As the title indicates, I'm trying to find books where the exposition of the main course of thought is done entirely or mostly in outlines of proofs, or as exercises with or without hints. I'm trying to force my reading to be more "active", and I think that such a book would be good training-wheels. No particular topics, but preferably something on the introductory level. I'm particularly interested in basic Differential Geometry and/or Algebraic Topology related topics right now. (I have "undergraduate level" background in Real Analysis, Abstract Algebra, Linear Algebra and General Topology, and I'm trying to get started with Algebraic Topology and Differential Geometry for my own personal and educational enrichment.) Texts on other topics would be welcome for later reference. Much thanks.


I just finished reading (and understanding) Steiner's proof of the isoperimetric inequality. His proof (which is sadly incomplete) seems to rely much on the fact Steiner symmetrization preserves area and decreases perimeter. The only resource I've found which provides proof of those two things are Andrejs Treibergs' beamer slides , however ,they are confusing as he is a bit heavy on notation. Can someone please point me in the right direction on where I could find an alternative resource providing detailed proofs. If possible, a proof if you don't know a resource would be highly appreciated.


Person A says to B: "I thought you were smarter/stronger/better/more trustworthy than that. I should have known better." I was thinking it would be termed passive aggressive, but after reading up on it, that doesn't seem right. I'm tripped up on how focused on the Person A the two sentences are. Typically an insult would be "You're not very smart" and that would be that. But this is "I'm disappointed in myself for thinking better of you." Is there a good term for this sort of insult?


I have been using Windedt with Texlive since many days. But I stopped using it for a while, and when I now compile a tex file, its giving error "The system cannot find the file specified". I don't know wats wrong. I have used this before. I tried re-installing Winedt, no use. I have recently changed the System Path for installing some software. Has it affected in anyway?? Please help me.


Please have a look at these two definitions: Chevalley-Eilenberg complex Koszul complex (German Wikipedia) Both are from Wikipedia pages on Lie algebra cohomology, and they look rather similar. Since I'm not very solid on the notation, I'm unsure about what's the difference between them. Furthermore, since both Wiki pages are about Lie algebra cohomology, and no page defines the "other" chain complex, I'm not even sure whether they are different at all. What's the essential difference between them?


Does a definition need to provide a unique or near-unique description or can non-unique descriptions also be categorized as definitions? For example: Is the statement "An apple is a fruit" a definition of apple? Is the statement "A car is something that gets people from place to place" a definition of car? Are these simply descriptions (or categorizations) of the objects in question or are they also poor definitions?


I've got a sci-fi based Physics question that involved Dark Energy... So I've been watching a lot of Doctor Who recently, and I'm very interested on how his "Tardis" is bigger on the inside. Here you can see the Tardis looks small on the outside. But inside, it's much bigger. So here is what I've been thinking. What if you got a box (like a Police Box), and somehow filled it with Dark Energy! Im hoping that the Dark Energy would expand space inside the box, thus making it "bigger on the inside". My question is, could this actually work? Thanks for your help. (This is just a fun question btw)


Just as the title says: is there a formal name for a convex polygon on a sphere, of which the vertices are connected not by great circle but by small circle segments? My end goal is to intersect two (or more) of such shapes, and find the area of the shape where the two small-circle polygons overlap. It would therefore be helpful to get to know some basic and/or special properties of such shapes, what equations apply, etc.


I'm trying to get the facts straight here. Suppose I'm throwing a ball with no angular momentum. It collides with the ground and Newton's third law tells us that a force opposite to the gravity will be applied to the ball at collision. During collision, the momentum will be converted to potential energy and then become kinetic energy again. Am I right to say that some of the kinetic energy can be converted to angular momentum, thus making the ball rebounce not straight up? (assuming some friction)


It's just a casual observation, so I wanted to check it: A regular spring, when not completely compressed, looks an awful lot like a sine wave. The idea of a circular shape stretched out in the third dimension also seems to support it. But I can find no mention anywhere that such a spring actually is (or isn't) shapes like a sine wave. I don't want to tell my pupils this unless it is more than just an observed similarity. So is a stretched spring, if seen from the side, shaped like a sine wave?


I want to read particularly about diophantine Analysis and Elementary Number Theory from a novice level. The books which I found on net: A Guide to Elementary Number Theory by Underwood Dudley Diophantine Analysis by Robert Daniel Carmichael Diophantine Analysis by Jorn Steuding Help me with good suggestions as I think M.SE. has people from novice to professional users. Thanks. P.S. I have searched M.SE. for this type of question, I didn't find any. If you know someone, you may give the link.


What are some quantities often used to measure the sparseness of a graph? For example, in a graph, with the number of vertices fixed, the smaller the maximum degree is, the more sparse the graph is. With the maximum degree fixed, the bigger the number of vertices is, the more sparse the graph is. For another example, the more sparse the adjacent matrix is, the more sparse the graph is. Then it leads to another question: what are some quantities to measure the sparseness of a matrix? Thanks!


I would like to know if a discontinuous local martingale with paths of finite variations almost surely is a martingale. I feel that it should be the case but can't find a straightforward argument. As noted by Did in its comments below, the continuous case follows easily from the widely known fact that continuous local martingales have paths of almost surely infinite variations. So I modified the question to make it more specifically about discontinuous local martingales. Best regards.


Prove that a) the medians b) the altitudes and c) the angle bisectors drawn from the base angle of an isosceles triangle are congruent. Looking at this problem, I feel that this statement is just false. If you have an isosceles triangle given in the following link Link. Then clearly these are not congruent. Is there something in the problem that I am missing? Thanks


I was told that a dielectric slab inserted into a capacitor connected to a battery (constant voltage) will be repelled, because the energy stored in the capacitor increases when the dielectric is inserted, due to increased capacitance. What is the physical origin of this force? The attractive force by a constant charge capacitor can be explained by fringe field, but why are the two cases different in terms of forces?


What is the official rule when mentioning a city/ state in a sentence? I always thought there was a requirement for a comma between the city and state, however is there also supposed to be a comma afterwards? Tim also enjoys football games, jumping on his motorcycle and being a guide for the Fredrick, Alabama, restaurant scene. I personally think it looks clunky, however I'm told that it is opinionated and that the above is correct. Should the sentence be written like this? Tim also enjoys football games, jumping on his motorcycle and being a guide for the Fredrick, Alabama restaurant scene.


I am new to studying abstract algebra (and math in general). I've been reading Gilligan and Pinter's books. I am trying to improve my understanding by doing exercises. However none of the books I am reading seem to come with exercise solutions. Is there an abstract algebra book with lots of exercise with solutions? I am confused as to why none of the math books come with complete exercise solutions. How do people verify that their answers are right.


Using packages like minted (thanks to Pygments), LaTeX can generate beautifully highlighted/colored source code. Literate programming tools such as noweb and cweb can generate beautifully typeset and readable documentation. Ideally, I would like that the docs generated by noweb and cweb have high quality syntax coloring, and this seems like a problem that someone on either community may have tackled. So here is the question: Does anyone know how to get noweb and related tools to use minted or a similar LaTeX package for syntax coloring?


I saw in my abstract algebra textbook that defines the gcd of a polynomial over a field (i.e. the coefficients of the polynomial is taken from a field). My question is that what happens if the field is not a real numbers? I mean intuitively, I always think of it as real numbers. Is there anything wrong in taking of it this way. Can someone give me an example of a polynomial taken from other fields such that thinking of it as real numbers will result in a misconception. Thank you.


Is there any text that I can use as a short reference for the standard techniques for solving basic ODEs? I currently have been using Boyce and diPrima as my ODEs text, and it is far too wordy for my taste. I'm also not too interested in expositions of applications of physics or phase plane analysis, as I have other books for that. Basically, I'm looking for something short that can quickly remind me how to use techniques like integrating factors, series solutions, etc. which I keep on forgetting.


I have two matrices(the first one is mxm, while the second one is nxn, m>n). They store data pertaining to human speech. The second matrix contains a data segment that acts like an acoustic "signature". I need to find where this data occurs in the first matrix. Which mathematical tool(s) or method(s) can I use to get it done? I know convolution can help me out but is this the best(more importantly fastest) way?


For each relation, determine which of these properties are present: reflexivity, symmetry, antisymmetry, and transitivity: I know the definitions of each of the properties but unclear as to how to apply them to each relation. {(a,a), (a,b), (b,a), (c,c), (d,d)} {(a,a), (a,b), (a,d), (b,a), (b,b), (b,d), (c c), (d,a), (d,b), (d,d)} {(a,a), (a,b), (a,c), (a,d), (b,b), (b,d), (c,c), (c,d), (d,d)} Does anyone know what I would need to do to solve this?


I'm looking for an adjective that describes that tasks should be approached "one-ly". To give some context: [Task] must be performed in one way and exclusively one way. or There must exist precisely one method of performing [task]. I am looking for an adjective that describes this "one-ness". The adjective does not need to specify what the method is, it should merely prescribe that the task should be approached in precisely one way. The closest word I could think of is "uniform".


I just came across the term quantifier-free first-order formula, I first thought that might be similar to a propositional formula, but then after a closer evaluation I realized there are more concepts in first-order logic then just the quantifiers. I believe the difference is the following. A quantifier-free first-order formula can still contain: Function symbols Predicates where as a propositional formula does not have these concepts. Is that correct? Did I miss something?


Please compare He shrugged. and He shrugged his shoulders. Is there anything else that can be shrugged, besides shoulders? To me it sounds like duplication when used in this way. I'm aware of constructs like "He shrugged it off." but that's not what I'm interested in, and it also implies the use of shoulders, doesn't it? So why the need to specify the shoulders as an object? [edit: I also find "he shrugged his eyebrows" but that's rather rare.]


Does the expression "to add another dimension to the situation" imply that the situation has become more complex? In Arabic we would say something like "adds another dimension to the situation that has become more complex". But I know English tends to be very economic and implicit. So if I wanted to express that idea, would "adds another dimension to the situation" be enough? Or do I have to say that the situation has become more complex?


The statement of the Cayley-Hamilton Theorem is fairly straight-forward. I now know how to find characteristic polynomials from a given matrix (or at least a matrix with certain properties that I am unaware of!). I know that the eigenvalues of the matrix are roots of the polynomial. But what does having such a polynomial mean? Wikipedia says that the characteristic polynomial "...encodes several important properties of the matrix...", but once we have switched to "matrix form" of the equation, what can we conclude? In other words, what does the Theorem do for us, besides allowing us to say, "Hey, I know a matrix solution to this polynomial"?? Is there an abstraction of this in abstract algebra (rings, fields, etc.)? Thanks for your time.


Is there a name for the practice of composing sentences in such a way that they don't contain proximate consonants which cause difficulty when read aloud? For example: Under this criteria, "dogs and cats" is preferable to "cats and dogs," because the consecutive d s in the latter are difficult to pronounce clearly and distinctly and will force the speaker to either interrupt the flow of speech or elide the first d.


During a sunny day the walls of my house warm up (no surprise). My question: how much of this warming up (if any) comes from visible light? I associate infrared with thermal energy. If my house was floating in space (to prevent any thermal exchange with its surroundings) and I installed a giant infrared (and UV) filter between it and the sun, would it still warm up (compared to its rest temperature in full darkness)?


Does a matrix with all non-negative, real entries have all non-negative, real eigenvalues? Where might I find a proof of such? Ideas: Perhaps we can multiply a prospective eigenvector so its biggest entries are positive, and then show that it is a contradiction for it to have a negative eigenvalue? I am currently looking at the Perron-Frobenius theorem on Wikipedia, but it seems not to mention this issue. (I suspect my conjecture is not true.)


I think the title says it all. I am planning on giving a talk in a few weeks about the Banach-Tarski paradox and I have some pdfs found online which describe the paradox a little but I am looking for a solid reference which covers the construction from A to Z and on which I can extract the main ideas for my talk from (I understand the ideas beneath the paradox, I am just looking for a formal proof with no details excluded,i.e. a well-structured document). Anyone has a reference in mind?


For instance, would you rather say "It became increasingly hard" or "It became increasingly harder"? From my understanding, both are possible, but their meaning is slightly different. The first simply means that it became "harder". The second literally refers to an increasing rate of getting harder. Though in practice, I believe the second is still used to express the exact same as the first, while sounding somewhat awkward.


I am looking for a specific word to describe a concept such that its meaning is linear and one-way. Such is the case for the title of this question. X would be of this concept and Y would not, Y would be the opposite of this concept. For example, all automobiles are motor vehicles but not all motor vehicles are automobiles. I cannot seem to find any word to relate to this. Any advice/guidance is very much appreciated.


Traditional English prescriptive grammar teaches that these two verbs, to see and to hear, when describing their sensory nature, should never be used in the progressive aspect of tenses. Thus I am seeing you/I was seeing I am hearing you/I was hearing you would be incorrect. But in spoken English, whether it be in the movies or on television, this usage seems to me to become ever more often used. I am seeing you for I see you now I am hearing you for I hear you now When did this usage become so universally accepted? Is it still considered colloquial English? Or is it already considered standard English?


I would like to create a table (cf. attached figure) and use curly brakes to summarize some of the cells. I did a lot of research and found this solution (Curly brackets around a table), but it doesn't help me much as it is only math content and I would like to enter text only. Does anyone have an idea about how to solve this problem elegantly? Cheers


I realize there are questions on the correct usage of "but" and "however". In this case, I am concerned with correctness in a formal context. I have heard it said that however should be used in formal mail communications instead of but because the word "but" has a negative connotation. Which should I use in, say, an email to a client? Examples: We understand the issue you have pointed out, but we cannot consider it as a defect in the application since the current behavior was requested by your team. We understand the issue you have pointed out. However, we cannot consider it as a defect in the application since the current behavior was requested by your team.


As I read on Wikipedia, the Rutherford atomic model is not correct according to classical electrodynamics, as it states that electron must radiate electromagnetic waves, lose energy and fall onto the nucleus. I don't understand this explanation. It is clear to me that with given acceleration directed to nucleus and proper speed, electron can move around the nucleus. I don't understand explanation about energy, but I understand that there must be some force directed to nucleus. Also this force must not be constant because if it is, a larger speed could keep electron moving around the nucleus. So what is that force? Why does this explanation on Wikipedia and on other resources operate with energy, not with force?


I'm reasonably happy with the dictionary functionality in TeXworks but I am struggling to see how I can 'find' all the spelling mistakes in my (painfully long) tex file. I was hoping for a Wordesque spell checker in which I can scroll through each of my errors and accept or ignore the suggested changes. I could install another editor that has this functionality but would prefer not to do this if possible.


I am wondering if the verb expect is used as a stative verb in the following sentence: I entered the classroom and [to expect] to see some students but instead I found a note which said [...] Would it be possible to say "I was expecting"? Or is the verb stative in this context? Because then we can't use it in a continuous form, according to my grammar book.


Can anyone please explain why light reflects at the boundary between two regions with different impedances? This sounds very simple but I got confused when I tried to think of how light and atoms interact with one another at the boundary. This question can actually be generalised to the reflection of all types of waves. I have to admit I have no understanding of the microscopic detail of reflection.


I believe I am stuck with only these two options, but would appreciate an alternative usage. I am somewhat limited on sentence structure as this application is part of a mail merge document. The purpose of the business is to own and operate The Tutoring Center franchise location. OR The purpose of the business is to own and operate a The Tutoring Center franchise location. Since the business will be a franchise (one of many), I feel the article "a" is necessary. Help is much appreciated!


What are the desirable pre-requisites to be able to learn model theory well? In particular, it seems that connections to algebra are used heavily especially as examples. I would like to know if a good grasp of algebra is essential for a deeper understanding of model theory and whether a good grasp of analysis and topology would compensate for lack of the former. Thank you.


I have started learning some basics of Category Theory.Currently i am trying to understand the concept of Direct limit and Inverse Limit but i find this concept really hard to digest. Could some one explain me with some details the good idea to think about Direct Limit and Inverse limit in practice? I have knowledge of subjects like Module Theory,Galois Theory,Commutative Algebra(at the level of Atiyah Macdonald),Basics of Lie Algebra and of course elementary topics like Group theory and field theory etc.


I'm interested in using TikZ to draw rooted trees whose leaves should all be on the same level, pretty much like this: Please don't pay attention to the fancy style and colours, I'm only interested in having all leaves at the same level. Anyone knows how to do this? I've looked into the manual and the appropriate section on Texample, but had no luck so far. UPDATE: a while after this question was answered, I found out about the LaTeX newicktree package, which allows you to draw such trees in an extremely simple way, just by specifying them in the format people in phylogenetics already use. This is obviously much simpler than resorting to TikZ, but thanks anyway to everyone who contributed to helping me.


My understanding is that dark matter cannot be (or is at least highly unlikely to be) an exotic form of any known particle. On the other hand, articles about particle accelerators seem to say that the Higgs is the last piece missing in the Standard Model jigsaw puzzle. If dark matter is determined to be some form of new particle, what are the certain implications? Might such a discovery "stand to the side" of the Standard Model or would it certainly change the foundations? (Forgive my extremely lay understanding and vocabulary -- please feel free to correct my mistakes.)


If I pick up a rock and set it on a ledge above my head, I do work in the process. The work I do is termed "potential energy". We know how to recover the energy (i.e. let it fall back to earth). However, while resting above the surface the energy is said to be "stored in the gravitational field", presumably meaning the space between the elevated rock and earth. Suppose we perform this experiment on the moon where we know the space between the elevated rock and surface consists of an empty vacuum of space which contains absolutely nothing. How can empty space store energy? That is, how can "nothing" have any properties whatsoever?


I was writing an email and faced the following situation: I am coming to Oxford this summer. If you are still around, I would be glad to hang out with you. Now normally we don't construct the conditional statement like the one above, where a present verb is followed by would. But if I change the second one to will it kind of becomes very direct and that's not what I want. So is the construction above still correct or should I go with something else? If it's correct, could someone explian the technical details behind it?


Im studying old exams and came across this one Question: a. Find a (discrete time or continuous time) random process that is wide-sense stationary (WSS) but not strict-sense stationary. b. Also, is it possible for a strict-sense stationary random process not to be wide-sense stationary? Answer: a. A sequence of uncorrelated random variables with common expected values and common variances constitute a WSS discrete time process, but is not strict-sense stationary if the random variables are not identically distributed. b. A seqeunce of independent identically distributed random variables with infinite variances constitute a strict-sense stationary discrete time process that is not WSS. a. Can anyone give a simple example of such a process? b. Our course litterature says WSS processes are always strict-sense stationary?!?


I remember being presented a mathematical puzzle some years back that I still can't solve. The problem is defined as follows: We have two points on a plane, and using only a compass, how do we find other two points, so that all four of them would be vertices of a square? I'm not sure whether the first two points were supposed to be vertices of the same edge of a square or not, so solutions to both variants are welcome.


I know the word uxorious that is used to describe a husband who dotes on his wife excessively. What is the corresponding word for a wife who loves her husband dearly? I cannot pull anything out of my memory... and I feel it is very harsh on men if such a word weren't there! Example: He was an uxorious man who assiduously took care of all of his wife's needs. She was a __ woman who assiduously took care of all of her husband's needs.


I don't know if the title of this question is accurate or not. My vacabulary of grammar is very limited. May you could understand me by the following example: She told me that the earth is moving around the sun. She told me that the earth was moving around the sun. Which one is correct? Should the tense in that clause always agree with the tense in main clause?


A lot of text books mention that one of the reasons that classical mechanics failed to explain atomic and subatomic processes is that electrons which accelerate should release energy in the form of electromagnetic radiation, which would lower the atoms overall energy level, but this does not happen. One place where I discovered this, for example, is in the description for the Bohr model. What I don't understand is why everyone takes for granted the fact that the electron is accelerating. I thought the electron orbits the nucleus at a, more or less, constant velocity. Are people referring to specific situations when the atom is excited? Furthermore, I was under the impression that electrons already travel at the fastest allowable speed, the speed of light.


Okay, I am in no way a mathematics student but I happened to be writing a program on a map with coordinates of locations displayed on the map. My problem is that the points could be scattered on the map with an indefinite boundary and I need to find a boundary (rectangle) around these points and thus I need to get a formula to form an algorithm so that the program will find the optimum boundary itself. I hope this is understandable and thanks in anticipation of any response.


I'm a high school student. The head is hydrophilic, the tail is fatty acid, in other words hydrophobic. Here is the thing I don't understand, all textbooks state that water is repelled by hydrophobic tail, why? The hydrophilic head is composed of polar molecules, and water is polar, so they will attract each other. But why hydrophobic tail will repel polar molecule? Isn't hydrophobic tail is just composed of non-polar molecules?


Consider this situation: You sent an application for a job, then the company boss sends you an email, and tells you if you are in town, he likes to meet you in person. Is it correct to say? "It could have been a great pleasure to meet you, but unfortunately I am currently in Russia." I'm not sure what tense I should use, and I'm pretty sure the above sentence is incorrect. At the same time, I want it as much as possible in a formal and polite way.


I am trying to use IEEETrans to write a scientific paper in two column format. The issue is that one is not allowed to place figures in the left-hand side column of the first page. Figures should start on the top of the right-hand side column of the first page or alternatively, on the second page (any column). Is there any way to restrict the placement of floats to avoid using the left-hand side column of the first page?


In sports, we have the term "Commentator's curse", (humorous) The supposed propensity of a player to blunder after having his/her talents pointed out by the commentator. [Wiktionary] Is there a phrase that emphasizes that the sportsperson performed brilliantly soon after the commentator's talk highly about their achievements? For instance, a Cricket player hitting a six immediately after the commentator says that he has the highest number of sixes for his team. Or a Football (Soccer) player scoring from way outside the D-box just when the commentator says that the player had scored from a similar distance in the previous game. It might be termed as "pure co-incidence" but I'm just looking for something humorous/ emphatic as "commentator's curse" itself.


I know that massive gravitational bodies will curve the path that light travels. I think that quantum optical mediums also bend light. I am still confused of whether quantum optical mediums actually slow down light or if they just use absorption, re-emission, scattering, and delaying to create the illusion that light slows down in a medium. I do not see a fundamental difference between the two theories, because both describe "optical" phenomena of light. I am not sure if gravitational optics is a real field of study or a made up name. My question is there a mathematical equivalence between gravitational optics and quantum optics or are they mathematically incompatible theories?


Most of the references I found online simply note that "see you later" is a farewell or parting phrase but nothing discussed when it is appropriate to use the phrase. Is it acceptable to use "see you later" in each of the following circumstances? When you will be seeing the person again later that day When you will be seeing the person soon (e.g., in a few days) When you will be seeing the person again at some point in the future (e.g., next year at Christmas) When you are suggesting that you want schedule something specific Whenever you say goodbye, regardless of whether or not you will ever see them again


I am reading up on splines and as a beginner I have a basic question - Does it make sense to say - "I will fit a cubic b-spline to the data". As b-spline is just a representation of spline in terms of its bases. I think, a more accurate statement will be - "I will fit a natural/not-a-knot/clamped/etc. cubic spline to the data and present results in terms of its basis." My question arises out of my limited understanding of relevant concepts. Could someone please confirm this. Many thanks!


At the moment I'm a bit baffled. What sentence part is "visitors" in "I'm not allowed visitors"? I would not call it an object or a subject complement as "to be allowed" is no linking verb. The only idea I have is to say the sentence is elliptic and the full form would be "I'm not allowed to have visitors". Here "visitors" is object of the to-infinitive. So is it possible to say "visitors" is object of a to-infinitive (to have) that was deleted?


As far as I understand, the incident photon interacts by photoelectric, Compton scattering or pair production with the electrons of the crystal (NaI(Tl) in our case). The electron that emerges from these interactions generates a number of photons in the UV or visible which are proportional to the gamma ray energy. Then, the light produced can be read out with photodiodes, photomultpliers, avalanche photodiodes, silicon photomultipliers, and other photo sensors, right?. So, how are those electrons converted into UV or visible photons?


I've read here that microwaves are blocked by the holes in a microwave window because the holes on the door are small enough to prevent the microwaves from passing through. However, if wavelength is only the distance between successive crests of a wave, why should that decide what a certain wave can fit through? Why is amplitude not the deciding factor and is there a way to explain this visually?


Projectiles containing delicate elecrtronic equipment may be damaged if they are subjected to high accelerations. For this reason, such projectiles may be fired from guns with long barrels but not from guns with short barrels. -Explain why a projectile fired from a long-barreled gun is subject to less acceleration than a projectile fired from a short-barrelled gun if the range is the same in both cases. This question has been a nightmare for me when i was in highschool and until now i cant have a good answer your help would be appreciated.


I am new to LyX so I'm sorry if this question is really naive. I'm using LyX on a MacBook and I'm hoping to use the AEA document class on it. Since it is not available automatically, seems that I need to somehow install it on my own. All the help pages I can find talks about layouts and stuff that I, as total layman, have no idea what they are. Can somebody please show me how I can do it with simple language?


I'm reading about how a "straight line" depends on the geometry of space. While I think I understand this, the example people give is: "Imagine a straight line on earth connecting two cities. It's actually curved because the earth is curved". My response to that would be "Well no, because a truly straight line would cut through the earth. Like a straight string would cut through an apple". So question: If two people took an infinitely strong and light string, stood on opposite ends of the earth, and pulled almost infinitely tightly, would the string curve around the earth a few feet above, or would it cut through to the core?


I am currently reading the following paper by F.Takens: Multifractal analysis of dimensions and entropies. This paper discusses two different measures. One is generalized entropies and the other is generalized dimensions, however as far as I can see it doesn't discuss any association between the two. Can somebody please in a qualitative or [even better] quantitative give me an explanation about the relation between the two?


Our course slides offer the following definition: "A parametric probabilistic model is a set of probability distributions indexed by a finite-dimensional parameter vector." This description defines "parametric probabilistic model" in terms of "parameter", which isn't particularly helpful. But it brings to mind some vector of indices which tells you where to find a row (or rows) in a table. It is not clear to me whether "parameter" in this sense is related to my CS understanding of a parameter (i.e., a "knob") or even if a model's parameters have something to do with its random variables (e.g., are the parameters fixed values of a random variable?). Furthermore, what does it mean to "parameterize" a model?


I have this question on a homework assignment. I sat down with two other people for a long time and we derived the alternating harmonic series example, but I don't think that's valid because the question explicitly asks about sequences and not series. Note that it's for an analysis class and so far we've covered open and closed sets and balls, preimages, and cluster points.


I've read through answers to similar questions, and I haven't been able to find a satisfying answer. I've heard it said that in order to make out something, you need to use light with a wavelength at least as large as the object. In what way can you mathematically derive this heuristic? Moreover, does this apply generally? The other day I was thinking in the context of sound waves and how fruit bats use them to resolve insects- is this a reason why they use high frequencies?


Question: What are some interesting or useful applications of the Hahn-Banach theorem(s)? Motivation: Most of the time, I dislike most of Analysis. During a final examination, a question sparked my interest in the Hahn-Banach theorem(s). One of my favorite things to do is to write a math blog (mlog?) post about various topics so that I can better understand them, but I know very little about Hahn-Banach and a quick google search didn't seem to point to anything neat. I was interested in seeing what you all liked (if anything!) about the Hahn-Banach Theorems. Also, I can't seem to make this a community wiki, but I think it ought to be one. If someone could either fix this, I would appreciate it! (If not, please delete this!)


is it possible to use participial prepositions in the following sense, or am I creating a dangling modifier. Moreover, is the punctuation correct? Considering his broken leg, Peter was not able to walk anymore. Peter was not able to walk anymore considering his broken leg. Understanding Einstein's statements is neccessary, since Peter wants to graduate in Physics in order to get a good job. Since Peter wants to graduate in Physics understanding Einstein's statements is neccessary, in order to get a good job. Can participial preposition be used as subordinating conjunctions? Maybe a native could help me out here. Thanks!


Special relativity theory says simultaneity is relative, meaning that different observers will not agree on what happened first and what second. Does it then make sense to say that looking at distant stars, we see them how they looked "billions of years ago" and not how they look now? Does it make sense to talk about what these stars look like now? How do we define this "now" if simultaneity is relative?


So there is this kind of building located in the centre of a fork in the road (maybe in streets too). I don't know how to put it into words, but it looks like this: Or this: Is there a single English word for that kind of building? If it is a house, is there another name for that? In my native language (Indonesian), it's called Rumah Tusuk Sate. Which if translated bluntly into English, it's Skewer House, which sounds a bit weird to me.


Possible Duplicate: Why the use of 'would' in the following When we tell a joke or narrate a story in simple present can we use 'would' in some cases? For example: He goes up to this man and punches him in the face. The man looks straight into his eyes and tells him that he will take revenge He goes up to this man and punches him in the face. The man looks straight into his eyes and tells him that he would take revenge


According to this extract from Wikipedia, the pronunciation of the term piste meaning "a ski run of compacted snow.": varies slightly in English, with British English using a long "e", (e.g. rhymes with "beast"), and American pronunciation using a short "i" (e.g. rhymes with "list"). Actually, references such as the Cambridge Dictionary or the ODO suggest that the pronunciation is the same in both dialects, that is, with a long "e". Does the AmE pronunciation of "piste" really differ from the BrE one, or are both versions correct?


I am finding hard to understand application of inclusion and exclusion in Matching problem? Suppose that n- n male-female couples are at a party and that the males and females are randomly paired for a dance ? what is the probability that none of the couple are paired together? I don't understand why we have to take Union then use inc-ex principle to solve it? Can some one help me to understand ? Thank you


I'm struggling to find a way to express the idea that it's possible that something was done before something else was done. I'm not sure if I'm just tired, but the idea is this: In the present perfect, you would say "it has been done before". When you add "may" it becomes "it may have been done already". In the perfect past, you would say "it had been done before". How does that sentence look like when you add "may" to it? "It might have been done before"? "It may have been done before"? Or even "it may have had been done before", as clumsy as it sounds? None of these sound particularly correct, so I'm at a loss as to what the solution here is.


If we consider two spins, we can think of the spins as being either parallel (up|up or down|down)or anti-parallel (up|down or down|up). Or we can think of them as being in the triplet or singlet configuration. Is one description more correct than the other? Or is it just a matter of choice between two basis sets? It would seem to me that using T/S is correct because it accurately reflects the symmetry needed in the wavefunction.


Find the distribution of U=min(X,Y) where X and Y are independent random variables and both exponentially distributed with parameters lambda and mu respectively. The only headway I have made is that P(U< u)= P(X< u)P(Y< u) by considering the joint distribution of X and Y and the fact X and Y are independent; is this right? If it is then U isn't a 'named' distribution function so is just stating the cdf enough to answer the question?


I've noticed "At this stage" preceding delivery of the negative to the reader. What's the reason for this? "At this stage, unfortunately (for you), we won't be proceeding further with your application." "At this stage, your application has been declined." "At this stage, travel arrangements will be at your expense." This is used when there are no other stages. Why do people use this adverbial phrase? do you think it helps the reader to deal with emotional pain?


In Thomas Harvey's Elementary Grammar and Compsition he says: A complex sentence whose subject or predicate only is a clause, need not be separated into principal and subordinate clauses in analysis. Later in the section he gives the example, "That he is very sick, is evident," parses the sentence, then gives a sentence diagram. I'm curious how one would break this sentence into its principal and subordinate clause.


I am studying probabilites and the notion of poisson random variable was introduced in the class. But it seems to me that the introduction of poisson random variable is to provide a easy approximation of the binomial random variable conditioned that n is large and p is small. Besides, the preconditions in the poisson distribution that events are independent of each other seem to come from the fact that binomial random variable is composed by many independent bernouli variables. So I wonder if originally, poisson distribution was invented to model binomial distribution or was it invented to solve a particular kind of problem


For some pdf ebooks, Adobe Reader displays the cover page alone even though the display mode is "two page view". Other than the cover page, the rest of the pdf content is displayed as two pages at once, which is expected in the "two page view" mode. My question is: Is it possible to enforce this behavior if I use pdflatex, and if so, how could I achieve that? The document class I'm using is memoir.


In algebraic geometry, we often consider a complex algebraic curve, and in order to get some intuition, we often draw it on the plane as if it were a plane curve. In most cases it turns out that the plane drawing really helps us thinking of the curve for instance when counting the number of intersections. Then I wonder, to what extent the plane drawing is legitimate (or not misleading) to "represent" the complex algebraic curve?


I have noticed that when certain tube-lights are switched 'ON' , they make certain noise corresponding to their "blinking". In blinking, there are alternate periods of the tube lighting up and then going out. The "sound" is heard at the instant the tube lights up. I can not properly describe the "noise" or the "sound" I'm referring to, but it seems as if the tube-light is struck gently with something. It feels as if the gas molecules inside the tube-light are striking its inner surface. Can anybody explain this ?


In this question... Why does a photon colliding with an atomic nucleus cause pair production? ...I asked why a photon colliding with a atomic nucleus can become an electron and a positron. The answer that I thought was most illuminating explained that a photon spends some of its travel time as a particle-antiparticle pair of an electron and a positron. If it strikes the nucleus at the right time, this pair will be separated. It was explained that this is because 'Quantum Electrodynamics allows it'. Why does Quantum Electrodynamics allow a photon to exist temporarily as an electron and a positron?


In batteries, what exactly do we mean by negative and positive charges? My understanding is that the negative charge of the anode is basically an atom with an extra electron in the last orbital and a minus electron in the last orbit of the positively charged atoms in the cathode? And so the electrons keep flowing and interchanging because the atom is trying to reach a stable state. Is this correct or am I wrong? If my understanding of positively/negatively charged atoms is correct, what do we mean by saying that the electron in itself has a negative charge?


Most books on strongly correlated electrons claim that when the number of itinerant electrons is small and the screening length is large, that the system becomes "strongly correlated", (i.e. the independent electron approximation breaks down). If this is the case, why isn't doped silicon (or some other semiconductor/insulator) a strongly correlated electron system at low temperature where there are no thermal carriers? What are the critical ingredients that make a system unwilling to abide by the independent electron approximation?


I am trying to draw an activation stack. I am really bat at TikZ, so I hope you can help me out. As you can see, there are four boxes stacked one on top of the other. Each box has a name, put on the right, and is made of an upper part, where variables and numbers are displayed, and a lower part which is divided in two halves. The left half has to contain an arrow pointing to one of the boxes placed under it, the right half contains only a few characters. The box at the bottom hdoesn't have a pointy arrow, because it doesn't have to reference anything. I really hope you can help me out, I really appreciate it.


A lot of textbooks and exam boards claim that light incident at exactly the critical angle is transmitted along the media boundary (i.e. at right-angles to the normal), but this seems to violate the principle of reversibility in classical physics. How would a photon or ray travelling in the reverse direction "know" when to enter the higher refracting medium? It can't know, so I conclude that such light is simply reflected? Is this correct?


My question is regarding effect of Sun's gravity on earth. I want to know that if Sun's gravity can change Earth's landscape in long duration (i.e. billion of years) or not? Means if earth is dead planet and all other planets are not present in the solar system and Earth is as near as Sun as mercury currently orbiting. And also suppose Earth's surface is not smooth (because of hitting with other small bodies at regular interval) then can Earth's surface can be smooth in long duration by gravity of Sun or not?


Let A and E be opposite vertices of an octagon. A frog starts at vertex A. From any vertex except E it jumps to one of the two adjacent vertices. When it reaches E it stops. Let a(n) be the number of distinct paths of exactly n jumps ending at E. Find a(n) Is it right if I conclude that number of steps has to be an even number ? Also I don't know how to go further with that except I think recursion is involved. P.S. I'm having trouble solving questions in probability and combinatorics where recursions are involved like finding expected value ones. Can you provide me suggest me some links/books through which I can understand it better. Thanks


I have a question that I am just curious about. Two principles: Magnetic fields and Magnetic susceptibility Supercavitation is the use of cavitation effects to create a bubble of gas inside a liquid large enough to encompass an object traveling through the liquid, greatly reducing the skin friction drag on the object and enabling achievement of very high speeds. Can a magnetic field hold a state of supercavitation if a magnetic field could be placed inside of the bubble of the supercavitation?


Some people love creating or fostering the creation of events that are noisy, chaotic and fun, such as parties, arguments, reunions, etc. Sometimes there's a negative connotation to this (like in provoking an argument), but sometimes there is not (like in organising a big, noisy party). The word I'm looking for would be the English equivalent to the Spanish armadanzas, just in case this is helpful. Any ideas?


Earlier, I asked here whether one can have spontaneous breaking of the Lorentz symmetry and was shown a Lorentz invariant term that can drive the vacuum to not be Lorentz invariant. How relaxed are the assumptions in the CPT theorem? Can one have the theorem to be valid for a Lorentz invariant theory with a vacuum that breaks it? Can someone point me to the original and alternate proofs of this theorem? Also, is there any connection between CPT invariance and renormalizability?


Why is it said that electric current always flows from higher potential to lower potential? It's said that current flows from a positive terminal to a negative terminal, but I've read that it is actually the [negative] electrons that flow from the negative terminal to the positive one. Rather isn't it that the protons don't even move? My question here is in reference to electric circuits with batteries generally used in textbooks.


"Half of the time I was working from about eight in the morning until eight at night, and the other half I was incapacitated by excruciating pain." This is what I have currently. Is this correct or is there a better way to approach this sentence? To be specific, I am wondering about my comma usage, and whether or not more commas should be added, however, all tips are welcome. Thank you.


I learned a lot from prof Tao's notes and books because unlike many authors, he seems to prefer writing more words, explanations and intuitions rather than just mathematical formulae. His approach is also very pedagogical, and quite often, when talking about one concept, he also talks about similar concept in other fields, like group theory, differential geometry, etc. (due to his very broad areas of expertise). This makes me understand (superfically) how mathematical concepts are related to each other. I would like to know if there are some books or blogs on probability theory, stochastic processes, Monte Carlo and convex optimisation, with similar writing style and approach. Many thanks!!!


I'm new in this study and I don't know much about the foundations of mathematics, so I have a question. If I'm doing category theory, and I need to talk about "small categories" , "locally small" and etc, I need to have some set theory foundation like NBG where I can talk about this things, right? But, if I want to take the category of all categories, I will have some problem in this context with NBG as foundation?


I'm looking for a foundational book that builds up ideas like transcendental functions. For example, how the trigonometric functions are truly defined when plotted as continuous functions. I believe Shilov briefly touches on this in his "Elementary Real and Complex Analysis", but I'd like a more broad and in-depth treatment - book that, in general, deals with mostly functions (and maybe continuity/their limits). Meant for a freshman mathematics major. Books that are entirely dedicated to this topic are preferred over others wherein this topic is only treated in passing.


We all know that it uses less energy to go by wheeled transport than walking if the terrain is smooth and there are not too many hills. We also all know that when going up steep hills we get off whatever wheeled transport we are using and walk. To make things concrete, let us consider a child's scooter. How can one calculate the steepness of the hill at which point you would use more energy to scoot than to walk?


The phrasal verb "sit on sth" means "to be a member of a group of people". Well, according to Longman, "to sit in / on sth" means "to be a member of a committee, parliament, or other official group", the same thing. I don't know why my phrasal verbs book "says" just "sit on sth" and doesn't say the "in" too. Perhaps Parliament can only be used with "in" with this meaning, I don't know. When I found the meaning of "sit on sth" for the first time, I realised that I could say "To sit on a Parliament" and not "To sit in a Parliament". Is my phrasal vebs book incomplete? Can "sit on a Parliament" be correct? I hope you understand my question.


I was wondering how one would actually calculate how much oxygen would dissolve into water given the necessary initial conditions, and what those initial conditions would need to be. I assume they would be pressure, and initial concentration, but I really don't know where I would go from there. Clearly air and water have different concentrations of gases and liquids, despite having been in contact for thousands of years. And once in water, is oxygen still considered gaseous? I assume it is, but why is it called gaseous-what quality of it deems it a gas despite being surrounded by liquid?


I'm designing a space ship for a comic I'm writing. I was just finishing up the first sketch when I thought whether such a design would actually be possible. Hence my question: A spaceship is located in deep space. A propulsion system (let's make it a rocket) is attached to the side, and thus not in one line with the ship's center. Would this spaceship travel in a straight line, or rotate?


Let S be the intersection of diagonals in a cyclic quadrilateral. Let p be a circumcircle of a triangle ABS and it intersects BC in M and q is a circumcircle of a triangle ADS and q intersects CD in N. Prove that M, N and S are collinear. I tried proving that vectors NC and DC are the same, and also that vectors CM and CB are equal but nothing seemed to work. I'm probably not thinking right.


Question: Why is ionic lattice energy inversely proportional to the radius of the atom? Most heterogeneous covalent molecules are polar to some extent. The degree of polarity, or the dipole moment, depends on the difference in electronegativity difference between the two atoms. The larger the dipole moment, the higher the ionic character. What I know: Electronegativity decreases as you go downward in a group, however, the size increases, usually, as you go downwards in a group. Thus, ionic character will increase upon going downward, but the ionic lattice energy will decrease? This seems contradictory. Is this true? And if it is, why is it so?


As far as I know, a magnetic field can only be produced by a moving electric charge, or from a particle's spin (this is how a permanent magnet works, all the spins are in the same direction) What is strength and direction of the magnetic field of a stationary electron at the origin with spin oriented straight up? I suspect a function of cylindrical coordinates is most convenient.


I would like to know the status of the following problem: Given a simple graph, is there a walk traversing each vertex at least once and each edge at most once? (I am asking for a complete trail, a sort of cross-breed between Eulerian and Hamitonian walks, in a way.) In particular, is that problem known to be solvable in polynomial time, or NP-complete? Or is it suspected to belong in NPI? More generally, what is known on it? Variants: One could also ask for a circuit rather than a trail, and/or consider oriented graphs instead. Thanks in advance!


Sometimes I'll look at a word for too long and I become convinced that it's spelled or pronounced incorrectly. This happens most frequently with "spider" (I think that should be pronounced "spih-dur" [like "spitter"], not "speye-dur") and happened just now with "finger" ("fin-gur", not "feen-gur"). There's a similar variant to this (discussed here), but that's related to a word losing it's meaning. To me, the word just looks incorrectly spelled, or I convince myself that everyone's pronouncing it incorrectly. Is there a difference between what I'm describing and Semantic Satiation? If there is a difference, does this one have a name?


I'm currently studying Single-Variable Calculus independantly through MIT OCW. I can only focus on one course at a time independantly since it takes up so much time, and I really want to study Linear Algebra next instead of Multi-Variable Calculus. My question is simply this: after understanding Single-Variable Calculus, would I be able to continue into and understand Linear Algebra, or should I do Multi-Variable Calculus first?


I've just been wondering about how atoms and molecules can be quantum mechanically thought about, and I have a question. It is often said that intermolecular bonding is purely "electrostatic". I hope what this means is that it is easier to consider molecules as point charges that are attracted, but in reality, there is some sort of orbital superposition, but the small distances between different charges within a molecule don't really matter. Is it correct to say that the essential physics is captured by considering the particles as classical Coulomb charges?


Are binary systems (in case of stars and other celestial bodies) more favorable than independent existence? I've been going through an article regarding pulsars, where it was stated that 'many pulsars are found in binary systems.The companion of pulsars have been found to be normal stars, planets, white dwarf stars, neutron stars and even another pulsar. So what are the criteria to form binary systems ? I know that their sizes should be comparable, and that leads to the equilibrium of the gravitational attraction between the bodies, etc. As it is a known fact that neutron stars and pulsars are quite heavy and dense, how are they able to co-exist as binary systems with other stars, planets, etc.? Search this @ http://outreach.atnf.csiro.au/education/everyone/pulsars/


Source: gradestack.com This is a problem I am trying to solve for a long time. But still not able to proceed. After spending some time, I got a doubt whether this question is correct. Because, in a rhombus, diagonals bisects each other. Here PA=PC. That means, P is the center point of diagonal AC. So PD must be equal to PB which is not the case. Why I am wrong here? Please give pointers in how to solve this problem. thanks.


I am interested in why many small animals such as ants can lift many times their own weight, yet we don't see any large animals capable of such a feat. It has been suggested to me that this is due to physics, but I am not even sure what to search for. Could someone explain why indeed it is easy for smaller objects/lifeforms to support several times their own weight, but this is harder as objects/animals become larger?


A set of r marbles is selected from an infinite supply of red, blue, white and yellow marbles. A selection must satisfy the condition that either the number of red marbles is even and the number of blue marbles is odd, or the number of white marbles is even and the number of yellow marbles is odd. Does it mean a selection is invalid when both red is even and blue is odd as well as white is even and yellow is odd ? Do we discard this case while counting the number of selection ?


I am looking about similar triangles and I always see the word corresponding but I always forget to include it in the definition. I am trying to think of an example I can use to show the importance of the word corresponding. So here is the definition: Ex: "If the measures of the corresponding sides of two triangles are proportional then the triangles are similar." So if the word corresponding is not in the definition: "If the measures of the sides of two triangles are proportional then the triangles are similar." but what is so important about the word corresponding? Any ideas?


Most of the proofs of the Cauchy-Schwarz inequality on a pre-Hilbert space use a fact that if a quadratic polynomial with real coefficients takes positive values everywhere on the real line, then its discriminant is negative(e.g. Conway: A course in functional analysis). I think this is somewhat tricky. Moreover I often forget its proof when the pre-Hilbert space is defined over the field of complex numbers. Is there a more natural proof (hence it's easy to remember) which is based on a completely different idea?


Autorefractors are being used by eye opticians for eye diagnosis. I searched internet for articles and wiki page as well but I wasn't satisfied. I am interested to know how do they actually work. How the machine is able to focus sharp on retina automatically. How do they figure out spherical/cylindrical aberration for human eye. I even asked optician as well but he tricked out of this position saying it does all happen automatically. LOL! A detailed answer would be much appreciated. Please don't hesitate to write any technical details or mathematical derivations. I would love to know how these machines actually work. Thanks in advance!


I received an e-mail including the following sentence: I am not asking for a facetious grade change, just one that would allow me to pass. What the writer means is that the request isn't frivolous or petty. This use sounds wrong to me, but the literal dictionary definition doesn't contradict it (the dictionary definition includes, for instance "flippant"). My question is whether this is a use typical in some situation or dialect I'm not familiar with, or if it's actually completely nonstandard.


The product of nonempty sets is nonempty. I am fascinated that such a simple and seemingly intuitive statement can lead to rather astonishing results such as the Banach-Tarski paradox or the solution to this riddle. I am also intrigued by the seemingly innocent results that rely on AC (the existence of algebraic closures, any ideal is contained in a maximal ideal) and I wonder if I am missing some intuition to see how truly remarkable they are. My question: What are other examples of seemingly magical results whose proofs rely explicitly on AC, and what are examples of seemingly innocent results that rely on AC that upon further examination turn out to be fairly remarkable themselves?


Let G be a finite planar graph, then there is a natural walk around the outer (i.e. the unbounded) face of G. It might happen that a vertex v is visited more than once by this walk. Proof that this is a cut-vertex. This is a task which might be obvious at first glance, but I wasn't able to do it rigorously, i.e. by citing any of the used arguments from a book and proving everything else. I would also be satisfied with a book which I could cite. Thx.


I am looking for a good mathematical rigorous introduction to solid state physics. The style and level for this solid state physics book should be comparable to Abraham Marsdens Foundations of mechanics or Arnols mechanics book for classical mechanics or to Thirrings Physics course for quantum mechanics. Any recommendations? Edit: As a reaction to Peter Shor's comment, I try to narrow the scope of the question a bit and give some more specific subareas of solid state physics I am in particular interested in: semiconductors and applications the quantum hall effect superconductivity


Is there any way to tell how many clusters there are with respect to all the roots of a polynomial? Specifically, I'm after the multiplicity of each root but since I would like to work in floating-point arithmetic I'm afraid I have to deal with clusters. I don't mind any method of finding out: be it by numerical-iterative means during the convergence, a priori/a posteriori guess, maybe some matrix method would help..? If I have to set some small disk radius, that's ok, too.


Example: The three picked up their chopsticks and started on the food. The scene seemed strangely normal to Naomi. It was as if everything were OK, and her mom were perfectly healthy. In fact, she looked more lively and energetic than Naomi remembered. The whole illness issue seemed like a distant, bad dream. Something that belonged to another reality. Still, Naomi couldn't take the thought out of her head, so she decided to ask. If I changed still for yet, would the sentence mean the same thing?


As I've been trying to wrap my head around the principles of decoherence and quantum behavior I am left wondering why fundamental particles are 'allowed' to exhibit quantum properties even in ideal conditions ( close to absolute zero and in a 'box'). If a particle/photon behaves in a probabilistic superposition state as seen in the double slit experiment then wouldn't we expect the multiple coincidental states of the particle to interact with each other and thus cause decoherence? This would in turn lead to an innate instability of any system above a zero point energy thereby making quantum properties not identifiable at all..but indeed we do have verifiable and replicable evidence of quantum mechanics.


This will depend on whether he's suitable for the job. This will depend on whether he's suitable for the job or not. This will depend on whether or not he's suitable for the job. It is still not defined whether we're following that approach. It is still not defined whether we're following that approach or not. It is still not defined whether or not we're following that approach. "Or not" doesn't really seem to be needed to complement "whether". Why do people use it then? Is it redundancy and nothing more? Or is it for emphasis? Or are there cases when "or not" is required for the sentence to be grammatical?


Please see the sentences: I scheduled to stay after school with you today, but yesterday I was assigned a detention for today too. I scheduled to stay after school with you today, but yesterday I was assigned a detention for today also. The sentence ending in also sounds better to me, but I am not a native speaker, and I don't know it it's correct to use also at the end of a sentence. Which one is the correct form?


Wikipedia defines the notion of a pure set as follows: a hereditary set (or pure set) is a set whose elements are all hereditary sets. Why does this definition make sense? It seems to be circular. Also, wikipedia says: The inductive definition of hereditary sets presupposes that set membership is well-founded (i.e., the axiom of regularity), otherwise the recurrence may not have a unique solution. Why does the definition sometimes not have a unique solution? Is the problem the existence or the uniqueness? Can you give an example of a situation where the recursive definition from above does not have a unique solution in a setting where we don't assume regularity?


The original sentence is "The Spaniards brought horses to the Americas." I recently have made a final-term test question like this: Q: Where did the Spaniards brought horses? A: ______________________________________(Make sure to use the 'it-cleft') Not a few students answered like this: "It was the Americas that the Spaniards brought horses to." But, I think that only "to the Americas" is a right answer. Does the preposition 'to' need to be next to the noun? So, Separating 'to' and 'the Americas' seems to be awkward. What do you think about it? I'd really appreciate it if you could help me...


We have terms like 'high culture' and 'high art'. They are used to describe things deemed of a higher quality, or held in higher esteem, than products of popular culture. We also have terms like 'art music' and 'art film'. What I'm looking for is a similar term to describe literature. Like high literature or art book or art novel, none of which I've heard being used. For example, we can say: John wouldn't like Transformers. He only watches art films. But what about: Harry wouldn't like Tom Clancy. He only reads ..... (what?)


I'm looking for a way to truncate outliers in a time-series graph. Some context: I'm plotting two different metrics on a single graph. It's important to understand what portion of "Metric A" takes up respective to "Metric B". "Metric B" can at times spike to a high value thus increasing the Y-Axis scale which then minimizes the visibility of Metric "A". Are there any good formulas that help normalize this data or visual treatments that can make the graphic honest as well as accessible? See screenshot for more context. The blue area is "Metric B". Thanks!


Context: I'm programming a system for exams, each exam has people on it, in the industry we would call one of these people a 'Delegate'. Unfortunately in the programming language c# 'Delegate' is a reserved word and I can't really use it without my code looking terrible. I need a word to represent a person who will be on an exam. A colleague suggested ExamSitter but I think that might have been tongue-in-cheek


How does quantum electrodynamics actually explain HOW reflection occurs on a microscopic scale? Note that Feynman's QED lecture series/book is not sufficient, as he only assumes that light DOES reflect ('turn around and go back') in order to expound his path integral theory. My question is why does light have the propensity to turn around in the first place. Is it just the absorption and re-emission of photons, and if so, why does it happen so uniformly (i.e. on a shiny thing, entire scenes are reflected near-perfectly). In essence, why are flat things shiny? Are all the molecules arranged at exactly the same angle?


I'm not very advanced in LaTeX. I don't know what e.g. XeLaTeX or LaTeX->dvipdfmx are. I only use ShareLaTeX and used to use Overleaf. I need to be able to insert thumbnails in a document which when clicked would display their bigger version in a popup or something. This gif is a perfect example of what I want to achieve. Is it possible to achieve something like that in one of the mentioned sites? If so, how? (if there's a way to achieve it online but on a different site, I'm still interested)


According to Wikipedia, the past tense (and past participle) of the verb to output is either output or outputted. Are these two forms entirely interchangeable? Or do they have certain nuance in meaning or context (e.g. in programming one of the forms is preferred)? Instead of correct results, my program output/outputted garbage. Which one should I choose? My logical sense tells me output is better because it derives from put, but intuitively I tend to use outputted when I speak and don't have time to think.


So one of the exercises I am doing is to prove (or disprove) that 'Every compact set on a metric space is bounded'. Verbally, I can 'prove' this by simply stating: "If the every compact set on a metric space is not bounded, then there exists an infinite number of open covers, and if something is compact, there are only a finite number of open covers. This is a contradiction, and thus every compact set on a metric space is bounded." Two questions: is this argument right? And if so, how to I learn how to represent this formally?


I've heard people using this idiom, such as "each day is better than the next", or "you hope that each experience you have is better than the next" (heard this one on a TV show not long ago), apparently in a positive way. However, if taken literally, I find the meaning very negative - if the current thing is better than the next, then the next thing must be worse, so things would just keep getting worse and worse. How do you explain this phrase/idiom?


Linkin Park sings "wash the poison from off my skin" in Castle of Glass. Shawn Mendes sings "ripping all the skin from off my bones" in Mercy. As far as I can remember these are the only two occasions where I have heard that expression. To me it sounds like it should be "off from" (if at all). Is this normal, or did the artists just do it to get another syllable for their rhyme?


I have been on and off of this problem for three days and need to present the proof tomorrow. I am thinking that because I know for any element in the additive group of integers modulo n the order for that element is the ratio of n and the greatest common factor of that element and n then I can say,[...chirp, chirp, chirp...] and my mind goes blank. What can I say? I have also tried using the division algorithm to say the order of any element in the group can be written as a multiple of n and some other integer but I am coming up blank. There is something I do not fully understand and would like for you to point out my oversight. Thanks.


"My female cousin working for a finance company was dismissed. Disappeared along with her job were her confidence and smiling face." There is a very complicated system in Chinese for naming different relatives. For example, in Chinese, different words are used for a female cousin and a male cousin. Also, the word for an elder female cousin is different from the word for a younger one. I'm having some trouble writing this sentence in English. I used "female cousin" in the first part, to translate the word that actually means "elder female cousin" in Chinese, but it still sounds awkward. I suspect that the "female" may be redundant too, because in the second part I use the pronoun "her".


I am trying to implement Microsoft Excel's GROWTH function in JavaScript. This function calculates predicted exponential growth by using existing data. What makes it tricky is that it must work with multiple sets of known_x's values. I could not find any reference equation. Any suggestions? The part that might be relevant is the reference equation. I cross-posted the question here because someone suggested it in response to the same question originally posted on stackoverflow. Thanks in advance for your help.


I'm interested in writing a numerical integrator to solve the motion of systems of pendulums. For example, a simple case would be the double pendulum. The motion can be quite complex in general. Is there a way to determine if a system of pendulums is going to produce stiff DEs before actually trying to integrate them? Usually the masses and lengths of the pendulums are considered to be identical for the classic double pendulum system, which I suspect makes the system quite tame. But I have nothing to back that up.


I am fairly new to this so apologies for informal terminology. After I discovered what space filling curves are, I came to the conclusion that any point in any number of dimensions can be represented as a single number along a space filling curve, given that the curve covers enough space. I also imagine that it is possible to rewrite any operation on these points in terms of movements along the curve. Is this correct to assume?


I've found plenty of blog posts and papers where the authors claim that the Higgs mass divergence (usually presented with a momentum cutoff) doesn't show up under dimensional regularization. Unfortunately I've never seen a book or paper which explains this with any more detail: it's usually just stated as a fact. Is there some specific reference that people can point to for this? I'm well aware that dimensional regularization just hides the hierarchy problem until physical particles are added, but I want to include a reference to motivate the introduction of new particles without going to through dimensional regularization.


I need some assistance with a calculation I'm trying to do. I have access to surgery waiting times data (date arrived, date completed) and I want to figure out the expected waiting time for a new patient - this should take into account the number of patients already waiting and those who have completed (I think). I've tried looking at the average time to wait for the past three hours but this doesn't seem the right way to do it (too simple I think). I've also thought about perhaps looking at the average wait time for the same day of the week for a previous week...would that be sensible? There are so many variables to consider but I need something relatively simple...


People on Reddit often comment "I lost it" while quoting the funniest part of a joke to highlight how funny that part is. As I don't speak English much, I am not sure if this is Reddit-specific, but I guess not. Where does this come from? Is it related that laughing from a joke is similar to "losing your mind"? That seems far-fetched. "I lost it" would intuitively mean "the joke stops making sense here" (i.e. "I lost track of the joke here"), which is the exact opposite of what it actually means. What is the etymology of this phrase?


I've searched the question in the forum, and found the following conclusion : - "In electromagnetism, electric flux is the rate of flow of the electric field through a given area. Electric flux is proportional to the number of electric field lines going through a virtual surface. But, It doesn't make intuitive sense to me, that electric flux is just something that we can count like no. of lines, I don't think electric field is passing in lines literally, so what exactly is Electric flux? and Rate of flow suggest that it's moving like water flow in river. These ideas are difficult for me to grasp, please clarify.


I've heard mentioned in various classes that neutron stars, like superconductors, are described by BCS theory. I know that in superconductors a key element in forming cooper pairs is a net attractive force between the electrons which would normally repel one another. That attractive force is accounted for via lattice vibrations (phonons) created and "absorbed" by electrons. So my question is: what provides the attractive force between neutrons? just gravity? If it is true that neutron stars follow BCS theory, by what means was someone able to verify that?


I have to submit some paperwork for the approval of an activity. However, I have already got approval, the paperwork is only necessary to submit the specifics (the date, people involved, and other details which have already been approved) for sign-off. What is the word (or expression, but preferably a single-word) to describe this kind of action? Examples: I already have approval, so submitting this paperwork is a ____ act. I already have approval, so the paperwork is just ____. It's not "futile" as it's still necessary, I'm not "going through the motions" as I really mean it and my heart is in it, so how can I describe this paperwork?


I've been studying the representation theory of groups from Tung's "Group Theory in Physics." I understand Young symmetrizers of different Young diagrams are essentially primitive idempotents in the group algebra of the symmetric group and then all inequivalent minimal left ideals as well as all inequivalent irreducible representation can be obtained. However, the construction seems unintelligible to me, while the property of Young symmetrizers is so striking. What is the idea behind the construction?


What is the meaning of the text quoted below? In the physical world, if a system is described by an equation that is first order in time, the system is general dissipative (has energy loss). If the equation is second order in time, the system may be non dissipative. Such a system has time-reversal symmetry. Can somebody explain what it really means to be first order and second order in plain English?


I'm thinking of those old U.S. wartime movie-theater newsreels that celebrated victories, but combined with aspersions and taunts on the enemy. "We're great and those guys are idiots." Not necessarily in film format like a newsreel; maybe a printed publication. Example: The Mouth of Sauron Magazine, the ??? of Mordor, celebrates its victory at Minas Morgul and continues to mock the armies of Minas Tirith.


Several people I know were good in mathematics when they were in high school and they loved it but when they joined a university (specializing in mathematics) they felt mathematics is hard and that they were somewhat deceived because this wasn't the type of mathematics they loved and joined the university to learn. How can we overcome this problem in contemporary curricula? and why there isn't a universal mathematics curriculum? (I mean mathematics is not country-dependent like languages or history)


After reading this answer about the gravitational force I wondered how that applied to the electric force, since both have the same basic form - product of the masses or charges over distance squared. If I was in a charged metal-walled elevator that was "falling" in an electric field, would I be just as oblivious as my friend who was in another elevator falling in a gravitational field?


I asked a question on one of the stackexchange sites and one user edited a sentence I made from "We were relocating so we...." to "We were moving house so we....". As an english speaker, I have never heard of "moving house" in context of relocation in a sentence before. Presently I am even doubting if relocating is correct in the context of : "Moving all your belongings in a house you onced rented into a newly rented house"


In many mathematics texts I've seen "ordered n-tuple" appear, and in such texts, there isn't any mention of just "n-tuple". So I'm wondering: are there really cases where one writes "n-tuple" and somehow it's not ordered? If not, I'm thinking the "ordered" in "ordered n-tuple" is really not necessary and can be shortened to just "n-tuple". Please let me know if there is significance is writing "ordered n-tuple" that I'm not seeing.


I had to edit my question because I think it led to misunderstanding. For me, the definition of "immunity" is quite different from "impunity", and I know the differences. But what I don't know is that is the law the thing that draw the line? It seems to me they both refer to exemption from punishment. for example, a police officer can kill a criminal, whereas I cannot. Another example is that a diplomat raped and tortured two women but nothing happened. But if a man who has a powerful and wealthy father did that, the word impunity would be said. So, here is the question: Does only the law define the differences?


Which is the better verb to use with data: feeding or entering? Furthermore, which is more common in the literature of the field and which do people who work in the field say more often? Are they used for different kinds of input systems? If the data are automatically continuously input by a machine (e.g., the Mars probe), is feeding data used, but if the data are individually input by someone at a keyboard (e.g., entering data into a user database), is entering data used?


I want to use beamer (Madrid style) for a presentation I'm giving next week, but I am forced to use the institutional title page. That means that the whole background is filled by an image and the title itself cannot be in a coloured box. In the worst case I will have to pdftk the title page onto the presentation, but I'm sure there must be a way to force LaTeX to do it. I've tried using a PlainFrame with a TitleGraphic, but I can't get the image to stretch across the whole page the title itself will still be in a coloured box Any ideas???


My teacher made an example to explain DFA, it was about paths (URL paths), the rules were as follows: S ::= / S ::= /O O ::= [a-z] O ::= [a-z]R O ::= [a-z]S R ::= [a-z] R ::= [a-z]R R ::= [a-z]S Examples of paths could be: /foo, /foo/, foo/bar and so on. However, I don't understand why you would need the R rules since they are equal to the O rules. Can I write it without the R? If not, why?


The problem is, that I fail to unambiguously understand this phrase. There are two ways in which I can understand it (and a number of similar phrases): I may never be able to do this = It's impossible for me to ever do this ('never' negates 'may') I may never be able to do this = It's possible that I will never be able to do this. ('never' negates 'be able') Which is the correct one?


According to Newton's laws, if net force is zero on a particle then the particle is at rest or in uniform motion, and if it is not zero then it is accelerating. So when a car moves on the road, friction helps it to accelerate. If the acceleration is zero, (i.e) it is moving with constant velocity, the net external force must be zero, so friction is zero. Then with friction equal to zero how is the car moving on the road?


I have to do a final project for my PDE subject and last year I did one about Game Theory (specifically, Prisonner's Dilemma and Snowdrift game) for my ODE subject, which the rest of the students enjoyed and which my teacher thought it was a fun approach to the contents of the subject. I want to do a similar thing, but every Game Theory problem I have seen only involves ODEs. Until now we have seen first order PDEs (both linear and non-linear) and second order (hyperbolic, parabolic and elliptic) PDEs. Also wave equation and heat equation. What would be some fun PDE related problems to discuss? (I hope I put the right tags, because it is the first time I ask a soft question).


The title gives away pretty much the entire question except for the fact that the class of functions I am interested in is a subset of twice continuously differentiable functions. I think that if an odd function (defined on the whole real line) is concave on the positive real line, then it is convex on the negative real line. If it is supposed to be convex everywhere, then it is linear on the negative real line. The argument goes also the other way. So it is linear everywhere. So maybe I don't even need the monotonicity property. This is the first question I am posting here that has no equations. So I hope that is allowed. Is my reasoning correct?


Do you know if the concept of chemical potential can be properly defined for nucleons in the nuclei? I mean, if I can picture the nuclei like an interacting gas of nucleons, then may I think of a chemical potential for nucleons, similarly to the case of an electron liquid. Of course the nature of the interactions is different for the two cases. My idea is related to the liquid drop model of nuclei.


So imagine you're at the beach; you go into the water and the moment you enter the water you stop hearing anything from the outside world. The same happens vice-versa: your friend shouts at you from inside the water but you only hear the bubbles rising to the top. So the questions are: Why does that happen? Would it also happen if you used a hydrophone?


I am writing a title and I was wondering whether I can skip the second article just like you would say a pen and pencil. Can I say, "A cap and tie for Zed," or must it still be "A cap and a tie for Zed?" Have been searching the net and books for some answers but have been unsuccessful. Would appreciate a quick response from some Grammar Genius! Need clarification here to finish up my work. Thanks ya.


The usual solution to fix lines that overflow is to rewrite it. However, that hardly works with bibliography entries that experience the same problem. So what can I do to fix it? I can't rewrite anything since the information needs to be "as-is". I can't provide a minimal example because if I take the entry that overflows in my document, then the problem fixes itself as the font, font size and margins change. EDIT Even though I've accepted an answer, please submit more answers if you have a good approach for addressing this issue. Maybe it can assist someone else who have the same problem and don't want to resort to using "ragged right".


I have studied in a course several algorithms to integrate ODE's numerical: Runge-Kutta, Predictor-Corrector methods, Taylor... However the teacher failed to show which is the best for every particular situation. The only thing I know is that implicit methods are appropriate for stiff systems. But how do they compare Runge-Kutta (of any order) with predictor-corrector methods or with Taylor method? Which is best for each situation? Heuristic answers based on experience may be good enough!


I have a paper copy of some old LaTeX notes, but I lost the source file. If I scanned them as a .pdf, does any OCR software exist that could convert the scanned .pdf to LaTeX source (.tex)? I also have a paper copy of some old notes created using Microsoft Word's Equation Editor. While I am considerably less hopeful, I ask the same question -- does an OCR software exist that could convert a scanned .pdf to .tex? Thank you for your help.


I came across a problem where I have to find out the longest path in a given graph. I have list of edges ( eg.{AB, BC} ) which states there is an edge between vertices/nodes (A,B,C). Now i want to figure out the longest path possible (not repeating the vertex) such that it covers maximum nodes starting from any vertex/node. What can be the best way to solve this? I have to implement this as a program. I looked up google for Minimum Spanning Tree, Dijkstra's Alogorithms , hamiltonain path( which i think suits- but not sure ) and many more. but can't figure out what would suit best for this problem. Any help or reading references would be much appreciated.


Here is an example sentence, written by a pupil of mine: Through the British Empire, which resulted out of Britain's urge to build up its economy, Britain was connected to a lot of different countries The sentence clearly has several issues, but I am only interested in one of them here. I would only use "result from", as a native British English speaker. My pupil has one American parent and one German parent, and lives in Germany. Would "result out of" be acceptable in written, academic American English? Can it be said at all?


What are the possible applications of Countable Infinite Sets and Power Sets in areas that are not strictly mathematical? Also I want to know the significance they carry. What was not possible before the concepts of countability and power sets were introduced and what became possible afterwards? How did the introduction of these concepts change our thinking and outlook and the areas they affect? Please explain in plain language, possibly with examples, as I am from a non-mathematical background. Thank you in advance. PS: The applications need not be in strictly practical fields either (as Asaf Karagila keeps pointing there aren't any). Please help!


What is the best word or way to describe a person who just has to try and do everything themselves because they think, either rightly or wrongly, that only they are good enough to do the task in hand? The type of person I'm thinking of will attempt to take over any situation. So if you're having a BBQ only they are good enough to cook the food or if you're planning a day out then they decide only they are good enough to drive you there.


My understanding of irony comes from the movie "Reality Bites": It's when the actual meaning is the complete opposite from the literal meaning Frequently people use the term incorrectly, applying it where the actual meaning and the literal meaning are surprisingly similar in unintended ways. A perfect example of the incorrect usage is the picture below. It's definitely not irony, so what word can we use? (Just in case the link has broken, it's a picture of a building with a plaque saying "George Orwell Lived Here". Pointing out in front of the building is a security camera.)


Why is it true that the lower sums of f with respect to some partition is less than the lower integral (which is the supremum of the lower sums) I think what I'm confused about is the difference between a lower sum and the lower integral, how can the lower integral be the supremum of the lower sums when a lower sum is a single number?


Gravitational force is mediated by graviton exchange. If I am standing outside a black hole, I can of course feel the attracting force towards the black hole. This should correspond to gravitons mediated between the matter inside the horizon and myself; but then these gravitons should cross the horizon from the inside to the outside. My question is: how is this paradox precisely solved? I guess a starting point for an answer is that these gravitons are off-shell, much like in QED where photons exchanged between electrons that feel each other are virtual. But still, this confuses me a bit - is there some references explaining this in detail?


I'm writing an essay for a grad school application detailing my unique characteristics as an individual. I am leaning towards words & phrases like "thirst for knowledge" and "earnest curiosity". I feel these phrases are underselling this attribute though - I feel so genuinely interested and invested in how things work and how people think, and thus how others perceive stimuli in comparison to my response. Is there a word for this constant absorption of external data in order to think critically and formulate theories for why the world works?


I am trying to understand the following question: "How much experience do you have gathering data for web analysis?" I understand the concept of "web data analysis", and I understand "gathering data from the web for analysis", but the above question is not clear. Perhaps I am being too literal or picky, but the question sounds like I would first gather data, then put it on the web, and then analyze it, which is not logical. If you have any ideas, could you please provide a specific example of what kind of work would satisfy the question. For example, maybe the question means I would download Apache Server logs from a website, import into Excel, and perform analysis?


Is there a word to describe the co-opting of a well-known phrase in the course of regular conversation or writing? For example, a sentence regarding death might refer to "shaking off the mortal coil" without mentioning that the phrase is taken from Shakespeare (perhaps assuming that the audience will be familiar with the reference). If it's not an exact quote, this could be called a "paraphrase" of Shakespeare. But what could it be called if it is an exact quote? Example: What adverb could be employed to complete this sentence: The phrase "shuffled off the mortal coil" was made famous by Shakespeare, and is used by this author ______y.


The book < Geometry and the Imagination > (written by David Hilbert) introduces a property of a Quadric Surface without a proof. Property : The cone consisting of all the tangents from a fixed point to a quadric cuts every plane in a conic, and the points of contact of this cone with the surface form a conic. Moreover, the quadrics are the only surfaces having any of these properties. It was easy to prove the property itself, but i found it difficult to prove that it is a sufficient condition for a surface to be a quadric. (the statement starting with "Moreover, ...") I would like to know the proof of this statement.


Ok, so I am trying to computationally model a Carbon Nanotube FET and I need to know a lot about the actual equations and PDEs(Partial Differential Equations) involved while modeling such a material. I had taken a course in material science in my sophomore year , but that just covered the basics of CNTs and its applications. I would really like to know if there are some good books out there that give an in depth understanding about CNTs ( structure, properties, behavior in presence of Fields, Equations and PDEs, Fabrication and applications, etc)


I am having a hard time figuring out how to configure TortoiseHg to do colored word diffs for my TeX and LaTeX documents. I found the following two threads How to use Mercurial for version control of text documents? Good strategy for line breaks with paragraphs of LaTeX source but they don't go in detail how to to set this up on TortoiseHg. I was wondering if anybody in the TeX/LaTeX community has figured this out already?


Given an urban area such as a shopping mall, is there a statistical model that estimates how often during a working day the area becomes depopulated and what is the maximum time of depopulation? By depopulation I mean that the number of people per unit area becomes zero. In particular I am interested to know the answer to this question: Can we say that typical urban areas are usually populated (during the working day) and there is an upper-bound for the depopulation time? any article that exposes some research regarding this subject would be useful.


In the following paragraph, is it appropriate to use the term "transversely" to describe something that has the opposite effect? Tests have shown that the lower the range, the more likely that a submatrix will be singular. Transversely, the larger the range, the least likely that a submatrix will be singular. If not, is there another word that I can use instead? I know that, in this instance, I cannot use the word "conversely" as this means something totally different.


This was an experiment I saw in my son's workbook. It said to mark out the top of your forehead and the bottom of your chin on a mirror using a whiteboard marker. Then slowly move backwards, and investigate what happens to the size of the reflection subjective to the two marks made. It actually got me quite flabbergasted. I always thought the reflection would get smaller as you moved away from the mirror. Why is this?


While explaining the quantum adiabatic theorem recently, I appealed to a thermodynamic analogy: when slowly contracting the walls containing a classical gas, the relaxation timescale can be taken to be fast, so the system remains always in thermal equilibrium. By analogy, the quantum system remains always in a stationary state. The hidden assumption here is that stationary states (eigenstates of the Hamiltonian) are in some sense attractors of the system: that a system in some energy-superposition will, over time, relax to a single energy eigenstate. Is this actually true? It seems like maybe it isn't, since for example coherent states of the harmonic oscillator are stable (?). Maybe it's true in neighbourhoods around the stationary states only?


She will (or already has) leave for Jamaica soon. She will leave (or already has left) for Jamaica soon. I'm uncertain which of these (if either) is more correct. Should I simply not use this language construct, opting for this instead: She will leave for Jamaica soon, if she has not already. I find the previous constructs, despite being awkward, do a better job of conveying the uncertainty of tense. Is there a better way to construct the parenthetical version to avoid tense conflicts?


As I understand: Accelerating electrons generate electromagnetic waves. An emitting antenna have an alternating current (electrons are moving) which generates an electromagnetic waves. The electromagnetic waves reach the receiving antenna and makes the electron inside move. Greate, the communication is done. However, since the electron in the receiving antenna are moving too, the receiving antenna is generating an electromagnetic wave too. How is this not affecting the incoming wave? Does it have a different wavelength?


I'm looking to communicate the idea that performing a task would be less costly (more than just financially; technically, or when risk is considered) if you start from scratch or anew, instead of attempting to alter or change some existing object or entity. The closest phrase I can think of that satisfies this is "it may be cheaper to build a new house than to renovate an old one". What other sayings exist?