I am sure there has to be a more precise word to describe a person that is not giving up on her beliefs no matter what other says. You could say consistent in her beliefs, but I am looking for something more elaborate. An example of such a person might be Margaret Thatcher. Thanks


What is the origin of the idiomatic expression rub someone the wrong way? Is it correct to use the idiom in reverse, i.e. rub someone the right way, possibly meaning to calm or to please?


I know of at least three equivalent formulations of QM: The "normal/standard" one, dealing with Hilbert spaces and state vectors. The Feynman path-integral formulation. The Wigner-Weyl phase space formulation. My question is: what is the usual name given to the first, "normal" formulation that everyone learns as an undergraduate?


I know the differences in the meaning of word "trash" and "garbage" but how about "take out the trash" vs. "take out the garbage"? Can both these expressions be used interchangeably? What is the difference in meaning if any?


My textbook states that "Laser communication is much faster than microwave communication." But, how can that be? Both are electromagnetic waves with different frequencies but, how can the speed be different? Or Is the statement referring to any kind of processing speeds involved?


I might be mistaken, but I feel odd calling the following "wooden handrails", since handrails are supposed to be made of metal. Is it OK to call it "handrail", "guard rail", or a "wooden handrail"? Or is there a better term for it?


Today, my teacher proved to our class that every convergent sequence is a Cauchy sequence and said that the opposite is not true, i.e. Not every Cauchy sequence is a convergent sequence. However he didn't prove the second statement. Is there an example or a proof where Cauchy sequence is not convergent?


Is there any good word/phrase/idiom for that feeling of scorn when you're accepted by someone/something after getting rejected the first time? I remember a phrase being quoted by Nathan Fillion in Castle when Richard Castle's daughter is rejected to Stanford and later gets in. Still can't remember the lines, just the situation.


I'm fairly well-versed in calculus but I would like to explore beyond calculus. I have looked into the basics of some topics in higher mathematics such as group theory and abstract algebra and they intrigue me. I am wondering if there are any recommended methods or resources I should use to learn more about these topics. And is there any recommended starting point?


I'm a complete beginner at LaTeX, and I'm trying to typeset a complex layout. How should I go about building a page like the one shown below. Edit I was able to replicate the layout using minipages with the multicols package. The images were embedded using the graphicx package.


I came across several forums and articles saying that criteria is plural and criterion is singular. Some gave me the impression that criterion is used to denote a set of rules. What is the correct use of these two words?


I have somehow managed to lose all my document classes and hence cannot compile anything. I have tried re-configuring and re-installing to no avail. I am on a university computer and cannot edit the program files.


"If they would have been painted this afternoon, the walls would be completely dry by tomorrow evening." The quoted sentence is ungrammatical. I believe it has something to do with tense, but I do not believe that the 'if...would be' structure requires parallelism. So what is wrong with this sentence? Thank you.


Is solving problems quickly an important trait for a mathematician to have? Is solving textbook/olympiad style problems quickly necessary to succeed in math? To make an analogy, is it better to be a sprinter or a marathon runner in mathematics?


I have had real difficulty with permutation/combination questions in probability and statistics texts. What I have real difficulty with is transforming word problems into mathematical form to solve. Are there any recommendations for a good text that might help me to do this? Or any suggestions for how to tackle these sorts of problems. Thanks


Is there a word that works as a milder version of strenuous? Strenuous is defined by Oxford as something "requiring or using great effort or exertion." I'm looking for a word with a similar definition except for the "great" part.


I am confused on how to properly describe an all expense paid vacation. Is it an all expenses paid vacation or an all expense paid vacation, and are there any hyphens between all, expense or expenses, and paid? I ask because it seems to be all over the place on google search.


What exactly is meant by Lorentz invariance? Is it just an experimental observation, or is there a theory that postulates it? What quantities do we expect to be Lorentz invariant? Charge? Charge densities? Forces? Lagrangians?


I've heard many people use the term "goose bumps"; in my family, they were "goose pimples," but I don't know if this was peculiar to us, or if others also use it. My wife's family's saying for this was "Chilly bumps." Are there others? Is one considered the "most correct"?


I've read a few different explanations of how hovercrafts hover, and they all mention a low-pressure cushion of air. This confuses me though: If there is low pressure under the hovercraft, why doesn't it get sucked to the ground? Also, if air is being forced into the space under the craft, why is this air cushion not high-pressure?


It is known that in general, a mixed state can have multiple pure state decompositions. However, it has a unique eigendecomposition in the absence of degenerate eigenvalues. What is the special significance of this eigendecomposition over other pure state decompositions for the same density matrix?


As the title suggests, Can the effects of a person's mass upon the local gravitational field be detected and measured remotely? I am aware any mass produces and effects gravity but couldn't find anything in my searching if it is possible or theoretically possible to detect this effect remotely.


I used to pronounce 'detail' as ['di:teil] with accent on the first syllable. However, these days I hear people (and on the radio) say [di'teil] with accent on the second syllable. Is my pronunciation incorrect? Is there a regional preference for this?


I'm a high school teacher and someone asked me this in my class, and to be honest I'm quite stumped! I haven't done any high level math in such a long time, and I'm really not sure how to approach this. Is the solution even approachable to a highschool student? Thanks for the help.


I am going to learn some math about functionALs (like functional derivative, functional integration, functional Fourier transform) and calculus of variation. Just looking forward to any good introductory text for this topic. Any idea will be appreciated.


That would not have happened if John had completed his work. That would have not happened if John had completed his work. The former seems correct. The latter doesn't seem incorrect. Are there any cases where one ought to use the latter instead of the former?


For example I've written: "I have an interest in rock climbing and am a member of the mountaineers Society." My friend thinks it has to be: "I have an interest in rock climbing and I am a member of the mountaineers Society." which I think sounds bad. Which is right?


What is the past perfect of "have"? Is it "have", "had", or "had had"? And if it is "had had", doesn't it sound weird and awkward? when and how do I use it? Thanks.


I have a problem that I need to prove using induction. Prove that a surjective function has at least as many members in its domain as it does in its codomain. Do I begin by using the axiom of choice? Thanks.


In Arabic, the expression "Summer cloud" is usually used to mean that "this is not going to last for a long time" or "it won't last as long as you think it will". Is there an equivalent phrase in English?


So I'm essentially trying to find an explicit description of the smallest subfield generated by a subset of the field. I know that if s is an element of the subset, we must also have its additive and multiplicative inverse. Is there a succint way of describing the subfield set-theoretically? Again, the intuition seems clear, but I'm struggling with a formal description...


This morning I heard the word "constitutionality" being used by a journalist with regard to the debate over the legality of health care reforms here in the US. This grates on my British ears as I would simply use 'constitutional' in this context. Does 'constitutionality' have a different meaning? Is this a recent Americanism?


When measuring blood viscosity, the literature claims that we generally use a cone-and-plate viscometer. Why is this; is there any way to explain this mathematically in terms of the shear rate, etc? Does it have to do with the fact that the shear rate is constant in this device? Why won't it work with a parallel-plate viscometer?


What does the phrase "begging the question" really mean? And does it even matter if I use it correctly? Almost everyone just uses it as a synonym for "posing the question" these days.


Is there a good template for software release notes? I typically cover new features, bug fixes, and enhancements. I'm currently using HTML but its ugly and not easy to distribute. Ultimately, I'd like to produce a single PDF.


I've been going over the algebraic topology part of Munkres and this question has stumped me. If we have a complete metric space that is not compact, must it be simply-connected (path-connected plus trivial fundamental group)? My intuition tells me no, but I can't come up with an example.


This may be trivial, but I want to be sure I understand correctly: Is it true that the interior of a simple polygon is always a simply-connected subset of the plane? I.e, is it eligible for the Riemann mapping theorem?


I am wondering if there's an English expression for making someone fall in love with you. In Hungarian there is a term that could be translated into something like "fooling someone into yourself" - meaning you make the person fall deep in love with you, and it is implied that it's thanks to some deceitful or dishonest behavior.


I'm trying to get some intuition, I think it almost makes sense, since you can think of the area of a square is length multiplied by height. The volume of a cube is length multiplied by height, multiplied by depth. I don't understand how to find the volume of the hypercube. What's the correct intuition?


I recently became interested in the solution to Hilbert's tenth problem, in reading about the succession of results that lead up to the proof I came across the notion of recursive sets and recursively enumerable sets. The distinction between the two sets is very subtle. I need some help in answering the question in the title. Thank you in advance.


What is it called when a character knows something the audience doesn't? For example, if the character was stating something obvious like "today is your birthday", saying it only to inform the audience, what is that called?


I know that the game which is called football in Europe is called soccer in the U.S. But I wonder to what extent this differentiation is strict. What do people from England call their favorite game in conversations with Americans? Is there a misunderstanding in this case?


When reading certain books I will encounter names of places or people that have been abbreviated. An example is in Catherine Hutter's translation of Goethe's "The Sorrows of Young Werther": A few days ago I met a man called V., an ingenuous fellow with a very pleasant face. Why is this done?


Let's say I'm trying to describe two things that change. The first changes often and the second changes less often. Is there a better word than "volatility" to use in the following scenario? Is it even proper in this use? The first thing has "high volatility" The second thing has "low volatility"


I feel like when two conjectures are inconsistent with one another, it's a clear sign of our misunderstanding of deeper mathematics. I was wondering if anyone knew of a comprehensive list of conjectures that contradict other conjectures.


An ice cream shop sells ice creams in five possible flavours. How many combinations of three scoop cones are possibles?[Note:The repetition of flavours is allowed but the order in which the flavours are chosen does not matter.]


Consider the following sentences: If I had my own place, I could do whatever I want. If I had my own place, I could do whatever I wanted. She said I could do whatever I want. She said I could do whatever I wanted. Which ones are correct and why? Is the answer different in a conditional, or does that not matter?


I am using sharelatex (great tool) but I haven't been able to find a way to show the ruler (blue numbers on the sides of the paper) that comes with the CVPR template. I was wondering if someone has had the same problem and how did they solve it.


"It's precisely in contrast to the ordinary that the resurrection stands out." Could someone explain what the author meant by "in contrast to the ordinary"? Could I replace "in contrast" in this case with "opposite"?


I'm fascinated by the fundamental questions raised by the Double Slit Experiment at the quantum level. I found this "Dr Quantum" video clip which seems like a great explanation. But is it scientifically accurate?


I'm following my first course in field theory and the professor began, like many books do, by introducing the scalar field. However, I am a bit hesitant about the physical idea of fields. My question is: what is the physical meaning of the fields? Why they are introduced? I read the introduction of the books of Peskin and Weinberg but I'm not satisfied.


I have an electric stove, and when I turn it on and turn off the lights, I notice the stove glowing. However, as I turn down the temperature, it eventually goes away completely. Is there a cut-off point for glowing? What actually is giving off the light? Does the heat itself give off the light, or the metal?


In Flemish we have a saying "Vijgen na pasen". Translated: "figs after Easter". It means a solution comes too late to be of any use. What is the English equivalent for this? Some googling gives me "Closing the barn door when the cow has bolted", but the explanation seems to point more towards the wrong solution for a problem.


I am an undergraduate physics student, and I wanted to study about interacting particle systems. I have studied probability theory, linear algebra, and statistical mechanics. I know basic measure theory and stochastic processes, but have very little knowledge of algebra. What more do I have to know to study interacting particle systems? (Please give names to some reference books if possible)


I know that the zeros of analytic function (with one variable) over complex plane are isolated. However, I am not aware about the structure of the zeros set of analytic functions over complex plane with several variables. My question is: How I can understood this structure.


According to this Fritinancy entry, the demonym for Norfolk, England is "North Anglian," rather than "Norfolker" or "Norfolkite," for historical reasons. What about Norfolk, Virginia, in the United States? I suppose you could argue for the same historical reasons, since it was named after Norfolk in England, but that just feels wrong to me.


I am using sidewaysfigure for many of my figures. When I print out my document, for reasons I don't understand, the figure captions are on the inside and not the outside. Is there a quick way to flip all the sideways figures without affecting anything else?


Does the current acceleration of the universe imply that our universe is open? If the universe is closed, from the Friedmann equations, the acceleration of the universe wouldn't be possible, would it? (Of course, except for the very early inflation era.)


I know how to use a .bib file to add BibTex and use that in the main document. Is there a way I can include the bib-file in my latex document and not use an external bib-file?


Do magnets (permanent) become weaker as they are exposed to para-magnetic objects? I was thinking about this after seeing this. I am buying a magnet and wish to know if a magnet (permanent) loses its magnetic domain structure or alignment when exposed to para magnetic materials.


If I understand correctly, scientists were able to find strong evidence for the existence of quarks by using deep inelastic scattering. If quarks (hypothetically) are composed of preons, could we use similar methods to detect the preons? If so, how would we have to modify the scattering experiments used to detect quarks?


I have a funnel and I notice when I try to pour powdered sugar through it, it blocks up unless I tap it. If I put sand in the funnel, it goes through. Is there a theory of funnels that can predict what materials will pass through a funnel of given dimensions?


If I project a circle over the Z-axis, I'll get a cilinder. If I project a square over the Z-axis, I'll get a parallelepiped. If I project an ellipse over the Z-axis, I'll get a... whatsitsname? I can't find the name of this object, and I can't believe it doesn't have one to begin with.


If I write a paper, is it acceptable for me to use a title like "analysis of PDEs related to blah blah blah" instead of "analysis of partial differential equations related to blah blah blah"? I intend on using in the abstract the words partial differential equations and writing (PDE) in brackets next to it. Is this reasonable? Thanks.


In my high school and college math classes, I've always seen graphs drawn with either no arrows on the ends of the axes or with arrows on both ends. A colleague recently argued with me that this was crazy and arrows are always placed on one side of each axis to indicate the positive direction. Like so: Which style is more prevalent?


In a normal double slit experiment, I'm told that sunlight doesn't produce a visible interference pattern because there is no stable phase relationship between the two slits. However, sunlight bouncing off a CD does produce interference-based rainbows, so sunlight can interfere if the slits are close enough. How close do the slits need to be to see interference when direct sunlight falls on them?


I'm sorry if this was asked before but with all the stars and dark matter and all the other stuff, curently in the Universe, what's the avarage temperature of the Universe? Is it like extremely high or extremely low?


Let's say we have a given wavefunction and we want to find a particle that will fulfill the properties for that wavefunction. How can we do that? Is it possible? I was thinking of using Schrodinger's equation.. would that work? (please tell me if there is anything wrong with this question so I can fix it!)


I need to find a word which means "the town or city in which one lives". Words like "residence" or "domicile" appear to have this meaning, but because "residence" and "domicile" can also mean "the home in which one lives", my readers will be confused. Is there a word which only refers to the geographic location?


The standard Bernoulli Equation has three terms on each side, a kinetic energy term, a potential energy term and a pressure term. I've never seen an extension of this to also include a rotational kinetic energy term. Do things get too crazy when fluids rotate for a simple extension to the equation?


I don't know why, but despite having read the definition over too many times to count, its one of those words that just doesn't click. I see it being used in sentences and have never been able to fully understand it either. Could someone help explain it, and use it in a sentence? Thanks!


I think I should use broke because it is in past tense. Am I correct? If incorrect, can you explain to me briefly which case is correct? I was thinking about punching him and breaking his teeth. I was thinking about punching him and broke his teeth.


I would like to get into a career that uses alot of applied math. I took a numerical analysis course in undergrad and liked it, so I plan to self-learn numerical methods for PDEs. Other than the MIT OCW, are there any good textbooks or lectures notes that can be viewed online? Particularly those that are geared towards engineers/scientists, since I'm not into theorems/proofs


I read here that a major ingredient in Whitney's strong embedding theorem and later Smale's celebrated h-cobordism theorem is the Whitney trick. Can someone give an intuitive description of the trick? To be more specific, I would be happy to know why and where the trick was applied ? Thanks.


So I am learning LaTeX and I find it slow to test changes. So I'd edit the .tex file, then go to command line, and run pdflatex, wait for it to complete, and then open total commander, and then open the .pdf file. Does anyone know a faster way to test changes? (WYSIWYG?)


I'm looking for a word to describe the final flight of an aircraft (or other vessels if there is no direct equivalent) such as the final flight of the space shuttle. Does anyone know if a word like this exists? Thanks!


We have a project at school to develop an idea about how we can generate electricity. My question is: do collisions between matter and antimatter particles generate heat, so that one could harvest it using turbines?


I want to show that the domain of any partially defined recursive function is equal to the range of some ( totally defined ) recursive function. I haven't understood which is the difference between a partially defined recursive function and a totally defined recursive function? Could you explain it to me?


Why are measurable functions called "measurable"? What exactly is being "measured"? For measurable sets, I can intuitively understand that the measure of the set "measures" how large is the set. Thanks for any help!


I have recently falled into an error because I didn't use % when I was supposed to. My question is now: When do I have to use % at the end of the line and when should I leave it out?


Can we add an uncountable number of positive elements, and can this sum be finite? I always have trouble understanding mathematical operations when dealing with an uncountable number of elements. Any help would be great.


I have seen this phrase do not pass go a couple of times reading Internet forums, but I don't remember figuring it out in context, as I've never played the game Monopoly. What does it mean?


Under normal lens operation, a beam is sent through the centre of the lens along the optical axis (ie perpendicular to the lens's plane). What happens when a beam is sent through a lens at an angle to the optical axis? Does it simply exit the lens at the same angle?


You get taught about matrices and how they work but nobody ever tells you WHY they work in the way that they do. What was the idea that sparked the creation of matrices?


In crystallography, we always speak about the direct and reciprocal lattice spaces. One property about these spaces in mentioned here, as follows: Each point (hkl) in the reciprocal lattice corresponds to a set of lattice planes (hkl) in the real space lattice. Can anyone give a proof for this statement?


What I am asking is this: Why can't a body be solid, then solid-ish, then solid-like, then liquid-like, then liquid-ish, then liquid, then vapor-like and then vapor? Why is there a rigid temperature boundaries between solid, liquid and vapor? Why doesn't water simply change "states" in a continuous manner?


I was wondering what is the difference between I wasn't knowing and I didn't know? If I say, I wasn't knowing, I am talking about something unknown in past, the act of not knowing is finished, it means that I know it now, but before it was unknown to me.


Suppose I have a graph and I calculate the eigenvalues of the adjacency matrix and find that there are some number of zero eigenvalues. Do zero eigenvalues have any significance? Also is there a good way to interpret the inverse (when it exists)? In some cases the inverse is called the Green's function. Thanks!


Verbs can be conjugated to past/future tenses. Nouns can be pluralized. Adjectives also have comparative and superlative forms. For example fast, faster, and fastest. What is the word that describes converting between the base form and the comparative and superlative forms?


I was wondering where the noun punk stems from. Obviously, it's used for members of a certain subculture, but has the word been in use before the invention of said subculture and been adapted for it, or has it been made up specifically for it?


Say I have a rigid body in space. I've read that if I during some short time interval apply a force on the body at some point which is not in line with the center of mass, it would start rotating about an axis which is perpendicular to the force and which goes through the center of mass. What is the proof of this?


I've been doing some maths work using the rate of flow of liquids. I've used various models for the flow and various methods to integrate these models. The one thing that is confusing me is the difference between constants of proportionality, and constants of integration?


With today's computers having several gigabytes of memory, why does TeX still have such limited capacity, and why are modern TeX installations not adjusted for higher capacity by default? Why is adjusting TeX capacity only recommended as a last resort?


State the domain and range of the following graphs. Express each domain using interval notation and express each range using set-builder notation. I think I'm beginning to understand using interval notation and set builder notation, but I'm still having trouble with the endpoints.


British English makes the distinction between 'practise' (verb) and 'practice' (noun). Based on this, I would judge the following sentence as incorrect: In practise, computers often crash. Nevertheless, I see it frequently. Could 'practise' possibly be a verb in this phrase?


I read the definition of a hereditarily compact space in topospaces.subwiki.org, it says: A topological space is termed hereditarily compact if every subset of it is compact in the subspace topology. I don't understand how this is any different from saying: every subset is compact. Can someone explain the difference?


Excerpt: In developed countries, running water is available everywhere and air-conditioning is expected not just in our homes but in our cars. We don't realize that these are luxuries. My sentence: The author mentions "running water" and "air-conditioning" as examples of things we take for granted. Are the quotation marks required?


Imagine a body in a tub which is in a lift. When the lift is stationary, the body is floating. If the lift accelerates upwards with a constant acceleration, what will happen to the buoyancy? Will its value change or not? I do not think it shouldn't. But my friend says it will. But why?


I have researched a little on loop quantum gravity (LQG), but all I got were very ambiguous explanations about loops and nodes that all end with mathematical equations. If someone could give me a non-mathematical, clear idea of LQG, I would really appreciate it.


Is it possible to formulate classical electrodynamics (in the sense of deriving Maxwell's equations) from a least-action principle, without the use of potentials? That is, is there a lagrangian which depends only on the electric and magnetic fields and which will have Maxwell's equations as its Euler-Lagrange equations?


I have seen the phrase used in this form or as a template for other rhetorical questions - e.g., "what's an honest economist to do?"; "what's an honest business owner to do?";"what's an honest Nigerian to do?" I cannot find any reference to its original use in literature, but was asked recently "who said that?" - and didn't have an answer.


I have tried to find this out through google and searching this site with no luck. Basically, are the terms 'law' and 'identity' interchangeable in Mathematics? What is described as 'logarithmic identities' in one place is referred to as 'log laws' elsewhere. Similiarly, 'index laws' and 'exponential identities'.


I'd like to create a custom design for my beamer presentations. How could I achieve this? I have neither found any resources for that nor does the official beamer documentation talk about it. Are there any good resources to start with?


What's a word for an inaccurate artistic representation that doesn't retain all the qualities or features of the original subject? For example: There is a painting of a king but it doesn't have the expected semblance. It looks like him but there are oddities, like an elongated nose, or bigger eyes and ears.


The energy required to remove an electron from the surface of a metal is less than that required to remove an electron from a free atom. Why?The electron is bound to the nucleus in an atom by electrostatic forces. How is an electron 'free' on a metal surface?


What is the difference between the following sentences? Even in those days he played golf on Wednesday. Even in those days he played golf on every Wednesday. Even in those days he played golf every Wednesday. In a non-progressive sentence, which adverb phrase (in those days, or every Wednesday) is used to refer a serial state (habitual)?


I've long been familiar with crenulated edges, but until today I wasn't aware of crenelated. Looking at the definitions, they seem to describe the same kind of feature, that of notches on an edge that are more rounded in contrast to, say, serrated. What are the differences in nuances? When would be a "wrong" time to use one over the other?


Each shop in a town has an odd number of customers and each pair of shops shares an even number of customers. Prove that there are at least as many customers as there are shops. Any hints are appreciated.


Can someone summarize why an ambient space isn't needed to measure curvature when parallel transporting tangent vectors or vector fields along a curve on a Riemannian manifold? How do we define the vector's direction and magnitude without one?


Is it possible to say that the directional derivatives of a function f at a exists but f is not differentiable at a? If so, why? I cannot get the intuition about it. Could someone please elaborate on this point a little bit? I am self studying mostly, so I need to discuss these trivial matters with someone :) Thanks in advance!


I am really bummed out to find that the term "strict monomorphism" is already used to mean something else. Can anybody console me with the knowledge that there is another name I can use for a monomorphism that is not an isomorphism?


I'm searching for the word for someone who believes in the preservation of other people's cultures. Does anyone know of a good word for this? I don't think "anthropologist" is a good word, as that indicates the study of the cultures, but people can have this belief without studying them.


Nowadays, the "white" in "decent white folk" can refer to race. But did it always refer to race, or did it have another meaning? I tried looking at Google NGrams, but it has very few hits.


The majorant criterion says if a series in a Banach space has a convergent majorant, then it converges absolutely. My question is, what if a series in a Banach space has a convergent minorant, does it converges? Or, what if a series in a Banach space has a divergent majorant, does it diverges? Thank you for help.


I would like to combine the words "event" and "explorer" to "eventexplorer". But I am actually not sure if that works. Can I use this combination to describe something like "people who discover new events"? English is not my mother language, therefore the question.


Possible Duplicate: Continuous footnote numbering How can I achieve that footnotes have a global numbering scheme, and not a per chapter numbering? I am using the book class from the koma-script. I am also using the footmisc package.


The following is a multiple choice question in an English test: The employers prepared, with all due _______, for a conference with the Trade Union. A. caution B. concern C. certainty D. consideration I feel A ("with all due caution") is ok, but I am not sure.


Is there a common phrase for the opposite of reaching a settlement in a lawsuit? That is, is there a common phrase for going to court and having a judge reach a decision?


Seeing how infinite the universe appears and out of all of those stars, planets, galaxies, there must be other life forms. Mathematically, the odds are very good. Is there a mathematical equation to determine the chances of other life forms in the universe?


I'm looking for a geometric interpretation of this theorem: My book doesn't give any kind of explanation of it. Again, I'm not looking for a proof - I'm looking for a geometric interpretation. Thanks.


I don't know whether this even makes any sense, but if 'observation' can be considered as 'recieving and reading information', can an act of observation (of a system) change (increase or decrease) its entropy?


Many books on plasma physics (Chen, Goldston, Lieberman) say that quasineutrality must be satisfied for the matter in question to be a plasma. Yet, we know that non-neutral plasmas exist. So why do so many books make this claim?


Some verbs are followed by ing, e.g. I enjoy swimming. We can't say I enjoy to swim. Likewise, some verbs are followed by to, e.g. I decided to make a plan. Which particular verbs are followed by ing and to? Can you please provide a list for that? Moreover, which verbs can be followed by both without having the meaning changed?


Consider the vector space V=Fun(X,F) where F is a field. Where Fun(x,F) is the set of all functions X->F. How do you show that there is a zero element in V? And how would you describe the additive inverse?


I know that if the second and third row can somehow be manipulated to zero, then the matrix will have a single pivot. However, I'm not so sure about a matrix having no pivots, because that just means the entire matrix is filled with zero. Thank you!


This question actually has two parts: a. Prove that every subspace of a topological space with the discrete topology has the discrete topology. b. Prove that every subspace of a topological space with the trivial topology has the trivial topology. If I can do part a, part b will be easy. What do I need to demonstrate, exactly, to show that a is true?


I am looking for a book that explains Linear Algebra, where it is build from axioms to higher level of Linear Algebra. It does not have to be a book on elementary level. As example from other fields, Tao's Analysis would be perfect example. Do you know if there is a book which is in similar manner as Tao's Analysis? Thanks


We all know that the universe is governed by four Fundamental Forces which are The strong force , The weak force , The electromagnetic force and The gravitational force . Now, is there any relationship between Electromagnetism and gravity?


I am trying to figure out which bibliography style this is: I am using the natbib package with the apalike bibliography style, which is what I want for the right column, but I am missing the left column (author year). I think this is a nice addition for easier reference.


I have been asked to construct this plot in maple for an analysis assignment. I have been given no other instructions on how to do this. I am not familiar at all with defining or plotting these kinds of functions in maple. Any help would be greatly appreciated as I have been struggling with this for a few days now. Thank you!


In the bible, "he" and "his" are capitalized when they are used to refer to God. Is this grammatically correct? If so, what rule do we have that allows these words to be capitalized?


From what I understand, superposition is when two states exist in all of their possible forms simultaneously until the moment of wave function collapse, when they essentially reduce into a single state. I'm having trouble demonstrating what a superposition state is using bra-ket notation (a simple example), and I don't really understand where the probabilities for collapse come from?


In the following sentence, what is the function of "not, as one might assume, in English"? For example, is it some type of clause that modifies "wrote"? The Irish author Samuel Beckett originally wrote his most famous play, Waiting for Godot, in French, not, as one might assume, in English.


A metal ring is placed in a magnetic field. The ring has a gap in it though. As it falls through the magnetic field, is it true that it still experiences an induced emf and eddy currents but not a proper current?


Recently a program gave me this text in a dialog box: "All purchases have been downloaded for this account." While I understand its meaning, splitting the subject (the noun and its attributive phrase) makes it awkward in my mind. "All purchases for this account have been downloaded" seems clearer to me. Is there a grammar rule to cover this?


One can easily prove that every covariance matrix is positive semi-definite. I come across many claims that the converse is also true; that is, Every symmetric positive semi-definite matrix is a covariance marix of some multivariate distribution. Is it true? If it is, how can we prove it?


Is it correct to end adjacent words in 's to show possession? For example "My neighbour's dog's ball is always in my yard", or should it be "My neighbour dogs ball is always in my yard" or "My neighbour dog's ball is always in my yard"?


I am trying to craft the following sentence in a way that sounds a lot less cliche. Any ideas? "The ability to write has been a gift that has gotten me through the hills and the valleys of my life"


I'm planning to do an internship and I usually address my supervisor as Mr. X, however, he signs his emails with his first name is that means I can address him with his first name ? Note: I haven't met him yet.(if that's makes a difference)


recently I sat for an exam where I was asked this question - Concurrence means all of the following except: A. Agreement B. Accord C. Consensus D. Coincidence E. Harmony What could be the correct answer ? I opted for D. Coincidence. Was I right ?


Does anyone know anything about how the meaning of "just about" came to have opposite meanings in the UK and North America. For example, in the UK, The team just about won. means that the team won, but it was close (ie The team barely won.). However, in North America, it means that the team almost won.


I have found some information on Wikipedia concerned with XeTeX and the unicode-math package. My question is, if I include the unicode-math package in my document, is it enough to have LaTeX installed and run the Unix latex command on my file, or do I need to have XeLaTeX installed as well in order to transform my source file into a .dvi/.ps/.pdf file? Thanks.


I am relatively new to TeX/LaTeX and so far learnt all about it using online documentation or question-and-answer websites like this one. Should I consider buying a LaTeX guide book though? What would be your advices for doing so?


I'm trying to prove the Arrow's Theorem is not true when there are two candidates, however I'm having trouble trying to prove that there is no dictator. I have suggested that in a majority rules voting system, unanimity and IIA are satisfied but I have no idea how to prove that it isn't a dictatorship.


The phrase "Fire Away", meaning "Ask me questions", appears to be a metaphor stemming from an old military term involving discharging firearms (source). However, "Away" is generally a directional term, yet in the phrase "Fire away" it seems to mean "at will" or "with abandon". How did it get such an unusual meaning? Was this a sarcastic instruction perhaps?


I have MathML that I need to render in my LaTeX document. Is there a recommended way of doing so? So far, from googling, I've found MathParser - a Java converter. But I was hoping there might be a better way to do it.


It is well known that the running time of the simplex algorithm depends on the diameter of the polytope induced by the constraints. Is there any non-linear optimization technique that also has this property ?


I used TeXstudio last year and I had a lot of trouble with referencing and producing a bibliography. Reading around online, this is a widespread problem. Which Latex editor is best to use to make it easier to produce a bibliography?


So far, I've come across several examples of frieze groups, but I've not yet come across an understandable definition of what they are. I've also been asked questions that ask me to state the isometries preserving said frieze groups and I don't really understand how I should know what these are either. Could someone please explain this to me? Many thanks.


I am looking for a generic word for a person who is being visited. The opposite of visitor according to thesaurus.com is host. That's not what I'm looking for. A couple examples: an employee being visited by a guest; a student being visited by their parent at school. I can't think of anything except "visitee" and I couldn't find it officially used anywhere.


I'm having trouble finding typical quantities in fiber optic communication. In particular, what kind of powers are generally used (or what is the minimum that fiber optics receivers can detect effectively)? What frequencies of light are generally used, and what are typical sampling rates?


Recently a colleague of mine said to me that the "Guys who are switched on really annoy me?". Since I am not a native speaker, I do not know what he meant by "switched on". Can you fine folks shed some light on this? Thanks!


The quantum fields are operator valued distributions. In some sloppy books like Peskin and Schroeder the Euler-Lagrange equation are used to get the equations of motion. What does it mean to take a derivative with respect to some operator?


Which is correct? I would like to request you to refrain from shouting. I would like to request of you to refrain from shouting. I would like to request from you to refrain from shouting. Something else?


Somewhere on the internet I've read about a guy ordering a coffee in the name of "Bueller". Apparently, the cashier called out that name repeatedly. I've looked up the term but it gave me no clue as to why it's so hilarious. Is that a cultural reference or a linguistic one?!


Almost all of the orbits of planets and other celestial bodies are elliptical, not circular. Is this due to gravitational pull by other nearby massive bodies? If this was the case a two body system should always have a circular orbit. Is that true?


I heard that, at the Quantum level, events can happen out of order making causality invalid. Thus the future can happen in the present and the present in the future. Is this true?


I know the Schwarzschild event horizon is a null surface generated by null geodesics. But what does that actually mean in terms of the path of a light ray that reaches it? Does that mean the geodesic trajectory of light on the surface will be along the surface? Which way will the light ray go? And what equation shows this?


So I'm guessing it has to do with temperature of surface water, temperature of surrounding, surface area, humidity,... I always wondered if there was an equation to describe evaporation, never learned one in high school.


From: Moduli space we see that moduli of curves is a very algebro-geometric topic. It is easy to understand its relevance and importance in algebraic geometry. But the mind boggles when we try to imagine how on earth such a topic from pure and abstruse mathematics is relevant in physics. I will be thankful if somebody can give some explanation.


I'd like to ask about parity of baryon. When I search a parity section of textbook, it only explain about parity of meson, not baryon. And I can't find experimental method for parity determination of baryons. Am I missing something? Or is there special reason for baryon?


Why is the inner product of a character with an irreducible character a non-negative integer? I can see that by properties of the inner product it will be non-negative but I cannot see why it would be an integer?


I have to find the curvature and torsion of a curve (parametrised by arc length), given only the Binormal vector. Whilst I understand how to find these if I have the curve, I cannot for the life of me work out how to go in this direction. Any help would be appreciated


Hi there I was wondering if someone could please help me? I understand the solution of this until it gets to the Polynomial Estimation Lemma part. What is the Polynomial Estimation Lemma and how can it be applied? Many thanks in advance.


Does anyone know what the chromatic number of a graph chosen randomly on n vertices is, as n tends to infinity? I mean, almost all graphs have chromatic number greater than any fixed k. But in terms of estimates, is O(n), O(log n), O(sqrt(n)) a good estimate for the chromatic number of large random graphs?


Possible Duplicate: How to superimpose LaTeX on a picture? Take this image for example: How can I add labels to an image as done above without manually drawing a line and adding text at the end of the line?


I am looking for a nice compact book in abstract algebra (especially group theory) which develops the material by asking questions the reader must answer. An example of what I'm looking for is Combinatorics through guided discovery.


I am trying to understand the following sentence from a legal document. Can anyone explain me what 'commutation' means? Immovable property presently sold is free and clear of all seigniorial dues having commuted and the price of commutation has been paid.


The two-slit experiment is a classic example of how measurements can affect the behavior of particles. This seems reasonable because, to my knowledge, the measurement is "active" in that it adds energy to the system. But how is this reconciled with macroscopic observations (such as looking at Schroedinger's cat) that are "passive" in that they are collecting information already there without affecting the system?


I have heard that the verb go used to be wend in olden days. I am curious if there is any historical or other explanation why the past form of wend, i.e. went, is still in use while the simple present and past participle forms are gone. Any idea or link to some resources which deal with this fact?


Are there specific rules / conventions at play when creating demonyms? Or are they merely formed organically over time - the most popular winning out? There are many suffixes to choose from, but I cannot find concrete guidelines as to which is proper to use in which instance.


Ok, so I got the answer to part i), but however, I'm not so sure how to get the answer to part ii). The answers say its an ellipse and they specified the equation, but I can't understand how they came to that conclusion. Can anyone help me out, please :)


The Oxford Dictionary of Difficult Words defines vindictive as follows: having or showing a strong or unreasoning desire for revenge. What is a word for a having desire to be destructive or to break something?


I want to turn off printing of DOIs in my BibTex bibliography in Lyx. When I try to add the option doi=false under Document settings -> Bibliography -> Bibliography generation -> Options the bibliography is not generated anymore (i.e. question mark for citation and empty bibliography). How can I set this option via Lyx ?


How can we know that North Korea and Iran (to name a few) are exploding nuclear weapons if no inspectors have ever been granted access to suspected nuclear sites in these countries? How can we passively detect a secret detonation of a nuclear warhead? What are the telltale signs of a nuclear detonation?


I know that Edmond's Blossom algorithm can be used to find a minimum weight perfect matching for a given graph. However, is there an efficient algorithm known that can find all the minimum weight matchings. Or one that can sample uniformly from them all?


I've been reading Lee Smolin's Life of the Cosmos. Great book and it makes a lot of sense that the conditions in black holes are the same as conditions at the big bang. Question is, has his theory about Cosmological Natural Selection been disproved as the wikipedia article states?


Loosely defined, sabermetrics is the study/usage/development of baseball statistics. I get the "metrics" part of the word, and I know what a saber is. But, what do sabers have to do with baseball? My only idea is that the baseball bat is kind of like a saber, but that seems to be a stretch.


I just finished reading the Wikipedia article on the Cauchy condensation test. I understand the trapezoidal view, but apparently "the 'condensation' of terms is analogous to a substitution of an exponential function". Can anyone explain what is meant by this?


How would you describe someone who fritters away money on inexpensive, trivial things, like coffee, but refuses to spend money on expensive, necessary things, like a decent new laptop (buying a lousy one instead)? Is there an expression? I want to say "penny foolish and pound wise", except that "pound wise" is inappropriate here, as skimping on the large outlay is a false economy.


I have heard of the "Big Rip" as one theory for the eventual end of the universe. If the speed at which the universe is expanding, and all the matter inside it is moving away from all other matter at an increasing rate, would the matter eventually have to move at the speed of light? Or would it have to slow down?


People often refer to the country US as America and to the people from the US as Americans. As far as I know, that's the only case in the world where a continent's name is used for a country's name (let me know if I'm wrong). Why does that happen?


I'm looking for the proof of that: d(x,y) = d (y,x). I know that I have to use the "non-negativity" and "triangle inequality" but I don't know how to combine them to get the result.


In discussing Doppler effect, we use the word "apparent frequency". Does it mean that the frequency of the sound is still that of the source and it is some physiological phenomenon in the listener's ears that give rise to the Doppler effect?


Could you please let me know if the following excerpt of a sentence is grammatically correct (specifically the preposition after "effort"): "Should there be anything that you feel we are missing, your insight would be invaluable to our effort in improving our products(...)" An explanation on why it is/isn't correct would also be highly appreciated. Prepositions are tough. :(


I am having trouble reducing the footer space. I have a list of contents that is printing below the page number appearing in the footer. How do I solve this issue? I tried setting the footskip length but it didn't work.


Is there an English equivalent to this familiar saying used in India: Don't speak unless you can improve silence. The saying loosely means it is better to be silent than prattle on about something.


I read that the canonical commutation relation between momentum and position can be seen as the Lie Algebra of the Heisenberg group. While I get why the commutation relations of momentum and momentum, momentum and angular momentum and so on arise from the Lorentz group, I don't quite get where the physical symmetry of the Heisenberg group stems from. Any suggestions?


This question may be redundant and I apologize in advance but I am really having a hard time to digest the notion of proportional to in mathematics. Kindly, can someone simplify the idea of it and when we can say proportional to something. Thank you for your help.


I am currently studying Multivariate Calculus (Larson and Edwards book). I want to do a project in computer science to see some nice applications of things I am learning. Any specific source of papers/journals/books? thanks


Since I am reading some stuff about weak convergence of probability measures, I started to wonder what is the dual space of the space consisting of all the finite (signed) measures (which is well known to be a Banach space with the norm being total variation). Is there any characterization of it? We may impose extra assumptions on the underlying space if necessary.


I see many people using "Dear" while addressing people in email. But, I feel "Dear" is more intimate word in English and should be used with only relations. How does the word "Dear" is used at all places irrespective of relationships?


I'd like to customize verbose biblatex citations to exclude URLs (and DOIs) while keeping them in the bibliography. Is there a simple way to accomplish this? While I'm at it, I'd like to get rid of the "In: " that the verbose styles use (in both the bibliography and citations), and put the URLs and DOIs on separate lines in the bibliography.


According to Feynman in this youtube video the photons that are released when a tree burns are sort of those that were trapped during photosynthesis. Are these the exact same photons produced by the Sun or do photons get destroyed / created somehow?


If I drag an object across a surface and a force due to friction acts on the object which is equal and opposite to the force I apply, the net work done on the object is zero. So where does the energy come from to heat up the object and it's surface?


Given the standard topology, is there any relationship between dense, uncountable sets and isolated points? For example, the set of irrationals is both dense and uncountable and contains no isolated points. Just something I've been wondering working my way through real analysis.


I am trying to build something like the two following pictures, using TikZ (I am still learning): The pictures are just an illustration and there is no need of the shading. Anybody can help me?


I'm British. I am editing a document, and I was going to correct a use of "all together" where the author clearly meant "altogether" (as in "entirely"). But then I realised this might just be a British distinction... is it? Do Americans accept "all together" as a valid alternative to "altogether", i.e. "entirely"? (The document is for an international audience.)


Possible Duplicate: A word for two very different things juxtaposed? What is the word to describe when two ideas (often contrasting) are placed next to each other to enhance the situation or idea being presented? I believe it could describe the placement of two words or ideas in a poem, or two melodies in a song, or two objects in a piece of art.


Since baryons (e.g. protons, neutrons) are composite particles it should be possible to split them apart. If so, is it then possible to extract useable energy out of the splitting of baryons in analogy to nuclear fission?


I've been having an argument with a colleague about this sentence, could you please let me know which one of us is correct: There are no shortage of applications for our product in this space. She is convinced that are should be replaced by is, and I think it should stand as it is. Thanks for your help!


A pendulum clock ticks slower on top of a mountain and ticks faster at ground level, but an atomic clock ticks faster on top of a mountain and ticks slower at ground level. Gravity affects pendulum clocks in the exact opposite way to atomic clocks. Which clock is giving us the correct time? Are they both wrong?


It's the wall that a suspect stands in front of when a mug shot is taken. I don't know what to call it so it's hard for me to find information about the wall itself.


Most references that I have come across refer to the Schottky Barrier in the setting of Metal and Semiconductor Interface. Would it be correct to use the term Schottky Barrier to refer to the mismatch in energy levels in other settings (e.g. semiconductor-seminconductor)?


I'm studying for an exam and came across this problem in the book. Any ideas how one would go about solving this problem? Determine the production vector x that will satisfy demand in an economy with the given consumption matrix C and final demand vector d. Round production levels to the nearest whole number.


By Godel's second incompleteness theorem, a consistent theory (to which the theorem applies) cannot prove itself consistent. I learned that it's also impossible to have a pair of consistent theories each proving the consistency of the other. But I can't see how this follows from the second theorem. Or is there something more involved?


If you have a set of points on a hemisphere, how do you find a point on that hemisphere that has the minimum total great circle distance to the points in the set.


I was boiling penne and rigatoni pasta in the same pot (both are hollow cylindrical pasta of approximately the same length, but rigatoni has a noticeably larger diameter) and almost all of the penne pasta managed to find its way inside one of the pieces of rigatoni. Why does this happen? Below is a picture of my dinner showing an example of this.


I don't understand the concept of asymptotic variance. Given the mle of a probability function, the likelihood function and the random variables how do I find asymptotic variance? What exactly is asymptotic variance? I'm finding this concept very confusing.


I had thought that Jew was gender-neutral, until I heard somebody who was asked if their mother was a Jew and responded, "of course not; she's a Jewess". Is Jew a gender-neutral form, or does one need to distinguish between Jew and Jewess?


I noticed that the Kelly Criterion resembles a ratio between the mean and variance in a continuous probability distribution. Now the mean and variance are important values in portfolio optimization (Modern Portfolio Theory). Is there some relationship between the two since both seek to maximize returns and minimize risk?


The wild flowers looked like a soft orange blanket ______________ the desert. A. covering B. covered C. cover D. to cover I chose C. I thought that 'looked' is a past participle and 'cover' would be the predicate. Why is 'looked like' the main verb and not a past participle?


When asking how someone wants their tea, they reply: "lots of milk and sugar". Is that to be interpreted as "Lots of milk and sugar" or "lots of milk and (lots) of sugar" according to the rules of English?


I want to prove this in the infinite dimensional Hilbert space case. What is the easiest way to go about this (What do I need to know, what theorems do I need,etc). My aim is to show every normal operator admits invariant subspaces.


Let R be a Commutative ring with unity, such that R[x] is UFD. If R[x] is a PID then it is a Eucledian Domain? Is the last statement about being eucledian domain correct?


There is no other harsher critic than yourself. I'm really stumped on this one. The more I read it the less correct it sounds. I think the word harsher is making the sentence sound fairly off putting. Other variations include: There is no harsher critic other than yourself. Any help would be greatly appreciated!


I'm looking for a word that could describe a person who is very gullible and easily trusts people. My specific example is Fortunato from Edgar Allen Poe's story: The Cask of Amontillado Edit: the character's name is Fortunato.


I'm making several assumptions, not sure if any are correct: there is a black hole at the center of a galaxy the black hole is eating the galaxy Eventually the galaxy will be gone, right? Has this been observed? Do we know what happens afterwards? Posting here since astronomy got merged into physics


https://en.wikipedia.org/wiki/File:Hopf-bif.gif Does anyone know how this animation was produced? I could make it by stitching together snapshots (what I'm doing) but this seems primitive, especially for how nice this animation looks. Any ideas?


I haven't been able to find a proof of this result online. Is there any proof that is relatively simple to understand? In particular, I want to also show that the upper triangle matrix that is similar to the complex one has diagonal values that are the eigenvalues of the complex one.


I was looking to apply a color gradient to some text (i.e. Chapter / section headings) and I searched through the Internet but was unable to find a minimum working example. Is it something that can be done in XeLaTeX? I decided to ask here, thinking someone in the community might have already done similar to that.


Suppose I have a sphere. Inside the sphere I have an inscribed cube. What I am interested in is finding out what is the latitude and longitude (or coordinates) of a point on the sphere which will be projected on a cube's face given the coodinate of a point on one of the cube's faces. Does anyone have any equations for this?


I understand that rectifying antenna (rectenna) is supposed to convert electromagnetic energy to electric current however I do not understand how it's really working. I do get that it's kind of like how transformer works, but beyond that I am clueless. Could someone explain this to me?


What does "characteristic wave" mean in the context of plasma waves? For example, when propagating parallel to the plasma magnetic field, the characteristic wave is circularly polarized. Does this mean that only a circularly polarized wave can propagate in this direction? Could an elliptically polarized wave travel in this direction instead?


Is there a word for days you don't work? In portuguese you have the term "dias de folga" that means the days that you would normally work but will not because the company gave you those days to rest. One is said in portugues to be in "folga". Is there a correspondent word or term in english?


Is there an overview available for Arabic fonts in LaTeX / XeTeX? Ideally with output examples, so that it becomes clear how they compare. So far, I'm aware of Geeza Pro, Amiri (and its variants), Sheherazade and, Al Nile. But I assume there is much more out there.


Well, since the electron is going around the nucleus, it must have an angular momentum (plus its spin). And since it is charged it must create an electric field. If I have understood it correctly it is the interaction with this field that is causing the so-called Anomalous Zeeman effect? Is this correct? And does does this have any other consequences?


According to Wikipedia, an open formula is a WFF without quantifiers. I have read that a propositional function is the same as open formula. Are both of these statements correct? Is it true that one can't have a propositional function with quantifiers?


Of course we can always say "the sun is up", but is it acceptable to say "the sun is risen" just as we use verb "to be" before other past participles like "she is gone"? Or should we say "the sun has risen"? Does it make any difference?


Why the current is the same at all of the positions in a series circuit? although there are different voltages at different positions of the circuit. What i know is that as the electron passes one resistor, the force of attraction from the negative terminal is less. Doesn't this affect the current?


What's the difference in spirit between "gallimaufry" and "farrago"? Are there any sentences where using one or the other would lead to a subtly different meaning? Both are defined in dictionaries as "a confused mixture", or a synonym of "hotchpotch". I'm afraid they're too close to just be able to compare their dictionary definitions.


I am interested in applications problems of Linear Algebra. If somebody can recommend me an interesting book, or papers about, I will appreciate this. Thanks. (I need this for a course of Linear Algebra for Engineering where I am the lecturer).


Does anyone know the font style used in Kato's Perturbation Theory for Linear Operators? Better yet, does anyone know of a TeX package or style file that replicates the layout and style used throughout Kato's book?


One of Twitter's error messages reads: Something is technically wrong. Of course I understand what they mean, but am I wrong to interpret this as "strictly speaking, there is something wrong" instead of the intended "there is a technical issue"? What would be the ideal, succinct way to phrase this, maintaining simplicity and clarity?


This is a very simple question but I wasn't able to find a question like it and I wasn't a math major. Did tangent originate in trigonometry, geometry, or calculus? I remember using tangent lines in geometry, trig, and calculus but I'm curious which came first. http://www.mathopenref.com/tangentline.html


G is a planar graph. E is an arbitrary edge of G. "There exists a planar drawing of G where E is on the unbounded face of the drawing." Why is the above statement true? Any help is appreciated.


I am reading Wilkie Collins' The Moonstone, and a prominent character in the story has the name of Betteredge. My question is (since I like to imagine the dialogue in a British-English book as if spoken with a British accent), would "Betteredge" sound more like "Better edge", more like "Bedderitch", or is it pronounced completely differently?


Does "so that" and "for ... to" have the same usage? For example: I bought this sweater so (that) you can wear it. and: I bought this sweater for you to wear it. Is there any difference between the two expressions or do they have the same meaning?


Can anyone give a specific example of a diffeomorphism and also of composing a function with a diffeomorphism and how this helps mathematics as a whole? In other words, how does this fit into the grand scheme of mathematics? Thanks


In the context of "Which station should I get off?" (asked when you're on the train), what would be another way to say this, without using "get off" (which has other connotations) and still colloquial (either American or British English)? I thought about "exit" or "alight". Are there other, better ways to say this? Thank you.


I'm familiar with the concepts of group isomorphism, ring isomorphism, and graph isomorphism, but it's never been presented to me what an isomorphism is in general: given any X, what is an X isomorphism? Informally, I understand isomorphism as "preservation of structure", where "preservation" is domain specific. Is there a formal definition?


I'm using the align package for multiple-row equations. Some rows the equations are just a bit longer than the textwidth (I'm using double column). I only want to reduce the fontsize of those particular rows, not the whole align environment. How do I do that?


I don't need to list all the proper divisors; I just want to compute their sum. While checking and summing up all proper divisors isn't an issue for small numbers, it becomes significantly slower for larger numbers. Any suggestions? Thanks!


While trying to locate the etymology of the Tamil slang, OB, I ran across the following claim: This entirely Tamilian term, strangely enough, has its origins in the term "off-beat" used in the British Army to refer to retired officers or soldiers. I'd like to know if there's any truth to the claim that "off-beat" was a British Army term for retired officers.


In graph theory what is the difference between isomerism and isomorphism? I found a post somewhat similar to it but couldn't understand my problem from that. So I asked again specifically asking my question.


The following is an excerpt from Chern's Lectures on Differential Geometry: I don't see how the proof shows the other direction of the set inclusion. Would anybody explain the logic in the "furthermore" part of the proof? The following are definitions of the notations:


I use it all the time since I work with databases, but every time I write it somewhere with spell check I get the squiggly line below it. I've seen other people spell it with an "s" instead of a "z" but neither have an entry in the Merriam Webster dictionary. http://www.merriam-webster.com/dictionary/denormalized http://www.merriam-webster.com/dictionary/denormalised Is this just technical jargon or am I misspelling it?


I think the both answer is "yes". How can I prove that Schwartz space is closed under multiplication ? Because if I know that, it is easy to see that being closed under convolution is satisfied. Help me please. Thanks


I am creating a poster using beamerposter, and I would like to include Feynman diagrams generated using feynmp. However, the native scaling of feynmp is indented for letter-size documents, and is far too small for a poster. Is there a way to scale the diagram to poster-size while keeping its features properly proportioned?


I am starting a chapter on divisibility in commutative rings, and I was wondering if there was a way to translate theorems about gcd to lcm and vice versa. I know the concepts are considered "dual" in some sense, so perhaps the theorems relating to them are also dual.


I am unable to see the correctness of this statement. It seems the author has considered this statement trivial and hence has not given any proof of this statement. But I am unable to prove it.


I've seen a lot of work on homomorphisms of modules, but is it possible to construct a homomorphism as a module itself? For example, how would you define a specific module structure on a set of homomorphisms, for example, a dual space.


Which of the three articles (a/an/the) would the blanks in the following sentence take? It is not unusual for ____ editor to tamper with _____ writer's manuscript. I think it should be 'an' and 'the' in the first and second blank respectively, but my colleagues think otherwise. Please suggest.


I would like to know whether it is correct or not to use the word "scope" to refer to a book's section, subsection or paragraph. For instance We will address these matters in the next scope.


A very common type of convergence in probability theory is 'almost sure convergence'. I don't understand why this type is used at all. In principle, we should always be able to substitute it by a uniform convergence almost sure due to Egorov's theorem. So why do people in probability theory refer to this weaker concept? If anything is unclear, please let me know.


I'm kind of confused as to how I should punctuate the following sentence: A land as old as time and whose history defies any simple description. The author added a comma before "and", but i'm not convinced. Can "whose history defies any simple description." stand on its own as a sentence?


I've seen the following (e.g. here): I've learned a bit about groups and I could give examples of groups, but when reading the given table, I couldn't imagine of what a magma would be. It has no associativity, no identity, no divisibility and no commutativity. I can't imagine what such a thing would be. Can you give a concrete example of a magma?


I saw a passage "this doesn't mean to get riches and honors." 'rich' is an adjective but 'riches' is a plural noun according to the dictionary. Are there any other examples where an adjective becomes a noun by adding suffix '-s' or '-es'? or 'riches' is the only case?


I wrote a text about our company's services and added that we also can offer products tailored to the needs of our clients. One of my colleagues insisted that the expression was wrong and I should say that we offered tailor-made products. Is there actually a difference between the two expressions generally or in the context I described?


I have found many places that list the various rules on using hyphens in math, but nothing to explain why we have the rule. I have some students who are asking and I would like to be able to give them an answer instead of saying it is just the rule.


I'm teaching a class with a very gentle introduction to Markov chains (the class is an alternative to taking calculus). I'm not a probabilist, so I haven't spent any time thinking about these objects. Does anyone know of any very compelling non-mathematical examples of Markov chains, or, at least, of non-mathematical things that can be well approximated by Markov chains?


Adherent ("a person who follows or upholds a leader, cause, etc.; supporter; follower") is the word I am considering. For example, would a group of people who support and fight for freedom be called adherents (ignoring the obvious "freedom fighter")?


Assume a scenario in which you talk with someone and then he puts your integrity/reliability "in doubt". Meaning: He doubts that you are truly reliable/innocent/trustworthy. So you want to say to him a phrase that means "Do you really put my integrity in doubt?" or something similar. I'm quite certain that this kind of phrase exists, but I don't remember what it is.


For example if a student wishes for a storm because he wants his classes to be suspended. His main intention is to have no school but not necessarily to have a storm. Not sure if this makes any sense. I'm pretty sure there's no word for it but I don't know because English is not my first language.


A student in my class mentioned his approach to proving isomorphism via the rank-nullity theorem rather than showing an inverse's existence. I didn't quite understand how, but perhaps someone here might be able to explain...


When creating an unconstrained optimization problem from an equality constrained one, the usual way to build the Lagrangian, is by adding a term consisting of a multiplier, multiplied by the equality constraint. Are there problem instances, where it makes better sense to square the equality constraint and then use that in the unconstrained problem?


I use LaTeX and want to create a bookmark which opens a file but I want to specify the page no. I have looked at texdoc bookmark but cannot get it to work. Anyone any ideas?


It seems, to me at least, that most Statistics textbooks focus on the Statistical methods and techniques, or on the mathematics behind them. Would you recommend me some textbooks (or any online source) that discuss the "logical" reasoning behind the techniques? Thanks in advance!


I have been introduced to algorithms, computability and computational complexity (as part of my minor in CS). What are some mathematical topics that I can tackle with the new perspectives I acquired from these ideas of computer science? Could you suggest some reference books (preferably self-contained and accessible to an undergraduate)?


Fourier says that any periodic function can be represented like a infinite sum of sine functions with their appropriate periods,amplitudes and phases. My question is: is it possible to represent the periodic function like a sum of other periodic functions like a square wave function or some other shape, or they must be the sine functions?


What experiments prove that heat can be "trapped" by a layer of material with suitable optical properties, such as carbon dioxide? I've read a little on Wikipedia but I've got only some names, not the experiments.


What is the difference between ago and before when they are both used as adverbs in the following sentences: I saw him seven days ago. and I had seen him seven days before.


Could somebody please provide a sketch of a proof of the fact that the Tor functor commutes with direct limits? I have been trying to show that the Tor of a module with the direct limit of a family of modules satisfies the required universal property, but it seems too complex.


Sometimes when I speak with my fellows at work, I start my sentences with "so" or "um". I don't know if this a bad behavior in business speak or not? If so, how can I get rid of those filler words?


From what I can glean, it'll and I've exist as standard contractions, but I am unsure of whether it'll've either exists or is acceptable. "It will have" should be able to be reduced to "it'll've", shouldn't it?


We say that if any particles from outside fall into the black hole, they eventually hit singularity. Then why not particles already inside the black hole are at singularity? Or are they? If yes, then why does a black hole have a finite size?


What is the meaning of the phrase "new strands of research" in the following sentence: The last two decades have also seen the emergence of new strands of research on multilingualism which have incorporated critical and poststructuralist perspective from social theory and embraced new epistemologies and research methods. Is there any synonym for this phrase or for the word strand in this context?


Given any figure with four vertices and four straight edges, prove that one can construct a perfect parallelogram by connecting the midpoints of such figure. This to me is a very fundamental and interesting geometry problem. How would I begin to prove this?


What actually causes a rocket to move? Is it the pressure in the rocket engine or the amount and velocity of mass that is being ejected out. The reason I am asking is, I found these two explanations for a rocket motion. Are they same or different. If same, is there any correlation between them?


I have a long definition for Picard groups, but it is too abstract for me to really understand and apply. Could someone give an example or two on how to calculate the Picard group for a quadratic integer ring? Also the relevance of the exact sequence?


I am comfortable with the way physicists use differentials as elements of area/volume. I know the (algebraic) formal definition of differential forms, but it makes no intuitive sense, especially since it is not immediately compatible (to me) with the physicist POV. How do the two fit in?


I know what anti-matter is and how when it collides with matter both are annihilated. However, what about anti-photons? Are there such things as anti-photons? I initially thought the idea preposterous. However I am curious because, if anti-photons don't exist, then anti-matter could theoretically transfer its energy to normal matter - through the mechanism of light. Is it right?


I am taking a course in Mathematical Physics Junior Level (Undergraduate). We are working from Arfken's "Mathematical Methods for physicists", but i am finding trouble with it specially in the determinants and matrices chapter, since it is not detailed enough. any suggestions for a more detailed book?


Which one should I use: Are you going somewhere nice on holiday this year? or Are you going anywhere nice on holiday this year? The presence of the word "nice" bothers me a bit and deters me from using "anywhere".


I'm trying to calculate an integral with respect to a complex value. I just want to know if I can estimate the integral using the residue theorem separately for the real and imaginary parts of the mentioned value or I cannot at all use this method here.


By Cantor's intersection theorem I know that a sequence of nonempty compact sets which are nested has nonempty intersection. But how can I use that to prove that arbitrary intersection of compact sets is compact?


Let's say I am having a telephone conference with rest of the team, and somebody asks me "Dude, are you there?" How should I reply? Yes, dude, I am there. or Yes, I am here? It is difficult for me to see the difference.


To what branch of mathematics does the Hodge conjecture belong? I'm aware that it's very advanced, but what kind of prerequisites would one need to understand those problems? Can you suggest some good texts for a senior undergraduate/beginning graduate?


I'd like to write a program that uses the exact (down to the second) amount of time from one new moon (or full moon) to the next. Yet, I am told that this period is irregular. Yet, it seems to be predicted in a number of places (ie, by the navy). How do they do this? http://aa.usno.navy.mil/data/docs/MoonPhase.php


For example a situation where you are on the phone with a friend and he is with a person that you know. You would ask him to say hi for you. The french equivalent would be "Passer le bonjour" I am looking for formal and informal ways to say that.


I'd like to label my nodes with text that has a 'halo' around it of a different colour. Is this possible? The reason is that my label text covers different parts of my diagram with different background colours, so no single colour will make the label legible. I'd ideally like the text to be black, with a white halo.


Each (finite-dimensional) Lie algebra has exactly one simply connected Lie group associated to it (up to isomorphism). What can we say about all other connected groups with the same Lie algebra ? Thank you in advance


When a car is traveling round a banked track as fast as possible, it has a tendency to slip up the slope. Opposite in the case when the car travels slowly and has a tendency to slip down. Can someone please give me an intuitive reason as to why this "tendency to slip up or down" occurs.


Possible Duplicate: What word means what many people think 'ironic' means? While irony actually means 'the opposite of what is expected', it is often misused in a context to make it mean 'an unexpected and amusing coincidence or similarity'. Is there any word that does actually mean this?


Given a parallelogram with congruent diagonals, you are asked to prove that the parallelogram is a rectangle. Would saying: A parallelogram must be a rectangle if the diagonals are congruent. be a valid proof for this question?


I'm developing a web based application that allows people to send email to many customers, for advertisement purposes. What's the correct name for that? newsletter or mailshot? EDIT: to avoid unnecessary answers like "spam" please note that customers subscribe to the service first.


We know that if the infinite sum of a multiplicative function is absolute convergent, then the sum can be expressed as infinite product and the infinite product is absolutely convergent. Does there exists a multiplicative function such that the infinite sum is divergent, yet its Euler product is still absolute convergent? Thanks a lot


I know that a smooth manifold is a topological manifold whose transition maps are smooth. Must the coordinate maps also be smooth? Must they be diffeomorphisms? MathWorld seems to think so, but I do not understand why it follows from the definition that this must be the case.


Since strings are extended objects, is all angular momentum in string theory essentially "orbital" angular momentum? Or is there still a kind of intrinsic angular momentum assigned to a string? Either way, is there anything that prevents the "intrinsic spin" of a particle represented by a string from being arbitrarily large?


Say I had a parameterization of a surface and I wanted to determine if the surface was closed, to determine the applicability of divergence theorem. My math professor said a surface is closed if it does not have a "boundary", such as the sphere or the torus. How would I determine this mathematically? Is there a specific property that is satisfied?


I've taken up self-study of math. (How smart can that be?) I've just about finished a course in real analysis which spent a lot of time on metric spaces and some time revisiting calculus. I was thinking of trying abstract algebra. I would appreciate any book recommendations. Thanks in advance. Andrew


I am using chapters. I want to include one section (a page or two) that does not have a chapter number, and that does not increment the chapter counter. I've tried includeing it, without using chapter, and the section didn't show up at all. How can I include a chapter-less section in a chaptered document?


A very quick question: Is it correct to say "In case I won't be able to watch that"? If it's acceptable to say that, what type of sentence is it? UPDATE: As a clarification, I want to say "Can you give me that in case I won't be able to have it later". Probably, it's more a supposition than a conditional.


I'm trying to better understand the meaning of the phrase in the Skyy Vodka ad ("Vodka so filtered we even took the Russia out of it"). AFAIK the name Russia is never used with a definitive article, but I also doubt that Skyy promoters are uneducated. Can someone please explain why "the Russia"?


What I'm looking for isn't quite the same as a Pyrrhic victory, as the action isn't necessarily done to obtain victory. Rather, it is any action that will ultimately result in the person taking said action to pay great (preferably terrible) cost either along the way or afterward, whether or not they realize it in advance.


I am going to be participating in the high school division of the Purple Comet Math Meet. Since computers are allowed to be used on the competition, I was wondering if anyone can recommend software/programs that I can use to solve these types of questions. Thanks!


Which of these is correct: I didn't realize how much clothes you had on the floor I didn't realize how much clothing you had on the floor Any additional explanation about clothes versus clothing would be helpful. This other question discusses the meaning of the two words, but doesn't help with the grammar problem above:What is the difference between "clothes" and "clothing"?


In the following two sentences "did" and "were" seem to be conveying the same meaning. If that's the case, what is the difference between the two and which one is more preferred? If you studied, you would pass the exam. If you were to study, you would pass the exam.


Where does the word "bravo" come from? In Spanish, it can be related to bravery but I fail to see the connection with the congratulatory mode it's used today in spectacles and related activities.


I believe there is also a computing or physics theory that contains this word. (i.e. "Something Theory") It's definitely at least two syllables long and I'm pretty sure there's at least one 'b' somewhere in the word. Thank you!


In some physics theories like string theory we have notion of spatially extended quantum objects - strings, membranes etc. Assuming that such objects exist, how would they appear in experiments ? More precisely are there any thought experiments for determining whether a quantum thing being observed has spatial extension or not, and if it has spatial extension then what is its dimension, topology etc.


I have seem one proof here. However it involves concepts like exterior algebra. For positive definite matrix there is a proof for Sylvester's criterion by induction. I am wondering if there is any simpler proof for positive semi-definite matrix. Thanks!


I'm really tired of going through tons of wikipedia pages. Wikipedia is a good thing, but at one point one who wants to go deeper needs an ordered, comprehensive and formal treatment. Me at least. Therefore, I would like some recommendations for detailed and comprehensive books on plane geometry.


I'm looking for a word that would describe something like a modern phone that retains a rotary dial. The dial's design was originally functional, but phone's function doesn't rely on the dial any more - it's just an aesthetic choice.


I got these graphs from Wolfram Alpha. Can anyone please explain why the result differs by the order of operations for square roots? I thought the rule only applied to division, multiplication and subraction algorithms. Thanks.


What's a non-judgmental way to say a person or group of people are "red necks" in the sense they may hold some very conservative or old fashioned beliefs. For example, someone who doesn't believe autism is a real medical condition? I realize any phrase for it may cause offense, but I'm aiming for the least possible.


I want to separate my book into different volumes, I searched this site and there is no satisfactory solution. I tried to work around with a package called volumes http://mirrors.ctan.org/macros/latex/contrib/volumes/volumes.pdf However, after reading the manual, I still have no idea about how to use it, so can anyone just give me a concrete and compilable example.


How do you describe in a phrase the movement of something that you may see through the corner of your eye ? Like a slight movement that the corner of your eye may catch ?


My math professor has said that it would be useful to start with complex analysis before learning fourier series in the signals and systems course(I'm an undergrad EE). Do you agree with that and why ?! (I'm planning to learn complex variables from Brown/Churchil)


If there are two spheres (hollow and solid) with equal mass and radius and we want to find the hollow sphere without using any equipment. What's the best way(s) to recognize the hollow sphere and solid sphere?


Possible Duplicate: Impurity scattering temperature dependence Is there any temperature dependence of relaxation time in impurity scattering of conducting electrons? It seems to me that there is none. But, some people claim that there is. So if you could explain, how temperature dependence comes into play?


When working in a professional setting, especially in administration or technical fields, common words/phrases/acronyms get used quite frequently. I've noticed that the more they are used daily, the faster they are physically spoken and the more relaxed the syllables become in the word, often making it sound completely different than when first used/spoken. Is there a term for this phenomena?


Gravitational waves are continuously stretching and squeezing the shape of Earth. Using this fact, LIGO is detecting it by interference of light. But the Sun and Moon's tidal effects are also trying to stretch the Earth. Does this affect the detection of gravitational waves? If so, then how do they eliminate these effects?


We will have a lot of skills, but certainly, we also have limitations. I don't know whether it's right or wrong to place "but" and "certainly" together. Since "but" is conjunction, while "certainly" is adverb here. Should i change it into: We will have a lot of skills, but, certainly, we also have limitations. Can you give me some help?


What would the Heisenberg uncertainty principle and De Broglie wavelength be for a baseball that is not moving (i.e has zero velocity)? Also, since macroscopic objects like baseballs have extremely small wavelengths, would this mean a high frequency? If so, would this frequency not be dangerous as it would greater than gamma and X-rays?


The Airy disk intensity formula is given in the mathematical details section of the Wikipedia Airy disk article in terms of the Bessel function of the first kind. I am interested in the asymptotic behavior of the tail of this function. Is it exponential or powerlaw? If powerlaw, what power? Can you give a Taylor series expansion? TIA.


I know how this formula works and it is quite interesting actually but how would you prove this relationship? Through induction (seems difficult since there's no equation for prime numbers), but I'm not sure if there is any other way to show this equation. Any ideas?


I am wondering if I can use the word "compensate" to mean "adjust to correct" something in a document for a programming/mechanic related subject? E.g. Can "position compensation of target objects" mean "position correction of target objects"? Thank you in advance.


For every graph G, prove that (vertex cover of G) is less than or equal to (twice it's matching). I tried a couple of examples and it works but I can't follow a trend to build my proof. Does anybody know how to prove it?


I noticed that when I throw salt into a cooking pot and then mix, the salt collects in the center. As salt is denser than water, I would have expected it to go towards the border of the pot, and not in the middle. What is going on exactly there?


Today I realized that the word "Real" in "Real Estate" might be about "royalty" instead of "reality". English is a foreign language to me, so I don't really know the literal meaning and origin of the term "Real Estate". Might "Real" be an alternative term for "Royal"?


Consider the example: I am happy to wait. In some publications, the function of the infinitive is called 'adverb'. In others, it is 'complement of adjective'. Is there a difference in the naming of the function? I've always learnt that adverbs do modify adjectives, and I'm wondering why the name of this particular function is not the same everywhere.


I don't know how this phenomenon is called. Simply put, I can either put the noun to the end and place it's properties before, or I can use "of". What is the difference, is some form more formal and suitable for academic writing?


A is a closed convex set with non-empty interior. Does A must equal to the convex hull of its boundary? I know this is false when A is half space. But what about other sets?


All the games of the Fallout franchise start their intro with the phrase War. War never changes... I was wondering if this was an original phrase or was it from literature or some speech?


I'm reviewing for a math test this Tuesday and just want to make sure I'm doing things right. If someone could check my work that would be great. Here's the question (work below): Here's my work: Thank you in advance!


I want to know if this function is unif.conv. or not... Actually it is one of the my mid-exem problems. I got zero score for it.... What did I wrong in this problem? and What is the easiest way to see whether it is unif.conv. or not?


Why is it common in English to address animals as "it"? It's not an inanimate or abstract object, they have a gender and they're alive. A chair, idea, tree, rock, etc are an "it", but why does English speakers address a dog, cat, bird, fish, etc as "it"? Isn't male the "default gender" in English?


I have no idea what a good title for this question should be. I'm asking a database guy about a problem with our data. I'm considering phrasing it this way: Is there any other field or fields that have this property? Is "have" correct?


So I've stumbled upon this question in Grimmett and Stirzaker's text. I have their solutions manual, which starts off like this: The line above, where the statement is expanded into sums, is where I'm stuck. Would be glad if someone could elucidate it.


Possible Duplicate: Can a word be contracted twice (e.g. "I'ven't")? What is the correct way to contract "should not have", if there is one? "Should have" becomes "should've", "should not" becomes "shouldn't". Is "shouldn't've correct? It seems very strange to have two apostrophes in one word, but it also seems incorrect to skip an apostrophe where letters are omitted.


Suppose we created a vacuum and spinned a turbine inside it with some amount of force. According to newton's second law it will keep spinning as there is no air resistance, so why can we not make electricity out of it?


I'm not a native speaker so it might just be me finding this strange, but why is the auto in grand theft auto at the end? Shouldn't it be grand auto theft or something like this? I thought the expression described the crime of stealing cars?


This question about alien species and planets brought up something I've been thinking about on and off for years. We capitalize names of alien races like Vulcan, Timelord, Cylon (well, maybe not alien), Krell, Nox, Minbari, and so on, but we never capitalize human. So how did we end up capitalizing names for sentient species when we don't do that for our own race?


I am new to Latex. I have to write thesis for my MS. For that I search latex templates and find one template at http://web.mit.edu/thesis/tex/ I edit the template accordingly but not able to change the institute name. Can anyone help me changing the name. Thanks in advance.


Telescopes have angular diffraction limit depending on the observed wavelength and aperture diameter. I've read that it's possible to go beyond the limit for microscopes. But is it possible to do the same for telescopes too?


I have a memoir document, with custom settings for margins, headers and footers, etc. I just found out it is being printed single-sided instead of double-sided, and I was wondering if there is a simple command to treat all pages as odd pages. How can I tell memoir to format the even pages exactly as it would the odd pages?


Which mathematical results has Lewis Carroll, the author of Alice's Adventures in Wonderland, produced? Wikipedia is very vague with regard to this topic and gives us little more than a matrix identity connected with his name. I would be especially interested in his most important theorems on mathematical logic.


Keep on: "to continue doing something, or to do something many times." Go on: "a) to continue doing something or being in a situation. b) to continue without stopping". From Longman Dictionary of Contemporary English


I'm looking for the most colloquial synonym for the term 'first class', in the meaning of 'first gathering of a course'. My reason for looking for an alternative is that 'first class' is too easily read as 'first-class', as in 'top-notch'. Would 'opening class' be correct and normal-sounding? Thanks in advance.


Possible Duplicate: Preposition usage: on, in, and at I guess the difference is that when you say "I work on a project", it shows that your are the owner of the project, whereas "in a project" shows that you are the member of a project. Is my guess correct?


In the movie As Good As It Gets, there is this dialogue: Simon Bishop: Thank you, Melvin. You... overwhelm me. [pauses] Simon Bishop: I love you. Melvin Udall: I tell you, buddy... I'd be the luckiest guy alive if that did it for me. What does I'd be the luckiest guy alive if that did it for me mean ?


I am very familiar to the congruences in modular arithemtic, But sometimes I can see questions related to supercongruences but I couldn't find any information about it on google. Can someone explain what it is? and what is the difference with a classic congruence?


In geometry, the set of points within a circle is called a disk (open disk if it excludes the boundary, closed disk if it includes it). Is there a similar notion for squares or rectangles? "A filled rectangle that includes the boundary is called a closed _______." "A filled square that excludes the boundary is called an open ________."


The Hertzsprung-Russell diagram that categorizes star types uses the letter codes O, B, A, F, G, K, and M to indicate a star's temperature/color. Hottest (blue) is O and coolest (red) is M. What do the letters themselves relate to? For example, why is "O" used for hottest/blue?


Does anyone know of any sources that talk about (or know what topics I can start looking at that relate to) the mathematics behind poetry, e.g., ratios of syllables, equations for meter, frequency of different sounds made in the English language etc.?


There are several ways/methods to perform factoring. I am revising factoring at KhanAcademy, there are factoring by grouping, factoring special product and factoring difference of squares. Although, I can work on the exercise, but I do not really understand when to apply factoring by grouping.


I am very confused with a question I have found in relation to Excel. I am hoping someone can help me do this or at-least give me direction in which I can figure out how to do this. So far I don't even know where to get started.


we know that in a group G, if H,K be subgroups such that H is normal, then the product HK is also a subgroup. does the converse hold? i.e. if H is a subgroup of a group G such that for any subgroup K of G, the product HK is a subgroup, can we conclude that H must be a normal subgroup?