If you choose an answer to this question at random, what is the chance you will be correct?  A) $25\%$  B) $50\%$  C) $60\%$  D) $25\%$
I found this math "problem" on the internet, and I'm wondering if it has an answer:     Question: If you choose an answer to this question at random, what is the probability that you will be correct?      a. $25\%$      b. $50\%$      c. $0\%$      d. $25\%$   Does this question have a correct answer?
How do I restrict my kids' computing time? Access to our computer (not only to the internet) needs to be restricted for the accounts of my kids (7, 8) until they are old enough to manage this by themselves. Until then we need to be able to define the following:   the hours of the day when computing is o.k. (e.g. 5 - 9 pm) the days of the week when computing is not o.k. (e.g. mondays to fridays) the amount of time allowed per day (e.g. 2 hours)   In 11.10 all of the following that used to do the job don't work any more:   : for > 11.10 no more available through the ppa. : command line alternative, but from 11.10 removed from the repositories. : Looks great but repeatedly crashes to force restarting X-server. So we can't use or recommed this program at the moment.   Are there any other alternatives? (PHP) Is handling a form submission on the same page more/less/equally good as handling on a separate page? How can I debug my JavaScript code? Surjective morphism of affine varieties and dimension I'm trying to do the following exercise from the book An Invitation to Algebraic Geometry:     Show that if $X \to Y$ is a surjective morphism of affine algebraic varieties, then the dimension of $X$ is at least as large as the dimension of $Y$.   Could anyone give me a hint? I'm looking for a proof that avoids commutative algebra since it is not mentionned in the book at this point. Relationship between complex number and vectors What is the relation  between complex numbers and vectors?  I have read in some places "a complex number a 2-dimensional vector".  One can easily observe that $i\cdot i=-1$ in complex multiplication like the cross product of vectors.    Is  vector only the generalisation of complex number to n dimensions?   Is there any fundamental difference between those two? Elliptical orbits of planets As we know, the velocity of the planet changes as it orbits around the sun, both in its magnitude and direction. The centripetal acceleration causes the change in the direction of velocity but I want to know what is the tangential force that acts on the planet to change the velocity by its $magnitude$. The answer normally as to why the speed changes is: If  the  force  that  the  Sun  exerts  on  the  planet increases  (as  the  planet moves closer),  then  the acceleration   of  the  planet must  increase,  resulting  in a  higher  orbital  speed,  and vice  versa. But this force is not along the body but towards the sun. So such a force should only change its direction. Why Dumbledore kept mum until the very end of the "Deathly Hallows"? I have an ASUS E402S laptop with Ubuntu 64-bit 16.04 LTS installed. The main problem is that the OS does recognize the touchpad with basic functions, but additional options such as scrolling through pages or between open applications are non-existent.  This is what my xinput result is showing:  ⎡ Virtual core pointer                      id=2    [master pointer  (3)] ⎜   ↳ Virtual core XTEST pointer                id=4    [slave  pointer  (2)] ⎜   ↳ Logitech USB Receiver                     id=9    [slave  pointer  (2)] ⎜   ↳ Logitech USB Receiver                     id=10   [slave  pointer  (2)] ⎜   ↳ FTE1000:00 0B05:0101                      id=12   [slave  pointer  (2)] ⎣ Virtual core keyboard                     id=3    [master keyboard (2)]     ↳ Virtual core XTEST keyboard               id=5    [slave  keyboard (3)]     ↳ Power Button                              id=6    [slave  keyboard (3)]     ↳ Video Bus                                 id=7    [slave  keyboard (3)]     ↳ Sleep Button                              id=8    [slave  keyboard (3)]     ↳ USB2.0 VGA UVC WebCam                     id=11   [slave  keyboard (3)]     ↳ Asus WMI hotkeys                          id=13   [slave  keyboard (3)]     ↳ AT Translated Set 2 keyboard              id=14   [slave  keyboard (3)]   Apparently my touchpad's type is not shown in the list. What are the possible solutions to fix this issue? Finishing a proof: $f$ is injective if and only if it has a left inverse I've already done a lot of searching (in particular: ) to try to prove this statement:      $f: A \to B$ is injective if and only if it has a left inverse. [I'm going to also assume $A$ and $B$ are nonempty.]   My attempt so far:     $\Leftarrow$ Suppose $f$ has a left inverse and $f(a_1) = f(a_2)$ for    $a_1, a_2 \in A$. Then there is a function $g: B \to A$ such that   $g\left(f(a_1)\right) = a_1$ and $g\left(f(a_2)\right) = a_2$. Since   $g$ is well-defined, it follows that $g\left(f(a_1)\right) =  g\left(f(a_2)\right)$, and hence $a_1 = a_2$.         $\Rightarrow$Now   suppose $f$ is injective. Then $f(a_1) = f(a_2) \implies a_1 = a_2$.   Furthermore, since $A$ is nonempty, there is an element $x \in A$.   Define $g: B \to A$ by  \begin{equation*} g(b) = \begin{cases}  f^{-1}(b), &amp; b \in \text{Im}(f) \\ x, &amp; b \in A \setminus  \text{Im}(f)\text{.} \end{cases} \end{equation*} This mapping is   well-defined, since if $f$ is injective, $f^{-1}(b)$ is unique.   My main question: does this imply that if $f: A \to B$ is injective that any mapping $g: \text{Im}(f) \to A$ is bijective? Otherwise, I don't understand why we can even use $f^{-1}$ on an element. [Please read the link above for more details - in proof 1.] Installing Steam on Debian I have ran into some issues when it went down to install Steam on Debian. The console as well as Apper comes up with missing dependencies  and the package isn't going to install. Show that both mixed partial derivatives exist at the origin but are not equal $$f(x,y) = \begin{cases} \displaystyle \frac{xy(x^2-y^2)}{x^2+y^2} &amp; \text{if } (x,y) \neq (0,0), \\ 0 &amp; \text{if } (x,y) = (0,0). \end{cases}$$ I tried finding both mixed partial derivatives but they ended up being the same for that function. I must be failing to take into account something dealing with the fact that it is piece-wise. I still need to show the mixed partial derivatives exist. How can I do all of this? Punctuation of a Direct Quote Is any particular punctuation necessary when a quotation is the subject of a sentence? For example, in the sentence below, should there be a comma after export? Or anything else?     "Laughter is America's most important export" is a well-known quote of Walt Disney's. Zooming / Focusing certain parts of a chart in latex I am using pgfplot to draw graphs in latex. Now this is the problem I am facing:  \begin{tikzpicture}[scale=1] \begin{axis}[ybar,grid=major,bar width=2pt,width=8cm, height=8cm,xmin=1,xmax=540,ymin=1,ymax=240] \addplot [draw=black,fill=green]  table [x=size, y=cluster, col sep=comma] {sim-0.4-0.6.csv}; \end{axis} \end{tikzpicture}   It generates the following graph:    Since, I have values mainly between 0-50 on x-axis, I want to zoom there. After that, values comes around 500. S0, there are no graph points between 50 - 450. How should I do that? Use Data From Class Without A New Init Anything I should keep in mind before I start mixing and matching between D&D 3.5e and Pathfinder? Anything I should keep in mind before I start mixing and matching between D&amp;D 3.5e and Pathfinder?  Off the top of my head, the biggest difference is feats every 2/3 levels. Would it affect balance if I picked every 3 levels instead of 2? I think PF options are still pretty solid regardless... As I finished programming the MCU in the Arduino Uno board to receive some RF Signal and drive relays, I moved it into a standalone board I made with Eagle. The circuit is pretty basic, the MCU receives data from a RF Receiver, processes it and then turns on/off some pins that drive 5V relays (through transistors, of course). When I test it on the Arduino Board, you can see that the LEDs toggle ON and OFF whenever I send the data through the RF Transmitter. You send once, it toggles ON, you send another time, it toggles OFF and so on. This works for all the 4 PINS that it has programmed. In my standalone board, each of these four pins are connected to transistors that drive 5V relays. The problem is that only one pin seems to activate, and then it won't even toggle OFF, like you send the data once and you can hear the relay's coil go ON, but then if you try to toggle it OFF it just won't, it's latched ON and I have to unplug and plug the USB cable again to reset it. The ATMEGA328P has a 16MHz crystal, two 22pF caps for the crystal, and a 10K resistor that goes from the Reset Pin to VCC. I'm not sure why the MCU behaves so weird in my board. There's also another VERY weird detail I didn't mention. If instead of using the RF Receiver, I connect the Data Output pin of the PIC that drives the RF Transmitter directly into the Receiver board, that one pin I mentioned that works is able to toggle OFF and ON again, but still, the other pins just don't work. I've tested it not only by hearing the relay's coil but also by using a multimeter. Any help would be really appreciated.  EDIT:  Here is the Eagle Schematic of the board:    It's true that I really didn't concern about the noise, but since it was quite a simple project I didn't think that noise was going to attack so violently, plus  I don't really know how to protect the circuit from inductive noise. Thank you for answering fast!  EDIT 2:  Okay so here goes more detail:    Real photos:      The soldering seems quite awkward but I probed all the traces to check for continuity.  I could make some extra holes to, for example, add a 100nF decoupling capacitor for the MCU and so, but if you think I should enhance the schematic and add a lot more components, then I'll gladly remake the board.   About the supply, right now (for debugging) I'm connecting the Arduino Uno to a Cellphone Charger (Notebook won't supply enough current for relays), and I'm connecting those red and black cables to the Arduino Uno. I'm using that Cheap 433MHZ RF Receiver so if the circuit is so vulnerable to noise as you say, it may also be getting fouled up by the noise. Show that there is no solution to the congruence $x^2\equiv3\bmod5$.   I'm not sure if the fact that the only numbers divisible by $5$ are those ending in $0$ or $5$ will help&hellip; What security systems and processes are in place to prevent malicious third parties from hacking/compromising the security of code in the Debian mirrors, or to verify that the packages we get are in fact the ones the maintainers think they are? Ifstream read function doesn't load into vector Why is $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ dense in $ L^2(\mathbb{R}^n)$?