is composition associative?  How do we define this?   I thought I could counter example this with integers and mappings  $f_1(x)=1$, and $f_2(x) = f_3(x)=2x.$  But I couldn't get it work on paper only in my head which means I am probably wrong thx
My intent is to show that a composition of bijections is also a bijection by showing the existence of an inverse.  But my approach requires the associativity of function composition.  Let $f:  X \rightarrow Y, g:  Y \rightarrow Z, h:  Z \rightarrow W$ be functions.  $((f \circ g) \circ h)(x) = h((f \circ g)(x)) = h(g(f(x)))$, and $(f \circ (g \circ h))(x) = (g \circ h)(f(x)) = h(g(f(x)))$.  However, I am having problems in justifying that the two compositions, $(f \circ g) \circ h$ and $f \circ (g \circ h)$, have the same domain and range.  When I consulted ProofWiki, whose link is at the bottom, I got even more confused.  Specifically, for $(f \circ g) \circ h = f \circ (g \circ h)$ to be defined, ProofWiki requires that dom$g =$ codom$f$ and dom$h =$ codom$g$.  First of all, I think that it should be dom$g =$ range$f$ ....  Moreover, as you can see in the example below, you actually have to adjust domains and ranges of $f, g, h$ for the requirement to hold true.  Let $f: \mathbb R \rightarrow \mathbb R$ be $f(x) = 2x$, $g: \mathbb R^+ \rightarrow \mathbb R$ be $g(y) = ln(y)$, $h: \mathbb R \rightarrow \mathbb R$ be $h(z) = z - 10$.   Then $((f \circ g) \circ h)(x) = ln(2x) - 10 = (f \circ (g \circ h))(x)$, with dom$((f \circ g) \circ h) = \mathbb R^+$ = dom$(f \circ (g \circ h))$.  As a result, we need to set dom$f = \mathbb R^+$, range$f = \mathbb R^+$; dom$g$, range$g$, dom$h$, and range$h$ remain the same.  Am I allowed to do that?  This adjustment implies that when we say dom$f = X$, $f$ must be defined for all elements in $X$, but $X$ may not be the entire set of elements for which $f$ is defined.
Creating controls within a Loop I have have some code which adds new cells to a table and fills them with text boxes.   The way I've coded it so far works fine:          TableCell tCell1 = new TableCell();         TableCell tCell2 = new TableCell();         TableCell tCell3 = new TableCell();         TableCell tCell4 = new TableCell();         TableCell tCell5 = new TableCell();         TableCell tCell6 = new TableCell();         TableCell tCell7 = new TableCell();          TextBox txt1 = new TextBox();         TextBox txt2 = new TextBox();         TextBox txt3 = new TextBox();         TextBox txt4 = new TextBox();         TextBox txt5 = new TextBox();         TextBox txt6 = new TextBox();         TextBox txt7 = new TextBox();          tCell1.Controls.Add(txt1);         tCell2.Controls.Add(txt2);         tCell3.Controls.Add(txt3);         tCell4.Controls.Add(txt4);         tCell5.Controls.Add(txt5);         tCell6.Controls.Add(txt6);         tCell7.Controls.Add(txt7);          tRow.Cells.Add(tCell1);         tRow.Cells.Add(tCell2);         tRow.Cells.Add(tCell3);         tRow.Cells.Add(tCell4);         tRow.Cells.Add(tCell5);         tRow.Cells.Add(tCell6);         tRow.Cells.Add(tCell7);   As you can see there's basically 4 instructions getting repeated 7 times. I'm sure there has to be a way to accomplish this with just 4 lines of code within a FOR loop and having all the names dynamically assigned but I just can't seem to find anything that would point me in the direction of how to do it.  Something like the following is what I'm after:      for (int i = 0; i &lt; 6; i++)     {          TableCell tCell[i] = new TableCell();         TextBox txt[i] = new TextBox();         tCell[i].Controls.Add(txt[i]);         tRow.Cells.Add(tCell[i]);      }    Any help would be much appreciated. SEO repeated keywords in domain and url how to use dom php parser How can infinite information be theoretically encoded or stored in a single qubit? I've just gotten started with Nielsen and Chuang's text, and I'm a little stuck. They mention that theoretically, it would be possible to store an infinite amount of information in the state of a single qubit. I'm not sure I completely comprehend this.    Here's how I rationalized it: You take all the information you want to store, put it in binary form, and make it the real component of $\alpha $ or $\beta$ (the coefficients of the computational basis states).   Now I'm not sure if I've understood it right, but since it's still fuzzy in my head it would be great to get some kind of ELI5 explanation or possibly a more detailed picture of how this would, even theoretically, be possible.    Apologies if the question doesn't meet standards. I'm new to the forum and would be open to feedback regarding asking questions or answering them. How do I remove a directory and all its contents? In bash all I know is that   rmdir directoryname   will remove the directory but only if it's empty. Is there a way to force remove subdirectories? Topological/Geometrical justification for $\text{CFT}_2$ being special It is known as a fact that conformal maps on $\mathbb{R}^n \rightarrow \mathbb{R}^n$ for $n&gt;2$ are rotations, dilations, translations, and special transformations while conformal maps for $n=2$ are from a much wider class of maps, holomorphic/antiholomorphic maps. I was wondering to know if there is any topological or geometrical description for this.   To show what I mean, consider this example: in $\mathbb{R}^n$ for $n&gt;2$ interchanging particles can only change the wave function to itself or its minus. It is related to the fundamental group of $\mathbb{R}^n-x_0$ ($x_0$ is a point in $\mathbb{R}^n$ and $\pi_1(\mathbb{R}^n-\{x_0\})=e$ for $n&gt;2$) but this is not true for $n=2$.   I want to know whether exists any topological invariant or just any geometrical explanation that is related to the fact that I mentioned about conformal maps on $\mathbb{R}^n$. How to make the taskbar blink my application like Messenger does when a new message arrive? "Seen" status is earlier than the user's last post I posted what I think is a clever answer to:    But then I noticed the question was posted a while ago on Mar 2.  I wanted to see what the chance of the user logging back in and seeing my answer, but the  says the user was seen on Feb 19.  I thought this was supposed to be the date they last logged in? How could they have asked a question on Mar 2 if they haven't logged in since February?  Thanks Repeated ANOVA unbalanced data I did experiment on one group of rats, the group was tested during 8 days, this is the first stage. The same group was also tested again but this time the test phase was 7 days. I tried to do my analysis using ANOVA repeated measures but I had a problem with the days.. I want to know the difference in the 2 tests but the days are not the same number. I don't know how to handle it. If there is any way to do the analysis using repeated measures ANOVA.   I tried to use repeated measure anova within subjects effect. My first factor will be test with 2 levels (test 1 &amp; 2),second factor is day with the level of ?? Here is the problem! if I write 8 there will be extra day in test 2 and the analysis cant carry!! Triangular Factorials I came across a statement online and have been looking for a proof :  It states that 1, 6 and 120 are the only numbers which are both triangular and factorials.  Is there any way I can prove this? This claim looks too 'big' and I've tried to prove it but  I couldn't. Can anyone help me to prove this? How do I remove the LAMP stack so I can start over? I'd like to start over with my webserver and I'd like to reinstall all the LAMP packages, how do I do that? Placing math environments inside math environments I have a macro which appears throughout my document in various places:  \newcommand{\placefraction}[2]{     $\frac{#1}{#2}$ }   Sometimes it is placed with regular text. Sometimes it is placed within other math modes. However, the document won't compile if it is placed inside another math mode.  None of my math ever appears in displayed formulas, it is always part of the paragraph.   How can I have a macro which sometimes appears inside math mode, sometimes outside, but itself contains math mode? $X$,$Y$,$Z$ mutually independent implies $X+Y$ independent of $Z$ Supposing $X$, $Y$ and $Z$ and mutually independent real random variables, how can we prove that $X+Y$ and $Z$ are independent from the definition? If not from the definition, using $\sigma$-algebras?  I know that if $X$ and $Y$ are not independent this doesn't work so I know I have to make use of their independence in my proof but I can't figure out how. And it means that if I write the following, I'm making a mistake, and I can't find it:  $$\mathbb{P}(X+Y \leq k,Z \leq l)=\int_{\mathbb{R}} \mathbb{P}(X=x,Y \leq k-x, Z \leq l) dx=\int_{\mathbb{R}} \mathbb{P}(X=x,Y \leq k-x) \mathbb{P}(Z \leq l) dx = \mathbb{P}(Z \leq l)\int_{\mathbb{R}} \mathbb{P}(X=x,Y \leq k-x)dx= \mathbb{P}(Z \leq l) \mathbb{P}(X+Y \leq k) $$  PS: I have seen the proof here but it much more advanced stuff, kind of a sledgehammer... I am trying to use Stack Exchange Andoid App to use my accounts in Stack Overflow &amp; Area 51.   For the Area 51, there is only Area 51 Discussion. Even Area 51 Discussion isn't working in the App!   VERSION 1.0.51  When adding this site, it's showing the correct full name as in this screen-shot:     However, when I pin this site to my fav sites in Stack Exchange, it is showing: Unknown Site. When clicking it, it shows black screen then just quit the App!  Here is the screen-shot:       Is the site not yet capable with Android? or is it a bug in the App itself?  PS: this question isn't duplicate:   In the previous versions of the App, the site was opening but with Unknown Site name. However, in this version, it quits the app when pressing the area site! It isn't opening at all! This question is also to track this bug in the App. As it's about the site name issue &amp; the content as well. Following a global preference for , as of now, the preferences for enabling keyboard shortcuts and disabling the fixed topbar are network-wide* as well.  If you had either checked off on any site in the network, that setting has been preserved and applied to the whole network for your account already. Future changes will also apply to all* Q&amp;A sites.  Find them in their usual spot as previously on the  pages.    * With the exception of , which will support the fixed topbar after the . Difference between a space and a "wave" symbol (tilde/"~")? Prove that $\sqrt[n]{n!}$ is increasing and diverges I have to prove that $a_n$ is (strictly) increasing and diverges  $a_n = \sqrt[n]{n!}$ ;  n $\in$  $\ \mathbb{N}$  From sequence I see that $a_n$ increasing to infinitive.  $\sqrt[1]{1!}=1 ,\  \sqrt[2]{2!} \approx 1.41, \  \sqrt[3]{3!}  \approx 1.81 ,...,  \sqrt[n]{n!} $ How can I run Windows applications on a MacBook Pro? I have a friend that wants to buy a Mac Book Pro and he has some windows applications that he needs to run. He doesn't want to buy the applications again.  Can he run the Windows applications on his MacBook Pro? What ways to accomplish this are available? I am building a regression model and I need to calculate the below to check for correlations   Correlation between 2 Multi level categorical variables Correlation between a Multi level categorical variable and continuous variable  VIF(variance inflation factor) for a Multi level categorical variables   I believe its wrong to use Pearson correlation coefficient for the above scenarios because Pearson only works for 2 continuous variables.   Please answer the below questions   Which correlation coefficient works best for the above cases ?  VIF calculation only works for continuous data so what is the alternative?  What are the assumptions I need to check before I use the correlation coefficient you suggest?  How to implement them in SAS &amp; R? Identifying records in MC to delete I'm trying to delete a large number of records from SFMC to avoid being charged extra. I've run several different SQLs to help identify the records I don't need, but when I run the Contact Count report, it still shows approx 100K billable contacts and approx 60K distinct email addresses.  Here are the SQLs I've tried:   No Opens in past 5 days Leads/Contacts not in SFDC  SubscriberKey as Email Address Leads/Contacts where SubscriberKey is Empty Unspecified records (DE populated by MC Support)    These efforts reduced the count a good amount, but not as much as I was expecting.   Has anyone else gone through this process before and if so, what queries can you share that might help my case?