So, I have been solving a number of exercises involving contraction mapping. However, I am struggling to understand what exactly contraction mapping is at its most basic. All I really understand from the concept is that if we have two sets, then one specific point on the set is being mapped closer to another - I'm not sure if I have the right definition. Can somebody please explain this to me in the most layman way possible?


I was trying to understand the undecidable nature of the continuum hypothesis and came up with the following question: The set of circles with a rational diameter is countably infinite (with cardinality equal to the cardinality of integers). The set of circles with a rational circumference is countably infinite (with cardinality equal to the cardinality of integers). The cardinality of the union of these sets is clearly smaller than the uncountable set of irrational numbers, but why isn't it larger than the set of countable integers?


I often hear about string theory and its complicated mathematical structure as a physical theory, but I can't say that I've ever actually seen any of the related math. In general, I'm curious as to what the mathematics of string theory look like, can anyone point me to some references? In specific, I want to know if there is a fundamental equation in string theory that is assumed as a starting point for most problems, something comparable to Newton's second law in mechanics or the Schrodinger equation in QM?


Recently, I am studying the online course Effective Field Theory provided by MIT OCW. Prof. Stewart gives a nice picture to summarize the effective theories: As a newbie in this field (I only have a knowledge of QFT on the level of Peskin & Schroeder's book), I am quite confused with so many terminologies. What do they stand for and Why do we need these different theories? Could someone explain this picture in depth (especially on their advantages and limitations)? Any help will be greatly appreciated.


When a solid object moves in the atmosphere (e.g. a plane), in which direction does air fill up the "empty space" left by the translation of the object? Does this depends on the direction of movement of the object? (e.g. if the plane is moving horizontally, or is landing, or taking off) PS: I am aware of this question and I think this is not a duplicate of that one.


Here in Pittsburgh, we have lots of "Let's go Steelers!" (and some diehards who also say "Let's go Bucs!", but they're dying out). What does that phrase even imply? I assume it's similar to "Go Steelers", which I'm also not sure of the implications. "Go Steelers... to victory!" is a very strange way to phrase the sentence. Does anyone know where this phrase comes from?


I was trying to describe my friend as being someone who gets upset in the absence of immediate gratification, but couldn't think of a good word. A good example would be someone who comes back from the gym and takes off their shirt expecting to look fit after one session. Is there a word to describe someone that does this? The closest I could think of was "impatient", but that doesn't quite pin down the meaning.


I was wondering how this phrase sounds in English: She is a little bit shy. In my country the people say "a little bit" a lot, and sometimes they use it wrong. I have the impression that in US (or even in other countries where English is spoken), people prefer to be more specific saying "She is shy". Anyway, I don't want to be misunderstood, can you tell me if the above phrase sounds awkward, and in which situations I can use "a little bit" without problem.


I know that Galois Theory can be used to answer the following question: Why is there no formula for the roots of a fifth (or higher) degree polynomial equation in terms of the coefficients of the polynomial, using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of radicals (square roots, cube roots, etc)? Can real analysis be used also to answer this question (e.g using the intermediate value theorem, etc..)? Is there some sort of connection between real analysis and galois theory?


My sentence is as follows: I lodged a complaint with Acme Corporation, a private entity. Does it need to be changed into: I lodged a complaint with Acme Corporation, which was a private entity. I appreciate the meaning between the two sentences is slightly different. I would just like to know if the first sentence is grammatically correct. It looks correct to me but I have been told that only the second version is grammatically correct. Is there a 'rule' governing this situation that I can reference?


Does one need to invoke quantum mechanics to explain Casimir or van der Waals forces? I see that textbooks show derivations of van der Waal forces with no QM but the Casimir force is typically described within QM. Additional questions I have are: Is there a difference between van der Waals and Casimir forces? Are there distinct examples of these two forces in real life? Is there a way to prove a given force is van der Waal and not Casimir or vice versa?


This question on a different SE site asks the question of what the etymology of the word "Xenomorph" is, and the consensus, from the comments at least, is that it has none - that it was purely invented for that movie. My question is, is this true? Did this word purely exist as a creation of the movies, as a portmanteau of two separate words? Or is there an origint that dates back to before the creation of these movies? I realize this is the ELU stack exchange, so I don't expect a detailed explanation of the etymological origins movie-side, I'm simply curious if there is any evidence that this word existed prior to the Alien franchise.


I looked up the definition of a Joke on Dictionary.com which gave me this "Something said or done to provoke laughter or cause amusement, as a witticism, a short and amusing anecdote, or a prankish act." If someone had the intention to make a joke but failed to provoke laughter is it still a Joke, and if someone did not have the intention to make a joke yet it provoked laughter, does it count as a Joke.


I proved the following statement: The composition factors of every finite solvable group are isomorphic to cylic groups of prime order. I want to use this result to prove that every two finite solvable groups of the same order have the same composition factors. If I knew that for every prime number in the decomposition of the order of the groups there is a composition factor that is cyclic of that order I would obviously be finished. However, I'm not even sure if this statement is true and I don't have a good idea on how to proceed in case the statement isn't true. Help will be appreciated! :)


A recent paper authored by Niayesh Afshordi and Joao Magueijo is in the news claiming to challenge Einstein's theory on constancy of light speed. It says light might have travelled with a faster pace during the initial stages of Big bang which apparently sets aside the inflation hypothesis. My question is whether it actually invalidates Einstein's relativity - which depends on constancy of speed of light. Or is it only applicable to the very early universe?


I'm writing a thesis in a scientific area, and in a draft I wrote that it was "not possible to tell which set of data is more accurate", meaning that it was not possible to tell which was closer to the true values of the thing being measured. One of my supervisors objected to this language, claiming that either one is accurate or one is not - that it's no more possible to be more or less accurate than it is to be more or less pregnant. For the purposes of this thesis I will simply avoid this phrasing, but out of curiosity: is he right?


I have been reading up on the correct use of 'which' and 'that' but I am still struggling. I am unsure if 'must be taken regularly' is restrictive or not. Both 'which' and 'that' sound correct to me. Can someone please tell me which word is correct to use in the following sentence: Arava is a strongly recommended medication, which must be taken regularly. Arava is a strongly recommended medication that must be taken regularly. Thank you.


I am looking for a word that describes a personality/person (not in a negative but neutral/positive way) that doesn't care abut anything, has a neutral opinion about most things, is very care free/laxed, etc. The closest I could think of is narcissist but it's not quite right. Is there such a word? I guess an example would be like "Dave doesn't care if we have a salad, a burger, or a soup for lunch, he is a/very ________." Thanks.


Given a finite dimensional algebra, what is the exact relation between the indecomposable projective modules, and a general indecomposable module? In the case of an oriented quiver without cycles for example, it is easy to find the simples, and the indecomposable projectives. What does this tell us about a general indecomposable module? I guess the main question is, to understand the complete representation theory of a finite dimensional algebra (or let's say a quiver), what else do we need besides the simples and the projective indecomposables?


I'm looking for an online coupled oscillation simulation. The best I have got so far is this --- https://phet.colorado.edu/sims/normal-modes/normal-modes_en.html But I'm looking for something which has more options like changing the mass of the objects, changing the spring constants, cutting off the springs. Please suggest if you have come across better simulations. PS : I apologize if this question does not belong here. I need a coupled oscillation simulator for my work.


Are there other usages of the expression "don't ask don't tell" besides the one from its Wikipedia page? Don't ask, don't tell (DADT) is the term commonly used for the policy restricting United States military personnel from efforts to discriminate or harass closeted homosexual or bisexual service members or applicants, while barring those who are openly gay, lesbian, or bisexual from military service. Or is the expression also used in contexts not related to homosexualism in the military?


As part of a promotion a toy is put in each packet of crisps sold. There are eight different toys available. Each toy is equally likely to be found in any packet of crisps. David buys four packets of crisps. a. Find the probability that the four toys in these packets are all different. b. Of the eight toys in the packets, his favourites are the yo-yo and the gyroscope. Find the probability that he finds at least one of his favourite toys in these four packets. I'm stumped on b. Can someone explain how to do part b to me please?


A Group is an algebraic structure consisting of a set of elements together with an operation that combines any two elements to form a third element. The operation satisfies four conditions Closure Associativity Existence of Identity Existence of Inverse Intuitively I understand the purpose of Closure and Associativity Property. But I'm not getting the intuition behind Identity and Inverse. Whats the purpose of having these elements in a group. I searched everywhere and find only common definition which are straight forward. But nobody discuss why it's important. Can anybody explain it to me.


I recently saw a question today pertaining to Russian mathematics and I have a similar question but of a slightly different flavor. I've always heard that the Soviet Union had a history of producing influential mathematicians in geometry/topology and I'm curious if anybody knows of any worthwhile Russian texts on these subjects. I can speak Russian fluently and I'm interested in pursuing these fields and learning the perspectives of some of the old Soviet mathematicians. Would anybody know of such texts? Thank you.


What is the trajectory of a particle in a string or a rope that goes under the effect of a wave pulse. The illustration in this image is not what I am asking about, I just attached it in order to give you an idea of the case I mean. If you give me a picture of the trajectory, I'd be grateful. Thanks in advance.


I find typing `a word' and ``a word'' quite inconvenient, because I find it rather annoying to type ` and `` all the time. Instead, I'd prefer typing 'a word' and "a word" (which I often do when I don't pay attention, causing wrong displays). I was hoping the inputenc package would make this possible, but apparently it doesn't. Is there a solution for this?


It is well-known that in a Noetherian UFD, every height one prime ideal is principal. I was wondering whether this statement holds if one replaces 'UFD' with 'locally factorial domain'. I am aware of the fact that 'locally factorial' in general does not imply 'factorial', i.e., 'unique factorization' (though have never come accross locally factorial domains that are not UFDs), so I believe that there is a counterexample to this. However I cannot come up with any.


Can anyone tell the difference between these two phrasal verbs above? A soldier took the flag down from the flagpole. A soldier got the flag down from the flagpole. My father took the electric fan from the wall. My father got the electric fan from the wall. To me, these four sentences look exactly similar in meaning. In Merriam-Webster Dictionary, the phrasal verb "take down" is used as "disassemble" or "pull to pieces" :Take down the building, take a rifle down. Is it possible for the phrasal verb "get down" to be used as "disassemble"?


I've been told that HJB equations can be solved numerically. I know very little about the subject, could someone provide a couple of comments or a reference (ideally, one that is accessible for a layman) on: What are the main numerical methods used and, roughly, how do they work? What are the limitations of the methods? For example, does the complexity scale badly with dimension?


I'm doing some research for a book I'm writing, and, as the title implies, I'm looking to understand what happens when a supersonic jet sustains in-flight damage. Specifically, what happens to the occupants if the cockpit is breached? (I assume that this is why jet pilots wear masks?) Could a jet maintain super sonic speeds (for at least a short time) with such a breach? What would happen to a person in the cockpit without a mask if there was a cockpit breach at those speeds? Just how weird do things get at super-sonic speeds? Could a conventional airliner ever reach mach one in extreme circumstances? Many thanks!


First of all, how exactly do you entangle two atoms? I've heard about how they entangled atoms in quantum Teleportation experiments, but I don't get how they did it. I've heard that the atoms must "interact", but what exactly does that mean? Do they have to touch? Do they have to be within a certain distance - like a foot or an inch - from each other?


After spending some time researching about the centrifugal force, I now understand that it is needed in a non-inertial reference frame for Newton's Laws to hold true. However, I don't understand why we only feel the centrifugal force when moving in a circular path. For example, if you imagine yourself being spun around in a circle, then in your frame of reference you would feel yourself being pushed outwards. But since in your frame of reference you are stationary, the outwards force must be balanced by an inward pull (for you to remain stationary). So why would you only feel the outwards force but not the inwards force?


I am studying elementary particle physics and I am wondering what a parton level Feynman diagram is? My understanding is that partons are representations of the quark and gluon substructure of hadrons, and so I am assuming that any diagram that has a quark or gluon qualifies as a parton-level diagram, but I cannot find any reference to it in my notes or online. Could anyone explain (with examples preferably), what a parton-level Feynman diagram is?


There are many 'rules' on the net saying that a comma should be placed before the relative pronoun 'which' in a non-restrictive clause. (http://www.oxforddictionaries.com/words/relative-clauses) But would the sentence have the same meaning without the comma before the relative pronoun? Does omitting a comma before a relative pronoun change a nonrestrictive clause into a restrictive clause? For example: 'We ate the pizza, which was nice.' 'We ate the pizza which was nice.' If so does the same apply to relative pronouns other than 'which'?


I am wondering whether there are countably many or uncountably many spanning trees of the integer lattice rooted at a particular vertex, say the origin. In a spanning tree rooted at the origin there is a unique path from each vertex to the origin. There are countably many paths to the origin for each vertex, and each spanning tree is uniquely determined by the paths for each vertex in the integer lattice. There are uncountably many ways to choose a path for each vertex, but not every choice of paths will correspond to a spanning tree, so this doesn't help in determining whether there are countably many or uncountably many. Any help would be appreciated.


I'm taking an intro to proof via number theory class and I'm trying to prove that if a|b and b|c then a|c. So you write down there exist integers k and m such that ka = b and mb = c. Then you substitute ka in for b in the second equation and write (mk)a = c (by associative property). We know (mk) is an integer by the Closure property and so a|c. My question is, how do we know substitution is allowed (or is it not)? Is this something that can be proven from the Axioms of the Integers or something inherent in equivalence relations? Any insight would be awesome thank you!


The reason I ask this is because I was trying to draw multiple lines in pgfplots (I was trying to make a drawing showing how Riemann sums and integrals are connected). And after unsuccessfully trying to use a for loop for an hour or so, I found this solution, which was pretty unsatisfying since it's just a workaround for this particular case. And I thought that it would have been much easier with the power of Python. So, my question is: What do I lose by switching to matplotlib?


The difference between rational and irrational numbers is always stated as: rational numbers can be written as the ratio of two integers, and irrational numbers can't. However, why do mathematicians make a distinction between these two types of numbers? Why are integers special anyway, other than being historically significant? Is there any property that sets rational or irrational numbers apart, other than the way they are written in our number system?


I am interested in learning about duality in convex optimization. I am looking for something to read which is: Reasonably short. Fairly self-contained (if it is a chapter in a textbook, I would like to be able to read it without having first to read the preceeding chapters). Mathematically rigorous: everything is proved. Well-written, i.e., does not take ages to parse. The first one is particularly important for me - I am looking for something I can learn in a few days. The best I have found so far is the chapter on duality in Boyd & Vandenberghe. Does anyone have any other recommendations?


My recent assignment was to write an essay using a template my teacher had given me. The essay was supposed to begin with a phrase like 'It is common knowledge that' or 'there is no denying that', but I wanted to be more original and wrote the phrase 'It is no secret that', which I had heard used many times, instead. Later, my teacher told me the phrase was kind of obscure and most likely informal and that I shouldn't use it. So I want to find out if it is actually applicable and acceptable in an essay. P.S. I'm not a native speaker, so please don't judge me if I'm being stupid.


I have three sentences of which I'm not sure which one is grammatical, and why. It refers to the Eiffel Tower. It is used as the symbol of Paris and of France for over a century. It was used as the symbol of Paris and of France for over a century. It has been used as the symbol of Paris and of France for over a century.


After watching a movie in which levitating water was used as an illusion, I was curious to discover how it works. Unfortunately I cannot find any great explanations of the science behind the effect, only that strobe lights are what cause it. I want to know why strobe lights cause this effect. Here is a video of the illusion. It is my assumption that there is a steady stream of water, and the strobe lights are flickering on and off at a certain rate. Why then do I not see the entire stream of water being lit up, rather than individual droplets? What causes the droplets to appear to "move?" And why are there dark spaces between the droplets when there is apparently water there?


I'm wondering if there is a name for graphs such that if there is an edge between vertices A and B and a second edge between vertices B and C then there must be an edge between vertices A and C. My understanding is that this means that the graph is composed entirely of cliques. The graph need not be connected so it may contain multiple cliques, but there cannot be any edges between cliques. Is there a particular name for this structure? Thanks!


I am not a native speaker of the English language but have been living in United Kingdom for last couple of years. Once I was with my friend who was an Irish and I said "Its cold outside" and he said that I was wrong and should have said "Its cool outside". I am still not sure why I was wrong? Could someone really differentiate the use of word "cool" and "cold" in that case?


Is there a word that names an excuse made so as to not look stupid? Here is an example that actually happened: A guy walks up to a girl and starts talking to her. After talking for some time, the girl asks the guy to give her a kiss, but the guy refuses. So to this, the girl said, "I just wanted to test you if you are a good guy or a bad one."


This is probably a stupid question, but what makes the three magical elementary row operations, as taught in elementary linear algebra courses, special? In other words, in what way are they "natural" (as opposed to "arbitrary")? It seems that they're always presented in a somewhat haphazard manner ("these are the three legendary elementary row operations, don't ask why, they just are"). From what I understand, they satisfy some nice properties, such as the inverse of each being an operation of the same type, etc. But is there something that characterizes them, i.e. is there some definition of what constitutes "elementary" that's only satisfied by the three types of elementary matrices, and no other matrix?


There is a single obscure word whose very definition is when one says "I'm sorry for saying this, but" or "I hate to tell you this, but", and then proceeds to do the very opposite and attacks or lambasts the person they're talking to. For the life of me I can't find it and I've searched high and low. Can someone please help me? Many thanks and cheers!


I can't find anywhere the advantages of using Ridder's method over any other numerical method. My professor said that it specializes in solving a unique set of functions, but I can't find anything about it anywhere. When is Ridder's method more useful than something else and why? I realize it may sound like a silly question but I have to answer it for a project. Thanks!


Consider the set of words {one, two, three, four, five, six, seven, eight, nine, ten}. I want to write the associated partitions of this set under the relation R where two words are equivalent if they have the same number of letters. It's obvious one, two, six, and ten are all equivalent but I'm not exactly how to write the partitions with the curly brackets... Would it be {{one, two, six, ten}, {four, five, nine}, {three, seven, eight}}? Or would I write these partitions separately like {{one},{two},{six},{ten}}, {{four},{five},{nine}}, and {{three},{seven},{eight}}? What do the curly brackets actually say and what's the difference between the two ways I just wrote them?


As far as I know, the classical approach to special relativity is to take Einstein's postulates as the starting point of the logical sequence, then to derive the Lorentz transformations from them, and finally to derive the invariance of the interval and other consequences from the latter. I'm curious to know if it would be sufficient to take the frame-independence of the interval alone as the initial hypothesis to derive the other results (including the Lorentz transformations and the postulates). Is this possible, or do we have to prove that the interval is Lorentz-invariant first? If this approach is consistent, would it offer any technical advantage w/r/t the classical one?


Question: In Functional Analysis we can note things like: every closed subspace of a Banach space is Banach. In this case, what does "closed subspace" mean? Does this mean closed under the norm topology? Or does this mean closed in the sense that multiplication of scalars and addition of vectors is closed? Or does this mean closed with respect to limits? I'm reviewing this material and I realized that even though I have this in my notes a number of times I am unsure of what this actually is. I thought it was the second statement above, but the third statement makes the "every closed subspace of a banach space is banach" statement easy to prove.


Is the use of two or more successive relative clauses common or grammarically accepted in English? As in "The man who is sitting in the wheelchair and who has a broken leg. Or "The man who is sitting in the wheelchair and whose leg is broken." I know, of course, there are ways to shorten them, but I just want to know if it sounds grammatical to native speakers of English to use successive relative clauses, whether in spoken or written language.


My professor asked us a few questions in class and asked us to think about them. He's going to reveal the solutions on Thursday, but I want to understand it before he talks about it on Thursday in greater detail. If anyone could explain any of the things below, it would be appreciated. Thanks in advance. (and if you can't, don't worry- I can ask the professor after class on Thursday)


Suppose I have the sentence: "All apples are green." Although it is not a true statement, clearly it is a declarative sentence. Can any declarative sentence like this be made into an interrogative sentence (a question) merely by replacing the period with a question mark? The reason I ask is because the typical word order for such a question would be: "Are all apples green?" Notice that the verb has been moved to the beginning of the sentence. This is the way most of us would word the sentence when phrasing it as a question.


I was wondering if there is a non-geometric interpretation of the dot product (or the inner product more generally). That is, an interpretation that has no concept of length and angle. My motivation for asking this lies in how, in quantum mechanics, the inner product between the bra and ket vectors are often taken. However, these vectors do not represent spatial concepts, but are rather, a vector containing probabilities (which isn't a spatial concept). Thanks for any help!


What is the way to say that something "should be done" in the past? I know there is a "should have" structure, but it is used for situations, when something was to be done in the past, but it wasn't. E.g. "You should have called me yesterday. Why didn't you?" So, would it be correct to say "It should have been done" in a situation when something was to be done and in fact it WAS DONE?


I was explaining it's vs. its to someone the other day, and I said "None of the pronouns (his, hers, theirs, yours, its, whose, ...) has an apostrophe." Later I got to wondering whether that was really true, and sure enough fairly quickly found one that does: one's, as in "One's memory isn't what it used to be." Are there other pronouns that use an apostrophe?


It seems to me that "optical" relates more to the mechanics of light and vision, whereas "visual" is a much broader term. For example, hallucinations are classed as "visual" or "auditory", rather than "optical" or "acoustic". Other examples are visual puns, visual cliches, visual metaphors, etc. I notice on the Wikipedia page on optical illusions uses the terms interchangeably. Is there a semantic reason why the term "optical illusion" is used instead of visual?


The sentence I wrote is as follows: The idea behind this technique was that the human eye could not differentiate between the individual colors of each point, thereby causing the eye to blend the colors together to form a more complete image. My professor said that the "thereby" in that sentence should have a semicolon behind it and a comma in front, but as far as I know, a semicolon separates two independent clauses. I don't think "causing the eye to blend colors together ..." is a complete sentence, but she said it should be written as ";thereby,". Is she correct?


I'm having trouble putting the pieces together. In SM, neutral kaon oscillation is heavily constrained. This means, roughly, that the squark mass matrices have to be diagonal. And this is called universality of soft parameters. What exactly is universality and why do we have universality in this situation? Furthermore, is having diagonal squark mass matrices sufficient? In a lot of SUSY theories, they go one step further and assume the scalar masses are the same for the first and second generations (pMSSM for instance). Why is this additional step necessary?


Since it's Friday and others are playing games, here's another one to take us into the weekend. Everyone knows about those Russian nesting dolls (called matryoshka dolls, I believe). A small doll nests inside a larger, which itself is inside a larger doll, and so on. This game does the same with words. You start with a three-letter word, add a letter to make another word, and so on. Longest word wins. The catch? Each additional letter has to form a legitimate word. Here's one to start you off: bur burr burro burrow burrows Good luck.


Is there a word that describes a person who tries to get as much information as possible when making a decision for fear of committing a mistake that might affect both him and others in a negative fashion? The only word I could think of was indecisive, but it has a connotation that suggests hesitation and irresolution when making decisions, which is not what I'm going for, since I want a sense of absolute commitment when making the decision.


Which of the following books is better to prepare for a mathematical competition at the undergraduate level? The art and craft of problem solving (ACoPS) or Problem solving through problems (PSTP). I think that ACoPS doesn't have an abstract algebra chapter and PSTP doesn't have a combinatorics chapter, so, which one would you recommend me to use and why (I can't use both right now)? Thanks in advance!


I'm not sure how to phrase this question. I'm sure I could write it in terms of operators between Frechet spaces, or something like that. Let me apologies to any analysts in advance for my lack of rigor. I'm basically interested in "how many" elementary functions have elementary anti-derivatives. I realize that I ought to impose some conditions on the functions, e.g. differentiable, and on the domains of definition. It would be great if someone could give an answer which puts this question in to the correct language, provide sufficient restrictions and then state any known results. Sorry to be so wishy-washy!


Occasionally, I write letters that I actually print out and send by snail mail :). I'd like them to look as formal as possible. Archaic (pre-typewriter) Dutch customs require the sender's address to be in the top left corner and the recipient's in the top right. The place and date should be aligned with the recipient's address. The sign off ("Yours truly") should also be aligned with that. Of course there are also specific rules for the amount of vertical whitespace. Does anyone know of a good package that lets me customize all this stuff, preferably in a .sty file or something that I can include, so I don't have to do it again for every letter?


Are the following two statements written in correct grammar? What do they mean? Can they be written in more easily understandable manner? If yes, how should they be re-written? The student's pass will be issued no earlier than one month from the course commencement date. This letter will be valid for two months or till the course commencement date as indicated above, whichever is later.


In a formal letter addressed to one or more unknown recipients, "Dear Sir or Madam" is the customary salutation. As a German native speaker, who is used to "Sehr geehrte Damen und Herren", writing "Sir" before "Madam" sounds impolite to my ear and I feel the urge to change the order. I know that I must never directly carry over conventional expressions from one language to another, but it makes me wonder: Are there any differences between "Dear Sir or Madam" and "Dear Madam or Sir"? In particular, would the latter sound more polite or rather come across as outlandish?


Ok, so here is an example sentence. "I hoped to lose the race so I wouldn't have to run against him." Here's my dilemma. I was questioning whether a comma was necessary after "race" to separate the two independent clauses connected by "so," the coordinating conjunction, but then it occurred to me: Is "I wouldn't have to run against him" an independent clause, or is it dependent? If it were standing on its own, would it be considered a fragment? Can someone please explain?


I have been googling around, searching for the origins of the phrasal verb "to fall asleep" but so far I have found no references. I was wondering specifically why we use the verb "to fall" to characterize the process of going to sleep and my conjecture is that it somehow links in with the connotations of tiredness and fatigue that the verb "to fall" bears.


The Feynman Lectures on Physics probably constitute the most famous introductory physics textbook ever written. The level of intuition it provides is hard to beat; I first started reading it in middle school, and a decade later I'm still finding new insights in it. Yet when I recommend it to others, they often have sticker-shock. It is a three-volume book, after all. So is there any legal way to read the Feynman Lectures for free, on the Internet?


When passing someone in the hall, you might say "How are you?" as a greeting, but not meaning it literally, as you don't really require an answer. Is "How are you?" (or "How's it going?", etc.) an idiom when used this way? "Idiom" doesn't seem appropriate because that combination of words generally has the literal meaning. Is there another word for this? Sorry if I am not making this clear. If I can clarify in some way, please ask. Edit: Could someone explain why this question might have been voted down? Not complaining, just trying to learn.


I want to know which of the following would be the correct way to phrase my sentence, pasted below. I've come across this problem several times in my writing and want to settle this for good so that I don't always second guess myself: "When he learned of the sins he had committed, Oedipus blinded himself as punishment" or "When Oedipus learned of the sins he had committed, he blinded himself as punishment" Is one of these more grammatically correct than the other, or are they essentially the same? Is there a general rule for phrasing such sentences?


In Australian English there has always been a distinction between "pissed" (intoxicated) and "pissed off" (angry, irritated). I've noticed a trend towards the American usage where "he was really pissed" is now much more likely to mean the latter. I'm aware of the futility of resisting the natural evolution of language, particulary the juggernaut of American language imperialism via films and TV, but it seems a shame that "I did it because I was pissed" is now ambiguous. Is this a trend in other English speaking countries where the Australian usage was common?


I often try to distinguish between immediate-feedback communication, such as talking in person, on the phone, or by instant message, and delayed-feedback communication, such as email. (Or at least, delayed in the sense that it's socially acceptable to not respond right away.) I usually would say something like, "I'd prefer to discuss this in real-time," which usually gets the point across, but I'm wondering if there is a better way to describe this concept. ("Can we talk about this using an immediate-feedback medium?" sounds a little awkward.)


"What does {holding nose} P.U. mean?" my son asked me tonight. I told him I didn't know, and he laughed and said "It means stinky, Mommy!" Very funny and well said, but it left me wondering. A search of Wikipedia and Etymology Online came up wanting and my etymological dictionaries are packed in boxes. What does this abbreviation stand for and why does it mean stinky?


There is a term orbit insertion which means a maneuver performed by a spacecraft flying by a celestial body in order to become an artificial satellite. What word or phrase would you use to describe the result of this process? And from the following sentences, which are correct? The probe entered the orbit of Mars. The probe inserted into the orbit of Mars. The probe was inserted into the orbit of Mars.


What are some simple proofs that the points on an elliptic curve form an abelian group under addition? I am mostly looking for proofs of closure and associativity, since the other three requirements follow immediately from the definitions. Note that I am specifically looking for simpler proofs, since I am only still in high-school, so I have not learned about homomorphisms and isomorphisms and matrices yet. An intuitive 'proof' will also suffice (since again, it's only high-school level). Also, are elliptic curve points always an abelian group under addition? Or only over the rational numbers or over some finite prime field (and not over the reals)? I'm trying to write a paper/essay on elliptic curve cryptography, but I can't get these proofs to work.


Possible Duplicate: An idiom for deriving pleasure from another's suffering We often experience the following: We feel happy when our fellow classmates get a bad grade and not necessarily when we ourselves get a good grade. We feel excited when our opposing team plays really awful and not necessarily when we our team plays well. We feel pleased when one person we hate experiences some kind of misfortune and not necessarily when we ourselves get good luck. What is a word or phrase to describe the phenomenon above?


My highschool textbook states that Zener diodes are a special type of diode which is made of highly doped p and n junctions, and which can survive reverse bias - unlike normal diodes, which get damaged due to the excess heat produced in that mode. How can heavy doping help Zener diodes avoid damage in reverse bias? Is it that it somehow gets rid of the excess heat produced in the diode? How does it do it?


My textbook doesn't explain this very well, so what I want to know is: What is the purpose of parametric equations? What is a parameter? What is the advantage of these equations over a function y=f(x)? What do they essentially enable you to do? I find this topic to be impenetrable so a thorough explanation in simple terms would be very much appreciated. Thank you.


This question may sound dumb, (it will to me, hopefully, in a day or two!), but does the term phase difference apply only for sinusoidal waves? Wikipedia defines 'phase' as the following: Phase in sinusoidal functions or in waves has two different, but closely related, meanings. One is the initial angle of a sinusoidal function at its origin and is sometimes called phase offset or phase difference. Another usage is the fraction of the wave cycle that has elapsed relative to the origin. I'm confused about this specific part: Another usage is the fraction of the wave cycle that has elapsed relative to the origin Can this definition apply for any wave function, or just sinusoidal ones?


Is there a single word that denotes the belief that women are or should be inferior to men? For example, That ideology advocates murder, rape, slavery, homophobia, xenophobia and [insert the requested word here]. The word patriarchy comes to mind but I am not sure if it fits in the sentence above. I am also thinking of sexism, but it denotes prejudice based on sex, which, although in practice usually against women, can work both ways. Any suggestions?


I am reading about the going-up and going-down theorems in Atiyah & Macdonald's commutative algebra book. I'm wondering if anyone could give me some basic facts/examples to help me understand why these two theorems are so important. Neither of these theorems are actually used in Atiyah & Macdonald, and yet I don't find the statements to be so obviously remarkable that they should deserve a special name.


While I was writing a status update today on social media I decided to do some grammar-learning. The status update was "Happy Labor Day everyone!" and I am now wondering if I should place a comma between "Day" and "everyone?" My grammar skills aren't great, and this may be staring me right in the face as I'm thinking about it, but researching on the web and Stack Exchange hasn't brought up the case yet. Help would be much appreciated.


I noticed the following sentence, taken from Mark Twain's "The Awful German Language:" Fifthly, I would do away with those long compounded words; or require the speaker to deliver them in sections, with intermissions for refreshments. What is the grammatical function of the semicolon here? I would have expected a comma to be appropriate here, as to me it coordinates a dependent clause, and that would be solely under the jurisdiction of a comma, as I understood. I thought semicolons were for replacing conjunctions between independent clauses or avoiding confusion with commas when writing a list that requires commas for one of the points within. Thank you!


Find the volume of the solid that remains after a circular hole of radius a is bored through the center of a solid sphere of radius r > a. So in the picture it looks like a circle with a cylinder cut out of the middle. I am not even sure where to start with this. I know this has to do with integrals but I am not sure how to set this up to even get an integral. Any help would be greatly appreciated.


I was reading about different variations of nim game and I'm trying to find the winning strategy in one of them: there are n empty places on the circle. Two players are placing their coins on empty places. They can put their coin only on empty places which aren't next to a coin of the opponent. In one turn each player can place only one coin. The player who can't move loses. Who's got the winning strategy for each n? I think that for even n the second player will win, because he can always place his coins symmetrically to the first player's. But what about for odd n?


I would like to have a definition for non-modular lattices which clearly sets them appart from their modular counterparts, thereby focusing on their main distinctive feature. Besides, I would be very grateful to you if you could provide me with an explanation about orto-modularity, which Birkhoff and von Neumann allegedly considered to be the basis of quantum logic, as opposed to modular logics. Is there any correlation between the two types of lattices and this issue of orto-modularity? Thanks in advance.


Is a Topological Space an Algebraic Structure? I am referring to the Wikipedia definition: algebraic structure generally refers to a set (called carrier set or underlying set) with one or more finitary operations defined on it that satisfies a list of axioms. Examples of algebraic structures include groups, rings, fields, and lattices. More complex structures can be defined by introducing multiple operations, different underlying sets, or by altering the defining axioms. Examples of more complex algebraic structures include vector spaces, modules, and algebras.


This is an exercise from Alhfors Complex Analysis book- to show that an analytic function with a nonessential singularity at infinity must be a polynomial. It seems like it should probably be pretty straight forward, but I must be missing something. If it has a removable singularity at infinity then it extends to an analytic function on the Riemann sphere, and so must be constant by Liouville's theorem. What if there is a pole at infinity though? This was homework some time ago, and I never finished it :/ but have been thinking about it again recently. Thanks :)


What is the word for "stealing things during natural disasters"? For example, two guys are listening to the news and hear that there will be a huge typhoon tomorrow, and that there will also be no electricity. So, these two guys decide to steal things from a shop on the day of the disaster. Is there any single word for "stealing things during natural disasters"?


I'm looking for a reference for something I thought was easy and well known. There are (at least) two definitions of expander graphs. There is a combinatorial definition via edge expansion, and an algebraic definition using the spectral gap. Neither of these definitions require the graph to be regular. Now, I always thought that the Cheeger inequalities implied that these definitions were equivalent up to the constants. However, when I looked up the Cheeger inequalities it seems that they only talk about regular graphs. Is there a version of Cheeger's inequalities for nonregular graphs as well? In general, is it true that a family of (not necessarily regular) graphs is a family of expanders in the first sense iff they are expanders in the second sense?


I am using graphicx to typeset the figures for my PhD thesis. I have large files in .eps format. However, whenever I try to typeset the file (using pdfTeX), the console says "LaTeX error: File 'Figure' not found" It is in the same directory as my .tex file, and the file will typeset if I create it as a .dvi and then convert it to .pdf. I would rather not do this though, because I want to use hyperref.


Light as an electromagnetic wave can be polarized in different ways, e.g. linear or circular. As far as I understand it currently this can be compared to the spin direction of a propagation electron (spin angular momentum of light). Now I have learned that an electromagnetic wave can also have an orbital angular momentum, which is described for example in Wikipedia as a kind of shifted wavefront. While trying to get my head around these phenomena I was thinking about other wave-like objects in physics besides photons. Do all particles and/or excitations (e.g. a spin wave) exhibit orbital angular momentum and what are the physical consequences?


I'm writing an essay right now and I'm deliberating whether or not I should use Pagan gods instead of Greek gods (to provide variation in the essay). I've looked up the word pagan in the dictionary and it mentioned it was a derogatory term. Would it be acceptable to use it though in an essay or would it be inconsiderate for me to use that term?


I'm trying to phrase a sentence in which I'm wishing that something goes easily for someone, but can't get the wording to sound right, and not awkward. For example, someone may be studying for a test and I wish that their studying goes easy. These are the phrases I could come up with. I hope it goes easy for you. I hope the studying goes easy for you. I wish you the best in studying. I'm hoping the studying goes easily for you. Good luck studying. (I don't really like this one since it's a bit vague) Any other suggestions?


Does anybody know the source of this idiom or have an explanation of how it originated? I know it means that the speaker does not trust the person in question, but I want to know the etymology of the idiom. How did it mutate from something literal into this? Obviously trust is not measured in the same way as distance, so I want to understand the correlation in this context.


Is there any good reference for Poisson geometry/Poisson manifolds out there? I would like to give a deep look to the subject, but all I seem to be able to find are short chapters or interludes in text about symplectic geometry. Looking a bit around on the web, the only thing I found is this book. Has anybody read it? And if so, is it good or is there something better? For those left wondering: the book in the link above seems a good introduction. A more advanced text is given by the book Poisson Geometry, Deformation Quantisation and Group Representations (Cambridge University Press).


I'm not a mathematician, but I'm teaching a bit of algebra to some budding logicians, and introducing them to/reminding them of the notions of isomorphism, homomorphism, etc. I'd like to give them an example of an endomorphism which isn't an automorphism, so that they can see the point of there being a name for these separate concepts. I'd also like the example to be as simple as possible, ideally just with some infinite group. But it's proving to be harder than I expected to do this. Every candidate I've come up with turns out to be either an automorphism or not really a homomorphism in the first place. Suggestions, please? EDIT: Should have said explicitly from the beginning, I'm hoping for an example where the homomorphism is surjective.


Is the probability space associated with a compact group with Haar probability always a standard probability space? I recall seeing somewhere the fact that if the topology generating the Borel sigma algebra is not metrizable then the probability space is not standard, or something along these lines. If so then any non-metrizable compact group would show the answer to the above question to be negative, but I am unable to find any firm reference for the above fact. Any help with answering the above question would be much appreciated.


Please help me determine whether or not to include a comma at the end of the third sentence: "I won't be talking to my ex at the party tonight." -"But what if she talks to you?" "I won't be listening to her either." -or- "I won't be talking to my ex at the party tonight." -"But what if she talks to you?" "I won't be listening to her, either." Thanks!


Can anyone provide a word which describes a person who shows a lot of enthusiasm in the beginning but gets bored gradually and eventually leaves the task uncompleted? I looked at jaded, but it doesn't fulfill the criteria. Jaded seems to describe more of an overworked fellow, whereas the word I'm looking for would describe the nature of person. The word I'm looking for would describe a person who always exhibits this behavior.


I have a few questions about simple situations. What should I say to another person right after falling? a) I fell b) I have fallen What should I say to another person right after throwing a ball? a) I threw the ball b) I have thrown the ball What should I say to another person right after taking something? a) I took b) I have taken Please provide explanations for your answers.


I live in Thailand and we pronounce "w" and "v" the same. When I spoke to American people they told me that the "v" sound was different from "w". They told me to move my lower lip to the upper teeth, then say "v", but I don't get it. I want to know how to pronounce "v" in English. If you can suggest me how to practice it, that would be great!


I was looking at the ice formed in my refrigerator and found out there were a lot of air bubbles inside it. The shape of the air pockets seemed strange to me. There were many small spherical ones and the bigger ones were in the shape of vertical ellipses. I am not able to understand what's the cause of these different shapes of bubbles. can anyone explain?


The following is from the MIT-Harvard Tournament: You are trapped in ancient Japan, and a giant enemy crab is approaching! You must defeat it by cutting off its two claws and six legs and attacking its weak point for massive damage. You cannot cut off any of its claws until you cut off at least three of its legs, and you cannot attack its weak point until you have cut off all of its claws and legs. In how many ways can you defeat the giant enemy crab? (Note that the legs are distinguishable, as are the claws.)


I am a physics undergrad interested in stuff like dynamical systems, chaos theory etc. Is there ongoing research in these fields? I am talking about pure research and not applications to things like weather etc? I hope this question is appropriate for Physics SE. I asked this question, because I browsed through the websites of the physics departments of a few renowned universities (MIT, Princeton, Caltech) etc, but nowhere it mentioned research in these areas.


I picked up a copy of G.H. Hardy's 'Divergent Series' a few days ago. So far I love it, as I love the ideas associated with sequences and series, but I am finding it a bit difficult to understand. I assumed I knew everything I would need to understand it(Calculus I/II, etc.), but perhaps not. What would the possible prerequisites for understanding this book be? Also, does anyone have any tactics for getting the best comprehension/understanding of the concepts in this book? Thanks!


I know that the proper way to use "not too long ago" is: "Not too long ago, contractors used to build houses and sell them to dealers. It was the responsibility of the dealers to provide financing to potential home buyers." Instead of these two sentences, can I use: "If not too long ago, contractors used to build houses and sell them to dealers, who were responsible for providing financing to potential home buyers, today most builders also offer financing alternatives."? My question is whether "if not too long ago" is correct or not. If it's not correct, I would highly appreciate it if someone could explain why. Thank you, Maria


I created a table and it starts from the left and goes all the way to right, which looks really skewed. So I want to center align the whole table but couldn't figure out how to do it. I'm new to latex so I don't know how to use other plugin or macros, but I suppose there's a command for this basic functionality. Can anyone tell me?


I am working with certain input-output maps that can be thought of as large RLC-networks. I thought maybe this might be a place to get some thoughts/ideas/answers. My basic question is, given some large connected RLC network (all linear and ideal elements) and two ports in the network, say 'a' and 'b', can the transfer-function between the ports have non-minimum phase zeros? More generally, is it true that every transfer function representing an RLC-circuit network is minimum phase? I suspect the answer is true, but I am having a hard time proving it. Thanks!


After taking an introductions to proofs course and abstract algebra, I have been trying to study from Rudin's Principles of Mathematical Analysis. Unfortunately, I still find it very very difficult to read through Rudin let alone do the exercises. Sometimes I spend an entire week trying to understand a proof but get nowhere. At this point, would it be advisible to read a more introductory text in analysis or continue to work through Rudin? Or try reading some more abstract algebra?


I am studying English and I have a question. I found it in my text book written by some teacher who is not a native speaker. "Those who came first and in greatest numbers to make their homes on the eastern coast of North America were mostly from England." I do understand the meaning of this sentence, but I cannot understand why the word 'greatest', Superlative adjective, is not accompanied by the word "the". Is there any specific reason for that?


How do you parse the following sentence? The lamp by which she read the book grew dimmer. My friend says that "by which she read the book" qualifies "the lamp". So, it would basically mean "The lamp that she was using to read grew dimmer." For some reason, I comprehend the sentence as "The lamp that she was near while reading the book grew dimmer." Can it go both ways, or...? Also, about the usage of "grew dimmer" there, should it be "dimmed" instead? Thanks.


Look at the endings of the following words: blight, bought, breadth, brought, caught, delight, depth, fifth, fought, fourth, fright, freight, height, light, plight, taught, thought, width, ... As you may have already noticed, some of the words in that list end in ht, whereas the rest of them end in th. I have some questions regarding those endings: I. Do all words of the English language that end with ht end, actually, with ght? II. Is there any rule of thumb out there that helps one to recall (or determine) when the ending of a given word is th and not ht (or the other way around)? Thanks in advance for your insightful replies!


I'm well aware that bulwark refers to something used as a fortification or a defense. However, I was wondering if the word bulwarker is an acceptable word to use in English. I know it's possible to be a bulwark ("He built a bulwark upon the hill."), and I know you can bulwark something from something else ("I will do whatever I can to bulwark the attack."). However, I'm unsure if it is possible to be a bulwarker.


I am entering an interesting team based math contest called the purple comet, and quite a lot of questions on this contest involve Diophantine equations. For this contest, you are given a computer, and I was thinking of making a program that solved a linear Diophantine equation. The issue with this is that I can barely solve one by hand, much less make a rigorous algorithm to solve one. I am aware of Euclid's algorithm, but when the numbers get big, this method gets pretty inefficient, not to mention that it would be kind of a nasty thing to program. Are there any other smart algorithms out there for solving a linear Diophantine equation?


I was writing an essay when Word picked up an error in my writing; specifically, it wanted to replace "vast majority" with "clear majority." So my question is: is "clear majority" better, more eloquent, or clearer than "vast majority?" If so, how? A similar post here talks about the differences between similar expressions, but I want to know about the difference between "vast majority" and "clear majority."


When speaking about turbulence in fluid mechanics many times authors speak about "premultiplied energy spectra" but they never explain what really it is. I have searched for a definition but I did not manage to find it. I guess that is something related about the real energy contained in a region of scales when using a logarithm scale representation but I am not sure. So, what are these "premultiplied energy spectra"?


I have been looking around to find an answer to my question, but I was unable to find one that addressed my specific problem. I want to create the following sentence, but I'm not sure whether it's correct. "Not only the world has changed during this time but also we have in particular." I apologize in advance if I have overlooked an easy solution to this.


I'm looking for a term to accurately describe a person who can only read but cannot write. While I'm primarily concerned with people who have never learned to write, I would also be interested in any additional terms used to represent people who have lost the ability to write as a result of disuse. Both illiterate and unlettered imply an inability to read as well as to write.


I'm a bit confused. QFT is claimed to incorporate both Quantum Mechanics and Special Relativity. Therefore it should address the problem of non-locality caused by entanglement. However when I search for an answer on the Internet, I found nothing. I'm not complaining. But it seems that most people only use QFT to do some fancy particle stuff and forgot we should care more about the more fundamental stuff.


I've noticed that, when I'm driving and it's raining, the faster I go the more rain I get on the windshield and the faster I have to run the wipers to compensate. When I'm stopped at a red light I can just about turn the wipers off, but once we get moving again I have to turn them up so I can see clearly. Intuitively I would think that, assuming a statistically even distribution of rainfall throughout the local area, speed should not matter because for every raindrop I move into the path of, I'm moving out from under another one at the same time. But this is definitely not what I observe in actual driving conditions, so what's going on?


I am studying Linear Algebra. I have faced a problem to understand Symmetric Matrix with spectral decomposition. After I studied spectral decomposition, the next page in my book talks about a positive definite matrix and quadratic form. I am kind of lost what relationships are there between symmetric decomposition, a positive definite matrix, and quadratic form. Hope I can have some explanations. Thank you in advance.


According to Collins English Dictionary, and of course as everyone knows, a doula is a woman who is trained to provide support to women and their families during pregnancy, childbirth, and the period of time following the birth However, the definition in wiktionary says: A support person, usually female, who may not have medical or midwifery training, who provides emotional assistance to a mother or pregnant couple before, during or after childbirth. Is pregnant couple actually an acceptable term for referring to an expectant couple?


I'm about to wrap a course in partial differential equations. We've discussed the heat/wave equations and introductory Fourier Analysis. I'd like to do some reading into the field of thermodynamics. Would it be best to start with an introductory text given to most lower-division under graduates or should I begin with something with a little more meat on the bones? By meat on the bones, I mean a bit more rigor and explanation for the fundamental theorems and applications. If this is not clear enough please ask for clarification, I'll be glad to provide it.


I know that air pressure and temperature are inversely proportional. Now I saw in a book that "Atmospheric pressure decreases as we go higher and higher." But at greater heights the temperature becomes low, and so the air pressure would be high. But it is given atmospheric pressure decreases with altitude. I understand that air pressure and atmospheric pressure are different. But I can't understand how they are different.


Given two (or more) loans of different balances and interest rates and a single amount of funds available per payment period, is there a way to calculate the best way to split the available funds to minimize the amount of money repaid? I'm not really sure where to start to look into this type of question so if someone could point me in the right direction or provide an answer that would be great!


In my physics classes, I remember it being repeated a few times that gravity is a much weaker force than the other three fundamental forces, and being told this is an open problem in physics. However, I don't understand why this is a problem. I see no intrinsic reason why the four fundamental forces should have similar strengths in the first place. So why then is the fact that gravity is many orders of magnitude weaker than the other fundamental forces a problem? Note: I've attempted to read the Wikipedia article on the Hierarchy problem, but it is far too technical for me.


I'm reading a paper titled "Luminescence spectra of quantum dots in microcavities II Fermions" (link). In section III, the authors introduce so-called closing operators. I don't have enough background knowledge to understand the technique there. So it's natural for me to look for other literature or books on the topic. But it turns out the "closing operator" is defined only by these authors and I couldn't find any literature on this. Could you give me any key words which can lead me to related literature? Or you can point me directly to a literature.


Generally speaking when we say "our friend" we are referring to someone who is a friend of ours. However, I have noticed that at times these words are used sarcastically to refer to someone who is actually not a friend but an enemy/adversary. So with respect to the sentence below, is the term our friend used as a form of irony? Our friends the Saudis, who never persecute women, gays and other religious minority groups, have somehow managed to lead the UN Human Rights Council


I'm starting my last year of high school, and I will have to do an all year research on a mathematical topic. I'm a really passionate learner and I'm very involved in computer science. I am stuck on finding an idea that will both be challenging and fun. I would also really love to integrate computer science in it, since that's what I want to study afterwards. Thank you in advance.


Some dictionaries define a scalar as follows: A quantity, such as mass, length, or speed, that is completely specified by its magnitude and has no direction. -- The Free Dictionary However, it is my impression that in many contexts scalars can be signed, in which case their magnitude (their absolute value) does not specify its value. This definition is even used on a test question here. Is it true that this definition is inaccurate?


I'm currently taking a numerical analysis course. We are covering linear algebra topics, the gist of the first chapter of the course being solving systems of linear equations. The lecturer has introduced SVD decomposition, condition number of a matrix, LU decomposition and QR decomposition (using Householder decomposition). In the past I've been used to very rigorous, thorough and well-organized math courses. But I feel like what I'm learning here is shallow since there are very few proofs. So my question is Can you recommend rigorous textbooks or detailed lecture notes that cover the topics I mentioned above ?


So I'm trying to prove that the set of polynomials in C([a,b],R) is not open. I understand the definition of an open set, but I'm wondering how to find a point that is not contained in the interior if it's the set of polynomials. Also, Can a subset of a metric space ever be both open and dense? I'm thinking it can, only if the interior is empty. ? Would love help with these! Thanks in advance!


Why is a homogeneous function called homogeneous? When I ask this, I don't mean, "Show me how to algebraically manipulate a function whose input has been multiplied by a constant to get the original function multiplied by the same constant." I mean--why do we use the word "homogeneous"? That word in particular must have been chosen for a reason; what is it meant to communicate in this context?


I know how to solve linear homogeneous ordinary differential equations with constant coefficients using the differential operator D, by using this method. Is it possible to use a similar method (using the differential operator) to solve more advanced ODEs? I'm thinking of both more advanced linear ODEs, such as Euler-Cauchy differential equations, as well as non-linear ODEs. Are there any articles on the web on this topic, or even textbooks that use this method to solve ODEs?


One oddity I notice is that if a random or quasi-random data series such as price data is plotted, it is similar to a random walk, but if the same data series is plotted using candlesticks (open-close, high-low), there appears to be in many cases an undulating pattern. I assume that this is caused by some kind of aliasing. Is there a mathematical explanation for it?


I am an mechanical engineering student so I'm kind of ashamed to ask this question but I have a weak math background and am digging into some of my knowledge gaps. So my question is where are all of the constants of integration and why are they generally ignored? Is it simply because they are usually treated to be zero? I guess what I'm looking for is an intuition in regards to integration constants...i.e. how do I wrap my mind around why the integration constant is/was ignored, and how can I truly understand what is going on when these constants are ignored. Any insights or conversation is greatly appreciated. Thanks in advance!


I don't know where I've heard such expression. I try to explain (for sales) what our software tool does and I use this sentence: "It helps you to deploy apps in a very simple fashion". And then I describe the steps of how it works. I googled this expression and I found only one reddit post. Is it a widely used expression in the US? Will native speakers understand me? What about non-native speakers like EU folks, will they? Can I rephrase it as "in the simplest possible fashion"? Thanks!


My background : I'm currently in my junior year in college majoring in astrophysics. I have taken GR, and some intro cosmology here and there, but nothing in great depth. I had my heart set on taking a full physical cosmology class this spring. As luck would have it, I can't take the physical cosmology class due to scheduling conflict ! I intend to proceed to graduate school in astrophysics, so I would have to take the graduate level cosmology then, but in the mean time, I was thinking of teaching myself undergrad cosmology out of Barbara Ryden's book. My question is : any advise for undergrads teaching themselves this subject matter ? Are there any "go to" resources? is Ryden a good choice?


I was wondering if it's possible to construct a table in LaTeX which has a fixed header so when we scroll down the page of the output PDF, we can still read the column names and keep track of what the numbers in the table represent. I have seen in a post here at stackoverflow about how to do this with HTML and someone had linked to this example: http://www.mustafaozcan.net/en/demo/fixedtableheader/jquery-fixedtableheader-demo-en.html I would like to be able to do that in a PDF produced by LaTeX. Is that possible?


I was doing some calculations in MATLAB, and noticed a pattern that may be obvious to stats experts, but I didn't notice it before. If I have a time-series, and I remove a linear trend / detrend it, I find that the mean of the detrended data is zero, as expected. However, I was surprised to find out that the standard deviation of both the detrended data and original data is roughly the same. Is this what one should expect? Thanks!


I am just beginning to read about the use of "Concave Programming" methods and use of the Karush-Kuhn-Tucker conditions to identify the maximum value of a non-linear objective function subject to inequality constraints. The examples I have seen in the text I have at hand, all involve only linear constraints. Is this method equally applicable to situations where not only are there multiple constraints, but where one or more of those constraints are non-linear ?


Wikipedia has the following quote on John von Neumann: Stan Ulam, who knew von Neumann well, described his mastery of mathematics this way: "Most mathematicians know one method. For example, Norbert Wiener had mastered Fourier transforms. Some mathematicians have mastered two methods and might really impress someone who knows only one of them. John von Neumann had mastered three methods." He went on to explain that the three methods were: A facility with the symbolic manipulation of linear operators; An intuitive feeling for the logical structure of any new mathematical theory; An intuitive feeling for the combinatorial superstructure of new theories. And I am wondering what do 'logical structure' and 'combinatorial superstructure' mean in this context? Please explain these methods. https://en.wikipedia.org/wiki/John_von_Neumann#Mastery_of_mathematics


I am studying client's request arrival patterns on web and application servers. About web server's request arrival pattern I read that "The request arrival rate on web server follows Poisson distribution". And about application server I read the sentence "The request arrival rate on application server follows exponential distribution. Now kindly explain " is there any difference between Poisson and exponential distribution in the context of client's request arrival pattern on server". Is there any difference between poisson and exponential distributions?


I'm working on a project right now and part of the introduction is about energy, and I'm trying to introduce the notion of energy in an unconventional way. Right now, I am looking for examples of situations in physics and chemistry where energy plays a key role. Another way of putting it is: I am looking for questions where the answer is energy. I'm basically trying to motivate the concept of energy and demonstrate why it is important. Does anybody have ideas of seemingly fundamental occurrences where a true answer is provided by the notion of energy? Thanks!


When used as an expletive, the name Jesus Christ often gets an H inserted into the middle of it for some reason. I've heard lots of guesses about what the H stands for, the most popular one being Henry, but no one seems to actually know, and I've never gotten any good explanation of where it came from or why it became popular to insert it. Is there any documentation of when Jesus H. Christ as opposed to simply Jesus Christ came into currency as an expletive? Is there any authoritative explanation of what, if anything, the H was originally supposed to stand for?


In high school a while ago, I learned a bunch of random terms for words and phrases: synecdoche, antonym, oxymoron, palindrome, etc. There's one term that escapes me but I can give examples of it. I don't love my wife because she is beautiful but she is beautiful because I love her. We travel not to escape life, but for life not to escape us. What is the name of this term? Thanks in advance, it's killing me not being able to remember.


D-branes, as I currently understand them, are submanifolds of spacetime on which open strings can end with Dirichlet boundary conditions. On the other hand, type II string theory is a theory of oriented closed strings - the difference between IIA and IIB coming from the choice of GSO projection. Based on these observations, I have the following questions: If type II string theory is built from closed strings, why are D-branes considered in the theory? If open strings are included in type II, does the GSO projection act on the open strings? If so, how?


Trying to show that if a sequence converges, it either has a maximum, a minimum or both, I reached a dead-end. Assuming it is not constant, it is still bounded and its supremum and infimum aren't equal. Then I assumed that the supremum and infimum are not in the sequence. I want to show that there are two subsequences that converge to each of them but for that to happen I have to show they are accumulation points. I tried to use definition but failed. I know logically that following my assumption they have to be accumulation points but I can't derive it from the definitions. Any help?


I'm working on finding whether sequences converge or diverge. If it converges, I need to find where it converges to. From my understanding, to find whether a sequence converges, I simply have to find the limit of the function. I'm having trouble getting started on this one (as well as one more, but I'll stick to one at a time). I would appreciate if someone could explain how I should start this one.


This may not be the correct place to ask such a question. I have read a mathematical paper on multiclass total variation clustering. I wish to use the algorithm in the contents to compare with another clustering result. My question is: Is there a proper etiquette for asking the author(s) of the paper for their code? Is it rude to do so? Should I just try to figure it out on my own? Or is this something that is common? One reason why I want to ask is because I am short on time and may not be able to figure it out by my deadline. Any comments would be greatly appreciated.


When writing a book on popular culture, I'm using italics to introduce a film title into my paragraph. In the next paragraph I refer to the film title again. Should I use italics on the second occasion, or stick with normal text? I'm tempted to go for the latter - as over-use of italics would spoil the flow of the text, and the title has already been introduced to the reader. Thoughts?


This question may sound naive but still I haven't seen it being discussed in any book. Why do we parametrize surfaces with two variables as opposed to curves? Initially I thought that it is not possible to fill enough points in the space with one variable but there are space filling curves so it is certainly not the reason. Is it because we need to move in two directions?


So I've modeled the interactions between the sun and all the planets (and the interactions between the planets) using Verlet integration. I've used data from Wikipedia for masses, distance from the sun etc. I initialized the initial velocities of the planets via the critical velocity equation. This produces nice stable velocities. I'm unsure of how to calculate the initial velocity of the moon so that it stays in orbit around the earth.


I just learned from Wikipedia that coproduct of two (commutative) rings is given by tensor product over integers, and that coproduct of a family of rings is given by a "construction analogous to the free product of groups." Can the tensor product approach be generalized to an arbitrary family of rings? (Infinite tensor product?) I'm a little surprised that coproduct of commutative rings requires noncommutative structure (free group). Does someone have a reference which explicitly constructs the coproduct?


If I have a binary number W bits wide, initially all set to zero, and I repeatedly pick a random bit and toggle it from zero to one or vice versa, how many times would I need to do this to achieve maximum entropy? I hope I am using the term "maximum entropy" correctly -- what I mean is a point where the distribution of ones and zeros is as random as possible, and no amount of continued toggling will make the distribution any more random.


Any given square is always a rectangle, but a rectangle isn't necessarily a square, so squares and rectangles have a _ relationship. I've been noticing this sort of thing everywhere ever since I noticed that I didn't know a good word for it. I've been calling it a container relationship because one class of things is contained within a larger class of things. However, a "container" relationship doesn't sound very good and doesn't really convey the meaning on its own. A more elegant word would be helpful. Edit: A specific definition would be something like: a is always b. b is sometimes a.


I'm not a physicist, but I'm really interested in Nanotechnology. I've a question, and I hope that I can be provided by an answer. I was reading some articles about the one-dimensional CdSe nanostructures. My question is why it's called one-dimensional? Is it because of their geometrical shapes or something else? I truly appreciate any hint or a resource could lead to the answer. Thanks.


Let's say I have an isolated, adiabatic chamber which is divided into two parts with a massless, frictionless piston. One part has vacuum, while the other has a gas. In the beginning, they are at equilibrium. I release the piston, and the gas expands, thereby pushing the piston towards the end of the box. Since the process is a free expansion, there is no work done by the gas. One could thus say, that the gas applied no force whatsoever, on the piston. Then why did the piston move in the first place?


What does the expression "eat shit" represent in the following sentences? Eat shit, I'm not going to do your dirty work. Is this similar to "I dislike doing your dirty work"? Or does it mean "Go to hell."? Eat shit, I am not going to follow you. I found many different interpretations of this expression on Urban Dictionary, so I am quite confused. Please could someone clarify.


What I want to ask is what would happen if you split a single light ray (a photon) into two using a half silvered mirror? Would the emerging light rays have half the amplitude as the original one or half the frequency of the original wave? Basically will the frequency get distributed or the amplitude? EDIT I specifically mean a "Single" photon or a single light ray. For those who think a single photon cannot be split, you are wrong. A photon has been split by using special crystals. It is called Parametric down conversion.


If the text I am quoting is italicised or bold, does my quote need to include this special font treatment? If I want to emphasise something in the quote, can I make words italic? For example, suppose that the text I'm quoting is: I love apples. I really love them! Can I quote that as "I love apples. I really love them!"? Also, can I quote the beginning as "I love apples,"? Thanks!


I don't understand how to solve problems of conditional probability.Here's a problem which I fail to understand."Consider all families with two children and assume that each child is equally likely to be a boy or a girl.If such a family is chosen at random and found to have a boy,then what is the probability that it has another boy."Please help me to solve this problem.


I have question which I'm unsure of whether or not I am thinking about it right. If an object is sliding down frictionless incline and it then comes to the "foot" of the incline where it encounters friction, how far would it travel. The foot of the incline is horizontal. The only information given was height of the incline and the coefficient of kinetic friction. I used conservation of energy to find the velocity at the foot of the incline. After, I solved for horizontal displacement using kinetic energy = displacement x mass x gravitational acceleration x coefficient of kinetic friction. Can anyone offer some insight into whether or not this is a good approach? Thank you.


The following true/false question was posed: An isomorphism between to vector spaces can always be represented by a square singular matrix. This is not true. I know that in the case of finite dimensional vector spaces: A map is an isomorphism iff it can be represented by a square non-singular matrix. I would like to know whether it is possible to prove this in general (i.e for all vector spaces)?


Initially, when the current starts flowing the temperature of the filament is less and thus resistance is less. So, more current flows through the bulb and the power delivered to the bulb is more. As, time passes and the filament starts glowing at a steady rate its temperature has increased which in turn has increased its resistance. So, less current flows through it and thus less power is delivered. Is this the case? Please help me understand the situation.


I'm a programmer and I'm looking for a single word that works for the following scenario: I have a list of basic information about people, I call this list the 'user repository'. I need a word for "to view and change more detailed information about a person". I've thought of using words like refine and elaborate, but they don't seem to quite fit the idea that the data can be viewed and changed.


Suppose that a person buys a packet of sugar. On the way home, the packet falls off his hands and the sugar scatters on the ground. Nobody is expected to gather the sugar again, but that person does so; of course not for his own use anymore, but to donate it to the poor! Obviously, if the packet hadn't fallen down, he wouldn't have decided to donate it. It goes without saying that this kind of donation and charity is morally worthless and not considered a good deed. Is there any English idiom to describe this kind of charity or donation? Thanks in advance.


I tutor student in SAT Reading. One of the SAT tests has the following sentence: "With their dark dungeons and damp living quarters, medieval castles offered few of the comforts to be found in royal palaces." The student thought this must be wrong as "to be" seemed superfluous. To me, there's a subtle difference in meaning if the sentence was changed to "few of the comforts found in royal palaces." The only way I could describe it was that "to be" adds a more theoretical and abstract quality to the sentence. Is there a clearer way to put all this and the change "to be" makes? Thx in advance, J


This is something that crops up on the BBC a lot and irks me. For example, from a story today: One woman is critically ill and three others have been injured after they were stabbed near a supermarket car park in west London. To me, you are ill (or sick) if you have, for example, contracted a disease and using it in the sense of someone being injured is wrong. Google gives the definition of ill as "suffering from an illness or disease or feeling unwell." and I'm sure that a car crash or stabbing would certainly make you feel unwell, but I still don't think it's right. Is it?


So i am currently learning about double integrals and to be honest i am not really understanding how we compute the volume over general regions. Here is the proof provided in my text book. It all makes sense until the last part. Since we use Fubini's theorem we can change the order in which we integrate. Since that is true wouldn't that mean that the Fubinis Theorem applies to all double integrals? Since this is obviously not true can you point out what am i doing wrong? IF you need any more clarification/photos please comment.


I am applying for a new job; I currently work in my family-owned F&B firm. Now, in the job portal, there is a column where I have to mention my current company. I don't want to give the name of my firm as it is not well-known and also, it is not in the same domain as the job I am applying for. I have currently put down "Own F&B establishment" in the column. But the "own" part seems kind of odd to me. Is there any other word or phrase that I can use instead, which would indicate I am not just working there, but am a stakeholder.


It is right on the tip of my tongue, and I have searched the online thesauruses but I can't come up with it. What's the word that could fill in the blank: He came to the ___ conclusion that you must be willing to serve if you want to truly lead. The best I could come up with is "non-intuitive", but that just doesn't sound right. I know there's a better one. I want a word that means "this is not what you would expect", "this goes against your intuition". Something that approaches "nonsensical" or "irrational" but without the disparaging overtones, yet more forceful than "surprising".


I am struggling on this one question, where it is asking to define an XOR automata which is defined as an NFA and it is defined as the following: N accepts the string x if the number of distinct accepting states that can be reached by a path labeled with x is odd. I'm supposed to prove that any languages accepted by the XOR automata is regular and that any regular language can be accepted by the XOR automata. Can anyone explain to me how I may be able to go about starting this proof? Thanks.


My question is simple. I'm an undergraduate student taking Partial Differential Equations. I'm looking for a problem book with solutions. My intent is to just do problem after problem to practice for my final and our assigned PDE book doesn't have the answers, so there's no way of checking. Would anyone happen to know of a good PDE book that has problems and answers (or if possible a solution manual) that I can find online? The book we're currently using is: Basic Partial Differential Equations by David Bleecker and George Csordas. Thank you


I'm looking for an adjective that describes the fact that a ship is sailing at sea. Much as "afoot" describes a person being in the state of walking. I was hoping "asail" would be a word, but can't find it from Google. The word would replace the square brackets in the following example: In Plato's ship of state metaphor, a ship is [sailing at sea], helmed by a captain ...


The Large Hadron Collider (LHC) "remains one of the largest and most complex experimental facilities ever built" (Wikipedia); it may even be the most complex project in humankind's history (?). Such projects usually have impact beyond their original target and boost science and technology in a non-trivial way. I wonder, then, what kind of impact it has or might have on other science and technology fields, in particular on mathematics (if any), and what specific impact has mathematics had (or might have) on the LHC project (if any)? Cross-post on mathoverflow: Impact of LHC on math.


I realize that "kinda" is an informal form of "kind of". However, would "kinda" be appropriate in fiction or dialogue? Or would it be more acceptable to stick to "kind of"? This is for a fiction novel from first person view. Additionally, I found that when I write, I tend to use "kinda" for more informal sentences ("it was kinda freaky") and "kind of" for more serious sentences, I would use "kind of" ("I kind of knew how they felt"). Is this acceptable or would it be preferable to use "kinda" or "kind of" consistently?


We know that a Hilbert space is separable if and only if it has a countable orthonormal basis. What I want to ask is If a Hilbert space has an uncountable orthonormal basis, does it mean that it is not separable? Or equivalently, does it imply that the Hilbert space does not have a countable basis? I know that if a vector space has infinite number of linearly independent vectors then it cannot have a finite (Hamel) basis. But here we do not deal with Hamel basis but with a complete orthonormal set, do I cannot apply the usual techniques. Any ideas?


I am having lots of trouble with this series. I will upload a picture of what I did. According to Wolfram, the series converges by the ratio test. I also don't know how I would use a different test for such a series. Two of the limits weren't difficult, but obviously I have to do something else, since I am not getting the right answer. We haven't covered Stirling approximation if it applies here, so I'd like some help without using it. Thank you so much.


In chemistry we learned about the penetrating power of three common types of radiation: alpha, beta, and gamma. Alpha can be stopped by paper, beta is stopped by a sheet of metal (I think) and gamma is stopped by lead. In the same unit, we talked a lot about nuclear weapons, but not once discussed the neutron radiation that causes the chain reaction in nuclear weapons. What kind of penetrating power does neutron radiation have in comparison to the other three we learned about?


I only have a basic, mostly classical understanding of physics and am just starting to learn about photons. Am I correct in saying that, as a photon moves through space, the strength of its electric field varies? If I am correct, what is the qualitative nature of this field? Is it positive (would repel a nearby proton it was passing by) or negative (would attract a nearby proton it was passing by)? Does it switch between positive and negative? Or would it have no effect on a proton it was passing by (the field is neither positive or negative)? If the field is neither, how is it even possible for an electric field to exist without being positive or negative? What would such a field even mean?


I know what "begging the question" originally means, but I just can't make any sense of the idiom. The phrase really seems to have nothing to do with its own meaning. The original Latin phrase, petitio principii, is often translated as "assuming the initial point," which quite simply explains the practice. Does the phrase "begging the question" carry any meaning (related to what it's used for)?


I was wondering why it is almost impossible to find a geometrical explanation of why adding two linear equations helps us to find a solution of a system of linear equations? Am I right that adding two linear equations will result in an equation of a line which will pass through a point where two linear equations intersect? If it is right then I completely don't understand why such a crucial point in understanding how to solve systems of equations is never properly taught even at a university level.


According to the Bernoulli's equation, if velocity decreases, then pressure increases. I am trying to understand the Bernoulli's effect based on a situation. Suppose we have a stream of water. Let's assume it is an ideal fluid. Imagine the water flows out from a wider pipe to a narrower pipe. Since the area decreases, according to the Continuity equation, velocity of water molecules increase. This causes an decrease in pressure. I don't understand the last part. If water molecules' velocity increase, then their kinetic energy also increases. Wouldn't this causes more collision between pipe's wall and water molecules, thus giving higher pressure?


I hope this isn't an inappropriate question here! I'd like to ask the following (perhaps slightly ill-posed) question: why do we study prime ideals in general (commutative or non-commutative) rings? The situation is somewhat clear to me in some nice cases such as Dedekind rings, but even in slightly more general settings (e.g. general commutative Noetherian rings) the theory becomes rather horrible. What benefit do we gain from an understanding of the prime ideals of a ring? Thank you!


The following quote from Anthony Burgess has a comma after it: There is, in fact, not much point in writing a novel unless you can show the possibility of moral transformation, or an increase in wisdom, operating in your chief character or characters. But this quote from William S. Burroughs doesn't: There is in fact something obscene and sinister about photography, a desire to imprison, to incorporate, a sexual intensity of pursuit. When should you use a comma after "there is in fact"? Is it necessary to put a comma after "there is in fact"?


While learning about shock waves in an introductory Gas Dynamics course, it was revealed that normal shocks are unstable if formed in a converging channel. Even if the local conditions ostensibly require the presence of a shock in the converging section, the flow instead chooses to reinvent itself, moving the shock wave to a diverging section while simultaneously altering the upstream conditions. I can verify that this is a genuine phenomenon, but is there any formal explanation in terms of the underlying flow physics?


We are struggling to find the correct word(s) that describes a list of all messages in the system that are sorted by recency. Trending does not work because what we are displaying is not what is popular in the system, only the most recent. Recent does not work because it implies there is a limit to the number of messages we display. All does not work because it does not imply that they are sorted by recency. Any hints or suggestions are welcome. Thanks!


In an electron gun, the heating filament heats the cathode, releasing electrons by thermionic emission. I've read that "electrons are negatively charged particles and the positively charged cylindrical anode develops a strong electric field that exerts a force on the electrons, accelerating them along the tube". However, I don't think that this explanation is very clear, and I was wondering specifically how the "strong electric field" inside the cylindrical anode is able to accelerate the electrons?


My understanding is that in quantum mechanics the wavefunction may be expressed as a function or as a ket vector (composed of many orthogonal ket vectors). I'm not too sure about the further differences between these two approaches. For example, I am familiar with operating on the wavefunction as a function with the hamiltonian operator, but when you use ket vectors can you use the hamiltonian operator in the same way or do you have to find matrix representations of the operators? I would also be grateful for any links to websites or books explaining the different approaches.


What is the English word for "collecting money for a special event from a group of people"? For example, say some friends are planing a party. Each one has to contribute some amount of money to the event. What is the word or phrase that can be used to explain this kind of activity? I have found "fund raising" to have a similar meaning, but I feel it's more along the lines of "collecting money for some charity event". Is that right, and is there a better word I can use?


I'm trying to write a tagline for my live blogging service, and this is what I've come up with: Live blogs that are a breeze to write, a pleasure to read and are actually updated in real-time. Is that correct grammar or should it be Live blogs that are a breeze to write, a pleasure to read and actually updated in real-time. The completions 'a breeze to write' and 'a pleasure to read' both follow '... that are'. So why does 'are actually...' sound better to me? Doesn't that repeat the 'are'?


Given any problem, how do I know whether it is solvable using Dynamic Programming? For example: consider the rod cutting problem. Now, how do I know whether dynamic programming will give me an optimal answer? Basically my question is: how do I check if a problem follows the "Principle of Optimality" - that optimal solution of every problem involves the optimal solution of a sub problem. And are there any problems that don't follow this principle? (If there aren't, my original question becomes redundant, of course!)


In mathematics we often have statements like a x <= b, where a and b are constants and x is a variable. Now there may be variables satisfying the inequality (that is the statement is true) as well as variables violating it (in which case it's false). However I don't have any words in my vocabulary to express stricter statements: Is there a word to describe that the variable x satisfies the equality a x = b? What about the case that a x < b? I remember having read "x is a root of the inequality" meaning a x = b somewhere, but I can't remember where.


Let's say me and a colleague are going to a meeting and I want to make sure his responses to the questions raised are similar to mine, that we are in agreement and we will responde in a similar manner. Can I say: "Let's make sure our responses are aligned" or, "Please let me know your views cause I want my responses to be aligned with yours."


The words "opportunism" and "opportunist" suggest the ability to take advantage of situations -- but they assume that one is doing so for one's own benefit, which makes the terms pejorative. What is a non-pejorative word in English that means "have the skill and wisdom to recognize and seize opportunities that lead to broad gains"? I am looking for something that is part of the "art" of diplomacy and politics, for example. "Clever" is a somewhat helpful adjective, but there is no such thing as a "cleverist." "Entrepreneur" means is a person who is willing to take on risk and innovate, but lacks the sense of seizing opportunities as they emerge. It seems like an important concept that has no word. Any suggestions?