Numerical resolution of equation with trigonometric expressions
Solve simple trig equation
Are there any options of bash with which it is invoked? I  that the default shell that I use when I start Terminal (with Ctrl+Alt+T) is /bin/bash.   And running the built-in command  on  as:   type -p bash   returns the result as:  /bin/bash   Suggesting the same executable is run when I do Ctrl+Alt+T or say bash.  This shell is the same as an invocation of the command  with some of its options passed to it right? If yes, what are these options?   As an example, perhaps the -i option gets passed so as to make the bash shell interactive?   PS: Maybe  is related too, but I cannot seem to piece together all the pieces here. We are experimenting with running an OpenVPN server for our business. One question I can't seem to find the answer to is this:  When we generate keys for one of our users for them to use at home, can their use the same keys on their home laptop as well as their home desktop? Or do we need to generate separate keys for each user's client machine? How to get next/previous record in MySQL? P.d.f of the absolute value of a normally distributed variable I came across this question as an exercise, had a brief idea, but didn't know how to proceed.     Let $X \sim N(0, 1)$. What is the p.d.f of $|X|$ ?   I know the final p.d.f looks just like the right half of the original pdf, but extended vertically for a factor of 2.  Could someone please show mathematically what the p.d.f of variable |X| is going to be, and explain briefly? Thank you. Multiplexing UART I'd like to interface my main microcontroller's hardware serial port to more than one device at the same time. My understanding is that UART on my microcontrollers is based on an active-low 5V TTL.  (I know what I'm explaining here can be technically achieved by using more suited protocols, such as SPI and I2C, but my project requires to be only implemented using UART.)  Here's a pseudo-schematics chart of what I'm trying to achieve:         &ndash; Schematic created using   Please note that:   I don't want to use a second UART interface on the master, or any of the slaves. Only one slave at a time is going to transmit or send data on the line to the master. (e.g. If the slave 1 is transmitting data, slave 2 and 3 should remain idle.) Whatever the master is transmitting can, and should be received by all of the slaves. All of the slaves should be hot-pluggable, meaning they could be removed or inserted at any time. The RX and TX wires to the master are parallel between each slave, so no daisy chaining.   So, my questions are as follows:   Assuming the correct impedance is considered, I shouldn't need an additional circuit to transmit data from the master's TX line to any of the slave's RX line simultaneously. Is this correct? I understand that without current limiting resistors between slaves' TX line and the master's RX line, if any of the devices begin to broadcast data, the line between the devices will short circuit and I risk damaging them. Is this correct? Assuming the items 1 and 2 are correct:  a) Would just some current limiting resistors be enough to use?  b) Can I use 74-series logic gates instead of resistors? Between options a and b above, what route should I choose? For a reliable 115200 baud rate connection, will the switching frequency of a 74HC08 AND gate suffice to be incorporated?   EDIT: I'm well aware of other well-suited protocols such as SPI and I2C. There's a reason I'm using UART, and that's because the optiboot bootloader that is used on the master's mcu communicates over UART, albeit UART's obvious limitations.  Also, while daisy-chaining is definitely a clever approach, due to breaking the 4th requirement, it also breaks bootloader functionality.  The devices could also be USB-to-Serial chips, and using SPI-to-USB or I2C-to-USB would not be feasible in this project.  So, I'd appreciate if you could focus on the 6 questions listed here instead, and share your UART knowledge. Proving $\frac{a}{a^2-a+1}+\frac{b}{b^2-b+1}+\frac{c}{c^2-c+1}+\frac{d}{d^2-d+1}\le \frac{8}{3}$ given $a+b+c+d=2$ Let $a,b,c,d\in \mathbb{R}$ and $a+b+c+d=2$. Prove that $$\frac{a}{a^2-a+1}+\frac{b}{b^2-b+1}+\frac{c}{c^2-c+1}+\frac{d}{d^2-d+1}\le \frac{8}{3}.$$   We have $$\frac{a}{a^2-a+1}\le \frac{4}{3}a\Longleftrightarrow \frac{-a\left(2a-1\right)^2}{3\left(a^2-a+1\right)}\le 0,$$ (Right)  So $$LHS\le \frac{4}{3}\left(a+b+c+d\right)=\frac{8}{3}=RHS$$  When $a=b=c=d=\frac{1}{2}$  But also $a=b=c=1;d=-1$ i don't know how to solve it. Why does Stephen Hawking say black holes don't exist? Can we have a moderator only "flag for community team/developer attention" link on posts? My machine can't play encrypted DVDs on a fresh install.  How do I add this capability?  Another useful bit of information would be what programs are best for playing DVDs, once I'm able to do so.  See the .  Will I be able to play DVD movies from ? Does text in brackets invoke the "an" vs. "a" rule? Actionbar background is not being changed How were bullets able to dent Captain America's Shield? Just after Captain America's shield was introduced in Captain America: The First Avenger, Peggy Carter shot few bullets at it which created few minor distortions on it.                                  From  page of Marvel.Wikia,     Proto-Adamantium   This is the original Adamantium created by Dr. MacLain. Proto-Adamantium is the official name for the metal that makes up Captain America's Shield. It has never been recreated and is the absolute most invulnerable item in all of existence.    From  page of Marvel.Wikia:     The true shield was once slightly dented by an enraged Odinforce empowered Thor, using Mjolnir, but he later reconciled and pounded out the small dent.    How were few bullets able to dent Captain America's shield then? How to remove top and bottom whitespace of longtable? I am using longtable instead of tabular environment and noticed that it puts some additional whitespace at the top and at the bottom of the table. Is there some way to remove it (something less hacky than \vskip -\baselineskip before the table)?  Small piece of code for illustration:  \documentclass[12pt]{book} \usepackage{longtable}  \begin{document}  Linelineline  \begin{longtable}{l l} cell1 &amp; cell2 \end{longtable}  Linelineline  \begin{tabular}{l l} cell1 &amp; cell2 \end{tabular}  Linelineline  \end{document}   And that's what I am getting:    Common tabular environment does not have any additional margins, but longtable does. My professor was lecturing today and he made this statement which I was unable to verify.  (I worded it nicer)  There is no map which is both smooth and onto from $S^1$ to $S^1$$\times$ $S^1$.  When he said this he included the original "one can clearly see" statement which is usually when the most people seem to be confused in his class.  Thus, I was hesitant to ask why.  So, please no one lecture me on the fact that I should have asked anyway.  If it is really that obvious, I would rather look stupid here than in class :) What happens, if a rocket is filled with a vacuum instead of high pressured air? Suppose you put wheels under a compressed air tank    so that it can move horizontally to the right and to the left. Suppose there is a nozzle on the right hand side of the tank (in the picture on the backside, if you like).  If the tank is filled with high pressured air and you open the nozzle, the tank will move to the left. This is how rockets work and one can easily imagine that.   What happens when the tank is filled with a vacuum and you open the nozzle?  My first impression was, that the tank is moving to the right but someone told me, that it would be stand still. Is this true? What is an intuitive explanation for that? Display maximum possible count of sites in footer Get bytes from std::string in C++ I'm working in a C++ unmanaged project.  I need to know how can I take a string like this "some data to encrypt" and get a byte[] array which I'm gonna use as the source for Encrypt.  In C# I do    for (int i = 0; i &lt; text.Length; i++)     buffer[i] = (byte)text[i];   What I need to know is how to do the same but using unmanaged C++.  Thanks! When I use the MouseUp event, I can get it to fire with a mouse right-click. But MouseLeftButtonUp won't fire with either click!  &lt;Button MouseLeftButtonUp="btnNewConfig_MouseUp"  Name="btnNewConfig"&gt;     &lt;StackPanel Orientation="Horizontal"&gt;         &lt;Image Source="Icons\new.ico" Height="24" Width="24" Margin="5"/&gt;         &lt;TextBlock VerticalAlignment="Center"&gt;New&lt;/TextBlock&gt;     &lt;/StackPanel&gt; &lt;/Button&gt;   I know this is most likely something simple. Thanks for the help! Intuitive explanation of convergence in distribution and convergence in probability What is the intuitive difference between a random variable converging in probability versus a random variable converging in distribution?  I've read numerous definitions and mathematical equations, but that does not really help. (Please keep in mind, I am undergraduate student studying econometrics.)  How can a random variable converge to a single number, but also converge to a distribution? C1/D Visa - Mexico and US