I recently read an article in the NY Times called A Black Hole Mystery Wrapped in a Firewall Paradox. I really liked the article, but reading one quote immediately made me think of asking Physics.SE a question: From the material in the smoke and flames of a burning book, for example, one could figure out whether it was the Bible or the Kama Sutra; the same should be true of the fizz and pop of black holes, these physicists argued. So, Physics.SE, theoretically, how would you figure out what book was burned from the smoke and flames? I do not intend for this to be a silly question. If the idea of "information loss" inside of a black hole is so difficult for some physicists to come to terms with, I figure those same physicists should have a perfectly good answer to my question.


To me (an American), "what to study in college" sounds acceptable. Meanwhile, "what to study in university" sounds wrong. This suggests that these words have different grammatical attributes. This is shown somewhat in the example sentences on m-w.com: http://www.merriam-webster.com/dictionary/university I applied to several public universities. He lives near the university. http://www.merriam-webster.com/dictionary/college She teaches art at a local college. He graduated from one of the country's best colleges. She attended a business college. He attended college for several years, but didn't graduate. She dropped out of college. I went to Mount Holyoke College. When I was a junior in college, I spent a semester in Spain. the Edinburgh College of Art the London College of Fashion She is attending fashion college. Replacing the non-proper noun 'college' examples with 'university' doesn't sound right in many cases. He attended university for several years, but didn't graduate. She dropped out of university. When I was a junior in university, I spent a semester in Spain. Placing an article in front of 'university' does make it sound better. He attended a university for several years, but didn't graduate. She dropped out of the university. Meaning aside, what is the difference between 'college' and 'university' that suggest a different sentence structure to make it sound "better". While, Difference between "college" and "university" looks at the difference between the meanings, the question doesn't ask nor do the answers address the perceived grammatical difference.


The answer to the question "Could it be that Goldbach conjecture is undecidable?" claims that it is possible for something such as the Goldbach conjecture to be undecidable, meaning that assuming that it is true and assuming that it is false would both lead to no contradiction. But if it is undecidable, then, if we assume that it is false, it would mean that exists an even number that cannot be written as the sum of two primes. If a natural number exists, then it can be written down using a finite number of digits (any natural number is definable). This means that that number exists whether or not the conjecture is true, so if we assumed that it was true, there would be a contradiction, so it therefore can't be undecidable. What is the flaw in what I just said?


Because of English's lack of a gender neutral third person singular possessive pronoun, whenever the need for such a referent presents itself in the course of writing, we seem to be left with alternatives that are either cumbersome or otherwise awkward. There is the informal gender neutral "himself", and the informal singular "themself", and of course there is the more formal "himself or herself" which is both grammatically and politically correct but has the disadvantage of being incredibly annoying to write very quickly. Are there any other ways to truncate this expression, particularly (but not limited to) ways that stay within the bounds of standard correct English usage and grammar? For example, I thought of shortening it to "his or herself", but upon second thought this feels akin to what mathematicians would call can abuse of notation.


When calculating the ideal class group of a number field, it is common to start with the Minkowski bound, followed by decomposing finitely many prime ideals of norm less than that bound, and finding relations between these primes. Is there a way of avoiding the use of Minkowski bound in the computation of the ideal class groups? For example, could we use some exact-sequence (say) to show some isomorphism of the ideal class group with some other familiar group? Or maybe the Artin reciprocity isomorphism can aid us in this direction? Or even, per chance we can avail of some suitable resolutions for the computation of some cohomology groups? As to why one wants to avoid the use of Minkowski bound at all, I just think that the idea of this bound is quite analytical, and there might be a way of algebraically calculating the ideal class group. I googled and searched this site, but didn't find anything useful. The site I found that talks about the computation of the ideal class groups, either views the Minkowski bound as a fundamental ingredient of its arguments, or uses the class number formula for imaginary quadratic fields, which I would like to avoid as well. Any hints, references, or ideas are welcomed, thanks in advance. P.S. This it not to say that I want to avoid all results of the geometry or analysis, just that I want to know if there are any results in this direction.


This is a data sufficiency question - Q - How is A related to B? Statement I. B is the only son of D who is the daughter of A's father. Statement II. B is the father of C and is the only son of A's mother. A - I. If statement I alone is sufficient to answer the question. II If statement II alone is sufficient to answer the question. III If the data either in statement I or statement II alone is sufficient to answer the question. IV If both the statement together is not sufficient to answer the question. V If both the statement together is necessary to answer the question. MY ATTEMPT I think the answer will be Option I. From statement I, I got A is the uncle of B From statement II, I got A is the sister or the brother of B But I am not sure if I am right. Please help me solve this question.


I have a question related to this: Projective modules I'm trying to understand the "philosophy" of the statement, because it seems too similar to the statement "a module is free iff every element can be written uniquely as a finite linear combination of elements of a basis". Is this "projective basis" property saying this: a module P is projective iff every element in P can be written as a finite linear combination of some elements of P? We lose uniqueness in the expression as a sum: in the elements of P, in the elements of R, and in the number of terms (so the concept of "rank" wouldn't make sense). Is this all, or am I misunderstanding the statement? Any other intuition related to that property is also appreciated.


I learnt about an experiment to show that acceleration is proportional to force. It was done by placing a trolley (like a toy car) on a smooth track. At the end of the smooth horizontal track was a pulley connected to the trolley by a string which hung masses over the edge of the table. So the trolley experienced acceleration due to the weight from the masses. My problem is in what my textbook says to do next. They say you have to take a mass being hung over the edge, which results in less force applied to the trolley. But then they tell you to put the mass onto the trolley, to ensure constant mass in the system. The problem is in placing the mass onto the trolley. Surely the trolley's mass should stay constant throughout, since that is what we are measuring the acceleration, and not the system as a whole? I hope you can explain this to me, or if my book is incorrect. I'll expand on the experiment if you find my description confusing.


Once again, a problem encountered while marking German pupils' exams. We teach them the following rules: A present participle can be used to shorten an active relative clause: The boy who was driving the car didn't stop = The boy driving the red car didn't stop A past participle can be used to shorten a passive relative clause: Strawberries which are grown in California are delicious = Strawberries grown in California are delicious. While marking, I encountered several problems. For example, why does this not work: The girl who has black hair is in the corner NOT The girl having black hair is in the corner or That's the man who is happy to be here NOT That's the man being happy to be here Does this all have to do with: the verbs have and be? (But "The girl, being happy, phoned her friend") the continuous and simple forms? (But "We help people who live in ghettos = we help people living in ghettos") the tenses? Or what? I'm totally stumped by this problem and do not know how to explain the pupils' mistakes to them. The problem seems to occur mostly with the use of the present participle. Who can help with some explanations or even better, specific rules!


I recently stumbled upon an interesting quirk regarding words that are both nouns and verbs. They seem to all follow the same stress pattern. Here are a few examples: NOUNS I have a really long address. There is a huge contrast between winter and spring. Not a single object is blue. I'm not very good at creating produce. VERBS Make sure you address him properly. I try to contrast the two twins in my head. He will object to any change you propose. Produce the paper right this instant! Why do the nouns have stresses on the first syllable and the verbs have stresses on the last syllable? Is there a good reason for this, or is it just coincidence? These are just the examples I thought of - I'm sure there are more. There are also some "noun/verb"s that have the same stress: That was a huge surprise! Next time I'll surprise you! But I've yet to find a counterexample - one where the noun has an ending stress and the verb has a starting stress.


Measurement of a quantum observable (in an appropriate, old-fashioned sense) necessarily involves coupling to a system with a macroscopically large number of degrees of freedom. Entanglement with this "apparatus" takes care of the decoherence. It is often said (I can provide references upon request) that the remaining problem is the one of "selection", and this is the point where one invariably invokes something philosophically radical, like many-worlds interpretation. In the above (pretty standard) context, I am trying to make sense of the following observation. Looking at the measuring system from a statistical mechanics point of view, it seems that triggering a particular macroscopic outcome requires spontaneous symmetry breaking via a (thermodynamically) irreversible transition of the "apparatus" from a metastable to a higher entropy final state. My attitude is that "statistical mechanics point of view" is not far from "decoherent large quantum system". So, the question is: Is it fair to say that statistical irreversibility ("the second law") and quantum measurement irreversibility (the "wave-function collapse") are necessarily linked? Can this link be made more concrete (e.g., traced in details in a particular model)? Can you give references to approaches to the measurement problem that explore this connection?


I am studying for the AP BC Calculus Exam and I know about the free response questions from AP Central, and the Multiple Choice Collection. I was wondering if anyone here knew of where to obtain more problems? At least a collection of problems similar to those that appear on the BC Exam. I realize this is not a question about an actual math question, but I don't feel very confident in my math ability so I want more problems to solve. Ps. I don't have alot of cash, so I can't afford to buy barron's books or any test prep books. The local libaray doesn't have any -someone check the book out but never returned it, and the book store doesn't like it when you just sit there working out problems and you don't buy the book.


From what I understand, in simple terms, The definition of iteration : The act of repeating a process The definition of recursion : The act of repeating smaller process of the same problem It these definitions aren't too far fetched, it looks to me that a recursion is a type of iteration. But I am yet to find a reliable source to confirm it. So my question is, is recursion is a type of iteration or I am comparing apple and orange? The premise : In the process of learning computer programming, a book is introducing me to recursion. I understand the basics and I know how it works from a programmers perspective. But I don't understand why they aren't introducing recursion as a type of iteration. They are introducing iteration and recursion as two different concepts. Why so? (I've scoured through math.stackexchange and stackoverflow, but yet to find a clear explanation to my question.)


I don't want to re-invent the wheel here, and I know that there are a lot of good math libraries out there for all sorts of things; what I'm wondering is if there's one that generates its answers in LaTex? (Could be any sort of TeX; I'm not really familiar with which ones are used for what) Well, it could look like anything (just linear algebra, just calculus, just physics or chemistry, etc.) but it would be nice if it were an application with its own GUI that generates copy-and-paste LaTex code from different inputs and a whole set of operations at the user's disposal. It doesn't have to be that great, though. I wouldn't mind coding this myself, but before I embark, I'd like to see what my options are so that I know where to start.


I'm going to start self-studying General Relativity from Sean Caroll's Spacetime and Geometry: An Introduction to General Relativity. I'd like to have a textbook on Differential Geometry/Calculus on Manifolds for me on the side. I do like mathematical rigor, and I'd like a textbook whose focus caters to my need. Having said that, I don't want a exchaustive mathematics textbook (although I'd appreciate one) that'll hinder me from going back to the physics in a timely manner. I looked for example at Lee's textbook but it seemed too advanced. I have done courses on Single and Multivariable Calculus, Linear Algebra, Analysis I and II and Topology but I'm not sure what book would be the most useful for me given that I have a knack of seeing all results formally. P.S: I'm a student of physics with a mathematical leaning.


Four suspects were assembled in the director's office, having been accused of a devious crime: turning off the light switch during Mr. Buehler's business presentation. It was known that only one of the four turned off the switch. All four were friends, and the director's secretary overheard them plotting before they were brought into the director's office. They all agreed to tell the same number of false statements, although the secretary did not hear the agreed-upon number. Their statements are below. Who turned off the light switch? Joe: -Frank didn't do it. -I went to college with Felipe -I didn't do it Felipe: -I didnt do it -Joe didnt go to college with me -John didnt do it John: -We all agreed to tell one false statement -i didnt do it -Felipe did it Frank: -We all agreed to tell two false statements. -Felipe didnt do it - i didnt do it


I would like to know if there is any physical significance associated with the derivative of a quantity with respect to proper time or is it just a mathematical trick. Since proper time is measured in its "rest" frame of a moving particle, it seems to me that particle is not going through any dynamics and therefore time derivatives should be zero in the rest frame. I understand that we use derivatives with respect to proper time to keep things Lorentz invariant....but that sounds more like a mathematical requirement rather than something of physical significance. Example: what is the physical significance of four-velocity. I know four velocity is tangent to the worldline but I find it hard to remember through physical intuition. I always have to go through a book to find its definition. Kindly excuse my ignorance.


I am looking for a word that describes audio that does not contain words. For instance: John William's piece Duel of the Fates would be this, since they are just vocables for their musical effect, similar to the way any other instrument is used in that context. A recording of machine gun fire would be this, since there is no linguistic meaning. A song where someone is singing would not be this, because there are words with actual meaning. A book recording would not be this. Speaking in Tongues by the Talking Heads, and the main theme of Close Encounters of the Third Kind are sort of borderline, I'm really not sure which side they would fall on. Does anyone know of a word or phrase that would describe this category of sound?


I know that string theory is still under heavy development, and as such it still can't make predictions (or not that many predictions anyways). On the other hand, it is clear from the number of years this theory has been under development and from the large number of theoretical physicists studying it, that it is considered a good and viable candidate as a quantum gravity theory. So, what is the evidence that this is true? Why is it considered such a good candidate as the correct quantum gravity theory? Without wanting to sound inflammatory in the least, it has been under heavy development for a very long time and it's still not able to make predictions, for example, or still makes outlandish statements (like extra dimensions) that would require a high amount of experimental evidence to be accepted. So - if so many people believe it is the way to go, there have to be good reasons, right? What are they?


We've learnt that friction is the opposition of motion and that friction appears the instant a force is applied on an object i.e when an object is at rest (with no force acting on it) then there is no frictional force. The moment a small amount of force is applied, friction becomes a factor. Therefore, friction is just the "equal and opposite" force between two bodies. Now, let an object be accelerated to a velocity 'v'. Then, let the acceleration cease. Ideally, the object will come to a stand still. However, if the acceleration is zero, doesn't that mean that there is no force => there will be no "equal and opposite" force i.e frictional force. And, only if there is an opposing force will there be retardation. Obviously, my reasoning is flawed, if not then an object that has been accelerated to a velocity will continue to move at a constant velocity. However, I don't get where my reasoning is flawed. Please do help...


I probably have seen this happen at various times in movies set in eras where people were very obsequious to royalty. The action I am trying to find the word for is a motion of the hand in a kind of circular motion or spiral (generally towards and away from oneself) while bowing towards someone, often while slowly moving backwards. My first thought was "genuflect" although to me that invokes the image of someone making the sign of the cross (although the dictionary doesn't seem to mention this, or in fact any hand movement - it seems to be closer to bowing then the hand movement) and I am looking for a word without any religious connotations. My second thought was "flourish" as that seems to be technically correct to some degree in the sense of "a bold or extravagant gesture or action, made esp. to attract the attention of others", but this word seems to have too many meanings that might confuse, and I am not sure anybody would recognize what I was talking about unless I put a lot of context into the sentence with it, or maybe even if I did!


I'm attempting a novel approach to some tough integration problems. I'm using the idea of series expansions to help integrate. In other words, I will attempt to approximate integration by integrating the series expansion of an integrand, rather than direct integration or standard numerical methods. I believe I can approximate integration of a series very easily, compared to the other methods. However, there's a catch. I will use at least two different series expansions. One for the lower limit of integration, and one for the upper limit. Now, when I attempt to integrate these expansions, the constant of integration comes into play, and it's not obvious what it is. Since I am using at least two different series expansions, the constant of integration may differ for each expansion. So I'm wondering if there is an easy way to get the constants of integration without much more work. Any help, ideas, or suggestions are welcome. EDIT A few additional notes... I know ahead of time that the series will converge. I consider that I could integrate in sections, like quadrature, while still using the series to aid in integration. However, I am considering the idea of only using only the endpoints, with two different series. So the constants of integration would be different for each series. If I could somehow find them or find how they differ relative to one another, that would save me the trouble of breaking the integral into sections and using something akin to conventional numerical methods.


In an earlier post - Phonetic understanding of tongue twisters - a comment was made that "hyphens ...(are) ...not needed in speech, so they must be extraneous". The phrase prompting this assertion was 'state of the art'. What does it mean to say that hyphens are not needed in speech? No one would say state hyphen of hyphen the hyphen art, of course. But when I say "This is a state of the art paper on tongue twisters" I make a point of saying the words 'state of the art' as a group and slightly apart from the run of the words on either side. If I didn't, and spoke the words in the same rhythm as the rest of the phrase, the meaning can easily be lost (and the sentence is certainly harder to read meaningfully at first sight). There is good reason to use hyphens, or some other notational device, in such cases, isn't there ? This is a state-of-the-art paper on tongue twisters This is a 'state of the art' paper on tongue twisters This is a state of the art paper on tongue twisters


In my Computer Science class, we were introduced to the Average Salary problem, where a group of people want to determine their average salary, but they don't want anyone to be able to determine the salary of anyone else. I proposed a solution which I later looked up and found to be a fairly common one, wherein everyone writes down several numbers on separate pieces of paper that add up to their salary. The papers are then collected in a hat, totaled, and divided by the number of people. My professor said that he was looking for a solution that only involved direct communication as to avoid the use of "trusted hardware". However, he also told me that my solution was flawed because some information is unnecessarily revealed, and, I must assume for the sole purpose of tormenting me, he said we would go over it later when he revealed the solution to the rest of the class. He also told me my solution was still inadequate when I said that the numbers could be both positive and negative, and everyone was to submit an arbitrary amount of numbers. My question is not what is the ultimate solution to this problem, but rather, what is wrong with mine? What information could be revealed from arbitrary numbers that when added up equal the total salary?


I am still trying to get a good grasp on the motivations behind various concepts in Differential Geometry. But I am struggling to come to terms with how certain concepts have this added attribute of being coordinate independent? How does one identify such objects, be it a tangent space or a covariant derivative. How does one go about trying to prove that a certain geometric object is coordinate independent? How is coordinate independence a part of the "geometry" of a given surface or is it? P.S.: Actually is the concept of a coordinate system part of the intrinsic or extrinsic geometry? I think its the former, but sometimes embedded spaces tend to make me think twice. Edit: I would appreciate if the covariant derivative could be used as an example.


I am going through the chapter on compactness and completeness from Sternberg's Advanced Calculus and trying to build an intuition for what many of this topological properties mean, and which imply which. The book defines these concepts in the setting of metric spaces, but most of what I found online is in the about topology, and from what I see (correct me if I am wrong) it doesn't change the general picture much. I've made this diagram to see whats the relationship of the different concepts and have examples of each. I don't know if it is correct. For example, Is it true that a bounded complete metric (sub)space is compact (and therefore totally bounded)? Then why bother defining total boundedness? If you don't think anything else I wrote is a valid question stick to answering that, although pointing out any misconceptions I might have is appreciated. For intuition about compactness I've found this posts really helpful. It helps me to think that there are (at least) two different kinds of infinity: one in the sense of largeness (of which boundedness is the opposite), and another in the sense of denseness (of which discreteness is the opposite).


My question concerns the theory proposed in this classic paper by Misner and Wheeler. In the paper, the authors propose the idea of "charge without charge"--namely, that positive and negative particles might really be the ends of a wormhole, with field lines going into a mouth interpreted as a "negative" particle and the outgoing field lines at the other end as the "positive" particle. However, I noticed that the paper didn't mention whether or not the wormholes were traversable. If the wormhole was not traversable by an external material source (i.e., they have unstable inner horizons, have curvature singularities, etc.), then could the field lines technically travel through the wormhole's neck, or would they, too, be blocked? I know that the "charge without charge" idea isn't probable due to the tiny wormhole tunnels collapsing to form black holes, but my question also concerns larger wormholes as well. Thus, could field lines travel through a macroscopic non-traversable wormhole?


I am wondering what some applications of POVMs are in mathematics (or mathematical physics)? I am going through Berberian's 'Notes on Spectral Theory', which shows how we can write a normal operator as an integral over a spectral measure. Because it is not that much extra work, he treats operator valued integrals in generality, allowing for integration over a POVM. As it is however, I can't find any examples or motivations for integrating over POVMs. In quantum mechanics I have come across using POVMs to represent the most general form of measurement, but in that case a POVM is defined as a series of positive operators which sum to the identity. I suppose if you allow for a continuous range of results then this sum would become an integral, but is there anything else to it?


I'm having a bit of trouble explaining to a friend whether or not there's a big difference between the three modifiers in the title. Same and very on their own are different enough, but when combined, I find it difficult to draw a proper line on their meanings. Consider the following: I lived in the same house you're talking about. I lived in the very house you're talking about. I lived in the very same house you're talking about. Here, I understand there is a nuance in sentences one and two, though I have trouble explaining just what it is. "The very same" sounds like "the exact one", but wouldn't that be what "same" means anyway? Plus, that last sentence truly boggles the mind. How do you explain the grade of intensity expressed in sentence three? How do you explain each modifier?


I'm trying to translate a video on TED into my native language (Latvian). At the very start there is an expression I'm unfamiliar with - "animal warmth". I think I kind of understand the idea intuitively, but I can't think of any similar expressions in my native tongue (short of direct translation). It would be nice if someone could explain the concept to me, or give some more examples of usage. Here's the context: ... Because in my family, reading was the primary group activity. And this might sound antisocial to you, but for us it was really just a different way of being social. You have the animal warmth of your family sitting right next to you, but you are also free to go roaming around the adventureland inside your own mind.


Is there a single word, or commonly-used term, to describe the act of baiting another person into calling bullshit, when in fact you're not bullshitting? Conceptually, this either a sub-type, or the direct opposite, of bluffing, which is the act of pretending you have better cards than you actually hold, in order to scare your opponent into folding. The word I'm looking for will describe (metaphorically) the act of pretending you have worse cards than you actually hold, in order to bait your opponent into calling you. I'm looking for a word which implies "actively misleading" (as in lying); examples might be a basketball player faking left (in order to make his guard shift left, so he can go around to the right) or generally faking out. This is not restricted to gambling (just as "bluffing" is not restricted to poker), I'm only using gambling terminology to make my meaning clear.


About two years ago, I absolutely fell in love with mathematics. Since then, I have studied math almost religiously, absorbing everything I can about every subject I can. I have now established what I would call an understanding of most undergraduate topics, up to intermediate complex analysis, some abstract algebra, multivariate calculus, etc. I really want to get into a good college for mathematics - specifically MIT. As a middle schooler, I have plenty of time before I have to submit an application, and I really want to make the most of that time. You can't exactly put down "was pretty good at math in middle school" on a college application, so I was wondering what things I can do now to get a leg up in the future, that I could put down in a college essay or in my application. So far I've started a mathematical blog, and I am working really hard to get published by the time I finish high school. My question is - what else can I learn/do/create to give myself a head start in college and in employment as a mathematician?


I recently came across the following problem from Paul Zeitz's book The Art and Craft of Problem Solving. Given the image below, can you find a way to connect corresponding blocks (i.e. A to A, B to B, C to C), without having any of the connecting lines intersect one another? The question was an interesting one for me, because for the longest time I was convinced that it was impossible, and when I finally became acquainted with the solution, it took me quite a while to "accept" it. Granted, I am not the sharpest tool in the shed, but upon introspection I also wonder if I am being hindered by the "intuition" I have come to develop, and implicitly "accept". I wonder if it would be a helpful exercise to perhaps go through experiences that help me dismantle this intuition. The most accessible way I can think of of undergoing such a process would be by reading helpful books, given my limited resources. While I think problem solving books such as the one I am reading right now is good for this purpose as a side-effect of its initial intention ("teaching how to problem solve"), I wonder if there are books that are geared specifically towards deconstructing and examining "intution"? Prospective answerers, please attempt to answer this refinement of the question instead.


I am writing an essay on a book that I read where many of the characters are not human and have artificial intelligence instead. When I try to describe these characters, though, I find myself using "android", "machine", and "robot," none of which seems correct. The characters are not humanoid, so I am not sure if android is the correct word for them (although I think it was used in the book). I also tried using "inhuman" and "nonhuman", but I feel like those may be too vague. I feel as if calling the characters "machines" sounds too insensitive since they express many traits that humans do and my essay is about how they are very similar to humans despite not being human. Does anyone have good synonyms for "android" or "robot" for me to describe these characters?


I'm trying to do Young's double slit experiment at home. Note that I don't have a laser, only a torch. I could get a bulb or use a candle though, if it helps I built the slits by cutting into a black chart paper with a knife. I tried to build a setup by placing a single slit, double slits and a screen one after the other, and shine a torch through the single slit. I tried varying distance between single-slit/double-slit and double-slit/screen which did not help. I did not observe any interference pattern. All I got was two parallel bright fringes instead, like the ones you would get by shining a torch through two very thick slits. I think that is precisely the problem, that the slits are pretty thick. What is the optimal slit width I should have to observe an interference pattern? How do I build that with a chart paper? I have seen videos of this experiment online, where people use pencil leads and hair(!!), but they both use a laser. I don't want to use a laser(just because I don't have one, and I'm probably too lazy to go and get one!). Note that you may suggest using any other materials that might be easily available at home.


I'm going to be a teaching assistant and I'm currently looking for books/reviewed articles/journals written by mathematicians or people who taught mathematics (at a university level) about pedagogy and/or their experience of teaching mathematics. I know that these readings can't replace the experience of teaching but I think sharing experiences can't be bad for my future students. To be more precise about what I'm looking for, I consider the following questions very interesting : What mathematical concepts are difficult for most students and why ? What are the basic errors of a new teacher ? What is a good course in mathematics ? What is a good exercise session ? I know that there are plenty of different answers of this question and I'm looking for different opinions to build mine. Thank you PS : References in French or English only.


As written in the title, there's a specific word that is not too common in English that's used to describe the feeling you can get when you finally resolve your long-term disdain for someone, or reaching some sort of civility between you and them. I can't remember it but I would know it if I saw it. Example: He attained [a feeling of] ____ after talking to his estranged father after many years. I'm not looking for synonyms of peace or closure, and nothing on Thesaurus.com has helped me. It is a very specific, unique word. It's like closure, peace of mind, or inner peace, but I've only seen it used in the context of sunsetting a toxic, detached relationship. Edit: The actual meaning of the word I was looking for was peace of mind for any period of negativity. It's not necessarily unique to interpersonal relationships.


I'm a physics undergraduate student who always enjoyed math, and briefly studied it at a university but for various reasons (laziness, youth) gave up and changed 'majors'. But I always wanted to go through an undergraduate math course in my own time, unconstrained by class, etc. Now that I've passed all my exams I was thinking of doing something over the summer. I had a look at Terry Tao's free lecture notes from an analysis course he taught and I was absolutely shocked at how good they are. I love the verbosity and how he motivates every bit of information. From what I read, he wrote an Analysis textbook which I intend to get. My question is, are there any other similar (in the sense of their exposition) textbooks for subjects such as Topology, Algebra (Linear and Abstract - from my brief studies I've come to believe that I'm an absolute algebra antitalent, but I'm hoping it's because I didn't have anything else than fairly dry lecture notes to study from, and let's be honest, I didn't study very much) and of course more advanced Analysis, Probability and Statistics?


Some actions (such as generating the Table of Contents) require two passes of the TeX compiler: during the first pass, some data get written to an auxiliary file, only to be retrieved during the second pass. Here are a few TeX.SE questions that require two-pass solutions: Highlight referenced equation number Backreferences for equations (To be completed... feel free to edit if you come across such a question) Two-pass stuff has piqued my interest; I have a few questions: Can I write (append) custom data to an existing auxiliary file (e.g. .aux)? Is that even a good idea? If not, can I generate my own auxiliary file (with a custom extension) to store/retrieve some data? What are good sources for learning the basics of writing to & reading from auxiliary files?


In answering Do these matrix rings have non-zero elements that are neither units nor zero divisors? I was surprised how hard it was to find anything on the Web about the generalization of the following fact to commutative rings: A square matrix over a field has trivial kernel if and only if its determinant is non-zero. As Bill demonstrated in the above question, a related fact about fields generalizes directly to commutative rings: A square matrix over a commutative ring is invertible if and only if its determinant is invertible. However, the kernel being trivial and the matrix being invertible are not equivalent for general rings, so the question arises what the proper generalization of the first fact is. Since it took me quite a lot of searching to find the answer to this rather basic question, and it's excplicitly encouraged to write a question and answer it to document something that might be useful to others, I thought I'd write this up here in an accessible form. So my questions are: What is the relationship between the determinant of a square matrix over a commutative ring and the triviality of its kernel? Can the simple relationship that holds for fields be generalized? And (generalizing with a view to the answer) what is a necessary and sufficient condition for a (not necessarily square) matrix over a commutative ring to have trivial kernel?


People use the phrase "x strikes a chord with me" to address enthusiasm or personal movement. I know there is another question that addresses what this idiomatic phrase means, but I'm very curious as to where this came from and when? I've searched a number of English dictionaries in hopes that a definition of the idiom or simply the word chord would be affixed with the origin; I started with the Cambridge English dictionary and proceeded from there. I also tried many fruitless Google queries. If someone could point me towards a reliable resource, I'd have no problem doing further research. I know music is a very emotional endeavor, so I could see the connection there--considering chords are a significant component of music--but this is purely a personal inference. Does anyone know this idiom's origin?


They had cooks and drivers, and guards who occupied a gatehouse, armed with machetes. Seeing as I had regularly petitioned my parents for an electric fence, the business with the guards strikes me as the last word in quiet sophistication. - David Sedaris, Me Talk Pretty One Day I have three questions about this sentence. First, I think "Seeing as" in the sentence is working as a conjunction, but I only get the meaning vaguely. Are there other alternative conjunctions for it? Second, the context is that the author is envying his boy friend's family ["They"] for having had servants such as cooks, drivers, and guards. My question is, what does "the business" mean here? I think it is more like "the story about the guards". Is that correct? Third, I do not understand what "in quiet sophistication" means. I looked up the dictionary for the word "sophistication", but I think the words put together make a new meaning or something.


I'm writing up my Teaching Statement for an Assistant Professor position in the sciences. Because all I do is read and write science, I have no elegance in my writing at all! I'm trying to make the last sentence sound better: I have been fortunate enough to work with some great mentors in my life journey thus far. Integrity, compassion, and selfless care for students were role-modeled for me on a day-to-day basis, teaching me things far beyond academics. It has been a life-long goal to attempt to replicate them. I'm trying to express that it has been a life-long goal of mine to pass-on what great mentors in my life have done for me to future students. They took me under their wings and provided important role-modeling, perspective, and guidance in my life when I was a drifting early-twenty-something male. Is there a nice and succinct way of writing this? Is there a phrase or an expression that has the same meaning as the highlighted text in the provided context? Is there a word that could replace "attempt to replicate them" to make it more concise?


I was considering honorifics and I realized that sometimes we include and sometimes we omit a possessive in front of them. I was wondering if there was a formal rule for such? For example: Your highness, the French delegation has arrived. vs. Highness, the French delegation has arrived. Obviously, the your has been omitted here (or perhaps elided). But, there are other honorifics where this is never done. For example: when addressing the mayor or a judge (in AmE), you might say "Your Honor", but you'd never say, "Honor". Rather, you'd say Mr. Mayor or Judge in those cases. Is there a rule to this, or is merely that your has been elided in the above example, and it should have been written with an apostrophe: 'Highness, the French delegation has arrived. Note: I'm deliberately ignoring the honorifics that never carry a possessive: Mr., Mrs., Dr., etc.


It is uncanny how many books will insist that neither 'many' nor 'much' can be used in positive sentences. Have you got many pens? / Have you got much money? --> correct I haven't got many pens. / I haven't got much money. --> correct I have got many pens. / I have got much money. --> INCORRECT And yet, those same books will invariably have a text where - lo and behold - 'many' is used in a positive sentence! I can only guess that proper grammar rules are as dictated, but every day use has drifted considerably from the said rule. So I ask you: what is the real usage of 'many'. Has it become common in any type of sentence? Or are there situations when 'many' can be used in positive sentences and situations when it can't? Because I really don't know what to say when the students point at a text and say it isn't following the rule they are supposed to follow.


I'm interested in others' suggestions/recommendations for resources to help me acquire reading proficiency (of current math literature, as well as classic math texts) in German. I realize that German has evolved as a language, so ideally, the resource(s) I'm looking for take that into account, or else perhaps I'll need a number of resources to accomplish such proficiency. I suspect I'll need to include multiple resources (in multiple forms) in my efforts to acquire the level of reading proficiency I'd like to have. I do like "hard copy" material, at least in part, from which to study. But I'm also very open to suggested websites, multimedia packages, etc. In part, I'd like to acquire reading proficiency in German to meet a degree requirement, but as a native English speaker, I would also like to be able to study directly from significant original German sources. Finally, there's no doubt that a sound/solid reference/translation dictionary (or two or three!) will be indispensable, as well. Any recommendations for such will be greatly appreciated, keeping in mind that my aim is to be proficient in reading mathematically-oriented German literature (though I've no objections to expanding from this base!).


My current background in analysis is approximately the material in Folland's Real Analysis. I've also read the Analysis text by Lieb and Loss and I also took a graduate level class on complex analysis, which went up to Big Picard and some Nevanlinna theory. For my own amusement I've thought about furthering my knowledge of general analysis. I've heard wonderful things about Stein's book on Singular integrals and his Fourier analysis on Euclidean spaces. Would these be an interesting next step? I'm especially interested in learning more about harmonic analysis and especially learning enough to understand the modern language of these fields. EDIT: Here's maybe a more concise way of phrasing this questions: What's the core knowledge that every graduate student in analysis, regardless of specialization, at a top school is expected to know? What would be a reading list?


When you're taking a mathematics class, you usually know exactly what sections of a book you need to know, and you can focus your time on these important sections. However, when studying by myself, even when I'm trying to study the book as thoroughly as possible, I often feel tempted to skip sections of material (maybe a subsection of a chapter, a proof, or an exercise set). Yet at the same time, I don't want to skip it, fearing that what I want to skip might be something really important. Some reasons you might want to skip are: You might feel that you already know it well enough The proof or the exercises might be too difficult or boring The section might seem not very important For instance, when self-studying from Apostol's Calculus, I felt the need to skip the section on calculating errors of the taylor series for the log function because it seemed unimportant, and the 'rigorous' proof of the FTC which seemed completely unmotivated. What are your strategies for studying material in a thorough, complete way? When is it best to skip -- and how should you determine if the material you're skipping is important or not?


I want to refer to parties that are hosted for players. Which of the following phrases is grammatically correct? "Player Parties" "Players Parties" "Players' Parties" "Player's Parties" A sentence where I might use this phrase is as follows: "We host [...]" Intended usage: I want to use the phrase in a promotional clip alongside a party that is taking place. The clip will show the party happening, and the phrase will appear over the video to describe the event taking place in the clip. The parties are put on for players to provide an opportunity for them to socialize, meet other players and relax. This reference cites the two main uses of the apostrophe that we are all familiar with (the possessive and to indicate omitted letters). In the context I have used the phrase, do the parties belong to the players? Which of the four above phrases would best match my intended meaning? This is most certainly not a duplicate of the question related by Edwin, I am referring to the specific case I have outlined above, which is dependent on the correct meaning to be ascribed to this phrase, as per my intended usage.


First of all, I want to make clear what I'm NOT asking. I'm not hoping to do a rehash of the implications of nonstandard analysis on calculus. Rather, I'm interested in its use in "harder" math. I'm currently reading through Goldblatt's Lectures on the Hyperreals and working on the later sections, wherein he discusses ways of rephrasing other areas of math in nonstandard language (e.g. Loeb measures). I'm trying to understand what the purpose of this is. I understand that nonstandard doesn't get us new results, that is there's nothing we can prove in a nonstandard framework that we can't prove over old-fashioned ZFC. I also understand that generally nonstandard allows us to see the spaces we work in "more intuitively", e.g. Loeb measures allow us to see Lebesgue measure in a more finitary light, but I don't have much of a sense for what this more intuition looks like when we're actually trying to prove statements. So what is the use of nonstandard analysis in its broadest sense? To those of y'all who study/use/teach it, what do you see it as buying you over "standard" analysis?


First time I've asked on this Stack; I hope this is on-topic. I'm laying out a control panel. One of its functions involves an alarm, but under certain circumstances the alarm might be triggered repeatedly, which is annoying. So there's a button which prevents the alarm happening for a while, which is labelled "[problem] Alarm Silence" - not entirely grammatical, but I'm happy with it in context. The alarm warns of a potentially expensive mistake, so you don't want to accidentally leave it silenced. So you should manually press this button again when silencing is no longer really necessary, and the control system will also do that for you if it notices you leaving the specific situation that causes multiple alarms. And in any case after a certain length of time. Underneath the button is an explanatory note which currently says: Silenced when lit. Press to re-enable. Also re-enables automatically. I think that's clear, but I don't really like the word "re-enable". Can anyone suggest a good alternative, bearing in mind the limited space available on the physical panel? All the replacements I can think of - "activate", for example - imply that they will immediately sound the alarm. That's not what happens; re-enabling just allows the alarm to go off if it wants to, it doesn't actively cause it to sound.


I recently started to study problems with prolate spheroidal geometries, for which prolate spheroidal coordinates are most suited. In particular I have the advantage that the problem is axisymmetric around the spheroid major axis. While I'm used to Spherical Harmonics expansions and also to solutions of Laplace equation in terms of Spherical Harmonics I'm not used to spheroidal coordinates and spheroidal harmonics. Specifically i'm looking for some reference on spheroidal harmonics, and how to expand scalar functions in terms of spheroidal harmonics. Do you have any reading to suggest me? Perhaps a book? I couldn't find anything useful with a (rather) quick search on google. PS I am an engineer so I don't want to go deep into the geometry and mathematical details of spheroidal coordinates and harmonics, i only need a way to solve a biharmonic scalar equation in these coordinates Thanks in advance.


In my physics lessons, my teachers have always been keen to tell my class that Jupiter is considered a 'failed star' by scientists. Is this true? In my own effort I wondered if maybe this could just be being regurgitated from an outdated physics syllabus that still considers the Solar System to have nine planets. From that thought onward, through my research on the Internet, I haven't found people referring to Jupiter as such and people always call it a planet rather than a brown dwarf. Furthermore, it's my understanding that brown dwarfs possess more mass than Jupiter suggesting to me that Jupiter possesses too little mass for fusion to even be plausible. So am I correct in thinking that Jupiter is 'only' a planet, or are my physics teachers correct in saying it is a failed star (and if so, why)?


I have read this text about a man who has spent a terrible holiday (in the island of Thassos) due to the disorganisation of the travel company. In fact the text consists in the complaint letter that he wrote to the bloke of the company... I report the passage that I can't understand: Over the years I have been on many holidays to Greece and I can safely say that, until this year, all of those holidays were wonderful. For example, I once spent six weeks on Crete. I loved that holiday so much that I have returned every spring for the last four years. Could you please tell me based on which rule is it necessary to use the present perfect (that I have put in bold) instead of the simple past?


I remember sitting in on a conference talk by a person (possibly Rainer Blatt) doing research with trapped ions (or single atoms strongly coupled to light in an optical cavity), and the person showed a photo of the trap with dots of light from the fluorescence of the single atoms/ions. I thought the person mentioned you could see this with the naked eye b/c the optical coupling to the ion in the trap was so strong, but thinking about it now I'm not sure if this can be true and I can't seem to find any (obvious) reference to this in the literature. So my question: Is it possible to see light from a strongly coupled single atom or ion with the naked eye? If so can you point me to a reference (and hopefully an image of this as well)? Note: The best I can find is the image below from the Blatt research group taken with a CCD (details here). However it is not at all obvious that this would be visible to the naked eye, or if the exposure was just set very high on the camera.


I am standing on the surface of some planet. Gravity is described via General Relativity with some static metric (e.g. the Schwarzschild metric, so static means no time dependence, but the metric may vary from place to place). I send a blue photon up to my friend, who is x meters above me in some tower (we are both at rest relative to each other). He measures the photon and finds out it is red. We both conclude that a gravitational redshift occured. However, where did the energy go? In GR there is no gravitational energy so the photon did not trade "light energy" with potential energy. I found several threads about this, but often they viewed this topic from a cosmological point of view where the metric does depend on time and thus Noether does not work to argue for a conservation of energy. Arguments without cosmology used the explanation via potential energy (which is not a thing in GR, as far as I know). So, since the metric is still time independent the energy should be conserved according to Noether. What is going on? Edit: On the Einstein thought-experiment in the linked question: This does not explain why energy is not conserved from a mathematical or physical point of view. This could also be viewed as a reason why you can not turn photons into matter (and vice versa) without losing energy.


Trying to name things in a computer data model. People have a variety of name roles, such as legal name, maiden name, etc. "The Sultan of Swat" is a nickname or pseudonym of George Ruth. It seems to stand alone. You don't often see him called "The Sultan of Swat" Ruth. "Babe" is also his nickname, but is often used together with other parts of his name, as in "Babe Ruth". It's more like the "Bobby" in Robert "Bobby" Kennedy. I've seen "diminutive" as well as "appellation" used. Are the two kinds of nicknames actually different? And if so, what terms to use? Update. A few more examples for clarification: Is Malcolm X a sobriquet, pseudonym, or just his preferred name? If an Asian student at a western college takes a more western name is that a sobriquet or preferred name?


I find it hard to comprehend the law of conservation of energy. Allow me to explain my confusion. I understand that the law of conservation of energy states that energy is neither created nor destroyed. However, it has to come to a point in time where the origin of that energy is magically 'created'. How do we explain that? For example, you can say that the energy in a falling ball comes from a human lifting and dropping it. Of course, that energy comes from food that we eat, and so on, all the way to the Sun. I know that some of you may be able to explain how the sun gets the energy, etc., but you get my point. I can go all the way back until a single point where you can no long go back.


I was copy-editing a report at work and came across the following sentence: While sustainability in the transport sector was rated relatively high, the sustainability of the power sector was found to be weak. The grammar nerd in me says this should be: While sustainability in the financial and transport sector projects was rated relatively highly, the sustainability of the power sector was found to be weak. Because after all, it feels more natural to say "The project was rated highly", rather than "The project was rated high." But for some reason, I would feel more at ease saying "The project was rated relatively high." and not "The project was rated relatively highly.". What is it about adding a "relatively" to this sentence that makes it different, when the fundamental syntax structure does not change? Adding one adverb in front of another doesn't automatically turn the former into an adjective does it? So why do I feel like it does here? Is "relatively" somehow unique relative to other adverbs? (see what I did there!)


To be clear on this, I know what is the definition of an inner product space and some properties and theorems about them. What I am asking for is an intuition for this definition in the complex case. In the real case, the intuition (or at least one of them) is geometric: The inner product of two vectors is the length of the projection of the first to the second scaled by the norms of both vectors so that it is symmetric (modulo some details). In particular I (and everybody else) think of "inner product zero" as geometric orthogonality and of orthonormal bases as, well, orthonormal bases and so on. The question is, what should I think about when working with complex (or should I say hermitian?) inner product spaces? what is the "meaning" of the complex number associated to two vectors called their inner product? I will be happy to hear all kinds of answers. For example, what physical phenomena does it model or in what mathematical situations does in "naturally" appear. Answers that stress the "nice structure" resulting are also welcome, yet I feel that by itself it is a bit unsatisfying.


Sometimes, I got really confused by the use of the Present Perfect tense. Given the fact, that we don't have this structure in Russian, all we can is to base our knowledge on grammar rules. The rules are quite simple: Experience: I have been to London twice. Unfinished actions: I have lived in Moscow since I was born. Close connection to the present situation: I have just cooked dinner. However, when it comes to simple questions, all that grammar rules are not so obvious. For example, if I am not sure and want to re-ask, could I say something like Have you meant? or Did you mean? Another case: I've sent you the letter and I sent you the letter. Does the first mean that I have just done it and the second that it was some time ago? How do you use it?


I am modeling a closed natural circulation loop, filled with water. Some parts of the loop are heated, some are cooled and other are assumed adiabatic. As an effect of heating and cooling the density of water changes and so does the total pressure in the loop. My question is as follows: Is there a way to calculate the total pressure of the system in terms of, for example, mean density, mean temperature and total volume of the system? For air, the ideal gas equation should be a nice approximation, but it is not applicable for liquids. The problem becomes more complex when the water starts to boil at some point, then it is a two-phase fluid. I have seen answer to this question What equation of state is needed for liquid states? but it does not help in my case.


I am trying to the calculate the link budget for link between a ground station on Earth (with a particular latitude and longitude) and a rover at a particular location on the surface of Mars, either directly or through a satellite on Mars. Now, if I need to determine the link availability between the rover and the ground station, the first step is to determine weather I have a line of sight between the ground station and the rover. For this, the first step is to determine whether Mars is above our horizon or not and if so, for how long. This can be easily done using packages such as PyEphem or Novas. The next step would be to determine if the rover is actually facing Earth or is on the other side of Mars. It is this second step that I need to determine with reasonable accuracy, but have not been able to figure out how to so far. Later on I would need to include the satellites in the link path as well, but for now, I need to determine if I can get a straight line of sight communication between the rover on Mars and ground station on Earth. Any sort of help will be appreciated.


I can find good explanations of how the disjoint union topology is constructed, but I am confused about how things such as complements, boundaries, limit points, etc. are to be understood in this context. For example, suppose we have two spaces, P and M and create their disjoint Union X with the disjoint union topology. It would seem that subsets of P and M must then be subsets of X that are disjoint. However, do they need to be separate as well or could a subset of P have limit points in a subset of M? With what open sets would the limit points be defined? How about the closure or boundary of unions of subsets of P and M? It seems from what I have been able to find that you could not define an open set in X that did not already exist in P or M, so I am confused. Any clarification or a pointer to a relevant treatment would be greatly appreciated. Ernie


A nice little oddity which I thought I'd ask about. I stumbled across the delightful word 'Boustrophedon' in relation to the scanning actions of some printers (inkjet/dot matrix). I believe that this roughly derives from the notion of 'As the ox ploughs the field'. I mentioned this to a colleague who had a farming background. He stated that in older times, this isn't actually how an ox would have ploughed the field, owing to the direction the plough would have turned the earth. Apparently, the field would have been ploughed in spiral pattern to ensure that meeting edges of the plough lines would have their earth turned in the same direction, or 'like the spider builds its web'. Which got me thinking. Is there a colourful noun of some sort that describes this sort of pattern/action?


I just graduated and a mate moved into a flat (none of us are physicists by the way). So, were graduates, we got a new flat, and were broke. So, were now having a debate on how to keep a fridge cool... well, cooler. I suggest that a fridge full of water will keep the food colder, as objects get cooler when water evaporates, (like when we sweat). He says that the fridge just has to do extra work to keep the water cool, but I said it doesn't matter as it's already colder when it enters the fridge, and it's properties should mean that the overall effect, is a colder fridge. Another conundrum, if I AM right, is it more effective to store the water at the bottom, middle, or top shelves? Also, what size/shape should the water be in, should it be stored in glasses, bowls, or in the form of cloths, where the surface area can be distributed, a lot more widely.


Let's imagine standing on a shore, and dropping rocks into a lake. Each rock causes ripples to travel outwards. Now let's imagine there is a monstrous whirl pool somewhere out there in the lake gobbling up waves.. It's quite complex to picture, but In my imagination I could envision a clever enough person analyzing the waves coming back to the shore and deducing things about waves that "got lost" so to speak. Now I understand there was some great debate between Stephen Hawking and Leonard Susskind about information theory and black holes. I don't remember who won that debate, but my understanding is that either way black holes are supposed to do a pretty bang up job of destroying information. So either my simplistic universe is flawed, or maybe I'm over estimating how clever the rock guy could be.. but I don't think so. Let's picture one single rock throw. The outgoing waves at some point get destroyed and you have something like this: and then at some point the wave pieces that didn't get destroyed reflect back and I can't draw that good and it get's all messy, but surely a lot of information comes back to the shore guy right?


A hollow metal sphere is electrically neutral (no excess charge). A small amount of negative charge is suddenly placed at one point P on this metal sphere. If we check on this excess negative charge a few seconds later we will find one of the following possibilities: (a) All of the excess charge remains right around P. (b) The excess charge has distributed itself evenly over the outside surface of the sphere. (c) The excess charge is evenly distributed over the inside and outside surface. (d) Most of the charge is still at point P, but some will have spread over the sphere. (e) There will be no excess charge left. Which one is correct and why? I guess it is some kind of electrostatic induction - phenomena going on. Am I right? I understand that excess charge is distributed over hollow sphere and that negative and positive charges are distributed opposite sides, but don't know which one positive or negative go to inside surface.


The following paragraph has been extracted from the Wikipedia (Atomic orbitals): Simple pictures showing orbital shapes are intended to describe the angular forms of regions in space where the electrons occupying the orbital are likely to be found. The diagrams cannot, however, show the entire region where an electron can be found, since according to quantum mechanics there is a non-zero probability of finding the electron (almost) anywhere in space. Is the statement by Wikipedia correct? Since, there is a probability of finding electron at any distance from the nucleus, when the electron comes far from the nucleus, I will block it, so that it won't return to its parent atom. Am I not stealing the electron? I can steal even the electron of your body being in India, be careful! That's what we layman think from those statements. What's the actual meaning of the wikipedia statement?


I was watching "Cold Opening: Homeland Security - Saturday Night Live". I am supposed to translate the entire sketch for my next classes, but I really don't know what is the joke here. I only know that magenta is a color and that is all. Can someone explain to me what is funny about this? I really need that. Here is a part of the transcript where the joke is used: "Before we begin today's briefing, I wish to announce that, on the basis of change in the nature of Al-Qaeda chatter, we are changing the current threat level to Magenta. Let me repeat: the threat level is now.. Magenta. What is Magenta? It's a darker maroon. It's not quite an ox blood. It's more plum color than.. say.. a crimson. How serious is it? [ sighs ] I honestly don't have an answer for that."


This question is sort of in the spirit of this xkcd: The light we get from stars was emitted many years in the past, but the distances to stars which are bright enough to be visible to the naked eye are not that great, so the light we received likely wasn't emitted long enough ago that the stars would have undertaken significant changes. On the other hand, some bright stars are red giants, which are very bright, very far away, and pretty close to the end of their lives, so there is a higher chance that they have collapsed in the meantime. So: what numerical fraction of stars which are visible by naked eye are likely to have undertaken significant steps in their stellar evolution? Here I'm interested both in main-sequence stars evolving into red giants, giants undergoing collapse, and similar events. Similarly, how does this answer change if you increase the range to stars that are visible using a reasonable pair of binoculars? In case special relativistic effects are important, for the purposes of this thread, both the current frame of reference of the solar system and the rest frame of the galaxy are interesting.


This popular question about "whether an AC circuit with one end grounded to Earth and the other end grounded to Mars would work (ignoring resistance/inductance of the wire)" was recently asked on the Electronics SE. (Picture edited from the one in the above link) Though I respect the AC/DC experts there, I think (with the exception of the top answer) they are all wrong. My issue is that they all assume that AC requires a complete circuit in order to function. However, my understanding is that a complete circuit is necessary for DC, but not AC. My intuitive understanding is that AC is similar to two gas-filled rooms with a pump between them - the pump couldn't indefinitely pump gas from one room to another without a complete circuit (DC), but it could pump the gas back and forth indefinitely (AC). In the latter case, not having a complete circuit just offers more resistance to the pump (with smaller rooms causing a larger resistance). Is my understanding correct - can AC circuits really function without a complete loop? More importantly, what are the equations that govern this? If larger isolated conductors really offer less AC-resistance than smaller AC conductors, how is this resistance computed/quantified? Would its "cause" be considered inductance, or something else?


I'd like to learn formal math. Preferably, though not necessarily, starting with predicate logic/first order logic rather than higher order logic. I am trying to find resources (papers, books etc.) for doing this, but I haven't found anything I really like. There are lots of resources for predicate and first order logic, but most do not approach the topics in a very formal way. For example, many text don't seem to try to define what they mean by "variables" or mention substitution as an important concept. Tries to explicitly describe as many of the rules of the game as it can. Many texts bring up "truth tables" without having formal rules for what you're allowed to do with those tables. Does anyone have resources that fit these criteria? Edit: many of the answers are good and helpful, but I feel like I should add some clarifying remarks: Many texts mention that you can view math as merely manipulation of symbols. I don't doubt that this can be done, but I would like to see it done. A resource that explains the process of producing proofs explicitly in terms of manipulating symbols rather than in terms of functions, statements etc (at least without first defining these terms) would be helpful. I'd like to be able to pretend I was a person who didn't know any math and was just acting as a human computer, producing proofs. I'd like a resource that explains producing proofs like I was such a computer (not necessarily ONLY like that).


I understand that photons, even when traveling at the speed of light, cannot escape the event horizon of a black hole. Are gravitons and other virtual particles traveling at the speed of light also confined by event horizons? If so, it seems that the gravitational field created by the black hole would result only from the mass of the black hole beyond the event horizon, where gravitons are capable of escaping. As a result, would there would be a disparity between the apparent mass of the black hole due to its gravitational field on other celestial bodies and the total amount of matter contained within the black hole? Also, I was reading this question: Nature of gravity: gravitons, curvature of space-time or both?, which suggests that gravitons and curved space may be indistinguishable. However, if gravitons are bound by the event horizon it seems that a black hole would act differently based on whether gravity results from gravitons or curved space-time. The existence of bound gravitons would negate the gravitational field of mass within the event horizon, resulting in a significantly lower gravitational field outside of the black hole. Would this occur, or am I neglecting some effect of relativity upon the gravitational field?


I was teaching my young nephew some math the other day, and from discussing the typical sort of word problems he's encountering in class, I noticed that the "-th" suffix adds a distinct meaning to adjectives. For example: If a ship is long, it has length. If a woman is wide, she has width. If a person is strong, he possesses strength. If what I say is true, I'm speaking truth. A lumbering panda moving slow is full of sloth. Now, I've learned some linguistics from English L&U, and I'm guessing this "-th" suffix is an affix that changes adjectives into nouns. My questions are: What exactly is this "-th" suffix adding to the meaning? Secondly, does the "-th" originate from a separate word in Old English? Lastly, is there something to say about the vowel shifts that seems to be occurring in some of the transformations (e.g., strong going to strength) that somehow fits in with the ablaut system of strong verbs/weak verbs, that I learned of from the excellent responses to my previous question?


Consider a lone photon. As its frequency increases, its energy increases. Taken to the limit, a sufficiently-high-frequency photon could be a black hole unto itself. But the frequency of a photon is dependent on the inertial frame of the observer. Two observers could each observe this photon to be either above or below this critical frequency. Or, I could accelerate to "catch up" to this photon, red-shifting it until it is no longer energetic enough to be a black hole. So couldn't I at one moment observe a particle to disappear beyond the event horizon, accelerate until the event horizon no longer exists, and hence observe what happened to the particle after crossing that threshold? Is this in-principle possible? If not, why not? EDIT: to clarify, I am not asking how much energy-due-to-photon-momentum is required to create a black hole, I am asking: given that threshold energy, how does the event horizon appear to different inertial frames which observe the photon to be above/below this threshold?


Does an object possess specificity to or for another object? Every time I go to express this concept in writing, I struggle over which preposition is the more appropriate and more precise. This is dilemma is encountered all the time in technical scientific writing, for example in biology where one speaks of enzymes with specificity to/for a particular substrate. I can't find much of a consensus there: I'm just as likely to see one form as the other in articles and published papers. My trusty Google consensus search isn't of much help either: the phrase "specificity to" occurs just about as often as the phrase "specificity for" does. Although all of the examples I can think of at the time are biological in nature, I'm certain that there are others, so I'm asking this as a more general grammatical usage question. Do they mean something subtly different? Should one form always be preferred over the other? Of course, it may be that either is entirely correct. Given that preposition usage is highly idiomatic in all languages (and English especially), there may not be a rule that definitively resolves this question.


It is fine to say that for an object flying past a massive object, the spacetime is curved by the massive object, and so the object flying past follows the curved path of the geodesic, so it "appears" to be experiencing gravitational acceleration. Do we also say along with it, that the object flying past in reality exeriences NO attraction force towards the massive object? Is it just following the spacetime geodesic curve while experiencing NO attractive force? Now come to the other issue: Supposing two objects are at rest relative to each other, ie they are not following any spacetime geodesic. Then why will they experience gravitational attraction towards each other? E.g. why will an apple fall to earth? Why won't it sit there in its original position high above the earth? How does the curvature of spacetime cause it to experience an attraction force towards the earth, and why would we need to exert a force in reverse direction to prevent it from falling? How does the curvature of spacetime cause this? When the apple was detatched from the branch of the tree, it was stationary, so it did not have to follow any geodesic curve. So we cannot just say that it fell to earth because its geodesic curve passed through the earth. Why did the spacetime curvature cause it to start moving in the first place?


Which is correct? There are no comments. There is no comment. Which would you use for a web application, i.e. what to display when a blog post or an article has no comment attached? Actually, I am trying to fix an application that says: "There is no comments"! Would that ever be right? More generally speaking, it feels wrong to have a plural after the negative no/none or with the preposition without (see my previous question "Without reason" or "Without reasons"?). Those words imply zero, i.e. less than one, while plural is two or more. Yet, I know that phrases like "There are no comments" or "He is without friends" are common. It seems illogical to me. Are the majority of people making a grammar mistake when using such expressions, or else can you explain why this is correct?


The word "complete" seems to be used in several distinct ways. Perhaps my confusion is as much linguistic as mathematical? A basis, by definition, spans the space; some books call this "complete" -- though then the phrase "complete basis" is redundant. In physics/engineering, "complete" seems to be reserved for orthogonal/orthonormal bases -- which necessarily means not merely a vector space, but specifically an inner product space. A complete basis in this QM sense does more than merely span the space: the concept of orthogonality allows for Parseval's relation, non-overlapping projections, Gram-Schmidt, etc. Is it even possible to have a complete basis (in this QM sense) that is NOT orthogonal? Though complete in the sense of Hilbert space and Cauchy sequences seems to be a different use of the term, the convergence of sequences within the space seems not so far afield, conceptually, from Parseval. So is it really so different?


I was recently asked the question "How do you know when you've become a better mathematician/better at mathematics?," and I realized that at that moment I did not have a valid answer, since I have been using my performance on tests to make that judgement. After putting some thought into it, I would say that one could at the very least use the following criteria: Learning material of similar difficulty at a faster pace with the same level of retention and understanding. Ability to bring together a larger number of theorems/lemmas/etc to use in showing a result (as opposed to doing problems that almost follow directly from the theorems). While those might encompass a good amount of information that would indicate whether you have been improving, I know that at my level there is still a lot more for me to experience and learn. So I would like to pose this question to everyone here, how do you know when you have become better at mathematics?


If I have a normal distribution, the posterior for the variance is the inverse Chi-square distribution assuming the same is used as a conjugate prior. But what if my data has extra noise added so that the observed sample variance is the sum of the population variance and my extra noise variance? But then the poisterior for the variance is different. Is there a name for that distribution? You can't just subtract the noise term because you can end up with negative values. It is similar to the Skellam distribution of the difference of two Poisson variables in this way. I am really interested in this from a Gibbs sampler point of view. I would like to draw the variance from the conditional posterior if possible. If that isn't easy I can fall back on Metropolis Hastings, I suppose.


I own a company called Find My Bus Ltd that brands itself as Find My Bus. Yesterday we sat down and had a discussion regarding the name and all came to the agreement that it didn't represent the company as we originally wanted it to. Seeing three separate words in a name that when thought of without any context sounds quite funky, we decided it would be better to merge the words into one name. We aren't the first to do this, in fact it is becoming a trend to merge words into one name. For example: DigitalOcean, StatusCake. In my opinion both of those names look fine because they consist of two words. However, when you do this with Find My Bus you see: FindMyBus. Is it just me, or does it look wrong having three words merged and capitalized? Would it be better to use Findmybus or perhaps FindmyBus? Apologies if this seems like I'm running a poll, I'm not, I just simply would like to ask users with experience in the English language which looks the most appropriate for a company name.


Quantum numbers are supposed to denote every individual orbital. But if orbital shells are probability functions, then orbitals can't be definite, solid things. So in that case, there can be variation in the amount of energy given off when an electron drops between shells - it might, say, give off a tiny little bit more energy and drop to just below the orbital shell. Isn't this possible since orbitals are just probability functions - like "Here's where the electron probably is"? Not entirely sure where I was going with this, but I think the final question is, how come quantum numbers are only ever integers? Edit: My question is about why quantum numbers as taught in schools are always integers. "Orbitals" as predicted by the Bohr model are in fact clouds of electrons, probability functions about where an electron probably is rather than a definite statement about where it definitely is. That means there's got to be wiggle room about how far an electron can be from the nucleus. So does that mean that quantum numbers are an oversimplification, or just averages? Or am I just misunderstanding the whole "orbitals are just probability clouds" thing? Edit: Ugh. Right. I'm an idiot. I forgot to mention that I'm only talking about the principal quantum number, n, the one telling which orbital the electron's in.


How do issues of naturalness arise when regularizing QFT using dimensional regularization? I can only recall ever seeing naturalness arguments (hierarchy problem, cosmological constant problem, etc.) phrased in terms regularizing with a cutoff, where naturalness issues arise when physical quantities are quadratically divergent in the cutoff scale. Is it hard to see how the same naturalness issues are addressed using dimensional regularization? Are there some hidden assumptions involved in using dimensional regularization? Do you reach the same conclusions as you do using a cutoff, but only after also using the RG equations? I recall being told that when dimensional regularization is used to remove power law divergences there is additionally some optimistic assumption being made about the UV physics, but I don't know if that's correct or relevant to this problem.


Well let's start off with that I'm not a physicist but I'd like some thoughts on something I came across in my hometown. This guy: Is it possible that due to the electrical charge of magnets this guy can make the illusion that he can float ? Or is this probably a cheap trick that fools the eye ? I was standing there for quite some time watching the guy and he keep moving his feet. The resistance that he appeared to have was from a magnet force keeping him afloat. So after I passed this guy I did some physics searches on the web and the first thing that caught my eye was the electrical charge of magnets. So the question is : Is this related to the electrical charge of a magnet or a cheap trick ?


I am trying to understand the meanings of "covariant transformation" and "contravariant transformation" and how they are related. I have read the related Wikipedia article and still feel I cannot state, with mathematical precision, the definition of these terms. The Wikipedia article states that a covariant transformation, in the context of a vector space, is one that "describes new basis vectors in terms of old basis vectors". This is not a satisfactory definition unless, of course, no other transformations can be described as "covariant". I have seen however the word "covariant" being used to describe other sorts of transformations as "covariant". Namely, the "physicists" definition of co/contravariant transformations where components transform as such-and-such (which makes absolutely zero mathematical sense to me). This leads one to believe that co/contravariant transformations are always defined in terms of derivatives of coordinate changes and I don't believe this is the case. I understand what co/contra-variant tensors are, at least from a mathematical perspective, so this is not a question about the meanings of "contravariant tensor" or "covariant tensor"; indeed, These concepts have been well-explained here. My question then, in summary, What are lucid, self-contained and mathematically precise definitions of "covariant transformation" and "contravariant transformation"? A reference to such definitions would also work wonderfully.


I had the idea of, what if you ground up some magnets into a fine powder, what would happen with the powered, and how would it act? After some google searches, it seems that this isn't done very often, and that not much would come of the powder as the poles will mostly be misaligned. So my next question was, what (if anything) would happen if you ran some electricity through some magnetic powder? Would the poles align? Could the powder be manipulated? Could it's magnetism be manipulated? i.e If you made a magnetic powder trail in a small amplitude sine wave shape, and applied some current to it. Would the powder be able to move into a straight line, or possible break the connection (or even move at all)? I haven't been able to find much information on this topic (of electricity WITH magnets) and I am trying to learn more on this topic.


Prompted by comments to this question on English Learners (about "That's you done"), I've been searching Google Books for similar constructions of the general form that's [pro]noun adjective (for this context, I classify past-tense verb forms such as done, fucked, finished as adjectives). What I seem to be finding is that using "That's" in this way (not referencing anything in particular, just "whatever came before/caused the current situation") is a relatively recent phenomenon. I'm also getting the impression it's more common in BrE than Ame. So by implication, if the boss says to his secretary... "Just get those letters off in the post, and that's you done for the day." ...I should assume the boss is probably British, rather than American. Would my assumption be right? Can anyone shed more light on the usage? Is it the same as... "Here's me doing all the work while you just sit around waiting to be fed." (said by, for example, hard-pressed mother to idle teenage offspring)


The question is written like this: Is it possible to find an infinite set of points in the plane, not all on the same straight line, such that the distance between EVERY pair of points is rational? This would be so easy if these points could be on the same straight line, but I couldn't get any idea to solve the question above(not all points on the same straight line). I believe there must be a kind of concatenation between the points but I couldn't figure it out. What I tried is totally mess. I tried to draw some triangles and to connect some points from one triangle to another, but in vain. Note: I want to see a real example of such an infinite set of points in the plane that can be an answer for the question. A graph for these points would be helpful.


Recent observations of the accelerating expansion of the universe have been quantified and for the time being given a name as to the cause: Dark Energy. And from what I've read from other, similar questions is that Dark Matter is a pressure that is causing this expansion, although we don't know the details of the mechanism yet behind this pressure. But is there anything in our present theories of physics or observations that rules out gravity itself as the cause of this expansion? I'm thinking along the lines of an analogy: the nature of the strong nuclear force which, at close distance, tends to bind together nucleons, but at even closer distances repels them. Couldn't this repulsive force we observe, this dark energy, just be the effects of the gravitational force on a larger scale of space?


I have a subtle doubt about the physical interpretation of the mathematical definition of vector field as a derivation. In basic physics we understand a vector quantity as a quantity that needs more than magnitude to be fully specified, in other words, quantities with the notion of direction. This goes very well with the also basic mathematical definition that a vector is an equivalence class of oriented line segments. However, when we go to the study of manifolds, we see that a better definition of vector is to say that a vector at a point is a derivation on the algebra of smooth functions on that point. But then, we represent forces for instance with vectors, what's the interpretation of representing one force acting on a point by a derivation on the smooth functions on the point? I imagine that there must be some interpretation for that, but I didn't find what's it. Sorry if this question seems silly. I'm just trying to bring together those concepts. And thanks you all in advance.


I have a question about the appropriate usage the possessive personal pronoun "mine" over the possessive determiner "my". I have done some research and understood the differences between them, but my questions isn't quite about that. One of the main purposes of pronouns is to eliminate repetitions, yet I cannot be sure whether to use it or not if there isn't a possessive determiner such as my, your, her, their etc that establishes the noun being referred to. For example, I thought of the sentence below: Your dog attacked mine! ('Mine' used as a substitute for 'my dog', as 'your dog' established the noun in question) The above is no doubt a better construction than: Your dog attacked my dog! Which sounds somewhat awkward in comparison. However, if there isn't a possessive determiner preceding it in a sentence: A wild dog attack mine! It sounds jarring and not as natural as: A wild dog attack my dog or My dog was attacked by a wild one. I don't know if there is a grammar rule that would explain my question, but fellow memebers, is it still correct to use the possessive personal pronoun (such as mine) in the case above, or can it only be used if a possessive determiner (such as your) is used before it? Many thanks!


When I used to play video games, "all in" meant that you were devoting your army to a "do or die" effort. In other words, you were either going to win or lose in the next battle. The "all in" meant that you typically brought all of your soldiers to fight, holding none back. But years ago I was doing the Daily Jumble, and the hint was something like "After running a marathon he was _ __". This was before the web, so I had nothing to fall back on. It turns out the answer was "ALL IN" meaning "tired". I asked a friend and he said "that's a usage from the eastern U.S." I certainly hadn't heard it before then and I still haven't since. Is this a common meaning? Maybe some easterners can chime in?


In this sentence, would you use "to inform" or "for informing"? These findings are critical ______ future research Likewise, would you use "to understand" or "for understanding" in the following? These results provide a powerful framework _____ existing patterns. This is a question that comes up a lot in science writing: when do you use the "to" vs. the "for" version of a verb? Is there a word for these two different uses? And is there a rule for deciding which one is correct? They often get used interchangeably, but I can't find any hard rules for deciding between the two. People often use the "to" version because it sounds less passive, but I find it more ambiguous and a bit clunky, but I can't find any actual sources to say which is correct.


I don't get this joke: It smells like up dog in here, What's up dog? Nuuthin dog, what's up with you? (source) I understand that the person asking "What's up dog" is tricked into asking a question they didn't intend to ask. I also understand the meaning of the "What's up?" question. I don't understand why this exact question has been chosen. If anything, it sounds demeaning to the person who's making the joke and is referred to as dog by the receiver of the joke. If I were to make a similar joke, I'd replace dog with something like master, so that it'd sound like the receiver of the joke respects the person making the joke more than it might be in reality. Does the word dog in this context have some alternative meaning, perhaps? Or is this joke funny simply because the receiver of the joke asks a funny question, no matter what the question is?


How common is it for native English speakers to actively replace the past tense 'did' with the past participle 'done'? I used to think it was only really done in rather vulgar dialects, but I have increasingly heard this usage now by people who speak otherwise normal Received Pronunciation, or Standard American English: I done the report. You done it yesterday. He done it already. I can't help but interpret this replacement of 'did' as sounding somehow uneducated, but is it becoming a standard in some dialects? Is the word 'did' dying out? And do these speakers use the 'done' replacement when switching the word order around in a question; for example, would they say: Done you do it today? ... or even: Done you it today? ... instead of: Did you do it today?


In the study of quantum field theory, one may encounter S-matrix a lot. Recently, in the study of integrability, I encountered R-matrix formulation which I am not familiar with. First of all, the S-matrix is a scattering matrix which comes from scattering processes. In the context of quantum mechanics and quantum field theory, we often compute the S-matrix, and we use well-known formulas, i.e, the LSZ reduction formula On Wikipedia, they describe a R-matrix, as related with the Yang-Baxter equation (governing equation for integrability), and they add some comments that it is related with resonance. My questions are, then: Can you give me a clear definition and a governing equation for a R-matrix? How are S-matrix and R-matrix related to each other? Why is the R-matrix important in integrability (Yang-Baxter equation)?


I am confused about the usage of the words like and unlike in sentences. Like and unlike make me confused because I see them being used everywhere almost interchangeably, and to make matters worse I even saw commas making difference in meaning. Which of the following sentences is most correct, and least ambiguous? Unlike Xbox, Playstation can't play Halo. Playstation can't play Halo unlike Xbox can. Playstation can't play Halo, unlike Xbox can. Like Xbox, Playstation can't play Halo. Playstation can't play Halo like Xbox can. Playstation can't play Halo, like Xbox can. Playstation can't play Halo unlike Xbox. Playstation can't play Halo, unlike Xbox. Playstation can't play Halo like Xbox. Playstation can't play Halo, like Xbox. In case you don't know what I actually wanted to say in those sentences: I'm trying to say that Playstation can't run the Halo video game and Xbox can run that game. I probably should have tried sentences with no brands but I can't imagine any meaningful sentences at this moment which contains like/unlike.


I'm new to finite fields - I have been watching various lectures and reading about them, but I'm missing a step. I can understand what a group, ring field and prime field is, no problem. But when we have a prime extension field, suddenly the elements are no longer numbers, they are polynomials. I'm sure there is some great mathematical tricks which show that we can (or must?) use polynomials to be able to satisfy the rules of a field within a prime extension field, but I haven't been able to find a coherent explanation of this step. People I have asked in person don't seen to know either, it's just assumed that that is the way it is. So I have two questions: What is a clear explanation of "why polynomials?". Has anyone tried using constructs other than polynomials to satisfy the same field rules? Thanks in advance.


There are many books, written by highly decorated academics, which feature proofs that I can hardly comprehend in an acceptable amount of time. Roughly each week, it happens that I find myself having spent around three hours to convince myself of a single proof - usually, the proofs are from textbooks for undergraduates. Sometimes, I am horribly slow at finding detailed arguments which connect the presented conclusions (and which are necessary to convince myself that the conclusions are beyond question). Then, I feel that the authors take steps that are not small enough for me; and I wonder whether I should digest every proof (or merely use the theorems). I am not particularly gifted. Still, I strive to improve myself. Do you think that one can learn or train to understand proofs faster?


I want to typeset diagrams describing functions which take multiple inputs and return multiple outputs. These should look like boxes with labels in them like "f" or "g" and with some input wires coming in and some output wires going out. The diagrams I'd like to typeset will involve several such boxes with some output wires connected to some input wires. I can draw some examples of what I want if this description is unclear. I would prefer to be able to do this in xy-pic, for example using xy-graph somehow. Is this a good idea, or should I just learn how to do this in TikZ? Or is there an even better way? Edit: To be more specific, here are the diagrams I want (although their orientation is negotiable, e.g. if you show me how to do this with the diagrams going to the right instead of down that's great). I should have mentioned that I also want to label the wires.