The Catalan number $C_n$ counts the number of ways $n$ pairs of parentheses can be written without having a close-parenthesis too early, and it also counts the number of ways $n+1$ factors can be grouped along with an non-associative binary operator.  For example, $C_3 = 5$ and the permissible binary strings are (copied from Wikipedia):  ((()))     ()(())     ()()()     (())()     (()())  The ways to group four factors $a, b, c, d$ are:  ((a*b)*c)*d     (a*(b*c))*d     (a*b)*(c*d)     a*((b*c)*d)     a*(b*(c*d))  What is the bijection between these two sets? I know that both of them can be shown directly to be counted by the Catalan numbers, but I want to know what the correct map is from one set to the other.
The  lists various combinatorial objects that are described by them. I posit that there might be bijections between these various combinatorial objects. For some of them (like Dyck paths, correctly matched parentheses and paths from the bottom left to top right of a $2n \times 2n$ grid) they are quite obvious.   I was then trying to find a bijection between the number of Dyck words and the number of ways of associating $n$ applications of a binary operator to $n+1$ items (third one on the list). I attempted to do this for a simple case ($n=3$ which is the example provided in the Wikipedia article). However, couldn't find one after multiple hours. Is it reasonable to expect such a bijection will exist? If so, how do we go about finding it?    EDIT: In addition to the very nice answer by @Marc, the following page also helped me see the bijection:   "Let $P$ be the Dyck path and $f(P)$ be the binary tree. If you go up in the Dyck path, create a left child. Otherwise, go up one vertex until creating a new right child is possible and create one."    Here is one of my attempts:   Number of Dyck words with length $2 \times 3$ is $\frac{6 \choose 3}{4} = 5$. They are:  hhhttt; hhthtt; hhttht; hthhtt; hththt  And the number of applications of a binary operator among $3+1=4$ factors is:  ((ab)c)d;     (a(bc))d;     (ab)(cd);     a((bc)d);     a(b(cd))  Both combinatorial objects have been arranged in a way that there is some kind of ordering between them. For the Dyck words for example, if h equals +1 and t equals -1, then the order is lexicographical from the left to the right of the cumulative score along the sequence.   Now, the first and last characters of the Dyck words are always h and t respectively. So, we can ignore them. We are left with:  hhtt; htht; htth; thht; thth  I tried to start from the left of the sequence abcd and if I see 'h', merge the character with the one to its right. This approach didn't produce a valid mapping from the third Dyck word to the third binary operator precedence order.
How do I counter-knife an enemy? When you press down the right stick to counter the attack, do you just press it, repeatedly press it, or press and hold? I've been knifed a few times from the front and each time I try to counter I fail. Also, if you believe you are about to be knifed and hold down the right stick until you are knifed, will that counter an attack, or do you have to press it down just at the right time? Clicking links in pdfs opened with Okular opens Abiword (instead of the default browser) I work with Xubuntu 12.04 and use Okular (version 0.14.3) to view pdf files. If I click a web link in a pdf file (created with "pdflatex" using the "hyperref" package), "Abiword" (It's a word-processing software, like Libreoffice writer) is opened instead of my default browser google-chrome. How can I change this behavior? The settings in Okular do not seem to provide a solution.   (Note that I set google-chrome as the preferred application for web browsing under Settings -> Preferred Applications, so that's not the problem). So I've been experiencing this in the last few days: sometimes when someone replies to my comment, and I reply to the reply, then the "add/show 1 more comment" link appears immediately after I post my comment. However, when I click it, it disappears and nothing is shown.  I don't think it's that OP is posting two comments of which he deletes the second one, because this has happened to me several times, even when I was replying within 2-3 seconds.  I do, however, believe that this is more or less the same bug as , except that I don't have to reload the page in order the link to be gone, and I also seem to be getting the erroneous behavior only if I reply to a comment which is itself a reply to an earlier comment of mine.  Unfortunately, I wasn't careful enough to take a screenshot last time this happened, but here's what I am talking about for those who like visual demonstration better: Consider the real-valued function  $f(x)=x^r$ where $r$ is a real number.  (1) For what values of $x$ and $r$, is this function differentiable?  (2) How do you prove the power rule, when $f(x)$ is differentiable?  It would be nice if the proof is accessible to a 12th grade student.  EDIT: Perhaps, I have not been very clear about my question.  I am not satisfied with the "textbook proofs" I have seen.  Please established the existence (of the derivative) first before proving the power rule.  That is, please consider first question first.  This is the reason why I call the question a mathematical engineering question.  (If we can we bring high level math down to Grade 12 level, without loosing precision, then I call it good mathematical engineering.) My supervisor asked me to find out which distribution represents a particular situation.  I have a VoIP generator that generates calls "uniformly" distributed between callers. This means that the volume per caller distribution is "almost" uniformly distributed between a minimum and maximum. So by running a test with 10000 users and a min value equal to 30 calls per week and a max value equal to 90 calls/week i obtain that not all the users respect this limits: we have some users that generate &lt;30 calls and some other users that generate >90 calls. It is clear that the obtained distribution is not uniform.  The situation is this:    He said that i have to perform a sort of numerical process in order to find some formulas that could define this distribution. Initially, as wrote before,we wanted to obtain a uniform (min,max) distribution (the green area in the figure) but this is not the case as proved with chi-square test.  Moreover the curve in figure is not symmetric, the probability of call generation below 30 call/week and greater that 90 call/week are not identical (it is high for 90calls/week). The variability of the number of generated call increases with the increasing call generation rates.   "Actually implementation of this distribution is nothing but assigning different call rates in a range for users in domain which indicates implementation of several delta functions. As the call rates increases the variability of the generated calls also increases with the average call rate and this leads to the asymmetric behavior of the curve." [cit. from the Voipgenerator documentation]  Someone can help me?I think that now i cannot use Q-Q plot because i don't know which theoretical distribution i have to use in order to compare it with my empirical data.  Sorry if I have stressed with a similar problem a few weeks ago, but initially we thought we could change the implementation, but now we cannot. Hence i have to discover the type of the distribution i obtained and i don't know how can i do this. I have been running  over some C# code, and it keeps reporting that my using directives should be inside the namespace.  Is there a technical reason for putting the using directives inside instead of outside the namespace? Learning and using English I'm always confused about what word to use for referring to things that have been described by me a few sentences earlier: "that" or "this".   Confusion comes from the fact that only the equivalent of "this" is always used in my native language for such referring. But I've noticed that in English for such referring "that" is used as often as "this" (or maybe even more often).  Some examples for illustrating (just tried to google something appropriate to convey my idea better):     We assess local... the demographics of the local population. What are the natural traffic drivers in the area. Things like cinemas and pubs and retail and office and all that type of things and we now put together a bit of a matrix and actually give a weighted score to each of the things we know help our business. That helps us decide in a more scientific fashion.   or:     The problem of Cervantes' origin became after that into a tough matter. Some experts believed that the Cervantes from Alcázar, in the times of the Lepanto Battle, was in the age of a child, more concerned about gathering nests and that type of things than about fighting as a soldier.   and for "this":     I believe lot of people who involved fishery industries in Mexico Gulf are suffering now but as for economic issues, U.S government and other countries will support them and give an utmost response to it. Important thing is that we learn from this mistake and make sure this type of things will never happen in the future and protect nature environment thus we have to take this technology to get energy from water very seriously. I really hope this technology will be available and used for everybody as soon as possible.   Is there a rule describing proper usage of these words in cases like this? (or should I have written "..in cases like that"?) Rewriting Triple Integral in different order How can I rewrite, say: $$\int^1_0\int^{z^2}_0\int^y_0{f}dxdydz$$ In the orders $dzdxdy$ and $dydzdx$?  Would I have to solve this triple integral all the way or is there a simple way to do this? SQL Server nvarchar(max) vs nvarchar(n) affects performance What do I need to know about temperature and food safety? What general rules do I need to follow to keep my food safe? How do I know what temperature to cook something to, or whether my food is safe at room temperature? I can understand this requirement for the old PPC RISC systems and even for x86-64, but for the old tried-and-true x86? In this case, the stack needs to be aligned on 4 byte boundaries only. Yes, some of the MMX/SSE instructions require 16byte alignments, but if that is a requirement of the callee, then it should ensure the alignments are correct. Why burden every caller with this extra requirement? This can actually cause some drops in performance because every call-site must manage this requirement. Am I missing something? Update: After some more investigation into this and some consultation with some internal colleagues, I have some theories about this:  Consistency between the PPC, x86, and x64 version of the OS It seems that the GCC codegen now consistently does a sub esp,xxx and then &quot;mov&quot;s the data onto the stack rather than simply doing a &quot;push&quot; instruction. This could actually be faster on some hardware. While this does complicate the call sites a little, there is very little extra overhead when using the default &quot;cdecl&quot; convention where the caller cleans up the stack.  The issue I have with the last item, is that for calling conventions that rely on the callee cleaning the stack, the above requirements really &quot;uglifies&quot; the codegen. For instance, what some compiler decided to implement a faster register-based calling style for its own internal use (ie any code that isn't intended to be called from other languages or sources)? This stack-alignment thing could negate some of the performance gains achieved by passing some parameters in registers. Update: So far the only real answers have been consistency, but to me that's a bit too easy of an answer. I have well over 20 years experience with the x86 architecture and if consistency, not performance, or something else concrete, is really the reason then I respectfully suggest that is a bit naive for the developers to require it. They're ignoring nearly three decades of tools and support. Especially if they're expecting tools vendors to quickly and easily adapt their tools for their platform (maybe not... it is Apple...) without having to jump through several seemingly unnecessary hoops. I'll give this topic another day or so then close it... Related How does deleting work? What can cause a post to be deleted, and what does that actually mean? What are the criteria for deletion? SELECT * EXCEPT Is there any RDBMS that implements something like SELECT * EXCEPT?  What I'm after is getting all of the fields except a specific TEXT/BLOB field, and I'd like to just select everything else.  Almost daily I complain to my coworkers that someone should implement this...  It's terribly annoying that it doesn't exist.  Edit: I understand everyone's concern for SELECT *.  I know the risks associated with SELECT *.  However, this, at least in my situation, would not be used for any Production level code, or even Development level code; strictly for debugging, when I need to see all of the values easily.  As I've stated in some of the comments, where I work is strictly a commandline shop, doing everything over ssh.  This makes it difficult to use any gui tools (external connections to the database aren't allowed), etc etc.  Thanks for the suggestions though. I cannot see WiFi option in my Ubuntu 15.04. What should I do? Revert asap  lspci -nnk | grep 0280 -A2 output:-  03:00.0 Network controller [0280]: Qualcomm Atheros Device [168c:0042] (rev 30) Subsystem: Foxconn International, Inc. Device [105b:e09a]   rfkill list output:-  0: hci0: Bluetooth Soft blocked: no Hard blocked: no 1: acer-wireless: Wireless LAN Soft blocked: no Hard blocked: no      dmesg | grep ath [    9.545199] ath10k_pci 0000:03:00.0: pci irq msi-x interrupts 8 irq_mode 0 reset_mode 0 [   10.029684] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/cal-pci-0000:03:00.0.bin failed with error -2 [   10.029701] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/QCA9377/hw1.0/firmware-5.bin failed with error -2 [   10.029704] ath10k_pci 0000:03:00.0: could not fetch firmware file 'ath10k/QCA9377/hw1.0/firmware-5.bin': -2 [   10.029711] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/QCA9377/hw1.0/firmware-4.bin failed with error -2 [   10.029713] ath10k_pci 0000:03:00.0: could not fetch firmware file 'ath10k/QCA9377/hw1.0/firmware-4.bin': -2 [   10.029719] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/QCA9377/hw1.0/firmware-3.bin failed with error -2 [   10.029721] ath10k_pci 0000:03:00.0: could not fetch firmware file 'ath10k/QCA9377/hw1.0/firmware-3.bin': -2 [   10.029726] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/QCA9377/hw1.0/firmware-2.bin failed with error -2 [   10.029728] ath10k_pci 0000:03:00.0: could not fetch firmware file 'ath10k/QCA9377/hw1.0/firmware-2.bin': -2 [   10.029733] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/QCA9377/hw1.0/firmware.bin failed with error -2 [   10.029735] ath10k_pci 0000:03:00.0: could not fetch firmware (-2) [   10.029737] ath10k_pci 0000:03:00.0: could not fetch firmware files (-2) [   10.029739] ath10k_pci 0000:03:00.0: could not probe fw (-2) Trace of a commutator is zero - but what about the commutator of $x$ and $p$? Operators can be cyclically interchanged inside a trace: $${\rm Tr} (AB)~=~{\rm Tr} (BA).$$ This means the trace of a commutator of any two operators is zero: $${\rm Tr} ([A,B])~=~0.$$ But what about the commutator of the position and momentum operators for a quantum particle? On the one hand: $${\rm Tr}([x,p])~=~0,$$ while on the other hand: $$[x,p]~=~i\hbar.$$ How does this work out? Getting equation from differential equations I have:  $\dfrac {dx} {dt}$=$-x+y$  $\dfrac {dy}{dt}$=$-x-y$  and I am trying to find $x(t)$ and $y(t)$ given that $x(0)=0$ and $y(0)=1$  I know to do this I need to decouple the equations so that I only have to deal with one variable but the decoupling is what I am having trouble with  Do I set them equal to each other and then just move like terms to separate sides getting two different equations and then integrate? As far as I understand, one of the main claimed problems with initializing e.g. a feed-forward neural network (with several $\text{tanh}$ or $\text{ReLU}$ layers) with $W=0$ is that it doesn't break "network symmetry", meaning, backpropagation would propagate the same error through all such units (i.e. "nudging all weights in the same direction"). This is, I presume, undesirable because we would not be learning "different" calculations through different paths of the network.  However, I'm confused why that even matters in this case given the fact that if $W$ ever drops to 0, we will effectively be propagating no gradients at all through the network, since W=0 would multiply all errors from the output and prevent any learning.  Put another way, even if $W=0$ does not break network symmetry (wasting calculations and paths in the network) is it correct to say that if $W=0$ (e.g. by initialization) we are effectively killing gradients in a neural network, and thus no learning can take place? How closely can a game legally resemble another? "If" and "Whether" - Is interrogative IF always replacable by WHETHER? I am asking here specifically about if and whether  when they introduce embedded questions (or to be more technically accurate, subordinate interrogative clauses):  I don't know if Bob's here. I don't know whether Bob's here.  In the sentence above we can freely replace if with whether. There are some situations in which the reverse does not apply. We cannot always replace whether with if. For example in standard English, we cannot normally use if after a preposition:  The question of whether he is actually eligible didn't arise. *The question of if he is actually eligible didn't arise.  We cannot usually use interrogative-if directly followed by or not:  It's unclear whether or not he's a real elephant. *It's unclear if or not he's a real elephant.  There are several more examples. However, what I want to know is:  Are there any examples where we can use interrogative-if but we can't use whether?   Edit note: There is an if that we find in conditionals that can't be replaced by whether. However, this is conditional, not interrogative, if. Applying for a Canadian ETA after previous visits without one