darklight03/StackMIAsub huggingface dataset:
I'm not much informed about manifold but I should answer some questions about it. Based on the definition I have written an answer for the following question but I feel there is something wrong with it! Could you please help me? Q: Let M be a smooth manifold and suppose that we have an open cover for that. If S is a subset of M such that the intersection of each element of that cover and S is a submanifold, then S itself is a submanifold. A: Fix an arbitrary point p in M. This point belongs to an element of that cover and as we know the intersection of that element with S is a submanifold, so there exists a map around p such that satisfies the condition, so we are done!
I hope it is appropriate to ask this type of question. I'm in my second year as an undergraduate right now. While my problem solving skills have improved tremendously, I almost never tried to actually remember what I learned. I always had the attitude: "If I don't remember it, I will simply look it up." While this might be legitimate to a certain extent (especially when it comes down to definitions), I'm not sure where this leads to in terms of being able to see various connections much faster. Would you say that it is necessary to remember most of the things that you learnt so far? Of course it wouldn't do any damage at all, but one has always think about it in terms of efficiency, I guess. I mean, it's hard enough to learn mathematics itself - but remembering all of those things just to gain the ability of solving some problems faster? I'm not sure about that and would like to hear your opinion on it.
Timothy Gowers asks What is so wrong with thinking of real numbers as infinite decimals? One of the early objectives of almost any university mathematics course is to teach people to stop thinking of the real numbers as infinite decimals and to regard them instead as elements of the unique complete ordered field, which can be shown to exist by means of Dedekind cuts, or Cauchy sequences of rationals. I would like to argue here that there is nothing wrong with thinking of them as infinite decimals: indeed, many of the traditional arguments of analysis become more intuitive when one does, even if they are less neat. Neatness is of course a great advantage, and I do not wish to suggest that universities should change the way they teach the real numbers. However, it is good to see how the conventional treatment is connected to, and grows out of, more `naive' ideas. and gives a short construction of the real numbers as infinite decimals, then using that to demonstrate the existence of square roots, and the intermediate value theorem. What are other reasons for or against thinking of real numbers as infinite decimals?
In the last assignment given to me by my professor, there is a question which asks: In classical theory, it says that in presence of air friction, a pendulum continuously loses energy with time and its amplitude also decreases with time. However according to quantum theory, the pendulum loses energy only in discrete packets or quanta. This would lead to a decrease in the pendulum's energy in a step-wise manner. How can you show that there is no contradiction between quantum theory and the observed behavior of laboratory pendulums and springs? What I want to know now is: What exactly do I have to show mathematically to prove this result? I mean what kind of an equation will I have to derive or prove in order to answer this question? What will be the final equation after writing which I can say that "So this concludes that there is no contradiction." I am completely clueless about what to do, right now. I don't want the complete answer or proof however. Thanks.
In a laser interferometry experiment, we project a pattern of interference fringes onto a CCD sensor. For best results, we want good contrast between the bright and dark fringes, and we carefully compensate for various sources of noise - for example, by taking camera images with no fringes present, and with the laser turned off, and subtracting these images in the proper sequence. We'd expect therefore that the remaining signal should be highly linear, with the CCD signal at each pixel in direct proportion to the number of photons reaching it during the shutter time. What we actually find is that, as we vary the laser intensity and shutter time such that the average intensity across the image remains constant, with no pixels saturated, there is a definite "sweet spot" where the fringes are much more well-defined than at other settings. Either increasing or decreasing the laser intensity away from this point (with corresponding decreases or increases in shutter time) causes the fringe definition to deteriorate. I can't think of any reason why this should be. I know that in cases where the process generating the pattern has a time-constant (for example, using laser speckle interferometry to measure Brownian motion), there is an optimum exposure setting, but that shouldn't be the case in our system which is entirely static. So, what am I missing? I assume it's some property of the CCD sensor that I've overlooked.
I was reading a physics book by some author and I got a little too confused with the explanation he stated about magnetic fields. A magnetic field is a field of force produced by current-carrying conductors or by permanent magnets. Correct me if I'm wrong: A magnetic field is basically a region / space where magnetic lines / flux lie. Thus, any magnetic objects inside the magnetic field will be attracted or repelled depending on its charges. Thus, this means that if a magnetic object is inside the magnetic field, the object should feel either a pulling or a pushing force acting on it. So, if the current-carrying conductors will produce a magnetic field, then this somehow means that I am inside a magnetic field produced by the current-carrying conductors (microchips and the wires inside the computer). And I believe I have enough metals around the computer, why don't my magnetic metals not experience any force acting on it?
First of all, I'm genuinely sorry if this question isn't "serious" enough for this forum! A common cliche in movies and tv is that a very tough object (eg the villain) is frozen, and then hit with something, shattering into a million pieces. I've seen a demo of a flower being put into liquid nitrogen, then being crumbled, but a flower is a very delicate object to start off with. If I take a leg of lamb (for example) out of the freezer, I don't feel like it's in any danger of shattering into a million bits (unlike my foot if i were to drop it). So, is the whole "cold = brittle" thing just movie bullcrap? Or is there anything to it? Sticking with the leg of lamb example: is there a temperature to which a leg of lamb could be dropped that would make the leg of lamb prone to shattering? EDIT - i just realised that the question title could be read as "Is there anything which is rendered extremely brittle by extreme cold?". Obviously there are some things, eg flowers. Hence the title change.
I'm sure his has a general form and welcome a link to a duplicate, but as I don't quite know what to search for, here goes. Whilst answering a question on the Mathematics Stack Exchange, I found myself needing to say the mid point of one diagonal of a rectangle is coincidental with the mid point of the other and that that point is the centre of the rectangle. So I said: "The key is to prove that for all rectangles, the mid points of the diagonals are coincidental at the centre of the rectangle ..." It suddenly struck me that the correct form might have been: "The key is to prove that for all rectangles, the mid point of the diagonals are coincidental at the centre of the rectangle ..." I suspect this is related to "everyone took off his hat". Could someone explain the reasoning behind the correct form.
Possible Duplicate: What is the correct way to pluralize an acronym? I am helping a former intern ready their resume for distribution. The candidate used an abbreviation I was unfamiliar with: B.S.s in Physics, Computer Science, and Mathematics I am familiar with the student's educational background so after a moment of thought realized "B.S.s" was their attempt to indicate multiple bachelor of science degrees. That said I can see those unfamiliar with the student's background being confused. Unfortunately, I do not know how to provide the correct punctuation to indicate the correct grammatical number for multiple degrees in the same discipline. What is the correct plural form for multiple bachelor degrees? Additionally, are there variations for other degree levels: associates, masters, Ph.D, etc.? In addition to the abbreviation what's the correct way to indicate plurality for the unabbreviated form?
I have only taken a basic quantum mechanics course (this book, so you know where I'm coming from), but I've been wondering about something. If we set up a quantum system in a known state and take a measurement of two incompatible observables, we will get two real numbers. If we repeat this experiment multiple times, then we will get two lists of real numbers (each list corresponding to the measurements of one of the observables). Quantum mechanics allows us to predict the average and standard deviation of these numbers, but it does not allow us to predict the individual numbers. If I understand correctly, this is a fundamental limit of the theory. The data is essentially random. Is it correct to say that most scientists believe that no theory will ever allow the prediction of these individual numbers? Why do they think that? And secondly, is there any other property of those numbers that quantum mechanics predicts that I am missing (other than mean and standard deviation)?
I'm teaching a geometry course this semester, involving mainly Euclidean geometry and introducing non-Euclidean geometry. In discussing the importance of deductive proof, I'd like to present some examples of statements that may appear to be true (perhaps based on a common student misconception or over-generalisation), but are not. The aim would be to reinforce one of the reasons given for studying deductive proof: to properly determine the validity of statements claimed to be true. Can anyone offer interesting examples of such statements? An example would be that the circumcentre of a triangle lies inside the triangle. This is true for triangles without any obtuse angles - which seems to be the standard student (mis)conception of a triangle. However, I don't think that this is a particularly good example because the misconception is fairly easily revealed, as would statements that hold only for isoceles or right-angled triangles. I'd really like to have some statements whose lack of general validity is quite hard to tease out, or has some subtlety behind it. Of course the phrase 'may appear to be true' is subjective. The example quoted should be understood as indicating the level of thinking of the relevant students. Thanks.
According to Wikipedia: For most radioactive nuclides, the half-life depends solely on nuclear properties and is essentially a constant. It is not affected by external factors such as temperature, pressure, chemical environment, or presence of a magnetic or electric field. Have any experiments been done to test what effect neutrinos have on decay rates? From what I understand, it might be difficult to isolate pretty much anything from the constant bombardment of neutrinos from the sun and other sources. So, how can we be sure decay rates are not effected in some way by neutrino collisions/interactions? Update "The overwhelming majority [of neutrinos in the beam] will continue on past the lab, to infinity." This is from this article which describes how a beam is generated and detected at CERN. I need help understanding this: if the "overwhelming majority" of neutrinos cannot be stopped and the actual detection is considered an "event," we can't possibly say that we know what decay rates would be in isolation since we can only stop an extremely tiny fraction of neutrinos from a beam. Neutrinos are naturally and constantly coming from all directions at all times. The best we have is test results of what happens when we beam extra neutrons at those substances (which I'd be interested in seeing the results of).
About two years ago I watched some old Monty Python interviews. In one of them, Graham Chapman, a Brit, makes fun of Terry Gilliam (the only American) for his lack of vocabulary. He specifically cited a moment when the group flew over the great lakes and Gilliam said "there's a bunch of water". I found this amusing. But it's also stuck with me. And ever since, every American I meet with seems to have an affinity for saying "a bunch" to describe anything with a high quantitative value. This can be anything from purely literal (a bunch of parsley) -- though this doesn't bother me as much -- to the generally "many", such as "a bunch of candies" and almost sarcastically as in Gilliam's case (obviously several thousand cubic miles of water is a bit more than a "bunch"). But what is a solid alternative for these uses? Not that I'm looking for something to replace "a bunch" entirely. I find it can be useful and an endearing "Americanism". But I'd like to hear of some options. EDIT: Maybe I can make this more specific and ask this: What would an appropriate British expression be for the Gilliam scenario above? Maybe something like "A Considerable Amount"?
In these forums and elsewhere it is routinely agreed that "we do not have a theory of quantum gravity." My question is, how do we know that canonical quantum gravity is "wrong"? I understand that the theory is perturbatively nonrenormalizable, but doesn't that just mean that we can't apply perturbation theory to it? It seems the theory is nonperturbatively renormalizable. A theory being nonperturbative doesn't make it "wrong". My understanding is that in modern QFT nonrenormalizability is not anymore considered such a big deal, and that even the Standard Model is expected to be an effective theory modified by nonrenormalizable terms appearing at higher energies. So barring practical considerations, why is canonical quantum gravity considered "not a theory of quantum gravity?" EDIT: I am not sure that the term "Canonical Quantum Gravity" is correct. I am referring to what is called "Quantum Einstein Gravity" in this paper. If someone knows better please help me correct my terminology. I was not meaning to imply anything about LQG because I thought that LQG "brought more to the table" than just a nonperturbative approach to the same QFT, but I could be wrong about that.
First let me introduce my terminology: A "Mechanical theory": A theory which describes time-evolution of a particle or a system of particles regardless of the fields affecting the particle/system. e.g. Classical mechanics, Quantum mechanics, etc. A "Field theory": A theory which describes time-evolution of a particle or a system of particles with taking the effect of fields on the particle/system into account and also a theory which describes time-evolution of the fields themselves. e.g. Classical gravitation, Classical electromagnetism, Quantum field theory, etc. But I am always confused that "General relativity" falls into which category. I know "Special relativity" just applies a modification to "Classical kinematics" to build "Relativistic mechanics". But "General relativity" is talking about "Relativistic gravitational fields" while also talking about "Non-inertial frames of reference". Now this is my question: Is "General relativity" a field theory or it's a mechanical theory? If it's a field theory is there any other way to study non-inertial frames in the context of "Relativistic mechanics" without bringing any special field into play?
I'm a high school student and I'm trying to understand the concept of the Higgs boson. So I apologize ahead of time for any incoherence I may say. As per my understanding bosons are force carrying particle that are excitations of their respective fields. For example the photon is the boson of the electromagnetic field and an excitation of such. As per my understanding the Higgs boson is an excitation of the Higgs field, and also the Higgs boson is what directly interacts with other massless particles (quarks, leptons, etc). So in the attempt of trying to visualize it. Could it be said that the Higgs boson is like a wave made in the surface of a pool? Since the boson is an excitation in the Higgs field and a wave is an excitation in the water. Would this be a proper analogy? Additionally, how do this excitations in the Higgs field actually occur? How do Higgs bosons appear?
I'm looking for an alternative way of saying "You can't run before you can walk." This is equivalent to saying "you can't take on higher level things before you have mastered the basics". I am looking for either a rewording of the original phrase or a whole new phrase with the same meaning. I prefer that this not have any fancy words nor leave the reader scratching her or his head. I will use it in a semi-informal tutorial for future students of a programming class I am in. I use a short phrase at the beginning of each section which establishes the underlying theme in that section. This tutorial is meant to be something fun (and useful) to read, which is why I want something that catches the attention of the reader. For my purposes, there is nothing wrong with the original. I simply want something that is a bit more inclusive (some people can't walk or run). Thank you.
I have a question related to already shortened words and their plural forms. As I have seen on this site and have found in the dictionary, words like mas and pas are the plural form of the shortened words ma and pa (as in mother and father). However, it occurs to me that were I to write this, in order to clarify the meaning of what I was trying to write (because if you are like me, you read mas and pas and did a double-take), I would write it as ma's and pa's. Some words like cuz have a similar issue, although the pluralization makes more sense: cuzzes. Perhaps cuz's? I know that in the pluralization of single letters there is some contention regarding whether there should or should not be an apostrophe (A's or a's versus As or as). "I got a lot of A's this semester." just looks better to me, although I have no grammatical backing for this. I'm not sure if this is the same for these types of words as well. My question is this: Is there a precedent for using an apostrophe for words that are shortened to convey appropriate meaning, or is the convention simply to omit the apostrophe altogether and leave the word pluralized with an s?
I'd like to know what franchise meas as a verb in the following sentence: Catering in this school has been franchised to the company. The native speakers I consulted, both American, don't seem very sure of its meaning. I know the verb typically means (of a large company like McDonald's) to permit an individual or a smaller company to sell products or offer services in its name. But in the example above, I can hardly imagine a large "catering company" sold the right to sell its products in a school to a smaller company. Perhaps this is because I haven't heard of catering companies in connection with franchise agreements in my country. Could it be that it means the authority of the school gave permission to a company to provide catering? Could 'outsource' be used in this context to replace 'franchise'?
As an applied mathematics student coming from a small university, I have not had an adequate course in writing/formulating proofs for problems in advanced calculus/real analysis (my university has an advanced calculus course, not a real analysis course). In the fall I will take the first of a two part course in advanced calculus. I believe we will use Fitzpatrick's book, Advanced Calculus. So, in order to prepare for the rigor and proof writing that will be necessary in this course, is there a good book or pdf that will provide solutions to problems involving proofs? I've found plenty of books with tons of problems but finding solutions to check myself (or see if there is a more clever approach) has been difficult. I hope this isn't too broad of a question, maybe some of you coming from smaller universities will understand my dilemma.
I'm trying to translate this text to Polish and everything seems pretty clear to me, apart from the usage of the words "within" and "without". I presume it's some kind of technical vocabulary referring to the subject of jousting. Could someone explain to me what these words mean? Here are three excerpts from the text: At those jousts the noble ladies and damsels will give the knight who jousts best of those without a horn garnished with gold, and they will give to the one who jousts best of those within a white greyhound with a collar of gold around its neck. And the noble ladies will give a circlet of gold to the one who jousts best of those without. And one within that jousts best will be given a golden belt. And there will be given in the same field to whoever jousts best of those without a noble courser, saddled and bridled. And whoever jousts best of those within will be given a fine chaplet well worked with silk.
"Crack The Whip" is a game played on ice skating rinks where several individuals line up all facing the opposite end of the rink, and skate forward. When the group reaches the opposite end of the rink, the "point man" or the person on one end of the line stops, and everyone else pivots around him. Supposedly, the person on the opposite end swings around, and rockets forward at much greater speeds. However, I don't believe this is possible. Because the question remains, where would the extra speed come from? I was told that the momentum of all the people is transferred into the one person on the end. While there is a ponderous amount of outward force on the chain of people, I don't think it's possible to actually gain speed. And after watching my friends do it several times, I concluded the person on the end didn't seem to go any faster. Was it simply that the process was inefficient? Or is the theory even possible?
Even though I don't work at Google, the Google Styleguides have been very helpful for me in adopting consistent, readable style conventions for my code. Unfortunately, there is no Google Styleguide for LaTeX. Q: For LaTeX, does anyone know of something equivalent to the Google Styleguides? Quick searches of the web for LaTeX styleguides have returned plenty of styleguides emphasizing how compiled LaTeX documents should appear. I couldn't find anything emphasizing how the uncompiled .tex document should appear. I suppose if there weren't anything equivalent to the Google Styleguides, a standard, very cleanly written and commented .tex template would suffice. UPDATE: Over at StackOverflow, I've found a similar post asking about Ruby coding style guidelines. There are a number of helpful links provided there. I'm looking for something kind of similar.
I'm trying to figure out a kind of taxonomy of quantum phenomena. So far the categories I've come up with are (in historical order, with example phenomena): discrete quantities where continuous ones are expected (black-body spectrum, atomic spectra, Stern-Gerlach [spin spectra]), interference phenomena where trajectories are expected (double slit experiment), and long distance entanglement correlations (Bell's inequality phenomena). Am I missing anything major from this list? Did I incorrectly group something that should be in its own class? Can any of these classes be merged? As requested, how I would define classical phenomena: anything that behaves according to classical mechanics - objects that have (approximately enough) definite position and momentum that obey Newton's laws (even the relativistic versions), and waves that have infinitely variable amplitude, even when boundary conditions limit the frequency spectrum. Basically, I'm trying to get a clear picture of what the boundary between classical and quantum phenomena looks like.