It's always taken for granted that the real number line is perpendicular to multiples of $i$, but why is that? Why isn't $i$ just at some non-90 degree angle to the real number line? Could someone please explain the logic or rationale behind this? It seems self-apparent to me, but I cannot actually see why it is.  Furthermore, why is the real number line even straight? Why does it not bend or curve? I suppose arbitrarily it might be strange to bend it, but why couldn't it bend at 0? Is there a proof showing why?  Of course, these things seem natural to me and make sense, but why does the complex plane have its shape? Is there a detailed proof showing precisely why, or is it just an arbitrary choice some person made many years ago that we choose to accept because it makes sense to us?
So, floor is a function that converts a real number to an integer. It rounds down. This makes sense; however, what about complex numbers? I know that depending on the number, it can be split linearly. However, I do not know what $\lfloor i\rfloor$ equals. Is there a clearly defined definition of $\lfloor i\rfloor$?
Looking for sci-fi novel I read about humanity finding alien station on orbit of Earth I am looking for a novel that I'd love to re-read it with my wife (my 13-year old self must be proud of me I guess). Key Plot points The plot was about humanity finding an alien station on an orbit of Earth. (I'm not sure if it wasn't Moon itself). On this station there were thousands of ships ready to travel so there rose a company that paid people to get on those ships and explore the galaxy. The problem was humans didn't understand how to steer those ships so they had to rely on preset targets. Of course there was no telling where particular ship was to go and fair amount of them never returned. Chronology and other relevant information I read it as an early teenager back in early 2000s. It was Polish translation of a foreign (possibly English) story. I think (not sure) that the book wasn't new at the time. It was just one of those books you dig out in a library and take to see if it's good. I don't recall cover either and even if I did it'd be of the Polish version. Additional information could be that the narration was led along the psychoanalytics of the main protagonists where he relives his memories from inside the station and his journeys. Finding the maximum likelihood estimator (Theoretical statistics) Let $X_1, X_2, ..X_n$ represent a random sample from this pdf:  $$f(x|\theta)= \frac{3x^2}{\theta^3}, 0\leq x\leq \theta$$ (with 0 elsewhere)  Could someone explain how I would find the maximum likelihood estimator (MLE) of this pdf? I know I have to get the likelihood function L, then the log likelihood function, then differentiate, but I am getting lost in the calculations. Thanks! How do I get the local machine name? Where in the file system are applications installed? I have installed a few Android applications both with the adb install MyApplication.apk command and via a webserver (like an alternative to Android Market).  But where in the Android file system on my phone is the MyApplication.apk file placed? Are applications installed with adb install not placed in the same directory as applications downloaded from the Market?  I found some applications on /system/app but I can't recognize them as some downloaded from Android Market or the one I have installed with adb install.  Where on the system is applications that I have installed located?  My phone is not rooted, and I am running Nexus One with Android 2.2.1. proving combinatorics identity - $\sum_{k=0}^m{n-k \choose m-k}={n+1 \choose m}$ $ which echo echo: shell built-in command. $ which ls /bin/ls $ which cat /bin/cat   Why is echo not an independent utility like ls, ps, cat etc? Why is it shell specific? Any good reasons? If I look through the microwave window I can see through, which means visible radiation can get out. We know also that there is a mesh on the microwave window which prevents microwave from coming out.  My question is how does this work? how come making stripes or mesh of metals can attenuate microwave radiation yet allow visible radiation?  Looks like an electrodynamics problem to me with periodic boundary conditions (because of the partitions on the microwave oven window). Is it discussed in any textbook? I have a psd that has rounded rectangle. I need to create a new rounded rectangle with the same radius. How do I determine the radius used on that said rounded rectangle?   I feel like this is such an awful question but thanks in advance for answering. Is it possible to use `find -exec sh -c` safely? I'm trying to use find to echo 0 into some files, but apparently this only works with sh -c:  find /proc/sys/net/ipv6 -name accept_ra -exec sh -c 'echo 0 &gt; {}' \;   But using sh -c with find -exec makes me feel very uneasy because I suspect quoting problems. I fiddled a bit with it and apparently my suspicions were justified:   My test setup:  martin@dogmeat ~ % cd findtest  martin@dogmeat ~/findtest % echo one &gt; file\ with\ spaces martin@dogmeat ~/findtest % echo two &gt; file\ with\ \'single\ quotes\' martin@dogmeat ~/findtest % echo three &gt; file\ with\ \"double\ quotes\" martin@dogmeat ~/findtest % ll insgesamt 12K -rw-rw-r-- 1 martin martin 6 Sep 17 12:01 file with "double quotes" -rw-rw-r-- 1 martin martin 4 Sep 17 12:01 file with 'single quotes' -rw-rw-r-- 1 martin martin 4 Sep 17 12:01 file with spaces  Using find -exec without sh -c seems to work without problems - no quoting necessary here:  martin@dogmeat ~ % find findtest -type f -exec cat {} \; one two three  But when I'm using sh -c {} seems to require some kind of quoting:  martin@dogmeat ~ % LANG=C find findtest -type f -exec sh -c 'cat {}' \; cat: findtest/file: No such file or directory cat: with: No such file or directory cat: spaces: No such file or directory cat: findtest/file: No such file or directory cat: with: No such file or directory cat: single quotes: No such file or directory cat: findtest/file: No such file or directory cat: with: No such file or directory cat: double quotes: No such file or directory  Double quotes work as long as no file name contains double quotes:  martin@dogmeat ~ % LANG=C find findtest -type f -exec sh -c 'cat "{}"' \; one two cat: findtest/file with double: No such file or directory cat: quotes: No such file or directory  Single quotes work as long as no file name contains single quotes:  martin@dogmeat ~ % LANG=C find findtest -type f -exec sh -c "cat '{}'" \; one cat: findtest/file with single: No such file or directory cat: quotes: No such file or directory three    I haven't found a solution that works in all cases. Is there something I'm overlooking, or is using sh -c in find -exec inherently unsafe? How are answers with the same vote score sorted? When a new question is asked how are the answers sorted?  I just answered a question and there were two other answers.  All three answers have 0 votes and no edits.  Immediately after I answered my answer appeared first.  Now the order refreshed,  it appears they are sorted by reputation. Why is the picture not inserted? How can PCB trace have 50 ohm impedance regardless of length and signal frequency? Hmm, this seems to be just another question on line impedances.  I understand that when we say "transmission line" effects we talk about things like cross talk, reflections and ringing (I guess that is just about it). These effects are not present at low frequencies where the PCB trace behaves like an "ideal" transmission medium, more like we expect a wire to behave in our early school days.  I also understand that the 50&nbsp;ohm value comes not from the line resistance which is going to be very small and less than 1&nbsp;ohm. This value comes from the ratio of L and C on the line. Changing C by changing the trace height above ground plane or changing L by changing the trace width shall change the impedance of the line.  We all know that the reactance of L and C is dependant on the signal frequency as well. Now my questions:   Why should we not call this as line reactance only rather than line impedance? How can it be just 50&nbsp;ohm? It has to be signal frequency dependent right?  E.g 50&nbsp;ohm at 1&nbsp;MHz  Will the world end if I chose to do a 100&nbsp;ohm or 25&nbsp;ohm trace instead?  I know that while we like to say 50&nbsp;ohm as a magic number, it will be within some range around 50&nbsp;ohm and not 50.0000 ohm exactly.  Is there any time when the actual resistance of a PCB trace may matter? I am having trouble with the following problem.      Let $R$ be an integral domain, and let $a \in R$ be a non-zero element. Let $D = \{1, a, a^2, ...\}$. I need to show that $R_D \cong R[x]/(ax-1)$.    I just want a hint.  Basically, I've been looking for a surjective homomorphism from $R[x]$ to $R_D$, but everything I've tried has failed. I think the fact that $f(a)$ is a unit, where $f$ is our mapping, is relevant, but I'm not sure. Thanks Show that the set of all finite subsets of $\mathbb{N}$ is countable. Prime factors + number of Divisors I know that one way to find the number of divisors is to find the prime factors of that number and add one to all of the powers and then multiply them together so for example  $$555 = 3^1 \cdot 5^1 \cdot 37^1$$  therefore the number of divisors = $(1+1)(1+1)(1+1) = 2 \cdot 2 \cdot 2 = 8$  What I do not know (and can't seem to find when I look) is WHY this relationship exists or a formula which shows its proof.  Has anyone seen such a formula? Dynamic select list in the form (dependent dropdown) Asking feedback/rating for a mobile app? Fig.1: A typical prompt, asking user for feedback/rating in iOS Background Any serious smartphone users must have seen a message similar to Fig.1 above at least once. Variations of such prompt include: &quot;Please give us 5 star ratings if you love this app!&quot;, &quot;Rate us :)&quot;, &quot;Your rating helps improve this app!&quot;, etc. Basically, Fig.1 asks user to leave a (nice) feedback/rating for the app on its AppStore(or equivalents).  The Shorter Question  When and how often should a mobile app user be asked to leave a feedback/ratings?   The Longer Question #1. Which User-statistics? What are the significant user's behavioral statistics that should be based on to generate Fig.1? For example:  Number of runs since install/reinstall of the application Time elapsed since current application run Time elapsed since first application run Time elapsed since first application install Certain events (e.g. after certain number of stages are cleared for a game app)  #2. When to Ask? If multiple user-statistics are chosen to be calculated to generate Fig.1, specifically how should the user-statistics satisfy the condition of the event? For example:  Form a unified equation (e.g. X^2+Y^2=Z^2) with all significant user-statistics partaken fairly to trigger the event. Create (multiple of) &quot;absolute&quot; event triggers (e.g. every 10th app run; elapsing 2 hours since current application run). Both of the two. Generate Fig.1 whichever comes true first. Neither of the two. &quot;There's a better way!&quot;  #3. How often? What would be the optimal min/max frequency of Fig.1 generation? By optimal, I mean users are not to be bothered too much, yet developers gain maximum output of great feedbacks from users? #4. &quot;Don't bother me again :(&quot; Should users have an option to opt-out from receiving Fig.1 at all? #5. Universal User-statistics for All Mobile Apps? Should apps under different categories be based on different user-statistics? There are significant differences in size and type of user-statistics in apps under different categories. For example, a &quot;Flashlight&quot; app is not typically run for one hour straight, while a music/movie streaming app may be run for over one hour. #6. &quot;This app does not need more feedback.&quot; Should every app of all kinds of user base ask this message? If any, should any mobile app stop generating Fig.1 when target conditions are met (e.g. downloads count, active user count, feedback count)? #7. Feedback/Rating Before Deletion Should users be asked to leave a feedback before deleting as seen in Fig.2 below? Users are more likely to leave very critical review on app deletion, but this kind of feedback will be very helpful for the developers to strengthen their app's weakness for next version release.                                                                                                 Fig.2: Asking user for rating before app deletion How to make GRASS and SAGA tools capable to use in QGIS？ I installed QGIS, GRASS and GDAL by OSGeo4w package, and as I finish the installation, I want to use GRASS tool, I face this problem：    But I already installed GRASS and SAGA, how to make those tools in QGIS to be used？ Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $. Here is my problem: I've got a dialog with some parameters that the user can change (via a spinner for example). Each time one of these parameters is changed, I launch a thread to update a 3D view according to the new parameter value.  If the user changes another value (or the same value again by clicking many times on the spinner arrow) while the first thread is working, I would like to abort the first thread (and the update of the 3D view) and launch a new one with the latest parameter value.  How can I do something like that?  PS: There is no loop in the run() method of my thread, so checking for a flag is not an option: the thread updating the 3D view basically only calls a single method that is very long to execute. I can't add any flag in this method asking to abort either as I do not have access to its code.