darklight03/StackMIAsub huggingface dataset:
Starting from tomorrow, I will be tutoring some undergraduate students following a course in general topology. I am looking for examples motivating the importance of topology in mathematics which can be explained without too much difficulty using concepts of other areas of mathematics (or physics) they have already treated (those areas would be mainly analysis, complex analysis, linear algebra, a little graph theory, some numerical methods for maths, and classical mechanics, electromagnetism, special relativity, some QM and a little statistical physics for physics). I have tried looking around, but I have found little that would motivate me to follow such a course. Do anybody have some nice example? Note: I will of course explain to them that without topology they'll be able to do very little advanced mathematics (e.g. functional analysis, differential geometry, ...) EDIT: Ok, I gave as examples Tychonoff's theorem, Brower's fixed point theorem and the Jordan curve theorem. I would like to keep this question alive, for personal interest. What are interesting (not too hard) applications of topology in other areas of mathematics?
I have this sentence: I strongly believe that the first step in making the most efficient solution for any problem is analyzing it well. Would it be better to use either of the following? I strongly believe that the first step on making the most efficient solution for any problem is analyzing it well. I strongly believe that the first step at making the most efficient solution for any problem is analyzing it well. The context is as follows: First of all, I concentrate on understanding the big picture of any problem. I always try to recognize all the factors that have caused the problem. Then, I start planning the solution at a very high level in order to create long-term benefits. I strongly believe that the first step in making the most efficient solution to any problem is analyzing it well. In my opinion, "What to do" is much more important than "How to do".
I was watching a show on the science channel about gas giants; there is something I do not understand. I am not a scientist, so this may be obvious to some. I learned that there a three states of an given physical object; solid, liquid, and gas depending on how cold or hot the object is. An easy example is ice, water and steam from coldest to hottest. So the theory is that Jupiter has a super-heated solid and very dense core that is made up of hydrogen. How does a gas like hydrogen become a solid while being super-heated? Is it that the pressure is so much that the gas is compressed into a solid? If so how much pressure does it take to compress hydrogen into a solid? How does the heat play into the equation?
I've recently discovered that the following theorems require the axiom of choice to be proven: every surjective function has a right inverse. a real-valued function that is sequentially continuous at a point is necessarily continuous in the neighbourhood sense at that point. every vector space has a basis. When I revisited the proofs I was taught in first year, I was surprised that my lecturers had used the axiom of choice without declaring so. It seems strange that so much effort was dedicated to establishing that mathematics is a rigorous subject [indeed much time was spent on learning the field axioms, well-ordering axioms, Archimedean principle and (later) the completeness principle] but to ignore the axiom of choice. I am interested if there are reasons for omitting to mention the axiom of choice. Are there pedagogical reasons? Is it deemed too complicated? Is it more contentious than the other axioms? Question also asked at Mathematics Educators S.E.
I've used 'extraterrestrial' twice in a paragraph already, so it's starting to get repetitive... Edit: The sentence I'm hoping to prove is 'Humans have reached [insert word] scales', referring to the furthest we've went in the universe. By reaching I mean actual physical human presence, which excludes anything discovered or photographed by probes, and limits us to the Moon. Hence, the question is to find a word describing Earth-Moon scales. In response to the numerous suggestions below: Unearthly or cosmic scales is too grand and fails to convey what I'm gunning for - a sense of humility and a slight disappointment at the lack of human achievement in interplanetary matters ever since the completion of the Apollo program. Alien scales just sounds plain silly. Extrasolar, interstellar, or intergalactic scales are factually incorrect. So far, off-planet seems to suit my purpose the best for now.
I am trying to describe the evolution of a motion which is composed of smooth parts called "free flights" and instantaneous impacts. For example, consider a bouncing ball: its motion is a succession of free flights, separated with impacts (when the ball touches the ground). I would like to refer to two time-related quantities: the duration of free flights, and the dates of impacts. The word "time" can refer to a length of time, or an instant (a date). To raise the ambiguity, I have chosen the following terminology: "free-flight durations" describes a length of time between two impacts; "impact times" describes the date of impact. Is this correct and would "impact instants" or "impact dates" be better? I am open to other suggestions too, but I want to avoid "period" which I am already using to qualify repetitiveness.
Based on the following example: Local Area Network (LAN) You can say that LAN is the short form and the Local Area Network is the long form. What is the another word for "short form"? (Is 'acronym' a better word for replacing 'short term'? or 'abbreviation' would be a better choice?) What is the another word for "long form"? (Would 'backronym' a better word choice to replace 'long form'?) I am open for other word choice to replace 'short form' and 'long form'. Appreciate any help offer. Update I have create a table as follows: |------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| | Acronym | ??? | Meaning (?) | |------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| | LAN | Local Area Network | supplies networking capability to a group of computers in close proximity to each other such as in an office building, a school, or a home. | |------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| What would be the header name to replace the "???"? And also, would the header name "Meaning" be suitable?
I was wondering if the cardinality of a set is a well defined function, more specifically, does it have a well defined domain and range? One would say you could assign a number to every finite set, and a cardinality for an infinite set. So the range would be clear, the set of cardinal numbers. But what about the domain, here we get a few problems. This should be the set of all sets, yet this concept isn't allowed in mathematics as it leads to paradoxes like Russell's paradox. So how do we formalize the notion of 'cardinality'? It seems to behave like a function that maps sets into cardinal numbers, but you can't define it this way as that definition would be based on a paradoxical notion. Even if we only restrict ourselves to finite sets the problem pops up, as we could define the set {A} for every set, thereby showing a one-to-one correspondence between 'the set of all sets' (that doesn't exist) and the 'set of all sets with one element'. So how should one look at the concept of cardinality? You can't reasonably call it a function. Formalizing this concept without getting into paradoxes seems very hard indeed.
I'm struggling to find the words to convey the concept even now, but perhaps it would make more sense to illustrate by example. Certain communities like Reddit and Quora tend to be liberal-leaning and have other attitudes and opinions not entirely reflective of the population as a whole, in large part simply because they are populated by the kind of people who can find these communities. Your average computer engineer is much more likely to be familiar with Reddit and use it regularly than your average janitor or priest, and especially more than someone without reliable Internet access. Similarly, Wikipedia articles or tumblr blogs are unlikely to be written by your grandparents, and thus reflect a certain subset of content and attitudes by virtue of the people who use them most. I thought about implicit bias, but it doesn't seem to quite reflect the unintentional barrier of entry in these cases, although it's the closest I could find off-hand.
Suppose someone (like a boss, friend, cousin,.., to whom you can not say "no" easily) has a personal request for you which you find difficult to decline, but you cannot bring yourself to say no, because you feel embarrassed or too shy to say no (i.e. you do not feel free to say no), or just simply don't like them to feel offended or hurt. What is the expression or idiom that would convey this meaning: "to accept a request in this situation, unwillingly but under your own moral pressure or just out of shyness". I have found this idiom: "to put somebody on the spot", can I use it in this situation? For example: My mother-in-law asked me to accompany her to the market, and I was actually put on the spot by her request so I went shopping with her despite having a severe headache.
I'm a software developer (although math isn't my strong point). I've developed a device to monitor/control my clothes dryer by monitoring the intake air's humidity & temperature, and the exhaust's humidity and temperature. The plan was to compare the exhaust air's humidity against the intake air. Once the humidity in both was around the same, it would mean that no more moisture is evaporating from the clothes, ie: they're dry. Unfortunately its not that straight forward. the air coming out of the dryer is much hotter, and because the sensor is giving me relative humidity values, I cannot compare it to the intake humidity (which is room temperature). I'm looking for a formula that will allow me to compare the two humidities, taking their temperature into account. An "absolute" humidity I guess, or literally the amount of moisture in the air. I've tried a couple of formulas I've found online but I'm not getting readings that look valid. Note that the value doesn't need to be an official, absolute humidity value, it simply needs to allow me to compare the intake and exhaust humidities, at different temperatures (but everything else, such as air pressure, etc... being equal). (Please feel free to suggest a more appropriate tag)
My school requires that dissertations follow a set of sometimes odd formatting requirements. We have a latex class file which does a reasonable job at bringing documents into compliance with their requirements. There is, however, one requirement that I have't got a clue how to make happen: Footnote are supposed to be at the bottom of each page which can be accomplished by using footmisc. However, on the last page of a chapter the footnotes are supposed to be directly after the actual chapter text ends not at the bottom of the page. Now I can solve individual cases by adding vspace, or simply not putting footnotes at the end of my chapters. However, I'm wondering if there is a way to modify the class file to make this happen automatically.
On a chat channel today I was reading two people talk about some of the more popular movie formats and movie players available. One of the interlocutors said something that got me thinking. I will cut quick to my question. Is there a semantic difference between the following two phrases: ... the last movie I played ... and ... the movie I played last ... To my non-native English ears and eyes, I fail to see a difference. But reading these two phrases again and again makes me uneasy. The more I read them, the more I feel that the first phrase gives off a whiff that would suggest that the speaker/writer does not watch movies very often, while the latter would identify someone who watches movies all the time. What do you folks think?
Chasles' Theorem in its strong form says: The most general rigid body displacement can be produced by a translation along a line (called its screw axis) followed (or preceded) by a rotation about that same line. Now, Euler's Theorem simply says that any rigid body displacement can be decomposed into a rotation plus translation. This is easy to visualize. But what Chasles' Theorem says is something much stronger. Unfortunately, I am just not able to visualize it. Perhaps, I am comprehending it wrong. I mean how is it possible to have the axis of rotation and translation the same (or parallel) for the most general displacement. I mean, think of this case: A body is given a finite rotation about the X-axis and then a finite translation about the Z-axis. How can we find that "screw" axis along which both of them can be described?
Lets say I have the following sentences... Cake is really bad for you. It contains a large amount of sugar. It contains common allergens. It looks silly. Additionally, lets say I wanted to connect all three of these thoughts together in a similar manner... Cake is really bad for you. It contains a large amount of sugar. It contains common allergens. And, it looks silly. Now, I know that using an "And" at the beginning of the sentence is poor grammar. I also realize this is more of a paragraph structure, so I could do some word-smithing to make it work that way. However, this feels a lot like a series of items, and I would really love to treat it as such... Cake is really bad for you: it contains a large amount of sugar; it contains common allergens; and it looks silly. Is there a structure in the English language that provides for this? Is the above the proper usage? Thanks!
I'm taking a very computational course in partial differential equations. Because of this emphasis, I'm feeling very underwhelmed by the course, and have a lot of questions that really aren't answerable in the current state of affairs. My professor basically tells me to take an advanced course in real analysis for a rigorous treatment, but that's a long way away for me (at least two years), I was hoping that someone could answer this question here. In the course, we have only looked at three equations and some minor generalizations on them - the heat, wave, and potential equations. I understand these to be characteristic of larger classes of PDEs, but I don't know anything at all about them except sometimes I can separate variables. For each of them, the method has been identical. Separate variables, solve two (or sometimes even three) ODEs. Then by superposition, sum them up. Determine coefficients with Fourier sums or integrals. Wash, rinse, repeat. The question is this: How do I know that that is all the solutions? There is no existence/uniqueness theorem for PDEs. How can I know that without some more advanced technique for solving PDEs that I couldn't find others? Does it follow from existence/uniqueness of ODEs? What about for those larger classes of PDEs?
When using the interrogative pronoun, 'who', what would the possessive form be? 'Who checks X letterbox every day?'. I feel it ought to be 'his' but some people I know claim it should be 'their', which to me seems to contradict the singular form of the verb 'check'. 'Who checks his letterbox every day?' is what I would say naturally. 'Who checks their letterbox every day?' sounds a bit off to me. Looking on the internet doesn't really return anything useful, only the use of 'their' as a singular pronoun, which seems to be somewhat popular a topic. Note, this is specifically regarding the interrogative pronoun; I understand the debate about his/her/their/ones in other circumstances but I want to know whether the same can apply to 'who' or 'whom'.
I would be tempted to rephrase my question as : why do people seem to care only about the curvature of a connection on fiber bundles ? Indeed, the curvature gives the vertical part of the commutator of horizontal fields (the horizontal part is the lift of base commutator), while the commutator of vertical fields has nothing to do with the connection. So it remains, for me, aside from the curvature, to understand the commutator of a horizontal and a vertical field. In the case of principal bundles and principal (=invariant) connections, I understand it well, since we can use the action of the group. But what happens then in general for fiber bundles? Doesn't it give information on how the horizontal distribution vary in one fiber ? Thanks, Amin
My understanding right now is that an example of conditional independence would be: If two people live in the same city, the probability that person A gets home in time for dinner, and the probability that person B gets home in time for dinner are independent; that is, we wouldn't expect one to have an effect on the other. But if a snow storm hits the city and introduces a probability C that traffic will be at a stand still, you would expect that the probability of both A getting home in time for dinner and B getting home in time for dinner, would change. If this is a correct understanding, I guess I still don't understand what exactly conditional independence is, or what it does for us (why does it have a separate name, as opposed to just compounded probabilities), and if this isn't a correct understanding, could someone please provide an example with an explanation?
In Afrikaans, it is considered very disrespectful to use "you" ( "jy") when referring to someone who is above the level of a peer. Instead, it is expected that you use "u", which is a very respectful form of "you". Also you can talk in the third person "How is ma'am today" would be the equivalent. I cringe internally when I say "How are you" to someone older than me, because in Afrikaans it would be very rude. I was bought up to only ever refer to my parents in the third person. "how is mom today", "what is dad doing" when speaking in Afrikaans. The lack of English equivalent feels very wrong and disrespectful. What is the best way to convey this in English? I have been reassured that saying "you" to a parent isn't rude, and I understand that this can be cultural, but I'm particularly looking for what options English offers in this regard, as far as existing vocabulary, that convey respect. I'm in South Africa.
In a modern nuclear reactor for example a PWR there are multiple containment systems which prevent the release of radioactive material into the environment and shield the environment from the radiation. Here is a quote from the wikipedia article about this: The reactor vessel is the first layer of shielding around the nuclear fuel and usually is designed to trap most of the radiation released during a nuclear reaction. The reactor vessel is also designed to withstand high pressures. I think that the radiation in this part is shielded partially by the walls of the pressure vessel and partially by the water it contains. However how much percent of the radiation is shielded by the water alone and how much by the walls? I.e. how do the shielding effects of the walls and the water compare to each other (roughly) and why?
My brain immediately suggested the non-word "promisand", but I doubt I would be understood if I said that. What's a good word (or failing that, phrase) for the action or thing that was promised? This reminds me of Latin expressions like Carthaginem esse delendam (Carthage must be destroyed, lit., 'Carthage is a thing-which-is-to-be-destroyed') or perhaps that once-common mathematical expression QED quod erat demonstrandum ('that was a thing-which-is-to-be-proven'). Circumlocution or other forms are often possible: "He gave me the promised widget" or "He did what he promised he would". But sometimes it's useful to have a word that stands on its own. More generally, is there a good way to express the construction 'thing-which-is-to-be-X'? In Latin this is a gerundive, though it seems in my brief searches that the term means different things in English and other languages.
"One way mirrors" are used in interrogation rooms etc. The wikipedia article states that: A true one way mirror does not, and cannot, exist. Light always passes exactly equally in both directions. However, when one side is brightly lit and the other kept dark, the darker side becomes difficult to see from the brightly lit side because it is masked by the much brighter reflection of the lit side. Which I think I understand. What I find strange is that this surely can't be any deep principle of physics, as it's easily "violated"; you could just place a camera on one side and a TV on the other side, and it would be perfectly "see-through" from one side but not the other. Couldn't you imagine a sort of microscopic version of the above "violation"?
I am typing up formal invitations, and I want to say that transportation will be provided from Point A to Point B (but also from Point B back to Point A). In order to clear up the to-from/from-to confusion, I tried to use 'between' as follows: Transportation will be provided between Point A and Point B. Is it grammatically correct to use between in this manner? Transportation is really FROM Point A TO Point B -- although the car may travel between the two locations (i.e. not at either end point but in the middle), the transportation itself is to/from specific locations. Another option is to say Transportation will be provided to and from Point A and Point B. I am trying to find the most appropriate choice of preposition to accurately convey the meaning, and I was unsuccessful in determining this after searching myself.
In a sentence where we have two listed words that are hyphenated, we can omit the latter part of the first compound and still be grammatically correct: I don't believe we will ever find helium-based or hydrogen-based life forms. I don't believe we will ever find helium- or hydrogen-based life forms. However, if we have two (related) words which both end in the same suffix, can we still apply this notation? I've seen this used before, but I'm not sure it's proper: It doesn't matter whether the character is a protagonist or antagonist. It doesn't matter whether the character is a pro- or antagonist. More often than in writing, I hear this in spoken conversation, usually with an emphasis on the prefixes (i.e. "... a pro- or antagonist ..."), as though there is actually a hyphen in both words. I've read up a small bit on conjugation reduction here (thanks to search actually finding that term for me), but it doesn't seem to answer this particular question. Is the reduction of non-hyphenated words allowed within English grammatical rules?
I'm currently a PhD student in mathematics at a decent sized graduate school, but I've been questioning my desire to continue on and finish my doctorate after I achieve my master's, which will occur within the next year. I've been thinking about trying the actuarial exams so I might have a route to leave academia. I've never taken financial math, but I have taken a Calculus-based probability theory class, a mathematical statistics class, and two measure theory based probability classes. I've gotten at least an A- in all four classes. I've never taken a financial mathematics class, but I have taken a wide range of both applied and pure math classes. Based on the experiences of some people that have passed these two exams, given my background does studying for and passing both exams seem like a possibility for me? Thanks for any advice!