In the last assignment given to me by my professor, there is a question which asks: In classical theory, it says that in presence of air friction, a pendulum continuously loses energy with time and its amplitude also decreases with time. However according to quantum theory, the pendulum loses energy only in discrete packets or quanta. This would lead to a decrease in the pendulum's energy in a step-wise manner. How can you show that there is no contradiction between quantum theory and the observed behavior of laboratory pendulums and springs? What I want to know now is: What exactly do I have to show mathematically to prove this result? I mean what kind of an equation will I have to derive or prove in order to answer this question? What will be the final equation after writing which I can say that "So this concludes that there is no contradiction." I am completely clueless about what to do, right now. I don't want the complete answer or proof however. Thanks.


In a laser interferometry experiment, we project a pattern of interference fringes onto a CCD sensor. For best results, we want good contrast between the bright and dark fringes, and we carefully compensate for various sources of noise - for example, by taking camera images with no fringes present, and with the laser turned off, and subtracting these images in the proper sequence. We'd expect therefore that the remaining signal should be highly linear, with the CCD signal at each pixel in direct proportion to the number of photons reaching it during the shutter time. What we actually find is that, as we vary the laser intensity and shutter time such that the average intensity across the image remains constant, with no pixels saturated, there is a definite "sweet spot" where the fringes are much more well-defined than at other settings. Either increasing or decreasing the laser intensity away from this point (with corresponding decreases or increases in shutter time) causes the fringe definition to deteriorate. I can't think of any reason why this should be. I know that in cases where the process generating the pattern has a time-constant (for example, using laser speckle interferometry to measure Brownian motion), there is an optimum exposure setting, but that shouldn't be the case in our system which is entirely static. So, what am I missing? I assume it's some property of the CCD sensor that I've overlooked.


I was reading a physics book by some author and I got a little too confused with the explanation he stated about magnetic fields. A magnetic field is a field of force produced by current-carrying conductors or by permanent magnets. Correct me if I'm wrong: A magnetic field is basically a region / space where magnetic lines / flux lie. Thus, any magnetic objects inside the magnetic field will be attracted or repelled depending on its charges. Thus, this means that if a magnetic object is inside the magnetic field, the object should feel either a pulling or a pushing force acting on it. So, if the current-carrying conductors will produce a magnetic field, then this somehow means that I am inside a magnetic field produced by the current-carrying conductors (microchips and the wires inside the computer). And I believe I have enough metals around the computer, why don't my magnetic metals not experience any force acting on it?


First of all, I'm genuinely sorry if this question isn't "serious" enough for this forum! A common cliche in movies and tv is that a very tough object (eg the villain) is frozen, and then hit with something, shattering into a million pieces. I've seen a demo of a flower being put into liquid nitrogen, then being crumbled, but a flower is a very delicate object to start off with. If I take a leg of lamb (for example) out of the freezer, I don't feel like it's in any danger of shattering into a million bits (unlike my foot if i were to drop it). So, is the whole "cold = brittle" thing just movie bullcrap? Or is there anything to it? Sticking with the leg of lamb example: is there a temperature to which a leg of lamb could be dropped that would make the leg of lamb prone to shattering? EDIT - i just realised that the question title could be read as "Is there anything which is rendered extremely brittle by extreme cold?". Obviously there are some things, eg flowers. Hence the title change.


I'm sure his has a general form and welcome a link to a duplicate, but as I don't quite know what to search for, here goes. Whilst answering a question on the Mathematics Stack Exchange, I found myself needing to say the mid point of one diagonal of a rectangle is coincidental with the mid point of the other and that that point is the centre of the rectangle. So I said: "The key is to prove that for all rectangles, the mid points of the diagonals are coincidental at the centre of the rectangle ..." It suddenly struck me that the correct form might have been: "The key is to prove that for all rectangles, the mid point of the diagonals are coincidental at the centre of the rectangle ..." I suspect this is related to "everyone took off his hat". Could someone explain the reasoning behind the correct form.


Possible Duplicate: What is the correct way to pluralize an acronym? I am helping a former intern ready their resume for distribution. The candidate used an abbreviation I was unfamiliar with: B.S.s in Physics, Computer Science, and Mathematics I am familiar with the student's educational background so after a moment of thought realized "B.S.s" was their attempt to indicate multiple bachelor of science degrees. That said I can see those unfamiliar with the student's background being confused. Unfortunately, I do not know how to provide the correct punctuation to indicate the correct grammatical number for multiple degrees in the same discipline. What is the correct plural form for multiple bachelor degrees? Additionally, are there variations for other degree levels: associates, masters, Ph.D, etc.? In addition to the abbreviation what's the correct way to indicate plurality for the unabbreviated form?


I have only taken a basic quantum mechanics course (this book, so you know where I'm coming from), but I've been wondering about something. If we set up a quantum system in a known state and take a measurement of two incompatible observables, we will get two real numbers. If we repeat this experiment multiple times, then we will get two lists of real numbers (each list corresponding to the measurements of one of the observables). Quantum mechanics allows us to predict the average and standard deviation of these numbers, but it does not allow us to predict the individual numbers. If I understand correctly, this is a fundamental limit of the theory. The data is essentially random. Is it correct to say that most scientists believe that no theory will ever allow the prediction of these individual numbers? Why do they think that? And secondly, is there any other property of those numbers that quantum mechanics predicts that I am missing (other than mean and standard deviation)?


I'm teaching a geometry course this semester, involving mainly Euclidean geometry and introducing non-Euclidean geometry. In discussing the importance of deductive proof, I'd like to present some examples of statements that may appear to be true (perhaps based on a common student misconception or over-generalisation), but are not. The aim would be to reinforce one of the reasons given for studying deductive proof: to properly determine the validity of statements claimed to be true. Can anyone offer interesting examples of such statements? An example would be that the circumcentre of a triangle lies inside the triangle. This is true for triangles without any obtuse angles - which seems to be the standard student (mis)conception of a triangle. However, I don't think that this is a particularly good example because the misconception is fairly easily revealed, as would statements that hold only for isoceles or right-angled triangles. I'd really like to have some statements whose lack of general validity is quite hard to tease out, or has some subtlety behind it. Of course the phrase 'may appear to be true' is subjective. The example quoted should be understood as indicating the level of thinking of the relevant students. Thanks.


According to Wikipedia: For most radioactive nuclides, the half-life depends solely on nuclear properties and is essentially a constant. It is not affected by external factors such as temperature, pressure, chemical environment, or presence of a magnetic or electric field. Have any experiments been done to test what effect neutrinos have on decay rates? From what I understand, it might be difficult to isolate pretty much anything from the constant bombardment of neutrinos from the sun and other sources. So, how can we be sure decay rates are not effected in some way by neutrino collisions/interactions? Update "The overwhelming majority [of neutrinos in the beam] will continue on past the lab, to infinity." This is from this article which describes how a beam is generated and detected at CERN. I need help understanding this: if the "overwhelming majority" of neutrinos cannot be stopped and the actual detection is considered an "event," we can't possibly say that we know what decay rates would be in isolation since we can only stop an extremely tiny fraction of neutrinos from a beam. Neutrinos are naturally and constantly coming from all directions at all times. The best we have is test results of what happens when we beam extra neutrons at those substances (which I'd be interested in seeing the results of).


About two years ago I watched some old Monty Python interviews. In one of them, Graham Chapman, a Brit, makes fun of Terry Gilliam (the only American) for his lack of vocabulary. He specifically cited a moment when the group flew over the great lakes and Gilliam said "there's a bunch of water". I found this amusing. But it's also stuck with me. And ever since, every American I meet with seems to have an affinity for saying "a bunch" to describe anything with a high quantitative value. This can be anything from purely literal (a bunch of parsley) -- though this doesn't bother me as much -- to the generally "many", such as "a bunch of candies" and almost sarcastically as in Gilliam's case (obviously several thousand cubic miles of water is a bit more than a "bunch"). But what is a solid alternative for these uses? Not that I'm looking for something to replace "a bunch" entirely. I find it can be useful and an endearing "Americanism". But I'd like to hear of some options. EDIT: Maybe I can make this more specific and ask this: What would an appropriate British expression be for the Gilliam scenario above? Maybe something like "A Considerable Amount"?


In these forums and elsewhere it is routinely agreed that "we do not have a theory of quantum gravity." My question is, how do we know that canonical quantum gravity is "wrong"? I understand that the theory is perturbatively nonrenormalizable, but doesn't that just mean that we can't apply perturbation theory to it? It seems the theory is nonperturbatively renormalizable. A theory being nonperturbative doesn't make it "wrong". My understanding is that in modern QFT nonrenormalizability is not anymore considered such a big deal, and that even the Standard Model is expected to be an effective theory modified by nonrenormalizable terms appearing at higher energies. So barring practical considerations, why is canonical quantum gravity considered "not a theory of quantum gravity?" EDIT: I am not sure that the term "Canonical Quantum Gravity" is correct. I am referring to what is called "Quantum Einstein Gravity" in this paper. If someone knows better please help me correct my terminology. I was not meaning to imply anything about LQG because I thought that LQG "brought more to the table" than just a nonperturbative approach to the same QFT, but I could be wrong about that.


First let me introduce my terminology: A "Mechanical theory": A theory which describes time-evolution of a particle or a system of particles regardless of the fields affecting the particle/system. e.g. Classical mechanics, Quantum mechanics, etc. A "Field theory": A theory which describes time-evolution of a particle or a system of particles with taking the effect of fields on the particle/system into account and also a theory which describes time-evolution of the fields themselves. e.g. Classical gravitation, Classical electromagnetism, Quantum field theory, etc. But I am always confused that "General relativity" falls into which category. I know "Special relativity" just applies a modification to "Classical kinematics" to build "Relativistic mechanics". But "General relativity" is talking about "Relativistic gravitational fields" while also talking about "Non-inertial frames of reference". Now this is my question: Is "General relativity" a field theory or it's a mechanical theory? If it's a field theory is there any other way to study non-inertial frames in the context of "Relativistic mechanics" without bringing any special field into play?


I'm a high school student and I'm trying to understand the concept of the Higgs boson. So I apologize ahead of time for any incoherence I may say. As per my understanding bosons are force carrying particle that are excitations of their respective fields. For example the photon is the boson of the electromagnetic field and an excitation of such. As per my understanding the Higgs boson is an excitation of the Higgs field, and also the Higgs boson is what directly interacts with other massless particles (quarks, leptons, etc). So in the attempt of trying to visualize it. Could it be said that the Higgs boson is like a wave made in the surface of a pool? Since the boson is an excitation in the Higgs field and a wave is an excitation in the water. Would this be a proper analogy? Additionally, how do this excitations in the Higgs field actually occur? How do Higgs bosons appear?


I'm looking for an alternative way of saying "You can't run before you can walk." This is equivalent to saying "you can't take on higher level things before you have mastered the basics". I am looking for either a rewording of the original phrase or a whole new phrase with the same meaning. I prefer that this not have any fancy words nor leave the reader scratching her or his head. I will use it in a semi-informal tutorial for future students of a programming class I am in. I use a short phrase at the beginning of each section which establishes the underlying theme in that section. This tutorial is meant to be something fun (and useful) to read, which is why I want something that catches the attention of the reader. For my purposes, there is nothing wrong with the original. I simply want something that is a bit more inclusive (some people can't walk or run). Thank you.


I have a question related to already shortened words and their plural forms. As I have seen on this site and have found in the dictionary, words like mas and pas are the plural form of the shortened words ma and pa (as in mother and father). However, it occurs to me that were I to write this, in order to clarify the meaning of what I was trying to write (because if you are like me, you read mas and pas and did a double-take), I would write it as ma's and pa's. Some words like cuz have a similar issue, although the pluralization makes more sense: cuzzes. Perhaps cuz's? I know that in the pluralization of single letters there is some contention regarding whether there should or should not be an apostrophe (A's or a's versus As or as). "I got a lot of A's this semester." just looks better to me, although I have no grammatical backing for this. I'm not sure if this is the same for these types of words as well. My question is this: Is there a precedent for using an apostrophe for words that are shortened to convey appropriate meaning, or is the convention simply to omit the apostrophe altogether and leave the word pluralized with an s?


I'd like to know what franchise meas as a verb in the following sentence: Catering in this school has been franchised to the company. The native speakers I consulted, both American, don't seem very sure of its meaning. I know the verb typically means (of a large company like McDonald's) to permit an individual or a smaller company to sell products or offer services in its name. But in the example above, I can hardly imagine a large "catering company" sold the right to sell its products in a school to a smaller company. Perhaps this is because I haven't heard of catering companies in connection with franchise agreements in my country. Could it be that it means the authority of the school gave permission to a company to provide catering? Could 'outsource' be used in this context to replace 'franchise'?


As an applied mathematics student coming from a small university, I have not had an adequate course in writing/formulating proofs for problems in advanced calculus/real analysis (my university has an advanced calculus course, not a real analysis course). In the fall I will take the first of a two part course in advanced calculus. I believe we will use Fitzpatrick's book, Advanced Calculus. So, in order to prepare for the rigor and proof writing that will be necessary in this course, is there a good book or pdf that will provide solutions to problems involving proofs? I've found plenty of books with tons of problems but finding solutions to check myself (or see if there is a more clever approach) has been difficult. I hope this isn't too broad of a question, maybe some of you coming from smaller universities will understand my dilemma.


I'm trying to translate this text to Polish and everything seems pretty clear to me, apart from the usage of the words "within" and "without". I presume it's some kind of technical vocabulary referring to the subject of jousting. Could someone explain to me what these words mean? Here are three excerpts from the text: At those jousts the noble ladies and damsels will give the knight who jousts best of those without a horn garnished with gold, and they will give to the one who jousts best of those within a white greyhound with a collar of gold around its neck. And the noble ladies will give a circlet of gold to the one who jousts best of those without. And one within that jousts best will be given a golden belt. And there will be given in the same field to whoever jousts best of those without a noble courser, saddled and bridled. And whoever jousts best of those within will be given a fine chaplet well worked with silk.


"Crack The Whip" is a game played on ice skating rinks where several individuals line up all facing the opposite end of the rink, and skate forward. When the group reaches the opposite end of the rink, the "point man" or the person on one end of the line stops, and everyone else pivots around him. Supposedly, the person on the opposite end swings around, and rockets forward at much greater speeds. However, I don't believe this is possible. Because the question remains, where would the extra speed come from? I was told that the momentum of all the people is transferred into the one person on the end. While there is a ponderous amount of outward force on the chain of people, I don't think it's possible to actually gain speed. And after watching my friends do it several times, I concluded the person on the end didn't seem to go any faster. Was it simply that the process was inefficient? Or is the theory even possible?


Even though I don't work at Google, the Google Styleguides have been very helpful for me in adopting consistent, readable style conventions for my code. Unfortunately, there is no Google Styleguide for LaTeX. Q: For LaTeX, does anyone know of something equivalent to the Google Styleguides? Quick searches of the web for LaTeX styleguides have returned plenty of styleguides emphasizing how compiled LaTeX documents should appear. I couldn't find anything emphasizing how the uncompiled .tex document should appear. I suppose if there weren't anything equivalent to the Google Styleguides, a standard, very cleanly written and commented .tex template would suffice. UPDATE: Over at StackOverflow, I've found a similar post asking about Ruby coding style guidelines. There are a number of helpful links provided there. I'm looking for something kind of similar.


I'm trying to figure out a kind of taxonomy of quantum phenomena. So far the categories I've come up with are (in historical order, with example phenomena): discrete quantities where continuous ones are expected (black-body spectrum, atomic spectra, Stern-Gerlach [spin spectra]), interference phenomena where trajectories are expected (double slit experiment), and long distance entanglement correlations (Bell's inequality phenomena). Am I missing anything major from this list? Did I incorrectly group something that should be in its own class? Can any of these classes be merged? As requested, how I would define classical phenomena: anything that behaves according to classical mechanics - objects that have (approximately enough) definite position and momentum that obey Newton's laws (even the relativistic versions), and waves that have infinitely variable amplitude, even when boundary conditions limit the frequency spectrum. Basically, I'm trying to get a clear picture of what the boundary between classical and quantum phenomena looks like.


In the following statement, I am confused with the meaning of "whether" in the second sentence. Could you please advise which one is the meaning of this statement? a or b? The statement: "For the grant of the permanent Partner visa, you can be either in or outside Australia. This is the case whether you lodged your original application in or outside Australia." The meaning: a) You can be either in or outside Australia for the grant of visa and it is not important where you have lodged your application. b) If you have lodged your application in Australia, you have to be in Australia for the grant of visa and if you have lodged your application outside Australia, you have to be outside Australia for the grant of visa. Many thanks, Kourosh


An inertial frame of reference is described as being a frame of reference in which the first law of Newton (the law of inertia) holds. This means that all events as described with respect to this frame of reference must have a zero net force acting on it and therefore traces a straight line with a uniform non-translatory motion. But, I have read in some books, especially "Introduction to Special Relativity" by the well-known Robert Resnick, wherein his definition of an inertial frame of reference also refers to such a frame of reference as being an unaccelerated system. This is where I am confused. How can we describe a frame of reference as being unaccelerated if we occupy the frame of reference itself? No mechanical experiment conducted solely confined to a single frame of reference can determine the absolute motion of the frame of reference relative to another frame of reference. All that can be understood is that there is a certain uniform relative motion between frames of reference and no more. Is Robert Resnick saying that the inertial frame of reference is unaccelerated with reference to another frame of reference?


I am currently a senior in high school, and I have been studying mathematics for about nine or ten years now in my personal time outside of school. I am not familiar with academia or in general higher education, but I do know that I want to continue studying mathematics or something in a related area. I am having a lot of trouble pinning down which universities I should consider going to, I don't think I am even ready to start comparing, so despite the 'softness' and vague nature of this question I feel I don't really have any where else to start then by asking which universities should I be looking into for mathematics? I have read many articles that rank schools according to some criteria, articles with titles like "Top Universities for Mathematics" or "Best Mathematics Programs", but I really want to get the opinions of people who study mathematics frequently. Any school suggestions would be greatly appreciated. I am trying to gather a list so don't hestiate. I live in California and am willing to go anywhere on the planet, so location is not a problem for me.


What would be an idiom or word or name for someone that is an initial tester (like a beta tester). I am writing a speech for my younger brother's engagement and want to say how I have always been the first to try everything in my family--schooling, learning to drive, college applications, etc.--because I was the oldest son. My parents basically used me as a trial run to test the waters each step of the way in their parenting. They tweaked and refined the process with each of my two brothers. Now for his wedding, he's being sent out to scout for the proverbial mines for the first time. So I'm looking for a better term than beta tester to describe what I have been my entire life. I was kind of like a "lab rat", but that sounds horrible. Here's the line: "Throughout our life, being the oldest, I was sent out as the front line to test the waters." I really hate that line. I hope having a better term for "beta tester" would make that line make more sense and sound better.


Possible Duplicate: When is it okay to end a sentence in a preposition? So we've all heard the admonishments from our teachers not to end a clause with a preposition A plumber visits a wealthy estate to fix a clogged toilet. As the butler opens the door, the plumber barks out,"I'm here to fix the toilet. Where's your bathroom at?" "Please try to speak with more discretion. We do not want to disturb our neighbors with the details of our plumbing issues. And we most certainly do not end our sentences with prepositions, sir. So the plumber lowers his tone and says more cordially, "I'm here to fix the toilet. Where's your bathroom at, asshole." Anyway, back to the matter at hand. I have come under the impression that this is a rule of thumb to help the elementary student avoid mismatching case for the target of the preposition rather than a hard rule. For example by placing the preposition closer to its target, you avoid constructs like: "Who did you give the invitation to?" instead of the proper "To whom did you give the invitation?". Moving the preposition closer makes the incorrect case sound absurd. No one would ever say "To who did you give the invitation?" All of this introductory text leads up to this simple question: Is this phrase correct "Whom did you give the invitation to?" or is it still incorrect english even though we addressed the issue of case?


Background: I am an upper level undergraduate physics student who just completed a course in classical mechanics, concluding with Lagrangian Mechanics and Hamilton's Variational Principle. My professor gave a lecture on the material, and his explanation struck me as a truism. Essentially, he argued that the difference between the Lagrangian evaluated along the parameters describing the true path and the Lagrangian evaluated along parameters corresponding to a mild perturbation of the parameters by a function an(x), where a is a scale factor, is zero. Where exactly is the profundity in this statement? I understood it as "If we deviate the parameters away from the parameters that minimize the integral, and then take the limit as that deviation vanishes, the difference between the path described by these two sets of parameters is zero and the path must be the true path." Well of course this is true. What am I missing? Alternatively are there any decent texts that outline this principle at an undergraduate level?


I am a high school senior and I am interested in doing a math research. I hope someone can recommend areas or topics of research that are challenging, rewarding, and yet do not exceed my capability. (I acknowledge this is quite hard) My math background: a. I have done competition math (Elementary number theory and combinatorics, Euclidean Geometry, and Algebraic manipulation) and I'm fairly comfortable with proofs. b. I had my first courses in Multivariable Calculus, Differential Equation, and Linear Algebra (Familiar with fundamental concepts, basic techniques and motivations) c. I have learned a portion of Abstract Algebra on my own and in summer programs including topics like Lagrange theorem, Vector spaces, Polynomial Rings, and Morphisms. d. I don't have a good background in statistics and probability e. I have been exposed to Knot theory and Chaos theory f. I do have basic programming skills in python and Mathematica, and I can work with LaTeX. I really appreciate your help!


I am planning to take a graduate Geometry course next semester. The preliminary syllabus does not specify any textbook but has the following descriptions: Catalog Course Description: This course studies higher geometry including triangulations of polygons, Voronoi diagrams and Delaunay triangulation, algorithms in computational geometry, Euler characteristic of geometric objects, conics, elements of differential geometry of curves. Topics Covered: inequalities, Helly's theorem, power of a point, inversion, Voronoi diagrams and Delaunay triangulations, algorithms in computational geometry, Euler characteristic of geometric objects, conics and their affine and metric classification, elements of differential geometry of curves, polyhedra. I would like to study early taking advantage of my downtime. Is there any textbook that you would like to recommend for my self study? Thank you very much for your time and pointers.


Fluid dynamical instabilities are present in many different everyday things. The famous tears in wine for example are a classical example of a Marangoni effect, where surface tension gradients due to evaporation cause an instability. I recently came across an instability that occurs when you have just emptied a glass with yoghurt drink and have only a film of the drink left on the walls of the glass. A tear-like pattern emerges over the time of about a minute that looks like shown below: I have been thinking what can cause this effect. It seems somewhat similar to a Marangoni effect, but given the ingredients of the yoghurt drink I think we can safely rule out evaporation on this timescale. Another potential effect explaining this would be viscous fingering, but for that to happen I believe the gas phase has to be driven, which it isn't here. Or perhaps it is a 'surface-version' of a Rayleigh-Taylor instability? Or maybe it is related to the non-newtonian nature of the yoghurt drink? To summarize: I don't know which instability I am looking at and I would love to know what causes this pattern on the glass with yoghurt drink! Does anyone know what is causing it?


There's an old debate going on in the guitar community about how much does wood choice and body shape affect the sound of an electric guitar. No one denies that there's a difference acoustically (how the guitar sounds unplugged) because in this situation it's the wood and body shape that amplify the sound made by string vibrations, but when we're talking about the sound as it comes from the pickups, things get much more uncertain because there are a lot of variations even in supposedly identical guitar parts, and accounting for all of them for the purpose of doing an experiment where the only difference between two guitars is the wood is difficult (I certainly haven't heard of an experiment that was satisfying enough, but feel free to prove me wrong). I'm interested in a way to circumvent these practical difficulties using a theoretical explanation: Since the pickup only sees the string's vibrations, the question basically becomes "does body shape and wood make enough difference in the way a string vibrates that it changes the sound in a noticeable way", and this sounds like something that may be possible to figure out mathematically, or at the very least should be much easier to test because there are less variables involved. My question is: Is what I'm describing possible to calculate/test, and has this been done before?


Let's say I am at the train station and I missed the train, I still see it driving off. I would naturally say to myself: 'Damn, that was the train I was hoping to get.' Would that be wrong? If not, why is the past progressive used here? I have been taught that you use the past progressive when you are either talking... about a longer action that was interrupted by a shorter action about actions that were happening at the same time about an action that was in process at a specific time in the past However, my sentence doesn't follow any of the rules above (at least I don't see that). Another sentence that I can't gramatically explain... Context: You told your friend David that Fred would not behave good when he is drunk before going to a party at which Fred gets drunk and then behaves bad. Later you say to David 'See? That's what I was talking about.' What's the grammatical explanation of this? Can anyone give me a rule why these sentences work? Thank you in advance!


My friends and I were playing a game where you roll dice and you bet money on what picture it's going to land on and I began reasoning with myself that if I tallied up what pictures the dice landed on that the ones it has not landed on were more likely to come up in the next roll of the dice. So I began a process of observing two rounds and tallying up the results and then on the third round betting on the pictures that have not come up yet hoping that they would make their appearance on the next roll. Although I do not have any mathematical proof to verify this. Does anyone know what this phenomenon is called if it is even possible or if it is possible? Edit: I figured it out its called "Gambler's fallacy"


Is "family" both plural and singular? or would I have to say families for the plural form? For example, which of these is the best option: "A majority of those whose family were unaware of their sexuality..." "A majority of those whose families were unaware of their sexuality..." "A majority of those whose family was unaware of their sexuality..." Edit: Here are some full sentences to give some context. Participants whose famil(ies) were aware of their sexuality were predominantly feminine and identified as gay, homosexual, drag queen, or a combination of these identities. All those who assumed their famil(ies) knew about their sexuality were gay identified and a majority identified as feminine. A majority of those whose famil(ies) were unaware were masculine and identified as non-gay, straight, down low or did not identify with any label.


I'm reading Chris Hecker's third article on rigid body dynamics http://chrishecker.com/Rigid_body_dynamics Quoting... "More importantly, if our collision detector supplies us with a 'normal vector' for the collision (denoted by n, and pointing toward body A by convention), we can define the 'relative normal velocity' as the component of the relative velocity in the direction of the collision normal." Which he defines as vAB . n where vab is the relative velocity of points A and B and n is the normal vector for the collision. I read Understanding Dot and Cross Product which explains that the dot product gives the length of one vector in the direction of another, which I think is what is being applied here, but I'm having a really hard time visualizing what is going on, specifically what the component is. Can anyone help explain what this component is and how using the dot product helps identify it?


I would like to better understand how neutrino oscillations are consistent with conservation of momentum because I'm encountering some conceptual difficulties when thinking about it. I do have a background in standard QM but only rudimentary knowledge of particle physics. If the velocity expectation value of a neutrino in transit is constant, then it would appear to me that conservation of momentum could be violated when the flavor eigenstate at the location of the neutrino source is different from that at the location of the interaction, since they are associated with different masses. For this reason I would think that the velocity expectation value changes in transit (for instance, in such a way to keep the momentum expectation value constant as the neutrino oscillates), but then it seems to me that the neutrino is in effect "accelerating" without a "force" acting on it (of course, since the momentum expectation value is presumed constant, there may not be a real problem here, but it still seems strange). Any comments?


I've recently learned about ultraproducts, but the source I learned from almost immediately after the definitions restricted to talking about countably incomplete ultrafilters. I know that the existence of countably complete ultrafilters is a large cardinal issue, but aside from this, is there a reason to focus on the countably incomplete case? Do ultraproducts (or even just ultrapowers) with a countably complete ultrafilter behave very differently from the countably incomplete ones? It would be great to have an example of the kinds of differences that happen, ideally in a fairly down-to-earth setting (maybe groups, or fields, or graphs?). The few places I've seen talk about countably complete ultrafilters all seem to be taking ultrapowers of models of ZFC, which is a bit much for me to grasp at this point. Since I just want to see the differences, it's fine with me if some set-theoretic hypotheses are needed to make the examples work.


While my physics teacher was explaining pseudo forces to us he gave the following example : An elevator is accelerating upwards. In it there is a bob strung up by a rope. There are two observers, A in the elevator and B outside of the elevator, on the ground and not accelerating. Due to the action of gravity and the lift's acceleration the rope breaks and the bob falls. Does it do so at the same time for the two observers? The falling of the bob will be registered only when an observer sees a change in its position. This will happen earlier for observer A in the elevator compared with observer B outside. Question: At a particular time instant, is it possible for the rope (which holds the bob) to be both broken and taut for different observers?


Is there a word to describe someone who uses complaints to indirectly brag about themselves? An example would be "I hate going to concerts because people start singing and because I have perfect pitch it irritates me." Perhaps another example might be "I don't like that video game. It's too easy and I get bored." The complaint would be in context, like the discussion is about concerts or the game in question, but the person uses it as an excuse to highlight something about which they want to brag. I don't think I'm looking for narcissism, as it's not necessarily that the person is trying to talk only about themselves, but rather that they specifically use a negative complaint to mask the fact that they are bragging. Is there such a word to describe this behavior?


I think prejudice is too general. The definition Google gives me for prejudice is: "preconceived opinion that is not based on reason or actual experience." - it doesn't specify that this preconceived opinion is due to membership in some group (or, more specifically, perceived membership in a group), although it seems to have that connotation, so maybe that is what I should go with. I think bigotry is too strong. For bigotry, we have (wikipedia): "Bigotry is a state of mind where a person strongly and unfairly dislikes other people, ideas, etc. " Bigotry sort of connotes hatred, not just bias. I want a way to describe the fact that a particular statement reveals an attitude that is unfairly biased against a culturally significant group, which may or may not be intentional or malicious.


In one of the books on algebraic topology (I don't remember exactly which one) there was an exercise to build an example of two topological spaces having two continuous bijections between them which are not homotopy equivalent. To be honest, this exercise confuses me a little because, as I understand, each pair of homeomorphic spaces is homotopy equivalent by construction. On the other hand, the existence of bijective continuous mapping between spaces automatically provides their homeomorphism (correct me if I'm wrong). Thus, in this logic, if we have two continuous bijections between topological spaces this will inevitably lead to the homotopy equivalence between them. I guess, however, that there is a simple counterexample related to the discrete topologies which breaks such a reasoning (see, for example, this: Is a bijective homotopy equivalence with bijective homotopy inverse a homeomorphism?), but I have certain difficulties in discovering it. Is there any suggestion?


One quality or trait that many employer looks for in a leader is the ability to not only perform well yourself but to also elevate the performance of others around you. For example, Steve Nash was an elite point guard and an all-star that made the people around him better. He made Shawn Marion and Amare Stoudemire all-stars and afterwards, when they were no longer teammates, they were not able to play at their all-star levels. You could say that Steve Nash is a great leader but it does not inherently imply that he also makes the people around him. Is there a specific word that describes this? A phrase to describe it would be, "he brings the best out of others", but I cannot think of a word to describe this ability/trait.


I was driving to work this morning when this question occurred to me. I was going up a clover-leaf entrance ramp to the highway. The person in front of me was lazily floating the outside of the curve, whereas I always tend to hug the inside of the curve. Hugging the inside of the curve tends to lead to an apparently higher speed... and I soon had to apply the brakes to avoid hitting the floater in front of me. This made me wonder, why does hugging the inside of the turn yield an apparently faster speed? Is it simply because the inside is a shorter radius and thus you traverse it more quickly, thereby appearing to go faster (something I know track racers take advantage of, thus the expression "he's got the inside track")? Or, it occurred to me that it could be similar to the way a figure skater performing a pirouette speeds up as she pulls her arms and legs inwards. So as you continually pull your car inwards on the curve it actually does increase its velocity. Does the second concept actually come into play here, yielding an actual faster velocity? Or is it simply the first concept, yielding only an apparent faster velocity? (assuming our cars are identical weight, engine power, etc.)


Given the equations of two spheres, how would I find the equation of any plane tangent to the two spheres? I tried something, but I realized that it failed, and I am not sure where to go from here. I have only basic knowledge of cross product, dot product, etc. and have not yet taken calculus. My attempt: I know the centers of the two spheres. I pick any point on the surface of the first sphere. I find the vector from the center of the first sphere to the point I selected. I then scale the center of the second sphere by the vector I just found divided by the radius of the first sphere and multiplied by the radius of the second sphere. Then, I construct the vector from the point I chose on the first sphere to the point I found on the second sphere. I take the cross product of this vector with the vector formed by the centers of the two spheres. I use this as the normal vector for my plane and plug in to get its equation. I noticed by experimentation that this does not work. Is there a way of solving this problem in a similar manner to what I tried above?


Back when I studying the time independent perturbation theory of the Hamiltonian in quantum mechanics, I remember reading that there are only three problems in physics with exact solutions: the free particle, the harmonic oscillator and the hydrogen atom. It was further stated that any given problem (even a simple pendulum) could have its solution determined by applying a small perturbation to the exact solution of another similar problem, i.e., it would be an approximate solution. So I was left wondering: are the solutions to all [a] real life scenarios (where you don't go around neglecting friction, using symmetries, etc.) obtained through perturbation theory? That is, are they all approximations and not exact solutions? In that case, would it be fair to state that physics is inaccurate to a certain degree? [a] With the exception of the aforementioned ones.


I'm a little out of my depth here... I'm trying to understand quasiparticle tunnelling in superconductor-insulator-superconductor junctions. Many books use the "semiconductor model" to explain this: (source: wikimedia.org) These diagrams show the available quasiparticle states (with a large band gap due to the formation of Cooper pairs), the filled states, and the empty states. My question with these diagrams is: shouldn't all the electrons exist as Cooper pairs? I assume that the lower band is filled with quasiparticles, since Cooper pairs would all be at the same energy level and quasiparticles obey Fermi-Dirac statistics, but I don't know where they're coming from. Also, why is there an energy gap in the quasiparticle energy states? I understand that this gap corresponds to the energy needed to break Cooper pairs, but I don't understand why would you need to break Cooper pairs to raise the energy of quasiparticles. Or is this "semiconductor model" not fully representative of the physics?


I was working on a lab in class drawing Free Body Diagrams. The problem required we drew an FBD of a ball that is in the motion of being thrown. I drew a slightly diagonal line labelled Applied Force, a vertical line straight down labelled Gravity, and a line opposite of Gravity labelled Normal Force. I was told that there is no normal force at all in this situation, and that the only two forces are gravity and applied force. I was confused because I thought that the ball was being held up by the hand (acting as a surface) which provided a normal force while still being pushed by the hand in the positive direction resulting in an applied force. So in this situation, would the ball have any normal force at all acting on it?


I'm a huge Pulp Fiction fan, and the following is one of my favorite scenes, but it also irks me. (source: IMDB) Jules: [Jules shoots the man on the couch] I'm sorry, did I break your concentration? I didn't mean to do that. Please, continue, you were saying something about best intentions. What's the matter? Oh, you were finished! Well, allow me to retort. What does Marsellus Wallace look like? Can a question be a retort to something? I see retort defined as (dictionary.reference.com): to reply to, usually in a sharp or retaliatory way; reply in kind to. to return (an accusation, epithet, etc.) upon the person uttering it. to answer (an argument or the like) by another to the contrary. All the above suggest some sort of a reply. But can a retort be a question, or even a counter-question ?


Break comes to a close, and you, a renowned mathematics professor, step into a grand lecture hall to deliver the first lecture of the semester on topology. This is an introductory course. Half of the students cannot even pronounce homeomorphism. As you look around the room, a bead of sweat works its way across your brow. All you can think of is the possibility that the entire class will fail, and you will be mocked by the other professors. Then you take a sip of water and pull yourself together. You pick up a fresh (but not too fresh) piece of chalk, write your name across the board--effectively marking your territory--and address the class. How do you introduce a class of undergraduate students to the field of topology? I am looking for a creative, but precise explanation of the field and the most fundamental topological concepts. Diagrams and metaphors are welcome.


Can a sentence like this: "I don't know who the first man that made such and such thing in such and such place was," be grammatically correct if we don't put "was" at the end of the long phrase, that is, if we write: "I don't know who was the first man that made such and such thing in such and such place"? I can see in Google Books examples that in such cases the verb is often put after the wh-word, but I don't know if there is a grammar rule to support this. Some examples: "We do not know who was the first man who ascended above a poor and humble people to become Egypt's first king ..." "... we do not know what was the ultimate judgment of the various members of the community ..." "I do not know who was the first to suggest a connection between the problem of free will and the breakdown ..." "I do not know what was the date of this change in me, nor of the train of ideas ..." "We do not know what was the primitive text from which Codex Bezae derived its Latin or its Greek ..." "We do not know what was the practice in the days of the monarchy, but the story of Athaliah shows ..."


I'm writing a paper about an algorithm that I have developed. Just for illustration, I will say that the method name is "quicksort". My question is about the usage of the in the following context: This paper proposes quicksort, a novel and fast algorithm. The advantage of quicksort is that... My question is whether I should use "The advantage of the quicksort..." or "The advantage of quicksort". I am also looking for resources explaining the usage of the in this context. [Meta] Usually, to check if a certain sentence is correct, I search Google using wildcards. However, in this case, the correct answer is depends on the context. I have also tried to find a answer in the following book but without success: Science Research Writing: A Guide for Non-Native Speakers of English.


I'm entering my second year of undergrad (majoring in mathematics), and I've found that I am really bad at Linear Algebra, but very good at Calculus and Differential Equations. I'm hoping to venture onto Sci. Computing/Applied Maths, but I'm worried my inadequacy (as quite personally, unfortunate lack of interest) for Lin. Alg. will prevent me from being successful in topics such as Numerical Analysis, Algebra, as well as Scientific Computing. Does anyone in the applied maths field/experience with applied maths have any advice on what I should do? That is, what else is there like Calculus/DEs that will help me in this field? Or do i just need to buck up and get on my Lin. Alg. horse in order to get remotely close to where I want to go? I appreciate any and all input.


I was looking for a synonym of spontaneous, and voluntary naturally came to my mind. In an attempt to understand the difference between them, I tried to google spontaneous vs voluntary. To my surprise, nothing really interesting popped up from search results. Then I decided to look them up respectively. As expected, voluntary is listed as a synonym of spontaneous according to many online resources, and vice versa. However, to my great shock, involuntary is also listed as a synonym of spontaneous by major online dictionaries. Although the fact does not necessarily imply voluntary and involuntary are synonyms, I continued to look further into voluntary vs involuntary due to confusion. I found that a voluntary action is something that is done voluntarily or with meaning to do so, while an involuntary action on the other hand is done automatically. Now I'm even more confused. The word automatically just reminds me of spontaneously, convincing me to believe involuntary is indeed a synonym of spontaneous while voluntary seems less so. Could someone please justify or explain the contradictions mentioned above? I'm totally lost.


Suppose we have a contact process on a finite lattice. I'm asked to give a heuristic argument for the fact that the extinction time for the contact process is exponential in the size of the lattice when it is in the supercritical phase, and logarithmic in the size of the lattice when it is subcritical. The supercritical phase means that on the infinite lattice, the infection never goes extinct almost surely. I really don't know why this is the case. I get that the extinction time in the supercritical phase will grow rapidly, because if you make your lattice larger, the number of infected nodes grows. Because all these nodes are infected and they infect other nodes quickly (because of the supercritical phase), the other nodes will stay infected much longer. But is there any reason why this should be exponential (or logarithmic in the other case)?


Motivation: If I start with the group axioms and drop the requirement that I have inverses, I get the monoid axioms. If I proceed to drop the requirement that I have an identity, I get the semigroup axioms. If I then drop the requirement of associativity, I get the magma axioms. If I drop the operation, I get the set axioms. A map preserving the monoid structure is a "monoid homomorphism;" a map preserving the semigroup structure is a "semigroup homomorphism;" etc. Question: Now suppose I start with the topological space axioms and start dropping conditions. Do the resulting sets of axioms have names? What about the maps preserving such structure -- do they have names? In particular, what about the smallest case of a sets equipped with some subset of their powersets, together with functions such that the preimage of a designated set is a designated set?


There is a challenge involving a lemon floating in a jug of water which seems impossible to beat. I've noticed it in several pubs of Edinburgh. The challenge is as follows: There is a jug half filled with water. Floating in the water, there's a lemon. The lemon doesn't touch either the bottom nor the edges of the jug. The challenge is to successfully balance a coin on the lemon. Modifying, moving, or more generally touching the lemon are not allowed. Any attempt to balance a coin on the lemon seems to result in the lemon flipping over, and the coin to sink in the water. Why is it so hard to balance the coin while it's extremely easy to balance a coin on a lemon set on a table? How do you beat the lemon challenge?


I've been using OpenOffice and/or MS Word to do my college assignments, but since I started this new discrete math course, I'm finding myself very annoyed with the math formula options in those programs. I learned about LaTeX and figured it would be my best option. Since I really just want to get my college assignments done with it and not use it for printing books I've written or anything, I decided to go with LyX. It has a quick and easy preview option, and isn't so different from what I'm used to. I still can't find a way to make the fancy, cursive script "U" that represents universe or universe of discourse that is used in set theory. Is there a specific package I need, or did I just miss it somehow. Also, if I do need a new package, how is that even done with LyX. Anyone with assistance has my gratitude.


My wife was discussing pudding consistency this morning and used a sentence along the lines of, "I only like the pudding you make". I blinked and asked if she really liked the pudding I make and she replied, "No, I mean the pudding you make, you know, like if I were to make some pudding". She was using the "you" in a sense of a hypothetical person, I guess, like saying, "That's what you do in a crisis" when commenting on a riot scene in the news. You're not using "you" to refer to the person you're speaking to, but rather a form of general humanity. Anyhow, is there a term for such a usage such that you might be able to answer, "Oh, sorry, I was using 'you' in the [term] context, not referring to you specifically"?


I'm new to QM so excuse my naivety. I was watching an online MIT QM course that described the double-slit experiment (with electrons) when it occurred to me that I have a question. In the video, the lecturer just drew a picture of a solid wall with two slits and then showed pictures of the interference patterns generated by shooting a single electron at the slits. Fine enough, the electron interferes itself, which is beautifully explained by the wave function. However, aren't the atoms in the wall with the slits also described by a wave function? I mean, can we even meaningfully draw a picture of a 'slit' if it is roughly at the scale of an electron? Aren't we supposed to be dealing with a wave function there too? Looks to me the wall with the slits is treated as a 'classical' object (you can touch it, feel it, smell it) while electron is treated as a quantum object. But that simply cannot be the case. Question: how does the wave function that describes the atoms around the slit 'know' to interact with the wave function of the electron? Does it collapse? The reason I ask this is because when the electron does not make it through the slit, it must have collided with one of the atoms: but wouldn't collision imply that two particles were at the same place at the same time? Doesn't that require wave function collapse? I'm confused ...


I'm trying to understand the connection between the wave model and the particle model for light. It's understood that the energy of a photon is given by E=hf, but from my understanding of fourier analysis, the only kind of wave that has a precise frequency is a plane wave. The plane wave is an idealization, since no real wave permeates all of space and time. So imagining a more realistic EM pulse, the frequency spectrum will have some kind of spread depending on the shape of the pulse. Is the pulse a single photon? Or is it a collection of photons, each with different frequencies? In the photoelectric effect, it's usually described as a single photon with sufficient energy being absorbed, kicking the electron out of its orbit. Let's imagine the pulse is symmetrically centred around the frequency with energy exactly equal to the metal's work function. What exactly happens to such a pulse? Will the whole pulse be absorbed, since its average frequency has energy of the work function? Or will the half of the pulse that has the higher frequency be absorbed, leaving the rest to reflect or what-not?


Whenever I add milk to my morning coffee I often enjoy watching the patterns which are created. These patterns have a striking resemblance to certain fractals and my question is, "Why?" Oh dear, that is never a good question, so let's try "Why shouldn't they?" The obvious observations are that the water is very hot when the milk is added, so we expect the "coffee particles" (non-milk part) to have a higher velocity relative to the "milk particles". Secondly, the coffee part, having been recently stirred, is often still rotating (clockwise in my case). Finally, the image we observe is a projection of some distinct portion of a "top layer" of the milk and coffee mixture. Add diffusion into the mix and it really seems like some crazy stuff should go down! But I'm afraid I'm not satisfied. Why physics.SE... why?


I read about this law / property a couple of months back, but I've forgotten what it's name was and I can't seem to find it by Googling. I was hoping someone could give me the name for this property. If I recall correctly, it was named after same famous mathematician like Gauss or something... More detail: This site was basically describing how you can make a long piece of metal, paper, etc. stronger by bending it along its long axis. This way, it is less likely to collapse along its length when upright. An example of this property was grass blades, which are able to stay upright due to the fold / bend along their long axis. If someone knows the exact name of this property, please do tell me!!


I am familiar with basic quantum mechanics and I know that there is no concept of 'force' in quantum mechanics, unlike in classical mechanics. Problems in quantum mechanics are solved by writing down the Hamiltonian for a system, and trying to solve for the various eigenvalues. Some of the first problems that are taught to students learning Quantum Mechanics are the harmonic oscillator problem, and the Hydrogen atom problem, where the Hamiltonian takes the same form as a classical system. Since moving over to quantum mechanics requires one to lose several ideas that have been built up while learning classical mechanics, why is the potential found in quantum mechanics problem of the same form as classical mechanics? The potential for the hydrogen atom, for example, is classical in origin and is derived from the Coulomb force. How is this direct usage of the potential, which is purely classical in origin, justified by the theory?


I'm in doubt in the application of Gauss' Law to find electric fields when the charge distribution is symmetric. Well, first of all: I know how to find the magnitude of the field - we just enclose the charge distribution with a gaussian surface on which the electric field will not change it's magnitude, and then using Gauss's Law we can write it in terms of the total charge inside and the area of the gaussian surface. My problem is: how do I find the direction of the field? I mean, in a spherical symmetric distribution it's easy, because we know what's the vector that points radially outwards (it's simply one of the unit vectors from spherical coordinates). But what about a cylindrical symmetric distribution ? Would I need to use the unit position vector of cylindric coordinates ? In the general case I would need to switch to more appropriate coordinates to write the field ? Is there a general way of treating this ? Sorry if this question is to silly or too basic. I'm just trying to understand how to use properly this law.


Good afternoon, I'm trying to learn about Fuzzy alone, I was using some texts on the Internet about it, but it was very difficult to learn. I want to learn about Fuzzy set as shaper of uncertainty. Operations with fuzzy sets. Characterization and properties of fuzzy sets. The Zadeh extension principle. Fuzzy numbers. Fuzzy relationships. Fuzzy relational equations. Systems based on fuzzy rules and the Mamdani inference method. Fuzzy controller and an application. Compositional rule of inference. Inference method of Takagi-Sugeno-Kang. Approximation properties of fuzzy systems. Measures and fuzzy integrals. Introduction to fuzzy dynamic systems. I found these books on web -Fuzzy Mathematics: An Introduction for Engineers and Scientists, John N. Mordeson, Premchand S. Nair -Fuzzy logic with engineering applications, Timothy J Ross -Fuzzy sets and fuzzy logic, George J Klir Does anyone know any of these books? I am totally beginner in this matter, and unfortunately none of these books has available solutions. If anyone knows of any video material on this topic I'm grateful too.


After spending almost one year on this site, I've realized that my knowledge of mathematics is not deep enough. I love mathematics, i mean I am obsessed with it. I have a masters degree in mathematics from a reputed university in India and now I want to pursue PhD in this subject. It's been one year since completion of masters degree and I am unable to find a place in any university for PhD (I failed at couple of entrances). So my question is, is my love and passion for this subject will help me to do research or to do research in mathematics one need an unusual brain? By unusual I mean who can instantly catch the logic and solves a problem, which I can't. I am slow. So I need some advice on this? How should I prepare myself? Thanks. Edit: By this question, I am not asking about which career I should choose. I know I want to do PhD. What I am asking is, "How one should prepare before starting a PhD?" What are the basic points I should be working on? Simply just reading many things would not help I guess. So if this question can be reopened and some one give me a pointer then it would be helpful.


I'm writing a paper about classes of formal languages, and I'd like to have a diagram showing their heirarchy. Something similar to this (from Wikipedia:) Is there a LaTeX package which, I can give the relationship between the classes to, and have automatically generate and lay-out a venn-diagram for me? The idea is that I would give it the list of classes, and some relations between them, i.e. which were contained in each other, which were disjoint, which were incomparable but not disjoint. Containment is assumed to be transitive (i.e. if A is in B, B is in C, then A is in C). Is there a way to do this programatically, or a package to do it automatically? The idea is that I'm continually adding new classes to the diagram, and would love if it would automatically update itself every time I discovered a new relation.


This sounds like a daft question, but I'm serious. Translation and rotation are clearly different -- the symmetry between them is broken by Newton's Laws. But in the Lagrangian/Hamiltonian frameworks, they look extremely similar! The Lagrangians for free rotation and free translation are exactly the same, up to the replacement of some letters. Working entirely with the Lagrangian framework, it's unclear when and where the symmetry breaking happens. Despite this, there are many clear asymmetries between translation and rotation: There is absolute rotation, but not absolute translation. (At least, I believe this is the orthodox position.) In space, starting with zero linear and angular momentum, it's possible to change your angular position but not your translational position (you can turn yourself around, but can't move your center of mass). In quantum mechanics, free particles can have continuous values of linear momentum but have quantized angular momentum. I know why the third point holds: localization causes quantization, and the set of possible angular positions is compact, while the set of possible positions is not. In fact, I feel like this is the only difference, a priori, between translation and rotation. In layman's terms, if you keep rotating, you'll get back to where you started, but if you keep translating, you won't. Is it possible to use this reasoning to extract the first and second bullet points above? If not, what exactly is the difference between translation and rotation?


It's my first time to write latex with Mac os (or in fact, using Mac os in general) and unlike with Windows I'm having some trouble viewing my output files. I'm using texmaker. I would like to have either dvi or pdf viewer constantly open and auto-refreshing when I save changes or convert them to latex. I was told that Skim application should do the job, but I always have to close the dvi/pdf and re-open it for Skim to refresh my changes. Any suggestions or advices how to handle this properly? This might seem like a small problem, but when I save and view my changes constantly it is really painful (and time consuming) to always close the pdf/dvi viewer, re-open it and then scroll to the correct location. Thanks for all the help in advance.


Basic books dealing with the interaction of X-rays with matter ussually don't mention anything about the polarisation, but I read somewhere that X-rays scattered in matter are linearly polarized, specially those scattered at right angles of the incident rays. If I remember well, the reason was discussed considering the unpolarized X-rays as a mix of classical electromagnetic waves polarized in all the possible directions: each polarized wave produces electrons oscillations in the direction of the polarization, and this oscillation leads to the emission of (scattered) waves with the same polarization but with maximum intensity in the plane perpendicular to the polarization. But we know that X-rays are not classical waves but quantum entities and they can be scattered by different mechanism: Rayleigh scattering, Compton efect...so, is the statement about the polarisation of the X-rays true? Is it valid for both types of interaction (Rayleigh and Compton)?


This was explained to me many years ago, by a physics teacher, with the following analogy: "If someone on the beach wants to reach someone else that is in the water, they will try to travel as much as they can on the beach and as little as possible on the water, because this way they will get there faster." I'm paraphrasing of course, but this is as accurate as I recall it. This explanation makes no sense to me. Was he telling me the light knows where it is going? It wants to get there faster? It chooses a different direction? (No need to answer these questions, this was just me trying to understand the analogy.) My attempts to clarify the issue were without success and many years later I still don't know. Why does light change direction when it travels through glass?


I am attempting to write a program that will compute the average amount of a particular product produced when randomness is involved. Let's say that I am trying to produce some widget. Whenever the production process for this widget is started, it is not guaranteed that the process will be successful. So let's say that the probability of successfully producing a widget is P(s). However, you only start with M number of materials, and whenever a widget is successfully created, then C materials are used up. Similarly, whenever a widget fails to be prodeced L materials are used up. So we know the following: P(s) = probability that production process will yield a widget. M = Amount of starting materials C = Amount of materials used if widget successfully created. L = Amount of materials lost if failure to make widget.</code> The problem I am having is in trying to compute the average number of widgets created (and similarly, the average number of failed widgets) given different values for P(s), M, C, and L. How can I come up with an equation that gives me the average successes and failures to create widgets?


Starting from tomorrow, I will be tutoring some undergraduate students following a course in general topology. I am looking for examples motivating the importance of topology in mathematics which can be explained without too much difficulty using concepts of other areas of mathematics (or physics) they have already treated (those areas would be mainly analysis, complex analysis, linear algebra, a little graph theory, some numerical methods for maths, and classical mechanics, electromagnetism, special relativity, some QM and a little statistical physics for physics). I have tried looking around, but I have found little that would motivate me to follow such a course. Do anybody have some nice example? Note: I will of course explain to them that without topology they'll be able to do very little advanced mathematics (e.g. functional analysis, differential geometry, ...) EDIT: Ok, I gave as examples Tychonoff's theorem, Brower's fixed point theorem and the Jordan curve theorem. I would like to keep this question alive, for personal interest. What are interesting (not too hard) applications of topology in other areas of mathematics?


I have this sentence: I strongly believe that the first step in making the most efficient solution for any problem is analyzing it well. Would it be better to use either of the following? I strongly believe that the first step on making the most efficient solution for any problem is analyzing it well. I strongly believe that the first step at making the most efficient solution for any problem is analyzing it well. The context is as follows: First of all, I concentrate on understanding the big picture of any problem. I always try to recognize all the factors that have caused the problem. Then, I start planning the solution at a very high level in order to create long-term benefits. I strongly believe that the first step in making the most efficient solution to any problem is analyzing it well. In my opinion, "What to do" is much more important than "How to do".


I was watching a show on the science channel about gas giants; there is something I do not understand. I am not a scientist, so this may be obvious to some. I learned that there a three states of an given physical object; solid, liquid, and gas depending on how cold or hot the object is. An easy example is ice, water and steam from coldest to hottest. So the theory is that Jupiter has a super-heated solid and very dense core that is made up of hydrogen. How does a gas like hydrogen become a solid while being super-heated? Is it that the pressure is so much that the gas is compressed into a solid? If so how much pressure does it take to compress hydrogen into a solid? How does the heat play into the equation?


I've recently discovered that the following theorems require the axiom of choice to be proven: every surjective function has a right inverse. a real-valued function that is sequentially continuous at a point is necessarily continuous in the neighbourhood sense at that point. every vector space has a basis. When I revisited the proofs I was taught in first year, I was surprised that my lecturers had used the axiom of choice without declaring so. It seems strange that so much effort was dedicated to establishing that mathematics is a rigorous subject [indeed much time was spent on learning the field axioms, well-ordering axioms, Archimedean principle and (later) the completeness principle] but to ignore the axiom of choice. I am interested if there are reasons for omitting to mention the axiom of choice. Are there pedagogical reasons? Is it deemed too complicated? Is it more contentious than the other axioms? Question also asked at Mathematics Educators S.E.


I've used 'extraterrestrial' twice in a paragraph already, so it's starting to get repetitive... Edit: The sentence I'm hoping to prove is 'Humans have reached [insert word] scales', referring to the furthest we've went in the universe. By reaching I mean actual physical human presence, which excludes anything discovered or photographed by probes, and limits us to the Moon. Hence, the question is to find a word describing Earth-Moon scales. In response to the numerous suggestions below: Unearthly or cosmic scales is too grand and fails to convey what I'm gunning for - a sense of humility and a slight disappointment at the lack of human achievement in interplanetary matters ever since the completion of the Apollo program. Alien scales just sounds plain silly. Extrasolar, interstellar, or intergalactic scales are factually incorrect. So far, off-planet seems to suit my purpose the best for now.


I am trying to describe the evolution of a motion which is composed of smooth parts called "free flights" and instantaneous impacts. For example, consider a bouncing ball: its motion is a succession of free flights, separated with impacts (when the ball touches the ground). I would like to refer to two time-related quantities: the duration of free flights, and the dates of impacts. The word "time" can refer to a length of time, or an instant (a date). To raise the ambiguity, I have chosen the following terminology: "free-flight durations" describes a length of time between two impacts; "impact times" describes the date of impact. Is this correct and would "impact instants" or "impact dates" be better? I am open to other suggestions too, but I want to avoid "period" which I am already using to qualify repetitiveness.


I was wondering if the cardinality of a set is a well defined function, more specifically, does it have a well defined domain and range? One would say you could assign a number to every finite set, and a cardinality for an infinite set. So the range would be clear, the set of cardinal numbers. But what about the domain, here we get a few problems. This should be the set of all sets, yet this concept isn't allowed in mathematics as it leads to paradoxes like Russell's paradox. So how do we formalize the notion of 'cardinality'? It seems to behave like a function that maps sets into cardinal numbers, but you can't define it this way as that definition would be based on a paradoxical notion. Even if we only restrict ourselves to finite sets the problem pops up, as we could define the set {A} for every set, thereby showing a one-to-one correspondence between 'the set of all sets' (that doesn't exist) and the 'set of all sets with one element'. So how should one look at the concept of cardinality? You can't reasonably call it a function. Formalizing this concept without getting into paradoxes seems very hard indeed.


I'm struggling to find the words to convey the concept even now, but perhaps it would make more sense to illustrate by example. Certain communities like Reddit and Quora tend to be liberal-leaning and have other attitudes and opinions not entirely reflective of the population as a whole, in large part simply because they are populated by the kind of people who can find these communities. Your average computer engineer is much more likely to be familiar with Reddit and use it regularly than your average janitor or priest, and especially more than someone without reliable Internet access. Similarly, Wikipedia articles or tumblr blogs are unlikely to be written by your grandparents, and thus reflect a certain subset of content and attitudes by virtue of the people who use them most. I thought about implicit bias, but it doesn't seem to quite reflect the unintentional barrier of entry in these cases, although it's the closest I could find off-hand.


Suppose someone (like a boss, friend, cousin,.., to whom you can not say "no" easily) has a personal request for you which you find difficult to decline, but you cannot bring yourself to say no, because you feel embarrassed or too shy to say no (i.e. you do not feel free to say no), or just simply don't like them to feel offended or hurt. What is the expression or idiom that would convey this meaning: "to accept a request in this situation, unwillingly but under your own moral pressure or just out of shyness". I have found this idiom: "to put somebody on the spot", can I use it in this situation? For example: My mother-in-law asked me to accompany her to the market, and I was actually put on the spot by her request so I went shopping with her despite having a severe headache.


I'm a software developer (although math isn't my strong point). I've developed a device to monitor/control my clothes dryer by monitoring the intake air's humidity & temperature, and the exhaust's humidity and temperature. The plan was to compare the exhaust air's humidity against the intake air. Once the humidity in both was around the same, it would mean that no more moisture is evaporating from the clothes, ie: they're dry. Unfortunately its not that straight forward. the air coming out of the dryer is much hotter, and because the sensor is giving me relative humidity values, I cannot compare it to the intake humidity (which is room temperature). I'm looking for a formula that will allow me to compare the two humidities, taking their temperature into account. An "absolute" humidity I guess, or literally the amount of moisture in the air. I've tried a couple of formulas I've found online but I'm not getting readings that look valid. Note that the value doesn't need to be an official, absolute humidity value, it simply needs to allow me to compare the intake and exhaust humidities, at different temperatures (but everything else, such as air pressure, etc... being equal). (Please feel free to suggest a more appropriate tag)


My school requires that dissertations follow a set of sometimes odd formatting requirements. We have a latex class file which does a reasonable job at bringing documents into compliance with their requirements. There is, however, one requirement that I have't got a clue how to make happen: Footnote are supposed to be at the bottom of each page which can be accomplished by using footmisc. However, on the last page of a chapter the footnotes are supposed to be directly after the actual chapter text ends not at the bottom of the page. Now I can solve individual cases by adding vspace, or simply not putting footnotes at the end of my chapters. However, I'm wondering if there is a way to modify the class file to make this happen automatically.


On a chat channel today I was reading two people talk about some of the more popular movie formats and movie players available. One of the interlocutors said something that got me thinking. I will cut quick to my question. Is there a semantic difference between the following two phrases: ... the last movie I played ... and ... the movie I played last ... To my non-native English ears and eyes, I fail to see a difference. But reading these two phrases again and again makes me uneasy. The more I read them, the more I feel that the first phrase gives off a whiff that would suggest that the speaker/writer does not watch movies very often, while the latter would identify someone who watches movies all the time. What do you folks think?


Chasles' Theorem in its strong form says: The most general rigid body displacement can be produced by a translation along a line (called its screw axis) followed (or preceded) by a rotation about that same line. Now, Euler's Theorem simply says that any rigid body displacement can be decomposed into a rotation plus translation. This is easy to visualize. But what Chasles' Theorem says is something much stronger. Unfortunately, I am just not able to visualize it. Perhaps, I am comprehending it wrong. I mean how is it possible to have the axis of rotation and translation the same (or parallel) for the most general displacement. I mean, think of this case: A body is given a finite rotation about the X-axis and then a finite translation about the Z-axis. How can we find that "screw" axis along which both of them can be described?


Lets say I have the following sentences... Cake is really bad for you. It contains a large amount of sugar. It contains common allergens. It looks silly. Additionally, lets say I wanted to connect all three of these thoughts together in a similar manner... Cake is really bad for you. It contains a large amount of sugar. It contains common allergens. And, it looks silly. Now, I know that using an "And" at the beginning of the sentence is poor grammar. I also realize this is more of a paragraph structure, so I could do some word-smithing to make it work that way. However, this feels a lot like a series of items, and I would really love to treat it as such... Cake is really bad for you: it contains a large amount of sugar; it contains common allergens; and it looks silly. Is there a structure in the English language that provides for this? Is the above the proper usage? Thanks!


I'm taking a very computational course in partial differential equations. Because of this emphasis, I'm feeling very underwhelmed by the course, and have a lot of questions that really aren't answerable in the current state of affairs. My professor basically tells me to take an advanced course in real analysis for a rigorous treatment, but that's a long way away for me (at least two years), I was hoping that someone could answer this question here. In the course, we have only looked at three equations and some minor generalizations on them - the heat, wave, and potential equations. I understand these to be characteristic of larger classes of PDEs, but I don't know anything at all about them except sometimes I can separate variables. For each of them, the method has been identical. Separate variables, solve two (or sometimes even three) ODEs. Then by superposition, sum them up. Determine coefficients with Fourier sums or integrals. Wash, rinse, repeat. The question is this: How do I know that that is all the solutions? There is no existence/uniqueness theorem for PDEs. How can I know that without some more advanced technique for solving PDEs that I couldn't find others? Does it follow from existence/uniqueness of ODEs? What about for those larger classes of PDEs?


When using the interrogative pronoun, 'who', what would the possessive form be? 'Who checks X letterbox every day?'. I feel it ought to be 'his' but some people I know claim it should be 'their', which to me seems to contradict the singular form of the verb 'check'. 'Who checks his letterbox every day?' is what I would say naturally. 'Who checks their letterbox every day?' sounds a bit off to me. Looking on the internet doesn't really return anything useful, only the use of 'their' as a singular pronoun, which seems to be somewhat popular a topic. Note, this is specifically regarding the interrogative pronoun; I understand the debate about his/her/their/ones in other circumstances but I want to know whether the same can apply to 'who' or 'whom'.


I would be tempted to rephrase my question as : why do people seem to care only about the curvature of a connection on fiber bundles ? Indeed, the curvature gives the vertical part of the commutator of horizontal fields (the horizontal part is the lift of base commutator), while the commutator of vertical fields has nothing to do with the connection. So it remains, for me, aside from the curvature, to understand the commutator of a horizontal and a vertical field. In the case of principal bundles and principal (=invariant) connections, I understand it well, since we can use the action of the group. But what happens then in general for fiber bundles? Doesn't it give information on how the horizontal distribution vary in one fiber ? Thanks, Amin


My understanding right now is that an example of conditional independence would be: If two people live in the same city, the probability that person A gets home in time for dinner, and the probability that person B gets home in time for dinner are independent; that is, we wouldn't expect one to have an effect on the other. But if a snow storm hits the city and introduces a probability C that traffic will be at a stand still, you would expect that the probability of both A getting home in time for dinner and B getting home in time for dinner, would change. If this is a correct understanding, I guess I still don't understand what exactly conditional independence is, or what it does for us (why does it have a separate name, as opposed to just compounded probabilities), and if this isn't a correct understanding, could someone please provide an example with an explanation?


In Afrikaans, it is considered very disrespectful to use "you" ( "jy") when referring to someone who is above the level of a peer. Instead, it is expected that you use "u", which is a very respectful form of "you". Also you can talk in the third person "How is ma'am today" would be the equivalent. I cringe internally when I say "How are you" to someone older than me, because in Afrikaans it would be very rude. I was bought up to only ever refer to my parents in the third person. "how is mom today", "what is dad doing" when speaking in Afrikaans. The lack of English equivalent feels very wrong and disrespectful. What is the best way to convey this in English? I have been reassured that saying "you" to a parent isn't rude, and I understand that this can be cultural, but I'm particularly looking for what options English offers in this regard, as far as existing vocabulary, that convey respect. I'm in South Africa.


In a modern nuclear reactor for example a PWR there are multiple containment systems which prevent the release of radioactive material into the environment and shield the environment from the radiation. Here is a quote from the wikipedia article about this: The reactor vessel is the first layer of shielding around the nuclear fuel and usually is designed to trap most of the radiation released during a nuclear reaction. The reactor vessel is also designed to withstand high pressures. I think that the radiation in this part is shielded partially by the walls of the pressure vessel and partially by the water it contains. However how much percent of the radiation is shielded by the water alone and how much by the walls? I.e. how do the shielding effects of the walls and the water compare to each other (roughly) and why?


My brain immediately suggested the non-word "promisand", but I doubt I would be understood if I said that. What's a good word (or failing that, phrase) for the action or thing that was promised? This reminds me of Latin expressions like Carthaginem esse delendam (Carthage must be destroyed, lit., 'Carthage is a thing-which-is-to-be-destroyed') or perhaps that once-common mathematical expression QED quod erat demonstrandum ('that was a thing-which-is-to-be-proven'). Circumlocution or other forms are often possible: "He gave me the promised widget" or "He did what he promised he would". But sometimes it's useful to have a word that stands on its own. More generally, is there a good way to express the construction 'thing-which-is-to-be-X'? In Latin this is a gerundive, though it seems in my brief searches that the term means different things in English and other languages.


"One way mirrors" are used in interrogation rooms etc. The wikipedia article states that: A true one way mirror does not, and cannot, exist. Light always passes exactly equally in both directions. However, when one side is brightly lit and the other kept dark, the darker side becomes difficult to see from the brightly lit side because it is masked by the much brighter reflection of the lit side. Which I think I understand. What I find strange is that this surely can't be any deep principle of physics, as it's easily "violated"; you could just place a camera on one side and a TV on the other side, and it would be perfectly "see-through" from one side but not the other. Couldn't you imagine a sort of microscopic version of the above "violation"?


I am typing up formal invitations, and I want to say that transportation will be provided from Point A to Point B (but also from Point B back to Point A). In order to clear up the to-from/from-to confusion, I tried to use 'between' as follows: Transportation will be provided between Point A and Point B. Is it grammatically correct to use between in this manner? Transportation is really FROM Point A TO Point B -- although the car may travel between the two locations (i.e. not at either end point but in the middle), the transportation itself is to/from specific locations. Another option is to say Transportation will be provided to and from Point A and Point B. I am trying to find the most appropriate choice of preposition to accurately convey the meaning, and I was unsuccessful in determining this after searching myself.


In a sentence where we have two listed words that are hyphenated, we can omit the latter part of the first compound and still be grammatically correct: I don't believe we will ever find helium-based or hydrogen-based life forms. I don't believe we will ever find helium- or hydrogen-based life forms. However, if we have two (related) words which both end in the same suffix, can we still apply this notation? I've seen this used before, but I'm not sure it's proper: It doesn't matter whether the character is a protagonist or antagonist. It doesn't matter whether the character is a pro- or antagonist. More often than in writing, I hear this in spoken conversation, usually with an emphasis on the prefixes (i.e. "... a pro- or antagonist ..."), as though there is actually a hyphen in both words. I've read up a small bit on conjugation reduction here (thanks to search actually finding that term for me), but it doesn't seem to answer this particular question. Is the reduction of non-hyphenated words allowed within English grammatical rules?


I'm currently a PhD student in mathematics at a decent sized graduate school, but I've been questioning my desire to continue on and finish my doctorate after I achieve my master's, which will occur within the next year. I've been thinking about trying the actuarial exams so I might have a route to leave academia. I've never taken financial math, but I have taken a Calculus-based probability theory class, a mathematical statistics class, and two measure theory based probability classes. I've gotten at least an A- in all four classes. I've never taken a financial mathematics class, but I have taken a wide range of both applied and pure math classes. Based on the experiences of some people that have passed these two exams, given my background does studying for and passing both exams seem like a possibility for me? Thanks for any advice!


I am writing tourist information for a city that has areas known for similar shops (fabric, jewelry, musical instruments), similar services (spas, funeral, automotive), and similar industries (textile, software, manufacturing). I am currently referring to all of these areas as districts (e.g. the fabric district, the funeral district, the manufacturing district). Unfortunately, this usage could potentially cause confusion because this city also has a number of governmental districts (similar to "boroughs" in NYC). Also, there is some ambiguity between places where things are made and where they are sold. For example, the fabric district could be easily confused with the textile district. Two questions: Is there a better general term than districts for areas that known for similar things? Are there better specific terms in the case that those areas are known for stores, services, or industries?


I have this combinatorial assignment problem: K candidates apply for a job. There are R referees available to review their resumes and make a recommendation. Suppose that we would like M referees to review each candidate (M < R). How would you assign candidates to referees (or, conversely, referees to candidates)? There are two important cases: (a) K > (R choose M) and (b) K < (R chooses M). Case (a) actually reduces to case (b), so we only have to consider case (b). Of course, there are some constraints that make the assignment a bit challenging. We would like to have an even distribution of the number of candidates reviewed by each referee. We would also like to have some randomness or "mixing" in the assignment such that it is probable for any candidate to be assigned to any M-plet of referees. Is this an instance of a well known problem in combinatorics? Any hints or references to algorithms is appreciated.


Quite a few times now, a waiter or shop assistant has asked me: Will that be fine? I've noticed that I've only ever heard Indian English speakers use this turn of phrase. To my (British) ear, it sounds unidiomatic: I would always ask Will that be OK? expecting the answer Yes, that's fine. I'm intrigued to know what's going on here. Am I right in my assumption, from my own experience, that this is common in Indian English but not British English or (I think) American English? I've been trying to analyse it to work out why there would be a difference, and I'm wondering whether it's something to do with stereotypical British reserve. The British question/answer would go something like this: Q: Will that be OK? [Subtext: of course, I wouldn't dream of suggesting that my poor efforts could ever be positively fine: mere acceptability is all a worm such as I can hope for.] A: Yes, that's fine. [Subtext: I wouldn't want to be so rude as to confirm his suspicion that it's merely acceptable. I'd better make it clear that his efforts are unrelentingly fantastic.] This seems a plausible enough reconstruction to explain why Brits like me are so unassuming, but it wouldn't really explain why Will that be fine? isn't also idiomatic in American English. Is this prevalent only in Indian English? If so, can anyone explain why?


Like many programming books, there are mathematics books which do not provide exercises. Although similar in theory, how can I come up with exercises that will help illuminate a subject? The difference here is that when I pick up a programming book, I generally have an application in mind for whatever subject it is. This is not necessarily true when I pick up a mathematics book because I am not sure what type of questions I should ask. For example, when I first saw the definition of a measure space, I was not sure what kind of questions I should ask myself so that I can get more intuition about the subject. It was not until I saw multiple examples and solved many exercises that I learned what sort of questions were investigated in the book I read. Gaining such a skill where I could take a definition and extrapolate meaning is something I strongly desire. Is this sort of skill something that everyone learns as they gain more mathematical maturity, in a natural fashion, or is this something that has to be sought out?


What does the word cousin mean when used as a verb? By context I take it to mean that someone is putting someone else on or being difficult with someone else. For example, in The Dark Tower (Stephen King) series Wizard and Glass, a character, Eddie, is pressuring another character, Roland, to tell a story of his youth and of the troubling things that happened to him. Roland has been reluctant to do this until now. When Eddie reminds him of his promise to tell them, he responds thusly: "Would you think that I was cousining", he said, "if I asked for one more day to think of these things?" In other examples I have taken it to mean lying or being decietful. Spoilers below: Later in Wolves of the Calla, Eddie shoots the "eyes" of a robot named Andy, effectively blinding him. Andy beings frantically yelling for help, interspersed with "You cousining bastard!". Eddie previously had lied and tricked Andy into entering a confined space where he would be easier to deal with. Regardless of instances, the word appears to have a pretty negative connotation of falsity or deceit. I would really like to know a more accepted definition, although I suspect that this use of the word is wholly Stephen Kings doing.


I have a general question when it comes to deciding if an infinite series is convergent or divergent. The tests im familiar with are ; Ratio test, Direct comparison test, Limit comparison test, Root test and the Integral test. My question is if there is any way to tell what test is appropriate to start with just by looking at the series. At the moment I usually follow my own pattern and systematically try different tests. What I do is: Divergence test to see if the series is divergent. If its unsuccessful Ratio test, if unsuccessful Limit comparison test etc Which has been working fine, my concern is that it may be very time consuming if the first tests are unsuccessful, and might not be very efficient during exams. So basically: Is there a way to determine what tests are appropriate by just looking at the series? ( I have found nothing like this in my textbook, all the examples simply jump straight into the "correct" test).