darklight03/StackMIAsub huggingface dataset:
The following paragraph has been extracted from the Wikipedia (Atomic orbitals): Simple pictures showing orbital shapes are intended to describe the angular forms of regions in space where the electrons occupying the orbital are likely to be found. The diagrams cannot, however, show the entire region where an electron can be found, since according to quantum mechanics there is a non-zero probability of finding the electron (almost) anywhere in space. Is the statement by Wikipedia correct? Since, there is a probability of finding electron at any distance from the nucleus, when the electron comes far from the nucleus, I will block it, so that it won't return to its parent atom. Am I not stealing the electron? I can steal even the electron of your body being in India, be careful! That's what we layman think from those statements. What's the actual meaning of the wikipedia statement?
I was watching "Cold Opening: Homeland Security - Saturday Night Live". I am supposed to translate the entire sketch for my next classes, but I really don't know what is the joke here. I only know that magenta is a color and that is all. Can someone explain to me what is funny about this? I really need that. Here is a part of the transcript where the joke is used: "Before we begin today's briefing, I wish to announce that, on the basis of change in the nature of Al-Qaeda chatter, we are changing the current threat level to Magenta. Let me repeat: the threat level is now.. Magenta. What is Magenta? It's a darker maroon. It's not quite an ox blood. It's more plum color than.. say.. a crimson. How serious is it? [ sighs ] I honestly don't have an answer for that."
This question is sort of in the spirit of this xkcd: The light we get from stars was emitted many years in the past, but the distances to stars which are bright enough to be visible to the naked eye are not that great, so the light we received likely wasn't emitted long enough ago that the stars would have undertaken significant changes. On the other hand, some bright stars are red giants, which are very bright, very far away, and pretty close to the end of their lives, so there is a higher chance that they have collapsed in the meantime. So: what numerical fraction of stars which are visible by naked eye are likely to have undertaken significant steps in their stellar evolution? Here I'm interested both in main-sequence stars evolving into red giants, giants undergoing collapse, and similar events. Similarly, how does this answer change if you increase the range to stars that are visible using a reasonable pair of binoculars? In case special relativistic effects are important, for the purposes of this thread, both the current frame of reference of the solar system and the rest frame of the galaxy are interesting.
This popular question about "whether an AC circuit with one end grounded to Earth and the other end grounded to Mars would work (ignoring resistance/inductance of the wire)" was recently asked on the Electronics SE. (Picture edited from the one in the above link) Though I respect the AC/DC experts there, I think (with the exception of the top answer) they are all wrong. My issue is that they all assume that AC requires a complete circuit in order to function. However, my understanding is that a complete circuit is necessary for DC, but not AC. My intuitive understanding is that AC is similar to two gas-filled rooms with a pump between them - the pump couldn't indefinitely pump gas from one room to another without a complete circuit (DC), but it could pump the gas back and forth indefinitely (AC). In the latter case, not having a complete circuit just offers more resistance to the pump (with smaller rooms causing a larger resistance). Is my understanding correct - can AC circuits really function without a complete loop? More importantly, what are the equations that govern this? If larger isolated conductors really offer less AC-resistance than smaller AC conductors, how is this resistance computed/quantified? Would its "cause" be considered inductance, or something else?
I'd like to learn formal math. Preferably, though not necessarily, starting with predicate logic/first order logic rather than higher order logic. I am trying to find resources (papers, books etc.) for doing this, but I haven't found anything I really like. There are lots of resources for predicate and first order logic, but most do not approach the topics in a very formal way. For example, many text don't seem to try to define what they mean by "variables" or mention substitution as an important concept. Tries to explicitly describe as many of the rules of the game as it can. Many texts bring up "truth tables" without having formal rules for what you're allowed to do with those tables. Does anyone have resources that fit these criteria? Edit: many of the answers are good and helpful, but I feel like I should add some clarifying remarks: Many texts mention that you can view math as merely manipulation of symbols. I don't doubt that this can be done, but I would like to see it done. A resource that explains the process of producing proofs explicitly in terms of manipulating symbols rather than in terms of functions, statements etc (at least without first defining these terms) would be helpful. I'd like to be able to pretend I was a person who didn't know any math and was just acting as a human computer, producing proofs. I'd like a resource that explains producing proofs like I was such a computer (not necessarily ONLY like that).
I understand that photons, even when traveling at the speed of light, cannot escape the event horizon of a black hole. Are gravitons and other virtual particles traveling at the speed of light also confined by event horizons? If so, it seems that the gravitational field created by the black hole would result only from the mass of the black hole beyond the event horizon, where gravitons are capable of escaping. As a result, would there would be a disparity between the apparent mass of the black hole due to its gravitational field on other celestial bodies and the total amount of matter contained within the black hole? Also, I was reading this question: Nature of gravity: gravitons, curvature of space-time or both?, which suggests that gravitons and curved space may be indistinguishable. However, if gravitons are bound by the event horizon it seems that a black hole would act differently based on whether gravity results from gravitons or curved space-time. The existence of bound gravitons would negate the gravitational field of mass within the event horizon, resulting in a significantly lower gravitational field outside of the black hole. Would this occur, or am I neglecting some effect of relativity upon the gravitational field?
I was teaching my young nephew some math the other day, and from discussing the typical sort of word problems he's encountering in class, I noticed that the "-th" suffix adds a distinct meaning to adjectives. For example: If a ship is long, it has length. If a woman is wide, she has width. If a person is strong, he possesses strength. If what I say is true, I'm speaking truth. A lumbering panda moving slow is full of sloth. Now, I've learned some linguistics from English L&U, and I'm guessing this "-th" suffix is an affix that changes adjectives into nouns. My questions are: What exactly is this "-th" suffix adding to the meaning? Secondly, does the "-th" originate from a separate word in Old English? Lastly, is there something to say about the vowel shifts that seems to be occurring in some of the transformations (e.g., strong going to strength) that somehow fits in with the ablaut system of strong verbs/weak verbs, that I learned of from the excellent responses to my previous question?
Consider a lone photon. As its frequency increases, its energy increases. Taken to the limit, a sufficiently-high-frequency photon could be a black hole unto itself. But the frequency of a photon is dependent on the inertial frame of the observer. Two observers could each observe this photon to be either above or below this critical frequency. Or, I could accelerate to "catch up" to this photon, red-shifting it until it is no longer energetic enough to be a black hole. So couldn't I at one moment observe a particle to disappear beyond the event horizon, accelerate until the event horizon no longer exists, and hence observe what happened to the particle after crossing that threshold? Is this in-principle possible? If not, why not? EDIT: to clarify, I am not asking how much energy-due-to-photon-momentum is required to create a black hole, I am asking: given that threshold energy, how does the event horizon appear to different inertial frames which observe the photon to be above/below this threshold?
Does an object possess specificity to or for another object? Every time I go to express this concept in writing, I struggle over which preposition is the more appropriate and more precise. This is dilemma is encountered all the time in technical scientific writing, for example in biology where one speaks of enzymes with specificity to/for a particular substrate. I can't find much of a consensus there: I'm just as likely to see one form as the other in articles and published papers. My trusty Google consensus search isn't of much help either: the phrase "specificity to" occurs just about as often as the phrase "specificity for" does. Although all of the examples I can think of at the time are biological in nature, I'm certain that there are others, so I'm asking this as a more general grammatical usage question. Do they mean something subtly different? Should one form always be preferred over the other? Of course, it may be that either is entirely correct. Given that preposition usage is highly idiomatic in all languages (and English especially), there may not be a rule that definitively resolves this question.
It is fine to say that for an object flying past a massive object, the spacetime is curved by the massive object, and so the object flying past follows the curved path of the geodesic, so it "appears" to be experiencing gravitational acceleration. Do we also say along with it, that the object flying past in reality exeriences NO attraction force towards the massive object? Is it just following the spacetime geodesic curve while experiencing NO attractive force? Now come to the other issue: Supposing two objects are at rest relative to each other, ie they are not following any spacetime geodesic. Then why will they experience gravitational attraction towards each other? E.g. why will an apple fall to earth? Why won't it sit there in its original position high above the earth? How does the curvature of spacetime cause it to experience an attraction force towards the earth, and why would we need to exert a force in reverse direction to prevent it from falling? How does the curvature of spacetime cause this? When the apple was detatched from the branch of the tree, it was stationary, so it did not have to follow any geodesic curve. So we cannot just say that it fell to earth because its geodesic curve passed through the earth. Why did the spacetime curvature cause it to start moving in the first place?
Which is correct? There are no comments. There is no comment. Which would you use for a web application, i.e. what to display when a blog post or an article has no comment attached? Actually, I am trying to fix an application that says: "There is no comments"! Would that ever be right? More generally speaking, it feels wrong to have a plural after the negative no/none or with the preposition without (see my previous question "Without reason" or "Without reasons"?). Those words imply zero, i.e. less than one, while plural is two or more. Yet, I know that phrases like "There are no comments" or "He is without friends" are common. It seems illogical to me. Are the majority of people making a grammar mistake when using such expressions, or else can you explain why this is correct?
The word "complete" seems to be used in several distinct ways. Perhaps my confusion is as much linguistic as mathematical? A basis, by definition, spans the space; some books call this "complete" -- though then the phrase "complete basis" is redundant. In physics/engineering, "complete" seems to be reserved for orthogonal/orthonormal bases -- which necessarily means not merely a vector space, but specifically an inner product space. A complete basis in this QM sense does more than merely span the space: the concept of orthogonality allows for Parseval's relation, non-overlapping projections, Gram-Schmidt, etc. Is it even possible to have a complete basis (in this QM sense) that is NOT orthogonal? Though complete in the sense of Hilbert space and Cauchy sequences seems to be a different use of the term, the convergence of sequences within the space seems not so far afield, conceptually, from Parseval. So is it really so different?
I was recently asked the question "How do you know when you've become a better mathematician/better at mathematics?," and I realized that at that moment I did not have a valid answer, since I have been using my performance on tests to make that judgement. After putting some thought into it, I would say that one could at the very least use the following criteria: Learning material of similar difficulty at a faster pace with the same level of retention and understanding. Ability to bring together a larger number of theorems/lemmas/etc to use in showing a result (as opposed to doing problems that almost follow directly from the theorems). While those might encompass a good amount of information that would indicate whether you have been improving, I know that at my level there is still a lot more for me to experience and learn. So I would like to pose this question to everyone here, how do you know when you have become better at mathematics?
If I have a normal distribution, the posterior for the variance is the inverse Chi-square distribution assuming the same is used as a conjugate prior. But what if my data has extra noise added so that the observed sample variance is the sum of the population variance and my extra noise variance? But then the poisterior for the variance is different. Is there a name for that distribution? You can't just subtract the noise term because you can end up with negative values. It is similar to the Skellam distribution of the difference of two Poisson variables in this way. I am really interested in this from a Gibbs sampler point of view. I would like to draw the variance from the conditional posterior if possible. If that isn't easy I can fall back on Metropolis Hastings, I suppose.
I own a company called Find My Bus Ltd that brands itself as Find My Bus. Yesterday we sat down and had a discussion regarding the name and all came to the agreement that it didn't represent the company as we originally wanted it to. Seeing three separate words in a name that when thought of without any context sounds quite funky, we decided it would be better to merge the words into one name. We aren't the first to do this, in fact it is becoming a trend to merge words into one name. For example: DigitalOcean, StatusCake. In my opinion both of those names look fine because they consist of two words. However, when you do this with Find My Bus you see: FindMyBus. Is it just me, or does it look wrong having three words merged and capitalized? Would it be better to use Findmybus or perhaps FindmyBus? Apologies if this seems like I'm running a poll, I'm not, I just simply would like to ask users with experience in the English language which looks the most appropriate for a company name.
Quantum numbers are supposed to denote every individual orbital. But if orbital shells are probability functions, then orbitals can't be definite, solid things. So in that case, there can be variation in the amount of energy given off when an electron drops between shells - it might, say, give off a tiny little bit more energy and drop to just below the orbital shell. Isn't this possible since orbitals are just probability functions - like "Here's where the electron probably is"? Not entirely sure where I was going with this, but I think the final question is, how come quantum numbers are only ever integers? Edit: My question is about why quantum numbers as taught in schools are always integers. "Orbitals" as predicted by the Bohr model are in fact clouds of electrons, probability functions about where an electron probably is rather than a definite statement about where it definitely is. That means there's got to be wiggle room about how far an electron can be from the nucleus. So does that mean that quantum numbers are an oversimplification, or just averages? Or am I just misunderstanding the whole "orbitals are just probability clouds" thing? Edit: Ugh. Right. I'm an idiot. I forgot to mention that I'm only talking about the principal quantum number, n, the one telling which orbital the electron's in.
How do issues of naturalness arise when regularizing QFT using dimensional regularization? I can only recall ever seeing naturalness arguments (hierarchy problem, cosmological constant problem, etc.) phrased in terms regularizing with a cutoff, where naturalness issues arise when physical quantities are quadratically divergent in the cutoff scale. Is it hard to see how the same naturalness issues are addressed using dimensional regularization? Are there some hidden assumptions involved in using dimensional regularization? Do you reach the same conclusions as you do using a cutoff, but only after also using the RG equations? I recall being told that when dimensional regularization is used to remove power law divergences there is additionally some optimistic assumption being made about the UV physics, but I don't know if that's correct or relevant to this problem.
Well let's start off with that I'm not a physicist but I'd like some thoughts on something I came across in my hometown. This guy: Is it possible that due to the electrical charge of magnets this guy can make the illusion that he can float ? Or is this probably a cheap trick that fools the eye ? I was standing there for quite some time watching the guy and he keep moving his feet. The resistance that he appeared to have was from a magnet force keeping him afloat. So after I passed this guy I did some physics searches on the web and the first thing that caught my eye was the electrical charge of magnets. So the question is : Is this related to the electrical charge of a magnet or a cheap trick ?
I am trying to understand the meanings of "covariant transformation" and "contravariant transformation" and how they are related. I have read the related Wikipedia article and still feel I cannot state, with mathematical precision, the definition of these terms. The Wikipedia article states that a covariant transformation, in the context of a vector space, is one that "describes new basis vectors in terms of old basis vectors". This is not a satisfactory definition unless, of course, no other transformations can be described as "covariant". I have seen however the word "covariant" being used to describe other sorts of transformations as "covariant". Namely, the "physicists" definition of co/contravariant transformations where components transform as such-and-such (which makes absolutely zero mathematical sense to me). This leads one to believe that co/contravariant transformations are always defined in terms of derivatives of coordinate changes and I don't believe this is the case. I understand what co/contra-variant tensors are, at least from a mathematical perspective, so this is not a question about the meanings of "contravariant tensor" or "covariant tensor"; indeed, These concepts have been well-explained here. My question then, in summary, What are lucid, self-contained and mathematically precise definitions of "covariant transformation" and "contravariant transformation"? A reference to such definitions would also work wonderfully.
I had the idea of, what if you ground up some magnets into a fine powder, what would happen with the powered, and how would it act? After some google searches, it seems that this isn't done very often, and that not much would come of the powder as the poles will mostly be misaligned. So my next question was, what (if anything) would happen if you ran some electricity through some magnetic powder? Would the poles align? Could the powder be manipulated? Could it's magnetism be manipulated? i.e If you made a magnetic powder trail in a small amplitude sine wave shape, and applied some current to it. Would the powder be able to move into a straight line, or possible break the connection (or even move at all)? I haven't been able to find much information on this topic (of electricity WITH magnets) and I am trying to learn more on this topic.
Prompted by comments to this question on English Learners (about "That's you done"), I've been searching Google Books for similar constructions of the general form that's [pro]noun adjective (for this context, I classify past-tense verb forms such as done, fucked, finished as adjectives). What I seem to be finding is that using "That's" in this way (not referencing anything in particular, just "whatever came before/caused the current situation") is a relatively recent phenomenon. I'm also getting the impression it's more common in BrE than Ame. So by implication, if the boss says to his secretary... "Just get those letters off in the post, and that's you done for the day." ...I should assume the boss is probably British, rather than American. Would my assumption be right? Can anyone shed more light on the usage? Is it the same as... "Here's me doing all the work while you just sit around waiting to be fed." (said by, for example, hard-pressed mother to idle teenage offspring)
The question is written like this: Is it possible to find an infinite set of points in the plane, not all on the same straight line, such that the distance between EVERY pair of points is rational? This would be so easy if these points could be on the same straight line, but I couldn't get any idea to solve the question above(not all points on the same straight line). I believe there must be a kind of concatenation between the points but I couldn't figure it out. What I tried is totally mess. I tried to draw some triangles and to connect some points from one triangle to another, but in vain. Note: I want to see a real example of such an infinite set of points in the plane that can be an answer for the question. A graph for these points would be helpful.
Recent observations of the accelerating expansion of the universe have been quantified and for the time being given a name as to the cause: Dark Energy. And from what I've read from other, similar questions is that Dark Matter is a pressure that is causing this expansion, although we don't know the details of the mechanism yet behind this pressure. But is there anything in our present theories of physics or observations that rules out gravity itself as the cause of this expansion? I'm thinking along the lines of an analogy: the nature of the strong nuclear force which, at close distance, tends to bind together nucleons, but at even closer distances repels them. Couldn't this repulsive force we observe, this dark energy, just be the effects of the gravitational force on a larger scale of space?