I hope your answer for this question will help me understand a lot of things. Thank you guys.
$dx$ appears in differential equations, such us derivatives and integrals.  For example, a function $f(x)$ its first derivative is $\dfrac{d}{dx}f(x)$ and its integral $\displaystyle\int f(x)dx$. But I don't really understand what $dx$ is.
"Its" as a Possessive Pronoun Since its can be both determiner possessive pronoun and nominal possessive pronoun, an example of its as determiner possessive pronoun would be:     We saved this question for last because of its complexity.   Because complexity is a noun, so its must be a determiner possessive pronoun in this sentence. I'm looking for an example where its is used as nominal possessive pronoun in a sentence as above. Extract URLs from text in PHP Relay: higher voltage allowed at lower current? Is it safe to use a 30VDC/1A relay for a voltage of 60V but a current of &lt;100uA? E.g. this one: Is there a spec that the id of elements should be made global variable? 3D software like GeoGebra Does it exist a free interactive geometry software, like GeoGebra, which works for 3D geometry? I would be able to draw spheres, great circles, and so on. Can I still use logistic regression if some variables are not applied to all the observational units? Some of the variables are questions that are not being asked to every single person in the questionaire depending on how they answered previous question(s). So would I still be able to use a logistic regression if some of the questions (the independent variables) are only measuring the responses of SOME of the respondents? How can I represent multiple pins at the same location on a map? Cannot Select Linked When Sync Selection Is Enabled Using shell's read command with live editing functionality (readline like) Is there a standard (POSIX) way of asking the user some data from within a shell script, with  for example, while allowing live edition of the text being typed (what readline does)?   I know bash has read -e varname that allows for the person launching the script to use keyboard arrows for exemple, to edit or correct what has just been typed without deleting the last entered characters with backspace.   However, read -e is . And still, it is pretty cumbersome to delete all that has been wrote if you realise you made a mistake at the beginning of your long sentence... Why are the even and odd Regge trajectories degenerate? This is an old classic which I don't think ever got a clear answer. The Gribov-Froissart projection that gives the relativistic version of Regge trajectories treats even angular momentum differently from odd angular momentum. The trajectory functions in general are separate for even and odd angular momentum.  But in QCD, the odd trajectories interpolate the even trajectories in every case--- the two trajectories are degenerate. Is there a way of understanding why the even-odd trajectories are degenerate? Is is a symmetry argument? Can you find a natural system where they are not degenerate?  A quick google search revealed this reference, which doesn't answer the question, but gives an experimental signature for the degeneracy:  , There is probably excellent data by now on this. jQuery :first vs. .first() Hide details of a chapter from TOC and list of tables, figures and algorithms This is solely a reference request. I have heard a few versions of the following theorem:     If the joint moment generating function $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ whenever the expectations are finite, then $X,Y$ are independent.   And there is a similar version for characteristic functions. Could anyone provide me a serious reference which proves one or both of these theorems? Do the topologies with the same convergent sequences coincide? Do the same sequential convergence in two topologies the result are the same topologies? Let $X$ be topological  space with topologies $\tau$, $\tau'$. Let $(x_{n})\in X$ and $x\in X$ and $(x_{n})\rightarrow x$ with $\tau$ if and only if $(x_{n})\rightarrow x$ with $\tau'$. Do the two topologies on $X$ coincide? One very convenient property of $\mathbb{R}$ as a ring is that there is an upper bound for the degree of irreducible polynomials in $\mathbb{R}[x]$, as      If $f\in\mathbb{R}[x]$ has degree larger than $2$, then $f$ is reducible.   However, the proof as I know depends highly on the fact that $\mathbb{C}$ is algebraically closed, and the very nice property: $z\in\mathbb{C}$ is in $\mathbb{R}$ if and only if $z=\bar{z}$.  This makes a generalization to other integral domains rather difficult. So the problem is      For what kind of integral domain $R$, we have a finite upper bound on the degree of irreducible elements in $R[x]$?   Some most familiar examples are ruled out: in $\mathbb{Z}[x]$, $\mathbb{Q}[x]$ and $\mathbb{F}_{p}[x]$, there is not such a bound. Unfortunately these exhaust all integral domains about which I have a working knowledge.  Another result might help is , which says if we can find a prime ideal $\mathfrak{p}$ in $R$ such that $\mathfrak{p}^2\neq\mathfrak{p}$, then by picking $a\in\mathfrak{p}^2\backslash\mathfrak{p}$ we have an irreducible $a+x^d$, where $d$ can be arbitrary, and hence the upper bound is not possible.  So we only need to focus on domains where $\mathfrak{p}^2=\mathfrak{p}$ for all prime ideals. This seems to be a quite strong restriction but I am not sure what to make of it.  Can someone give a hint?  Thanks! Across the Stack Exchange network you may leave comments on a question or answer.  How do comments work? What are comments for, and when shouldn't I comment? Who can post comments? Who can edit comments? How can I format and link in comments? My comment doesn't contain some of the text I typed in it; why? My comment appears to be long enough, but I can't post it because it's too short; why? Who can delete comments? When should comments be deleted? Why can't I comment on specific posts even if I have enough reputation to comment? When can comments be undeleted? What are automatic comments? How can I link to comments? Anything else I should know about comments?  Also see the FAQ articles on  and . How do I check if an integer is even or odd? Typeset an = with an ! above I've seen an equals sign with a small exclamation mark on top being used to say "equals because of data that was given in the problem", for example for a boundary condition or initial value in a differential equation.  A rough picture of what I mean:       ! f(x) = 4   How do I typeset this in LaTeX? I've tried $=^!$ (which doesn't produce the desired result) and looked through the "relations" section of the  without success. "It looks straightforward, but actually it isn't" In , I asked about proof of statements which are simple but incorrect.   Here, I ask about statements which seems, at a first glance, straightforward, but if we try to write a proof, we can see it's much harder than it looked. So I expect the answers to contain:   the statement; why it looks easy to prove; why actually it isn't. Java ReentrantReadWriteLocks - how to safely acquire write lock? I am using in my code at the moment a  to synchronize access over a tree-like structure.  This structure is large, and read by many threads at once with occasional modifications to small parts of it - so it seems to fit the read-write idiom well.  I understand that with this particular class, one cannot elevate a read lock to a write lock, so as per the Javadocs one must release the read lock before obtaining the write lock.  I've used this pattern successfully in non-reentrant contexts before.  What I'm finding however is that I cannot reliably acquire the write lock without blocking forever.  Since the read lock is reentrant and I am actually using it as such, the simple code  lock.getReadLock().unlock(); lock.getWriteLock().lock()  can block if I have acquired the readlock reentrantly.  Each call to unlock just reduces the hold count, and the lock is only actually released when the hold count hits zero.    EDIT to clarify this, as I don't think I explained it too well initially - I am aware that there is no built-in lock escalation in this class, and that I have to simply release the read lock and obtain the write lock.  My problem is/was that regardless of what other threads are doing, calling getReadLock().unlock() may not actually release this thread's hold on the lock if it acquired it reentrantly, in which case the call to getWriteLock().lock() will block forever as this thread still has a hold on the read lock and thus blocks itself.  For example, this code snippet will never reach the println statement, even when run singlethreaded with no other threads accessing the lock:  final ReadWriteLock lock = new ReentrantReadWriteLock(); lock.getReadLock().lock();  // In real code we would go call other methods that end up calling back and // thus locking again lock.getReadLock().lock();  // Now we do some stuff and realise we need to write so try to escalate the // lock as per the Javadocs and the above description lock.getReadLock().unlock(); // Does not actually release the lock lock.getWriteLock().lock();  // Blocks as some thread (this one!) holds read lock  System.out.println("Will never get here");   So I ask, is there a nice idiom to handle this situation?  Specifically, when a thread that holds a read lock (possibly reentrantly) discovers that it needs to do some writing, and thus wants to "suspend" its own read lock in order to pick up the write lock (blocking as required on other threads to release their holds on the read lock), and then "pick up" its hold on the read lock in the same state afterwards?  Since this ReadWriteLock implementation was specifically designed to be reentrant, surely there is some sensible way to elevate a read lock to a write lock when the locks may be acquired reentrantly?  This is the critical part that means the naive approach does not work.