of creating convection cells, carrying heat to the surface and gradually erasing the boot prints and tracks left by the astronauts by stirring the dust particles surrounding them? Does the Earth create gravitational forces within the interior of the moon, even though I appreciate they may be a low order compared to the extreme example of Jupiter on Io?


As you know the active noise cancellation technology used in many application such as protection of aircraft cabins and car interiors to reduce engine noise also some headphones use this feature to reduce unwanted ambient sounds. Actually I don't know the details of how this feature works, but I do know that it has input (microphone) that receives noise waves, control unit that invert the incoming noise waves and output (speakers) to emit inverted noise waves and therefore they will cancel each other. What I'm not understanding is that If we considered the sound waves are traveling in air, then the inverted noise waves emitted from the speaker will reach the listener before they combine with the original noise waves so they cancel each other while the inverted noise waves are traveling in a solid-media (components of the microphone, speaker, etc.) faster than the original noise waves traveling in air. I'm I missing something or there is another feature they use to adjust the speed of sound in the active noise cancellation?


I am an undergraduate CS student but I love mathematics and spend most of my time doing and reading math books. I realize that it's important to get into the habit of reading papers and journals so it will be easier to think of ideas for projects and research. I'd like to know how I should start reading papers and what papers are good for beginners? The areas of mathematics I really like are Discrete Mathematics, Combinatorics, Number Theory, Mathematical Induction, Problem Solving, etc. I like things like Calculus too but I feel papers on Calculus would be too difficult to understand. Also, are there any particular efficient methodologies for reading papers? Is knowledge better gotten from books or from papers? Note: I want to say that there are already threads asking which papers every mathematician should read, and which every computer scientist should read. The purpose of this thread is slightly different. It isn't asking which standard papers everyone should know. It's asking which papers allow for an incisive entry into deeper knowledge of the subject. P.S. : For the benefit of anyone who sees this thread later, I did find a wonderful journal called Crux Mathematicorum'' dedicated solely to problem solving! They allow free access to their back issues on their website. Other good journals I found wereParabola'' and ``Pi in The Sky,'' both of which may be read online for free.


I am trying to make a computer game, which involve radioactive hazard. The game is still in its early stage, and I want to add mechanisms which are as realistic and as fun as possible. So I'm trying to understand how radioactivity is "absorbed" in the air. For example (please excuse any physic mistake), if you have some alpha radioactivity, it will be absorbed by a very thin material, like a sheet of paper. For Beta, it's more like an aluminium sheet, and for gamma you're dead already. But how does the radioactivity weaken in the air ? Is it absorbed by random elements and the energy is lost in them ? I guess the explanation is similar to why Wifi just doesn't work at great distance, but then I'm not sure how it happens. Feel free to give any text explanation, however I'm open (even if a bit afraid) of mathematical explanation. I'm pretty sure there is a theorem or something about this, this would be even better (although explanation won't hurt).


I have a basic enough question. Assume that one has one of those ideal see saws, i.e. the teeter-totter pivots on point source, the balance is of uniform mass etc. Now assume that one places an object of mass m on one end. This will force the balance to tilt on side. Now suppose that one adds another mass of m on the other side. What happens? It seems that everyone agrees that the the balance would move to the stable equilibrium, that is the middle. What I don't understand is, what is the Force that moves the teeter-totter to the middle position? Once the second mass has been placed on the other side, the two net forces acting on the pivot is the same. And since the distance is the same, the torque is the same. So what force is it that moves the balance to the middle? And if no such force exists then every position of the balance is valid if the two weights are equal, then how do those mass balances work?


In a previous question, I learned that in order to detect an object in space, what matters is how much electromagnetic radiation it is giving off, and what sources of EM radiation the sensor can pick up. Given that the sensitivity of the sensor over the spectrum is a parameter for detectability, I would like to learn if some wavelengths of EM radiation are more difficult to detect than others, over increasingly longer distances in space. From my experience in biology, I know that longer wavelengths of lights from a source can be "pierce" deeper into tissue than shorter wavelengths (a bit counter-intuitive!). In fact, that's the premise of "two-photon microscopy" -- using wavelengths of light with such low energy that fluorescent dyes need to absorb two low energy photons in order to be excited. So, the probability of that happening is lower, but there are many benefits, including the fact that longer wavelength light can penetrate deeper into tissue. Does the physics behind such concerns at the microscale also affect detectability at astronomical scales? Is some EM radiation easier to detect than EM radiation with other energies? Can EM radiation of certain energies "travel" longer distances through imperfect, real space?


Good evening, I'm writing my thesis in Latex, with which I already had the pleasure to work for some other projects, but I'm now facing a problem I don't know how to solve. I'll try to get to the point as straight as possible: let's say I have a chapter of simulation results, which includes a huge set of images. This chapter is subdivided itself in some sections, among which the images are distributed. Now, what I would like to have is a way to have all the images related to section A inseted before section B starts and so on. I don't mean I want to precisely insert them where I want (the most of them are set of subfigures, difficult to be fit anywhere), it would be enough to be able to tell the compiler to put them all before the next section starts. Do someone of you have any suggestion on the matter ? Thanks a lot in advance Michele


This is kind of a big squooshy question (or series of questions), which I will try to cast in a more precise form. Apologies if I don't succeed. Context: I'm an amateur set theory/category theory enthusiast with special interest in stratified set theories. My knowledge is substantial in odd spots, but not very well rounded. Because of the ubiquity of talk of models in the literature, it seemed like a thing I should learn, but I'm having a difficult time because I feel like I don't grasp certain informal aspects of "model theory in practice", even though I technically understand about the first half of David Marker's "Model Theory: An Introduction", for instance. Questions: What are the big questions one wants to answer with models that would either be unapproachable or extremely painful without it? I gather questions of consistency are important ones, but are there other main ones? When does one want to know about the second order properties of elementarily equivalent models, and why? Are there even general answers to these questions, or am I asking things I would have to ask, say, another set theorist about to get anything informative? I hope this isn't too poor a question; I'm flailing a bit to articulate beyond "it's cool, but WHY???"


Looking at Cohen's success at proving independence of the Axiom of Choice and the Continuum Hypothesis, I was wondering if it was possible to mechanize forcing techniques for the purpose of proving independence (and hence undecidability) of conjectures. It would seem to be useful to test any sentence you're trying to prove or disprove with an automated theorem prover for independence. I'm imagining some automated forcing technique that constructs a model of ZFC plus the conjecture and its negation to prove it isn't a theorem of ZFC (or whatever foundation being used) rather than spinning forever on an undecidable statement, since the incompleteness theorems indicate that any statement in a sufficiently strong formal system can be undecidable. Its not obvious to me how to automate such a method, as its probably quite a bit more complicated than just implementing resolution, but it would be nice to have automated reasoning go down three paths of searching for proof of a conjecture, proof of its negation, and proof of its independence.


If I write My car can go pretty far and it gets good mileage I have combined two independent clauses to create a compound sentence. I might just as easily write My car can go pretty far. And it gets good mileage. But if I write Surprisingly, my car can go pretty far and it gets good mileage, meaning that it is a surprise my car has both of these attributes, then the independence of each clause seems diminished, because one without the other is not surprising. In other words, I cannot write Surprisingly, my car can go pretty far. And it gets good mileage. Is there a way to describe this difference? Is there an overriding term for two or more independent clauses that actually do not mean the same thing when not joined?


I was watching a BBC sitcom. The scene is set in a wedding ceremony. In the opening of the speech of the father of the bride, he tends to be humorous, and thus he says: "Welcome to the wedding of Laura and Paul, whether you're friends, family or freeloaders, loved ones or loathed ones, people we like or people we had to invite, and whether you're here for a free meal or a free drink, people who wouldn't have missed this special day for the world or people who had nothing better to do. You are all welcome." I am baffled by his using "wouldn't have missed." According to the context, I reckon what he means is more or less "people who wouldn't like to miss this special day for the world." But if he means exactly like that, why does he use the perplexing "wouldn't have missed?" Alright, he might be trying to be witty, implying "people who thought if they came to the wedding, they wouldn't have missed this special day after the wedding." Is my interpretation right? Even the the implication is like that, I was still wondering how it sounds like in a native speaker's ears? For me, a non-native speaker, all I received was only full of confusion. Not humorous at all. If I were in that wedding, I would get lost and stuck in this "wouldn't have missed" and miss out a bunch of the rest of the speech.


If a conformal mapping, e.g., a Schwarz-Christoffel mapping, maps the real line (from left to right) to a polygon, which is traced out from left to right, why is the upper half plane mapped to the interior of the polygon? Is it a preservation of orientation argument? I.e., since the upper half plane was to the "left" of the moving point in the z-plane (moving along the real axis), then the image of the upper half plane will again be to the left of the curve, which is also going from left to right, until it closes at the final vertex to form the polygon. I know that one can use a test point and then make a connectedness argument, but if the integral (or any tricky conformal mapping) were hard to evaluate, such as an elliptic integral, I don't feel that this is a wise move to make, and that there could possibly be a better way to argue why the image of the UHP is inside the polygon. Thanks,


So, I know that this question may be tough to answer, but I am asking this question in all seriousness, and I don't consider myself a newbie... Lately, I am trying to find a way to "generalize" my approach classical mechanics. Specifically, I need a system that can always guarantee a solution to any problem. In a sense, just like mathematicians like to generalize stuff. For example, if you get a system of linear equations, you can always solve it by using the Gauss method of substitution... I often have the problem of seeing why the solution provided by the textbook is right, but I don't see why my solution is wrong... I'm guessing this is the motivation behind my quest. I know a field as big as mechanics is hard to cover with one or two principles, but I need something to always help me set the correct system up, the rest being "just math". So is there a method that yields a uniform system? I have heard that Lagrangian mechanics can provide something like this, but I'm not sure whether to take it up... Can anyone post their experiences with that? Lastly, I would add that I am no stranger to advanced math topics. I like linear algebra and multivariable calculus... from my experience learning new math has always enhanced and deepened my understanding of physics. Don't hesitate if you have to offer something... maybe a book or similar resource...


Today my laptop battery died while I was finishing off a sticky note and so naturally I just went 'Oh, bollocks!'. After all, it didn't warrant one of the "big six"; it was more a "damn" moment. So I opted for "bollocks". I didn't think it was rude; it was just fun to say. I was around a colleague at the time and they gave me a bit of a look (and, frankly, an 'Excuse me?!'). They told me that it's just another way of saying "bullshit", yet based on their response, obviously not a euphemism. (I'm sceptical though. I know the dictionary says "balls", but it didn't say anything about "bullshit".) I've always thought that "bollocks" is just one of those great British words that are relatively harmless but fun to say, like "bugger" or "sugarbush". So, is it actually rude? Or is it just rude to some people? (If it's relevant, I live in Australia.)


I understand the phrase when it is used literally in this example. He's ill and doesn't know what he's saying. I perceive it to mean that a person is struggling with an illness that prevents him from thinking clearly and understanding his words. However, there have been times where the usage seems more vague to me, particularly when it is directed at an able-minded or sound-minded person. He doesn't know what he's saying. I don't think you know what you're saying. Okay. I think there are some possibilities. A person uses the phrase as a personal attack on the other person's intellect. A person uses the phrase to discredit the other person. A person is implying that the target person is oblivious to a connotation or perception of the target person has said. A person uses the phrase to expose evidence that the target person is a dilettante. Perhaps, the phrase means each one in different contexts, or perhaps, it means a combination of them?


Nearly four years ago, upon hearing of the observation of time dilation in two optical atomic clocks at an elevation one metre apart, due to acceleration towards earths centre of gravity by Chou, C. W.; Hume, D. B.; Rosenband, T.; Wineland, D. J. I wondered where the point would be, at which time dilation caused by acceleration toward the earth gives way to time dilation caused by acceleration around the earth, via orbital velocity. (I mean the point at which the two effects are at equilibrium with each other.) I asked a friend who is a physicist, he got back to me almost immediately, fascinated by his back of an envelope calculations that this point is the Clarke orbit, or the geosynchronous orbit. If for instance the rotation velocity of earth were a slightly different speed, the result would be different, and no relationship apparent. I think there may be a very interesting relationship waiting to be revealed. Is there a physical or mathematical relationship? And if so, what is it? Not being a mathematician (I'm an artist) I'm not capable of doing the math, but this has fascinated me ever since.


While studying for my thesis (in dynamical systems) I've encountered multiple times with the concept of nuclear operators and nuclear spaces, often linked with the works of Grothendieck. For example, when studying the generalized transfer operator (or Ruelle operator) for the Gauss Map, Dieter Mayer points out that this operator is in fact nuclear (On the thermodynamic formalism for the Gauss map). While I can understand the definition of a nuclear operator, I still cannot get the real importance of being nuclear of order zero. Usually I'm interested in spectral gap properties for transfer operators, but is there any implication of the nuclear property? Also, any reference for nuclear operators and Fredholm kernels would be appreciated, since trying to learn directly from Grothendieck's works has been really difficult for me. Thanks in advance.


I don't really understand the reason why wave theory of light fails to explain the blackbody radiation. My textbook says the Planck's quantum theory explains blackbody radiation. It says "If we assume, he said, that radiation is emitted in packets of energy instead of continuously as in a wave, then we can explain the black body spectrum." So, the problem in the wave theory is that energy is absorbed or emitted continuously, not in multiple quanta. But why is continuous emission or absorption a problem when we deal with black body radiation? I was taught in class (or maybe I interpreted it this way) that if energy would be continuously radiated, then the intensity of radiation must increase on heating the black body and wavelength of light would stay same. But from experiment, wavelength changes. Hence it fails to explain it. But assuming that what I wrote above is correct, why can't the wavelength change? And even in Planck's theory, why can't simply the number of quanta emitted increase and frequency be constant, so it gives radiation of same wavelength? Please note that I have tried my best to look this up on the Internet, but all I see is explanations for back body radiations in terms of Planck's theory. If you find this question not framed well/not acceptable in its current form, please leave a comment so that I can edit it, before downvoting it to close it.


Not all questions start with Wh- words, so why don't we start a question with a question mark to make it more obvious that it is actually a question? For instance, when I'm reading a book which has a long interrogative sentence at the bottom of the right-hand page, it often isn't until I turn to the next page that I realize it is a question. Not that this represents too much of a problem, but from a beginner's perspective it must make it harder to interpret the words of the sentence in the correct context. Similarly, why don't we start exclamatory sentences with an exclamation mark? I know this is a trivial question, but I'm wondering whether some kind of historical or colonial/imperial decision might explain the absence of these marks in the situations I describe. P.S. I am a student programmer, so pardon my ignorance / funniness


We are regularly taught in high-schools and universities that, according to General Relativity (GR), gravity is nothing but a manifestation of space-time curvature (which, in its turn, is caused by matter and energy). However, GR is still only a model, which hasn't been challenged by experimental evidence/precision thus far. E.g., in wiki one might find a lot of alternatives to GR, some of which agree with observations not worse than GR (e.g., Brans-Dicke theory). There are theories which describe gravity not in terms of curvature, but in terms of torsion - but in reality gravity cannot be both at the same time! Besides, as far as I understand, curved space might be described as a curved surface in non-curved space of a higher dimension. So my question is: do I miss something and there are strong model-independent reasons to believe that gravity is geometry, or is it just that authors in most textbooks and articles imply that this is a model-dependent interpretation, without saying it explicitly?


Suppose the big bang did create equal portions of matter and antimatter and they exist right over top of each other, such that there is a constant inversion going on. Since a matter and antimatter universe would be identical, we wouldn't know whether we're living in one or the other; therefore, isn't it possible that at any given time we could be in one or the other, or oscillating between the two at a very high frequency, perhaps even at the smallest possible unit of time? Could such frequent inversions be responsible for what we call "inertia", since disturbances could cause a slinky-like reaction where the matter and antimatter pair are annihilating and recreating each other or swinging around each other in a tandem dance across space? Could such a phenomenon be responsible for quantum phenomena such as particles having no definite location, since they would be constantly coming into and out of existence in slightly different locations, with fairly regular offsets giving rise to an apparent quantum field? Wouldn't that explain entanglement as well? And if such a thing is occurring throughout the universe everywhere, perhaps it explains inflation as well, since it wouldn't be an expansion of space, so much as a shrinking of the size of every single particle in existence as the constant oscillation between matter and antimatter loses energy somehow...


My daughter is in high school and is working on a science project. She has really enjoyed writing the paper in LaTeX. Now, she has to make a tri-fold presentation of the major points. In the past, that has meant PowerPoint printed onto letter-size paper then pasting those pages (one slide per page) onto the board. How can we resize beamer to have at least the same aspect ratio as letter-size paper? Ideally, we would have the same dimensions, also, but that is not so important as we can scale at print time. I have used beamerposter in the past, but that generates just one page, I think. Any suggestions are appreciated. The default is to simply use beamer as-is and scale to as close to letter size as possible when printing.


There are a few tools that will automatically retrieve BibTeX-formatted references from MathSciNet. BibDesk is offered with TeXShop, but only works on a Mac. Are there similar tools that work for Unix/Linux or for Windows? There are a few applications written as stand-alones that get the job done: mscget (requires python), bibupdate, bibget (a shell script). JabRef can retrieve references from a number of archives, but MathSciNet is not listed among these resources. Some potential tools are mentioned in this answer to What are good sites to find citations in BibTex format?, listed under "Reference managers that allow BibTeX export/import". MathSciNet is not included among the archives covered, although it is mentioned as a resource from which BibTeX-formatted references can be downloaded manually. Note: I work for Mathematical Reviews / MathSciNet.


I have previously done a course on group theory and now I am doing a reading course on category theory. So as an interesting exercise I have been asked to write an exposition of group theory for someone who already knows category theory but doesn't know any group theory. I have been given the liberty to decide how I build the theory. I already have a vague idea of what is to be done. However, I would like to hear ideas about what should be done. So I solicit advice on things I should emphasize, the ways I can exploit the given familiarity with category theory for a more economic presentation and/or a exposition through "the path of least resistance". And please feel free to also mention any tips or precautions. Please refer me to material along this line. Thanks!


I often have problems with article words referencing to its correct referent. In the following statement for example: The touching scenes in the film leaves the audience a heavy feeling, encouraging them to consider charitable acts and the role it plays in our society. My intention is to have the word "them" linked to the audience, and the word "it" linked to the charitable acts. I am quite confident that the word "them" does link to the audience. But I am worry that the word "it" may not link to the charitable acts. I find that there is possibility the reader may confuse "it" referring to the touching scenes in the film instead of the charitable acts. Are the article words in the sentence that I have constructed above correctly linked to its referents? In general, is there a rule to watch out when identifying references in long and complex sentences?


I am currently studying English and as such enjoy reading English books from time to time; for instance I have recently been reading the fifth book of A Song of Ice and Fire since the French version was not to be released anytime soon. As I read it I often came across a grammatical form I had never encountered previously, this form being the usage of "for" to introduce propositions as per this example: "We must learn english, for it is one of the most widely-spoken language." My question if the following: is this form correct (I do think it is), and if it is then is it still used today? I really like this usage of the "for" preposition but still, I'd like to be sure it can be used before actually using it (in class or elsewhere). Thanks! EDIT: Upon further searches, now knowing that "for" is in that case a conjunction, I found out that this form is perfectly correct - though it is more of a literary form.


I am discussing with a friend (over email) the pros and cons of various interview styles. At some point in the email I write: "Ideally the candidate will demonstrate X." Then in the next sentence I want to describe, by comparison, what action by the candidate would not be ideal. I was initially tempted to write: "What would be bad would be if the candidate did Y" but the bold-text "would be" sounds wrong as used here. Instead, I feel I should write something like: What would be bad is if the candidate did Y" because I'm saying that [the action that would be bad] = Y, as opposed to [the action that would be bad] would be Y. In other words, I know that the action Y would be bad. Neither phrase sounds perfect though. I would appreciate any explanation of a 'correct' phrasing! Also I am from the UK, but living in the US, so I am interested in usage in each place (if there is a difference). Finally, I apologize if I have mis-tagged this question - it seems to me to be a question about use of either the conditional tense or the subjunctive mood, but this could be wrong.


My question is basically this, if I am only able to measure the total electric field and the magnetic field at a few discrete points in space and time, is it possible to separate the convective and the inductive electric field? Here is the background. Consider the geomagnetosphere. There is a background geomagnetic field. There is a convective background electric field. Now picture a bunch of charged particles in addition with various energies and velocities so they are guided by the fields and in return modify the fields and so on. There are also other oscillations in both fields introduced from outside. So now we have an induced electric field. I then throw a satellite in there which reports the total magnetic field vector and the total electric field vector but these are only discrete values in space and in time and I have a very small sample of the entire magnetosphere. Is this even theoretically possible? It seems like at least some theoretical work must have been done on this. If someone can nudge me in the right direction or point to some references, it will be a great help.


The Boltzman approach to statistical mechanics explains the fact that systems equilibriate by the idea that the equillibrium macrostate is associated with an overwhelming number of microstates, so that, given sufficiently ergotic dynamics, the system is overwhelmingly likely to move into a microstate associated with equilibrium. To what extent is it possible to extend the story to nonequilibrium dynamics? Can I make concrete predictions about the approach to equilibrium as passing from less likely macrostates to more likely macrostates? (Wouldn't this require us to say something about the geometric positioning of macrostate regions in phase space, rather then just measuring their area? Otherwise you'd think the system would immediately equilibriate rather than passing through intermediate states.) Can the fluctuation-dissipation theorem be explained in this way? Edit: Upon a bit more poking around, it looks like the fluctuation-dissipation theorem cannot be explained in this way. The reason is that this theorem discusses the time-independent distribution of fluctuations in some macroscopic parameter (e.g. energy of a subsystem) but, as far as I understand, it does not describe the time dependence of such a parameter. In particular, I'd really like to understand is if it's possible to explain things like Fourier's Law of thermal conduction (that the rate of heat transfer through a material is proportional to the negative temperature gradient and to the cross-sectional area) with a Boltzman story. According to these slides, it's surprisingly hard.


After having switched computer I had to reinstall TeXLive and TeXMaker on my Mac OS X Yosemite, however, after having done this I whenever I try to compile with (pdflatex) TeXMaker I get the error: log file not found. I am aware that this means that TeXMaker is not able to find my TeX distribution, but what baffles me is that if I compile manually via the terminal with pdflatex test everything works like a charm. I have tried to reset the settings (even though they should be after a clean install), without luck. If I first compile via the terminal and afterwards via TeXMaker, I do not get an error, but it does not compile either. Any suggestions? EDIT: I tried to remove the installation and install it using macTeX in stead of the general installer for Unix systems. Of some reason this helped, even though it have installed the same version in the same location on my harddrive.


I am searching for the "most natural" definition of a (geometrical/space) point as an element of "something" in mathematics (I am trying to design a small computational geometry library on strong mathematical basis). For example, for a vector, it's easy: a vector is an element of a vector space, end of story. Same for a tensor: a tensor is an element of the tensor product of vector spaces. But how to define a point in the same way? Can a point be defined as an element of an affine space or as an element of a topological space? If both are true, what are the difference between the two types of points, and what would be the most natural (it's subjective) approach to define a point in geometry? Moreover, do other approaches exist (a point is an element of XXXX)? EDIT: I know that a point is an axiom/primitive notion but to design the library I need to make a choice. And I want to know the best option...


This is a question I have been wondering about the last few days, and I'm still not sure if it should be posted here on in the music section, so please excuse me if it's misposted. Hatsune Miku is a world-wide popstar, taking news and filling concert halls. What makes her unique is that she is not a human person. She is a VOCALOID, a program. While browsing the interwebs I started to realize I was in conflict with myself. What would be the correct way refering to Miku: "What is Hatsune Miku?" or "Who is Hatsune Miku?" From my perspective I see it as "What" defines an object while "Who" defines a living persona, an identity, person you can refer to. Miku efficiently takes both spots as she is originally only a created persona, but have also performed live on concert. Thanks in advance.


It'd be nice to be able to style words or sentences or subsections based on arbitrary metrics. For example, it might be useful to highlight words based on their length - bolder highlighting for longer words, so that you can zoom out of the document, and get a quick picture of where it is getting complex. Another example along the same lines might be to highlight paragraphs based on their readability score. For the last example, this might involve something like converting the paragraph to plain text, calculating the score (externally?), mapping the score to a colour, then applying the color to the paragraph. Is it possible to do something like this on a document-wide basis? ie. not having to add a command around every paragraph? Can the length of a word/sentence/paragraph be easily calculated within tex, or should this be done externally?


I've read questions like this one, this one, and this one which ask why we use complex numbers instead of real numbers, and that's one half of the question, but the other half, which I've never seen conclusively explained, is why we use complex numbers as a wrapper at all. From what I understand, there's nothing special about complex numbers in this case; they're simply a container (a struct, for those familiar with programming) for a pair of numbers which describe magnitude an phase. Is there a mathematical or convenience reason why, traditionally, FFT uses complex number systems for this purpose? Is there a historical aspect to it? Is there any reason why we shouldn't just describe the magnitude and phase as a vector or some other similar structure?


Reading this explanation, I've understood that the divergence in computation of Casimir force on two parallel conducting plates is because of an unphysical model of ideal conductor, which makes EM field vanish on its boundaries. This looks much like unphysical model of rigid body in relativistic mechanics. Still, zeta regularization appears to somehow correctly remove the divergence and give sensible results. This must mean that effectively, it replaces our "too ideal" conductor with a "moderately ideal" one. So, the boundary conditions for EM field are no longer as simple as zero. So, my question is now: what boundary conditions does zeta-regularization implicitly impose on the EM field instead of the original ones? Does such change of boundary conditions in fact mean that the force will depend on which conductor the plates are made of?


Why isn't this possible? I've seen the ducted fans, it works (even if it is very rough). I've also seen the magnetic boards that hover above a special magnetic-reactive surface. But that isn't true hovering in the sense of usability as you cannot leave that surface and you cannot surely expect a proper surface to be everywhere so your hoverboard is quite ineffective as the ATV that we've always wanted in a device like this. So my question is this: why can't a device with a strong powerplant charge electromagnets that are facing eachother to repel, hover? I have had this idea since I was a child but no one really can tell me why or why it won't work. I do not understand physics that well and I am sure there is an obvious answer. It makes no sense to me why it won't work.


If I have an X amount of randomly generated positive numbers, what type of algorithm could I run to find the following: Precisely where the largest difference exists between two of the numbers? How much is that difference? Where does the smallest gap exist? How much is that difference? Generate a list of all differences and how many occurrences there are of each difference It has been a while since I have taken any sort of math class, so pardon the potentially inaccurate tags. Ideally I'm looking for a way to generate a list of all the differences, and then place those differences into a descending order. Once the list of all the differences into a descending order is created, we still need to be able to identify where in the original set of numbers, each particular difference takes place.


I am writing a story with a character who was a prince born albino in a very superstitious country. His Uncle intensely feared killing him, as he believed it would release a demon inside of him. This myth becomes the only thing that saves his life when his kingdom is overthrown as a child. He learned that having people falsely fear him keeps him safer than trying to dispel the rumors. I need a term for someone who encourages his own awful reputation, so that he can win through fear without having to actually be fearsome. Think of Captain Shakespeare from Stardust as being a good example. The word in a sentence may be used much like the words illusion or facade. I am not necessarily looking for a noun or an adjective in particular; just a word, or perhaps a short collection of words to explain or describe the concept. Several example sentences include: "He was known to ____" "His ____ was falling to pieces".


I am in a bind, trying to wrap up my thesis in two weeks time. My university requires me to use their citation style, which they only provide for endnote or zotero. Since all my citations are in bibtex (which I know, and I have no time to experiment with other bibliographic formats or programs, trying to make them collaborate with latex) I wonder if there is any possible way to convert a citation style language (csl) style file into a bst file that I can use. That would make life so much easier. I have googled and searched the forums that I know of, but all answers I have seen are discouraging. They are however not up to date. If someone knows of a procedure to accomplish csl->bst style file I would be highly indebted.


I am just going to give an example of what I mean using Skolem's Paradox. I don't want to get into Skolem;s Paradox itself or its "resolution." Skolem's showed that in first-order formulations of ZFC, whether some set A is countable depends on what is in the model. For example, take a model M of ZFC (assuming there is one). Let M satisfy the statement "S | S is countable." This means there exists a bijection from S to {naturals} in M (i.e. there is a particular set of ordered pairs in M). Now, remove all and only those bijections from M and call this new model M'. Assume M' is still a model of ZFC. Is S still countable? No. Countability is relative to the elements of M's domain. How how does a mathematical object, e.g. a countable set, relate to the real world? Is it in the real world? Is is instantiated in the real world? What is meant when mathematicians say (e.g. as Cantor's Theorem says) that uncountable sets exist? That is, what is the relation of math-objects to the real world?


I decided to study Euclid for fun. I have Oliver Bryne's edition. I also want, as much as possible, to construct the figures myself, to get a deeper understanding. How did people traditionally do this? I have a compass, and a ruler. So far I've constructed the first three propositions from book one. However, it's not clear to me how I ought to draw the fourth proposition, or whether it's only meant to be understood. The later propositions use the earlier propositions where equal line lengths were drawn using circles. If I want to use those same deductions to construct later propositions, should I simply copy the line length with a ruler? Surprisingly, google didn't turn up much guidance for this project. I'm assuming earlier generations of pupils would have drawn Euclid, no?


From looking at undergraduate mathematics programmes it's quite apparent that mathematics degrees are demanding, one could even say the work load is grueling. However I'm certain that there are things that one could do to prepare in advance for the rigours of such a degree. What i'd like to know is what foundations must be in place so that the experience of learning mathematics at university is an enjoyable one. Enjoyable in the sense that if you're exposed to a new topic you aren't floundering and you can dive straight in and enjoy the exposition and the process of learning, without having to go backwards plugging in numerous gaps and addressing other deficiencies in your knowledge. I'm certain that a good grounding in pre-calculus mathematics and calculus are a prerequisite but aren't all that's needed. What are the things that must one know in order to have a solid grounding in mathematics, with the aim of studying mathematics at a higher level? Edit: Let's assume it's a quite a demanding degree programme: MIT, Harvard, Cambridge, etc.


I was thinking deeply about figurative language today, and I read a sentence that must be an example of a specific type of figurative language, but I didn't remember learning about it and couldn't find it on a reference I use. The sentence, from Street Love by Walter Dean Myers, is: Could it even withstand the voltage of / His mother's shock? Myers builds the metaphor of voltage off of one definition of the word shock, while also calling the other to mind, since the mother would be surprised/upset at this event. It seems to me that there ought to be a word for this! Any ideas? I found a similar example, too. It was used as an example of parallelism on the site I was referencing: She liked sneaking up to Ted and putting the ice cream down his back, because he was so cool about it. It ins't a metaphor, but it similarly uses both meanings of the word (cool, in this case). I thought it might be helpful as a clarifying example.


As I understand it, there is a large contingent of physicists who believe that the measurement problem is "solved" by decoherence, without, for example, needing to postulate the existence of "many worlds." Yet at the same time my understanding is that in the decoherence picture there is only unitary evolution of the wave function, and that while the appearance of collapse is explained, the global superposition of states still in fact exists, and whether or not multiple states within the universal wave function observe the same appearance of collapse (but to different eigenvalues) is a question that is left completely unaddressed. Therefore my reading of the decoherence picture is that it is virtually identical to an Everettian approach, except that it purposefully ignores an obvious interpretational consequence of its description. Is this true, or do decoherence-based approaches somehow argue that there really is only a unique observer within the universal wave function that observes a collapse to unique eigenvalues, and that there is some form of symmetry breaking that allows this to happen at the expense of all the other potentially conscious components of the universal wave function?


My colleague and I are working on our style used in communications, and we have a question around whether it's acceptable to use "you're" or if we should always write "you are". Our style guide says: Our tone is conversational, honest, thoughtful, but never familiar or using slang terms The specific example we're looking at would be in an email from an IT department of a company to employees with instructions on installing a new system. We want a sub-heading at the end to conclude the message. And so the heading would read either: You're ready to use [product name]! You are ready to use [product name]! Which better meets the requirements of our style guide? Is you're too familiar? Slangy? Is you are too stilted to be conversational? Does it convey thoughtfulness? Could one form be more appropriate than another in different circumstances? There has been discussion on using contractions in formal writing (of which this question has been tagged as a duplicate), but I feel our style guideline is more nuanced than just "formal".


I cannot hear the distinction between certain sets of vowel sounds. Normally the words in each of these sets (and of several others) all sound identical to me: Don, Dawn; marry, merry, Mary; ah, awe; cot, caught; ferry, fairy. If the speaker's accent heightens the differences between them I might be able to tell them apart, but cannot tell which word is intended by which of the different pronunciations. I can do that only from context. This may not be uncommon in American English; maps of regional variations in pronunciation suggest that about half of the country pronounce Don and Dawn the same way. This would explain why I hear those names the same way, but only if I lived in that part of the country. What is this trait called? I wish I knew.


In any sizable system, the number of equilibrium states are much, much greater then the number of non-equilibrium states. Since each accessible micro state is equally probably, it is overwhelmingly likely that we will find the system in an equilibrium state. However, for a closed system, one that does not interact with any external system, the number of micro states is fixed. Therefore the entropy is fixed. At any given time, the system will be in one of its micro states. Even if the system is in a micro state that does not look like equilibrium from a macroscopic point of view, its entropy will remain the same, since entropy is a property of all the micro states, not just a given micro state. So, are the number of accessible micro states (and hence entropy) of an isolated system constant?


There is a mother, Ann, who has a stereotypical 'good' personality: she is religious, a teacher, and a very generous lady. Her daughter, Emily, isn't the nicest lady: she whines a lot, insults people, and is ignorant to her family history. My professor asked to mention in my essay that Emily's personality was an 'un-improvement', as compared to her mother's. I can't find a way to say this without sounding awkward. What I have so far is: "Emily's personality shows a degraded structure as the generations of her family trickle down." I already know that I do not want to use that sentence: it sounds awful and confusing, in my opinion (I can bet that you had to read it twice before understanding what I was trying to say). Can anyone else think of a way I could say this?


When I've attended math competition discussions, I've often heard people remark "oh, this is a famous problem" or say that it's similar to one. Most of them I've actually never heard of before. Competition books tend to have a vast list of past problems, and I haven't been able to sift through and tell which ones are supposedly well known and which ones aren't. Contest math isn't especially a priority for me, but it would be nice to at least know of certain problems I should be familiar with. Could someone give a list of some problems and technique that are part of the "folklore", so to speak? I'm interested in any competition topic, but to narrow the discussion, let's try to emphasize competition problems in calculus/analysis (sequences, series, integrals, etc.)


I was wondering if there are some general definitions for direct product and for direct sum, for example in category theory or in set theory, so that the concepts for vector spaces, Abelian groups, rings can be unified, or in other words, the common features of those specific concepts can be abstracted? In particular, the following quotes from Wikipedia (direct sum and direct product) seem to make attempts to reveal their relation to Cartesian product in set theory and (co)product in category theory, but also say that the relation is not always true. one can often define a direct product of objects already known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions. one can often define a direct sum of objects already known, giving a new one. This is generally the Cartesian product of the underlying sets (or some subset of it), together with a suitably defined structure. More abstractly, the direct sum is often, but not always, the coproduct in the category in question. Thanks and regards!


I'm interested in learning about algorithmic trading, particularly in bitcoin. Looking at this chart, I can see that I could simultaneously offer a bid that was slightly higher than the highest bid, and an ask that was slightly lower than the current lowest ask. Whenever anyone bought or sold, that would mean that I would always be one of the people they bought/sold from/to. This would allow me to make a profit equal to the gap between the two. The problem I'm having is in calculating the risks. As far as I can tell the variables involved are: Variables out of my control Gap between highest bid and ask offered by others Average price paid for "pot" of BTC that I'm trading with Some measure of the volatility of prices over the preceding period (Risk) How much volume would move the market by a given amount higher or lower Variables within my control Maximum exposure in terms of money Maximum difference in ratio between GBP reserve and BTC reserve Size of the gap between my bid/ask prices (out from the exact centre as percentage of total gap) I'm struggling to figure out how to model this effectively though. I studied Computer Science and have a basic grasp of probability theory, but this is a bit beyond me. Any help, or pointers to the "proper" formula to model this would be greatly appreciated.


Of course a lot of people misunderstand the primary meaning of the adjective "moot" -- "open to question" or "argued about but not possible to prove" -- using the word only in the expression "moot point", which rightly or wrongly is often taken to mean "not worth debating". As a result the word is apt to cause confusion, even when properly used. So is there a better word to use for the minor meaning of "moot" -- "not worth talking about" or "no longer important or worth discussing"? A term which might be used in the sentence "The question of what color to paint the ceiling is _____ since we won't be adding on after all"? (One advantage of "moot" in this sense is that it's short and abrupt, which serves to enhance the meaning. Except that the wrong meaning may still be "heard".)


This might appear to be a bad question to some but i can't figure out the answer. While reading about an electric circuit, i found this on the web: Over the course of time, one could think of positive charges moving from the positive plate through the charge pipe (wire) to the negative plate. That is, positive charge would naturally move in the direction of the electric field that had been created by the arrangement of the two oppositely charged plates. As a positive charge leaves the upper plate, the plate would become less positively charged as illustrated in the animation at the right. As a positive charge reaches the negative plate, that plate would become less negatively charged. So my question is that why does only the positive charge flow? Since positive and negative charges attract each other, it seems to me that the negative charge can also move towards positive charge. Am, I wrong at some point? If the question is unclear, ask me to edit it in the comment.


Very soft question I admit, but it's something that's been bothering me for a while. I've been thinking that being self taught has the problem of accreditation. You can't evaluate a mathematician in a vacuum. You need an accredited mathematician to decide whether or not someone else is also a mathematician worthy of accreditation. Well, who evaluated the other mathematician? Other accredited mathematicians. It's sort of like becoming a member of an exclusive club. We put the job of accreditation on our universities. But what if some person was discovered, off-the-grid so to speak, who had taught themselves mathematics from library textbooks. How could such a person evaluate themselves? How do you know if you're making progress when you study? It's tricky. It's like language learning. Do I speak German more fluently now than I did yesterday? I've no idea. Who can say? It's like playing with Lego. How do you know if you're getting better with Legos? You build more complicated things. But who's to say one person's Lego helicopter is better than another's Lego Enterprise? What's the goal with Legos? Is there one? Should there be one? I know already that this question will be deleted almost immediately, but I think these are important questions and many people visiting this site are in fact self-taught and I'm sure these questions show up as massive roadblocks. Thanks for reading.


In English, a vector is said to have two properties: a length and a direction. The possible directions correspond to half-lines out of the origin (so that, eg, up and down are different directions). In many other languages, a vector is said to have three properties: directions correspond to lines (so that a vector pointing up and one pointing down have the same up-down direction), and a third property determines which way the vector is pointing along that line (up or down in our example). This may seem strange at first, but it's actually very useful to separate these concepts in mathematics. In fact, I need to do it in something I'm writing right now, but I can't find the right English words for it. Firstly, I need a word to indicate that third property. Is there any accepted term for it? In the wikipedia article on vectors, there is a picture where it is labeled as "sense", although the term does not occur anywhere in the article itself. Secondly, I might need a clear way to indicate which sense of the word "direction" I'm using (line or half-line). This does not need to be a single word, but it should be as clear and unambiguous as possible. It would be even better if any official references for this usage could be found. This problem is bound to have come up before in the translation of foreign scientific literature.


I saw on wikipedia that the shape of a raindrop is explained by using Laplace pressure. But why? Since the drop is in motion, we shouldn't be able to use an hydrostatic law, am I wrong? The bigger a drop of mercury is, the lesser spherical its shape is. Ideally, if we consider only the surface tension, the drop should be spherical. What energy should you take into account in order to understand the problem? I know the surface tension tends to minimise the fluid surface. I also know it will tend to minimise the potential energy of the drop. So we should take into account the gravitational potential energy. But how can we take the minimum of two different constraints (fluid surface and gravitational potential energy)? Is there a potential linked to the surface tension?


I was pondering about EM Waves and fields and felt that there is an inconsistency in the physical picture of EM waves that I have in my mind. For example let us consider a charge at rest . Now lets say we want to test the electric field due to this charge Q at a point P. So to do this we need to place another charge at that point and find if we can feel any force due to that charge. Now I think for Q to exert a force on test charge or an electric field to be established at that point there must be EM waves propagating from the charge Q to the test charge. However there are no accelerating charges which are the prerequisites for EM waves to be generated . So this brings in an inconsistency in the physical picture of EM waves I have in my mind. I believe that there is some misconception I am having and I am positively looking forward to replies to clear it out.


After a period in a .tex file, within a paragraph, should I use a space or a new line? I.e. like this: Lorem ipsum dolor sit amet, consectetur adipiscing elit. Donec rhoncus vestibulum metus ut suscipit. Sed suscipit sem ligula, iaculis dignissim turpis ultricies vitae. Curabitur interdum lorem fermentum tellus blandit venenatis et et nisi. Ut elementum interdum nulla, at dapibus urna consequat ut. Nulla sodales in nulla et semper. Proin et consequat lacus. Nam faucibus, dui sit amet sodales cursus, nunc neque mattis eros, a pharetra purus elit quis tortor. Or like this: Lorem ipsum dolor sit amet, consectetur adipiscing elit. Donec rhoncus vestibulum metus ut suscipit. Sed suscipit sem ligula, iaculis dignissim turpis ultricies vitae. Curabitur interdum lorem fermentum tellus blandit venenatis et et nisi. Ut elementum interdum nulla, at dapibus urna consequat ut. Nulla sodales in nulla et semper. Proin et consequat lacus. Nam faucibus, dui sit amet sodales cursus, nunc neque mattis eros, a pharetra purus elit quis tortor.


'George hasn't turned up,' said Nick. 'A nuisance his having to go back to Plymouth last night. He'll get over this evening sometime or other, I expect. In time for the dance anyway. I've got a man for Maggie. Presentable, if not passionately interesting.' A faint roaring sound drifted in through the window. 'Oh! curse that speedboat,' said Lazarus. 'I get so tired of it.' 'That's not the speedboat,' said Nick. 'That's a sea-plane.' 'I believe you're right.' 'Of course I'm right. The sound's quite different.' 'When are you going to get your Moth, Nick?' 'When I can raise the money,' laughed Nick. 'And then, I suppose you'll be off to Australia like that girl-what's her name?' 'I'd love to-' 'I admire her enormously,' said Mrs. Rice, in her tired voice. 'What marvelous nerve! All by herself too.' 'I admire all these flying people,' said Lazarus. 'If Michael Seton had succeeded in his flight round the world he'd have been the hero of the day-and rightly so. A thousand pities he's come to grief. He's the kind of man England can't afford to lose.' (Peril at End House, Agatha Christie) I suppose Moth here refers to an expensive brand/thing but can't figure out exactly what. The phrase in question is bolded.


Can anyone explain the essential difference between situations where nouns are used in a general sense and where they are used to indicate a specific type or instance of something? Specifically, I want to understand when to use a definite article and when to use no article at all. Murphy (fourth edition) give as an example of the specific type: When was the telephone invented? and as an example of the general type: Doctors are usually paid more than teachers. In some instances, i get a sense of why the noun is treated as it is, but not always. Moreover, in the above examples, it isn't obvious to me that the author means "telephone" exactly as a type of communication, and "doctors" may be defined as a particular profession (not in general, but rather precisely this class of workers who are paid better than others).


I've just gone through every google search I can think of without success. Is there an archive, database, list, or other compilation of LaTeX style files? I've found plenty of style files specific to certain journals or conferences, but are there any 'generic' themes out there? I'd like to make my document look a little different from whatever the out-of-the-box LaTex default, and more than just changing the font... I'm no pro, and don't have the expertise, time, or motivation to write my own style file. Also, I assume there are plenty of style files out there, analagous to the different 'Themes' or 'templates' used by microsoft products. Ideally there would be an image/example so that i could see what the style made the document look like. Where are they?


This is a phonetics question. I am teaching English as a Second Language. In phonetics, we all know the "i" in "think" is a "short i" sound. Additionally, the "i" in "bit" is a "short i" sound. However, when many of my students pronounce "think", "link", "stink", and so on - they pronounce the "i" in those words in an unnatural fashion (as if they are taking the "i" directly from other "short i" words and putting it into these words). To my ear, and my pronunciation, the "i" in "think" is far different from the "i" in "bit", even if it is short. In the same way, the "i" in "-ing" is different from the "i" in "bit" (or "sit", "grit", "fit", etc.). I believe this has a lot to do with the "n", but I'd like some technical explanation regarding the nuances of the system and how I can explain different pronunciations of a "short i" sound (instead of just having them listen and repeat).


We know that neutrino eigenstates are not mass eigenstate and this therefore produces neutrino oscillations. This is, however, deduced from the fact that the neutrino of one flavor produces the corresponding partner lepton of the same flavor (electron neutrino only produces electrons, etc.). My question is : how do we know that this is indeed the case experimentally? I know that the standard model predicts that it is the case, but how can we verify that, if the only way to tell that there was a neutrino of a given flavor is to detect the corresponding partner ? EDIT : To give some precision about the kind of answer I'd like : Saying : we know the flavor of a neutrino by the lepton it produces is not enough. In that case, we could imagine that a lepton does not always produces the corresponding neutrino (say, an "electronic neutrino" is by definition a neutrino that creates electrons when it interacts with a nucleon, but we could imagine that an electron could produce half of the time a "muonic neutrino" (always producing muons) and half of the time an "electronic neutrino"). I know that's not what predicts the standard model, but has it been verified experimentally ? If so, how can we do that, as we don't know what is produced before detecting it... I ask that because we could imagine that it could be another way to "explain" at least part of the neutrino oscillations physics.


"You just won the lottery? Chapeau!" This is the first time I have seen such usage in English. Literally 'Chapeau' means 'hat', but the intention (that I get from the internet) is something like 'My hat is off to you' or 'Hats off!' or 'Congratulations!' or 'Mad props!' or 'Good for you' or 'Cheers'. The primary question is: What is the provenance of this in English. And secondary questions: - at what point did it enter English? Is it direct to British English from across the channel? Or is it North American only, borrowed from French-Canadian? - And what is the feel of it? Is it mostly ironic (as 'good for you' often is)? - Is there a difference in English usage between the French singular 'chapeau!' and the plural 'chapeaux!' (or is it meaningless distinction in English). I've seen both usages.