I'm interested in the eigenmodes of the membrane for various mediums, such as vacuum, air, water, etc., which impose a damping effect on the membrane. This cannot be done by merely changing the value for the wave propagation speed, c, because this value belongs to the string/membrane itself, and not to the medium around it. Can the influence of medium be incorporated in the wave equation for vibrating string/membrane?


What is the word to describe a person who popularized a sport? The person does not need to be one of the original players nor even currently active. An example is Arnold Schwarzenegger popularizing competitive bodybuilding. There were many superstar bodybuilders before him, like Eugen Sandow, and he has been inactive for a decade. Arnold is, no doubt, the reason why many people started bodybuilding.


My English teacher just asked us to write a random sentence in English. Off the top of my head I wrote "I like to eat apples and bananas". She highlighted "apples" and said: "man, this is blatantly wrong". "Uh, what's wrong, M'am?" "It should be bananas and apples. English people always enumerate things in reverse alphabetical order. Always. Just like when they enumerate parts of the body, they always converge to the heart. Always. We do that automatically, without even thinking about it. Not doing it is wrong". Granted, I'm not a native speaker, but I've never heard anything like this before. Bullshit or what?


Possible Duplicate: Electrodynamics textbook that emphasizes applications I am a graduate student in applied mathematics and I am looking for a concise introduction to Maxwell's equations / basic principles of electromagnetism. (I have enjoyed the book by Purcell, Electricity and Magnetism, as an undergrad but I have forgotten most of it). I would like something that covers the conceptual details in a quick way and not necessarily a big textbook. You may assume that I have the necessary mathematics background. I am looking for something that will explain the physics, i.e not a huge textbook with intricate calculations but a short book to get a feel for what is going on.


So, I finished my undergrad with a degree in applied mathematics, but when reading some graduate level texts and/or papers, I often find myself struggling. I eventually get there, but I often feel like I lack the intuition necessary to be able to come up with concepts on my own. I feel like I'm just missing some pivotal step in the journey to mathematical maturity. Does anyone have any books/references/advice? PS: If this means anything at all, as I was not a mathematics major, I did not take analysis or abstract algebra.


The first thing that comes to my mind when I hear the term marketing is a word that is related to advertising, business and commerce. A colleague once said to me that she went to do some marketing. I came to realise that she meant shopping for grocery in the supermarket, only after dwelling further onto the subject. But this is something that is rarely used in such context based on my own personal experience. My question is: Is the usage of the term marketing in the literal sense still appropriate in the English speaking community?


Say I had a plastic water bottle that was slightly crushed, I could uncrush it (mostly) by covering the opening with my mouth and blowing air into it hard. Would it be easier to do this if the bottle was full of water? or while empty (full of air)? Why? My thoughts are that it would be easier if it was full of water because water doesn't compress as easily as air and so more force will be applied to the walls of the bottle. Is this correct thinking? Are there other factors involved?


Let's say you have a garden hose connected to an ordinary water tap which is opened fully. If you pinch the end of the hose, water leaves the hose at a higher speed (and this can be useful while watering plants, to reach pots which are further away). However when a tap (with no hose connected) is opened only slightly, water flows out at a low speed, possibly even in drops. The actions of pinching the end of a hose and of almost-closing an open tap seem similar, so why the difference in behaviour?


The cosmic microwave background that we observe uniformly around us is usually explained by assuming that our universe is the surface of a four dimensional sphere. That way the uniformity makes sense since there is no center. My question is if this is true then what is the explanation that describes the fact that the farther we look into space, the further we look back in time. I can't perfectly picture this and see how it would coexist. Help me out.


In my research, I make use of a tool which provides a platform for researchers to investigate phenomenon X in a variety of ways. I want to frame it this way in a paper that I am writing: "Using the --- platform provided by the tool for investigating X, we develop and illustrate a novel method to assess aspect Y of X". I am looking for an adjective to fill ---. "Versatile" comes to my mind. But I feel like there might be better adjectives to use. I don't mind restructuring the sentence if that makes it sound more professional. Any help is appreciated!


Given the sentence, We discourage people from committing crimes by using law enforcement, religion and education. I see two possible interpretations: [We discourage people by using law enforcement, religion and education] from committing crimes. We discourage people from [committing crimes by using law enforcement, religion and education]. Of course the second interpretation may not make sense in this particular case, but generally speaking both interpretations are equally valid. So my question is, how to rearrange the sentence for it to be well-structured and have only one clear interpretation?


I was reading Frankenstein and I've noticed that the word after the exclamation mark usually isn't capitalised (unless it's a noun). Some of the quotes I've found: Alas! who is safe, if she be convicted of crime? and Alas! to me the idea of an immediate union with my cousin was one of horror and dismay. and Great God! what a scene has just taken place! Why the word after the exclamation mark does not begin with a capital letter? Has the rule for using exclamation marks changed and when did it change?


I was waiting on a red light the other day and was wondering. If I'm in my car, not moving and I see a car that's going to hit me from behind. Would I (my body) be safer if I put on the break or if I put the car in neutral? I assume there's no car in front of me and there's no posibility to be hit by an other car. Also, the car crashing into me is going at a good speed.


What is an English phrase to describe a video that is not running smoothly, but rather has interruptions in it, becoming sort of like a GIF? This typically happens due to technical problems such as low memory or CPU power, which is why I'm afraid I can't think of a way to demonstrate exactly what I mean (e.g., I cannot paste a link to a video like that). I am not necessarily looking for an adjective; any phrase will do. I would also like to include another related question: what's a word/phrase for almost the same situation but with sound rather than video? I.e., when the sound comes and goes (e.g., over the phone).


I need to construct a sentence, in which I'm referring to a feature of each of the animals in a given species. I don't quite know what the possessive of species should be, both in singular and plural. Example: (Talking about a single species with fuzzy ears.) The species' ears are notably fuzzy. Is the above correct? What if I had to mention multiple species all having fuzzy ears?


Here are the examples of compound subjects: Everything on the bed and everything in the closet was organised in under an hour. Everybody who witnessed the shooting and everybody in the room were interviewed. Anyone on the soccer team and anybody on the basketball team was eligible for the scholarship. Should I also use was instead of were in the second example? Or the first and the third examples are incorrect that I should use were instead of was.


Not all equations of motion admit a Hamiltonian. Several questions and answers on this site concern this correspondence, for example Hamiltonian or not?, When can an autonomous system be written using a Hamiltonian?, and What are the necessary/sufficient conditions for a system to be Hamiltonian/non-Hamiltonian?. My question is who was the first to establish that not all equations of motion can be described in a Hamiltonian fashion? Hamilton himself? Someone else? In which paper or book was this lack of total correspondence first exposed? Who has then developed the details of this correspondence? What is the current state of affairs? (From Hamiltonian or not? I gather that it is still a matter of trial and error to obtain a Hamiltonian from the equation of motions?)


Does every convex, nonempty, body have a supporting line at every point on its boundary? This seems intuitively true to me, but I am having a hard time figuring out how to prove it. I was thinking of using a slicing argument. That is, cut a line through the body and then we have a set of curves which are convex. So more generally, do we need to prove that every convex function has a supporting line at every point?


What are some interesting applications of the Mean Value Theorem for derivatives? Both the 'extended' or 'non-extended' versions as seen here are of interest. So far I've seen some trivial applications like finding the number of roots of a polynomial equation. What are some more interesting applications of it? I'm asking this as I'm not exactly sure why MVT is so important - so examples which focus on explaining that would be appreciated.


I am trying to conform to the British practice (specifically Oxford Style Guide) and I am a little confused which to use to mark a nickname: Andrew 'Andy' Johnson Andrew "Andy" Johnson I know following the rules I should use single inverted commas, however the first one looks a tad weird to me. I have seen the latter option a lot on newspapers and business cards but not the former. I am going to use that for email signatures and names on business cards.


I am trying to find a verb which can be used to indicate that someone is acting pedantic. I first considered "pedanticise", but, having found only one source for this, I thought I'd broach the question here. Edit: To clarify, I am defining pedantic to mean: excessively concerned with minor details or rules; overscrupulous. -- Google Or: overly concerned with minute details or formalisms, especially in teaching. -- Dictionary.com An example of this word used in a sentence would be, "Because Ellie spent the whole day ____ing over the grammar choices in her emails, she didn't get any work done!"


There is a phrase in Malay that goes "prepare an umbrella before the rain", meaning one must be prudent and proactive of future challenges by making all the preparations necessary. I would like the English equivalent of such a phrase. Thanks in advance, and apologies if this has been asked before! (I checked as much as I could) EDIT: What I'm looking for is the act of preparing a solution to a specified problem before the problem happens. So anticipating rain, one prepares an umbrella.


It is generally assumed that there is no limit on how many bosons are allowed to occupy the same quantum mechanical state. However, almost every boson encountered in every-day physics is not a fundamental particle (with the photon being the most prominent exception). They are instead composed of a number of fermions, which can not occupy the same state. Is it possible for more than one of these composite bosons to be in the same state even though their constituents are not allowed to be in the same state? If the answer is "yes", how does this not contradict the more fundamental viewpoint considering fermions?


I keep hearing from native speakers the phrases like these: There is a lot of cars (books, hotels) There is a couple of cars (books, hotels) There is five (ten, etc.) of cars (books, hotels) There is a few of cars (books, hotels) Nevertheless, somebody told me that the phrases like above are incorrect for sure and there should be there are instead of there is. Whom could I believe?


When someone makes an assertion, the distinction between "how did you know" and "how do you know" seems to be that "how did you know" implies that the person in question is correct in their assertion. "How do you know" is normally an inquiry into the person's credentials, and often expresses that the assertion is incorrect and/or ungrounded. Does anyone know what the reason for this distinction is? Why does simply changing the tense of the verb change the implication so strongly? I suspect that, linguistically speaking, the "did" might be a different tense than the past tense, but I'm not sure what it is, or I could be entirely wrong.


I know, that if a polynomial function fails to have any real root, then, by the fundamental theorem of algebra, it must possess at least a pair of complex roots having non-zero imaginary part. The real roots are points of intersection of the curve, with the horizontal axis, on the real plane. My problem is in visualising location of complex roots. Can I make use of Argand Plane in visualising?


I found that some theories about quantum theory is similar to Fourier transform theory. For instance, it says "A finite-time light's frequency can't be a certain value", which is similar to "A finite signal has infinite frequency spectrum" in Fourier analysis theory. I think that a continuous frequency spectrum cannot be measured accurately, which is similar to Uncertainty principle by Hermann Weyl. How do you think about this?


I study cognitive neuroscience and I periodically run into physics related questions in the context of neuroimaging technologies. My question specifically refers to electric and magnetic fields that can be measured by electroencephalography (EEG) and Magnetoencephalography (MEG), respectively. One interesting difference between the EEG and MEG signal is that unlike the electric field, the magnetic field is unimpeded by differing conductances across brain, skull, scalp and other tissues. I was wondering if somebody could explain what differences between the two fields account for these phenomena.


I know that many questions on topics similar to this one have been asked before, but I have read many of them and still not able to solve my problem. You could attribute it to my poor command of English. My question relates to two weird sentences, and I want to know which sentence is correct: I am so hungry, it feels like my tummy is eating myself. or I am so hungry, it feels like my tummy is eating me.


Based on my very basic understanding of the Johnson noise, it's not just a DC phenomena, but should change with frequency in a system, where there is a frequency dependent, real component to the impedance. I think both the skin effect and eddy current losses contribute to the effective resistance of an inductor, so Johnson noise should increase at higher frequencies. Am I being dumb?


I had a confusion in the diffraction experiment about which I was taught in school. According to the formula we Derived The width of the central maxima decreases on decreasing width of slit and vice versa. But what i wonder is what happens to the max intensity of central maxima in these cases? I asked two physics teachers and they gave me opposite answers-one said it increases on decreasing slit width which i feel is right but the other said it would be the same as light is of constant wavelength(monochromatic). What do you think? Please explain:)


As a result of the Ehrenfest Paradox, the geometry of a rotating disc is non-Euclidean. However, while reaching this conclusion, we assumed that "the radius doesn't undergo Lorentz contraction", because "the radius is always perpendicular to the velocity vector", which is equivalent to : "in a circle, the radius is always perpendicular to the tangent at that point." But this is an Euclidean assumption (We have to use the parallel postulate to prove it.) Therefore it doesn't work in Non-Euclidean geometries, and we shouldn't be able to use it.


Good evening everybody, I'm stuck with the following problem from an old exam in Linear Algebra. One is given two Eigenvectors with corresponding two Eigenvalues and told that the trace is negative. I figured out a third orthogonal eigenvector and tried to get to the matrix via diagonalisation, using a parameter alpha for the third eigenvector. Then I set the trace of this matrix equal to the sum of the known eigenvalues plus alpha. But my method does not work out; it does not give me a negative value for alpha. Where am I going wrong? How could I make this work? Thank you so much.


For example in the models for holographic superconductors we can calculate the conductivity. Also there is an energy gap. I can understand that it describes a superconductor. However I have also heard people talk about superfluidity in holographic QCD. I believe they are talking about the condensates such as the pion or the chiral condensate. Where is the superfluid in holographic QCD (quark gluon plasma maybe?) and how do I know that it is a superfluid?


I would like to produce some diagrams (preferably using TikZ/PGF) using the visual notation given in Enterprise Integration Patterns by Hohpe & Woolf. I'm aware of stencils from the above site, based on Visio, but I'd rather be able to compose them directly in my LaTeX documents. Is anyone aware of PGF library packages that I can use, or failing that how I can create reusable symbols for use in a TikZ/PGF figure?


Even though I can find this command hiding somewhere on the internet, I though it might be useful to have it up here for the LaTeX community in general. How does one run TeX from the command line interface (Terminal) in Linux? Are there any required parameters? What if the command is part of a bigger script, how does the script know when the TeX processing is done to continue with any subsequent actions needed? Is there a clear manual for this somewhere? A starting point might be: pdflatex [options] filename.tex latex [options] filename.tex xelatex [options] filename.tex


If I understand correctly, axioms are those statements that we assume to be true, instead of proving to be true. I have seen that in topology theory, various axioms of countability and separation axioms seem to be definitions of some concepts, instead of being assumptions. So I was wondering if "axioms" in topology theory are really axioms? If no, is this kind of naming rules common in all other branches of mathematics? Thanks and regards!


Imagine I have a vertical straw, and that it is partially filled with water. If I were to decrease the radius of the straw at the bottom end, eventually surface tension would allow the water to remain in the straw (i.e. the force of gravity would not be enough to overcome the surface tension and for water to flow out of the straw). What I'm curious to understand is for a certain (small) radius at the bottom, how much force/pressure would be required at the top of the straw to overcome surface tension and force the water out of the bottom of the straw at a certain (volume flow) rate. Any help would be greatly appreciated!


If we have two complex numbers, in polar form, as the numerator and denominator of a fraction, and we are asked to write them as a single complex number, is there an easier way to deal with them rather than the usual procedure? (By usual procedure I mean first expanding them by writing the value of each term and then realizing the denominator, etc.) Thank you.


Why is Erbium Doped Fiber Amplifier (EDFA) preferred over Erbium Doped Waveguide Amplifier (EDWA)? The question has been asked from an engineering point of view, but obviously I would also be interested in Physics of it. Is it just because of the difficulty of constructing a waveguide, or is there something more? I understand the individual benefits and limitations of a fiber amplifier and a waveguide but cannot understand the preference in this case. http://en.wikipedia.org/wiki/Optical_amplifier#Doped_fiber_amplifiers


I have been using TikZ to draw game-trees for my game-theory lectures (mostly using the automata package) but I could not figure out how to draw an arc across the edges the leave a given node. When we want to denote that a player has a continuum of actions available at a node, we usually draw an arc across the edge leaving such node. For example see the arc crossing the edges that leave the first node in this picture: Any suggestions on how I could try to implement this in TikZ? Thanks!


We have a car dealer around here with an ad that reads "No Fake Lures". Now, as I understand the concept of a lure, if it doesn't attract attention, it's not a lure. And if a lure does attract attention, it's a lure, so no matter what is used to make the fake lure, it's a an actual lure. Therefore, it's not possible to have a fake lure. What do you call a statement like "No fake lures"?


I've been looking around about LCD monitors, and how they polarise light. When a pixel needs to be black, the light is "twisted" so that it can't go through the polarising sheet in front. What happens to this light? Does it relfect back into the screen? Surely that would mean the screen would get quite hot if you leave it on a black image for a while? (but I've never noticed that happen). What happens to the energy (from the electricity used to create the light) when it's blocked by polarisation.


I'm currently converting my thesis into LaTeX, and i have an issue: being an IT thesis, I have tons of code quotes. The problem is not about the blocks of code: they are not that many and i can wrap them with {verbatim} tags. The problem is about the millions of inline single terms which should be formatted the same way. Is there a way, for example, to define a macro or shortcut or whatever which, is i select one word and click it, automatically wraps it with {verbatim} (or equivalent) tags? Hope I've been clear. This could really spare me hours of tedious work, thank You in advance!


I am writing a novel and am constantly trying to figure out how to describe the action when someone gives a sort of small cough or a quick "huh" in agreement. I suppose it would be the non-word equivalent of "I'll say" or "You think?" To be specific, I'll use it in a dialogue. "He must really hate Ian, then." Clarissa [insert word here]. "Yes." It's usually used when the speaker is trying to convey that the subject is much more serious than their conversation partner is taking it.


There are many kinds of integrals, with the most famous being the Riemann integral which is taught in elementary calculus classes. The motivation behind the Riemann integral is to find the area between curves by using rectangles of infinitely small width. When i first studied calculus, i was fascinated by this formulation and thought it was indeed ingenious. My question is: what are the limitations of the riemann integral? What could be the link or the motivation to be able to "create" new ones like lebesgue, or darboux?


Heard an English teacher claim that: "Dogs is not my cup of tea" is correct; whereas "Dogs are not my cup of tea" is incorrect. The explanation was that the verb form of 'to be' must agree with the singular noun 'cup' and not the plural noun 'dogs'. Checked Google on this and it appears that this is an extreme minority opinion at best. Nonetheless, can the teacher's argument be authoritatively refuted, or must this person be allowed to continue impressing pupils with their dubious insider's knowledge of the English tongue?


I took my son to a science museum where they had a solenoid oriented vertically with a plastic cylinder passing through the solenoid. An employee dropped an aluminum ring over the top of the cylinder when there was no current going through the solenoid. Then they turned on the current going through the solenoid and they aluminum ring went flying up and off the top of the solenoid. What law of electro-magnetics causes the force on the aluminum ring?


While studying Electrostatics, I was wondering about whether a charged sphere gains or loses mass while just adding electrons? If it is possible then the negatively charged sphere will have more mass than positively charged sphere, but according to me I think that since the mass of electron is so negligible then there must be no change in mass just because of electrons. So is it possible or not ?


For a paper I'm writing, I need a bit of second order logic in order to convey a few steps in an inference. I've completed the introductory and intermediate/advanced logic classes at my school (second order logic wasn't included in the curriculum, unfortunately). I use predicate logic in most of my classes, so I'm comfortable with it. I'd like to learn just enough to properly notate the steps of the inference. Is there a concise tutorial on the subject available online?


As far as I know, whenever we talk about a black hole, we refer to a static or stationary solution to Einstein field equations. I know the formation of a black hole can be discussed by the numerical GR, but I have never saw an analytical solution which can describe the formation of a black hole. So do anybody know that there actually exists such solution which has power to describe the formation of a black hole? Even the simplest case, for example, the spherical symmetric collapse of radiation into black hole can satisfy me very well.


In Chemistry, an amount of energy has to be supplied for a reaction to occur. This energy, known as the "activation energy", breaks up the bonds between molecues in the substance. It is equivalent to the total bond energy of the reactants. However, in high school I learnt that the energy required to start a nuclear reaction is the difference between the binding energy of the reactants and the binding energy of the products. Why is it that the minimum required energy is not the binding energy of the reactants, similar to a chemical reaction?


Which are the differences between Polar curve and Nyquist plot in System Control Theory? I wasn't able to figure it out myself. Are they the same thing? As googling for these two concept returned similar plots, but from the theory I can't find any similarities between them, or I am not able to understand how they are related to each other? Edit: Even though I edited this after more than two years from posting it, I just realized that the terminology that is used at my university, Nyquist plot and Polar Curve mean the same thing.


I am curious about the vibration of metal objects as a sign of an impending lightening strike. This is apparently a common occurrence; so much so that a quick google search will yield many pages instructing people to head for shelter if their keys (other other small metal objects) start to vibrate in the midst of an electrical storm. One climbing site even mentions that a vibrating rope is a sign of a lightening strike. Can someone explain the mechanism behind this? My guess is that it a piezo electric effect, though this contradicts my (limited) understanding that atmospheric electricity is static. Thanks :)


I collected the following "top eight" text books on computability (in alphabetical order): Boolos et al., Computability and Logic Cooper, Computability Theory Davis, Computability and unsolvability Hermes, Enumerability, decidability, computability Hopcroft et al., Introduction to Automata Theory, Languages, and Computation (thanks to Bill Province) Kleene, Introduction to Metamathematics Minsky, Computation Sipser, Introduction to the Theory of Computation(thanks to Prajwal Kansakar) I know it's opinion-based, but which important text books did I miss?


I would like to get the formula on how to calculate the distance between two geographical co-ordinates on earth and heading angle relative to True North. Say from New York to New Dehli , I draw a straight line THROUGH THE EARTH - as they were two points in space. How can I calculate that angle from say New York to New Dehli if I was to draw a straight line through the surface of the earth . What kind of mathematical calculation/formula would be involved in order to do that?


There are three Sakharov conditions of baryogenesis: CP symmetry violation. Baryon number conservation violation. Thermodynamical equilibrium violation. In SM the first one is satisfied by CKM phases, the second one is satisfied by instanton-like transitions for zero temperature and sphaleron-like transitions for non-zero temperatures. As for the third one, I've thought that EW phase transition provides violation of equilibrium, but after reading Gorbunov and Rubakov book "Introduction to the Theory of the Early Universe: Hot Big Bang Theory" I've started to doubt. They write that this transition is of second type one; thus it doesn't provide sufficient thermodynamical equilibrium violation. So can EW phase transition provide baryogenesis?


I'm interested in the following question: I want to visit where my grandmother was born. To me it seems like a noun clause because I could replace the clause with a noun. For example: I want to visit Ireland. Is this the correct way to check? The definition I read for an adverbial clause states that Adverbial clauses of place modify the main verb in the sentence and provide information about the place that an action takes place. This definition has confused me about the difference between a noun clause and an adverbial clause. How about the following sentences: We can go wherever you want. You can show me where it is. Thanks for your help. Patrick.


I can calculate the derivative of a function using the product rule, chain rule or quotient rule. When I find the resulting derivative function however, I have no way to check if my answer is correct! How can I check if the calculated derivative equation is correct? (ie I haven't made a mistake factorising, or with one of the rules). I have a graphics calculator. Thanks!


I am teaching English to my cousin, but I am not sure how to explain phrasal verbs correctly. For example "take off". I explain it as two words but a single entity. When I ask her to name a verb in the sentence The plane is taking off, I expect to hear "to take off", not "to take". Do I understand this concept correctly? Update: In the sentence "She put her hands around her hear and squeaked." Is "put around" a phrasal verb or there are separate verb and preposition?


Does the demonstratives refer to the next word or previously mentioned statement? The medieval center is Prague Castle and St. Vitus Cathedral. An evening view of these illuminated landmarks is one of the most memorable sights in Europe. In the above sentence, does these refer to "Prague Castle and St. Vitus Cathedral" or to the "illuminated landmarks"? I'm a little confused about this. A clear explanation would be great. Also, what's the difference between demonstrative determiners and demonstrative pronouns?


Full disclosure: I'm taking my first complex analysis course as a graduate student and the title of my question looks like a dumb question to me. In any case, there's a problem in my book that deals with a sequence of "holomorphic polynomials" converging to a "holomorphic polynomial". Is this just a redundancy or is there some weird world where certain polynomials aren't (complex) differentiable?


Imagine two entangled quantum particles in the singlet state, one held by Alice and the other by Bob. Alice and Bob are both widely separated. Bob measures his spin in some axis and finds it to be up (say). And that puts his particle now into the prepared state of spin up. Entanglement means Alice, if she were to do a similar measurement would find spin down. But assume Alice does not actually do any measurement. So does Bob doing his measurement mean that Alice's particle has immediately and spontaneously collapsed into a prepared state of spin down? Or is the state of her unmeasured particle unchanged? The literature, and popular press, are both vague and inconsistent on this seemingly important point.


About a month ago I posted this question about eigendecomposition of a special matrix; and I received an answer. This is not my main field of study and I came to this as part of the proof for a method I developed to address a bioinformatics problem. Now I am interested to learn more about this area. Could anyone suggest some references and relevant literature that I can read. Moreover, I may end up publishing my method. How should I acknowledge and cite the stackexchange users/website ?


Suppose you have a set A which has the same cardinality with real numbers R, which means |A| = |R|. Also suppose that you have a finite set B , which of course has finite cardinality. Also suppose A is a proper subset of A union B, I mean there are some objects which are not an element of A but they are an element of B. Now is the cardinality of A union B greater than the cardinality of real numbers, also if so is this cardinality of A union B equal to cardinality of the power set of real numbers ?


I often find myself in the situation of wanting to capitalize the phrase "Enterprise IT" due to its frequency of reference as the title of my industry. It feels appropriate when I write: "In Enterprise IT ..." "The Enterprise IT industry..." I would not be surprised if my capitalization is off here. Could one of you help me with a definitive answer to the argument?


Forgive me if this is a non-question, but I could not find anything regarding this. I only know parts of GR, and I am not familiar with the math. Since energy and matter density curve spacetime, does this mean that when entropy increases, spacetime curvature flattens? Hypothetically, if the universe is destined to reach total entropy and be in equilibrium, would this mean that the spacetime curvature of the universe be mostly flat?


I am interested in the history of functions. Why did Euler introduce them? When and why did they become central to mathematics? I know the second question has something to do with the famous Fourier-Cauchy dispute over the heat equation, but don't have any sense of the details. By the way, I'm asking because I would like to give a short, historically informed explanation to high school students why functions are important.


When a trap is set up to be active, is it "sprung" or "unsprung"? I'm confused by the ambiguity of the verb "spring". That is, a trap that is set up and active could be "sprung" in the sense that a spring is compressed and ready to launch, but it could also be "unsprung" in the sense that it hasn't yet sprung to action because its spring force has yet to be released. And after a victim walks into the trap and causes it to spring, the same ambiguity (in my mind) seems to apply.


My dad's an author (English books with some Arabic) and until recently has been using WordPerfect for DOS for all his writing. He's been moving to Word and hates the lack of control he has compared to WP. I'm looking for options and LaTeX seems to be a good one. Here are the things I'd like to be able to do: Arabic inline text Support for chapters (as separate files) Support for ebook output (epub/mobi) Cross platform environment (Mac and Windows) Some way to move from WP and Word Is this realistic with LaTeX, and what would be the best way to start for a newbie?


I've read 'Axiomatic Set Theory' by Patrick Suppes, and one thing I've noticed throughout is that he seems to be obsessed with definitions, and he tries to allow for urelements. Is this standard for ZFC? I thought in general when we say 'set' in ZFC we really mean 'pure set', and so avoid having to worry about individuals. In addition I've never seen such a fuss over definitions in any other mathematical book I've read, is this something I should get used to in Set Theory? If this is not standard, can anyone direct me to a book similar to Suppes' which builds (from the axioms) all the usual set theoretical structures needed for other areas of mathematics that is?


I quite frequently use a word that sounds like "shtook", to mean, trouble with the law or other authorities, as in, "You'll be in dead shtook if you do that" or "you'll be in real shtook if you don't finish your homework", but I have never seen it written, and don't know how it is spelt. Can anyone shed some light on the matter. A search like this doesn't get very far; the real dictionaries don't recognise it and the Urban Dictionary here has only a typically obscene interpretation not in the least bit like I remember it.


I am undergraduate in economics. As you may know, most prestigious departments in economics now require their aspirants to have taken Real Analysis (and Linear Algebra, too) before entering their programs. I have time to take these courses but don't know whether any college will give admission in their math program to an econ undergrad. I don't know whether distance learning courses will help me. Can anyone suggest me how to take math courses? Your replies will oblige me.


I am learning Fourier transforms, Z transforms etc. in Digital Signal Processing and I can work easily with integrals. However, I don't understand how a Fourier transform converts time domain signals to frequency domain signals or why it is invertible. I also have a very basic understanding of linear algebra, not suitable for fourier analysis. I looked up books on fourier analysis but they don't explain anything I don't already know. So should the next logical step be to learn spectral analysis? Also, which books would you recommend for linear algebra? I already know matrices, eigenvectors, but not Hilbert spaces.


I was watching a TV show, and this girl said: I just wish that once, I'd bring a guy home that they actually liked. Shouldn't it be like this? : I just wish that once, I'd bring a guy home that they actually like. Didn't she mean 'I wish they'd like my future boyfriends'? I think this is OK: I just wish that once, I brought a guy home that they actually liked. But the way she said it sounds strange to me.


I am writing an article for a magazine about my visit to my village. I want to use a better word or phrase for 'my village' or 'my native place'. The following is the part 'When i heard of stories of hardships and suffering from my grandparents I realized how fortunate I was to live in the city I now call home. When the day came to visit "my village/my native place", I felt uncertain and nervous..' And better word or phrase to make the content more appealing?


In Euclidean space, There is a classic Theorem claims that: The length of every rectifiable curve can be approximated by sufficiently small straight line segment with ends on the curve. Now, The question is that: Is there a similar thing happen on plane of constant curvature? I.e., Can the length of every rectifiable curve on the plane of constant curvature be approximated by sufficiently small geodesic line segment?


In any cubic meter of space in our solar system there is predicted to be some amount of dark matter. Also in a cubic meter of space is a known average amount of cosmic ray energy. What is the ratio of cosmic ray energy to dark matter energy in our solar system? I'm curious if they are on the same order of magnitude or far from it.


Imagine two charges A and B separated by some distance. Charge A emits a photon which is absorbed by charge B. Is the recoil momentum received by charge A always equal and opposite to the momentum gained by charge B? Is this true both for static Coulomb fields and radiation fields from accelerating charges? I suppose there is no momentum "left over" in the EM field after the interaction so that all the momentum lost by A is absorbed by B. Is this how it works?


I have always thought it acceptable to say and write, "She looked at me disgusted." However, I know some consider it ungrammatical, saying it must instead be either "She looked at me disgustedly" or "She looked at me, disgusted." Please give me your opinions and explanations as to whether any of the foregoing sentences must be deemed grammatically wrong. Briefly, my own analysis is as follows. It is correct to say, "She looked disgusted." So why would it be incorrect to say "She looked at me disgusted"? (She was disgusted while looking at me--or, she was looking at me while disgusted.)


In "Pragmmatic Programmer" by David Hunt and Andy Thomas, there is a sentence: Do you ever watch old black-and-white war movies? To my best english grammar knowledge, present perfect tense should be used here: Have you ever watched black-and-white war movies? Is the version of the authors correct? If so, what's the difference between these two variants and what is the appropriate way to use each? PS. I checked this sentence in the translation of the book in my native language and it was translated exactly as if the orginal sentence was in present perfect form.


I read here that there is a general rule to write an adjective order. But I didn't find any explanation if the rule has a specific order for colours, especially for primary colours. This may sound stupid but I'm just wondering. I mean is it preferable to say: red and white flowers Or: white and red flowers Or is there really no grammatical rule to obey?


I heard an interesting argument from a colleague recently that went something like this. Whenever we are using an axiom scheme, we are essentially choosing one of the instances of this scheme, and hence, whether or not we include the axiom of choice in our axioms, we are implicitly using some kind of choice principle to choose that instance. My gut feeling is that this argument seems fishy, but also interesting, and I lack the expertise to give a good answer. My question is whether this argument holds or not, and whether it makes a difference if the axiom scheme is uncountable. I realise that the question is somewhat vague, but I hope there can be some interesting answers anyway.


An isomorphism indicates that two structures are the same, using different names for the elements. Therefore it's obvious that every (algebraic) property of the first structure must be present in the second. However, homomorphisms only indicate that the two structures are "similar", so it's not quite as obvious that every property will be preserved. Yet all the properties I've ever seen are preserved under homomorphism: commutativity, cyclicality, solvability... What are some examples of properties of algebraic structures not preserved under homomorphism? Feel free to use any algebraic structures you like, but I'm particularly interested in your garden variety structures: group and rings, say.


This is potentially a very stupid question but I'm going to ask it anyway. With all these huge buildings such as the Abu Dhabi Mosque, where an unbelievable amount of materials such as marble was moved from one side of the earth to the other, is it possible that if we 'shift enough stuff' that we could change the earths centre of gravity, and potentially alter its orbit?


There is no "action at a distance" in nature. Attraction of a piece of iron by a magnet, attraction between distant electric charges of opposite sign, have to be mediated by something. The virtual particles are proposed as an explanation. If they have an observable effect, it seems like they must exist. If so, why do we call them virtual? It can't be that they have short lifetimes, as there are short-lived particles (resonances) that we don't call "virtual." On the other hand, some people say that the so-called virtual particles only exist on paper, or that they are just a technique to help us handle our equations. So what is the meaning of the word "virtual" here? Do these particles really exist, or not?


Why it is said that the Microchannel Plates can be operated in vacuum? What is the maximum pressure in which it can be operated? Also, while it is not operating, should it be kept in vacuum? Is this because the semiconducting property of the channels can be affected by the atmospheric pressure? Or while operating the gas molecules can be ionized and burn the detector?


A friend of mine corrected my sentence but I couldn't understand it. Just hoping someone can explain it properly for a non English speaker. My sentence is: "Finally your passport GOT ready for pickup" My friend (native English speaker) corrected it to: "Finally your passport IS ready for pickup" Or "Finally your passport got processed and is ready for pickup" Why can't i use "got ready" in the above sentence just like in sentences "We got ready for bed", "Alpha team got ready to play" Thanks in advance.


I'm looking for a single word to indicate a point in time which is either the beginning or end of an event. Edit, for a better example: A bell rings at the start a round of boxing. A bell rings at the end of a round of boxing. Assuming the bell rings at no other times, we can say the bell always rings at the BLANK of a round. Currently, for BLANK, I'm using the phrase "start or end." It feels clunky. I'd like to replace it with a single word, if possible.


I couldn't clearly understand the phrase "credential mongering" as used in an article in the New Republic about the Ivy League. The extract, from "The Trouble With Harvard" by Steven Pinker, is as follows: Admission to the Ivies is increasingly seen as the bottleneck to a pipeline that feeds a trickle of young adults into the remaining lucrative sectors of our financialized, winner-take-all economy. And their capricious and opaque criteria have set off an arms race of credential mongering that is immiserating the teenagers and parents (in practice, mostly mothers) of the upper middle class. Is it to do with attaining qualifications by any means? Or is the phrase itself imprecise? Thank you!


This suggests that the current MLDA is not effective. However, the estimates in this paper challenge that claim and show that the effects of the current MLDA is statistically significant in all cases we observe. This suggests that the current MLDA is not effective. However, the estimates in this paper challenge that claim and show that the effects of the current MLDA is statistically significant in all cases that we observe.


I'm using latexdiff to highlight the differences between two versions of a tex document, using the default style which is UNDERLINE. This style formats added text in blue with a wavy underline, and removed text in red with a strikethough. I'm interested in formatting the output such that added text is in blue but without the wavy underline (and removed text is in red as before). Is there a command for achieving this? I wasn't able to figure it out from the man page. Another way would be to modify the CCHANGEBAR style, which formats added text in blue and removed text in red but without the strikethrough, to add the strikethrough to the removed text (my question is how).


I'm looking for an equation that looks like the image below. I am comfortable constructing functions that pass the vertical line test, but as this is a relation, I am not really sure how to start. Would this need to be defined implicitly? An explanation on how you constructed it would be great as well! I should note that the scale does not matter, just the general shape.


Group theory is all about symmetries. Can this be seen from the axioms defining a group? Or equivalently can the group axioms be motivated from this point of view? Of course one can look at several examples and check that the group axioms are fulfilled, nevertheless this doesn't make clear why the axioms have to be precisly like they are. Any ideas would be much appreciated.


A few years ago I read a short little article about how big our eyes would have to be to observe microwaves (or any long-wave radiation for that matter). I don't remember enough about the article, or know enough about observing EM waves to reconstruct the equations, and I can't find the article again. So my question is simply this: is there some equation or rule of thumb that says if you want to observe ('see') a wave of length x, then your observing tool ('eyes') have to be such and such a size?


I am currently doing a one semester course on groups and rings where we have learned about (so far): Definitions of groups, subgroups, cyclic and normal subgroups, the symmetric group, homomorphisms, isomorphisms, The Correspondence Theorem, Product and Quotient Groups. As of yesterday's lecture we learned about the First Isomorphism Theorem and a little bit about rings. By the end of the course we should have done rings, endomorphisms, The Orbit-Stabilizer Theorem and subjects which I am not sure about. I am wondering if this would be sufficient to start Atiyah Macdonald; I have opened the first few pages and it looks hard. For those who have done it, what do you think are the prerequisites before doing this? Perhaps something like Herstein's Topics in Algebra? Thanks.


I've noticed people use (in speech) the word godness for "feminine god", e.g.: Oh my godness! However, in classic texts it is goddess, e.g. Shakespeare's "King Lear": Hear, nature, hear; dear goddess, hear! Suspend thy purpose, if thou didst intend To make this creature fruitful! Into her womb convey sterility! It is my understanding that godness can be an urban slang for "divinity", but not for an impersonated deity. Is godness just a mistake?


I am trying to understand pure and mixed states better. If I have N quantum particles in an isolated system. The many-particle state is a superposition of the product of single-particle states by the appropriate statistics (bosons, fermions, or distinguishable). Would this state be still considered pure since there is no interaction with the environment? Does that mean isolated systems or microcanonical ensembles are always pure?


I have studied that topologically equivalent metrics produce the same open and closed sets. They also produce same compact and connected subsets. Does it mean that topologically equivalent metrics have same open, closed or compact subsets? In what context topologically equivalent metrics differ from each other? Intutively, how can we explain topologically equivalent metrics? Can we say that two topologically equivalent metrics have same topological properties? Thanks and regards


If you were going to teach you kids programming and asked me what book to use as a guide, I would recommend you either Java programming for kids or Python for kids. But what if I want to teach kids math including derivatives, integrals, differential equations, set theory, etc? What book can provide me with a road map I can use (assuming kids are smart enough to learn what I give them)?


I am a beginning graduate student with (almost) no background in algebraic geometry. I would like to learn the proof of the Riemann hypothesis for curves over finite fields, including all prerequisites. I am looking for books to get me there. I am not necessarily looking for the quickest way, but rather for self-contained well-written books that will get me to this result. Nevertheless, I prefer to be focused and study only the needed prerequisites in the adequate generality (in particular, I'm focusing on the case of curves, rather than more general varieties). (my knowledge in group theory, field theory, and algebraic number theory is equivalent to what appears in J.S. Milne's notes, but as mentioned - I have almost no background in algebraic geometry).


This question is related to another question. If we have a Riemann surface with punctures of negative Euler characterstisc, how can one define a complete hyperbolic metric? I know that in this case the universal cover is the hyperbolic plane and it has a complete metric. Do we project this metric to the puntured surface? If so, why is it complete? I will deeply appreciate if somebody gives an example or a good reference.


I'm not sure of the name of this problem so haven't really been able to research it. I have a complete weighted graph with a start and end node and n distinct sets of nodes (lets call them red, blue and green), each with m member nodes. I need to find the shortest path from start to end and must pass through exactly one red, one blue and one green node; is there an algorithm for this? An extension would be that I need to find the shortest path while visiting blue first, then green, then red (again, exactly once each); is there one for this?


Let's say for example: Two people sing the same note (frequency) and volume (amplitude) together. Why is it that the two persons sound louder than they would individually? I would imagine that since the traverse waves generated by the two persons would randomly be louder or softer, based on the phase shift between the two sound waves. But that hypothesis is obviously untrue since in the real world a choir is clearly louder than a soloist.


What is the exact word for someone who always pretends to be nice to someone's face but makes fun of them behind their back? When they are talking to you, they'll treat you very well, like a good-mannered person. They care a lot about their image in front of others and show a positive image of themselves in front of an audience. But they are not actually good in their heart. Any word except hypocrite.


How do you say to someone that you will reuse a sentence (or a joke) you've just heard from them, as-is, because you liked it a lot ? In Italian we say "Questa me la rivendo", that translated is "I'm gonna resell that one" (ri-vendo / re-sell is the idiomatic part, the other words could be recombined / replaced). I can't find anything in the web, so... what is the English version ?


I'm looking for a noun for a person to convey two things: their particularly keen appreciation of aesthetics, and their ability to create aesthetically appealing objects. For example, Steve Jobs. A man with a famously keen sense of what is beautiful, at every level, be it user interface design, the connectors on cables or the stairs at retail stores. The closest I've got is "aesthetician", but that conveys more the study of aesthetics, rather than the appreciation and ability to craft aesthetic perfection. Any ideas?


I work for a client at the rate of X USD (United State Dollars) per hour. Recently, he accidentally paid me at the rate of Y USD per hour. Y is greater than X. I notified him of his mistake and he replied with the following sentence:- Go ahead and keep the difference to be applied to the next round of work you do for us. What does this mean? Should I accept the payment at the rate of Y USD per hour or correct it to X USD per hour?


Recently I've wondered about two idioms which have a strange relationship. Come Hell or high water and Lord willing and the creek don't rise Grammatical accuracy, alternative formulations, and questionable folk etymologies, and literal meanings aside, why do these two phrases (often used interchangably) have such different implications? My thoughts so far have centered around the former being an expression of an internal locus of control (i.e. I will make this happen) and the latter of an external locus of control (i.e. I hope this won't not happen). Why does the more apparently positive formulation reference Hell while the less (certainly) positive one mentions the 'Lord?'


I always have doubts whether to use a singular or a plural noun when I refer to certain peoples. For example, we say Americans, Italians, Brazilians, Russians and Austrians. But we say The British, The English, The Portuguese, The French, The Spanish, The Chinese, and The Irish. When it comes to those nationals I rarely hear of, I'm never sure. These would include people from countries such as Azerbaijan, Armenia or Kenya. Is there any rule as to whether we should use singular or plural ?


I have come across many hand dryers that attempt to dry your hands really fast after you wash them. Here are two of them: XLERATOR http://www.exceldryer.com/ Dyson Airblade http://www.dysonairblade.com/homepage.asp So I guess I have a ridiculously high standard cause I think even these are too slow. Would it be possible to create a large static electric field to attract the water molecules off of your hand? Can someone offer some ideas that would remove the water off of your hand using some type of electric field generated by charged plates or something?


I am a high schooler who really likes math, and I am interested in pursuing it in my undergraduate years. I have a basic facility with proofs, and I am currently exploring several different areas of math (analysis, graph theory, and topology). I am interested in seeing if I can contribute to mathematics even if it has no applications, mathematical or otherwise, however most areas of math don't seem to have any "low hanging fruit" so to speak, so I'm looking for a different, more obscure area where I can learn all that is known, and maybe contribute myself.


Prove or Disprove: If V is any vector space and with an orthonormal basis B for a subspace W of V, then V has an orthonormal basis C with B as a subset of C. First Question: Can you have an orthonormal basis for any vector space or can you only have an orthonormal basis for inner product spaces? Second Question: Does it also make a difference whether or not V is finite-dimensional or infinite-dimensional?


I always find it difficult to discuss the meaning of a word because I don't really have a definite meaning of word in my head. Cook refers to the verb (to cook) but it can also refer to the noun (a cook). Would you say cook is one word with multiple meanings or that the verb and the noun are separate words? Is a word simply the arrangement of the letters?


I'm trying to come up with some math problems (word or otherwise) that get to the meaning of adding zero, but I'm getting stuck because it seems just too simple to me. I have come up with questions like "John is having a birthday party and he invites five friends over but nobody shows up. How many people are at the party?". While this question gets to the problem at hand, it might be too simple to understand the meaning of adding zero. This problem also suffers from the possibility that first graders will answer like so, "Well there would be three people at the party because John's mom and dad would be there too." Any ideas?


As the title says. For example, if I take a fat metal wire with non-negligible thickness, and then wind it into a spiral, but such that none of the parts touch one another (there are no topological 'holes'), then if I add a charge to the wire, would it equally distribute over the wire? Or would the charges from neighboring 'loops' of the spiral interact with one another somehow, similar to a capacitor almost?


Outside of your field of research / application, how much of your undergrad education have you retained? Thought experiment: How would you fare today if handed old exams from your introductory topology / number theory / differential equations / whatever classes? No studying or preparation. I've always wondered how much of this material one should expect to have internalized, and how much is acceptable to forget over time and need a reference aid.


In Mac OS X, the home/end keys per default throw you at the beginning/end of the document instead of the beginning/end line. I have changed my system key bindings to get the latter, "usual" (Windows- and Linux-like) behavior. However, texmaker seems to ignore this setting. It is horribly distracting when I have to search for my editing position in a longer text each time after unconsciously hitting the end key. I did not find these commands in the "configure texmaker" dialog. Is there a way to persuade texmaker to respect the system key bindings, or can I modify the setting in texmaker individually?


On the new Astronomy.SE site, I was having a short discussion on one of my answers. The basic discrepancy was; can MACHOs like black holes/brown dwarfs/neutron stars be termed "dark matter"? My reasoning is that these objects do not radiate EM radiation on their own but they do gravitate, and thus constitute a small part of the total dark matter in the universe. I agree that there is a lot of dark matter which doesn't In other words, can the term "dark matter" be applied to nonradiating (or faintly radiating) bodies which still participate in the electromagnetic interaction (baryonic or otherwise)? Or is it necessary for all dark matter to not interact electromagnetically?


It is quite common to hear in newspapers and television about the increasing temperature, i.e. global warming. But I am interested in something else. How does global warming affect the internal energy of the earth, or else how does this change the gravitational potential energy of a point mass and the earth? This is a open question and everyone should feel free to make any plausible assumptions in estimating the increase or decrease. I thought of global warming as causing the expansion of the atmosphere, possibly changing mass distribution of Earth's atmosphere.


In my school and university I was taught to say "Not at all" or "Don't mention it" in response to "Thank you!". Now I rarely hear these phrases used, but rather something like "You're welcome", "It's OK", "My pleasure", or "No problem". My real life conversation experience is very poor. I often listen to some English learning podcasts, and watch some films in English. So I listen to dialogs, which are probably not from the modern real life world. How do native English speakers tend to respond to "Thank you!" now? What I should care about, when choosing from the available options?


Are there any big controversies in contemporary mathematical research? Other domains contain big controversial research topics (for example string theory in physics). The specific nature of mathematics however, makes me suspect that there isn't much room for any serious disagreement, or at least less room in comparison..? Is this intuition right, or completely ignorant and absurd? --- This is not a question about mathematics per se, but more a question about the people doing it. I hope the question is still appropriate in this stackexchange. ----


If for some reason I want to create the lightest document (in terms of the weight of the output file, not the font), what would be some good practices? Specifically in terms of: Packages: do they add significant weight to the file? Are there some which are particularly heavy? Graphics: should they be EPS, PDF? Some other format? Is TIKZ code always lighter? Classes: do some add more bulk than others? (Some other important things I'm sure I'm missing...) Of course I could just start trying to "play" with the above mentioned elements, but I was hoping to get an informed answer, not just the result of trial-and-error experiments.


From The Assassination of Jesse James by the Coward Robert Ford: Ed Miller: I was with a girl once. Wasn't a squaw, but she was purty. She had yellow hair, like uh... oh, like something. Dick Liddil: Like hair bobbed from a ray of sunlight? What does bobbed mean in this context? I guess it means "cut", but I would like a native speaker to verify.


I am aware of the idiom like a fish out of water. What intrigued me is an article using like a fish takes to water. Teo Zhen Ren, the swimming sensation from Singapore, took to swimming like a fish takes to water. Is like a fish takes to water a valid usage? Is it an idiom? I thought the usual phrase was like a duck takes to water.


Is there a systematic treatment of (finite dimensional) manifolds with corners in the literature which carefully introduces all usual differential topological notions (submanifolds, embeddings, etc.) and which includes proofs of the usual statements in geometric topology like the existence of collars or isotopy extension theorems in the generality of manifolds with corners? Most of the common textbooks treat the case without corners nor boundary and mention the case of boundaries. Some of them take care of boundaries more closely, but I am not aware of a detailed reference covering the situation with corners.


I have to translate a Washington Post article on Trump but I cannot understand one part: It's post-policy politics. It's about having the right feelings instead of the right proposals. Now, that doesn't mean that we shouldn't point out when he's contradicting himself or when his plans don't add up. But it does mean that we should probably be a little less credulous about this supposed policy pivot or that change in message. There has to be one for that to happen.


It's logical to think that the time it takes a microwave to heat the food would be proportional to the mass heated. But since a microwave is based on dielectric heating, I think that if you increase the mass of food there will be more water, which will heat the food faster (due to thermalization). Is this reasoning right? Is there an optimal quantity of food to heat and the time it takes?


If I had a rigid body, such as a wardrobe, so that its centre of mass is above the ground and I tilt it slightly. I have read that the normal force of the floor on the wardrobe must be equal and opposite to the weight of the object as we have no translational motion. However if I let the wardrobe topple over, its centre of mass is now lower. How can the centre of mass have changed positions if the normal force and weight are equal and opposite to oppose any translational motion?


Why doesn't a plane wave solution represent a single photon? And what is meant by the quantum-mean field being zero? EDIT: This post is an extension to a previous post I made asking about the photon in QFT. I was asked by the person helping me to start a new question so he can respond in the form of an "answer" post and thereby keeping all "answers" on topic. The link is here Quantum State of Photon Question


The mechanical advantage is given by MA = Load/Effort which is quite universal (at least in my short experience with them). However when i reviewed over the mechanical advantage of simple machines, especially levers , the formula for mechanical advantage stunned me, instead of MA = Distance of Fulcrum to Load / Distance of Fulcrum to Effort it was instead MA = Distance of Fulcrum to Effort/ Distance of Fulcrum to Load Does this not contradict what the MA formula is? Thank you for your time


There must be a name for this sort of after-the-fact non-argument. Sorry, this is the only way I could find to describe it. Debbie finds a kitten. Kitten has been burned over half its body, but it's alive. Debbie says, 'The kitten wants to live!' and takes it to a vet. Bob says, 'The kitten is suffering and will continue to suffer horribly, the kindest thing would be to euthanize her.' Debbie refuses; the kitten continues to suffer for months but in the end manages to survive. Bob says, 'The kitten suffered unnecessarily.' Debbie then says to Bob, 'The kitten would now be dead had it been up to you. You have no respect for life.'


In a crystalline solid each atomic level 'splits' into n levels (n = number of atoms in the system). When the number of atoms is large each level becomes replaced by a band of closely spaced levels. In a semi-conductor we have an empty "conduction band" and a fully occupied "valance band". Conductivity arises because electrons get excited to the conduction band. Question: Why can't electron in the valence band freely move around and therefore conduct electricity? My question also applies to metals where the conduction band is already half-filled. What's special about this conduction band that allows electron to move around freely?


According to the popular Solar System formation models, what are the bounds on the Gas Giants' cores (note I am only interested in the bounds for our Solar System). For example according to Wikipedia on Jupiter Assuming [a core] did exist, it may have shrunk as convection currents of hot liquid metallic hydrogen mixed with the molten core and carried its contents to higher levels in the planetary interior. This suggests that a core would shrink over time, which would put an upper bound on the size today. Are there other bounds? I would like numbers together with some reasons for those numbers.


I have two books on algebra where one makes the definition of isomorphism to be a bijective homomorphism, while the other makes it as an injective homomorphism. I am doing the exercises from one book and learning from the other (not optimal I know), and so I am wondering what definition is considered the convention. If both are conventionally okay, is something lost by removing the surjective part of the bijection and if so what? I realise it's a rather uninteresting question, but if someone has the time I would appreciate the answer. Thank you!


I'm trying to draw a chain of arrows going under and above each other, as depicted: The issue isn't labelling the arrows, I can easily do that, but I don't know how to get them to curl, have the arrow head in the middle (ideally, though not necessary), and have one on top of the other. Any help is appreciated. If a solution exists without TikZ, I would prefer it. For those interested, the above is an eigenvector of the Cartan generators of a Lie algebra acting on a state as a raising or lowering operator, used in the roots and weights method to classify all semi-simple compact Lie algebras.


I'm having trouble with a problem in Propositional Logic Using induction I am supposed to show that if a well formed formula (wff) X has no repetitions of sentence letters then X is invalid. The hint in the back of the book says "Instead of trying to show directly that every wff without repetition of sentence letters has the feature of PL-invalidity, find some feature F that is stronger than PL-invalidity (i.e. some feature from which PL-invalidity follows), and prove that every wff has that feature." What does invalidity follow from?


It is well known that the Lagrangian of a classical free particle equal to kinetic energy. This statement can be derived from some basic assumptions about the symmetries of the space-time. Is there any similar reasoning (eg. symmetry based or geometrical) why the Lagrangian of a classical system is equal kinetic energy minus the potential energy? Or it is just because we can compare the Newton's equations with the Euler-Lagrange equation and realize how they can match?


After a hot shower, the mirror in my bathroom steams up. When I try to clear it with a towel, it immediately refogs. Yet once I use my hair-dryer, it will clear the fog and the mirror will stay clear. I'd like an answer to the mechanics behind this difference. I suppose it got something to do with the heat, and maybe the refogging is only delayed due to it. But that is just me speculating, and I'd like a concrete answer on this phenomenon.


I can see this is true for the sum of two roots of unity with some basic trigonometry (the resulting argument is the half the sum of the original arguments, and so must also be a rational multiple of Pi) but trying to extend this method to the sum of more than two doesn't seem to work. Is there some other way of showing this? It seems related to this question: Sums of roots of unity but I'm not sure if that can be extended to this.


I am studying Hyperbolic Geometry. At this part, I have proved that semicircles and straight lines orthogonals to the real axis are geodesics in the hyperbolic plane. But how I proof that this geodesics are uniques? That it does not exist others geometric places between two different points that minimizes the hyperbolic length? I have seen a topic related to this, but I do not know nothing about Riemannian Geometry. If someone could indicate to me some references to this proof(or the idea behind) I will be very grateful!


I need to show through a proof that the set of nonnegative numbers is denumerable I know a set is denumerable if its members or elements can be put into an order and counted. I am supposed to show this through a proof as well. I was considering using Induction possibly but I'm not sure if this is a viable or feasible method to use. Any help is great appreciated


I have a rather large text file that I'm trying to convert into a table. I'm pretty new to Latex, so I was using this other answer from this site as a guide to help me: save table on a separate file and I got some great results, but I realized that my table needs to be split up across many pages, which they say to use longtable to achieve. I don't completely understand where you would add that in or which part you would replace with that, and I was hoping that someone could explain that to me. Thank you!


We know that gravity affects time. More the gravity, slower the time, and vice-versa. Now consider a hypothetical condition where two people are walking next to each other. One person is subjected to a lot of gravity and the second person to a lot lesser. How would first person see motion of second person? Since time would be running slower at first person's place and faster at second person's, would the first person see the second person to be moving faster than we usually see? Like a video is being fast forwarded?


In the phrase 'in the field instead of behind a / the desk', would you use 'a' or 'the'? It seems to me that 'the' is the more sensible option, as English has other expressions of the 'the N' form which denote an activity associated with the object referred by the noun, e.g. the bar as signifying the legal profession, the pen as in 'the pen is mightier than the sword'. However, a search on Google shows that 'in the field instead of behind a desk' exceedingly outnumbers the other version.


According to general relativity, inertial mass and gravitational mass are the same, and all accelerated reference frames (such as a uniformly rotating reference frame with its proper time dilation) are physically equivalent to a gravitational field of the same strength. Refer: Einstein's thought experiment of a Physicist in an Accelerated Box. I find that the 'How' part of this question is somewhat easy to understand Mathematically but hard to visualize physically and on the 'Why' part I am totally blank.


A chain of some mass, forming a circle, is slipped on a smooth cone. If we consider an infinitesimally small section of the chain, a component of gravity will try to accelerate it along the surface of the cone. So there must be some tension in the string that prevents the acceleration. What will be the direction of the tension on this section of the chain? Will it be towards the height of the cone, parallel to the ground?


In mathematics, we talk about tangent vectors and cotangent vectors on a manifold at each point, and vector fields and cotangent vector fields (also known as differential one-forms). When we talk about tensor fields, we mean differentiable sections of some tensor power of the tangent or cotangent bundle (or a combination). There are various natural differentiation operations, such as the exterior derivative of anti-symmetric covariant tensor fields, or the Lie derivative of two vector fields. These have nice coordinate-free definitions. In physics, there is talk of "covariant derivatives" of tensor fields, whose resulting objects are different kind of tensor fields. I was wondering, what is the abstract interpretation of the general notion of a covariant derivative in terms of (tensor products of) tangent vectors and vector fields.


Let's say you wanted to calculate, for example, the average (i.e. mean) cost per day for a hotel to accommodate a tourist. If only aggregate cost data is available, is dividing the average cumulative cost for one person's stay by the average person's duration of stay an acceptable method? Would using the median cost and median duration of stay be better? Worse? The same? I know the median is less susceptible to outliers than the mean, so if both the median and the mean work the same I'd rather use the median.


Richard Feynman showed that Quantum simulation on a Turing machine will have an exponential slowdown. If that is so, does this put quantum simulation outside of P (complexity class)? I thought quantum simulation was polynomially possible on quantum computers, but there is still no proof that BQP is strictly bigger than P. So either quantum simulation has not been shown to lie outside of P, or it is not in BQP. I can't seem to find the answer.


So I was wondering about the event horizon on a black hole. And wondering if the point of no return for radio waves vs gamma rays would be different. I guess the logic being, since gamma rays have more energy than radio waves, their point of no return might be different. But I'm not sure if gravity has the same affect on all frequencies of electromagnetic radiation. I'm sure the answer for this would lie in gravitational lensing, ie are different colours of light lensed by the same amount? or do higher frequencies lens less?


I was watching a Physics TV show, When someone called Alex Filippenko said that when there was the Big Bang, the Space extended at a speed faster than speed of light. He said that it wasn't against the Theory of relativity because space isn't a particle and can go faster than speed of light. So I wanted to know if that is a fact? And otherwise how could the light go further if space around wasn't going at least as fast as light?


I've been reading my Stewart Calculus book and I honestly find most of the coverage of sequences and series easy to grasp (excluding power series, Taylor and Maclauren since we haven't covered those just yet). However, when the book guides me to a test to use to solve a problem I don't have an issue, but if I'm given a bunch of problems and told to choose a test to use I honestly have no idea where to begin. How can I intuitively know how to attack a given problem? I realize the thought process is quite similar to integration, but I still feel like integration is much easier than determining convergence and divergence of a sequence/series.


I have been interpolating cubic splines to some data, but it is now clear that I need my curves to be monotonic. Wikipedia and StackExchange sources describe how to impose the monotonicity condition while fitting with cubic Hermite splines. However I need my splines in B-spline form and I'm not sure how to translate from Hermite form to B-spline form. Is there a method to do this monotonic interpolation so that the resulting curve is explicitly in B-spline form (i.e. a list of knots and B-spline coefficients)? If not (and this might warrant its own question) how does one take a spline in Hermite form to B-spline form?


I would like to write a sentence to thank a number of colleagues. I would like to express the idea that it was a pleasure to spend time with them. Which of the two sentences below is better? I also take this opportunity to thank a number of colleagues with whom I had the pleasure of spending time these past two years. I also take this opportunity to thank a number of colleagues I had the pleasure of spending time with these past two years. Can you think of an improvement?


if I make a circuit with a battery and a capacitor (with a dielectric inside), how it is possible to get a current in the circuit? If electrons go from one pole of the battery and they arrive to one plate of the capacitor they cannot flow to the other plate of the capacitor because the plates are separated by a dielectric and this dielectric doesn't have free charges, they are bound charges. How can I explain a net flow of electrons in this circuit? Thanks.


I have been told that industrial mixing machines (say, for cake batter) switch directions periodically, first stirring in one direction, then the other, because this mixes the material more thoroughly. I imagine (but don't know for sure) that stirring in only one direction will tend to create helical structures in the mixed material, where each helix is more or less uniform but two helices might be quite different from one another; and that switching directions tends to break up and mingle these helices. Is this at all correct? Is there a way to quantify the effectiveness of different methods of stirring? If so, how much better is it to stir in alternating directions, and how often should one switch directions?


I am thinking of a (greatly simplified) computer simulation of a universe that followed something like Newtonian rules. Inside the simulation are A.I.s that are made from those same rules, and can only use those rules interact the world around them. Would there be some fundamental limits on what those A.I.s could work out about their universe, like their own version of an uncertainty principle? Sorry for phrasing this question in such a convoluted way. If anyone recognises what I am asking, and can point me in the right direction that would be appreciated.


I quoted the following from a pamphlet: Please read the instructions carefully before filling out the application form. The application will be returned to you and the registration may be delayed, if the information is not filled in completely. Does "filling out" equal to "filling in"? The words in and out have opposite meanings, how can they produce almost equal meaning in these phrases filling in and filling out?


I have not idea how to call these things, but I just saw a paper with the following figure and I was wondering how one would go about doing it in LaTeX. The closest thing I saw on TeXample.net was the Ford Circles example, but that's for a very specific case. I would like to see a way use this as a plotting device. At this point, I am not even sure how to do it in "normal" programming language, so perhaps the question is ill-placed. At any rate, I'd love to hear what people think!


Is there a term for when a person is getting really irritated/frustrated by someone, but they don't want to yell, so they do that thing where they exhale sharply through their nose? Say, for example, in customer support the operator has to explain and re-explain a procedure over and over, but the caller can never seem to get it right. To be clear, I'm talking about the act of blowing air out of the nose when angry/frustrated as opposed to breathing an exasperated sigh out of the mouth. It's just air; there's no vocalization or loud sound like a snort.


Is there any difference between the mass defect and the mass deficit? I have read that the mass defect of a nuclide is never negative and have also been told that the mass defect is the same as the mass deficit, which can be negative. Is it purely a sign convention? As a result of this is the binding energy of a nuclide ever negative?


While studying about band theory of semiconductors, I observed that when the electrons were excited from the valence band to the conduction band, they left behind holes in the valence band. From my existing knowledge, I believe that the valence electrons alone occupy the valence band which tells me the valence band is negative. For the holes to exist, the valence band has to be neutral. So, why is the valence band neutral? If my reason for the valence band to be negative was wrong, I would like to know the reason.


I've been watching this little HBO show by the name of Game of Thones. Perhaps you have heard of it. In the most recent episode the following line is used (and also happens to be the title of the episode): Now their watch is ended This sounds foreign to my ear. I would be inclined to say "Their watch has ended". Is there a subtlety in the English language that alters the meaning of the sentence by changing "is" to "has" and vice versa in the context of a simple sentence such as the above?


I have an upcoming statistics exam and I'm studying it on my own. I was recommended Hogg's Introduction to Mathematical Statistics, but I didn't find it helpful. I just want a book which covers basic mathematical statistics and statistical inference: stochastic convergence, estimation of parameters, testing hypothesis, etc. all at an elementary level (suitable for a math undergraduate), with good introduction to the required background for such concepts. Can you provide me a good source?


I was discussing about particle accelerators with my friends and it came to my mind whether it is anyway possible to make a table-top accelerator(accelerator that can fit on a table). I asked this to my professor and he mentioned something about plasma field accelerator.He said that charged particle passing through a plasma created by laser pulse can get acclerated. How can plasma accelerate charged particles? What will be the maximum energy we can achieve using such accelerator?What is the progress towards the development of such science? Will it be really a table-top accelerator?


I have developped a little tool in matlab to fill directly some .tex files and then compile them in a pdf file (I use the command pdflatex to compile). It is well working on my computer because I have MiKTeX installed. Now I would like to make a installer (using Inno Setup) to share this tool but I need to include the compiler in the setup. Does someone know if there is a way to only install pdfTeX for the furure users of my tool or do I need to require them to install the whole MiKTeX?


This is more of a general question rather than anything specific but I was just wondering if someone could point me toward resources which discuss singularities in a PDE rather than in an ODE (by singularity I mean as in https://en.wikipedia.org/wiki/Singularity_(mathematics)). I did a bit of work with singularities in ODEs, but it seems working with singularities in PDEs is harder, and I can't seem to find any material on it. I have a first order PDE so it shouldn't be too difficult (I think) to work with singularities with it, but seeing as how I'm from a more probability/stats background I can't be too sure about it... Thanks in advance.


I've looked up both and seen the answer to "why do we say 'soup up.'" Having always thought myself that the phrase was "to soup-up," I've been lately surprised to see a significant amount of usage of "suped-up" with claims that it's from the term "to supercharge" or "super" and that "soup" is the misspelling. Is there a definitive word on whether "suped-up" is a real term for "supercharged" or if it's a misspelled variant of "souped-up"?


I hear people say things like "inside a black hole the laws of physics are not valid" or "there can be parallel universes with different physical laws" or "before the big bang there was nothing". Can physics really make such predictions? Is not there a logical issue here, namely, is it possible to predict a system where all physical theories break, by using only validated/well-established/working theories of physics?


Is there a difference between "headquarters of" and "headquarters for"? It is the headquarters of many branches. It is the headquarters for many branches. It is the headquarters of the party. It is the headquarters for the party. Intuitively, "for many branches" seems correct and "of the party" seems correct. I can explain the conceptual difference between branches and party, but I can't connect my intuition to any rules. Any thoughts on this? Thanks.


I am working on a project whose object language is in non-monotonic logic. Since the project involves reasoning about the models, I am thinking of translating a non-monotonic problem into a first-order logic, use model theory in first-order monotone logic to solve the problem, then translate the results back to a non-monotonic logic. However, this seems cumbersome. Could you please provide me with a book or paper reference on model theory in non-monotonic logics. Preferably, classical topics such as Compactness theorem, Incompleteness theorems and Stable theories should be included.


I'm interested in the semantic implications of using the words quote and estimate in a business scenario. Here's the situation: When someone wants to purchase a service that I provide, they can fill out a form on a Website with information about the work to be done. Based on that info, they are instantly given a predicted price. This is intended only to be a ballpark. Once I manually review the work order, I will contact them with a personalized, much more accurate guess at the price. My instinct is to use estimate for the ballpark and quote for the more precise guess, but I'm interested in both your opinions and the more subtle distinctions and implications of the two words.


I want to say something like: The system stores the crazygonuts data separately from the data feed. I think this is wrong (maybe I am wrong in that), but I'm not sure exactly why. One alternative I thought of: The system stores the crazygonuts data separate from the data feed. I checked some dictionaries online, and I did not find separate listed as an adverb, so this also seems incorrect. I want to keep the verb (stores) in there. Is there any way I can keep this same sentence structure without sounding silly? If possible, I would like to use the word separate (or something sufficiently close, like distinctly); or is this formation doomed from the start?


Most modern texts spend some time deriving the LSZ reduction formula that connects S matrix elements to time ordered field correlation functions. It seems essential, and really helps clear up what you are calculating. Yet some earlier texts and even some modern texts (e.g. "Student Friendly Quantum Field Theory" by R. Klauber) seem to skip right past this, working everything out in the "interaction" picture. It seems there must be something going wrong with this latter procedure, but I am not quite able to put it together.


I'm trying to read about anabelian geometry and obviously the things to start with is the algebraic (etale) fundamental group. Every now and again I encounter authors talking about the arithmetic fundamental group, especially when they're talking about curves. Are these really the same thing, except the term algebraic fundamental group is used when talking about arbitrary schemes and the arithmetic fundamental group when talking about curves?


Possible Duplicate: [Singular] Is/Are [Plural]? I'm currently writing my master thesis on Bitcoin and I'm not sure which version of this sentence is correct: "The first most important part of the Bitcoin infrastructure are all applications that communicate with the Network." "The first most important part of the Bitcoin infrastructure is all applications that communicate with the Network." Or perhaps both of them are wrong? EDIT: Finished sentences.


Possible Duplicate: Unbreakable block I have a block of text (possibly comprising many paragraphs) near the end of a page. The page break is being inserted in the middle of this block--how do I prevent that? (How do I tell LaTeX that this block of text needs to always stay together on the same page?) This question should apply to all formats, but in my case I am using the memoir class, and am willing to include any necessary packages.


Currently i am reading about bandgap engineering in zinc oxide. I read that using external dopents likes magnesium or cadmium we can increase or decrease bandgap of zinc oxide,thus giving unique optical and electrical properties to zinc oxide. But how adding an external dopent causes changes in the bandgap of zinc oxide. I tried to search research article regarding this but was unable to find a mechanism for this phenomenon. Advance thanks for your help


If the coproduct of a family of objects of a Poset (seen as a category) is the least upper bound, who is the coproduct of a family of objects of a Preorder (seen as a category)? My intuition tells me that is the lub again but I'm having trouble writing the proof formally due to it is my first time with category theory and I'm not a mathematician, can you help me?


I am a physics undergrad and thinking of exploring quantum information theory. I had a look at some books in my college library. What area in QIT, is the most mathematically challenging and rigorous? From what I saw in the books, most topics were just simple linear algebra. I am looking for an area which is mathematically richer, and uses maybe more concepts from theoretical computer science, number theory, discrete maths, algebra, etc. Classical cryptography is an area on the interface of maths and TCS which uses many areas of maths such as number theory, algebra, elliptical curves. Is the quantum cryptography also rich in mathematics? What are the prerequisites? If not, please could you suggest some areas that I are mathematically rich in QIT?


Is the phrase "item is ignored for deletion" grammatical and idiomatic? The context is a software program. I have a list of items to be deleted from a database, and if an item from the list is not in the database, I want to display a message saying that the item is ignored. Since this message can appear among other unrelated messages, I want it to be as informative as possible but laconic.


Can the word dear replace expensive, as in "That new T.V is too dear"? The dictionary says so, but I was completely unaware that it had that connotation. I want to use it in writing because it's a shorter, simpler sounding word with a regular comparative/superlative, dearer and dearest, but it doesn't sound idiomatic at all. No one round these parts would say that. Does anybody else in the English-speaking world say this? I tried searching a corpus but all results came back with dear used as a term of endearment.


I am beginning to learn quantum mechanics. Since determining the position of an object involves probing by electromagnetic waves and since i have read a simple derivation of Heisenberg's uncertainty principle which says that uncertainty in momentum of a quantum particle is proportional to the momentum of light (or e.m.) used, I am beginning to wonder whether the theory of quantum mechanics is solely based on measurement using electromagnetic waves? In that case it is not a fundamental law. If we measure the position of a particle using its gravitational field change, does then the uncertainty exist? Pardon me if my question sounds silly.


I was wondering if a library of mathematical expressions in Latex was existing on the web (I don't know if the library term is appropriate) which would help us copy and past them. It would save a lot of time when using, for instance the normal law... PS: I'm not sure of the tag below, tell me it's not the right one, I will change it right away.


A question was recently asked on Musical Practice & Performance, asking what physical properties of silver would contribute to the sound of percussion instruments. It is likely to be off-topic there; I wondered if any of you physicists/material scientists would be able to offer a humble musician information on this site. It would also be interesting to know what contribution solid-silver or silver-plated components would make to brass/woodwind instruments, and what properties of silver are responsible for this.


In many forums, I have seen people keep saying to avoid repetition of the same word in a paragraph. But in the sentence like below, how do you avoid the repetition? Suddenly, the dog stands up and stares at something in the hallway. A shadow of something enters the living room from the hallway. The dog continues to stare towards the hallway. I have to repeat the word hallway three times, and see no way to avoid it. Why and how should I avoid?


I am looking for a word or expression, that describes the specific distance in the space that lets the observer appreciate and completely understand the object of observation. For example, when one is in a museum and stands too close to an impressionist painting, one can only see the brush strokes but not the flowers. But if one stands at the right distance, one can easily recognise the flowers and the whole garden. Is there an expression that describes the right distance to appreciate fully a given object? Thank you very much! anak


I am starting to write my master thesis in latex. So far I have only written article-like texts, so this is my first big document. I would like to start with some good general good practices. Specially since I might go into academia and pursue a PhD. What are good principles to keep in mind?. For example, do you have one single huge tex file, or do you have several smaller files for each chapter?


This was something I understood when we first went over it in class, but my text doesn't really cover it and I cannot find a direct answer online: If I use, say, the midpoint rule on a contour map to get the approximate value of an integral how can I determine if it is an over-approx. or under-approx.? Thanks! I should clarify that I am working with contour maps and double integrals- not single integrals.


I am studying for the Berkeley Math Tournament (BMT) and I was wondering if anyone had any ideas on what I should study to prepare. This is my first math competition and would like to have as much practice before hand as possible. Also, taking a look at last years competition there are a lot of problems that I am unsure of where to begin approaching them, if anyone has any references that would help me learn the ideas and mathematics behind the problem it would be greatly appreciated. Thank you!


Mass appears to attract other mass, irrespective of position or orientation. That is, gravity is attractive no matter what. Contrast that with electrostatic forces which could attract or repel depending on the charges involved, or a voltage between two metal plates which has a specifically directional force on electrons travelling through it. How do we know gravity attracts in all directions at once? What experiments have been done to confirm this?


There's a word, I think that starts with a 'p', that describes doing something to solve a situation before it happens. For example, if a hurricane is approaching a store might take __ action to prepare for the flood of people that might come. In a fictional world, an organization might take __ action against crime based on some foreknowledge of a person's future actions. Edit It's not "preparatory", "precautionary", or "preventative".


I'm looking for a word that is used quite often to mean something that feels real like you can reach out and grab it. Like it has substance. It's used most often around me to refer to user interface or game design. Like when a UI has weight and feels like it has substance behind it you say it's ............... I originally thought of... Visceral Corporeal Ephemeral (although I was looking for an antonym of this) But none of them are correct. But it is a word with the same "al" ending I think.