darklight03/StackMIAsub huggingface dataset:
British English often employs mass nouns where American English would only employ count nouns. Count nouns are nouns which take pluralization and numerical quantifiers like 'many'. Mass nouns can't be pluralized and take volume quantifiers like 'a bit of'. For example: I like sport. I like drink. I pay tax. I've got toothache/earache/stomachache/backache. I eat mashed potato/scrambled egg. I play with lego. In American English, we would render these: I like sports. I like to drink / I like to have drinks. I pay taxes. I've got a toothache/earache/stomachache/backache. I eat mashed potatoes/scrambled eggs. I play with legos. These facts are documented here, here and here. But this difference is not mentioned on the Wikipedia page comparing British English to American English. My question is simply what other nouns which are standardly count in AmE are often mass in BrE? Further, are there any broad categories of nouns (for example, food nouns) that tend to be count in AmE but tend to be mass in BrE? Or does the difference only exist for particular and isolated lexical items? (The first link I provided says that many foods are not conventionally massified in BrE, for example 'refried bean'). Here are two final caveats: First, I know that any count noun can be turned into a mass noun by simply putting it in the syntactic position of a mass noun. That's why "I ate (a bit of) chicken" is acceptable in both AmE and BrE. But I am asking about the obvious differences between AmE and BrE regards their conventional uses of nouns as mass and count. Second, I know these types of list questions are hard for single users to answer, and sometimes generate controversy on EL&U (for example, this one on kennings generated some controvery). Still, I'll upvote any responses that make a novel contribution.
An elementary fact that people learn about mirrors is the law of reflection, that the angle of incidence of a light beam striking the mirror (as measured with respect to a normal) equals the angle of reflection. Does this law also hold for a mirror that is moving? Consider a square mirror that is moving at speed v in a direction perpendicular to the mirror. (You can think of the mirror as starting in the xy-plane and moving in the positive z direction of a Cartesian coordinate system.) As the mirror approaches a certain observation point, another person shines a laser beam of frequency w at the mirror so that the beam makes an angle A with the normal to the mirror. (You can think of the beam as lying in the yz-plane.) What angle and frequency will you measure for the reflected light beam? Does the law of reflection still hold? Do your conclusions change if the mirror moves parallel, rather than perpendicular, to its plane (say in the y direction if it starts in the xy-plane)? Note: The large mirror of the Hubble space telescope is an example of a mirror in motion as it orbits the earth. From your analysis, do you think the users of the Hubble have to take into account the motion of the mirror when measuring properties of its images? I know that this question could be according to the .SE standards be too broad. But I request you not to close this question, I would really appreciate people if this question is answered rather than flagged or closed.
"You do research in mathematics! Can you explain your research to me?" If you're a research mathematician, and you have any contact with people outside of the mathematics community, I'm sure you've been asked this question many times. For years now, I've struggled to find a satisfying answer. I think an ideal answer to this question should: be accessible to someone who hasn't studied math since high school build intrigue and wonder honestly, albeit vaguely reflect your research only require a few sentences (Of course, these guidelines will change depending on the audience and venue. For example, speaking with an engineer over a meal allows more time and technical language than would speaking with a stranger on a bus.) I study the representation theory of algebraic groups and Lie algebras over fields of positive characteristic, so I usually say something along the following lines: I work with two algebraic objects that are closely related called algebraic groups and Lie algebras. These objects can act on spaces (like three-dimensional space) by transforming them in a nice way, and I study these actions. One aspect of my work that is especially challenging is that I use number systems in which a chosen prime number is equal to zero. Honestly, based on my guidelines above, I think this response is poor, but with so much to communicate in such limited terms with such limited time, the task seems nearly impossible. Using my guidelines, how would you describe your own field of research? Or, if my guidelines are too strict, how would you deal with this question?
So I am reading the essay "On Some Verses of Virgil" by Michel de Montaigne (translation by Donald M. Frame) and I came across this particular sentence. Is there any ugliness in doing wrong that can dispense us from the duty of confessing it? The choice of vocabulary here is, honestly, quite confusing to me. Here is the sentence again with the preceding paragraph for context: The diseases of the body become clearer as they increase. We find that what we were calling a cold or a sprain is the gout. The diseases of the soul grow more obscure as they grow stronger; the sickest man is least sensible of them. That is why they must be handled often in the light of day, with a pitiless hand, be opened up and torn from the hollow of our breast. As in the matter of good deeds, so in the matter of evil deeds, mere confession is sometimes reparation. Is there any ugliness in doing wrong that can dispense us from the duty of confessing it? It's been a long time sense I've had this much difficulty understanding the exact conclusion a sentence was trying to reach. My first instinct is to assume that the sentence is somewhat synonymous with: Is there any ugliness in doing wrong that can [spare] us from the [ordeal] of confessing it? But then a friend of mine offered an interpretation more to the effect of: Is there any [wrongdoing so exceptional] that [it will] [render our confession truly unprecedented in the context of humanity]? Meanwhile, another translation (found on Project Gutenberg) renders the sentence thus: Is there any deformity in doing amiss, that can excuse us from confessing ourselves? I don't have much confidence in either his or my interpretation, and this other translation isn't clearing things up for me much at all. What is Michel de Montaigne saying?
I am writing a formal document. It is a petition to appeal an administrative law decision. The decision is jam-packed full of sloppy mistakes of every kind imaginable. (The guy who wrote it couldn't even manage to spell the name of my town correctly in the title of the document.) I list all the errors and inaccuracies in my appeal petition, with specific references to pages in the hearing transcript and exhibits. But I want to preface the list with something that means the following: A number of the problems noted might seem rather insignificant, but I will list them all, to show the review officer how incredibly sloppy the decision is. OR: what a slapdash job the hearing officer did in analyzing the case and writing up the decision. Only I can't say "incredibly sloppy", I need more formal language. I need a more subtle way of saying sloppy. I don't mind making adjustments in my sample sentence to accommodate a different part of speech or whatever. Note, the following question is somewhat related but doesn't. A word for not paying attention to detail, causing sloppiness. One idea: A number of the problems noted might seem rather insignificant, but I will list them all, to show the review officer the shocking lack of care taken in analyzing the case and documenting the decision. Another idea: The hearing officer's slapdash [or: haphazard] analysis and write-up resulted in erroneous references and errors of fact that are too numerous to list in the space allowed. I will only be able to provide corrections for a portion.
I have used classical Lagrangian mechanics for quite a while, and what I like about it is that everything can be derived from a very small number of geometric principles. There are just three things you need to "take on faith": That configuration space should be endowed with a Riemannian metric given by mass; That forces arise from scalar potentials on configuration space; Hamilton's principle of extremal action. The last is perhaps most objectionable: why should the universe want to evolve in a way that minimizes the action? I have no idea, and my understanding is that nobody else does either. But if I assume these hypotheses, everything else follows mechanically: equations of motion, Noether's theorem, etc. What I would like to do is to learn quantum mechanics, starting from similar geometric principles. For example, I might take as a key principle that configurations should be probability distributions over configuration space, rather than single points, but this doesn't lead to quantum mechanics; I also need (for some reason?) to instead take configurations to be complex-valued functions over configuration space. But even then I don't see how to get anything like Schroedinger's equation from a (modification of) Hamilton's principle. I did some searching and I've found that there is indeed some variational foundation for quantum mechanics (and Feynman seemed to have championed this formulation) but the articles I've seen so far assume I already understand quantum mechanics. My question: Is there a good reference that builds quantum mechanics from scratch, based on variational and geometric principles? If it's easier to instead start with a more general theory (e.g. quantum field theory) let me know as well.
For instance, "A politician must be able to think quickly on the spot. He or she must also have no qualms about lying." I know some people who use "they", but as that both sounds and is ungrammatical, I'm wondering if there is any other concise and non-awkward alternative. Edit: Apparently many do consider "they" as correct. This surprised me, because I know someone who actually wrote a grammar book, who told me in no uncertain terms that "they" could not be used (in fact, I'd asked them - ha! - this question before I posted it here). And a quick look online found legitimate sources advocating both positions. I suppose it's one of those issues, such as whether the previous sentence was incorrect for beginning with a conjunction, where it really just depends who you ask. Regardless, to me personally, "they" in the above example sounds clunky at best and incorrect at worst. So I suppose my question can be rephrased as, 'is there any alternative to 'he or she', other than 'they'?" Secondly, to whoever tagged this as "politically-correct": acknowledging the existence of genderqueer or transgender people isn't a political issue at all, until those made uncomfortable by people unlike them make it one. Trying to avoid excluding and thus unconsciously discriminating against a group of people in your writing isn't born out of a need to be 'politically correct' - it's born out of basic human decency. Finally, yes, in the example above, I could use a comma and avoid the need for a pronoun at all - but the question I'm asking applies to instances where one may not want to do so (i.e. the sentences are long and stringy enough as it is).
I'm writing some copy for a marketing campaign that promotes the use of a software testing tool by demonstrating how the results of the tool provide you with a greater wealth of information on which to make business decisions. In relation to this, I'm using the term "informed decisions" and my initial wording included the phrase "help you make more informed decisions". The phrase "more informed decisions" is one that is common to me and from a Google search appears to be common in the context in which I am using it, that is to denote that with additional evidence a decision can be more informed than without. I considered whether this phrase runs the risk of being misinterpreted or being confusing due to a slight ambiguity. The phrase "more informed decisions" could be interpreted as meaning decisions that are more greatly informed, or it could be interpreted as a greater number of informed decisions (e.g. here is one informed decision, here are more informed decisions). I then considered the phrase "better informed decisions". I feel this has the same intended meaning but removes the above ambiguity. It could refer to decisions that are better informed or it could refer to informed decisions that are better. Both are on subject and would be fine to me. A quick Google search suggested that the phrase "more informed decisions" is significantly more commonly used than "better informed decisions". I have a feeling that "better informed" is more grammatically-correct than "more informed" although I have no formal basis for this. Out of the phases "more informed decisions" and "better informed decisions", is there a more correct form?
I was reading this interpretation from this site, where these lines are noteworthy enough to talk for the fact that this interpretation doesn't actually talk about many-worlds: These are the "many worlds" in question, although it should be clear that the label is somewhat misleading. People sometimes raise the objection to the many-worlds interpretation that it's simply too extravagant to be taken seriously--all those different "parallel realities," infinite in number, just so that we don't have to believe in wave function collapse. That's silly. Before we made an observation, the universe was described by a single wave function, which assigned a particular amplitude to every possible observational outcome; after the observation, the universe is described by a single wave function, which assigns a particular amplitude to every possible observational outcome. Before and after, the wave function of the universe is just a particular point in the space of states describing the universe, and that space of states didn't get any bigger or smaller. No new "worlds" have really been created; the wave function still contains the same amount of information (after all, in this interpretation its evolution is reversible). It has simply evolved in such a way that there are now a greater number of distinct subsets of the wave function describing individual conscious beings such as ourselves. The many-worlds interpretation of quantum mechanics may or may not be right; but to object to it on the grounds that "Gee, that's a lot of worlds," is wrong-headed. I'm not really understanding the reason "It has simply evolved in such a way that there are now a greater number of distinct subsets of the wave function describing individual conscious beings such as ourselves." What is it saying? Can anyone explain me?
Firstly I'm not saying that I don't believe in Cantor's diagonalization arguments, I know that there is a deficiency in my knowledge so I'm asking this question to patch those gaps in my understanding. From my understanding of Cantor's Diagonalization argument, if you apply diagonalization to a mapping from one set of numbers to another, you will always obtain a number that is not in the mapping. So this works to prove that the reals aren't countable because if you have a mapping from the naturals to the reals then you can use diagonalization to obtain a number that's not in the mapping, and this number is a real obviously, so the mapping isn't a surjection. We're not allowed to assume that the mapping from the naturals to the reals is a bijection to begin with. But when people explain why the diagonalization process doesn't produce a rational from a mapping from naturals to rationals we are allowed to assume that the mapping is a bijection to begin with? In the questions asked here: Why does Cantor's diagonal argument not work for rational numbers? The answers says: To be precise, the procedure does not let you guarantee that the number you obtain has a periodic decimal expansion (that is, that it is a rational number), and so you are unable to show that the "diagonal number" is a rational that was not in the original list. In fact, if your original list is given explicitly by some bijection, then one is able to show just as explicitly that the number you obtain is not a rational. Why are we allowed to assume that the original list is a bijection? Is there some way to prove that the mapping from the naturals to the rationals is a bijection that is not susceptible to diagonalization? If we can assume that the mapping from naturals to rationals is an undiagonalizable bijection why can't we do the same for the mapping from naturals to reals?
According to path integral theory, it can be assumed that a particle travelling from a point A to B (in an experiment that has not been designed to detect the path) takes all the possible paths from A to B. (Please note that I use the words 'can be assumed' because I know that there is a debate whether the particle really takes all the paths that are considered in path integral or not.) Now, energy of a particle is closely related to its momentum in quantum mechanics. So, in a double slit experiment, we can setup the detector at the interference screen so that it not only detects the position of the individual particles, but also their energy. This will not violate uncertainty principle because we will still not have the knowledge of the direction of the velocity of the particle, but only the magnitude of energy. Now, being able to detect the energy of particles in a double slit experiment will be interesting because then we will be able to find out whether the energy of the detected particle is equal to that of the input particle or not. If it is equal, then it would mean that the energy is conserved during the path. Yet, according to the path integral, the particle can be assumed to take all the possible paths. Since we know the initial and final energy of the particle, this means that we can only consider those paths in path integral where the energy fluctuates only in the middle of the path! So, is it theoretically possible to detect the energy of the individual particles at the detectors in the double slit experiment. If yes, then theoretically, is the energy conserved at the start and the end of the path. And most importantly, do you know of any experiment that has been done to detect the energy of the individual particles at the interference screen. Please give me the links to the paper discussing this.
I was looking up how to find relationships between Sobolev spaces and I came across this post on MO in which the first comment talks about a scaling procedure for understanding the relationships: To find the right values of k,p,r,a, I was taught to use scaling arguments. Take a nice enough singularity for the function inside the domain, and make it worse and worse, and compare how the norms change. This usually lets you solve for the appropriate parameter (and if it doesn't it tells you there is something special about that embedding). For unbounded domains you can do the same thing, and this often shows why certain embeddings cannot exist (handling singularities both inside and at infinity is hard). Other answers also mentioned this scaling approach, which I have never heard of, with one comment saying: Comparing the behavior of the different Sobolev norms for a compactly supported family of functions converging to a Dirac delta function is indeed the simplest way to figure out these inclusions. In general, understanding how things behave under rescalings is extremely useful but, as far as I know, rarely mentioned in print. It seems that you have to learn about it by word of mouth or stumble onto it yourself. Physicists and chemists also use this often and call it "unit analysis". So does anyone here know of this scaling approach? If so could you give a worked example for some specific function, so somebody (like myself) who is new to Sobolev spaces can learn how to apply it and gain a better understanding of these spaces?
In all textbooks the magnetic field around a wire carrying a current is found from Ampere's Law. However, I would like to know what the expression for the magnetic field is, using the full time-dependent Maxwell's equations (and how to get there): Hopefully this will give me some insight into how the field is set up via EM waves. Note that I am specifically interested in how the field according to Ampere's Law is actually set up in a time-dependent fashion via EM waves. For simplicity let's say I'm interested in the magnetic field created by a wire attached to the electrodes of a battery. When you turn the battery on, the magnetic field around the wire is rapidly created. How can I get a time-dependent solution for this? Feel free to suggest how the battery could be treated, how the electrons in the wire respond etc. (I suppose all of these things must be part of the model). Also feel free to treat a simplified problem, eg. we could do the equivalent problem for the magnetic field around an infinite slab of material if that makes the geometry easier. My primary interest is in how the field is propagated into the vacuum around the wire - I would like to see how the solution eventually tends toward Ampere's Law. EDIT: I will try to make the question easier to understand: We are often taught the magnetic field around a wire arises from Ampere's Law. This applies to a steady state situation only - i.e. it does not occur instantaneously. It does not tell you how the signal (whether that be voltage or current) originating in the wire actually gets out to the vacuum around the wire, as the current starts at zero and ramps up to some constant value. Presumably this happens via the generation of EM waves, because of course Ampere's Law is not valid on very short timescales. I would like to know what the solution is for the magnetic field around a wire when you are not in the steady state, which should reveal how the magnetic field propagates into the vacuum. Obviously the solution would have to correspond to Ampere's Law in the long time limit. I do not mind so much what device you use to get the current flowing in the wire in the first place. For example if you want to use a battery connected by a switch to the wire then fine. Or an AC generator slowly ramped up with some assumed time-profile would also be fine. In other words, if you would like to answer this question, you could choose whatever model you think best describes the scenario in which a wire is connected to a battery by a switch and the switch is turned on (or, as I mentioned some other, similar process). The wire can be any shape you choose. I just choose a "wire" because it's the most commonly used simple conductor, but if you would prefer to use for example a slab of conductor, that is fine, because my interest is in how the time-dependent fields converge to Ampere's Law.