Internals of Prime function
What is so special about Prime?
Additive function $T: \mathbb{R} \rightarrow \mathbb{R}$ that is not linear. A function $T:V \rightarrow W$ is additive if $T(x+y) = T(x) + T(y)$ for every $x, y \in V$. Prove that there exists an additive function $T: \mathbb{R} \rightarrow \mathbb{R}$ that is not linear.  My attempt: Let $T$ be the function $T: \mathbb{R}$ (over the field $\mathbb{Q}$) $\rightarrow \mathbb{R}$ (over the field $\mathbb{R}$). The set $\{1, \sqrt{2}\} \subseteq \mathbb{R}$ is linearly independent for the vector space $\mathbb{R}$ over $\mathbb{Q}$. Then, there must exist a linearly independent set $W \subseteq \mathbb{R}$ (over $\mathbb{Q}$) such that $\{1, \sqrt{2}\} \subseteq W \subseteq \text{span}(W) = \mathbb{R}$ (over $\mathbb{Q}$).  I have been told that the function defined as $T(1) = 1$ and $T(w) = 0$ for all $w \in \text{span}(W) \setminus \{1\}$ is additive but not linear, but I cannot see why this is? I can see why it is not linear, clearly $T(\sqrt{2} \cdot 1) = 0$ but $\sqrt{2} T(1) = \sqrt{2}$. But, why is $T$ additive? For example, $T(1+1) = T(2) =0$ but $T(1) + T(1) = 1+1 = 2$? Is there a mistake somewhere? Let $f$ be a continuous function on a set $E$.  Is it always true that $f^{-1}(A)$ is always measurable if $A$ is measurable?  I say no.  We know that $\bar{\psi(x)}:=\frac{\phi(x)+x}{2}$ where $\phi$ is the Cantor (or Cantor-Lebesgue) function and $\bar{\psi}:[0,1] \rightarrow [0,1]$.  We know this function maps a measurable subset of $C$ (the Cantor set) into a non-measurable set $W$.  A few observations about $\bar{\psi(x)}$.  It is strictly increasing and continuous, so it has a continuous inverse $\bar{\psi(x)}^{-1}$.  Thus $\bar{\psi(x)}^{-1}(W)=c \subset C$; that is, $\bar{\psi(x)}^{-1}(W)$ is non-measurable, but c is.  Is this correct? How to derive the formula for the sum of the first $n$ odd numbers: $n^2=\sum_{k=1}^n(2k-1).$ How to derive this formula?   $$n^2=\sum_{k=1}^n(2k-1).$$ It's essential that the documents (should)/(ought to) be destroyed immediately.   Which one is more suitable one as I know "ought to" and "should" are synonyms. Need help finding a good book on Riemann Geometry strptime() equivalent on Windows? Is there a good equivalent implementation of strptime() available for Windows?  Unfortunately, this POSIX function does not appear to be available.   - summary: it converts a text string such as "MM-DD-YYYY HH:MM:SS" into a tm struct, the opposite of strftime(). One of my bank accounts earned $1.50 for the entire year.  Because the amount is under $10 they did not send a 1099-INT form.   Is this bank policy or IRS policy?  Does that $1.50 need to be reported?  I'm not worried about the difference that this small amount makes on my taxes bur rather what is legally required. Are getters and setters poor design? Contradictory advice seen I'm currently working on a simple game in Java with several different modes. I've extended a main Game class to put the main logic within the other classes. Despite this, the main game class is still pretty hefty.  After taking a quick look at my code the majority of it was Getters and Setters (60%) compared to the rest that is truly needed for the logic of the game.  A couple of Google searches have claimed that Getters and Setters are evil, whilst others have claimed that they are necessary for good OO practice and great programs.  So what should I do? Which should it be? Should I be changing my Getters and Setters for my private variables, or should I stick with them? When can a (finite) group be written as the quotient of some other group by its center? So if I'm given $H$, when I can conclude there is a group $G$ such that $H\cong G/Z(G)$? It's easy to show that non-trivial cyclic groups are not of this form. More generally, any group with the property that all finitely generated subgroups are cyclic (e.g. $\mathbb{Q},\mathbb{Q}/\mathbb{Z}$) cannot be of this form. Are there are any others? I have a list of requirements for a program, but I don't know what program meets these criteria. That's about general-purpose software, for example, not a programming IDE or statistical workbench, questions about which would certainly go to respective sites.  Is this question appropriate on Stack Overflow? Where should I ask about it? On Super User or elsewhere? How can a single slit diffraction produce an interference pattern? How can a light passed though a single slit produce a similar interference pattern to the double-slit experiment? How does the diffracted wave produce the points of cancellation and reinforcement, if there is only one wave? When to use "respectively"? I have been wondering what it means when people use "respectively" in, before, and after sentences. For example:     We are looking for a babysitter to pick up and supervise our kids ages   6 and 3, respectively, on Monday and Friday.   I don't know if this person used it correctly, but if they did what does it mean? If they didn't use it properly what would be the proper usage?  Thanks in advance. No CUDA and GPU option ETS() function, how to avoid forecast not in line with historical data? I am working on an alogorithm in R to automatize a monthly forecast calculation. I am using, among others, the ets() function from the forecast package to calculate forecast. It is working very well.  Unfortunately, for some specific time series, the result I get is weird.  Please, find below the code i am using :  train_ts&lt;- ts(values, frequency=12) fit2&lt;-ets(train_ts, model="ZZZ", damped=TRUE, alpha=NULL, beta=NULL, gamma=NULL,              phi=NULL, additive.only=FALSE, lambda=TRUE,              lower=c(0.0001,0.0001,0.0001,0.8),upper=c(0.9999,0.9999,0.9999,0.98),              opt.crit=c("lik","amse","mse","sigma","mae"), nmse=3,              bounds=c("both","usual","admissible"), ic=c("aicc","aic","bic"),             restrict=TRUE)   ets &lt;- forecast(fit2,h=forecasthorizon,method ='ets')      Please, you will find below the concerned history data set :   values &lt;- c(27, 27, 7, 24, 39, 40, 24, 45, 36, 37, 31, 47, 16, 24, 6, 21,  35, 36, 21, 40, 32, 33, 27, 42, 14, 21, 5, 19, 31, 32, 19, 36,  29, 29, 24, 42, 15, 24, 21)   Here, on the graph, you will see the historical data (black), the fitted value (green) and the forecast(blue). The forecast is definitely not in lines with the fitted value.  Do you have any idea on how to "bound" the forecat to be "in line" with the historical sales? Where to start looking up if there's something wrong with cck field variables? in short: Something in my zen-based custom theme is keeping a couple of cck fields from being display on my page. Basically my question is where to get access to the variables that are supposed to contain the html for the cck fields (which are missing when I was looking them in firebug). Below I explain why I think it's looking up the variables that might bring me closer to a solution  So I made this custom node type (events) with cck fields added. Also made some content using those fields. Added some css and it all displayed fine. Later the taxonomy terms below the title and some metadata bugged me so I followed along with a 'theming cookbook' pdf book that taught me how to make a node-events.tpl.php and simply deleted php variables like $terms. This worked great.     Obviously this also means that all this time, the view permissions of those fields have been checked and also that the display options under 'display fields' have been set to standard all the time. I also ruled out that the issue could be anything else than theme related as my cck fields do show up in a stock Drupal theme. The theme for my site is a zen-based custom theme.   So then even later, after I made a view for this node type and started adding css to that view I noticed at some point that my image field and two other fields are missing. It's hard to tell if they disappeared because of things that I did when creating the view and adding css (which would surprise me) InDesign: How do I automatically align the bottoms of different columns of text boxes to the same height as the one with the most text? In InDesign, I have some filled text boxes, sized by columns, with the one on the left being three columns wide.   The wide text box is most likely to contain the most text (it's an invoice template).  I will use the 'Fit frame to content' button on the Selection Tool toolbar for automatically vertically resize it to fit the containing text.  How do I make the other three smaller text frames automatically adjust to the same height as the big (and longer) text box? Making adult decisions "is" or "are" really not fun: Which is correct? Embedding images explodes generated PDF size What are the differences between executing shell scripts using "source file.sh", "./file.sh", "sh file.sh", ". ./file.sh"? Have a look at the code:  #!/bin/bash read -p "Eneter 1 for UID and 2 for LOGNAME" choice if [ $choice -eq 1 ] then         read -p "Enter UID:  " uid         logname=`cat /etc/passwd | grep $uid | cut -f1 -d:` else         read -p "Enter Logname:  " logname fi not=`ps -au$logname | grep -c bash` echo  "The number of terminals opened by $logname are $not"   This code is used to find out the number of terminals opened by a user on the same PC. Now there are two users logged on, say x and y. I am currently logged in as y and there are 3 terminals open in user x. If I execute this code in y using different ways as mentioned above the results are :  $ ./file.sh The number of terminals opened by x are 3  $ bash file.sh The number of terminals opened by x are 5  $ sh file.sh The number of terminals opened by x are 3  $ source file.sh The number of terminals opened by x are 4  $ . ./file.sh The number of terminals opened by x are 4   Note: I passed 1 and uid 1000 to all these executables.  Now can you please explain the differences among all these? I opened the PC in the morning, and this is what I got:      As is visible the icons of the UWA apps were no affected and the start menu also is unaffected.    When Control Panel is in category view the entire page is empty and in large/small icons the name is visible, no icons.  I deleted the iconcache.db in %USERPROFILE%\AppData\Local folder. Following the instructions here: . But to no avail and now windows is not rebuilding the cache.   I also ran sfc /scannow but no changes took place.  What should I do?  (01/01/2016 8:00PM GMT +5:30) Edit: I tried opening the system properties and I got 0x8000ffff error and this:        Then I closed the popup and pressed the OK button on the System Properties Windows and I got 0x8007057A error:      Opening the Image Backup Control Panel page I again got an 0x8007000E error:        I tried using system restore via advanced startup but once again I got 0x800FFFF error.  P.S.: I did try restarting, didn't work.