I am not an expert on figure exports and, although I have been searching around quite a lot, I can't find the answer to my problem. During the writing of a journal article, I had the idea of using psfrag to replace text on my .eps figures generated with Matlab. That works fine, as long as the main text .tex file is compiled as a whole. The problems start when one tries to generate the figures separately (since almost all journals require the figures to be uploaded as separate files). I have been trying to find a way to compile each figure into an .eps file with the psfrag labels added, so that I can submit these files separately. Maybe my problem is ill-posed to begin with (for example, even if one manages to compile a single eps file with the psfrag substitutions, how can one control the relative sizes of figure and text?). If this is the case, I would appreciate if someone could provide a correct workflow for ending up with .eps figures whose text has the same font with the article. Note: the only solution that I have managed to find and seems to be what I want, is the fragmaster.pl script, which however I do not how to operate in windows.


I'm trying to fill out a survey that asks me about features that should or should not be included in a smartphone app. The actual questions are confidential, but it's in the style of a sentence like this: "You can control the temperature of the room you're in" followed by these options (of which you can choose one) Assuming I'd choose "I would like it" for the second row. Does that mean find it positive that the feature is not part of the app, or rather that if it's not a part, I'd like it to become one? Similar with the "expect" answer. Would I expect the feature not to be part of the app, or would I expect it to be and miss it if it's not a feature? I was sent the link by a "noreply@foo.bar" mail, so I don't know whom to ask. I'm passionate about the topic, though, and would like to answer this as precise as possible.


A video shows two spherical objects (made up of particles) collide. The title refers to them as two planets colliding. When the objects are indeed as large is planets (however large that might be), the video seems sped up. What if the objects are smaller than that, like the size of marbles? Would they still move as slowly (relatively to each other) as if they were planets? So, to put it differently, when scaling such a situation down, would they move faster relative to each other? Could the video be showing 'real-time' progression when they would be marbles? Updated question If there are two situations that both have two spherical objects in them that have the exact shape etc. (so both situations only differ in proportion), and you would look at both situations at the same time, would one be progressing quicker than the other? Disclaimer: I have limited physics knowledge. Please help me improve this question if it is unclear or hard to understand.


I'm spending too much time thinking about this problem : I need to show that the shrinking wedge of circles which is path connected, locally path connected ,doesn't have a simply connected covering space . hatcher gives a condition for a space to be semi locally connected it's about the induced injection between the fundamental group of a neighborhood U and the fundamental group of our space (here we need the opposite of that) . I've asked my doctor about it and he told me that I can't use it since our space is "locally path connected " and not semi locally ..Can anyone help me prove that this X : shrinking wedge of circles does't have a simply connected covering space ? I supposed it had , then i took a loop in my space X ,I lifted it up to a loop in the simply connected covering space , this loop would be homotopic to the constant loop, my intuition is to project it down , and prove that since my space is not simply connected this loop can't be trivial but I'm feeling there's something missing or wrong .. Any help is appreciated.


This is something probably very basic but I was led back to this issue while listening to a recent seminar by Allan Adams on holographic superconductors. He seemed very worried to have a theory at hand where the chemical potential is negative. (why?) For fermions, isn't the sign of the chemical potential a matter of definition? The way we normally write our equations for the Fermi-Dirac distribution the chemical potential happens to that value of energy at which the corresponding state has a occupation probability of half. And within this definition the holes in a semiconductor have a negative chemical potential. It would be helpful if someone can help make a statement about the chemical potential which is independent of any convention. {Like one argues that negative temperature is a sign of instability of the system.} Also isn't it possible for fermions in an interacting theory to have a negative chemical potential? Also if there is a "physical argument" as to why bosons can't have a positive chemical potential? (Again, can an interacting theory of bosons make a difference to the scenario?) And how do these issues change when thinking in the framework of QFT? (No one draws the QCD phase diagram with the chemical potential on the negative X-axis!) In QFT does the chemical potential get some intrinsic meaning since relativistically there is a finite lower bound of the energy of any particle given by its rest mass?


Is there a way to selectively disable a specific ligature in XeLaTex? I cannot see how. The selnolig package seems to do what I need but I don't even know how to use Lualatex with my Texshop setup on Mac. Specifics: I am using XeLaTex, new to it and trying to figure my way around and set up some standard templates for my own use, e.g. academic papers. I am mostly using Linux Libertine O. I have the same issue with Adobe Caslon Pro. Why: The "Th" ligature bothers me, particularly in titles. I never see it used in print. See, for example, the New Yorker magazine which uses Caslon, most ligatures are set but never Th. I never see it in books either. (I know this is picky).


If a body becomes charged by losing electrons for example, it will experience a braking force when it is accelerated due to radiation called Bremsstrahlung radiation. Part of the energy used to accelerate the charged body will be emitted as radiation. It should therefore take more energy to accelerate the charged mass than the energy required to accelerate the body without the charge. The Larmor formula calculates the non-relativistic power radiated by the acceleration. There is also a relativistic derivation. The charged body will appear to be more massive due to this effect. It can be explained by the fact that part of the energy is going to kinetic energy and part to radiation. Nevertheless, F=ma should imply a higher mass so are we dealing with a more massive body?


Why can't you create energy out of nowhere For me it's not enough that a smart guy for hundreds of years said so. Some scientists and religious people said that the earth was flat - untill someone said it wasn't. And we know that it isn't flat, we can prove it. I can prove it. Do we just say we can't create energy out of nowhere because Newton said so? Is that our argument - and that no one has proven wrong yet? I think you know some kind of mathemagic that proof that Newton is correct; could you please explain it to me? What about Vacuum Quantum Fluctuations and the Casimir Effect? Is that just a mathematical estimation? Or where do the energy come from? In my (non scientist) eyes it comes like - from nowhere. Is a Zero Point Energy Module possible? I have found this article. Actually all my questions are based on this article. It would be nice if you take a look on the article and try to explain against it. In this article on a no-name-blog there is the claim, that a lot of scientists (phds, professors and so on from well known universities) say that such a "free energy generator" is possible, and that it was build. But because companies would lose money, it will never come to society. There is a lot of fancy words in it, a lot of PhD and other titles and sources on like Nature.


Below is a review by music critic Robert Christgau of a Flaming Lips album. I've pasted the entire review, but my focus is on the first two sentences. Primarily, is it idiomatic to use "couldn't" here? Because unless I'm wrong, isn't the point of what he's saying that you don't have to hate Sgt. Pepper to think it "could" do with a little ribbing, etc.? You don't have to hate Sgt. Pepper to think it couldn't do with a little ribbing, travesty, desecration. In fact, you could love it as much as I do and think that. As hilarious sobersides from multiple generations charge indignantly that the Lips and their various beards fail to "interpret" the songs, all three modes of deconstruction are in play on this grand hoot of a fore-to-aft remake. Highlights for me include a theme statement that gains meanings it never had from its attendant distortions, a creaky "When I'm Sixty-Four," Miley Cyrus so sweet on "Lucy in the Sky," and Julianna Barwick adding just what "She's Leaving Home" cries out for--a female voice. Only "Fixing a Hole" truly fizzles. As for "A Day in the Life," yeah--the original rocks. A-


M-W has the following definition for mixed metaphor: a figure of speech combining inconsistent or incongruous metaphors Hence a requirement is that a 'mixed metaphor' contains more than one metaphor. Eric Lippert comments in another thread: "mixed metaphor" is more commonly used to describe the result of accidentally combining two metaphors in a way that does not make sense as a whole. "For me it was stormy in the great sea of life, but then I came to a crossroads." He does not claim that this is a necessary condition, and I'd agree. I've just written in another thread: ' "John is a real tiger" works, but "That lion is a real tiger" is best avoided.' Each of these two statements contains a single metaphor. Does any definition of 'mixed metaphor' apply to the inappropriate (incongruous juxtaposing of tenor and obviously related vehicle) metaphor in the second sentence?


Else-site, I got into a discussion with a member where the thread turned accusatory and defensive because I said a programming language had tricky details that you learned over time. I did not mean that it was a bad thing, only that there were features less traveled that really enhance the effectiveness of code. One of the Definitions of 'tricky' given by M-W is "requiring skill or caution" and the other two are even more negative. Even "difficult to do or deal with" leans negative into the "It shouldn't be that difficult" realm. Yet difficult is not always bad. Sometimes it just is, mastery of any topic is difficult, and sometimes just knowing the subtle features is enough, it's a sign of a skilled practitioner to leverage these subtleties, and it's those I call tricky. Am I off on my usage, and what resources exist to try to divine these usage subtleties? I'm a native speaker, yet I can't always explain why I shade word meaning a certain direction that is not supported by a dictionary.


I came across a question where i was asked to find the force exerted by the walls of a frustum shaped bucket on a liquid inside it . I found the force exerted by the liquid on the walls of the bucket by concepts of hydrostatic force and thought that by newton's third law both the forces will be equal and opposite . But the answer was totally different . I checked the solution in that simple force dynamics (newton's second law) was used considering the weight of the liquid , the normal force by the floor of the bucket, the atmospheric pressure and the force by the walls. Then since the liquid was at equilibrium all the forces were vetorially equated with each other . I am convinced with this solution but i also don't see any problem with my approach . So how should it be ?


How do you see non-academic mathematics? I have an impression that the academy has still a quite significant prestige and is thought to be the safe-guard for "real science". That is, to verify that those that have the most experience in science, can produce science and have the "blessing" of the academy, in order to deviate from informal publications made outside of academia, whose content cannot be guaranteed and that do not necessarily go through peer-review. However, esp. in mathematics, there's no real reason why mathematics cannot be produced anywhere. I.e. academy does not add much to the process of doing mathematics. Social connections to like-minded people perhaps. Then there's the internet, which makes all information pretty accessible. So do you think there's a place for "open source, non-academic mathematics"?


So basically, I was trying to find a good answer to the question of how light interacts matter. Namely on the quantum level what causes matter to appear transparent, reflective, opaque, etc I came upon the answer with respect to transparency here, but I feel that the concept isn't fully explored here and that something is missing (particularly in the coloured light talk). The jist is that light passes through some matter because the energy gap between the matter's electron's low and excited state exceeds the energy of the photon, and thus the light is not absorbed and the photon passes through the matter. Could someone provide more detailed information about transparency, and information about reflection etc. So I guess for the purposes of forming this into a question I'll simply ask why some materials reflect light and look like mirrors while other materials reflect light with less precision? (thought I get the feeling it has similarly to do with basic mechanics and the regularity of the electron distribution)


I live by Internet, as both a user and a developer. This dual role gives me a chance to observe. Every time I try to ask something in communities like StackOverflow, I always pick words carefully and provide sufficient information, hoping that my questions will be answered in a short time. But even so, sometimes my questions still get ignored. Maybe StackOverflow, or even the whole Internet, is too big a world, filled by floods of information. Small people try to fit in, but end up being swallowed by Twitter and Facebook, without even a sound. So, enough of complaining, I just want to know: Is there a word or a phrase to describe this phenomenon where people find themselves lost in the the great Information Age, ignored by others?


I've been reading the book "Geometric Mechancis" by Darryl Holm and the in the first chapter he treats geometric optics. There the author talks about light rays and those light rays looks like trajectories as of particles as we consider in Classical Mechanics. The first thing that the author state is Fermat's principle that seems to define one action and then determine the path that light follows (i.e. the light ray) being the one which extremizes the action. In all of that discussion, it seemed to me that geometric optics is then all about treating light not as a wave, but rather as a collection of particles. Is that it? In geometric optics we should think of light as a collection of particles? In that setting, a light ray is just the path followed by one such particle, or is it composed by many particles?


This question stems from a disagreement between me and my girlfriend. During the hot summer months, we like to open up the windows during the early morning to cool down our flat a bit. One of us thinks that every window should be opened fully, since that will ensure maximum airflow, the oter swears that the actual area of open window is not a huge factor. We both agree though, that at least in most situations, opening the windows fully is the best option; the question is rather. All other things being equal, how will doubling the area of one open window affect the airflow in a flat? If it helps at all, we typically also have a fan or two running in the flat and we have windows at opposite ends of the flat, which are all open.


Let imagine a tunnel that connect two distant places at the globe (eastern-western or north-south) There are a lot of posible "distances" or metrics, defined by maps, routes, "as the crow flies", etc.. but none of those distance can be shorter than the distance of the tunel. So if two trains travels at same speed, one inside the tunnel and other above in the surface, the one on the tunnel will reach first. If this is possible, then perhaps it's possible to have differents coexisting metrics with differents dispositions or topologies, within the same system. Of course that if we describe a space-time metric surrounding a sphere, then "holes" in it would change the metric (just because it's not a sphere anymore). But it's strange for me that making a hole we could change in some way the space-time shape. In an extreme case. Could be an euclidian space of same dimension be build within a non-euclidian space? I would like to have a view from people familiar with general theory of relativity, thanks


The hippopede is historically famous because Eudoxus used its properties in the first mathematical model of planetary motion. He nested concentric spheres rotating at different inclinations to each other, and had the motion transfer from outer ones inward, the planet was attached to the equator of the innermost sphere. With three or more spheres he achieved trajectories with backward loops ("retrogradations") similar to those observed for the planets. But with just two (see animations here and here) the trajectory is the figure eight shaped curve obtained by intersecting a sphere with a cylinder touching it from the inside (see interactive graphic illustrating that it really is the same curve). Clearly, under any composition of rotations a moving point has to stay on a sphere, but why is it also confined to an off-centered cylinder?


I was reading G.I.Taylor's Single - Photon Double Slit experiment. Now, at a time only a single photon gets emitted. What is the probability of it at a certain point of the screen to hit? The answer is written rather abruptly: Although the photon can hit anywhere, the probability is more, where the brightness(bright fringes) i.e. Intensity is more. Now, this means intensity existed before photon actually hit the wall? How can it be possible? When photon hits the screen, the energy gets transferred to the screen. So, after photon collides, intensity does come in discussion. But if we accept the answer, it is like saying that energy came somehow on the screen before the photon; after that photons would strike there where the intensity of the energy is more. So, who is actually transferring the energy: photon or the waves? If it is wave that transfers energy as being evident from the above quoted argument, then what photons are doing actually ?? What are they meant for if they are not responsible for transferring energy??


Let's define a "hand-incalculable problem" as a mathematical problem that can not be solved by available human calculation power (using only writing materials and utensils) at a specific date and geography, during lifetime of the person who posed the problem. Given a specific date and problem can we strictly label the problem as "hand-incalculable"? Edit: We assume that algorithm to verify the solution is known. Edit: I am looking for a general classification similar to P/NP/NP-Complete which is used in computation theory. I wonder if we could define such classes for mathematical problems as HC(hand-calculable)/HI(hand-incalculable)/HI-Complete at any specific date. Even an absolute classification based on available human FLOPS and not related to a specific date, would be nice. Edit: HC: Hand-calculable at a specific date. HI: Hand-Incalculable at a specific date. HI-Complete: Hand-incalculable no matter how far mathematics will advance in future; consider a dynamic programming problem that needs millions of FLOPS of calculation power to be solved in reasonable time (during lifetime of the person who posed the problem).


I'd like to know if there have been attempts in solving the full problem of the dynamics of a classical hydrogen atom. Taking into account Newton equations for the electron and the proton and Maxwell equations for the electromagnetic field produced by these charges one obtains a higly non-linear set of coupled equations. In such a nonlinear system could some feedback effects between proton and electron take place so to make possible a stable dynamics (or at least a dynamics unstable on such long time scales longer that we can consider hydrogen to be stable)? In this way the system's stability already obtained through quantum mechanics could be reproduced by a full classical approach! P.s.: Please, as I know of the great successes that quantum theory has had since its birth, try not to answer the question telling how quantum mechanics wonderfully solves the problem. P.p.s.: I'm also aware of the fact that electron should lose energy and that this should cause it to fall on the proton in a very short time, so please try to avoid also this argument. I asked this question to understand if the oversymplifing hypothesis', which are fundamental in solving this problem (neglect proton's motion and, as a consequence, magnetic effects) and are quiet ubiquitous in physics, wouldn't mask the potential richness that could arise from mathematical complexity.


He was born and brought up in this village, but after staying and studying abroad his childhood friends seem backwards to him; he _ this place now. Or He had lived in this city all his life but whenever he gets a chance he starts counting the flaws. He thanklessly _ his homeland. It might even apply to a situation like say you party every weekend but if somebody else does the same you criticize them. Note: Read title: act of crticizing one's own homeland/city/country I can use words like criticize, betray or badmouth in my sentences, but I am looking for more like an idiom or phrase or even a proverb, that may or may not fit into my sentences but explains the intent. The intent here is being a hypocrite to an extent, yes but there's an addition of viewing one's own homeland or people as backwards(where I don't think that person has changed, they are just trying to show off). In hindi there's a saying "jis thali me khaya usme ched kiya" which means destroying the plate in which you are served. I'm kinda looking for an english analog for this proverb. I thought "don't shit where you eat" would be the close but it was hilarious to find out it's meaning and of course that it's nowhere close.


Whenever I write TeX/LaTeX documents, I always find that annoying because I can't really focus on the content, It's not that I don't like TeX or what it does for me, it's just the fact that the document quickly becomes something way too verbose. I was thinking about adopting an intermediate syntax, something more "dry", that can help me focusing on the content and only use LaTeX later on in the process to create the last pieces of the layout and the paging. So far I was able to identify rst ( reStructuredText ) and docbook, needless to say I have no experience with neither of the two, but I have a few requirements while asking for your help: support for math formulas support for include external resources like snippets of code or images (vector images for the most part) The kind of documents that I would like to write while adopting this "dry" solution are small books and articles for the most part, so often times I need to separate things into chapters or paragraphs and I would like to have a solution that is modular and flexible.


First and foremost I want to say that by no means am I a physics guru, but I still ponder this question. While I can understand from my current understanding of physics that a perpetual machine is yet,by current understanding, unattainable, does that mean that there is no hope for a free source of energy? Confined to the forces of the Earth we are bound by gravity, a constant form of energy acting on all, but what is the reasoning for not attempting such a feat in the midst of space where the levels of interference with the generation of energy(again from my current understanding) is so minimal? I understand that there would technically be friction on a molecular scale with particles bouncing and colliding with each other, and other potential space debris. However, other than that what is stopping us from achieving the goal of this, or am I simply missing information?


So I decide to self-study the real analysis (measure theory, Banach space, etc.). Surprisingly, I found that Rudin-RCA is quite readable; it is less terse than his PMA. Although the required text for my introductory analysis course was PMA, I mostly studied from Hairer/Wanner's Analysis by Its History (I did not like PMA at all). Although I said readable, I do not know if I actually understand whole materials as I am middle of first chapter, and I already have topology background from Singer/Thorpe and Engelking. I actually like Rudin-RCA, but I am not sure if I am taking great risk as many experience people seem to not liking Rudin for learning... Is Rudin-RCA suitable for a first introduction to the real analysis? Is it outdated? What should I know if I decide to study Rudin-RCA. I am not planning to read the chapters in complex analysis as I am reading Barry Simon's excellent books in the complex analysis.


I have the following sentence: Increase your rating as you develop your coding skills while you evaluate the skills of other developers. However, I now realize that this sentence is a bit off since it doesn't communicate properly. Some feel that this sentence sounds challenging since some imply that you're competing and working with other programmers. I am trying to make it feel more inviting and feel less like a challenge but an opportunity to not only give back to the community but also improve your own skills. I am trying to remove the potential stress that goes into evaluating someone or letting x number of people evaluate you since you're already judged by your manager. I was thinking that maybe changing the tone of this sentence to be less a declaration but more of an open suggestion. Any help would be greatly appreciated.


If an angle is the measure of distance between to points (Edit: Ok, admittedly bad phrasing. A measure of rotation between two intersecting lines, or points, etc.), is there such a thing as a zero degree angle? I asked a math nerd friend about this years ago and he still hates me for it but couldn't prove it either way at the time. Is this merely pedantic or delusional thoughts regarding definition of "angle" (Edit: based on answer below regarding Euclidian geometry, this all now seems very likely a colossal misunderstanding) or am I missing something? I do not think zero degree angles exist. Is there proof of this either way? Edit: A better question would be why Euclidian zero degree angles were thought to not exist, or why he avoided them?


Suppose someone comes up with a great original idea. Some time later, someone else comes up with exactly the same idea without knowing that idea been conceived already and even without being influenced in any way by the other person's thought. In other words, both of them make the discovery totally independently and not at the same time. For example, Leibniz and Newton invented Calculus this way; Cook and Levin established NP-Complete theory this way. The idea from the very first inventor is no doubt considered original. But what about the same idea independently conceived by others later in time? I have no problem calling the latter original, although it seems less original than the very original one. Would there be a word or phrase that can readily convey the state of being "less" original?


Specifically, I am referring to the traditional gesture some female service staff members seem to do, when demonstrating their work or their readiness to receive orders. Also, this is the gesture Alice Cullen performs in Twilight movies on several occasions (if I remember correctly). I could not find a photo with this exact gesture. It is similar to the one shown in the first photo, but the hands are held higher with demonstrativeness, tension, sharpness of movement and gesture itself (possibly with emotion and a smile), and military-like demeanor. The main difference with with the second photo is that in the gesture I refer to, the hands are held in the front and not in the back; also, the way the hands are interlocked is different (more like in the first picture).


I have a circle moving on a spherical surface. If the camera angle is fixed, is there a way to calculate the original circle's offset from the centre given the distorted ellipse formed when the circle moves to the edge of the sphere(as seen from the camera)? I've looked into Listing's plane and the Tissot directrix(because I'm working with eyeball rotation), but neither seems to provide a clear solution to this. I've also looked papers regarding eyeball roation but they seem to focus on the physiological aspects rather than the math of the movement. In other words: In the picture above, given one of the outer nine images as well as the middle one, I want to calculate how much the circle(i.e. iris) has moved/rotated from its position in the middle picture. I assume it has to do with comparing the distorted ellipse with the circle, but I can't seem to find out how. Is there some formula for this?


Reading the wikipedia article about the particle in the box, there is this image: Animations from B to F show wave function of a particle in a box starting from ground state up to excited states. The animation C shows wave function behavior in the first excited state and at the middle point both real part (blue) and imaginary part (red) of the wavefunctions are zero all the time. Does it mean that particle will never ever be found at that point? Similarly, for second excited state in the picture D we have two points where this happens, so does that mean that there are two points in space where particle will never ever be found? I can't really tell from the picture if this continues for higher excited states. Is there some law that says for the n-th excited state there will be n points in space where probability of finding particle is zero?


I've seen nice graph showing relative activity of each isotope in Chernobyl fallout: Could anyone suggest similar graph or raw data but for absolute isotope activity for the case of nuclear explosion fallout? (for some boosted fission bomb) So that one could see how much gamma, beta and alpha-active particles one should expect over time, and particle energy. PS. This graph is for damaged nuclear plant fallout, but I want absolute graph for nuclear explosion fallout. Explosions have different isotope content due to much shorter duration. Update: I see that graph might be hard to find. Maybe someone have isotope content i.e. how much % of each isotope? Or at least how it is supposed to be different from Chernobyl fallout. Then I might be able to simulate decay by myself.


What's a word that can be used to describe an author's portrayal of a scene as normal even though the content of what is described is innately disturbing/unnatural? He cut open the dog's underbelly and we all took turns examining its insides, which were just as we had expected them to be. To get over the disappointment it was suggested we head out for drinks later, which I was forced to decline... The writer's description of the scene is word. The writer words the scene. This disgusting scene is worded by the writer. Earlier I was able to think specifically of the word, but now I can't even remember whether it was an adjective/verb, so sorry for being so open in my example usages, but hopefully this contains enough information for someone to propose something.


I have a question about a Quantum Mechanics machine that should interact with the past, I can't find any flaws with it, but I just want to make sure. The machine has a lot of facts about history, some true and some false, and the machine knows which one each fact is. It measures the spin of an electron, and if it has an up spin, it tells a true 'fact' and then a false 'fact', but if it has a down spin, it tells the a false 'fact' and then a true 'fact'. It does this several times so it will probably have both at least once. So until you check which one, the past is in a sense, in a quantum state of being one thing or another. If this all stems from incredible stupidity, I'm really sorry, but if it doesn't, thanks in advance. EDIT: To specify, the part where it 'interacts' with the past, is when you check which spin it is, so you can use a coin to do the same thing. In other words: each fact is true and false at the same time, so checking collapses and 'changes' the past.


in Dire Straits "Sultans of Swing" what is the meaning of these two lines: In the first verse: You get a shiver in the dark It's been raining in the park but meantime South of the river you stop and you hold everything A band is blowing Dixie double four time You feel all right when you hear that music ring As I understand blowing Dixie means something like fooling around. But what is double four time? And then later: And Harry doesn't mind if he doesn't make the scene. He's got a daytime job, he's doing alright. He can play the honky tonk like anything, Savin' it up for Friday night. What does it mean? Is he playing really good but saving it up or doesn't do it very well? Thanks.


As is the case with most students (I suppose), I seem to have lost touch with the do doing some "basic" math -- calculating integrals, playing around with matrices and doing some linear algebra etc. -- as I have proceeded through my undergraduate program. To be honest, I didn't practice a whole lost of questions when I took these introductory courses, so I presume that's one source from where this issue stems. I'm a student of physics with a mathematical leaning. Ideally, I don't want to go through a mathematical methods textbooks for physicists/engineers since they're pretty superficial (at times) and are more like "here's-a-result-now-go-solve-problems." At the same time, I don't think I have the time to sit through very long textbooks, such as James Stewart's Calculus. Any suggestions of textbooks, given my aforementioned concerns, where one can "practice" calculus, linear algebra, complex analysis etc. to stay in touch with the practice of being able to solve in questions in closed form.


Is there a way to tell TeX to avoid breaking the page after the first word of a sentence? ... ... He was quite dead. Apparently his neck had been broken. The lightning flashed for a third time, and his face leaped upon me. I sprang to my feet. It (text continues on next page) And then you have to turn the page for the rest of the sentence. It's not in a line on it's own so it can't be penalized like an orphan line. Can TeX be told, to resolve these by, say, breaking the page before that first word? Note: Although the answers given below are very informative, the general consensus had been that the best practice is to leave this for the proofreaders to spot, and then fix manually.


I got a question about the usage of generalization of collocations. There is a word such as "bird of prey" to refer to birds eating animal flesh. And one way to generalize reference of a noun is to put "the" in front of the word, especially for animals, plants, recent inventions; the lion is the king of the animal world of Africa / the smartphone is nowadays a required item. Then my question is whether this kind of expression below is also okay to show generalization; The bird of prey is a bird that eats animal flesh. For other nouns, such as "man of steel," when someone says "the man of steel," a native English speaker would think some particular or specific man of steel since there are several possible possibilities; Stalin, or the Superman in the recent movie, etc.


I am not very familiar with the quirkiness of relativity, and I was wondering how to explain this situation. If a beam of light is shining at some object at some distance from the origin of the beam, does relativity imply that photons in the beam lack time flow and, thereby, never actually experience contact with the object? Wouldn't the photon observe no time passing in the surroundings, immediately at the source of the beam? Could it tell it was moving? However, we see it make contact, so does that mean that what is observed might never actually ever happen in another reference frame? Also, when we see the beam of light traveling at c, how can we record the beam's movement? Shouldn't the beam's photons appear to be in some timeless state?


The closest description I can think of is a "drama queen/king", except that tends to describe people who like being in a drama. Instead, I want a phrase describing people who like to pick up on stories such as the following: questionable decisions of others controversies immoral acts and then they get on their soapbox and passionately rant, uttering phrases like "it's disgusting", "it's a joke", "anyone who agrees with them should be ashamed". Perhaps the descriptive term for that person might need to include that they are often immune to reason on subjective topics, and that the rants cause displeasure in the listeners, since frequent rants can wear listeners down, as the tone is loud, highly opinionated and of a very negative nature. The reason I ask is I want search terms to Google to find techniques for dealing with such a person, and protecting one's self from their effects. Edit: it is not really about whether someone finds faults in others, but the manner in which they frequently rant about issues, either in others, or institutions, or the nature of things.


The area/arc-length is given by an integral or the integral defines the area/arc-length is one the first things we learn in Calculus, but that is done in the Cartesian coordinates, next one moves to polar coordinates and the area is transformed/redefined by a new integral using the Jacobian. My question is : Since area and arc length are invariant of coordinate system, is there a way to define them other than in Cartesian coordinates and transform them from coordinate system to coordinate system with the help of Jacobian? In other words is there a way that area/arc-length is defined independet of coordinate system and then according to the structure of coordinate system it's integral form is reached, without tranforming between coordinate systems, but for a given coordinate system it is derived. (and without the use of Jacobian to move between coordinate systems)


I have a pair of shoes, which seem to isolate me from the ground. In effect I'm gathering static charge and every time i grab an aluminum door handle, that current discharges and that hurts. Ouch. I invented a way to workaround that: when I'm going to touch door handle, first I take my Skeletool (a stainless steel multitool) and touch the door handle with it. Sometimes I even see the small spark and hear the discharge and then I can touch the handle myself unharmed. But what interests me is: why discharging through the multitool does not hurt? This is an electrical current flow what hurts and the current flows the same way even if I hold the tool in my hand. Tool has surely less resistance than human body, so it shouldn't change anything. But it does not hurt :) Why?


Please refer to the figure attached. Consider a normal force is acting on the top of sphere. A constant coefficient of friction causes frictional force throughout the sliding. I want to know after this sphere slides (pure sliding no rolling) for sometime and assuming that it wears as it slides, what should be the shape of sphere after sliding? The one shown in (a) or (b)? In short, I want to know whether the worn side of sphere will be a straight line or a curved one? What will happen if a sphere slides against a sphere? Also, I will highly appreciate if someone can refer to some good papers / books about this. Edit: Additional assumptions Lets assume that hardness of both surfaces is the same. Also assume that material is removed but is not attached to any of the surfaces (no adhesive wear). The phenomenon under consideration is abrasive wear (but no accumulation of wear debris). If a lubricant is in circulation, it is easy to realize this kind of wear


This question pertains to programing but is really a math question. I am building an application that draws a line graph, similar to a stock line graph. The problem is I am not starting with a known set of numbers. The values will come in based on an altitude value. So, when I start the graph I have no idea what the max and min values of my data set will be, because they max might not come along for some time. Like I mentioned, these values will be added to the graph when I receive and elevation value but I don't know these until they are received from my device (iPhone). Is there a formula used to calculate a line graph for unknown values? How can I know what x and y values to use for each new value received? And how can I calculate the min and max of the graph?


I want to discuss with my boss a part of this code - so I need a name for it. items = array.map(function(item){ return item.item; }); I want to say "This usage of the same identifier twice is _______________ and confusing." They might respond "No this is _________ which is fine". Is tautology an accurate word to use there? the saying of the same thing twice over in different words, generally considered to be a fault of style The use of item in the function call e.g., function(item) could be named anything you want, so instead you could have: items = array.map(function(data){ return data.item; }); What is this called if not tautology? Edit: A non-programming example, would be naming a street Road Street, or worse naming an area in London, London (perhaps New York New York would also work).


In several articles and books, I have read that "the magnetic force on or due to a small element of a circuit is equivalent to two or more of its component elements, provided that the current remains same". i.e. the magnetic force on or due to AB(in red) is the same as the magnetic force on or due to its component current elements (in black) However this doesn't make sense to me because in a closed circuit, the vector addition of current elements is zero. It means the resultant current element of a closed circuit is zero. Hence the force on or due to a closed circuit is zero. However this is not the case (a closed circuit experiences and exerts magnetic force). Then how can "law of vector addition of current elements" be valid?


I don't know why, but since recently my Emacs is behaving strangely. As usual I compiled using C-c C-c to call LaTeX, then there were errors and it asked me to type (as usual) C-c '. but when I did it just commented out the whole paragraph, which is the same effect as C-c %. C-c % still works the same and typing ' produces the same letter, i.e., '. so I somehow have my doubts that it is related to the keyboard, but not out of the question, of course. I can't recall having done any setting changes or changes to my setup since the last time it worked perfectly, but i am not completely sure. I'm running Emacs under Ubuntu, Latex/MP Fly Ref are the automatically hooked modes. Does anyone have an idea?


Dictionary definitions of `bonny' admit to chiefly British (or even Scots), but give no further hint of the possible tinges of this word. Bonny (adj.) means attractive, fair; fine, excellent [M-W]. Perhaps I perceive it as slightly archaic, hence this post. Is this word (still) functioning in the meaning given, and what are the caveats of its usage? Is it the teensiest bit pretentious, or otherwise out of the ordinary? If not, what might be typical usage? (I know about bonny, bonny banks of Loch Lomond. I know about blithe and bonny ladies. I know about one wife or another being `very bonny, thank you'. These, and a perhaps a few other titbits I acquired somewhere, more or less defined the word for me, and I'm finding it difficult to shift away.)


I am creating a program but facing problem with proper language for comment string in program. My program performs some operation. This operation typically always generates some data. (meaningful data if operation went as expected or error data if operation didn't go well). Now this program returns these two things to user: data, True - if operation generates meaningful data data, False - if operation generates error data I want a comment string at the start of program which mentions what this program is returning to user, and I am not able to come up with a single word which will define True/False nature of this data. Things I have on mind for comment string: "Returns data and True/False based on result of operation" (UGLY) "Returns data and success nature of operation" (WUT?) "Returns data and boolean result of operation" (Not good) "Returns data and ....." (I am not good at this really)


I got into a discussion with my physics teacher about the speed of light and I asked What if an object with mass was to lose mass as it gained speed-- would that allow for an object to eventually reach the speed of light or surpass it if possible through that thinking? My teacher didn't really have an answer to that and I was wondering if there was any law of science that prevented an object with mass to become mass-less. Because I know this may be simple minded in thinking but if an object with mass must have infinite energy to reach the speed of light and at that point will have infinite mass, what if an object was to reduce its mass until it had none? Would it then not need infinite energy and be able to move at or past the speed of light in theory?


What cases can "y'all" work in? A prior question asks about the 'proper' usage of "y'all", but it and its answers only address nominative case (all examples are nominative). I think that there are some cases where "y'all" doesn't work as is. For example, nominative: "Y'all come back now, ya hear?" accusative: "I'll ring y'all up tomorrow after the fish fry." possessive: "Bring y'all's swimsuits. The pool will be open." (or "y'allses". yes, I find this is questionable usage) vocative: "Hey y'all! Where's the keg?" but instrumental/dative/indirect object?: nothing sounds right. ? I'll bring the BBQ over to y'all. ? This party is all for y'all. These don't sound right to me. The alternatives that sound right to me would be: I'll bring the BBQ over to you all. This party is all for you all. Can anyone confirm my usage? Has there been a study/paper on this? I ask because most pronouns in English have forms for different cases, so it is not given that "y'all", though a synonym for "you" which only has a different form in possessive case, would or would not have a different form.


What word best describes the experience of starting something, and ending up, before you know it, far deeper into it than you had ever intended, or far more embroiled, involved, or even having finished or done it irreversibly. An example might be, say, opening a book out of curiosity, and finding yourself, the next thing you know, dumbfounded to have finished the whole book, almost as if it had happened by some driving involuntary osmosis or in a blackout. Example sentence with blank for desired word: Having opened the tome with the intent of reading only the first paragraph, she found herself, in what seemed like hardly an hour, staring at it's back cover almost as an alcoholic might awaken from a blackout to an empty liquor cabinet, and wondering at the _________ way some deeds are done in life, almost as if they are done to you.


I watched this video: popped water balloons And as you see when the needle pops the balloons the water stays in the same shape as the balloon for a few fragments of a second. The force impacting the water after the balloon has been popped must be gravity, and since the water stays in the same shape for a while must mean that the water "doesn't know" that it's gravity it should "listen" to now during that time. What does this say about the speed of gravity? How slow/fast is it and what happens with larger/smaller objects? One thing holding the water together could be air pressure, but what happens with if you do the exact same thing in vaccuum? Will it stay in the same shape for a few moment or will it dissolve at the "speed of gravity"? Let's say it stays the same, what would that mean for earth if the sun disappeared? Would we feel the gravitational reaction way after the light went out?


I've just got wondering if this sentence is grammatically correct: You are who I love. This is what I am thinking: Let's focus on the who clause, then you can find that the missing element from this clause is actually an object after the verb love. Therefore, who should be changed to whom, which is an object relative pronoun. You are whom I love. Now, the whom clause has no problem, but you can see that whom clause works as an object. However, there is the place at which a subject complement should be. So, I should either change the whom clause back to a who clause or make you into an object. You is whom I love. It's you whom I love. Whom I love is you. After all that, I came this far with these three sentences and I think the most appropriate sentence is the second one, but I'm not sure if my assumption is right; the way I fixed the sentence, is it right?


I will greatly appreciate your help with this question. I use beamerposter and want all the blocks (across various columns) to be of the same height (= max height of among all other blocks). I came across your solution here: How to modify columns/column environments so they resize automatically to the largest column ? However, in case of beamerposter, blocks can be located in different columns. And this trick of placing blocks inside columns and count the height does not work here. I need somehow to fetch all blocks of all columns of the poster and compute max height (and set each block height to this value, maybe a certain parameter for blocks ?). I imagine there should be a way to re-use a solution from here: How to modify columns/column environments so they resize automatically to the largest column ? by nesting blocks into special dummy columns to count them and to adjust their height. I tried to play around this idea but failed in my tryings. Can you please suggest a solution here? I will greatly appreciate your help with this question. Thank you / John


i have problem to determine if the word bake is transitive or intransitive verb from the Phrase "Samantha can bake at any time of the day." i am reading from a book called English grammar understanding the basic it say Action verbs that act upon something are called transitive verbs. Action verbs that do not act upon something are called intransitive verbs. If a verb (in any of its forms) can be put in one of the following slots, it is transitive: (a) What did you_____? (b) Who did you______ ? If a verb cannot be put in one of these slots, it is intransitive. If a verb (in one of its forms) can be put in one of the following slots, it is transitive: (a) He _____ something. (b) He____ someone. If a verb (in one of its forms) can be put in the following slot, it is intransitive: He____ by following the grammar i can say "he bake something" or "what did you bake" it look transitive for me


Since we have working models for forces "combining" or being describable via a single framework at higher energies (such as the "electroweak" force and the aim of GUTs), does electromagnetism split into an "electric" force and a "magnetic" force at lower energies, or is this simply a misnomer in the English language born of the order in which the behaviors where discovered and related? As an aside, the main reason I ask is because I have heard "spin" (the quantum number; which I might add in hindsight was badly named if considering the laymen such as myself) described as a "magnetic moment", and something entirely disconnected from "electric charge" (another quantum number); to my limited knowledge, there is no known meaningful correlation between the two properties (though obviously I might have just not read about/understood such). I'm a confused layman, and any answers will be much appreciated!


Looking around trying to find questions concerning the intuition behind discrete/indiscrete topologies, I haven't found much towards the essence of what these particular topologies imply about the space (and why it is meaningful). I realize that in a discrete topology, every set is both closed and open, as is the case with the indiscrete topology, however that the topologies are disconnected and connected respectively. That this somehow tells us about distinguishability between points; in the case of the discrete topology we have that every set of points can be at least disconnected from other sets of points, and that in the indiscrete topology, this is not the case. Now the difference between the neighborhoods then would be then that for the Indiscrete topology, all the neighborhoods include all other points, but that in the discrete topology each point has a neighborhood which doesn't include the other points. Am I correct so far? But I know there is much more, could someone expound on the intuition what is going on, maybe even including things such as levels of "seperated-ness", and perhaps what happens in between the Indiscrete and Discrete Topologies? Thanks, Brian


If I fill a plastic ziploc-shaped bag with water, the cross section profile should be sort of teardrop shaped (assuming we ignore the edge effects of the bag being sealed on the sides as well as the top and bottom). The bag should "sag"/get wider until to get the center of gravity as low as possible. Initially, getting wider will let more water towards the bottom but eventually this is offset by the bottom of the bag moving up (because the sides are fixed length). Is there a common function that describes the shape the cross-section of the bag makes? I would guess the bottom is a parabola, since gravity likes to make parabolas. Then I would guess the top is linear because its under tension. But I have no idea what the transition region might look like and whether you could put those two together into a nice function.


Are the slang usages of "bud" (mainly meaning cannabis, and occasionally clitoris, from what I understood) prevalent enough that I should avoid using them in a product intended for international availability? Example sentences which could be printed: "John has created a bud here!" "Share this bud!" (ouch) "Create bud" I would appreciate very much if you could state which part of the world you're from when answering this question. Some context: The lexical field surrounding a service I'm designing revolves around trees, plants, and nature in general. My team and I are in the process of selecting names for the various parts of the service, and we are struggling with one: It should depict something with potential, which is not yet fully developed, but holds the bases for future interactions. Seed would not be relevant, as this part of the service is sprouting from something we called a Tree, and I feel it would generate confusion about which one generates the other. Bud came to mind, and seems a good choice. it fits wery well with the rest of our service part's names. However, it does not seem to be a very commonly used word (much less than "tree", "branches", "leaves", "roots" and so on), and I'm afraid this could heighten the chances of it to be read in an alternate meaning. This item could be named a Leaf as well, which conveys a less accurate meaning, but possibly avoids this problem.


Is there a logical story behind this phrase? Because when looked at from a naive perspective, giving somebody their own medicine sounds like a kind thing to do as it would only treat their illness. The phrase only makes sense in the context of someone who knowingly sells poison as medicine, which is a rather rare thing (or was it common in the old times?). The most popular origin story I could find about "dose/taste of one's own medicine" is a tale where a cheat sells a medicine that does nothing, then gets sick and people give him his own medicine. And even with this background, it makes little sense, as the medicine doesn't sound like it harms him. Does anybody know the name of this story? Perhaps there's more depth to it.


The smallest possible cardinality of a base is called the weight of the topological space. I was wondering if all minimal bases have the same cardinality, and if every base contains a subset whose cardinality is the weight of the topological space? What aspects are common between a (smallest) base of a topology and a base of a vector space, besides the following similarity (open subset <-> vector, union <-> linear combination): every open subset is the union of some members in the base; every vector is the linear combination of some members in the base. Note that a base in a vector space is also a base in the linear matroid. Not sure if we can have some nice structure like matroid for a topological space to understand its (smallest) bases. Thanks and regards!


My daughter said to me this morning (the context is irrelevant): Er, it's all wet! The interjection I have written here as Er was synonymous with Yuck. Its wetness did not cause great happiness. But what's the right way to write it? The problem with the way I've formulated it above is that it looks like the Er of (genuine or mock) hesitation, synonymous with Um. In fact I think that would be as natural a reading of the sentence as the Yuck reading. This would significantly change the overall understanding, and change the response from disgust to something like confusion. I could try Err, but I don't think that's any less ambiguous. And of course I could write Yuck, which would convey the right sense, but direct quotation ought to respect the original wording. (I have a feeling that this is a British expression, so I'm tentatively tagging as BrE, but I'm uncertain about this. It's supported by noting that the answers to this question don't mention Er at all.)


Let us suppose I am running on a street. When my eyes are open, I can see many things moving backward, and thus it gives me an idea that I am moving wrt those things. Not even this, even if I close my eyes during the run, I can really feel that I am moving. Now assume that I reach a world where there is absolutely nothing except my own body, and my eyes are closed. I am still feeling that I am running. The question is I am running wrt to what? Even if I open my eyes I would feel that I am still running, may be very fast. But visually I would not able to perceive my movement. Does it mean that my movement in space is an illusion? If not, then I am moving w.r.t what? There is absolutely nothing around me!


Or, I guess it could be worded, since when and why was it counted as part of a formal writing style to capitalize many general nouns? (After all, it's not German ...) This is also a trend in legal documents, however, I believe, attributable to the fact that certain terms (e.g. "Company") can have very specific definitions. Here is the second paragraph of the Declaration of Independence. We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.--That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed, --That whenever any Form of Government becomes destructive of these ends, it is the Right of the People to alter or to abolish it, and to institute new Government, laying its foundation on such principles and organizing its powers in such form, as to them shall seem most likely to effect their Safety and Happiness. Prudence, indeed, will dictate that Governments long established should not be changed for light and transient causes; and accordingly all experience hath shewn, that mankind are more disposed to suffer, while evils are sufferable, than to right themselves by abolishing the forms to which they are accustomed. But when a long train of abuses and usurpations, pursuing invariably the same Object evinces a design to reduce them under absolute Despotism, it is their right, it is their duty, to throw off such Government, and to provide new Guards for their future security.--Such has been the patient sufferance of these Colonies; and such is now the necessity which constrains them to alter their former Systems of Government. The history of the present King of Great Britain is a history of repeated injuries and usurpations, all having in direct object the establishment of an absolute Tyranny over these States. To prove this, let Facts be submitted to a candid world.


One formulation of Dilworth's theorem(for finite partially ordered sets) states that : There exists an antichain A, and a partition of the order into a family P of chains, such that the number of chains in the partition equals the cardinality of A. The above is an extract from this wiki page. However, I don't understand how the above statement is equivalent to the following formulation of the theorem. In any finite partially ordered set, the maximum number of elements in any antichain equals the minimum number of chains in any partition of the set into chains. The wiki article does try to explain this. Here is their explanation: Dilworth's theorem states that there exists an antichain A, and a partition of the order into a family P of chains, such that the number of chains in the partition equals the cardinality of A. When this occurs, A must be the largest antichain in the order, for any antichain can have at most one element from each member of P. Similarly, P must be the smallest family of chains into which the order can be partitioned, for any partition into chains must have at least one chain per element of A. I don't get why the statements "...for any antichain can have at most one element from each member of P" and "....for any partition into chains must have at least one chain per element of A" together imply the maximality of the antichain and the minimality of the number of chains. Any help in understanding why these are equivalent formulations will be appreciated. Background: I am mostly a problem solver. I am learning order theory because of its connections to Hall's marriage lemma and because it seems very interesting. I am not following any book, but am trying to learn some order theory online. So I am a total beginner in this area.


I'm looking for the name of the logical fallacy where intent or agency is assumed when in fact there is none. It's a common fallacy in my experience, but I can't seem to find it described specifically on any site that discusses logical fallacies. It seems to be a special case of apophenia, but Wikipedia and Google turn up nothing specific to an "apophenia of intent" or any related phrase. The best I've been able to find is this rather obscure blog post that names it the "agency fallacy," but I haven't been able to find any other sites that use that phrase. Does this fallacy even have a name? Here are a few examples: A child, playing with magnets, finds that they repel when held one way and attract when held another way. Seeing this, the child thinks "oh, those ends want to be together, but the other ends want to be apart." They have (unconsciously) read intent into a deterministic physical process. A picnicker gets rained on and thinks "Darn it, rain! You just couldn't resist ruining a nice day, could you?" Again, they have (unconsciously) attributed agency to a mindless natural process. Two people pass each other in a crowded hall. One trips and stumbles into the other. The other thinks "What a jerk! Why'd they do that to me?" They have mistakenly attributed intent (specifically, malice) to an action that was accidental. This is the case of the fallacy that Hanlon's Razor warns against. A socialist on a soapbox shouts "Capitalism is the source of all the world's evil! It's stealing our jobs and lining the pockets of bureaucrats!" Although the use of metaphor is more deliberate here than with the child or the picnicker, it is still misleading: capitalism can't do anything on its own, only the people who participate in it can.


I was given the following question for comprehension pasted below: What does John mean when he says that he would only win his own money? And I answered that John meant that Marry won't be betting any money, so he will be winning only his own money. Professor marked it incorrect. Can you please what exactly is the problem? I have marked the relevant information in bold letters in the comprehension passage. "Hullo," said John, "it's raining again. What a nuisance I Now we shan't be able to play tennis this afternoon." "No," said Mary, "I suppose we shan't. Let's stay at home and stick some photographs in our book, shall we?" "Do you really want to?" said John rather doubt. fully. "I thought you said you didn't like doing it because it was so messy." "Oh,' replied Mary, "I'll just watch you sticking them in." "Oh, no, you won't," said John. "You've done that to me before. You'll have to do your share, my girl I " "All right. Then what about playing cards? You're keen enough on doing that with your friends at the club." "But you don't know how to play," protested John. "That's all right. You can teach me, can't you? You taught me to drive a car." "Well, it's not much fun teaching someone to play cards. The whole point of playing cards is the chance of winning." "You'd be sure to win if you played with me, because I don't know how to play. Surely that would be better than playing with your friends, because you sometimes lose there." "But if I did win when I was playing against you, I would only win my own money." Mary laughed. Then she said. "All right, I think we'll just have to go to the cinema. There's a good film on at the Grand, isn't there?" "Not the Grand," said John. "The Splendid." "Oh, yes," said Mary. "I keep on getting them mixed up. The Grand's the one down by the river, isn't it?" "Yes," answered John. "After we've been here a few more weeks we'll know everything there is to know about this town: it's small enough. Well, we'd better go, or we'll be too late for the matinee. Have you got your umbrella back from Jane yet, or will you have to go on sharing mine?" "No, I got mine back. I came across Jane in a shop this morning. She was carrying my umbrella, but she put it down for a moment while she looked in her handbag for some money. So I picked the umbrella up and started to walk out. The shopkeeper was horrified. He said, 'Madam, that umbrella belongs to this lady!' 'No, it doesn't,' I answered, 'it belongs to me.' 'That's right,' said Jane. 'It does.' You should have seen the shopkeeper's face! He didn't know whether to believe his eyes or his ears-especially as Jane pretended never to have seen me before: you know how she loves a joke!"


What are some good ways to motivate the material on infinite series that appears at the end of a typical American Calculus II course? My students in this course are generally from biochemistry, computer science, economics, business, and physics (with a few humanities folks taking the course for fun) - not just math majors. I have struggled some in the past to motivate the infinite series material to these students. For one, it doesn't fit with the rest of Calc II, which is on the integral. Over the years I have "converged" on telling them that the main point of the unit is Taylor series and that the rest of the material is there primarily so that we have the tools we need in order to understand Taylor series. Then I illustrate some of the many uses of Taylor series (mainly function approximation, at this level). This approach works better than anything I've come up with thus far with respect to getting my students to care about infinite series, but I feel a little like I'm selling the rest of the material short by subordinating it to Taylor series. Does anyone have other ways of motivating infinite series that they would like to share? (Again, only a small percentage of the students in my class are math majors.) Background: The material in this unit typically consists of sequences, basic series (like geometric and telescoping ones), a slew of tests for convergence (e.g., integral test, ratio test, root test), an introduction to power series, Taylor and Maclaurin series, and maybe binomial series.


I am a pre-engineering student currently taking a Single Variable Calculus course at a community college. I recognize that my future success (or not so much) as an engineer will be based, in large part, on my capabilities with and understanding of Calculus. Therefore, I really, really want to master it like I've never mastered any subject before. I'm doing well in my class, and my instructor is great, but I am under the impression that this course and it's textbook (Calculus, Early Transcendentals by Stewart) do not delve quite as deeply into Calculus as I would like. Also, the textbook frequently introduces new techniques and concepts with little to no explanation. (Incidentally, I'm a self-taught software developer, so I am adept at learning new topics on my own. Learning Mathematics is, IMHO, quite similar to learning a new programming language.) So I'm hoping to find some really excellent Calculus textbooks that will give me deep insight into the topics of differentiation and integration (and any other topics my course may be missing). I've used Google and my school's library to search extensively, and I've found no shortage of Calculus textbooks. My problem is that, since I'm just now learning the basics, I have no way to know just how in-depth an advanced or in-depth book should go, or what important information my current textbook may be missing. I own a copy of The Calculus Lifesaver, by Adrian Banner, which is absolutely outstanding. If anyone reading this happens to be struggling with Calculus, this is the book to turn to. I also have been taking advantage of the Calculus courses in MIT's OpenCourseware. Calculus Revisited, with Herbert Gross, has been very helpful. His way of explaining the concepts just really "clicks" with me. So, with that said, I'm just hoping the experts in the community here can recommend some great resources (e.g. books, free online courses, or other media) to help me optimize my knowledge of Calculus. Thanks in advance!


I am new to the world of stochastic processes and Brownian motion, and am having a hard time finding easily digestible reference material (especially ones with well-worked out examples). So, in a way this question is more of a request for sources/references than a specific answer. To give a little context to where I am approaching this from, it is specifically from the class of statistics referred to as "group sequential testing" (though there are several similar or analagous names, such as "interm analyses", "conditional power calculations", "futility analyses", etc.). I am posting this here rather than on CrossValidated simply because cursory searches reveal that people here are more familiar with the basics of stochastic processes than seems to be the case over on the stats-specific StackExchange, so I feel that I will get more helpful advice/answers here. Anyway, the primary textbook for this field of statistical analyses is Statistical Monitoring of Clinical Trials: A Unified Approach, by Proschan, Lan, and Wittes. The basic mathematical framework for much of the book is Brownian motion; that is, the theory behind many of the group sequential tests outlined in the book is founded on the fact that standardized test statistics evaluated over time can be seen to satisfy the properties of Brownian motion. What I am trying to do is prove whether or not certain classes of non-parametric statistics (specifically the Wilcoxon rank-sum) satisfy these properties, and thus allow one to use this broader framework in that context. So, what I am looking for are sources or references with examples of demonstrating how some arbitrary process can be shown to satisfy the properties of Brownian motion. When I search this site or Google, I can find dozens of examples, but none of them are quite that helpful, since the are generally all based on proving that some function of a Weiner process is itself a Weiner process (e.g. this question or this one). These aren't terribly illustrative to me, since the proofs all rely on the fact that some element of the process under consideration is already given to satisfy the properties of Brownian motion, rather than "starting from scratch", so to speak. Now, I do know the basic steps for proving a process is Brownian motion (this question has them delineated rather nicely). However, I am having a difficult time finding any worked examples of these steps being applied to a process "from scratch" (which is especially critical since I am unfamiliar with a lot of the notations used in stochastic processes, which are often subtly different than those taught to us statisticians). Can anybody point me in the direction of some nice examples or references that would be helpful, here?


There are easy methods for discrete simulations of gas dispersion in two dimensions. If you take a large square lattice, each cell of which is assumed to contain at most one gas molecule, and you move the molecules from cell to adjacent cell at random, the large-scale results are in many ways a good simulation of gas behavior. In particular, even though molecules are individually moving in only the four cardinal directions, at a large scale the simulation does not show any bias to these four directions. If you start with a large square lump of material, it will rapidly diffuse into a circle. This property of the simulation is called isotropy. Long ago when I was involved in research in this area I was told that this worked fine for gas flow simulations, but not for incompressible fluids. If one wanted to make an isotropic two-dimensional simulation of incompressible fluid flow, I was told, one had to use a hexagonal lattice instead of a square lattice. I was told that one could deduce from the Navier-Stokes equations that that any simulation of incompressible fluid on a square lattice would necessarily be anisotropic. Is this correct? If so, how does the proof go? If the argument is complex, is it possible to get an intuitive idea of why compressible and incompressible fluids are different in this regard? What is a reference that would include the full proof? Addendum: Despite the answer I posted below, I still don't understand any of the details. I would be glad to award the bounty to someone who could explain it.


British English often employs mass nouns where American English would only employ count nouns. Count nouns are nouns which take pluralization and numerical quantifiers like 'many'. Mass nouns can't be pluralized and take volume quantifiers like 'a bit of'. For example: I like sport. I like drink. I pay tax. I've got toothache/earache/stomachache/backache. I eat mashed potato/scrambled egg. I play with lego. In American English, we would render these: I like sports. I like to drink / I like to have drinks. I pay taxes. I've got a toothache/earache/stomachache/backache. I eat mashed potatoes/scrambled eggs. I play with legos. These facts are documented here, here and here. But this difference is not mentioned on the Wikipedia page comparing British English to American English. My question is simply what other nouns which are standardly count in AmE are often mass in BrE? Further, are there any broad categories of nouns (for example, food nouns) that tend to be count in AmE but tend to be mass in BrE? Or does the difference only exist for particular and isolated lexical items? (The first link I provided says that many foods are not conventionally massified in BrE, for example 'refried bean'). Here are two final caveats: First, I know that any count noun can be turned into a mass noun by simply putting it in the syntactic position of a mass noun. That's why "I ate (a bit of) chicken" is acceptable in both AmE and BrE. But I am asking about the obvious differences between AmE and BrE regards their conventional uses of nouns as mass and count. Second, I know these types of list questions are hard for single users to answer, and sometimes generate controversy on EL&U (for example, this one on kennings generated some controvery). Still, I'll upvote any responses that make a novel contribution.


An elementary fact that people learn about mirrors is the law of reflection, that the angle of incidence of a light beam striking the mirror (as measured with respect to a normal) equals the angle of reflection. Does this law also hold for a mirror that is moving? Consider a square mirror that is moving at speed v in a direction perpendicular to the mirror. (You can think of the mirror as starting in the xy-plane and moving in the positive z direction of a Cartesian coordinate system.) As the mirror approaches a certain observation point, another person shines a laser beam of frequency w at the mirror so that the beam makes an angle A with the normal to the mirror. (You can think of the beam as lying in the yz-plane.) What angle and frequency will you measure for the reflected light beam? Does the law of reflection still hold? Do your conclusions change if the mirror moves parallel, rather than perpendicular, to its plane (say in the y direction if it starts in the xy-plane)? Note: The large mirror of the Hubble space telescope is an example of a mirror in motion as it orbits the earth. From your analysis, do you think the users of the Hubble have to take into account the motion of the mirror when measuring properties of its images? I know that this question could be according to the .SE standards be too broad. But I request you not to close this question, I would really appreciate people if this question is answered rather than flagged or closed.


"You do research in mathematics! Can you explain your research to me?" If you're a research mathematician, and you have any contact with people outside of the mathematics community, I'm sure you've been asked this question many times. For years now, I've struggled to find a satisfying answer. I think an ideal answer to this question should: be accessible to someone who hasn't studied math since high school build intrigue and wonder honestly, albeit vaguely reflect your research only require a few sentences (Of course, these guidelines will change depending on the audience and venue. For example, speaking with an engineer over a meal allows more time and technical language than would speaking with a stranger on a bus.) I study the representation theory of algebraic groups and Lie algebras over fields of positive characteristic, so I usually say something along the following lines: I work with two algebraic objects that are closely related called algebraic groups and Lie algebras. These objects can act on spaces (like three-dimensional space) by transforming them in a nice way, and I study these actions. One aspect of my work that is especially challenging is that I use number systems in which a chosen prime number is equal to zero. Honestly, based on my guidelines above, I think this response is poor, but with so much to communicate in such limited terms with such limited time, the task seems nearly impossible. Using my guidelines, how would you describe your own field of research? Or, if my guidelines are too strict, how would you deal with this question?


So I am reading the essay "On Some Verses of Virgil" by Michel de Montaigne (translation by Donald M. Frame) and I came across this particular sentence. Is there any ugliness in doing wrong that can dispense us from the duty of confessing it? The choice of vocabulary here is, honestly, quite confusing to me. Here is the sentence again with the preceding paragraph for context: The diseases of the body become clearer as they increase. We find that what we were calling a cold or a sprain is the gout. The diseases of the soul grow more obscure as they grow stronger; the sickest man is least sensible of them. That is why they must be handled often in the light of day, with a pitiless hand, be opened up and torn from the hollow of our breast. As in the matter of good deeds, so in the matter of evil deeds, mere confession is sometimes reparation. Is there any ugliness in doing wrong that can dispense us from the duty of confessing it? It's been a long time sense I've had this much difficulty understanding the exact conclusion a sentence was trying to reach. My first instinct is to assume that the sentence is somewhat synonymous with: Is there any ugliness in doing wrong that can [spare] us from the [ordeal] of confessing it? But then a friend of mine offered an interpretation more to the effect of: Is there any [wrongdoing so exceptional] that [it will] [render our confession truly unprecedented in the context of humanity]? Meanwhile, another translation (found on Project Gutenberg) renders the sentence thus: Is there any deformity in doing amiss, that can excuse us from confessing ourselves? I don't have much confidence in either his or my interpretation, and this other translation isn't clearing things up for me much at all. What is Michel de Montaigne saying?


I am writing a formal document. It is a petition to appeal an administrative law decision. The decision is jam-packed full of sloppy mistakes of every kind imaginable. (The guy who wrote it couldn't even manage to spell the name of my town correctly in the title of the document.) I list all the errors and inaccuracies in my appeal petition, with specific references to pages in the hearing transcript and exhibits. But I want to preface the list with something that means the following: A number of the problems noted might seem rather insignificant, but I will list them all, to show the review officer how incredibly sloppy the decision is. OR: what a slapdash job the hearing officer did in analyzing the case and writing up the decision. Only I can't say "incredibly sloppy", I need more formal language. I need a more subtle way of saying sloppy. I don't mind making adjustments in my sample sentence to accommodate a different part of speech or whatever. Note, the following question is somewhat related but doesn't. A word for not paying attention to detail, causing sloppiness. One idea: A number of the problems noted might seem rather insignificant, but I will list them all, to show the review officer the shocking lack of care taken in analyzing the case and documenting the decision. Another idea: The hearing officer's slapdash [or: haphazard] analysis and write-up resulted in erroneous references and errors of fact that are too numerous to list in the space allowed. I will only be able to provide corrections for a portion.


I have used classical Lagrangian mechanics for quite a while, and what I like about it is that everything can be derived from a very small number of geometric principles. There are just three things you need to "take on faith": That configuration space should be endowed with a Riemannian metric given by mass; That forces arise from scalar potentials on configuration space; Hamilton's principle of extremal action. The last is perhaps most objectionable: why should the universe want to evolve in a way that minimizes the action? I have no idea, and my understanding is that nobody else does either. But if I assume these hypotheses, everything else follows mechanically: equations of motion, Noether's theorem, etc. What I would like to do is to learn quantum mechanics, starting from similar geometric principles. For example, I might take as a key principle that configurations should be probability distributions over configuration space, rather than single points, but this doesn't lead to quantum mechanics; I also need (for some reason?) to instead take configurations to be complex-valued functions over configuration space. But even then I don't see how to get anything like Schroedinger's equation from a (modification of) Hamilton's principle. I did some searching and I've found that there is indeed some variational foundation for quantum mechanics (and Feynman seemed to have championed this formulation) but the articles I've seen so far assume I already understand quantum mechanics. My question: Is there a good reference that builds quantum mechanics from scratch, based on variational and geometric principles? If it's easier to instead start with a more general theory (e.g. quantum field theory) let me know as well.


For instance, "A politician must be able to think quickly on the spot. He or she must also have no qualms about lying." I know some people who use "they", but as that both sounds and is ungrammatical, I'm wondering if there is any other concise and non-awkward alternative. Edit: Apparently many do consider "they" as correct. This surprised me, because I know someone who actually wrote a grammar book, who told me in no uncertain terms that "they" could not be used (in fact, I'd asked them - ha! - this question before I posted it here). And a quick look online found legitimate sources advocating both positions. I suppose it's one of those issues, such as whether the previous sentence was incorrect for beginning with a conjunction, where it really just depends who you ask. Regardless, to me personally, "they" in the above example sounds clunky at best and incorrect at worst. So I suppose my question can be rephrased as, 'is there any alternative to 'he or she', other than 'they'?" Secondly, to whoever tagged this as "politically-correct": acknowledging the existence of genderqueer or transgender people isn't a political issue at all, until those made uncomfortable by people unlike them make it one. Trying to avoid excluding and thus unconsciously discriminating against a group of people in your writing isn't born out of a need to be 'politically correct' - it's born out of basic human decency. Finally, yes, in the example above, I could use a comma and avoid the need for a pronoun at all - but the question I'm asking applies to instances where one may not want to do so (i.e. the sentences are long and stringy enough as it is).


I'm writing some copy for a marketing campaign that promotes the use of a software testing tool by demonstrating how the results of the tool provide you with a greater wealth of information on which to make business decisions. In relation to this, I'm using the term "informed decisions" and my initial wording included the phrase "help you make more informed decisions". The phrase "more informed decisions" is one that is common to me and from a Google search appears to be common in the context in which I am using it, that is to denote that with additional evidence a decision can be more informed than without. I considered whether this phrase runs the risk of being misinterpreted or being confusing due to a slight ambiguity. The phrase "more informed decisions" could be interpreted as meaning decisions that are more greatly informed, or it could be interpreted as a greater number of informed decisions (e.g. here is one informed decision, here are more informed decisions). I then considered the phrase "better informed decisions". I feel this has the same intended meaning but removes the above ambiguity. It could refer to decisions that are better informed or it could refer to informed decisions that are better. Both are on subject and would be fine to me. A quick Google search suggested that the phrase "more informed decisions" is significantly more commonly used than "better informed decisions". I have a feeling that "better informed" is more grammatically-correct than "more informed" although I have no formal basis for this. Out of the phases "more informed decisions" and "better informed decisions", is there a more correct form?


I was reading this interpretation from this site, where these lines are noteworthy enough to talk for the fact that this interpretation doesn't actually talk about many-worlds: These are the "many worlds" in question, although it should be clear that the label is somewhat misleading. People sometimes raise the objection to the many-worlds interpretation that it's simply too extravagant to be taken seriously--all those different "parallel realities," infinite in number, just so that we don't have to believe in wave function collapse. That's silly. Before we made an observation, the universe was described by a single wave function, which assigned a particular amplitude to every possible observational outcome; after the observation, the universe is described by a single wave function, which assigns a particular amplitude to every possible observational outcome. Before and after, the wave function of the universe is just a particular point in the space of states describing the universe, and that space of states didn't get any bigger or smaller. No new "worlds" have really been created; the wave function still contains the same amount of information (after all, in this interpretation its evolution is reversible). It has simply evolved in such a way that there are now a greater number of distinct subsets of the wave function describing individual conscious beings such as ourselves. The many-worlds interpretation of quantum mechanics may or may not be right; but to object to it on the grounds that "Gee, that's a lot of worlds," is wrong-headed. I'm not really understanding the reason "It has simply evolved in such a way that there are now a greater number of distinct subsets of the wave function describing individual conscious beings such as ourselves." What is it saying? Can anyone explain me?


Firstly I'm not saying that I don't believe in Cantor's diagonalization arguments, I know that there is a deficiency in my knowledge so I'm asking this question to patch those gaps in my understanding. From my understanding of Cantor's Diagonalization argument, if you apply diagonalization to a mapping from one set of numbers to another, you will always obtain a number that is not in the mapping. So this works to prove that the reals aren't countable because if you have a mapping from the naturals to the reals then you can use diagonalization to obtain a number that's not in the mapping, and this number is a real obviously, so the mapping isn't a surjection. We're not allowed to assume that the mapping from the naturals to the reals is a bijection to begin with. But when people explain why the diagonalization process doesn't produce a rational from a mapping from naturals to rationals we are allowed to assume that the mapping is a bijection to begin with? In the questions asked here: Why does Cantor's diagonal argument not work for rational numbers? The answers says: To be precise, the procedure does not let you guarantee that the number you obtain has a periodic decimal expansion (that is, that it is a rational number), and so you are unable to show that the "diagonal number" is a rational that was not in the original list. In fact, if your original list is given explicitly by some bijection, then one is able to show just as explicitly that the number you obtain is not a rational. Why are we allowed to assume that the original list is a bijection? Is there some way to prove that the mapping from the naturals to the rationals is a bijection that is not susceptible to diagonalization? If we can assume that the mapping from naturals to rationals is an undiagonalizable bijection why can't we do the same for the mapping from naturals to reals?


According to path integral theory, it can be assumed that a particle travelling from a point A to B (in an experiment that has not been designed to detect the path) takes all the possible paths from A to B. (Please note that I use the words 'can be assumed' because I know that there is a debate whether the particle really takes all the paths that are considered in path integral or not.) Now, energy of a particle is closely related to its momentum in quantum mechanics. So, in a double slit experiment, we can setup the detector at the interference screen so that it not only detects the position of the individual particles, but also their energy. This will not violate uncertainty principle because we will still not have the knowledge of the direction of the velocity of the particle, but only the magnitude of energy. Now, being able to detect the energy of particles in a double slit experiment will be interesting because then we will be able to find out whether the energy of the detected particle is equal to that of the input particle or not. If it is equal, then it would mean that the energy is conserved during the path. Yet, according to the path integral, the particle can be assumed to take all the possible paths. Since we know the initial and final energy of the particle, this means that we can only consider those paths in path integral where the energy fluctuates only in the middle of the path! So, is it theoretically possible to detect the energy of the individual particles at the detectors in the double slit experiment. If yes, then theoretically, is the energy conserved at the start and the end of the path. And most importantly, do you know of any experiment that has been done to detect the energy of the individual particles at the interference screen. Please give me the links to the paper discussing this.


I was looking up how to find relationships between Sobolev spaces and I came across this post on MO in which the first comment talks about a scaling procedure for understanding the relationships: To find the right values of k,p,r,a, I was taught to use scaling arguments. Take a nice enough singularity for the function inside the domain, and make it worse and worse, and compare how the norms change. This usually lets you solve for the appropriate parameter (and if it doesn't it tells you there is something special about that embedding). For unbounded domains you can do the same thing, and this often shows why certain embeddings cannot exist (handling singularities both inside and at infinity is hard). Other answers also mentioned this scaling approach, which I have never heard of, with one comment saying: Comparing the behavior of the different Sobolev norms for a compactly supported family of functions converging to a Dirac delta function is indeed the simplest way to figure out these inclusions. In general, understanding how things behave under rescalings is extremely useful but, as far as I know, rarely mentioned in print. It seems that you have to learn about it by word of mouth or stumble onto it yourself. Physicists and chemists also use this often and call it "unit analysis". So does anyone here know of this scaling approach? If so could you give a worked example for some specific function, so somebody (like myself) who is new to Sobolev spaces can learn how to apply it and gain a better understanding of these spaces?


In all textbooks the magnetic field around a wire carrying a current is found from Ampere's Law. However, I would like to know what the expression for the magnetic field is, using the full time-dependent Maxwell's equations (and how to get there): Hopefully this will give me some insight into how the field is set up via EM waves. Note that I am specifically interested in how the field according to Ampere's Law is actually set up in a time-dependent fashion via EM waves. For simplicity let's say I'm interested in the magnetic field created by a wire attached to the electrodes of a battery. When you turn the battery on, the magnetic field around the wire is rapidly created. How can I get a time-dependent solution for this? Feel free to suggest how the battery could be treated, how the electrons in the wire respond etc. (I suppose all of these things must be part of the model). Also feel free to treat a simplified problem, eg. we could do the equivalent problem for the magnetic field around an infinite slab of material if that makes the geometry easier. My primary interest is in how the field is propagated into the vacuum around the wire - I would like to see how the solution eventually tends toward Ampere's Law. EDIT: I will try to make the question easier to understand: We are often taught the magnetic field around a wire arises from Ampere's Law. This applies to a steady state situation only - i.e. it does not occur instantaneously. It does not tell you how the signal (whether that be voltage or current) originating in the wire actually gets out to the vacuum around the wire, as the current starts at zero and ramps up to some constant value. Presumably this happens via the generation of EM waves, because of course Ampere's Law is not valid on very short timescales. I would like to know what the solution is for the magnetic field around a wire when you are not in the steady state, which should reveal how the magnetic field propagates into the vacuum. Obviously the solution would have to correspond to Ampere's Law in the long time limit. I do not mind so much what device you use to get the current flowing in the wire in the first place. For example if you want to use a battery connected by a switch to the wire then fine. Or an AC generator slowly ramped up with some assumed time-profile would also be fine. In other words, if you would like to answer this question, you could choose whatever model you think best describes the scenario in which a wire is connected to a battery by a switch and the switch is turned on (or, as I mentioned some other, similar process). The wire can be any shape you choose. I just choose a "wire" because it's the most commonly used simple conductor, but if you would prefer to use for example a slab of conductor, that is fine, because my interest is in how the time-dependent fields converge to Ampere's Law.