Proof for $\gcd$ being the smallest linear combination of $a,b \in \mathbb {Z}$.
Proving if $d_0$ is the smallest positive integer in $S$ then $d_0 = \gcd(a,b)$
I've been given a proof that shows the following:  If $f:[a,b]\to \mathbb C$ is a continuous function and $f(t)=u(t)+iv(t)$ then $$\left| \int_a^b f(t) dt \right| \leq \int_a^b \left| f(t) \right| dt$$ The proof begins by letting $\theta$ be the principle argument of the complex number $\int_a^b f(t)dt$ and there is one step in the proof I don't understand; can anyone explain to me why $$\int_a^b e^{-i\theta}f(t) dt=\Re\bigg(\int_a^be^{-i\theta}f(t)dt\bigg)$$  Would this not imply that the imaginary part of the left hand side is equal to $0$? I'm not sure why the LHS and RHS would be equal here. cardinality of all real sequences I was wondering what the cardinality of the set of all real sequences is. A random search through this site says that it is equal to the cardinality of the real numbers. This is very surprising to me, since the cardinality of all rational sequences is the same as the cardinality of reals, and it seemed fairly intuitive to me that if cardinality of a set $A$ is strictly greater than the cardinality of the set $B$, then cardinality of $A^{\mathbb{N}}$ should be strictly greater than cardinality of $B^{\mathbb{N}}$. It turns out to be false.  Some technical answers have appeared in this forum elsewhere but I do not understand them. As I am not an expert in this topic, could some one explain me in simple terms why this is happening?  Also is the cardinality of all functions from reals to reals also the same as the cardinality of reals? Well, I'm trying to find the Thevenin equivalent of this circuit:    As you can see, the only current source is a dependent source. However, the control variable is in a wire where, if nothing is connected between A and B, is \$i_\Delta = 0\$ A, and that makes the source 0 A. However, if I connect an impedance \$Z\$ to the circuit, I will have current, and a Thevenin equivalent. I don't know how to solve this problem. I've bee searching here for a while, and I found this:   . I can see how is solved that case. However, I still don't know how to apply that method to my circuit.  Any help about finding the Thevenin equivalent is appreciated n.n Does driving an microcontroller's GPIO when it is powered off weaken the chip? We are using the  in a project. The way our design is now, the IC's GPIO pins will sometimes be driven high (to 3.3V) or low when it is powered-off. Will this weaken the chip? The absolute maximum ratings state the voltage on any digital I/O pin is -0.3V to VSupply+0.3V. If the answer is no, could the IC be weakened during power up?  The LTC5800-IPM does not have the most detailed datasheet. If it is not clear for that device in particular, I'd be interested to know for CMOS microcontroller GPIOs in general. Prove that for any square matrix, an invertible matrix B exists, so that BA is triangular I'm given a matrix A, its dimensions are n x n.  I am required to prove that an invertible matrix B exists, such that the product of the matrices BA is triangular.  Any help? How do I clear the render border? In the 3d viewport I am using a border region to only render things inside the selected region. I did this with CtrlB. I cannot get rid of the region. How can I make the border disappear? I'm planning to visit a summer school in the Schengen zone. What kind of visa should I apply for? I will be attending a summer school which will last 10 days.  I want to know: for applying for a Schengen visa, is the purpose of my trip considered as touristic or study? How to convert coordinates from vertex to world-space? How do I convert from object-space to world-space?  I suspect it is:  vert_os = obj.vertices[foo] vert_ws = vert_os.getPositionFromMatrix(obj.matrixWorld)   but does the matrix also handle translations, or do I need to handle that separately?  Can someone link to the relevant documentation? Using a diode to ensure current flows in only one direction, without causing a voltage drop To power my microcontroller (ATmega8), I am using a ~5.4V voltage source. I want to ensure that I don't accidentally connect the voltage source in reverse, and figured a diode would be a nice way to accomplish this as from what I have learned so far, a diode allows current to flow in one direction, and blocks it in the other.  But what I also learned is that diodes create drops in voltage. I have a few of the typical diodes (1N4001, 1N4148, etc.), and would like to use them to achieve the aforementioned result without dropping the voltage as it would be too low to power the IC.  My question is, is there any way to do this with a diode? Or do I need some other component (if so, what would you recommend)? I have already proved Triangle Inequality $\lVert v+w\rVert \le \lVert v \rVert+\lVert w \rVert$ in vector space $\mathbb{R}^n$. However, I'm having difficulties proving the Reverse Triangle Inequality in the vector space.  I have started with   $$\lVert v \rVert^2=v \cdot v-w\cdot w+w\cdot w \le \lVert v-w \rVert+\lVert w \rVert^2$$ $$\lVert w \rVert^2=w \cdot w-v\cdot v+v\cdot v \le \lVert w-v \rVert+\lVert v \rVert^2$$  based on the Triangle Inequality but I don't know if it is correct. If this is right, I think I can go on with proving it the same way I would the normal Reverse Triangle Inequality. Forgive me for the unclear title. Some time ago, my HTC M8, which is rooted and runs on a  showed something weird. Firstly, in the Google Maps App a mysterious blue dot appeared, that captures the screen completely and makes it impossible to quit or use the application. It looks like this (I could make a screenshot)     When this happens in Maps, the only thing I can do is a hard restart with Power+Volume Up.  Since I have seen this occasionally in the camera app as well, I suspect it is not a bug of Maps itself. I know that the Venom ROM is not standard but maybe this is connected to some setting and anyone has seen this. Btw, the Venom ran smoothly for almost 3 years now.  Question: Has someone a clue what this could be and how I can get rid of it? I understand that the mechanics and design of a lightsaber aren't that complicated.  Do others besides the Jedi and Sith (or their minions) make or use lightsabers?  And if they don't, why not? Warn new users when they ask a question after a previous question is closed, downvoted, or deleted Another &quot;miséricorde&quot;:  Consider  (emphasis added):  I think its okay, please dont take any efforts for me. I personally do not feel the system helps ask good questions. I dont know why asking for a software in this case is bad. Basically I needed a command that would give me this data. There is no other way I could have asked the question. I could have changed the question just for the sake of changing. I feel I am better off elsewhere. Thanks for your very very prompt responses. Regards. Bye. – user1744649  We just lost another user due to inadequate guidance on what questions are allowed.  The  by the time they are banned has been proven incorrect. Given that , we need a better way to make sure that users can understand how to ask good questions and what types of questions are allowed. I suggest that we warn newcomers, in a clear but positive tone, when one of their questions has been closed, downvoted (to a score of -2 or lower), or deleted, so that they don't repeat the same mistake.  This is probably best done the next time the user asks a question, and should appear as an &quot;interstitial&quot; page prior to reaching the actual Ask Question form.  The page should not appear to users:  with at least 50 (or perhaps 75 or 100) reputation that have asked a good question since the last downvoted/closed/deleted question that have fixed a bad question and had it reopened/undeleted/upvoted.  The content of the page should be adapted based on the specific problem with the previous question.  I would imagine something similar to the following (example for a question closed as off topic on Super User):  Before you ask another question... One or more of your previous questions have been closed as off topic. The scope of on-topic questions is defined in the , as follows:  What kind of questions can I ask here? Super User is for computer enthusiasts and power users. If you have a question about …  computer hardware, computer software, or personal and home computer networking  and it is not about …  programming and software development, video games or consoles, websites or web services like Facebook, Twitter, and WordPress, electronic devices, media players, cell phones or smart phones, except insofar as they interface with your computer, issues specific to corporate IT support and networks, asking for a shopping or product recommendation,  … then you’re in the right place to ask your question!  Please make sure that the question you are about to ask is on topic as described above. If you need help determining whether your question can be asked on this site, you can ask for advice on  or . You may also want to edit your previous question to address any problems associated with it. To ensure a high standard of content quality and prevent disruption to the community, users who repeatedly post low-quality questions  Continue  The  doesn't address this issue as it is shown only to users with 10 reputation or less (rather than up to 50 and possibly higher) and does not address the specific problem with previous low-quality questions.  The point of this feature request is to provide a easily understood targeted message that actually teaches the user about what happened with the previous question he or she asked.  The EULA is generic and may not have conveyed the intended message clearly if the user still asked a low-quality or off-topic question. For reference,  currently has 143 reputation and the proposed message would probably have been shown after the first closed or deleted question. I know this can be a complex feature to implement, since it needs to be adaptive, but this should help eliminate confusion on what the rules are and minimize the incidence of good-faith users getting post-banned. Finn knows nothing about his family because he was taken while he was very little.   But a black main character makes me think (not being racist guys). And when he used the lightsaber I thought that he had a Jedi parent. Master Windu is the only black Jedi we know so is there any canon explanation to that? Is it right? What is the maximum size of a web browser's cookie's key? What is the maximum size of a web browser's cookie's key?  I know the maximum size of a cookie is 4KB, but does the key have a limitation as well? How could I detect (preferably with Python) duplicate MP3 files that can be encoded with different bitrates (but they are the same song) and ID3 tags that can be incorrect?  I know I can do an  checksum of the files content but that won't work for different bitrates. And I don't know if ID3 tags have influence in generating the MD5 checksum. Should I re-encode MP3 files that have a different bitrate and then I can do the checksum? What do you recommend? I want to quantify the relationship between two variables, A and B, using mutual information. The way to compute it is by binning the observations (see example Python code below). However, what factors determines what number of bins is reasonable? I need the computation to be fast so I cannot simply use a lot of bins to be on the safe side.  from sklearn.metrics import mutual_info_score  def calc_MI(x, y, bins):     c_xy = np.histogram2d(x, y, bins)[0]     mi = mutual_info_score(None, None, contingency=c_xy)     return mi Given a Lie group, how are you meant to find its Lie algebra?  The Lie algebra of a Lie group is the set of all the left invariant vector fields, but how would you determine them?  My group is the set of all affine maps $x \rightarrow A.x+v$ from $R^n$ to $R^n$ under function composition. Run command whenever someone joins a Minecraft server I'm trying, but something is wrong with my code in LaTeX: \begin{asy} pair dynkinMark(real x, real y, string s = &quot;&quot;, pen p = defaultpen, pair B = dir(45)) { if (s != &quot;&quot;) { dot(&quot;$&quot;+s+&quot;$&quot;, (x,y), B, p); }     else { dot((x,y), p); }     return (x,y); }  dynkinMark(0,1, &quot;1&quot;, dir(135)); dynkinMark(0,2, &quot;2&quot;, dir(135)); dynkinMark(0,3, &quot;3&quot;, dir(135)); dynkinMark(0,4, &quot;4&quot;, dir(135)); dynkinMark(7,1, &quot;1'&quot;); dynkinMark(7,2, &quot;2'&quot;); dynkinMark(7,3, &quot;3'&quot;); dynkinMark(7,4, &quot;4'&quot;); path p1 = (0,1)..(2,2.0)..(5,3.5)..(7,4); path p2 = (0,2)..(2,1.2)..(5,2.5)..(7,3); path p3 = (0,3)..(2,2.8)..(5,1.5)..(7,1); path p4 = (0,4)..(2,3.8)..(5,3.5)..(7,2); draw(p1, blue); draw(p2, heavycyan); draw(p3, mediumcyan); draw(p4, lightblue); dot(&quot;$12$&quot;, IP(p1, p2), dir(90), heavygreen); dot(&quot;$13$&quot;, IP(p1, p3), dir(90), heavygreen); dot(&quot;$23$&quot;, IP(p2, p3), dir(90), heavygreen); dot(&quot;$14$&quot;, IP(p1, p4), dir(90), heavygreen); dot(&quot;$24$&quot;, IP(p2, p4), dir(90), heavygreen); real y = 0.8; label(&quot;$s_1$&quot;, (IP(p1,p2).x, y), heavygreen); label(&quot;$s_2$&quot;, (IP(p1,p3).x, y), heavygreen); label(&quot;$s_1$&quot;, (IP(p2,p3).x, y), heavygreen); label(&quot;$s_3$&quot;, (IP(p1,p4).x, y), heavygreen); label(&quot;$s_2$&quot;, (IP(p2,p4).x, y), heavygreen); \end{asy}