I think the way that I've come to think about mathematics is becoming problematic and I'm wondering if I should abandon it. When I study mathematics, I find myself trying to compare the mathematical constructs, operations, entities, and even the basic terminology (which I have come to understand is incredibly elegant, precise, and deliberate) to real world, physical, even visible phenomena. I think under the pretense that the things I do in the mathematical world represent real, fundamental structures in this Universe. For example, the fact that terms can 'cancel' out in an equation has profound implications on the workings of the Universe and should be heeded and studied as such. In other words, I try to make sense of the things I learn in math classes by finding their analogs in the real word, because I assume they must have at least one. Thinking with this frame of mind has led me to appreciate mathematics in a deeply profound and beautiful way, and it's the mindset that I try to share with other people when explaining why mathematics should be studied and why people describe it as beautiful. When I learn something new in a math class, I try to understand and remember that these are not simply tedious equations and formulas that mean nothing and come from nowhere, but that they have real physical and, mostly, intuitive meaning. All that being said, I'm taking my first liner algebra course this term, and it's becoming harder to utilize this mentality, not simply because linear algebra deals with such things as infinite dimensionality which we obviously have no intuitive way of grasping or visualizing, but really just because the class seems more about computation and calculation than concept and philosophy. I worry that my thinking has led me astray, primarily because it becomes hard to focus on just doing sheer, brute force calculation without wondering and worrying about what these constructs really mean. This leads me to fall behind in lecture, take hours longer than is probably necessary on the homework, and add to an overall level of frustration that has been building for some time now because of it, which only clouds my understanding even more. My question is really more of a plea for advice. Should I abandon my way of thinking about mathematics as though it will become increasingly unhelpful in future courses and topics, or is linear algebra truly more about numerical gymnastics than tangible interpretation? Should I focus, currently, on simply learning the algorithms for computation now assuming that the philosophical groundwork will be exposed later on, after which the conceptual work that I'm looking for will yield itself? I'd really appreciate responses from the people that frequent this site. I've been nothing but overwhelmed at the level of quality, thought, and sincerity in the answers I've read here and throughout the conversations I've eavesdropped so far. Also, please direct me to similar questions if you know of any, and help me with the tagging of this question, as it is the first one I've ever asked on this site.


There's a poem in bahasa indonesia, titled "Aku Ingin (I want)" by Sapardi Djoko Damono, translated to english by John H. McGlynn. This is the english version: I want I want to love you simply, in words not spoken: tinder to the flame which transforms it to ash I want to love you simply, in signs not expressed: clouds to the rain which make them evanesce source What I want to ask lies on the last line, especially the word "make". This was originally being questioned in a blog post (in bahasa indonesia). The OP caught that make usage, and his post caught my attention, thus I asked it here to get more understanding of it. Seems like for Indonesian who reads the poem in bahasa indonesia, the last line interpreted as "rain that makes the clouds vanish". Just like the first verse interpreted as "fire that burns tinder to ashes". Thus, the word make on the last line should have be used by rain, that would make it makes. But in the poem's english version, make stayed make, make it looks like it's used by the clouds. Meanwhile on the first verse, both tinder and flame are singular, thus hard to know to which transforms is used. My question is, why is it make, without s? Is it a different interpretation, or is there some kind of poetic usage on that word, or is it a common usage for english poem (by english poem, I mean poem using english language)? Sorry for broken english, and in case I asked in a wrong place, please tell me where should I had asked.


I have never understood what measuring process (if any) is supposed to be continuously polling the quantum state of an unstable bound system subjected to decay via quantum tunnelling. The reason I reckon some kind of polling process should exist in the first place is the following: According to the QM postulates, the unitary evolution of such a system should by definition keep it reversible, so it is only when measured that a decay can be observed or not. But this would make the decay rate dependent on the measurement rate, while we well know that the decay probability is constant, and the decay deemed "spontaneaous". What am I missing? Edit: From the discussion in the comment section I gather I have not been clear about what I am asking here exactly. Let me try to reformulate the question. It is about how quantum tunnelling is supposed to explain the exponential dynamics of a decay process. I am not asking about the Zeno effect; quite the opposite, actually: why, in the absence of any measurement, do we have an exponential decay at all? I just do not understand at what point in the unitary evolution of the unstable system the tunnelling effect is spontaneously happening. The polling process I imagined is just a way to ask "why does the tunnelling effect manifest iself ?" because I cannot see how it can manifest without a measurement. Please do not infer that I am making up my pet theory here. I am only looking for a way to picture the situation, which at this time I don't get at all.


We have statements, and we have questions. A request made in the form of a statement has a question equivalent. But is that question equivalent implicit, or is it simply a rewording of the statement form? Take the following: "Please tell me why I sound like a sissy." This has the question form of: "Why do I sound like a sissy?" Is that question form implied within the statement form? I would argue it is not. http://www.merriam-webster.com/ defines imply as: "to express (something) in an indirect way : to suggest (something) without saying or showing it plainly" This definition can be interpreted to mean: Something implied must be identified with knowledge outside of what is only directly stated but based on what is stated - one must add context to what is stated. That may be stretching it a bit, but if one does not use any logic or outside knowledge to examine the meaning of provided information, other than that needed to identify the direct meaning directly present, one cannot identify anything that is implied. Here's an example: in the statement form of the quote I provided, one could say that it is implied that the individual requesting the information wants to know said information. It is implied because it is not directly stated in any of the words that the individual requesting actually wants to know; it is suggested, but not directly stated. For all we know, the individual is requesting for someone else. The reason I say that the question form is not implied is because we need no more information (other than the "why do", but that is simply part of forming a question) than what is presented in the statement. We have all the information needed to form a question out of what is provided, but we do not have enough information to state with absolute certainty that the individual requesting the information is the one who actually wants to know said information. Based on this, I would say that the question form of a statement is not implicit. Am I correct?


Recently I became very much intrigued by algebraic topology and am spending quite some time learning it. My reasons are three-fold: it's a beautiful theory; it gives geometric justification to (or perhaps rather an application of) many purely algebraic structures; and it has fascinating applications in quantum field theory and condensed matter theory. Nevertheless, what I am familiar with currently are just basics: various homology and cohomology theories, homotopy theory and some standard applications (Brouwer, Borsuk-Ulam, etc., etc.). While these are of course interesting of and by themselves (and I expect spending a great amount of time on understanding all of this properly), I guess it is more or less understood for some fifty years now, so supposedly people work on topics far more advanced than this (or at the very least they use far more advanced tools to understand standard but hard problems). So, I'd also like to know what the field is about from the modern perspective (some interesting problems and research topics, advanced tools, etc.) so that I can see a little where will the study of the subject lead me in the long run. Sorry if the question is too broad but I am not sure where else to look (I've more or less browsed through all general articles on AT at wikipedia and tried to search MO too). I've heard few magic words like K-theory, sheaf cohomology, various spectral sequences, etc. but I don't understand these at all yet; more importantly my motivation to learn these things is lacking since I have no idea how or when these magic words are used (although I am pretty sure they are used a lot).


I have been inspired by some sci-fi cannons that seem to operate by initially spinning up a projectile inside the cannon, and then suddenly firing the projectile out at high speed. Now, I am wondering whether it is possible for such a cannon to perform practically. So, from an energy point of view, it appears that this hypothetical cannon is able to do work on this projectile by spinning it up using a torque, and this work increases the rotational KE. Then, there is some unknown process that cause the transfer of rotational KE into translative KE, causing the projectile to fly off with a particular velocity. It would be appear that the spinning-up function of the cannon would be an interesting way of having the KE of the cannonball stored up and ready for firing. However, even though translative KE and rotational KE are both KE, the same type of energy, it looks like extra work needs to be done to transfer between both types of KE. To do this, by applying the principle of superposition, it looks like you would need to combine the effects of undoing the spin with a reversed torque, and then applying a force, in the direction of the cannon firing, in a short space of time. Does this apparent backtracking of doing work not seem inefficient, making the spinning-up of the projectile a waste of resources? Are there any real, practical mechanisms that work especially well in converting rotational KE into translative KE? In short, can there be any practicality in a cannon that initially spins up it's projectiles?


So, to make it abundantly clear what I am asking for, here is an example picture edited (poorly) in a drawing software: At left, you can see my current TexStudio look: the are where real text is indeed entered had its colors properly customized with a .txsprofile object. However, the menu toolbar, the tabs area and so on are still colored in light-gray, while I would like them to have a darker look (as in the image to the right). Is it possible to achieve that somehow? I am aware that plenty of questions have been asked here and elsewhere on the topic of TexStudio's colors. For example: How do I change the colors of the application interface in TexStudio? Dark theme for Texstudio How can I set a dark theme in TeXstudio? How do I change color settings in TeXStudio? However, none of them really addressed the main point at hand: how to customize the colors of TexStudio's menus, toolbars, etc (not the colors of the text-editor part of TexStudio). The colors of the text-editor of TexStudio can be modified directly at the Syntax Highlighting options in the Tools>Configurations menu. However, rarely something is said about the other parts of TexStudio. When a question pops up about this, it gets either no answers or very low-detailed reactions. For instance: How do I change the colors of the application interface in TexStudio? Change background colour of toolbars in TeXstudio I realize that what TexStudio is doing is probably just retrieving the operational system color for it's menus, tool-bars and such. My question is whether there is anyway to bypass that and customize such colors.


So I've recently seen a few people use the word "sufferer" to describe themselves having a certain mental disorder. I know that a person thinking that they are suffering a certain disorder may be quite subjective, but their usage is still questionable. The best exhibit I have for this question is somebody calling themself a "sufferer" of the mental disorder "misophonia". To save you a Google search, basically it means that you become pretty annoyed or even enraged at noises like people chewing ice, people chewing food loudly in general, and et cetera. I too have this disorder (if you can call it that) and I did not relate to the word "sufferer" at all, so I disagreed with the usage (not verbally; I didn't want to start an argument). To elaborate even more, I will include another example. I have seen people call themselves a "sufferer" of a disorder called "visual snow". Basically, this disorder causes static to cover one's vision in a way almost relatable to TV static (not really). It's very hard to explain and it is rare, but I suggest you Google it because it is interesting. Once again, I too have this disorder and have had it for my entire life. Seeing people describe their experience as "suffering" almost baffle me. I only notice my static if I am in a situation that has solid color (like pitch black or looking at a whiteboard) or if I purposefully pay attention to the static. It never obtrudes life in any way and many who tolerate this disorder will agree that it is not obtrusive. With all of this I do not understand why one would say they are suffering from a disorder that does not inflict physical, mental, and/or social harm. Each side of health is arguable, as neither of these technically cause harm on either end (referring to a commenter, you becoming enraged and punching a wall is not direct pain from the disorder "misophonia). So my main question is, after seeing that Google defines the "dated" usage of the word as a "tolerator", would the word "sufferer" be acceptable in cases in which one does not necessarily "suffer" anything? I'll apologize that this question is really subjective and may not have any answer, but really I am just trying to get a consensus. Thank you in advance!


This question isn't answered here: Differentiate between past and present just by pronunciation when word is followed by d- or similiar sound That question asks about what happens when the following word begins with a consonant in general. This one is asking about the various possibilities for an intervocalic tap if a following word is an unstressed function word beginning with /t/. It's also asking whether there's any possibility for regressive assimilation here. I read this in American accent book. I quote the text exactly how it is written in the book: The suffix -ed is not pronounced precisely when it is linked to another consonant. For example, mailed the sounds very much like mail the in the following sentences: I already mailed the letter. I will mail the letter. The suffix -ed is not heard at all when it is linked to /t/ or /d/. For example talked to sounds identical to talk to in the following sentences: I talked to her yesterday. I talk to her every day. Okay but then my question is, how do Americans distinguish between these two sentences: I try to call you. I tried to call you. In the first sentence the /t/ will be realised by a voiced tap because it occurs between two vowels. In the second will we get a regular [t] and then a voiced tap? Will we get a double tap or a double length tap? Or will we just get a single tap? Or maybe here we will just get two regular /t/s? Do native speakers differentiate between "I tried to call you" and "I try to call you" from the context?


For the sake of time, is it bad to just accept some fast paced class's theorem's (such as MIT's algebra class) as true even if you don't completely understand the proof or can't remember the proof off top of your head (after a while has passed?). I often find myself wasting too much time trying to memorize proofs when that's not the point of the class (and I can actually wait to memorize the proof later). Sometimes, I get caught up in one detail of the proof for hours and end up not having time to learn how to actually use the theorem and do the homework. Also, is it bad to gain a complete and working understanding of something after you take the class. I am not mentally capable of fully absorbing both what the class want's us to get and thinking about it enough to have a complete and sufficiently deep understanding of the subject all in one semester. However, I feel bad if I wait to think deeply about the class material until after the course is done, but I simply don't have enough time to fully understand some things during the semester. I always hear this advice on making sure you understand everything when you are studying to practice/do mathematics, but that seems not practical if you are taking four or more classes and struggling to make sure you understand what you need to for your other classes have other obligations to attend to.(and also if English is not your first language). I feel like a lot of the advice I hear is for native English speakers (I came to the U.S. when I was four, so I'm practically a native English speaker, but not when it comes to understanding things well the first time through in math, or at least making the understand be thorough by a native English speaker's standards.) The reason I ask this is because I can't say I fully understand something in English unless I can explain the proof verbally/descriptively to someone.


We make an important distinction between the topological insulators (which are essentially uncorrelated band insulators, "with a twist") and topological order (which covers a variety of exotic properties in certain quantum many-body ground states). The topological insulators are clearly "topological" in the sense of the connectedness of the single particle Hilbert space for one electron; however they are not "robust" in the same way as topologically ordered matter. My question is this: Topological order is certainly the more general and intriguing situation, but the notion of "topology" seems actually less explicit than in the topological insulators. Is there an easy way to reconcile this? Perhaps a starting point might be, can we imagine a "topological insulator in Fock space"? Would such a beast have "long range entanglement" and "topological order"? Edit: While this has received very nice answers, I should maybe clarify what I'm looking for a bit; I'm aware of the "standard definitions" of (symmetry protected) topological insulators and topological order and why they are very different phenomena. However, if I'm talking to nonexperts, I can describe topological insulators as, more or less, "Berry phases can give rise to a nontrivial 'band geometry,' and analogous to Gauss-Bonnet there is a nice quantity calculable from this that characterizes instead the 'band topology' and this quantity is also physically measurable" and they seem quite happy with this. On the other hand, while the connection to something like Gauss-Bonnet might be clear for topological order in "TQFTs" or in the ground state degeneracy, these seem a bit formal. I think my favorite answer is the adiabatic continuity (or lack thereof) that Everett pointed out, but now that I'm thinking about it perhaps what I should have asked for is -- What are the geometric properties of states with topological order from which we could deduce the topological order with some kind of Chern number (but without starting from a Chern-Simons field theory and putting in the right one by hand ;) ). Is there anything like this?


I've been thinking about infrared radiation and noticing more and more how the human skin seems actually pretty sensitive to it. You can easily feel a bonfire from several meters away, far away from where any convection would heat your skin. When you open the hood of your car you can feel the heat from the engine even standing back a step or two (away from the updraft of hot air). Now try this: hold the palms of your hands against eachother a couple of inches apart and keep them like that for a couple of seconds. Then slowly (to avoid wind cooling) lift the other palm so they no longer face eachother. Do you feel it? For me there's a noticeable difference in warmth. Is that the skin detecting black body radiation from other skin? This could be easily blind-tested with a friend; you hold your palm out and look the other way, then see if you can correctly tell when your friend's palm is near you palm and when its not. Maybe the human skin is even able to detect black body radiation from another human standing behind her? Kind of like a sixth sense. Could explain the sensation of "i knew someone was there". I've noticed also that when you stand close to a concrete wall that was heated by the sun, but the sun has just set, you can tell which direction the wall is just from the heat on your body. Is this all placebo or does it actually work that way?


Has anyone actually developed a Program Synthesis system for creating computer programs automatically from a non-procedural specification that is taken from a fairly robust system of specifications? For example, I might say "Check if X is a factor of Y." or "List all prime numbers between X and Y." and out pops a program or two written in PHP that does just that. I would actually pay money for an example that shows that the answer is yes. Let's say you have relations for less than and multiplication, and you give a wff of logic. So it can be (exists A)MUL(A,X,Y) where MUL(A,B,C) iff A x B = C, for the first specification above. For the second specification with output of a number (the prime numbers) instead of TRUE or FALSE, we could say variable N means output all values of N such that the wff is TRUE. I have heard the claim that the answer is yes, but articles give no real examples of its being used. Either there is no example, or what they call an example merely displays a program and claims that it was or can be generated by some system that is imagined or even described in detail, but the example lacks any details at all. Can someone give a complete example of a system, a specification, the resulting program(s) and step-by-step exactly how that program is automatically constructed? And if an example is given, could it be a simple program with only a few steps to create, rather than dozens of pages of formulas that would take months to go through and is suspect of being obfuscation hiding the fact that it is not genuine?


I was recently in an argument with a friend who - equipped with an apparent understanding of the etymology of the words lend and borrow - insisted that to lend an object required not just the temporary exchange of its possession, but also a geographical displacement. He compared the words lend and borrow to take and bring which involve a transition to/from one's locus, which are apparently linked (though I can't independently verify) to our subject words grammatically. I argued that to so strongly stipulate (as he did) that an exchange was not a lending because the relocation was not significant (ie; it did not leave his personally defined location), is foolish given the complete generality / ambiguity of the word location. He insisted he could not lend me the salt-shaker as it traveled from his hand to mine because we were both common to the location 'this house', though I remarked that we sat in different locations; our respective chairs. After much debate, we concluded that to require lending to constitute a relocation is a poor definition, since there are (according to him) obvious stipulations as to the definition of the location. So... is any of this actually correct? Does lending actually have any requirement for a displacement of the lent object? If so, what exactly are these conditions (or rather, if they are as contextually defined as we both probably expect, what is a more appropriate word than 'relocation'?)? (I finally remarked that language is defined by the understanding of its speakers, and that I'd personally never heard a relocation was required in the lending of something. I also think the phrase "lend me your ears" doesn't beckon for their displacement...)


Rational numbers, rational functions, and Gaussian rationals are examples of fields of fractions. In each of those cases, one knows what the quotients are long before one hears of the idea of constructing the field of fractions of an integral domain. One case where one (typically??) does not know of such a thing in advance is the field of "convolution quotients"---the field of fractions of a ring of functions of a real variable in which the "multiplication" is convolution. But convolution quotients will not be appreciated by students who just finished a first-semester calculus course last week. Is there some example one could mention to such students where they wouldn't think they already know what is meant by division of the objects in question? Later edit suggested by answers and comments posted so far: I had in mind two or three purposes. One was that I wanted to mention this topic a bit obliquely in something the students are to read, and that had to be really terse, so I can't do anything really involved. Less than an hour after I posted the question, this ended up being a parenthetical comment on the course web site that said: "(for example, why is it that one can `divide' one divergent series by another?)". Here I had in mind the ring of formal power series suggested by Chris Eagle, but of course I needed to ruthlessly avoid mentioning power series. A second purpose concerned possible future uses. Not only in courses: if we get some good examples here, I'd like to add them to Wikipedia's article titled "field of fractions". A possible third purpose was just the satisfaction of knowing more than one decent example (since the only one mentioned above that's "decent" in the relevant sense is convolution quotients).


Assuming that antimatter is matter with time arrow reversed, would it be right to say that matter beyond black hole event horizon then would become antimatter because of space and time axes exchanged? Would not black hole then appear like a nice universe consisting from antimatter that slowly expands as matter falls into it? I do not claim anything just want to find out how wrong the idea is. Although, it is not really related to the question but I would like shortly explain where from my crazy idea that matter can indeed move to the opposite time direction is coming. I think that there was no Big Bang but initially was space filled with matter fluctuating back and forth in time (field fluctuating between matter and antimatter). Since there was no real matter - matter and antimatter fluctuated from vacuum and annihilated chaotically and hence there were no state transitions (movement) which we perceive as time - there was no time, in fact there was no matter either - just vacuum. But at some random event indicated as 'shortly after Big Bang', CP-symmetry got broken, which caused antimatter to disappear and gave rise to the time which we since perceive as going forward (if by some other event antimatter had won our time would go into opposite direction - but this does not mean 'back in time'). CP-violation caused universe expansion, but there was no initial rapid expansion, since universe did not arise from a singularity, but from a homogeneous space, which is in sync with recent Cosmic background radiation observations.


One only needs to search MMA.SE, math journals, wikipedia, or god-forbid, n-cat lab, for keywords listed in the title, which can be extended with: uniform-, regular-, complete-, local-, partial-, non- (see below) &c&c, to be convinced that modified concepts are replete across maths, proliferating, and their diversity is likely accelerating. Shafarevich: "it is the destiny of mathematics to expand in all directions." This trend, coupled with the lack of standardized terminology, can make it difficult to compare results or in same cases even definitions. It seems clear that in general a modifier term doesn't categorically reveal whether the modified concept is a specialization or generalization of the underlying concept (eg, subset versus superset, or subcategory versus supercategory). In some cases the modified concept might not bear a sub/super relation to the underlying, for exmaple, co- and op- in category theory and universal algebra (what's the relationship of universal co-algebra to algebra or co-induction to induction?). So it appears we must be content with enumerating cases to discern the relation and then compare to see if a big picture emerges. Basic examples: Semigroups are generalizations of groups but inverse semigroups are specializations of semigroups. (Quasicrystals are crystals - this got the Nobel - but their symmetries don't satisfy the crystal restriction theorem, eg, translation invariance, so are not groups, but might be modeled by inverse semigroups [ML]). Quasimetrics are generalizations of metrics, but ultrametrics are specializations of the latter[VS] . Noncommutative geometry, Connes stresses, includes commutative geometry so it is a generalization. In the absence of an online OEIS-like database, would it be possible to crowd-source many more examples of mathematical concepts or categories noting sub/super (or other) relation to the underlying?


I think physicists can deal with this question best. I answered a question about "immortality" when some guy claimed I got it wrong that neurons die (I argued that even if you live a billion years you die slowly many times over because all your cells incl. nervous cells will have to be replaced, meaning you will regularly - but never suddenly - become a new person; also: continuous learning, and information storage is physical, and our brain has very limited capacity so memories will change), immortal means they are immortal. That got me thinking: Is it actually possible to have repair systems in each cell, or isn't it better to have a repair system on a much higher macro level and cell-level immortality is actually impossible? Background: The smaller the scale, the more events like quantum effects and Boltzman (energy) distribution (e.g. random atom movements, crystal structure defects, breaking bonds) dominate. Meaning small systems WILL break in unpredictable ways, so only large systems can live long and have reliable repair systems because the larger the system the less important those effects become. Summary: It is much better (or even possible) to have big long-living systems than small ones, because the inevitable repair system cannot be too small because it will be subject to random events of the micro-world. An "immortal" human would have a body-level repair system that repairs by letting broken cells die (we have that already) and build a new one, instead of having a repair system on the cell-level that would make cells immortal (too unreliable, won't work). Or in other words: The smaller the system the less likely a long life can be achieved. Note that I'm asking about complex dynamic systems, not static objects or information. Like living organisms or (complicated) machines, and about their chance to "live long".


There's a list of "New York" words and phrases that's been surfacing on the Web periodically for quite a few years. Not all New Yorkers speak like that, I assure you. Only barely-above-the-gutter white New Yorkers with Brooklyn roots, plus lower-middle-class Jews from Borough Park and wanna-be Italian mafiosi from Bensonhurst. I know one Italian limo driver who still says "youse" because (get this! ...) he wishes to preserve that particular brand of Brooklyn dialect. Here's the list: Filayda: A single phrase made out of two English words. As in, "I don't want this for now but I'll take it 'filayda.'" [I don't remember ever using the expression myself, but I do hear it often. Ricky V.]. Wawda: That which if not inhaled directly from the faucet is, if from a classier social set, sipped out of a glass. Naydivs: Local types born and reared in "Bronzvle." Tooner Samwidge: Luncheon staple. Mellid Cheese Samwidge: A menu choice if the luncheonette is out of tooner. [Somewhat archaic, both of them, since the disappearance of the luncheonette as a concept. R. V.] Kawfy: What washes down Mellid Cheese. Berle: Not Milton's last name but the method by which an egg is often cooked. Earl: What French fries lay in. Goil: The way people who say "berle" and "earl" pronounce "girl." Buzz: Large lumbering public vehicle usually located all in a clump or, if you're in a rush to be transported crosstown, at sporadic intervals. [An exaggeration, actually. R.V.]. Gazz: That which goes into the buzz. Fiff: The major avenue which separates Manhattan's East Side from the West. Sixt: The major avenue one block from Fiff. Ate. The major avenue one block from Seventh. [Despite the numerous moronic efforts to rename Sixt to "Avenue of the Americas" (eww); despite the fact that the ... uh ... official ... name appears on anything and everything that's mailed to or from Sixt, it's Sixt, and will still be Sixt a thousand years from now, one would hope] Ey: A manner of summoning those whom you would alternatively greet with "Whassup?" Bronnix: One of the land masses the Triboro connex. [The Bronx, actually: a neighborhood named after the Swedish immigrant Jonas Bronck and his family, the Broncks. The Triboro is the Triborough Bridge, built by the infernal Robert Moses. It was an automobile-only bridge back then; it has been (reluctantly) changed to automobile-and-bicycle and renamed (idiotically) Robert F. Kennedy Bridge, since. R.V.]. Lirracher: Stuff like what Dostoyevsky wrote. [Personally I don't know what this obsession with Dostoyevsky is. There have been infinitely better writers, including Russian writers. R.V.] Liberry: Where one goes to immerse in lirracher. Purdy: The view outside the Empire State Building. [This isn't specific to New Yorkers. R.V.]. Awfissa: Policeman. Cop. Patrolman. Sergeant. Dude in blue on horseback. Downashaw: Where Rockaway is. [Rockaway and Far Rockaway, actually, commonly known as the Rockways; the latter is the last stop on the "A" train; never take it if you can help it. R.V.] Ackrost: From here to there. [An exaggeration. R.V.] Monicker: Miss Lewinsky. Lannick: As opposed to Pacific. Fewcha: Follows the present. Ahkinsore: Clinton territory. Winda: What is always broken and what in the hot weather never rolls down in a cab. Payment. What one walks on so one doesn't get hit by a car. Innerestin: What this particular column should be unless you're somebody who demands quality. Youse: Second person singular. Yiz: Second person plural. Yizzle: A contraction to be used in the sense of, "Yizzle call me." Now most of this is very familiar to me as a New Yorker; I am, however, puzzled by a couple of things here. Specifically, I can't figure out what the original list's author had in mind when he wrote "Earl." I mean, yes, I get it, it's "oil." But ... hmm ... either I've never heard a New Yorker say it like that or I have a problem seeing the forest for the very familiar-looking trees ingrained in my psyche. Before you start pondering on this, let me remind you that the New York brand of English is unapologetically rhotic. Agressively rhotic. Defiantly rhotic. "Earl"? Really? Rhyming with "girl"? I don't get it. What am I missing? Bonus question: While I hear "youse" all the time, as Damon Runyon did nearly a century ago, I've yet to encounter "yiz." I don't get the joke. Care to explain?


This question related to Why are magnetic lines of force invisible? and is motivated by a comment of @BlackbodyBlacklight, based on that, the illustrating example may depend on that linked question as context to be clearly understandable. A remote magnetic field, in the sense that it is not at the location of measurement, could influence the location of measurement in some (possibly indirect) way that allows to derive information about it's structure. This is comparable to deriving information about a remote temperature profile based on properties of the local electromagnetic field, like when using a camera, or just seeing something glow. It might well turn out that it is fundamentally impossible to derive information about a remote magnetic field, (given some sensible constraints). In this case, an Answer should ideally explain why that is the case. What is described above is roughly comparable to human perception, which was the context where the question came up originally. Therefore, I will illustrate my initial ideas in that context in the section below: Establishing the context for the question (The biological aspects referred to are part of the illustration, not directly related to the question): The motivating idea was: "We can not see magnetic fields, but that may be because it was not important during evolution to acquire this capability." Could it be possible, in principle, to "see" magnetic fields? Now, if it would have been helpful during evolution - what kind of perception is possible purely from the physical side of the question - assuming "perfect evolution". The linked question asks about seeing magnetic field lines - so could something like eyes for seeing field lines have evolved? I assume not, so we do not need to go into details whether to see them on surfaces, as lines at a fixed distance, etc. (Feel free to make creative assumptions as needed regarding how to "see") What did evolve, in some birds and bacteria, is perception of the field of Earth in terms of direction of the local(!) field lines - something like "feeling north and south". The actual question, related to physics of magnetic fields, in comparison to phenomena for which human perception exists: What are the physical constraints? Seeing a magnetic field like a fourth base color would not work - there is no radiation. Something similar to spacial sound perception? Which would mean to measure from a finite set of "local" reference points to collect measurements on a given remote locatioin. Anything better than measuring a local field vector is certainly interesting.


I have a question about adjoint operator. I have known that bounded linear operator on Hilbert space has a unique adjoint operator, but I am wondering whether there is similar existence result about bounded linear operator on Banach space? Thank you.


I am learning measure theory this semester. The definition for sigma-algebra is "a collection of sets that is closed under complements and countable unions and intersections." I wonder what does it mean by "closed under complements and countable unions and intersections." Thank you so much for your help!


I know that "callipygian" means "having beautiful buttocks"... so I was wondering if there is an English word that means "having beautiful hair". I tried googling this but couldn't find anything so far.


I'm preparing for an exam and I can't seem to figure out the reasoning behind the answer to this question. Why do they use a chi-squared test? Can someone walk me through their explanation? Thanks.


I'm looking for a good textbook for an introduction to Stochastic Analysis, preferably one that focuses on rigour. I am familiar with measure theory and basic probability theory. The direction I am mostly interested in is stochastic differential equations.


I'm terribly confused on the concept of "rank of a linear transformation". My book keeps using it, but it doesn't clarify what it means (or at least I haven't been able to find it). Is it the same as the rank of the matrix? For example, if A is a mxn matrix, what would be the rank(A)?


I've recently been working through a lot of physics problems and a lot of them say to assume that the mass of the string used in a problem involving a pulley, for example, is negligible. Why is this important? What would happen if the mass of the string wasn't negligible?


How do I properly punctuate this sentence: "I prepared, packaged, and priced beef, pork, chicken, and seafood." I am trying to say that I did those three actions to those four kinds of meat, but I am unsure of what to do. Should there be a colon or semi-colon between priced and beef?


I'm not a native English speaker, and I don't understand the meaning of the phrase "in your general direction." I have found its use in the line from Monty Python and the Holy Grail: I fart in your general direction.


This problem is taken from the book Mathematical Circles by Dmitri Fomin, et al., translated by Mark Saul and published by the American Mathematical Society. Can anyone describe what the question actually means?


I'm reading a book on probabilistic robotics and it mentions that "this probability density function is quadratic in x." I haven't heard of the phrase "quadratic in x" before. Can someone explain what it means? Does it mean that the graph has a quadratic shape?


I don't want subsections to appear in the table of contents of my Lyx document. How do I turn them off? I went to Tools -> Settings -> Numbering & TOC, but it won't let me modify anything. What should I do?


I just read this interesting interview with Frank Wilczek and he talks a couple of times about gate symmetry, without ever defining the term. This isn't a term I've come across, and google throws up a blank. What is gate symmetry, and are there any good references?


I have a LaTeX document that contains proprietary information. I need to print a version of the document that has a box around the text with a disclaimer (something to the effect of "Proprietary information of company XYZ, do not redistribute without express consent'). How can I do this? If it matters, I'm using MacTex.


Density Functional Theory (DFT) is formulated to obtain ground state properties of atoms, molecules and condensed matter. However, why is DFT not able to predict the exact band gaps of semiconductors and insulators? Does it mean that the band gaps of semiconductors and insulators are not the ground states?


I'm not a particle physicist, but I did manage to get through the Feynman lectures without getting too lost. Is there a way to explain how the Higgs field works, in a way that people like me might have a hope of understanding?


Is it possible to establish that the lines joining the mid points of opposite sides of a quadrilateral bisect each other. I attempted using mid point theorem of triangles but I couldn't prove it


I am trying to find an English translation of Camille Jordan's work "Cours D'analyse". Only the French edition is on Amazon, so since this is a somewhat specialized topic, I thought perhaps someone in this forum might know. TIA, Matt


I am currently studying Electrical & Electronic Engineering. I wish to pursue Quantum Mechanics or Quantum Computing as my research subject. Is it possible for me to do my M.Tech. and then pursue my research subject? What are the prerequisites for studying these subjects? I would be grateful if you could help me.


Prove that if the real part of an entire function is bounded so is the imaginary part, without using Liouville's theorem. In particular, is there a way to prove this using the Cauchy-Riemann equations?


I'm sorry if I ask this question at the wrong place, but I don't know a better one. I am a Master's student and I am really interested in analysis, but I also want to get into AI. Does anyone know a natural way to combine these two interests? Thanks in advance.


The title basically states the whole question..I was trying to invoke the Mean Value Theorem on it but it hasn't worked..I was wondering if I'm supposed to solve it some other way. I just need hints, please. Thank you.


This has been bugging me. Why is the lower case letter "a" used to spell "abelian group" when upper case letters are used to spell the terms, "Gaussian Integral", "Cantor set" or "Cauchy sequence"? Don't know where else to ask.


As far as I can tell, if a function is holomorphic on its domain, then it's also meromorphic and vice versa. Can someone tell me what the difference between these two properties are (if any)? A counter-example and an explanation of why it's a counter-example would be nice.


I'm reading "The Portrait of a Lady" by Henry James, and I found the following two sentences. "I suppose that after a girl has refused an English lord she may do anything," her aunt rejoined. "After that one needn't stand on trifles." What does "one needn't stand on trifles" mean?


I am making a piece of software which has the ability to send out Emails and SMS messages. My boss has asked for both facilities to go under the same heading in a dropdown menu. But I can't think of a suitable work that encapsulates both. Any suggestions?


We've all seen that label on our passenger side mirrors that says, "Objects in mirror are closer than they appear." Why is this? Further, why does it only apply to the passenger side mirror, and not the driver-side or rear-view mirrors?


Show that Lebesgue measure can be expressed as a countable sum of probability measures. I'm trying to do something with the countable additivity property in order to show this, but so far nothing is working. I don't think this is supposed to be difficult, but I'm not seeing it, so any help you could give would be most appreciated!


I'm trying to compile a LaTeX template to a PDF, but it's not working. The template is available in this link (on the right side in Article Templates). I'm using TeXnic Center. Can anyone please try to compile this and let me know if it's working.


I am trying to use Egorov's Theorem in a proof. However, I only have convergence in measure of f_k to f and uniform integrability of f_k. How can I combine these two to get convergence point wise so that I can use Egorov? Thank you so much for your help!


Is the following sentence grammatically correct? "Symptoms of lactic acid build up in muscles include cramps, spasms, fatigue and sensitivity." Does "sensitivity" need to be qualified, as in "sensitivity to touch"? Can someone point out the "rules" that come into play here?


As a student in a highschool physics class, my teacher has repeatedly told me that photons are massless. Yet, I have also heard from other sources that photons have momentum. If photons were to have momementum, that would mean that they have mass as according to p = mv. Do photons really have mass? Also, how would this mass be calculated?


I heard somewhere that the capacitance of a superconductor is much higher than regular conductors, but I haven't heard or seen anything yet proving or disproving such. So whats the truth? And why?


A colleague asked me this question, and I couldn't come up with an answer that satisfied him, so I'm wondering if anyone can help: Why does a man with a short temper become a short-tempered man? In other words, why do you need the -ed at the end? Are there any special rules for this?


Is there a path connected topological space such that its fundamental group is non-trivial, but its first homology group is trivial? Since the first homology group of a space is the abelianization of the fundamental group, we are looking for a non-trivial group whose abelianization is trivial. Is there such a group?


I got edited on Stack Overflow because I used "I'm", "you're" and "I'd" instead of "I am" etc. Is it considered rude to use contractions like that in informal conversations on the internet? I would not use them in papers but, I thought on Stack Overflow it was okay.


This is probably a stupid question but I will have to ask it. If you had a set of N correlated random variables and knew the correlation matrix, can one compute the joint probability distribution of all variables? Does it make a difference if the correlation matrix was built using Pearson's rho or Kendall's Tau for example? Thanks, Bogdan.


My book says all extensions of finite fields are cyclic, but I could not find a proof (maybe I haven't looked hard enough). If it's straightforward, can you tell me why it's true? Thanks :)


I'm trying to solve this question: I didn't understand why the hint is true and how to apply it. I really need help, because it's my first question on this subject and my experience on this field is zero. I need some help. Thanks a lot


I do know Schreier's theorem, which states that a finite index subgroup of a finitely generated group is finitely generated. Other than this, I have no reason to suspect a positive answer to my question, other than it would be nice.


Could someone tell me if I've worked this out right? I'm unsure of the process, especially the final parts where I convert it to a sinc function. Please let me know if I've made mistakes anywhere else too.


The Shannon-Hartley theorem gives an expression for the capacity of a bandwidth and power limited channel. How would one formulate this theorem mathematically (rigorously)? I understand the formula given, but not how to formulate the assumptions made mathematically.


What is the standard way in latex to display certain objects (e.g. pictures, tables, etc.) side by side, instead of one below the other? I thought about using the tabular environment for this, but I'm not sure if there isn't a better way.


I am reading the book "Lebesgue Measure and Integration: An Introduction" by Frank Burk, and it says the following: If the Lebesuge outer measure is finitiely additive, then it is countably additive. How can I prove this?


Hello everyone, I'm trying to solve this problem, but I'm stuck... i don't quite understand the definition of the norm, If you guys can give me a better explanation, I would appreciate it, Thanks


It is well known that a local ring is a ring containing only one maximal ideal. I was wondering if there is a characterization (or any information) of the commutative rings such that all their non-trivial ideals are maximal. Thanks.


I am an undergraduate student doing a project on rooted trees. I was wondering if anyone would know any easy to understand references that explains Grossman and Larson's Hopf Algebra on rooted trees? In particular, I want to know how the comultiplication, counit, and antipode are defined. Maybe if there is anything that explains their paper 'Hopf-algebraic structures of families of trees'.


I've seen in a documentary that when a star collapses and becomes a black hole, it starts to eat the planets around. But it has the same mass, so how does its gravitational field strength increase?


Which one is correct? Centuries ago, people weren't aware that the planets revolved around the sun. Centuries ago, people weren't aware that the planets revolve around the sun. In my grammar book it has mentioned the first one as "grammatical correct sentence", but it has not done such a comparison.


What lessons do we have from string theory regarding the fate of singularities in general relativity? What happens to black hole singularities? What happens to cosmological singularities? Which points of view on string theory yielded results in this respect? String field theory? AdS/CFT? Matrix theory? I suppose perturbative string theory is not applicable in the vicinity of singularities.


I've always thought "to slumber" meant to sleep deeply or for a long period of time. Several colloquial but less reputable dictionaries agree with me (http://www.urbandictionary.com/define.php?term=slumber, http://www.yourdictionary.com/slumber) but Merriam-Webster and the OED disagree, defining slumber as "to sleep lightly" and "to sleep, esp. to sleep lightly; to doze or drowse," respectively. Which is correct in modern usage?


Question: Which of the following graphs correctly describes the variation of kinetic energy with time of a block when it slides down a smooth inclined plane from rest? The answer is C but I do not understand why. How do we know if the velocity increases as it slides down the inclined plane?


Why must a symmetric positive definite matrix must be invertible? I'm reading a proof of the Levi-Civita theorem in differential geometry but the author states this without proof and I haven't been able to prove it.


I was wondering if it's possible to produce the anarchy symbol (the circled 'A') in LaTeX. I tried googling "latex anarchist" and got nothing but merchandise and porn, and while I'm not complaining, I also wanted the anarchy symbol.


Is there a way to increase the font size in the source window of TeXworks? I guess I'm showing my age, but for me it is almost unusable. It's kind of annoying having to put my face so close to the screen to get any work done.


I'm trying to make a figure environment that can select between multiple images. I am not trying to do multiple subfigures but rather a single figure environment that can cycle through multiple images like a slideshow. At this point I don't have any particular format in mind. I am just looking to see if Latex has the ability to do this. Thanks in advance.


I read the following sentence in the book Options, Futures, and Other Derivatives: Treasury bond prices in the United States are quoted in dollars and thirty-seconds of a dollar. Here is my question: How should I understand the phrase in bold, namely, "thirty-seconds of a dollar"? I think the plural form "thirty-seconds" means several thirty-seconds of a dollar. Does it make sense?


noob question here. I'm interested in reading The Elegant Universe by Brian Greene, but I've read some reviews that say the science has been discredited. I'm a total layman, has the science been discredited?


I have a strong mathematical background and I am interested in the relationship between mathematics and music. I have found some introductory material on the web. Do you know any good books that will broaden my perspective of this subject?


While looking for something on this site I accidentally found a document class which is perfect for a scientific report with two columns layout. Unfortunately I forgot its name. Could someone point me out the document classes which implements two column layout without using multicol packages?


If the universe is undergoing inflation, and there is a minimum scale that things can exist at (the Planck length), does that mean that new Planck-sized domains have to be continuously popping into existence? If not, does that mean that the Planck length is constantly changing?


What do they mean? How should I use them? Which one is more appropriate to what context? I was talking to a colleague of mine and we couldn't get to a consensus about what should we say when referring to drinks, for example: A lemonade is made of lemons. or A lemonade is made with lemons.


If the title of a song ends a sentence, whether a statement or interrogative, where does the period or question mark go? For example, is it: The Beatles sang "She Loves You." or The Beatles sang "She Loves You". Is it: Did The Beatles sing "She Loves You?" or Did The Beatles sing "She Loves You"?


I'm having some trouble understanding the concept of negative work. For example, my book says that if I lower a box to the ground, the box does positive work on my hands and my hands do negative work on the box. So, if work occurs when a force causes displacement, how does negative work happen? Are my hands displacing anything?


How to prove Ampere's Circuital Law in case of any conductor. My text gives the proof of only the special case when the conductor is long and straight. I am trying to prove it, but haven't been successful. Any hints would be welcome.


I've recently come across the following sentence: Round the corner walked Hannah, and nearly bumped into Louise. The first clause sounds clumsy to me. I think the example above should be written like this: Hannah walked round the corner, and nearly bumped into Louise. Is it ever right to put the subject at the end of a clause?


When I was much younger, I remember the press always referred to the U.S. president using the title of the office: "President Nixon" was followed by "President Ford" then "President Carter". Now that seems to have fallen out of favor and the common reference is: "Mr. Obama". My question is: when and why did that change happen? Was it associated with a particular president?


Is there a sequential criterion for differentiability,just like there is one for continuity ? If not then,why so ? I'm studying undergraduate real analysis and haven't really come across one. Thanks in advance !


What does "sit in the back of the bus" in the following sentence mean? It has been taken from Harvey Milk's "The Hope Speech." The first gay people we elect must be strong. They must not be content to sit in the back of the bus. They must not be content to accept pablum.


The Ricci tensor is defined as the contraction of the Riemann tensor in its upper and the second lower index. I was wondering why it is defined this way. What happens if the Ricci tensor is defined as a different contraction of the Riemann tensor? Would it satisfy Einstein equations? Does the usual definition have any physical or geometrical meaning?


What is it called when the subject of a sentence is the same as the object of the previous sentence? For example: I'm going to Freddy's house. Freddy lives down the block. The block is paved with gravel. The gravel is jagged.


I am unable to comprehend why ammeter is connected in series and voltmeter in parallel in a circuit. My book doesn't give any explanation about it nor am I able to understand it from the internet. Can someone please explain this to me (a beginner).


Helium is a scarce resource, as it escapes the atmosphere over time. If we run out of Helium deposits, will it be possible to manufacture more helium through nuclear fusion or another nuclear process? If so, how much energy will be required?


Is it more correct to say 'The picture hung crooked on the wall' or 'The picture hung crookedly on the wall'? My instinct is that the first one sounds better to me as a native speaker, but someone else I was talking to was convinced that only the second is correct. What's better here?


Possible Duplicate: Should I put a comma before the last item in a list? What is the difference between tomato puree, paste, and sauce? OR What is the difference between tomato puree, paste and sauce? Do we need a comma before and in this case?


A Lie group is a group which is a smooth manifold such that the multiplication and inversion are smooth. When does a Lie group become simple? What is the difference between simple and semi-simple Lie group? Just want a quick answer here. Thanks


I'm working on an engineering project, and I'd like to be able to input an equation into my CAD software, rather than drawing a spline. The spline is pretty simple - a gentle curve which begins and ends horizontal. Is there a simple equation for this curve? Or perhaps two equations, one for each half? I can also work with parametric equations, if necessary.


How can I center the titles of the chapters in the frontmatter and the backmatter of a document written using memoir? Keep in mind that the mainmatter chapters should not be affected at all.


I keep coming across the term pseudocubic unit cell while reading about orthorhombic perovskite structures. No clear explanation is given in papers. Can someone please tell me how these structures are generated and what exactly is meant by them?


What does "sanding a doorstep" mean? Does it mean polishing the doorstep? Or filling it in with sand? I read this expression in Charles Dickens' book, The Haunted House where it says, I found the landlord of the little inn sanding his doorstep.


A person who makes desserts is called a pastry chef, but is there a name for the action itself of making desserts? "Cooking" is too general, and "baking" implies bread, cakes, etc. which is not quite what I mean. Any ideas?


What are the differences between "unto" and "to"? It seems that in many contexts where the word "unto" is used, "to" could be substituted and would be perfectly correct. It reminds me of flammable/inflammable, where "flammable" came into use because the "in" in "inflammable" caused people to think that it meant not inflammable. Is this a similar situation?


Is it possible that the universe is infinitely large and contains an infinite amount of mass that is distributed in such a way that gravitational force is never infinite? If so, is it possible that the infinite amount of mass that is outside of our light cone could affect us with gravitational force?


Which one is correct? I am a new graduate and have been teaching German for a few weeks now. I am a newly graduate and ... I can't decide whether to use "new" or "newly".


I have noticed the appearance of the phrase "not so much" in the language recently. It strikes me as both grammatically incorrect and humorous when used. For example,"Jim is very smart; his brother, not so much." Or,"That girl has a beautiful face; her figure, not so much." Has anyone else noticed this new colloquialism?


I'm having trouble understanding how to find an invariant to check if it's preserved, and generally how induction is used in proving the correctness of algorithms (binary search primarily, but others as well).


I'm doing a science project, and we're wondering if it is possible to compress pure carbon (C) to the point where it becomes diamonds? What would the process have to be and how much energy would this take? Has this been done and is this feasible?


Prove that any graph that can be obtained from the Petersen graph by adding one extra edge has a Hamiltonian cycle. So I've found that removing any vertex yields a Hamiltonian cycle -- I'm not sure if that's relevant or helpful, as I wasn't able to link that to the main question above. I thought I'd include that in case though.


Are both "an hour and a half" and "one and a half hours" correct? If so, is either more appropriate in different contexts? Example context: "The Superbowl starts in less than one and a half hours." "The Superbowl starts in less than an hour and a half."


Does anyone have a link to Hilbert's Original Proof of the Nullstellensatz, or know a book where it's printed? I'd be interested to see what it was like. I only really know the Noether normalisation and Zariski proofs. While these are both good, it would be nice to have it 'from the horse's mouth'! Many thanks in advance.


The term "ginger" is often used as a slang term for someone with bright red hair. But ginger (the spice) is actually a bright yellow in color. Where does this term come from, then?


I have been trying to understand clearly the concept of non locality, hidden variables, quantum measurement etc through research papers. I also read Quantum Theory and measurment by Wheeler and Zurek but I feel I've got nothing into my head. Please recommend some introductory books on above mentioned topics for beginners.


I have been trying to learn some basics of topology on my own, I have learnt the basic definitions. I have not been able to understand the proof provided in the text. Could anyone provide a clearer proof, or kindly explain me the outline of the proof provided here?


In a commutative ring (with unity), is it true that (a) any maximal ideal is a prime ideal? (b) any prime ideal is a maximal ideal? (b) is almost certainly false, because a maximal ideal is a stronger concept than a prime ideal, but I don't know of any example to give. And I'm not sure about (a).


An orthogonal matrix necessarily has orthonormal columns, and orthonormal columns necessarily give an orthogonal matrix. Also, orthonormal columns imply orthonormal rows. But how about the converse of the last statement? Meaning, do orthonormal rows necessarily imply orthonormal columns? Thanks


I wanted to know what is the typical strength of the magnetic field generated by the tip of a magnetic screwdriver, but couldn't find it anywhere on the manufacturers' webpages. So I was wondering, is it possible to estimate it from the sort of objects that a magnetic screwdriver can pick up?


I just want to confirm that I am right about these expressions. The "foot of the stairs" is the bottom of the staircase, and the "head of the stairs" is the top, correct? Are these expressions commonly used? Or is there a better alternative?


The speed of light in a medium is independent of motion of source relative to medium but it depends on the motion of observer relative to the medium. I don't understand why it is so.


I'm working on an app which displays status messages, like: "XY is now available" "XY is now offline" I'm wondering which of the followings is correct: "XY is now idle" "XY is now idling"


Is there a word or phrase to describe someone who is willing to do extra work in the short term to avoid work in the long run? I have seen "lazy" used for this, but I'd like to know if there's a more precise term.


In our daily life a lot of photons of visible light, infrared and radio etc move around us. We know that light is an electromagnetic radiation. So why doesn't that electromagnetic radiation affect a magnetic compass?


Is there a word or expression describing a phone call when the caller doesn't hang up, but also doesn't say anything (or at least nothing can be heard)? I'm specifically looking for a term describing such call made to scare someone (with the caller breath being heard). My initial thought was that it's called silent call, but according to Wikipedia it means something different.


"If matrices B and AB have the same rank, prove that they must have the same null spaces." I have absolutely NO idea how to prove this one, been stuck for hours now. Even if you don't know the answer, any help is greatly appreciated.


I am an absolute newcomer when it comes to quantum physics. So kindly suggest me a book which covers all the fundamental ideas, equations and all such stuff related to qp, so that i can master the basics and then move on to the advanced stuff.Thank you.


I have recently read the proof that the Brownian Motion and Fractional Brownian motion are almost surely Holder Continuous. I was wondering if this can be extended to a higher class of continuous stochastic process, is there a characterisation of such class? Is this class much bigger than the Brownian Motion/ Fractional Brownian motion? Maybe even the whole set of continuous stochastic processes?


I am studying finite element method.While studying i am confuse with numerical integration and interpolation.Is this two methods are same or different?. If they are different then is there any relation between them?.


I need some help understanding Gaussian mixture models. In particular, I am trying to find the relationship between GMMs and K means. What is the basic algorithm for GMM? I am not sure where the "clustering" comes in. Can someone give me a basic example as to how this actually works?


I'm currently prepping for some high school math competitions soon, and I was wondering if anyone knows any resources that are out there with an abundance of contest-math-related geometry problems. Geometry is definitely my weak point in contest math, and any input would be appreciated. Thanks!


How come the following sentences have the same meaning when they have different voices? I can be substituted for John. (Passive) I can substitute for John. (Active) They both mean "I can replace John."


I am studying a function whose Fourier transform is zero on a set of strictly positive Lebesgue measure and I need to know this: If a set has a strictly positive Lebesgue measure can we prove that it contains an interval? Help is much appreciated


Please describe how a vacuum flask/thermos works. How does the vacuum prevent convection/conduction/radiation? How does the lid with the curvy lines prevent either of the aforementioned heat transfers? If there are any other parts that may prevent heat transfer, please also list them.


I'm trying to show that the quotient of the Heisenberg group with it's own center, H/Z(H), is abelian. I'm not entirely sure what makes up this quotient group in the first place though... and I'm a little confused as to what quotients of matrix groups with multiplicative operators look like. Help, thanks


There has been a similar question before: How to convert a hexadecimal number to an octal number? But, in my case I need an Algorithm to directly convert a number from Octal to Hexadecimal and back without converting it to binary/decimal as an intermediate step. Is it possible?


Gentlemen, thought my practice in solving Gauss Jordan elimination questions, I faced this question and I got stuck in it for quite long time, and I couldn't figure out the answer. Could you lend me a hand in solving this question. Thank you!


Someone recently pointed out to me that most English-speakers will say "I saw a big brown spider," rather than "I saw a brown big spider". However, the second sentence has the same literal meaning as the first. Are there instances in which adjectives are non-commutative, so that permuting the order of the adjectives changes the literal meaning of what's being said?


I've read that attraction and repulsion between particles is caused by the exchange of virtual photons, and that virtual photons carry information. I don't understand how a virtual photon actually causes any attraction or repulsion, and how does it carry information anyway if it's "virtual"? Aren't photons an excitation of the electromagnetic field?


Why is the standard for implementations of the AES algorithm called "Rijndael"? Is it simply a name? Or has it some kind of meaning? I tried googling this one, but without any helpful results.


From the linear algebra books that I've encountered, they either discuss exclusively about finite-dimensional vector spaces, or assume that the reader already knows about infinite-dimensional vector space, Hamel basis, etc. What books explain the concept of infinite-dimensional vector space and its structures?


In Dutch we use it to refer to (the airtime of) tv-shows that start around dinner. Is there an equivalent to it? I suppose it's sort of an idiom, but probably too specific to be considered so.


I would like to measure the energies of particles emitted from a variety of ionizing radiation sources (alpha and beta), and then convert these energies to velocities. However, I am not too sure about how I might go about measuring these energies. Would this even be feasible? If so, how could it be done? How are these energies measured by professionals?


Are there any integral domains in which no nonzero prime ideal is finitely generated? (Other than fields, of course, where the condition is vacuously satisfied.) I asked a similar question the other day, but the solution there relied on using zero-divisors and that didn't really help clear up the situation I was considering.


If I have an ordered set X = {a, b, c} and another ordered set Y = {a, b}, I know that that Y is a subset of X but I also want to convey that Y is the prefix of X if that makes sense. Is there a name for that?


The Great Dark Spot is an anti-cyclone in Neptune. But unlike the Great Red Spot of Jupiter which lasts for more than hundred years, the Great Dark Spot exists for only one year or so. Why is it so??


I was attending my college re-union and a speaker just said that "Having an ebook reader is status quo". Apparently, it means that it is in vogue or in fashion. I do not think it is the correct usage. But I am a bit confused. Can someone clarify?


I know from reading about the gravitational waves detected by Ligo, that when an object has angular acceleration, it produces gravitational waves. I'm wondering if an object creates gravitational waves when only accelerating in one direction, however? I'm also curious as to how the nature of the waves would differ in this case.


Possible Duplicate: What is the correct way to pluralize an acronym? How would you make the plural form of an acronym? For example, if you have two of Nintendo's game console, the DS. Would you say, "I have two DSes," "I have two DSs," or something else?


I am going to send a email to a secretary of a CEO thanking her for scheduling a meeting. What would be a good phrase for the opening of email? Should I write: Thank you for scheduling the meeting or would it be more appropriate to write: Thank you for accommodating our request


Recently, I am interested in the Hecke algebras, but I'm not very familiar with it. Who can provide me some meterials about the history and the development of the Hecke algebras? Is there any good books or papers about Hecke algebras? Any help will be appreciated.


When you get an error in LaTeX, the message "Undefined control sequence" is printed. Why is it called "control sequence" and not "command", for example? How does the sequence work? Is this a TeX or LaTeX thing?


I'm an Android programmer and am working on a graphing calculator. I have been looking for a formula for sine and cosine to put in there. I have a decent understanding of mathematics but can not seem to find this formula. Any help would be great, thanks.


I am having difficulty in understanding about weighted linear least squares. Could anybody explain me instead of minimizing the residual sum of squares why we need to minimize the weighted sum of squares? Further, I want to know about the term weighted? Although I have gone through some wiki notes but I am not able to understand. Thank you very much for the help.


I just downloaded and installed MikTex and TexMaker today, and was wondering if you have any advice. I'm starting to read through various wikipedia tutorials, and other guides, but videos would be an immense help!


A and B worked together to complete a certain task in three hours. If A works alone, he can complete the task in five hours. If B works alone, how much longer will it take him to complete the task? I have the answer to this question, but I just don't get how that works. Please help me out.


Most important theorems in measure theory do not assume the completeness of measure spaces. Monotone convergence theorem, Dominated convergence theorem, and Fubini's theorem, to name a few. So I wonder if the completion of a measure space is necessary. In particular do we need to consider Lebesgue measurable sets which are not Borel sets?


When i put a pencil in the middle of a paper and rotate it very fast, whatsoever is written on it, will appear in concentric circles. What is the reason behind this phenomenon and what is it called?


The variational method, as it pertains to quantum mechanics, is used to approximate the energies of a particular system. Is it possible to use the same variational method to approximate the eigenvalues of other observables?


I have been searching in vain for statistics about the runtime of pdfTeX, LuaTeX and XeTeX. My impression is that (ordering by increasing runtime), LuaTeX > XeTeX > pdfTeX, but exactly how big is this difference?


In referring to a particular femme fatale, I described her "femme fataleness." That is "ungrammatical" (I believe) but got the point across. Is there a correct term for this particular attribute? If so, what is it?


I'm struggling to understand how to define multiplication and addition, now that I've been told that multiplication is not just repeated addition. It seems that the axioms for the two are identical, save that multiplication is said to not have an inverse for the additive identity. Doesn't this imply that multiplication cannot be defined without an addition?


I know that any Riemann integrable functions have at most countably many discontinuities, and similarly, any bounded functions with countable number of discontinuities are Lebesgue integrable. But is it necessarily true that any Riemann integrable functions have at most countably many discontinuities? Thanks


I see that you can use a Stern-Gerlach apparatus for massive particles that can be deflected, but that doesn't work for photons. What would you do instead to measure photon spin in a lab?


I would appreciate it if you could explain to me the difference between implementing and executing in the context of the following sentence: The implementing agency(ies) for this project is UNEP and the national executing agency is the Department of Environment.


Can you please clarify for me which is the correct answer to the following question: Do you often have visitors? a. Yes, pretty much every day b. Yes, quite often c. Not very often.


As the title says it, I want to know how people find the work function of a metal without knowing the threshold frequency. Yes I've already searched on Google but I didn't find anything really relevant. I just found how to calculate Work function with threshold frequency.


I was given a piecewise function, which happens to be continuous on some interval. It asks me to prove that it's Riemann integrable. I know that continuous functions are Riemann integrable, but is there a way to prove it for some specific function (without using that theorem that continuous functions are Riemann integrable)?


For any graph G, prove that the line graph L(G) is claw-free. I have a fairly good intuition for this one but it's hard to put into words. I really need help with this one! I feel that I should use the pigeonhole principle...


This integral comes from a physics book when calculating a field of an uniformly charged sphere (without Gauss' Law). It says that it can be done by partial fractions, but I cannot imagine how.


Question: Prove that at any given moment, there exists a pair of diametrically opposed points on the equator of the Earth that have the same temperature. Use the Intermediate Value Theorem. Now I've been kinda struggling with how to deal with the Intermediate Value Theorem and I have no idea where to even start on this one.


You can visualize a first-order Markov chain as a graph with nodes corresponding to states and edges corresponding to transitions. Are there any known strategies to visualize a second-order Markov chain, i.e. where transitions have two past elements and one future element?


In graph theory, what is the difference between a cycle and a simple cycle? My impression is that a simple cycle is the same as a cycle except that you cannot repeat vertices. Is this correct?


I have been searching in the literature, with no results, for the (Zero Field) Magnetic Susceptibility of BiSCCO, but I cannot find any article regarding that. Does someone have a reference that might be useful to me? Thanks a lot.


Consider a recursive Mergesort implementation that calls Insertion Sort on sublists smaller than some threshold. If there are n calls to Mergesort, how many calls will there be to Insertion Sort? Why? Wouldn't this depend on the threshold, or is there another way to look at this?


Does "feel committed to" require an infinitive or gerund complement? For example, which of the following is grammatical? I feel committed to following up on that. I feel committed to follow up on that.


Even if Lebesgue measure and integration can be defined without using the extended reals, it might not be a very helpful formulation. My motivation for asking this question is that I'm curious about in which ways Lebesgue integration is an extension of Riemann integration, which doesn't necessarily use extended reals.


The Cayley table tells us whether a group is abelian. Because the group operation of an abelian group is commutative, a group is abelian if and only if its Cayley table is symmetric along its diagonal axis. Sorry, but why is this true?


What is the difference between pull over and pull away? I am still trying to get used to American English. It seems like if I do not understand the driving vocabulary I am going to fail in the driving test.


What's the difference in the usage of these two adverbs continuously and continually? Here are some examples: a) She was told off for continually being late. b) It rained continuously for three hours this morning. c) The firemen worked continuously through the night. d) I'm afraid you continually make the same mistakes.


Does adding the word "ok" to the end of a statement imply a choice instead of a command or order? Example: "Wash the dishes now". as opposed to "Wash the dishes now, ok?" Does this weaken your statement?


The four major cities in the Netherlands, i.e. Amsterdam, Rotterdam, The Hague and Utrecht belong to the Randstad. What is the English translation for the Dutch word Randstad? I have found downtown but I wonder whether this is the correct translation.


I was wondering about discrete metric space. Is it a proper metric space? I am trying to find some example of non proper metric space. I am not sure whether discrete metric space is proper or not.


Possible Duplicate: Why is matter drawn into a black hole not condensed into a single point within the singularity? When we speak of black holes and their associated singularity, why is matter drawn into a black hole condensed into a single point within the singularity?


I've heard a lot about the CJK package, so I want to give that a try. There is, however, no documentation for that package (see the link), nor do the available documentation files at CJK's website tell me anything about how to use the package. So how do I figure out how to use this package?


In my physics textbook it describes the events at the beginning of the Universe. I'm confused about the order at a certain point. It says that at some point primordial helium is created, then it says that later atoms are formed. Isn't primordial helium made of atoms? Thank you :)


Will we ever come up with non-gender specific pronouns and possessive pronouns for the English language? It seems that there are many new words in the English language every year, so why not gender-neutral pronouns? This would solve many problems, in literature and in everyday life.


Which would be better to say? He reminds me of Dill from To Kill a Mockingbird. He reminds me of Dill in To Kill a Mockingbird. He reminds me of Dill of To Kill a Mockingbird. Also, which is more appropriate in a formal context?


Possible Duplicate: What are your favorite English language tools? I have a question regarding the English language that is too general for this site according to the faq. OTOH, I don't know where to look for the answer. Is there a list of sites, books, etc. that are commonly used to answer questions?


While playing around with cyclotomic fields, I started to wonder about taking the roots of unity in higher dimensional analogues of the complex plane. Are the roots of unity well defined in the quaternions, octonions, and other hypercomplex algebras? Are there higher dimensional cyclotomic fields, and if so do they have any interesting properties like unique factorization or interesting morphisms?


Do the Law of Sines and the Law of Cosines apply to all triangles? Particularly, could you use these laws on right triangles? That is, could you use these laws instead of the Sine=opposite/hypotenuse, Cosine=adjacent/hypotenuse, and Tangent=opposite/adjacent rules to solve right triangles? I can't find this stated in any of my textbooks, nor has my instructor said anything about it, which I find odd.


Is there any difference between the following two statements: I have to pay bills I have bills to pay Could you please tell us the difference between the above two statements and when to use them?


I've always found it irksome when people form sentences such as "I'm done my homework." It is of my understanding that you are never done something, but rather you are done with something. Am I right to be bothered by this?


I'm told at school that the Electromotive Force (e.m.f) of a battery equals the potential difference between the terminals of the battery when there is no current. How is that possible? How can there be a potential difference with no charge flowing?


How do you exactly define what is and isn't a dimension? I heard somewhere that it is "anything you can move through" but if that is right, why wasn't time and space considered a dimension before Einstein?


Connes's non-commutative geometry program includes an approach to the Standard Model that employs a non-commutative extension of Riemannian metric. In recent years I've heard physicists say that this approach does not hold significant interest in the physics community. Is this, in fact, the case? If so, why? I do not mean for this question to be argumentative, but instead would like clarification.


We were taught at school that if we want to make something plural, we should add "s" at the end. Later on, I learned that there are uncountable nouns. Not so much familiar with it. Can you please give some examples?


I learned how to use bibtex from the LaTeX wikibook, found here. They say that to compile a LaTeX file that uses bibtex, you need to call pdflatex once, followed by bibtex, followed by two more calls to pdflatex. I'm wondering: what's actually going on at each of these steps? Why do I need to run pdflatex and bibtex in that order?


If physics is the study of the physical universe, wouldn't that mean chemistry, biology, neuroscience, zoology, history, economics, sociology, etc, wouldn't they all be subfields of physics? I believe they are. If anyone can convince me otherwise with a good argument, the answer will be accepted.


I need to figure out this property of Hermitian / Orthogonal projections "A is a Hermitian projection if and only if it is an orthogonal projection" Your assistance will be highly appreciated. Thank You


Photons travel at the speed of light. Is there a known explanation of this phenomenon, and if yes, what is it? Edit: To be clearer, my question is why do photons travel at all. Why do they have a speed?


I'm having some problems determining which of the following is more grammatically correct: The season that I like the best is Winter or Winter is the season that I like the best I know that both are technically correct, but when would you use one over the other?


One of the requirements of my English final includes removing all prepositions from a previously written essay. I'm having trouble getting rid of prepositions like "to", "in", "of", and other common preposition that seem like a requirement in writing. Could someone point me in the right direction? Thanks.


I recently came across this proposition that the eigenvalues of a product of square matrices are invariant under cyclic permutation of the product order. Is there perhaps some group theoretic way of proving this proposition? I've tried a few cases and it seems to be true, but a direct proof has proven elusive. Or is a proof not so simple?


I've read that the characteristic function of a probability distribution always exists because it's bounded. However, the characteristic function is still Taylor expanded in terms of the moments of a given probability distribution. Given the the moments don't necessarily exist, why is it that the characteristic function still exists?


I'm trying to wrap my mind around the semantic differences between the words "Register" and "Apply". For example, are there any real differences between these two sentences: "I will apply for this event." "I will register for this event." What is the best way to think about the meanings between the two words?


From my experience, it seems that although unstable is more commonly used, instable is often preferred in engineering and scientific contexts, e.g. "aircraft instability", "instable algorithm". Are there any differences in the implied meaning of the two terms? Should unstable be preferred?


I saw a few groups on the web. A group's name is "Non alcohol ~" and another one is "Alcohol free ~". But I don't know the difference between "non alcohol" and "alcohol free". I think that "non alcohol" and "alcohol free" have the same meaning. Am I right?


In fluid mechanics the terms 'pure straining flow' and 'pure rotational flow' are often thrown around (see e.g. here). I have tried looking for a formal definition of these two but have not been able to find one (or in fact anything you might call a definition). So how can we formally define these two terms?


In the spin Hall effect, electrons with different spins accumulate in opposite directions without any external magnetic field. Broadly, the reason behind this is the spin-orbit interaction. But, the question is how? How does the spin-orbit interaction leads to this. A physical explanation will suffice.


Is there any connection between fluctuation dissipation theorem and Kramers-Kronig relations? They are often described together under linear response theory but I do not see any exact connection (like one being special case of another).


The force of gravity is constantly being applied to an orbiting object. And therefore the object is constantly accelerating. Why doesn't gravity eventually "win" over the object's momentum, like a force such as friction eventually slows down a car that runs out of gas? I understand (I think) how relativity explains it, but how does Newtonian mechanics explain it?


I am trying to find the equation for the form that is in the picture. Basically it is an infinitely extending roller coaster loop. I just can not find the magic words in Google. Any suggestions? What is it called? What is the equation?


When referring to a U.S. state in a formal document, is it correct to capitalize "State," or should it remain uncapitalized? For example: This school is accredited by the State of Maryland OR This school is accredited by the state of Maryland


I have a doubt about the Hermite Spline. Is it the interpolation with the minimum value of curvature among the possible interpolative functions between two points? Is it possible to demonstrate it? Thank you very much!


Dirac delta function can be defined in several ways. I know two definitions. One is as a distribution and the other is as a measure. I found many materials on the derivatives of delta function as a distribution. However, I couldn't find materials dealing derivatives of delta function as a measure. Could someone point me at any materials or explain it?


Would you please tell me which of these four sentences is correct one? I am so confused with them :( Action movies are three times as popular as comedies for men. Action movies are three times more popular than comedies for men. Men watch action movies three times as many as comedies. Three times as many men watch action movies than comedies.


I'm aware that the Milky Way has a dark matter 'halo' around it, presumably a spherically symmetric distribution. But I'm completely ignorant regarding the theories explaining dark matter... Is there any reason to not expect a star-sized object to also be made of dark matter? I know they'll be extremely difficult to detect, but I'm wondering if it's even physically possible to exist.


How many triangles in this picture: I know that I can just count the triangles to solve this specific problem. I would be interested to know if there is a systematic approach to doing this that can be generalized to larger diagrams of the same type.


I'm having trouble obtaining the answer for this practice-test problem. I'm taking the volume of the whole cylinder as if it weren't cut then subtract the portion cut off. But I'm not getting any of the answers below. Any hints will be greatly appreciated. Thank you all.


After listening of some lectures of Leonard Susskind about black holes, he mentioned that conservation of information is one of the foundations of physics. After searching the web I cannot seem to find how we came up with this theory. Could someone explain how we know this is true and/or how did we come to this conclusion?


According to a book I'm reading, the Fourier transform is widely used in quantum mechanics (QM). That came as a huge surprise to me. (Unfortunately, the book doesn't go on to give any simple examples of how it's used!) So can someone provide one please?


I want to know if I can determine the time where the sun's azimuth is at a given value. In other words, I want a function that takes the sun azimuth along with longitude or time zone and outputs the time. Is there anything like this?


After reading definitions of universal and k-universal (or k-independent) hash function families, I can't get the difference between them. Also, I couldn't find any examples of hash function families being universal, but not k-universal (it's written, that k-universality is stronger, so they must exist). Could you please clarify the subject to me, or give a good piece of literature/articles to read about it?


A recent arXiv article measures the variation of gravitational potential in a local region around the solar system, and from that it tries to infer the mass density. Are there any valid counterarguments to their conclusion, i.e: that there is no dark matter near our vicinity?


I rarely need the ancillary files TeX produces when compiling a document (.aux, .log, .out, .synctex.gz). In fact, I've only ever had to read the .log, and that's only a few times. Is there a way to write them to a separate directory, so that my working directory only contains .tex and .pdf files?


I'm looking for a package in which you can create nice bar and/or line charts in LaTeX. Currently I'm plotting the charts in Calc (OpenOffice), export it to a PDF document, crop it and then including it as graphics. I reckon there should be some more efficient way of doing this, preferably entirely in LaTeX.


In special relativity, when two observers move in respect to each other, each observes the other's clock tick slower. An observer floating in space far from gravitational fields, observes earth clock ticking slower; at what rate is the clock of the floating observer ticking as observed from earth?


Where does the word "crack" originate from in the phrase "Give me another crack at that"? Curious to know if it's in reference to driving horses? Perhaps a derivative of "craic" in Irish? Or in a structural metaphor about material splitting? Cheers.


Do you use a question mark when a sentence starts as a question but then turns into a statement? For example, "Can you explain this to me, because I don't understand the second part." "What did he just say, because I couldn't hear him over the noise of the television." I usually use a period but don't know if that is correct.


I'm trying to use different distance metrics like Euclidean, Manhattan, cosine, chebyshev among other distance metrics in my k-means algorithm to calculate distances between the data points and the centers. In what situation would one distance metric be more useful over the other in a clustering scenario? [Comparing all the above mentioned distance metrics]


Let I be an open interval that contains the point c and let f be a function that is defined on I except possibly at the point c. Suppose that lim |f(x)| as x->c exists. Give an example to show that lim f(x) as x->c may not exist. Not really sure of an example for this?


I'm modeling the flight trajectory of a golf ball, and using angular velocity to calculate the Magnus force. Currently, I'm assuming angular velocity to be fixed throughout the ball's flight. How accurate will this be? How much will the angular velocity of the ball be affected by air resistance?


I got really confused about the graph of the relationship between wavelength and intensity of black body radiation. What does the peak stand for? And what does the graph tell us? How can we analyze it?


How would I go about computing the volume of an ellipsoid with spherical caps removed and a cylindrical hole through it? I'm thinking about finding dV of a cross section (the ellipsoid has a circular horizontal cross section). Would this be the best way to go? I need to find if the volume is dependent on the radius of the sphere.


I'm looking for a proper English word to say "conversational partner". Is "interlocutor" a word an average person would understand? Are there any other words or shorter expressions which could describe a person you are having a conversation with, but who is not your friend and is basically unknown to you?


Is there a polynomial time algorithm that gives the extreme point as output for which objective function is minimized/maximized ? I am not looking for any solution that minimizes/maximizes the objective function, but an extreme point of the feasible region for which objective function is minimized/maximized.


I have read that in projectile motion both vertical and horizontal components are independent of each other but i don't get it that how it is possible i think that they are dependent. If they are not could someone explain with a mathematical relation.


On a yoghurt advert, the voiceover claimed that you have infinite combinations with it. However, given that there is a finite amount of matter, is it possible to have infinite combinations with the yoghurt or just a very large number of combinations?


In the Lagrangian path-integral formulation of QFT, an anomalous symmetry is defined to be a symmetry of the action which is not a symmetry of the measure of the path integral, and therefore not a symmetry of the partition function. How do we define an anomalous symmetry in the Hamiltonian formulation of QFT, where this is no path integral or partition function?


Speaker A: We want to ask you some questions. If you don't mind, of course. Speaker B: [He opens the door of his house] Sure, be my guests. Is the idiom being used correctly? If not, what would be a better alternative?


He missed out several important facts. He omitted several important facts. I think these two sentences are the same meaning. Is there a slight difference? A few days ago, an American native speaker explained to me that "omit" means intentionally take out whereas "miss out" means mistakenly take out. Is it true? Could anyone clarify the difference to me?


There are several PR firms, lobbying groups who specialize in mud slinging, making wild allegations on behalf of their clients. Is there a word that describes such a person whose job is to make allegations against others?


I looked all over but could not find anything. Is there a package out there that has a symbol/dingbat of a calculator? I know there probably isn't, but I've seen much stranger stuff out there before...


Suppose there are two light beams. One is red while the other is violet. The energy of both is the same. Which one of these beams has a larger number of photons, or is the number of photons relevant?