I am a computer science student that is struggling with a problem of mathematical nature. Thus far I have only studied calculus, discrete mathematics and linear algebra, but cannot figure out how to approach this problem. I tried Stack Exchange, but due to the mathematical nature they suggested me to ask here, so here goes: I am trying to create shopping lists from a collection of products, where the returned shopping list should be optimized for cost as well as to meet another condition. For example, let's say that I want to create shopping lists based on the energy content of the products. When the user enters a total sum, the returned shopping list should try to max out the kcal content while keeping the total sum at or around the sum specified by the user. I've gotten so far as to create the collection of products, and all products are stored as objects with fields holding nutritional values and price etc. The kcal-value is also stored as a member variable in each product's object. At first I considered looping through all combinations of products, sort out those that are way out of the price interval, and then return the one with the highest kcal content. But as the numbers of products available increases this soon becomes a non-viable option I think. I now wonder if there is any algorithm to solve this problem, if not, is there any way to easily implement this? I've understood that this is a problem of linear type, (discrete?, diophantine?), but that's about all.


As far as I understand it, quantum mechanics requires that a particle's position to be not specifically determined in space, but rather be 'spread' out through space, in the sense that we can only know the probability a particle is at a particular location. This can be visualised through the wavefunction. When we then try to measure the particle's position, (say by firing a high energy photon at it), the particle will turn out to be at some particular location, which corresponds to the wave function collapsing (I'm not too sure if this is the right use of the term). The particle could however be found in a large range of positions. Consider an air particle, which has an initial wavefunction (in black), which we then fire a photon at to determine the position of. The air particle could then be found at position A or found at position B, with roughly equal probability. The two circumstances however cause a slight disturbance, which 'propagates' through space. What I mean by this is this air particle's position and momentum will affect the air particles near it, which will affect the air particles near those, and so on and so forth. Through chaos theory, a small change in initial conditions will result in a very different outcome, so this single misplaced air molecule has the potential to change everything about Earth. In the above diagram, air particles are depicted by dashes, with their velocities depicted by the length of the dash. As can be seen, the two situations A and B lead to a 'propagation of disturbance' (the area in which the air particles are different between situation A and situation B) which is depicted by the black circle. What I'm interested in is how quickly the circle grown in size. At first, I thought that it should propagate at the speed of light. Imagine that our air particle is situated at the North Pole. An air particle situated at the south pole will have a wavefunction that is VERY NEARLY zero at the north pole, but it will still be finite(I think). For this reason, a small disturbance of the air particle at the north pole will result in a disturbance at the South Pole directly, at the speed of light. Another voice in my head however said that this was rather silly. The disturbance should propagate rather slowly, and should only propagate through the collision of air particles. Through the atmosphere it would travel at roughly the average speed of air particles in the atmosphere, and through the solid ground it would travel very slowly. Which of the two, if either, is correct?


I am by no means an expert in the realm of physics. I do from time to time, try to understand the concepts of modern physics and their applications. I came across this video that I am currently watching, and in the beginning it explains what would happen to someone if they crossed over the event horizon of a black hole. From my understanding, the time dilation is so great near a black hole, that if you were to cross over the event horizon someone viewing you from further away would never actually see you cross and you would appear frozen forever at the edge of the event horizon due to this time dilation near great mass. If this is true, in the sense that the entire existence of the universe outside the black hole would unfold while you were stuck at the edge of the event horizon due to time dilation (Please fact check me here, I would love to know if time does actually stop for you if you cross the event horizon of a black hole), wouldn't it be true that you would actually spend no time at all inside the black hole due to hawking radiation and it's effect on the evaporation of black holes? What I mean by this is that, apparently from what I've read, a black hole will evaporate and cease to exist at the end of some finite period of time in the universe due to it giving off hawking radiation. If this is true, and if the fact that you appear to be forever frozen at the event horizon due to time dilation is also true, wouldn't it seem from the perspective of the person crossing the event horizon that the black hole ceases to exist? I am very intrigued by this possible paradox that I was questioning and I would love to know more about it's ramifications, if there are any, from many of you whom are more advanced that I. Thank you in advance.


What determines whether something is a "noun adjunct" or just a garden-variety adjective? Does it matter in any meaningful way? Here is my hypothesis, but I can't find any authoritative source to back it up. I'm hoping someone here can weigh in more definitively. Classification of noun adjuncts is based subjectively on whether the word is in "common usage" as a noun. "Book" and "chicken" are commonly nouns and would be considered noun adjuncts in "book collector" and "chicken soup"; "yellow", not so much. Noun adjuncts may share some common qualities that differentiate them from other adjectives (like not being able to be inflected into superlative forms... one can't be a "book-est collector"). But they are still fundamentally adjectives and may even appear in the dictionary as such if the adjectival usage is common enough. What I've found so far... Wikipedia defines a "noun adjunct" as: an optional noun that modifies another noun; it is a noun functioning as an adjective. But English words are not decreed to be nouns or adjectives by some higher authority. So it seems strange that one can prescriptively conclude that something "is a noun" in the first place, let alone extrapolate that it "is a noun functioning as an adjective". The wiki article cites "chicken soup" as an example of a noun adjunct, but at least one dictionary gives a definition for "chicken" as: adj. (of food) containing, made from, or having the flavor of chicken So it seems that there are some differing points of view on how to categorize these words. This question was spawned from some discussion in this question, this question and this other question.


The past few months I have been studying astronomy and Integral Field Spectroscopy (IFS). What I want to do is to fit a galaxy kinematic model to data (ie: estimate the model parameters that give the best fit result). At the moment I extract the velocity and velocity dispersion maps from an IFS datacube but I am not sure how to deal with the Point Spread Function (PSF). What is more correct: Deconvolve the data with the PSF and then fit the model to the deconvolved data? Or convolve the model with the PSF and then fit the PSF-convolved model to the data? The first approach sounds computationally faster to me because only one deconvolution is involved, but at the same time it won't give the best result because deconvolution is ill-posed even if the PSF is known. Is that right? The second solution sounds computationally slower because I will have to convolve the PSF with the model for every single model evaluation, but it will give better results because the convolution result/solution is well defined. Is that right? The data sources I use for my experiments are products of some data reduction pipeline. Why the deconvolution of the PSF is not part of the data-reduction step? Is it because of what I mentioned above? ie: The deconvolution is an ill-posed procedure and it may affect (in a bad way) the data. I am not very familiar with the deconvolution procedure but so far I have found that the Richardson-Lucy technique is a method for deconvolving with a known PSF. Are there other better techniques that are proven to give better results?


Before getting into the question, here are some remarks: Given a single point charge, the value of electric field at the position of the charge is singular/undefined, which makes sense, since a particle cannot interact with itself. Given a discrete charge distribution, the value of the field at an empty point(i.e. no particles reside at that point), is the field contribution from all the charges. However, if the point of interest contains a point-charge, then the value of the field is the field contribution from all the charges except the charge that resides at that point. Now here's my inquiry: In the case of discrete charge distribution, it makes sense to speak of the value of the electric field at a given point, whether that point contains a point-charge or not. What about continuous distributions? My intuition says yes, we can; since one can think of continuous distributions as an extension of discrete ones, with the difference being that the former contains uncountably infinite charges. Therefore, to calculate the field value at any given point on the continuous charge distribution, one has to consider the field contributions from all the charges except the one residing at that point. Example: The value of the electric field at any point on an infinite sheet (plane) of charges should be zero. Since at any given point on the sheet, one can think of that point as being surrounded by infinite concentric rings, where the field contribution from each ring (by symmetry) is zero. So does it make sense to speak of the value of electric field on continuous charge distribution? Or is it not defined?


A rather simple question for liquids specialists I guess but I have hard time finding information about this. Here is my problem. I understand the ideal gas theory and the Maxwell's speed distribution. I see an ideal gas as small balls (mostly surrounded by void) moving around very fast and colliding elasticly with each others. If you want to be more precise, you use an interatomic potential such as the Lennard Jones potential that takes into account Van der Waals attractive interactions as well as the repulsive ones. You can define a kinetic (positive) pressure, kinetic temperature and molecular (negative) pressure with such a simple model. I think I understand that pretty well for now. On the other side, I think I understand cristals fine as well. I see them as atoms bounded together by springs in which waves can flow and each atom oscillates around an minimal potential energy position. I have seen how you can calculate cristal's thermal capacity using Debye's model. So for now I think I have an idea of how a solid behaves at the molecular scale. But what about liquids? I have read very interesting posts here about molecules velocity in liquids and I would be glad to have a more general view of what a liquid is from a molecular perspective. As I understood it, molecules in liquids also oscillate around a minimal potential energy position but they can also swap positions with each other. Is that correct? Are there any tabulated values of molecules swapping speed in liquids ? Concerning pressure. Should I represent pressure in liquids as a sum of a (positive) kinetic pressure due to molecules collision and a (negative) molecular pressure due to attractive interactions between molecules? Is this a good way of representing myself a liquid at a molecular scale? Is there a model explaining the relation between viscosity and molecules attractive interactions ? -----------EDIT--------- I got the answer about swapping molecules. Now this brings me to my question about pressure in liquids from a molecular perspective. Concerning ideal gases, pressure is due to molecules collisions. Does this still stand for liquids or is it more a question of "weight" exerted by molecules on each others? Does any one know a molecular pressure model for liquids? Thank you


We can speak of "microbes" or "micro-organisms," and I used to think that these terms clearly included viruses. And they are used this way by at least some other people; here's a website that refers to viruses as a category of microbe. However, I recently discovered that these terms are usually defined as referring to microscopic "life," and the definition of "life" with regards to viruses is a contentious topic. When dealing with disease-causing agents, we can use the word "pathogen." However, not all bacteria and viruses are pathogenic. All viruses must infect living cells to reproduce, so it is appropriate to refer to all viruses as "infectious agents" (this is what the Wikipedia article on viruses uses in its introduction). However, not all bacteria are necessarily infectious. The informal terms "bug" and "germ" do exist. While these don't seem strictly limited to pathogens, that seems to be their most typical use. But for a technical audience, does any term exist for which there is a consensus that it refers to both all bacteria and all viruses? If no single word exists, a short two-word phrase along the lines of "biological entity" would also be OK. (This phrase also comes from the "virus" Wikipedia article, and seems to be the best fit I've found so far. I can also think of a few others along these lines, like "microbial entity" which could be seen as a shorter equivalent to "microscopic biological entity.") Here are the most important criteria I'll consider when deciding whether to accept an answer: technical correctness: The term must be acceptable regardless of whether one considers viruses to be living or non-living, organisms or not organisms. "Micro-organism" does not meet this criterion because some people do not consider viruses to be organisms. positive scope: It must include all viruses and all bacteria. "Pathogen" does not meet this criterion because not all viruses or bacteria are pathogenic. negative scope: It should not include inorganic objects, whether microscopic or macroscopic. E.g. rock particles. It's OK if it includes protists or multi-cellular organisms, or prions and other "pro-life," or some combination of any of these. Other important criteria for me: established terminology: I'd prefer a term that is already in use to a neologism. If a neologism seems to be necessary, I'd like it to be linguistically well-formed and etymologically transparent. length: all else equal, I'd prefer a shorter term. grammatical number: I'd prefer a term that can be used in the singular to refer to a single species of viruses or bacteria, as opposed to an always-plural word or a singular mass noun.


I recently had an argument with a friend around the question "have you ever thought about something?" The question was asked in the context of exploring some life possibilities, such as buying a sports car or moving to a different country. The disagreement was around whether an affirmative answer to the question bears the hidden meaning that the something being considered is something that the person answering actively wants. To give an example: when asked "have you ever thought about moving to Sweden?", if a person answers in the affirmative, which of the following two meanings best describes their answer? They consider moving to Sweden a practical possibility, and they actually want to do so in the future. The thought has crossed their mind, but nothing can be inferred about whether they want or plan to move to Sweden in the future. Assume the subject is not currently living in Sweden :) Question update: Some of you answered that the meaning depends on other factors, such as tone of voice, body language, context, etc. As I said in a comment, to the purposes of this question, ignore such secondary conversational artifacts. They can always extend the range of meaning of any sentence or word, from the "proper" sense, to the complete opposite, such as when being sarcastic (e.g. "Would you like to go to Sweden?" "Yeah, right...") Consequently, the disclaimer "it depends on the tone of voice" can probably be applied to most answers on this site. To put it another way: assume you read the text, with minimal context. What's the meaning then?


Ok so I'm a programmer, I'm not a mathematician, I've got a minor in math but I didn't even do particularly well at it so please bear with some possibly really stupid thoughts. Just please try to explain to me why what I'm asking is stupid so that I don't keep making a fool of myself. I recently found myself thinking along the following lines: If you take the stand that it's solipsism to talk about anything that cannot be put to use in predicting an outcome then cannot you apply the same axiom to basic mathematical operations? For example does addition really have any meaning outside of the context of placing two objects in a box and predicting how many are inside? Yeah, I realize that's a dangerous question to ask a board full of mathematicians but I imagine you can ask this question about any axiom. The problem is that in reality if you put one thing and then another into a box it does not mean that there will be two things in the box. There will likely be two things of course but if the objects are point particles like electrons there's a chance there will be one or three, or a million electrons there. Heck, the same is even possible with apples, just incredibly unlikely. I don't have the mathematical wherewithal to think through what this would mean but intuitively it would seem that this might not have many implications for addition, but might for subtraction (unless you could somehow have negative amounts of particles which I won't rule out), and certainly for things like integration and derivation. I would assume that someone's done work along these lines before? Has anyone actually created a system around it? Was it useful? Are there any accessible books or articles about it? Just interested


This question was actually asked by Alan Munn in a comment to How do I create a LCM tree diagram?. I repeat (and self-answer) it here because the answer is too long to fit the margin :-) qtree is a well-known and heavily used package for drawing trees using the so-called bracket notation. (The bracket notation is especially familiar to linguists.) forest is my own, recent package for the same job. Due to the awesome power of pgf/tikz (in particular, the pgfkeys utility), which it is based on, forest is an extremely flexible package. ---From the feedback I got so far, including feedback from this site, I don't seem to be the only one who believes so. Which makes me happy. :-) Although both qtree and forest encode trees using the bracket notation, the exact syntax is somewhat different. While forest requires that each node (including its children) be enclosed in square brackets (like this: [node [child node] ... [child node]]), qtree relaxes this requirement in the case of terminal nodes (leaves): they can be separated simply by whitespace, like this: [.node leaf ... leaf ]. Furthermore, the packages differ in the encoding of node labels: as showh above, in qtree a bracketed (usually non-terminal) node's label must be preceded by a dot (.). (forest uses the same syntax as synttree; another package that uses (and extends) qtree's syntax is (obviously) tikz-qtree. As I mentioned in a comment in the above-mentioned question, I have decided for synttree-like syntax purely out of personal taste. I guess I found it more consistent.) As Alan pointed out, the difference in the syntax makes the potential transition from qtree to forest harder: nobody wants to throw away tons of trees (s)he has painstakingly drawn. Thus Alan's question: would it be possible for forest to support both syntactic dialects?


My good friend is from Pittsburgh and frequently uses the word whenever to mean the word when. I am aware this is a regional dialect and really wish to respect that, but it is causing numerous problems in our spoken communication. (I am also a native English speaker but am not accustomed to this usage of "when" and "whenever".) I have expressed my concerns to him and asked him to use a neutral dialect to improve communication, but he argues it is valid English, he doesn't understand the difference between the two anyway, he "doesn't have problems with anyone else understanding [him]", and that my misunderstandings are because I "have Asperger's and understand [his] speech literally". (I would guess if no one else has an issue with his speech, it's because he speaks English primarily with people who have the same regional dialect and non-native English-speakers and uses French and German for work. That said, perhaps everyone else does understand what he means without any confusion. When I ask for clarification, he gets irritated.) What should I do? Examples of such misunderstandings are below: Example: Whenever my aunt was about to die, she called me into the room and told me she loved me. I understood this as his aunt periodically became ill to the point where she was close to dying and called him into the room to say she loved him. (My background in healthcare makes this seem like a very plausible situation.) I responded to him in a way that reflected my understanding of the habitual nature of this. He was annoyed and said it was obvious that the aunt was about to die one time and that, as such, this calling-into-the-room was a one-time occurrence. Example: Whenever my sister was born, my dad fainted. It is obvious to me that his sister was born one time. In this instance, although I believe the better word choice is when, I can understand that his father fainted when his sister was born. Example: Whenever I moved to Germany, I lived in Berlin. I knew he had moved to Germany once for a (temporary, location-based) job. However, his statement surprised me, and I thought maybe I was wrong (and as a friend I wanted to learn more if he had actually moved numerous times), so I asked how many times he'd moved to/lived in Germany. He was equally surprised by my question, responded he'd moved to Germany once, and could not understand how there could be any confusion in the statement.


I have an object with incident light rays traveling away from this object. Some of these rays are traveling from the left-hand side through a simple lens (say, a double-convex converging lens). As these rays enter the lens, they are partially refracted, reflected, and absorbed. As these rays leave the lens, they are refracted even further, and eventually the light rays converge to a point (on a film screen or something). Since the light rays are partially reflected and absorbed, wouldn't the light rays that entered the lens at the edges (where the lens is thinner) be less reflected/absorbed than the light rays that entered the lens near the center (where the lens is thicker)? If the light rays converged onto a film screen, would the differences in the intensity of light caused by these reflections/absorption (which can be traced back to differing thicknesses of the lens at different points) cause any issues with the images? Would this ever need to be taken into an account by a scientist or student conducting an experiment involving data obtained from a camera? Also, for a given convex lens (made of a particular material, and with a specific curvature), can the intensity of light as it reaches the focus point be thought of as a function of "vertical" distance from the center of the lens? This thought comes from the fact that this vertical distance would determine how much lens-material the light travels through. Are there examples of such functions, or is there a way of coming up with one for a simple lens?


Recently I was asked to explain the difference between reflection and total internal reflection from a purely conceptual standpoint (no math). Let me explain what I already know. Reflection and refraction at the quantum level are the same thing. Light is a photon. A photon is a discrete particle that has wave characteristics (a finite wave traveling like a bullet). As the photon travels it collides with electrons in the matter of the medium it is traveling in. Depending on the energy of the photon and the allowed energy bands of the medium the photon cause the electron to jump up a level. If the photon is absorbed then the medium will increase its motion (at the macro scale increasing its temperature). If the photon is not absorbed it will be re-emitted (really as a new photon). I have read and watched Feynman's QED lectures and book and have a pretty good understanding of his process for determining how all these paths come together to give the net path of the photon. The general rule of thumb is that the photon wants to take the path which requires the least amount of time. I understand how this principle goes to explain refraction and reflection. What I don't seem to understand is why does one material seem to cause a higher percentage of refraction compared to another (metal vs. glass). What about the electron configuration of a the medium changes the net effect of the absorption and re-emissions of the photons? Is there a change in the probability of the photon being re-emitted in a reverse direction? Is there a farther distance the photon can travel before being incident onto an electron? This is the part where my understand breaks down. When you have hit the critical angle in a medium that refracts and the light completely reflects, are the photons moving is the same manner as they would be in a material that always reflects? How does this connect to the question in the previous paragraph? I know I have a bunch of mini questions embedded in answering this one larger question. Any help on any of the parts would be greatly appreciated?


A few months ago I was telling high school students about Fermat's principle. You can use it to show that light reflects off a surface at equal angles. To set it up, you put in boundary conditions, like "the light starts at A and ends at B". But these conditions by themselves are insufficient to determine what the path is, because there's an extra irrelevant stationary time path, which is the light going directly from A to B without ever bouncing off the surface. We get rid of this by adding in another boundary condition, i.e. that we only care about paths that actually do bounce. Then the solution is unique. Of course the second I finished saying this one of the students asked "what if you're inside an elliptical mirror, and A and B are the two foci?" In this case, you can impose the condition "we only care about paths that hit the mirror", but this doesn't nail down the path at all because any path that consists of a straight line from A to the mirror, followed by a straight line to B, will take equal time! So in this case the principle tells us nothing at all. The fact that we can get no information whatsoever from an action principle feels disturbing. I thought the standard model was based on one of those! My questions are Is this anything more than a mathematical curiosity? Does this come up as a problem/obstacle in higher physics? Is there a nicer, mathematically natural way to state the "only count bouncing paths" condition? Also, is there a "nice" condition that specifies a path in the ellipse case? What should I have told that student?


I have a long-term goal of acquiring graduate-level knowledge in Analysis, Algebra and Geometry/Topology. Once that is achieved, I am interested in applying this knowledge to both pure and applied mathematics. In particular, I am interested in various aspects of smooth manifolds, co/homology and mathematical physics. I have acquired a smattering of knowledge in all of these areas but feel that I need to become more focused to make make coherent progress. I have a very bad habit of picking up a book, reading a bit, working out a few details, and then moving on to other random topics in other random books. In doing this, I don't really feel like I accomplish much. To rectify this admittedly undisciplined approach, I have decided to select core source material from each of the three major areas listed above and focus on it until I have assimilated all the information in that material. For analysis, I have selected Amann and Eschers' Analysis, volumes I, II, and III. I made this choice because out of the analysis texts I have surveyed, theirs seems to be the most comprehensive and treats elementary and advanced analysis as a unified discipline. My basic strategy is to treat each theorem, example, etc. as a problem and give a fair amount of effort to proving before consulting the text. I think this is probably the best way to approach the material for maximum understanding but it requires a considerable amount of time. There are probably thousands of these sorts of "problems" among the three volumes. Ulitimately, I would like to end up with a notebook (which would probably number in the thousands of pages) that contains all of the details to all of the theorems completely worked out, as much as possible, with my own thoughts. Again, this seems like it will take forever and my time on this earth is unfortunately finite. I'm reasonably confident though that the production of such a set of notes would lead to at least a fair level of mastery of the material in question. Can anyone suggest an alternate strategy that might be more effective in terms of time but that would lead to a comparable level of mastery? It is also a problem that I might actually prove a fact completely on my own but then, a month later, might not be able to recall it in a time of need. What strategies are helpful for best ingraining this material (other than the obvious "Work lots of problems" approach)? Would appreciate any tips or pointers.


On StackOverflow.com I often find that people ask questions about problems that arise due to poor design choices (typically due to a lack of knowledge about the particular programming language). For example, the OP will make a choice at point A that is wrong, then in order to correct follow-up errors goes on to B, C, D ... and at point X (s)he gets stuck, and thus asks a question about X, when the solution to the problem is actually to fix A. Note that this is not limited to programming, but can be any project. Earlier, I came up with The Underwater House problem to describe a similar situation: Q: "I have this underwater house. I am having big problems with leaks and water damage. What is the best way to stop a leak?" To which the answer of course is: "The best way is to not build a house under water." When faced with such a question, I often feel the urge to name it, or create some classification, to let the OP know right away what the mistake is. The best way to state this that I have come up with is: "You are asking The Wrong Question." However, I feel that this is inadequate, and requires further explanation. Is there a more self-explanatory way to state this? Some simile, saying or phrase? Update: I felt that no answer really fits the bill better than "The Wrong Question", though "treating the symptom" was arguably the best answer. The amalgam "you're treating the symptom of a design problem", while dead on the money, is not as clear, concise and pithy as one would like. And sometimes not correct.


Related: How would a black hole power plant work? I have put a bit of commentary enumerating my confusions in parentheses I read in Black Holes and Time Warps (Kip Thorne), that quasars can generate their jets from four different processes. These all involved the accretion disk, but there was one which doesn't make quite as much sense. It was called the Blandford-Znajek process, and it involved magnetic field lines carrying current. The process was visualized in two ways. A black hole, with magnetic field lines, is spinning. In the first visualisation (viewpoint actually), the magnetic field lines 'spin' along with the black hole, and nearby plasma is anchored onto the field lines by electrical forces (where did the electrical fields come from?). The plasma can slide along the field lines but not across them (why?). Since the field lines are spinning, centrifugal forces will fling them up and down the field lines, forming jets. The other viewpoint is this, and it makes even less sense (to me that is, I haven't had a formal education in GR): The magnetic fields and the swirl of space generate a voltage difference across the field lines (Why? How?). The voltage carries current across the magnetic field lines (why are the field lines behaving like wires?). This current travels across plasma, which accelerates it, creating the jets. Now the main thing that doesn't make sense, is that magnetic field lines are behaving like wires. Why would they? I suspect the answer lies hidden somewhere in the equivalence of EM waves in different frames, but I can't think up any convincing argument from that side. If the answer involves GR equations, you don't need to solve it here (wouldn't make sense to me), but if you have to, just refer to the equation and what you did to it, along with the final result. Thanks!


I can't seem to find a specific answer to this anywhere. I understand that in a rocket there is a chemical reaction that causes gas particles to leave the rocket at high velocity. By Newton's third law, and the conservation of momentum, this caused the rocket to be propelled. What is missing is a physical explaination of what exactly causes this force on the rocket, as most answers annoyingly miss this final bit! People often use a skateboard-bowling ball analogy. However, in this analogy the person throws the ball and the ball provides an equal and opposite force on the person as it is thrown. The problem is, rocket doesn't 'throw' the exhaust out as it directly doesn't accelerate the particles - this is a result of a reaction. So what exactly causes the force on the rocket itself? Is it the case that in the explosion some gas particles collide with the rocket base, and the nossle is designed as to maximise collisions that will provide an upward force? I have seen people say this is wrong, or suggest it is right. Wikipedia says: 'About half of the rocket engine's thrust comes from the unbalanced pressures inside the combustion chamber, and the rest comes from the pressures acting against the inside of the nozzle' I want to be able to understand this in terms of particle collisions. I know there is a force due to the physical laws but people don't seem interested in the mechanics of the force itself. Back to the analogy: if someone where to throw a bowling ball over your skateboard, you wouldn't move, just as particles leaving a rocket, without collision of any kind, wouldn't cause the rocket to move (I know that's not possible, but hopefully makes my point). Hope this question makes sense.


Here is a traditional derivation of time dilation: There's a train with a lamp in the ceiling, moving at velocity v with respect to an observer. In the frame of the observer, the path taken by the light from the lamp straight down to the ground is actually diagonal because the train has moved forwards by the time the light hits the ground. Since the speed of light is constant, the time it took for the light to reach the ground must have been GREATER, because the distance traveled was the hypotenuse of a triangle whose side is the height of the lamp and whose base is the distance traveled by the train in the time it took the light to travel. That's the essence of it, math not included because it's not relevant to my question: This derivation works for light, yes, but it's based on the fact that the speed of light is identical in all frames, so applying the same procedure to a ball, say, would not work. In short: We calculated that light travel time has been dilated. How does this argument extend for ALL objects, not just light? Also: I have heard of answers involving light clocks (devices which measure time based on how long it takes light to move some distance), using the following arguments: Measuring time with a light clock shows that time clearly dilates. counter-argument: how do you know that the light clock is accurate then? Maybe other clocks would disagree, and time only dilates for light? If one uses both a light clock AND a variety of other clocks: The argument is that if you used both clock types and only the light clock went out-of-sync, you could tell that YOU were the one moving, so this violates the postulate of relativity (all inertial frames are equally valid; none are "THE" rest frame). counter-argument: this is okay with me if the person observing a difference is in the clock frame. But if they are not, relativity seems satisfied with the condition that, if a train observer and a "stationary" observer both have both types of clocks, each person sees the other person's clocks as out-of-sync with the other person's light clocks (nobody looks at their own clocks). I am aware of the experimental evidence that particle decays follow time dilation. I'd just like some evidence that it applies to all phenomena, rather than just the set which we have experimentally verified. Best would be a theoretical argument from Einstein's postulates. I am an undergraduate in my senior year, who has not yet taken General Relativity, so I would appreciate it if that were kept in mind in any explanation!


Context: I prepare my scientific documents using LaTeX and compile to a PDF. I often need to seek comments on drafts from collaborators who do not use LaTeX. Most of these collaborators use Windows OS. Assume also that the collaborator does not need to edit the document. They only need to be able to add comments to the document. Most would be familiar with the commenting system in MS Word, for example. I'd like to be able to give the collaborator some clear instructions about what software and system they should use to comment on the draft. This should involve free software and an easy to use interface. Question: What is a good strategy for getting comments on draft documents when the collaborator does not know LaTeX? Initial thoughts: I know Adobe professional allows you to add comments to a PDF. However, some collaborators don't have this software and it costs money. I could send the raw LaTeX to the collaborator. However, given all the markup, the collaborator may find LaTeX source a bit mysterious. UPDATE: After posting I noticed Andrew Stacey's answer to a similar question. Along with a number of other good tips (such as printing and getting paper comments), he mentions jarnal, xournal, and gournal as free cross-platform PDF annotators. I'd be curious to know whether experts have found them adequate for the above mentioned purpose and whether any of them are to be preferred. Others mention FoXIt. And yet others discuss the option of exporting to MS Word or Open Office and using the reviewing system within these programs.


I read everywhere the famous, but offhand statement that "the universe began from a single point" and this bugs me because that surely isn't true. My understanding has always been that this was an oversimplification for the sake of explaining it to children when it fact the universe is - and always has been - infinite. The observable universe has a size, but surely no one is seriously suggesting that the observable universe is all there is? That's just the farthest we can see, what with space expanding faster than light and what not. But surely the truth of it is that the universe goes on forever in every direction and has no "centre". Then it seems to me that the concept of everything start from a "point" is just silly. Infinite density, yes. But surely it still had infinite size? I had always thought it quite obvious that the universe "started" as an infinitely dense, infinitely sized, "clump" of energy which gradually expanded and cooled as the energy was distributed over a larger space (because space itself is expanding, not because the "point" expands). Am I at odds with the scientific community on this? Do people honestly believe everything start from a literal singularity? A single point? Or is that just the result of only talking about the observable universe which has a definite size? But the universe as a whole is infinite, surely, so how it possibly have come from a single point? I understand that if you assume the observable universe is all there is then when you extrapolate backwards in time you end up with a singularity, but the observable universe is just a bubble in an infinite universe.


The famous Riemann rearrangement theorem states that for a conditionally convergent real number series, we can rearrange the order of summation to make it converge to any prescribed number in the extended real line. In particular, this result sheds much light on the significance of absolute convergence, without which it would be quite dangerous to manipulate a convergent series. For improper (Riemann) integrals, we can also distinguish conditionally convergent integrals from absolutely convergent ones using analogous definitions. I'm wondering, however, if there also exists an analogous "rearrangement theorem" for improper integrals which reveals the essential difference (like eligibility for rearrangement, in the case of numerical series) between the two kinds of convergent integrals? Indeed, can we even define an integral version of "rearrangement"? Of course, one distinction I'm already aware of is that absolutely convergent integrals are also integrable in the Lebesgue sense while conditionally convergent ones fail to be. But this is not what I want, since it doesn't appear nearly as striking as what is exhibited in the series version. PS: as far as I can tell, one probable way to see why eligibility for rearrangement matters is how we define a valid expectation for a numerical random variable. We require expectations (numerical series for discrete random variables, and integrals for continuous ones) to be absolutely convergent, for if they were not, then they would undesirably depend on the "chronological order" in which we observe events, violating our basic principle that expectations should be stable and inherent in the random variable itself, rather than affected by how each event chronologically arises.


My questions concerns that classic train paradox, wherein there is a train and a tunnel of equal length, and the train is traveling and some fraction of the speed of light towards the tunnel. According to the Special Theory of Relativity, an observer outside the tunnel will see the train length contracted (Lorentz Contraction), whereas an observer inside will see the tunnel contracted. Additionally, suppose that there were doors at the ends of the tunnel and that the observer outside the tunnel closed both doors instantaneously when he/she saw that the train was completely inside the tunnel. The classic resolution of this paradox invokes the non-simultaneity of events, explaining that the observer in the train sees the far door close first, and then, once the train has begun to exit the tunnel, looks back to see the door at the beginning close. Thus, both observers agree that the train does not touch the door when they are closed for an instant. Now my questions. Why is it that the observer on the train sees the far door close first? It seems to me that the information coming from the far door would reach the observer on the train only after the information from the other door is reach. Under this interpretation, the observer on the train would observe the train getting hit by the doors. What if, by some means, this event can be explained in terms of a stationary observer too? Everyone always concludes that the train remains untouched by the doors, but really the only condition that needs to be met is that both observers must agree. Why can't they both agree that the train was hit? So, to summarize. Why is the door that is farther away from the train observed to close first? Why can't the other possible conclusion (both see a hit train) be observed?


I was reading on wavelets and it seems that its hard to find a precise mathematical definition of what this concept is. My confusion first arose due to Gilbert Stang's linear algebra book. In particular consider the following extract: It talks about how to change a vector from one basis to another but it never rigorously defines what a wavelet is (by the way, I did understand that extract I included, just not the concept of "wavelet"). From my understanding, some special basis are called wavelets (for some special reason). But which basis are we allowed to call wavelets? I would assume that they have something to do with linear algebra and oscillation/sinusoidal functions but I don't really see what the relation between the two is. To look for an alternative explanation I went to wikipedia and the initial paragraph starts as follows: A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases, and then decreases back to zero. It can typically be visualized as a "brief oscillation" like one might see recorded by a seismograph or heart monitor. Generally, wavelets are purposefully crafted to have specific properties that make them useful for signal processing. Wavelets can be combined, using a "reverse, shift, multiply and integrate" technique called convolution, with portions of a known signal to extract information from the unknown signal. With that description it makes me feel that wavelets are actually functions. However, I've had difficulty understanding this precisely, specially when trying to relate it to linear algebra. I guess I am having a hard time connecting the three, wavelets, linear algebra and their relations to sinusoidal functions (if there is any relation to them).


I have confused myself about the following variant of Maxwell's demon and I can't seem to find out where the energy went. Consider this: You have a chain (one dimension) of spins (up/down) with a nearest-neighbor coupling. Energy is minimized if spins are aligned. Let us say the energy difference between alignment and not-alignment is E. The zero temperature state is either all up or all down. If we heat the state up to a temperature T, some of the spins will flip with a probability given by the Bolzmann-factor, depending on the ratio T/E. So far so good. Now the finite temperature state has energy because the states aren't all aligned, but the distribution is thermal and it's no useful (free) energy. However, if you knew which spins are misaligned, you could selectively flip them. Let us say the system is such that you can flip them by shooting a photon with energy E at the spin. Eg, you shoot at the middle spin in a series of three. If it's up,up,up then the photon will be absorbed and you end up with up, down, up. If it's up, down, up, the photon stimulates emission and you get up, up, up plus two photons of energy E. If you have up, down, down, the photon doesn't change anything about the total energy. The same happens if you exchange all ups with downs. Now the thing is this: If you do not know which spins you have to flip, your chances of gaining or losing energy by shooting photons at the chain are the same. You just convert one thermal state into the other. But if you knew which spins to flip, you could topple them over selectively and get energy out of the system. Essentially, you extract it from the thermal bath that did heat up the chain. That's possible (I think) because you are using information to reduce the entropy of the system. My question is this: How do I see that the energy needed to measure the spin orientations in the chain is at least as large as the energy I can gain by flipping them selective once I have measured? It isn't clear to me why it should not be possible to measure them with some very low-energetic probe, eg measuring the local magnetic field with the Hall effect.


Consider an American actor who is tasked with mastering British Received Pronunciation for an upcoming role. If he has a talent for vocal mimicry, as many actors do, he should have no trouble picking up the "rules" of RP just from listening to people speak it: the non-rhotacism of the dialect, the aspiration of intervocalic t, the characteristic intonation patterns and prosody of RP, and so on. For the most part, he should have no trouble speaking RP like a native. Yet he would never in a million years figure out on his own that lieutenant should be pronounced "leftenant," for example, or that controversy is often pronounced with the accent on the second syllable, unless he hears those specific words pronounced. To my knowledge, there are no general characteristics of RP that account for the mysterious appearance of an f in lieutenant (or, if you prefer, there are no general characteristics of General American that account for its absence). You just have to know how those specific words should be pronounced, because you'll never figure it out on your own. Do linguists recognize a distinction between the "rules" of a dialect on the one hand and its individual pronunciation "quirks" on the other? Is there a term for this phenomenon? Is it considered merely a variation on regional preferences for certain words over their synonyms (e.g., rubbish vs. garbage), or is there something else at play here? (Disclaimer: I take no position on whether the BE or AE pronunciation of any of these words should be considered the "quirky" one; I simply note that one couldn't easily intuit one pronunciation just from knowing the other.)


Suppose I am conducting the Quantum Eraser experiment. The results of this experiment are easy to understand with the traditional quantum mechanical interpretation of a pair of entangled photons. Specifically, suppose that I am performing the "eraser" part of the experiment in which one photon is diagonally polarized so that when the entangled partner passes through the circular polarizers of the double-slit apparatus it cannot be determined through which slit it passed (thus reproducing an interference pattern). Now suppose (It's like supposition Inception! Supposings inside supposings inside supposings.) that I Lorentz boost into a reference frame in which the particle passing through the double-slit apparatus is received at the detector (and consequently has its polarization measured) before its entangled partner gets diagonally polarized. It appears that there are two possible outcomes: either (a) I disagree with the results recorded by an observer in the commoving frame of the experiment (i.e. I see no interference pattern.) or (b) I must conclude that the future polarization of the entangled partner somehow reached back in time and changed the outcome of my measurement. So which is it? Do I get different results or do I need to embrace a further layer of quantum non-locality? This is a tough one for me to understand because of its complementary experiment: Imagine putting the polarizer much farther away than the double slit apparatus so that there is no interference pattern created (because the entangled partner does not get diagonally polarized until after the measurement has already occurred). In that case, I could boost into a frame where an interference pattern should occur because the partner gets polarized first. (Or am I missing something? Will there be - or not be - an interference pattern regardless?) If the answer is that I get the same results as the commoving measurement, then why is the commoving frame preferred? Why is it its results that are maintained in all reference frames and not the (expected) results from my boosted frame (which should be just as valid)? Also, I think that this version of the experiment is fundamentally different from the Delayed-Choice Quantum Eraser experiment and Wheeler's thought experiment since both of those experiments are easily explained by representing the particle in a different eigenbasis (namely, as being in a superposition of the interfering and the non-interfering states - instead of being in a "collapsed" state of another observable). However, I am also interested to know whether I am wrong in this estimation. Are these experiments basically the same? Can conclusions about or explanations of one of them be generalized to the others? Edit: Because of the question in the comments, I've added a little additional explanation for why I think the experiments are basically different. Primarily, the difference comes from the fact that both Wheeler's thought experiment and the Delayed-Choice Quantum Eraser can be explained in such a way that apparent "cause" always precedes "effect." Causality is still violated, but at least its not giving us information about the future. For instance, if we didn't know whether or not the entangled partner beam would get polarized a year from now (in the quantum eraser experiment), we could predict whether it does or not today by determining whether or not there was an interference pattern at the double-slits. If there exists any boosted frame in which the beam gets polarized first, then that result must carry over to the frames where it gets polarized after. So "future-telling" can happen in the relativistic quantum eraser, but not in the other two experiments. For example, in the Wheeler experiment, we polarize the photon and then after the fact decide at random whether to measure its polarization or not. If we measure the polarization, we get no interference; if we don't measure the polarization, we do get interference. Often this is ascribed to a sort of "predictive" non-locality, but just as easily, we could say that the polarized state of the photon is actually a superposition of states (namely, interfering and non-interfering) and when we measure its polarization we collapse it into the non-interfering state (and if we don't, we collapse it into interference). Likewise for the delayed-choice quantum eraser: instead of thinking that the beam-splitter in the future decided the interference in the past, you could just as easily conclude that the interference (or not) in the past decided the outcome of the beam-split in the future.


In a previous Phys.SE question, Does a spaceship travelling at near lightspeed see the universe aging slow or fast?, the answer (which was followed by a proof involving co-moving reference frames) was given as The short answer is that yes, an astronaut moving relative to the cosmic microwave background would measure a shorter time since the Big Bang than an observer stationary wrt to the CMB. However, an observer in such a spaceship will consider the time of any object which is at the CMBR co-moving reference frame to be moving slower than itself. Is this not a conflicting result? For example, let's say the spaceship and a CMBR Earth communicate as they pass by each other. Each would have an estimate of the age of the universe, and each would have an estimate of the measured age of universe that the other would give, based on their own measurement of the age and the time dilation that they assume the other would experience. Here are the results The CMBR observer is fine - both his estimate of the universe's age that the spaceship would give and the estimate actually given by the spaceship match and are less than his own estimate of the universe's age. However, the spaceship expects the CMBR observer to have a lower estimate of the age of the universe because their clock is (according to spaceship observer) ticking slower than his own. What the spaceship observer does not expect is that the CMBR observer's estimate is larger than his own estimate of the age of the universe, yet that is what happens. How is this resolved without implying a preferred reference frame?


I tried to make rain with a bottle and a balloon but it failed. The bottle was small, only a couple inches wide and the threaded part that I had the balloon on was less than an inch. In particular it was the size of this hydrogen peroxide bottle: It was originally a hydrogen peroxide bottle but all the hydrogen peroxide was out of it when I did this experiment. I stretched the balloon a little bit and put it over the threaded part of the bottle. But right before that I put hot water in the bottle. I knew that some water was evaporating. Also there was regular air(including dust particles) inside the balloon(I did this outside during a dry spell during the summer). This gives plenty of opportunities for the water to condense to form a cloud in the balloon and eventually, rain. However I left it out there for several days and nothing happened. The balloon didn't inflate from the water vapor bouncing off of the balloon. The water didn't condense enough to be noticeable. Rain didn't form. I thought that maybe there was too little water and too little air so I went to a much bigger bottle(about a gallon in size). The balloon started to separate. I also had a control of an open bottle in the same conditions while doing this experiment. The water level did not change in the open bottle even when the humidity was low. So why didn't the balloon inflate if the fact that both the water and the air were hot should have increased the vapor pressure, thus causing the balloon to inflate? Why didn't the water condense into a cloud that I could see through the latex of the balloon if it were inflated? Why didn't the water get to the critical size for raindrops if there were plenty of dust particles in there for the water to condense on to form water droplets and eventually raindrops? Why didn't the water level in the open bottle change? Is there anything I can do to improve this rain in a balloon experiment besides having a source of heat underneath the water?


The principle conservation of energy is often taken as an obvious fact, or law of nature. But it seems to me the definition of energy is far from obvious, or natural: http://en.wikipedia.org/wiki/Energy lists lots of different types of energy. So if I want to apply this principle in some concrete experiment, I have to go through all the forms of energy and consider whether this form of energy is applicable to each particular entity in my experiment. This seems like a rather cookbook-oriented approach (and the wiki list doesn't even claim to be complete!). Now I wonder: ---> to what extent can these different energies be derived from some single simpler definition? For example, if my model is that everything is made up of atoms (I don't want to consider anything at a smaller scale, I fear that would muddle the discussion and miss the main point. Also, I'm considering only classical mechanics.) which are determined by their position, momentum, charge and mass (?), is there a clear and exhaustive definition of the energy of a such a system? EDIT: In light of comments and answers, I think I need to clarify my question a bit. Is it true that the electric potential and gravitation potential (for atoms, say) will explain all instances of conservation of 'energy' occurring in classical mechanics? If no, is there some modification of "electric potential and gravitation potential" above which will yield yes? My question is not really about mathematics - Noether's theorem for example is a purely mathematical statement about mathematical objects. Of course mathematics and my question are related since they both involve similar kinds of reasoning, but I'm ultimately after a physical or intuitive explanation (which is not possible using only mathematics since this involves choice of a model, which needs to be explained intuitively) or assertion that all these energies (chemical, elastic, magenetic et.c. (possibly not including nuclear energy - let's assume we're in the times when we did not know about the inner workings of atoms)) come from some simple energy defined for atoms (for example).


I'm nearing the end of the semester of an introductory-level complex variables class. (Very introductory -- it's the version of the class that's only required for engineering and physics majors, as it doesn't require two semesters of undergrad analysis that are prerequisite to the complex variables class for math majors, at my school.) One of the many fascinating things I've seen this semester has been, speaking in broad terms, the behavior of analytic function, and the way that a harmonic function and its conjugate 'synchronize' (for lack of a better word) to create analyticity. Despite the examples I've seen of harmonic functions being steady-state solutions to heat problems and showing up in descriptions of electric fields and whatnot, I feel like I lack any sense of what the 'harmonicity' of harmonic functions is all about. On a side note: I do recall one day, however, where I was working through an example having to do with the level curves of a harmonic function and its conjugate, where I believe the significance what they points of intersection where always orthogonal. This 'mesh' notion created, for me, a visual image of how the the two functions work together to give an analytic function its synchronized, predictable nature. (But, as with most things, I could be mistaken in my understanding of this; these weren't points being stressed in the book, and it was in a chapter later in the book than what we'll cover in the class.) So, my question is that of how one ought finish this statement:"I was considering a problem, and I intuitively knew the solution would need to be a harmonic function because the problem had the property..." I stress the word 'intuitively,' by the way. If you feel this misses the point of harmonic functions and how I should think of them, then by all means please answer however you feel is appropriate.


I'm quite perplexed by the notion of 'observation' in regards to the collapse of a particle's probability wave. Does a particle's wave only collapse when it is involved in a strong interaction (such as a collision with another particle, like bouncing a photon off another particle to determine the other particle's position) or does any interaction with another particle or field also cause this collapse? For example, I presume traditional detectors such as those mentioned when talking about the double slit experiment are devices that do exactly as I stated above and (in the case of the double slit) have a stream of photons (like a curtain) going from the top of the slit to the bottom which the electron (or particle being shot at the slits) has to pass through and therefore get smashed into by the photons. So given an environment where there was no other gravitational or electromagnetic influence, or where the effects of such have been taken into account, what would happen if you had a region of space within which a massive neutral particle 'is' (ie, it's probability wave fills this volume), and then shot a beam of photons across the bow of this region (so as to pass nearby, but not intersect), and then have a photo-sensitive plate on the other side to see if the photons passed straight by, or were pulled towards one direction slightly by the warping of spacetime caused by the massive neutral particle (gravitational lensing on a mini scale)? Does this even make sense? And if the photons were being pulled slightly towards the massive neutral particle, would the waveform collapse at this point, since if the mass of the neutral particle was known, the amount by which the photons path had been bent could confirm it's position couldn't it? Or perhaps take the above and use an electromagnetically positive particle as the target and an electromagnetically negative particle as the beam you shoot by it to see if it is effected by the electromagnetic field caused by the target particle if that makes more sense. I might be making some silly errors or assumptions here, I'm only a layman with no formal education or in depth knowledge, so be gentle :)


First I will have to explain my question. Look at the image below. This shows doppler shift when an object is moving horizontally to the direction of the wave. Keep the word 'horizontally' in mind. Now this happens because: I will quote Jim from his answer for Redshifting of light from a moving light source We all know that light is a wave, when you turn on your headlights and drive in reverse, the light is doppler shifted because of the motion of source. When not moving, each cycle of the light wave is emitted from the same position; it has a specific set of wavelengths. The distance between one crest of a wave and the next crest is equal to the speed of light, c, times the period of the light (which is determined by the oscillations in your headlights and won't change when you are in motion). When you drive backwards, the distance between one crest and the next becomes the period times c plus the period times your backwards velocity (approximately); the second crest is not emitted at the same location as the first, so it extends the wavelength. From your perspective, the emitted wave would not be red-shifted at all, but from a stationary observer's perspective it is. So now my question is, imagine a car which has a torch attached to one of its windows. The torch is switched on and the car begins to move. When the car moves, its movement is in the opposite axis from the propagation of the wave. So each crest will be released from a different location while the first crest is already on its way in a straight line. I will try to represent this graphically. The representation is very estimate. It just shows how would the light bend as each crest is released from a different location. Please explain this to me. Will the light actually bend? Why or Why not? Edit What I have concluded from the answers is that first a photon is emitted and then it continues as a wave and is in no way attached to other photons. Is this right? If I got this then I got the answer for my question.


I'm a math student, starting a PhD in the near future. My field of research will be mostly in the field of applied mathematics / numerics. Topics will deal with Kinetic Theory, Moment Equations, Fractional Diffusion, Spetral Methods. I think I have a solid background in numerical computing, especially for PDEs. Now for my Masterthesis I've dealt with numerical methods for fractional diffusion equations. Since (fractional) diffusion is related to Brownian motion / CTRWs, a lot of authors named mathematical finance as a field that is impacted by their research. The problem is, that I have absolutely no background in finance / econ whatsoever, but I would really like to get into this topic. I think it would open me a lot of opportunities to gain a little expertise in that field. I started to read some mathematical finance papers, referenced in the papers I encountered and noticed quite quickly, that I lack the non-mathematical background. The goal of this question is maybe a bit ambition. I would like to get to know the field of mathematical finance over the next three years. Start with the basics and then move quickly to mathematical finance with a focus on computing / simulation. Since I will do this on top of my work, I would also appreciate books that I can pick up every other day/evening and just read a little. What interests me espacially are processes that are related to Stochastic DEs, Brownian Motion, Ramdom processes. Especially topics that might rely on the same basics as diffusive processes / kinetic theory. For example, some buzzwords I encountered where stylized facts, options, derivatives (in finance). Maybe you could split up your recommendations in the categories finance/econ, general mathematical finance and random processes / SDEs in finance. If you have general remarks regarding my proceedings please feel free to contribute (for example, what do you think about visiting certain lectures offered at my university, broadening my mathematical background, relating my research to the field of finance, software I should get to know like R, etc.) Thank you very much!


Suppose I have plotted the body angles of a flying aeroplane.I have two such plots. One of a normal plane, and the other plot in which a primary sensor of the aeroplane is removed. So with the absence of a sensor you are expected to see an instability in their flight. Below are the two graphs--- The first one is the control flight, and the second one is the experimental (sensor removed.) Lets look at the second graph --- you can see that the roll of the flight is sinusoidal. Even the pitch is slightly. But the sinusoid fluctuation of the roll is more prominent. And I see this fluctuation in all my experimental graphs (sensor-removed) that I have obtained, and none in the control (normal) graphs. Now, I would like to quantify this fluctuation. One way of quantifying it would be to plot the mean and standard deviation of roll in both cases and show that the standard deviation is very high in the second case. But this is not correct because, the standard deviation might also be very high in the first case ---- this can happen when the flight rolls a lot, but does not necessarily fluctuate. It could roll in one direction, maintain that for some time, then roll in the opposite direction or the same one. Its not fluctuating, but standard deviation will be high. So, I am looking for another parameter, to quantify this fluctuation in the graph, something like a fourier transform perhaps. Something that would accentuate the instability bit. Suggestions people?


During an episode of Archer, he criticized a journalist's grammar for her misuse of the word 'child-murderer'. She meant one who murders children, and Archer argued in using the hyphenated form, she implied the accused man is a child who murders. Is this correct? I searched "child-murder" and "child-murderer", only to find columns eschewing the hyphen in nearly all cases. Instead, the columnists, relied on context as to whether they are referring to a child who murders or one who murders children. I, however, am purely interested in the proper use of the hyphen in this situation, as it could possibly extend to other situations as well. The trouble seems to arise from child not having an adjective or descriptive form. With 'teen', one does not run into this problem: Teenage murderer vs teen murderer However, if one uses 'adolescent murderer', it becomes unclear as to whether one means an adolescent who murders or... you get the picture. This problem arises from adolescent being both an adjective and noun; a hyphen can resolve the ambiguity. But once again, how should the hyphen be used? I found a similar question: What is the plural of 'only child'? I err on the side of only-children, in the event that 'only children' reads as 'just/simply/merely children'. Some suggested entirely new phrasing, while others say that the context is sufficient. I don't believe one should change his entire sentence when proper use of the hyphen can get his meaning across just fine, and even when context is suitable, proper grammar is still rule of law.


I'll try my best to explain my question with examples because I don't have much knowledge on the theory. So say that I had a block of water in air, a cube of water, that is dropping. If the cube wasn't too big, I believe that as it will drop, then the air from underneath it will rush to either side of it, as the water displaces the air on it's way down. Now lets change the situation. Say I had the cube of water, but now its suspended in the air in a tank (like a fish tank), as shown in the picture. Here, I believe that there is the weight force of the water acting evenly downwards, but the air pressure may be different, as the whole system is bounded by the walls. Logically, I believe that the air would then rush through the center of the water cube, forming a bubble, allowing the water to rush down to the bottom. Though this may be completely wrong, I'm not sure. So my question is, is that is there any possible shape for the water to be originally in, so that all the forces sort of balance so that there is no distortions in the shape, and that then the air will be compressed under the force? I understand that perhaps at the microscopic level, if we take into account the random motion of molecules, and other factors, that there will inevitably be some sort of inbalance, but macroscopically what would be the perfect shape? I'm not even sure if any of this makes any sense. Thanks for any help understanding.


Saw a question about faster than light travel... I still have the same question though none of the answers offered any resolution for me. It is so summarily assumed by all physicists and commentaries that exceeding the speed of light would turn back the clock. I can't see the relation. Doubling the amount of any speed halves the time taken to travel a given distance. Keep doubling the speed and that time is halved (or otherwise divided). Divide any quantity (time in this case) and you always end up with a fraction of it but never zero and certainly never a negative amount as would be the case for the causality conflict. So it seems to me that whatever speed one attains, there is always a positive time element in the travel no matter how tiny!! The speed of light is only unique to me in that it is the fastest observed speed but is otherwise just another speed quantity set by nature (just like the speed of sound etc) could it be that other elements in nature are travelling faster than light but we lack the means to detect or measure them (like the rebellious neutrino)? I also don't understand time as an independent element that can be slowed sped up etc. It seems to me that time is simply a relative measure of the ever-changing state of matter relative to other states of matter. If every thing in the universe stopped- that is all state of matter everywhere frozen, all electrons frozen in place etc wouldn't we observe that time had stopped? Isn't it therefore our observation of the changing state of matter around us that gives the perception (perhaps illusion) of time? I can therefore only understand time as a subjective sense of changing states relative to an observer! It should be the rate of change of these states that slow down or speed up (in relation to the observer or instrument) and not the universal rate of change or universal time that changes! It would also debunk any notion of time travel, as it would involve the manipulation of every particle in the universe to a previous of future state... Disclaimer.. I hate calculations, stink at them and have no idea what mathematical formulas are used to arrive at the accepted conclusions so I'm not trying to dispute any findings etc by the experts, just trying to align my lay understanding to their conclusions.


Everyone does know that the surface of a conductor is at equipotential during equilibrium. I was reading Feynman's lectures where I found this (bold)line: Suppose that we have a situation in which a total charge Q is placed on an arbitrary conductor. Now we will not be able to say exactly where the charges are. They will spread out in some way on the surface. How can we know how the charges have distributed themselves on the surface? They must distribute themselves so that the potential of the surface is constant. If the surface were not an equipotential, there would be an electric field inside the conductor, and the charges would keep moving until it became zero. This is a much good reasoning for the surface to be an equipotential one; if there were any region to be in higher potential, charges would flow towards them to neutralize and again make the surface equipotential. To understand his explanation, I thought of a positively charged surface that is not in equipotential status; so there would be an electric field which would prompt the free electrons inside the conductor to go there & nullify the field to make the surface equipotential, right? But what about the positive charges that are now inside the conductor? Okay, they would by repulsion move towards the surface. But what is the GUARANTEE that they would form the equipotential surface? What really happens when they go on the surface that compels them to make an equipotential surface?? [After all, you can't say:" since you are studying electrostatics, there must be equipotential region on the surface no matter what happens; that's it"-this is what my school-teacher said when I asked him.]


I'm thinking of writing a short story set on a version of Earth that is tidally locked to the Sun. I'm not exactly sure how to research the topic. Here's a number of questions about what would happen: How hot would the light side get? Are we talking ocean-boiling levels? I imagine that life would eventually flourish, given the massive constant energy source. Is this accurate? On that note, I imagine massive thunderstorms along all the coasts due to increased evaporation. How bad would they get? Would the ground ever see the Sun, or only rainfall? How cold would the dark side get? Is it conceivable that any life could still exist there? (Life has proven itself quite versitile in the past, i.e. life at the bottom of the ocean.) What wind speed would the twilight zone experience? I imagine the atmosphere would transfer heat from one side to the other, but would the wind speeds be bearable? In what direction would air flow? I hear that the oceans would recede into disjoint northern and southern oceans if the world stopped spinning. Would this also happen if the Earth became tidally locked? Would the Sun create a 'tidal' bulge in the ocean at the apex of the light side? Would this or the above dominate ocean behavior? Would we completely lose the magnetic field? Would life be able to survive without such shielding from magnetic radiation? Would the Moon eventually unlock the Earth? What state would the Moon have to be in for there to be both a locking between the Sun and the Earth as well as the Earth and the Moon? What other radical differences would exist between our Earth and a tidally locked alternative?


I was asking the other day how to find certain fonts of a document. Well at last it was easy. I came to the Mathematical Pi LT Std font and ITC New Baskerville Std results and for that part, I was pleased because I finally managed to get those fonts. So far, I have only used LaTeX for my documents and papers for the University and own projects and as most of us know, there are very few pleasant Math Fonts to use (at least for me). I have tried XeLaTeX and mathspec package to set the digits and latin to Baskerville and Greek to Mathematical Pi, but it turns out to be very clumsy when it comes to spacing and that stuff... and I ended up not liking the result. I have been checking the glyphs of Mathematical Pi and it contains a lot more than Greek Letters (obviously) but I don't know how to tell XeLaTeX to use for example the integral sign, relation signs, sums and everything else. I would like to know if there is any way to use Mathematical Pi font at full with any of the TeX systems and if that is not the case how to use that font other way, because I have been trying to use it in MS Word with its equation editor but I have not been successfull nor finding how to do it. Can anyone help me with this?, I'm sure there has to be a way since I have found that font in many books. Thank you all in advance.


It's not really clear to me how does QM attacks determinism. It sure attacks computability, which is a component of newtonian, naive determinism, but it's often claimed to destroy determinism itself (which says that we can't compute events, but they are determined anyway). A photon is both a particle and a wave; a particle doesn't have both speed and momentum defined values at the same time; some things are not a in a state, but in a superposition, until they are measured. It's cool, I get it. But aren't there a lot of macroscopical objects that can be described this way as well? A just published book is both a best-seller and a failure, until enough people buy it. It would be impossible to compute which state it is until the state itself 'happens'. I'm not sure I'm just referring to hidden variables. Where is the problem with observation exactly? It is obvious that the fact that a scientist decides to observe the particle as a wave is due to the fact that he's doing that experiment. And he's doing that experiment because he likes science, because it's written in his DNA or in his education. He's working because he needs to eat, because chemistry of his body tells him so, and so on. The impossibility to compute in advance which state is correct is due to the free will of the observer, in the exact same way in which the book is a best-seller or not (the decision of people to buy it). But it's just because it's too difficult to know everything: a God-like creature could perfectly know how many people would buy the book and also if the scientist decides to measure the photon as particle or as a wave at a given time. Is this just superdeterminism? (https://en.wikipedia.org/wiki/Superdeterminism) Looks like a pretty logical observation to me. What I am missing?


I have recently been studying continuous dynamical systems whose phase space can be divided into a number of regions. Inside each of these the flow is smooth, but there is a discrete jump in the flow just at the boundaries. In the mathematical description, the right hand side of the differential equation is different for different regions of the phase space of the dynamical variables. Note: I don't mean something trivial like systems which exhibit smoothness in different regions of physical space separated by boundaries, like differently heated gases in partitions, or water in contact with vapour etc. The different regions I mention are regions in the phase space of the dynamical systems. So imagine a set of continuous-time differential equations defining a flow which is segregated in its phase space into regions in which the evolution of the equations is piecewise smooth. I also don't mean phase transition. There is no variation of order parameter or bifurcations here. The piecewise smoothness exists in the dynamical phase space for a fixed value of the system parameters. I have been studying them in an engineering context of a mechanical device in which there is a sudden change in the velocity of a moving part when it hits something. But it struck me that such piecewise smooth systems should be found in many scenarios, from other areas of physics, maybe certain quantum phenomena, to biological systems that can be studied with the theory of dynamical systems. Some examples of the kind of systems I am looking for are: Quantum mechanics: the Muffin-Tin potential is a quantum model where the potential (the right side of the differential equation) is approximated to be piecewise defined. Classical mechanics: the hard impacting oscillator (oscillator with a rigid wall at an end restricting the amplitude, like the devices I was studying). Theoretical computer science: Hybrid automata and reachability problems which are further piecewise linear. I am curious to apply my understanding of the mechanical system to such systems. So, what are other dynamical systems in nature which exhibit piecewise smooth behaviour?


I posed a closely related question here but it received a tumbleweeds award. So I thought I would post it from a different angle to see if I can illicit at least some thoughtful comments if not answers. The modeling of many physical systems utilize the mathematical tools of calculus, by writing the relationship of physical quantities in the form of differential equations. Considering time dependent operations of integration and differentiation, the dynamics of a physical system may be expressed in terms of one form or the other. A good example are the Maxwell Equations which are often written in both differential and integral forms. Integral forms tend to express where the system has been up to where the system is at present while differential forms tend to express where a system is now and where it will be in the near future. So the two forms tend to imply a sense of causality. So this brings me to my question. Since we tend to observe a causal universe (at least at a macroscopic level) are integral forms a more natural approach to modeling systems? I'm using the word 'natural' in the sense that the nature of the universe tends to work one way vs another. In this case I'm saying nature tends to integrate rather than differentiate to propagate change. We can write our equations in differential form, solve them and predict, and they are useful tools. But isn't mother nature's path one of integration? I tend to believe this is so by my experience in simulating systems. Simulating systems in an integral form rather than differential form always seems to lead to better results.


Right now I'm writing an essay on Death in Venice, and I'm having trouble finding the right word or phrase to express how Aschenbach is parallel to the old man on the boat to Venice (both dress up to fit in with youth, some repetition of specific phrases in descriptions, youth from Pola vs youth from Poland, etc.). The closest word I can find to describe what is a "parallel". I see this word used a lot on the internet to describe (usually in TV shows) when something that one character did in a past episode is similar to something that a character (usually different, but I've also seen it used to point out character development via slight differences in the parallel). It kind of describes two completely separate scenes that have a much more powerful meaning when juxtaposed (usually very blatant mirroring, etc. to point this out to viewer/reader). In my case, it would be the scene with Aschenbach observing the man on the boat and the scene where Aschenbach applies makeup for Tadzio. Specifically, the man on the boat foreshadows the "endgame" for Aschenbach. In my essay, I've referred to it mostly as foreshadowing, but I think the depth of the connection that Mann makes warrants something a little stronger. The word foil came to mind, in terms of the intensity and how it's character specific, but obviously, it has the opposite meaning of what I'm going for. I did think about just using the word "parallel", but I when I looked it up, the definition for the word parallelism came up, and as a literary term it seemed from this definition, it seems that this word as a literary device refers to parallel syntax and a not to a broader similarity. I've come up with "symmetry" and "mirror" to describe individual aspects, but I was wondering if there is a proper term that encompasses the broad connection between the two characters.


I was looking at this question on SE and the answers seemed to say that the reason why matter doesn't expand along with space is because of forces like gravity, electromagnetism, etc. However, i feel like this has to mean the fields themselves don't "expand" along with space. Let me explain... From my understanding, the expansion of space is about the expansion of space itself, not about distances within space changing. For example it's about the space taken up by one meter increasing, not about the distance between entities increasing from one meter to two. If these assumptions are off please let me know. Keeping that in mind, look at redshifted light for example- This happens when the distance between crests of the light wave increases, changing the light's frequency. However, if space expansion doesn't increase distances within space, there should be no change of frequency, because there is no change in distance between crests. We do however see that the frequency does change with metric expansion. The only way i can think of to resolve this issue is to say that the expansion of space itself affects the wave. The only way for the expansion of space to affect the wave, is for the wave's field to not expand with space. The EM field "density" has to stay constant relative to the density of space (sorry for all the informal wording). Or in the case of gravity- if space expands, the distance (and therefore gravitational influence) between objects should remain unchanged. However, for gravity to keep mass from expanding along with the expansion of space, it would have to exert a force that isn't proportional to changes in space density. The behavior of gravity (relative to the distances that determine it's influence) changes based on the density of space. This would have to mean, like EM, the gravitational field does not expand with space. That's how we're able to deduce expansion, because the fields themselves are our reference points. This isn't about waves/excitations of a field, it's about the field itself, if that makes any sense. If this is true, how do we reconcile the expansion of space with fields remaining constant?


I play the flute as a hobby, and I've noticed that when playing middle D or E flat, one can interrupt the air column by releasing a certain key (which is near the middle of the air column), and yet have no effect on the pitch (though the quality changes for the better). I'll be putting a few diagrams here, since it is hard describing the situation in words. The black portions of the diagrams represent closed holes--basically "air cannot escape from here". The gray represents holes which are closed due to lever action, but need not be. Here's a diagram without coloring (all diagrams are click-to-enlarge): The mouthpiece is attached on the left, as marked in the diagram. The second key is just a ghost key connected to the first (and has no hole underneath it), so I'll just remove it from the diagrams: A few examples Alright. Normally, when playing consecutive notes, you make the air column shorter by releasing a key. For example, this is F: This is F#: And this is G: One can easily see the physics behind this, an unbroken air column is formed from the mouthpiece. The weird stuff Now, let's look at middle D and E flat: D: E flat: Here, the air column is broken in between. I feel that both should play the same note, that is C#: But they don't. I can close the hole, creating an unbroken air column in both cases, but the sound quality diminishes. A bit more experimentation (aka "what have you tried?") Reading this section is strictly optional, but will probably help I did a lot of experimenting with this key, turning up some interesting results. Hereafter, I'm calling the key "the red key", and marking it as such in the diagrams. When the red key is "closed", no air can escape and it forms part of the air column. If I play low D/E flat, I only get a clear note when the red key is closed. With it open, I get a note which has extremely bad quality, as well as being off-pitch. This is markedly opposite with what happens on middle D/E flat (mentioned above), there there is no change in pitch, and the difference in quality is reversed. Pictured: Low E flat (for low D extend the RHS of the black portion a bit more). Note that the fingering, save the red key, is the same for middle D/E flat Actually, this seems to be happening for all of the low notes--each one is affected drastically when the red key is lifted. Going on to the notes immediately after E flat For E, quality is drastically reduced when the red key is open. The harmonic (second fundamental) of E, which is B, is more prominent than the note itself. One can make the E more prominent by blowing faster, but this reduces quality. Red key closed gives a clear note, as it should. For F, a similar thing happens as with E. With the red key closed, it plays normally. With it open, you hear a medium-quality C (first harmonic of F), and no F at all. Blowing faster just gives a high C. The notes immediately below D have a fingering starting from no keys pressed (has to happen every octave, obviously). For the first few notes here, lifting the red key gives you C#, as expected. (I'm not explicitly marking the key red here, otherwise it'll get confusing what the correct fingering is) In C#, pressing the red key will obviously change the note ....to C: Obviously, lifting the red key here will get you back to C# One (half) step lower, we have B, which again goes to C# when the red key is lifted It gets interesting again when we play B flat. Lifting the red key here gives a note between C# and C And if we go down to A, lifting the red key gives us a C And a bit of experimentation with the trill keys (the actual holes are on the other side of the flute). Whereas messing with the red key for D and E flat produces no change of pitch, messing with the trill keys (which are the same size as the red key and are furthermore pretty near it) does. Hitting the second trill key while playing D gives E flat. One should note that this second trill key opens the hole closest to the mouthpiece. Note the visual similarity between this and the situation in the "the weird stuff" section Hitting the first trill key while playing D gives a note between D and E (the two trill keys are close to each other, you may have to see the enlarged version to get the difference) Hitting the second trill key while playing E flat gives a note between E and F Hitting the first trill key while playing E flat gives E flat (No diagram here, these two are the same as the last two, except that the far right edge of the black portion is closer) The Question Now, the red key(and the trill keys) are about half the diameter of the other keys. I suspect that this is quite significant here, but I can't explain it myself. My main question is, why does disturbing the air column as shown in the section "the weird stuff" not change the pitch? One has added an escape route for air, the column should then vibrate as if the remaining keys were open--that is C#. I suspect that the underlying principle is the same, so I have a few other related questions (optional): Why does the red key not change the pitch on D/E flat, but makes it go into the second fundamental/harmonic for E and above? Why does the red key change the pitch to notes which are not harmonics, instead close to C#(one of them isn't even part of the chromatic scale--it is between two notes) for B flat and A? The red key is pretty similar to the trill keys with respect to size and general location. Yet, using a trill key on D changes the pitch, whereas using the red key doesn't. Why is this so?


Having been an avid lover of Mathematics, it is my dream to become a mathematician one day. I have been learning some "Advanced Mathematics" (Real Analysis and some Abstract Algebra mostly, and a little bit of Linear Algebra). The first thing that anyone jumping from High School Math to proof based, rigorous math, should notice is how different they are from each other. Math that I am learning now is definitely not like the Math that I now do at school. This is the fist thing I realized when I started doing Real Analysis a few months back. Advanced Math, I noticed, is exceptionally beautiful and at times it is like art, utterly elegant and aesthetically pleasing. I don't know if others share this same feeling with me. I used to enjoy Math then, for sure, but nothing compared to the enjoyment I am having now. Because of the enjoyment I get while doing Math, I am pretty much sure that I would like to major in math someday and if possible go to a graduate school in mathematics. But this being said, I do have a small concern. Because Higher Math is so very different from grade-school math, I fear that as I dive in deeper and deeper into mathematics, I might realize that the Math I do then has changed so much that it was nowhere close to the Math that appealed me. How substantial is this fear? Is it legitimate? Because this community is full of Mathematicians, I figured this would be the best place to ask If you guys experienced such a feeling too? How and in what ways did you find math different from fairly lower level math that I am currently into. Any help is much appreciated!