darklight03/StackMIAsub huggingface dataset:
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).