So I had a test today, and one of the questions was:   "Give an example of a set, S contained in $R^3$ that is closed under scalar multiplication but isn't closed under addition"  And I was just wondering -- is there a systematic way to go about this rather than search for examples? Perhaps some sort of algebraic way to create such a set that satisfies these conditions without any further thought?
Could anyone provide an example of a nonempty subset $U$ of $R^2$ such that $U$ is closed under scalar multiplication, but $U$ is not a subspace of $R^2$
HTTP Callout from Triggers I have a trigger on the Custom Object that calls a class. That class makes an HTTP GET request to a Web Service.  However I get the following error:  Callout from triggers are currently not supported.  How can I get data to an external Web Service using Apex code?  However i can use @future method but it can only be support void and this method cannot return results, i need to be able to send data to external ervice and receive acknowlegements from it. How can i make the grass bend when this cube hits it?  Here's the picture: pgfplots: prevent single plot from being listed in legend In order to fill the area between two complex curves I use tables and the vertcat command (see ).   Now I don't want the plotted area to appear in my legend, just the two curves. Is there any command to prevent the yellow area plot from being added to my legend?  this could be a minimal example:  \documentclass{article} \usepackage{pgfplots}  \begin{document} \begin{tikzpicture}   \begin{axis}[area style,axis on top]     \addplot+[mark=none,fill=yellow,draw=none] {0.1*x^2} \closedcycle;     \addplot[mark=none,draw=red,line legend] {0.1*x^2};     \addplot[mark=none,draw=blue,line legend] {0.0*x^2};     \legend{red curve,blue curve};     % \legend{\empty,red curve,blue curve};   \end{axis} \end{tikzpicture} \end{document}   With this legend{...}-command the last plot is skipped, an additional \empty doesn't help, too.  Sadly, drawing the first to plots together with fill=yellow,draw=red is not an option for me. Does anybody know any inside hacks? As far as I know, there is no command shown in the manual. Thanks in advance. Nonstandard normal distribution Alignment of equals sign in multiple align environments I have a bit of a unique problem. Well, maybe not a problem because I'm really just curious about it, but...  I have a simple 8 digit calculator. It has +, -, x, /, and a constant operation function. It has no usable memory slot.  I need (lit. would like to know how) to calculate ln(x) on this calculator. Any series I have found lose a lot of precision since whatever is outside the 8 digit range is truncated without rounding. I have used Padé approximants with trig functions, and a continued fraction for exp(), but none of these two methods are adequate for ln(x), either because there is no way around the truncation error or it requires writing down many intermediate values, or it requires memorizing and punching in many constants.  Does anyone know any good way to calculate ln(x) (or any other base, since memorizing one constant is OK) to as many digits as possible as easily as possible on this calculator (bonus points to methods where the value on the screen is immediately, meaning you don't need to write it down)?  For that matter, if you happen to know some good ways to calculate other special functions, I am interested in that as well.  Thanks! Is Firefox 57 multithreaded? According to  on MDN, Firefox 57 finally runs things in different processes.  So is Firefox 57 multithreaded?  In particular, how many cores does it use at most? Say I open 20 tabs at once on a quad-core machine, will Firefox 57 launch as many threads, or at least as many as needed to max out my CPU?  (Chromium for instance spawns a number of sub-processes and very easily uses all the resources available. Firefox up to 56 in GNU/Linux is rarely seen going over 135 % of one core however many tabs I open.) How can I easily clone my entire system? I am running an Ubuntu 14.04 LTS and I experienced some issues with my hard drive. I bought a new HDD of the same size (1TB), and I intend to move all my data (softs, work, pics, etc.) on the new HDD.  What is the easiest way to do that? I am very new to LaTeX and I am writing my thesis. In a chapter, I want to show my codes, which are in Mathematica.   I use the listings package as follows:  \documentclass{article}   \usepackage{listings}   \begin{document}   \lstset{language=Mathematica}    \begin{lstlisting}   Do[i^j,{i,1,10},{j,1,10}]   \end{lstlisting}    \end{document}   However, the output is much different to what the code looks like in the Mathematica IDE (even if I use other languages in \lstset{language=&lt;name&gt;}):    I read some questions on the site but, in some of them, the font or color of the codes were not the same. Could you please help me to display something like the picture below in LaTeX?  If it is important, I am using the BiDi package. How can I prove the formula for calculating successive entries in a given row of Pascal's triangle? I've found in Wikipedia the formula for calculating an individual row in Pascal's Triangle: $$v_c = v_{c-1}\left(\frac{r-c}{c}\right).$$ where $r = \mathrm{row}+1$, $c$ is the column starting from $0$ on left and $v_0 = 1$.  Now, I've tried to do by hand and it works,  but I don't understand how to find this magic formula when I don't have access to Wikipedia...:) How do you get \mathbb{1} to work (characteristic function of a set)? Why is menstruation in wild animals not a disadvantage to organismal survival? When I watch wilderness specials for more than a few minutes, I notice a familiar pattern: predators are depicted as being alerted by the scent of blood. Wounded animals seem to make the best prey and predators seem to have a heightened awareness for the smell of blood.   It would seem that it would then be a disadvantage to evolutionary survival for female organisms to menstruate. How do females of a species not attract predators to the signal of menstrual blood?   Do female organisms in the wild actually menstruate? If so, is there a difference in the composition of menses that does not alert predators? How would this not at the very least give predators an idea of the general vicinity of the female?  If wild animals excrete their menses, do they have ways in which they are able to mask the evidence of its release or are they somehow internalizing the tissue so that they are not released?  If female organisms in the wild do menstruate, then why is this not a survival disadvantage? Lately my laptop (HP g42 247sb) screen is damaged, so I hook it up with an external monitor (LG something) and it works fine now.  The only annoying thing is I cannot navigate the BIOS menu for some tweaking because the BIOS not shown on the external monitor, instead, it only shown on the broken laptop screen, and  it only output to my external monitor when Windows/OS is logged-on.  So, is there anyway I can force output during BIOS/BOOT/POST to my external monitor?  Things I have done and didn't work:   Set my LG monitor as primary display on both window properties and Intel Graphics panel  Enter the BIOS (F10 key) and press the Fn+F4 key (change display output). Disable and uninstall my internal screen (broken laptop screen) using device manager and restart, but Windows (BIOS?) install it back on log-on. Closed lid/Magnet on sensor. Evaluation of $\int\frac{1}{x^2.(x^4+1)^{\frac{3}{4}}}dx$ Write down the sum of sum of  sum of digits of $4444^{4444}$ Let $A = 4444^{4444}$;  Then sum of digits of $A = B$; Then sum of digits of $B = C$; Then sum of digits of $C = D$;  Find $D$.  What should be the approach here? How do you maximize the cycle count of your battery?  There seems to be a debate between:   Leave plugged in 24/7 but do a full cycle once a month Charge to ~100%, drain to ~10%, repeat.   NOTE: Taking your   Which method is better and why? How much of a difference does it really make?  Here are some of the better sources I've found about battery behavior so far. Even still I feel they don't conclusively answer the question above. See my elaborations below:           Elaborating on Option 1  Now IF Apple engineers were smart and optimized for AC draw when plugged in to not use up your charge cycles, then it seem logical to assume that you are using effectively none of your 1000 charge cycles while plugged in. If that's the case, it would seem that the LiPo drop from holding at 100% would be vastly outweighed by the fact you're not using any charge cycles.  However, this is a purely speculative assumption and I could find no evidence to either confirm or deny.  If this is NOT true, and the AC adapter only goes to charging the battery, then there would be no difference. Your battery would drain to 99%, then back up to 100%, then back down to 99%, etc. Those micro-charge cycles would add up at the same rate as 100% -> 0% -> 100%, and you would get no gain. In this case the negative effect of holding LiPo batteries at 100% would outweigh everything else.  Elaborating on Option 2  There are lots of good reasons why Option 2 is the way it is:   Because of known LiPo chemistry issues, holding a 100% charge for a long time causes the battery to degrade Apple specifically recommends not to leave it plugged in all the time  Draining a battery all the way to zero all the time is bad for the cell (Which is why ~10% is used) Heat is a killer, and when plugged in AND charging, you get extra heat that causes damage.   There are a couple reasons why I'm challenging Option 2:   It seems silly that the most intuitive use case for thousands of people that use their macs as primary computers is wrong. If the hypothesis in option 1 is true, you're just burning unnecessarily through your finite charge cycles. It's a pain to have to remember to keep plugging in and plugging out and be worried about whether or not your device is plugged in. Is there a free site to play Yu-Gi-Oh? Submitting a form by pressing enter without a submit button is a simple way to script small GUIs.  Can anyone give a nice example with a button and a text widget. When the button is pressed should a shell command be executed and the output piped to the text widget.  If you have other nice and clean examples for useful tasks, please add them too. Proving that the set of limit points of a set is closed