Please make the proof as simple as possible; I'm still a beginner. Or rather, it's fine if you make it complex, but try to explain it using layman's language.
I've noticed something while trying to prove the properties of parallel lines, and the properties that a triangle has 180 degrees. To prove the properties of parallel lines, such as alternate angles, you need to use the property that a triangle has 180 degrees. To prove a triangle has 180 degrees however, you need to use the properties of parallel lines. This really bothers me because of how circular it is. They are both reliant on each other to be true, and do not logically show, without being reliant on each other, why triangles have 180 degrees, and why parallel line properties are true.   So what I hoping for here is a way to prove parallel line properties without using the fact that a triangle has 180 degrees, or a way to prove triangles have 180 degrees without using parallel line properties. This way, things will be logical to me, and make sense.   Thanks in advance.
What is the best way to repeatedly execute a function every x seconds? I want to repeatedly execute a function in Python every 60 seconds forever (just like an  in Objective C). This code will run as a daemon and is effectively like calling the python script every minute using a cron, but without requiring that to be set up by the user.  In , the solution appears to effectively just  for x seconds. I don't need such advanced functionality so perhaps something like this would work  while True:     # Code executed here     time.sleep(60)   Are there any foreseeable problems with this code? Sketching base characteristics of first order pde $$xu_{x} - uu_{t} = t$$   with intial data: $u(1,t) = t, −\infty &lt; t &lt; \infty$  So I want to sketch the base characteristics and I'm not too sure what I'm doing.  characteristic equations: $\frac{\partial{x}}{\partial{\tau}}=x$, $\frac{\partial{t}}{\partial{\tau}}=-u$, $\frac{\partial{u}}{\partial{\tau}}=t$  initial conditions:  $x(0)=1$, $t(0)=\xi$, $u(0)=\xi$  $x=e^\tau$ so $\tau=lnx$  Notice that:  $\frac{\partial^2{u}}{\partial{\tau^2}}=\frac{\partial{t}}{\partial{\tau}}=-u$,  This gives me the second order ODE  $\frac{\partial^2{u}}{\partial{\tau^2}}+u=0$  Solving I get:  \begin{align}   u&amp;=c_{1}\cos(\tau) + c_{2}\sin(\tau)\\ t&amp;=c_{1}\sin(\tau) - c_{2}\cos(\tau) \end{align}  From here I don't know what I'm doing? How long can I stay in the UK during a 1-year period? I am a Chinese citizen holding a HK SAR passport. How long am I able to visit the UK within a 365-day period?  My stamp is for 6 months. Could I leave within 6 months but re-enter again a couple of weeks for another 6 months? Are moderator tools, specifically flagging or voting to close going to be implemented in the Android app? If we have a dispute, we should at least have the flag option.  What about share option? If we would like to share a question, we could long press on the question title, and the question title, excerpt, and short URL copied to clipboard. Now that all SE sites have their own meta site, why doesn't Stack Apps have a meta?  Sure, Stack Apps is somewhat of a limited site (or rather, a site about a limited subject), so by some counts, it might not need a meta. However, I think there are some things that could be discussed about how Stack Apps should be run and there is no good place to do that now. Thus, I support the creation of a Stack Apps meta. Sets with no asymptotical density over $\mathbb N$ Let's consider the  defined by:      Take $ A\subset \mathbb N$; define the sequence $x_n= \dfrac{|A\cap[1,n]|}{n}$, and then if $\lim\limits_{n\to\infty} x_n$ exists, call it $d(A)$, the density of $A$ over $\mathbb N$.   My teacher said that the collection of sets that have a natural density is not an algebra, but I don't know why. I can't even find a set that does not have a natural density.  Please help me )= How does the comma operator work in C++?  For instance, if I do:  a = b, c;     Does a end up equaling b or c?   (Yes, I know this is easy to test - just documenting on here for someone to find the answer quickly.)  Update:  This question has exposed a nuance when using the comma operator.  Just to document this:  a = b, c;    // a is set to the value of b!  a = (b, c);  // a is set to the value of c!   This question was actually inspired by a typo in code.  What was intended to be  a = b; c = d;   Turned into  a = b,    //  &lt;-  Note comma typo! c = d; Let $A,B \subset \mathbb{R}$. Show that $\sup(A \cup B) = \max\{\sup A, \sup B\}$   Here's what I did so far:  Let $\sup A, \sup B$, and $\sup (A \cup B)$ denote upper bounds of $A,B$, and $A \cup B$, respectively. If $x \in A$, then $x \le \sup A$. If $x \in B$, then $x \le \sup B$. In either case, $x \in (A \cup B)$. So either $x \le \sup A$ or $x \le \sup B$. In either case, $x \le \max \{\sup A, \sup B\}$. So $\max \{\sup A, \sup B\}$ is the upper bound of $A \cup B$, which means $\sup (A \cup B) \le \max\{\sup A, \sup B\}$.  I do not know if there's a better way to go at it, but all I did was show the inequality. Can I do this, and show the reverse inequality? If so, how can I go about doing that? Can I use one of my attacks to cast a cantrip? In the D&amp;D campaign I am currently participating in, I am playing as a level 5 high elf champion (I actually just reached level 5 at the end of the last session).  As a High Elf, I got a free cantrip at first level, and chose the firebolt cantrip.  Here's what I'm wondering now: for the purposes of the extra attack feature, does firebolt (and similar cantrips) count as an attack? Special Relativity - Regarding the Simultaneity of Events During the Train Paradox My questions concerns that classic train paradox, wherein there is a train and a tunnel of equal length, and the train is traveling and some fraction of the speed of light towards the tunnel.  According to the Special Theory of Relativity, an observer outside the tunnel will see the train length contracted (Lorentz Contraction), whereas an observer inside will see the tunnel contracted.   Additionally, suppose that there were doors at the ends of the tunnel and that the observer outside the tunnel closed both doors instantaneously when he/she saw that the train was completely inside the tunnel.  The classic resolution of this paradox invokes the non-simultaneity of events, explaining that the observer in the train sees the far door close first, and then, once the train has begun to exit the tunnel, looks back to see the door at the beginning close. Thus, both observers agree that the train does not touch the door when they are closed for an instant.  Now my questions.  Why is it that the observer on the train sees the far door close first? It seems to me that the information coming from the far door would reach the observer on the train only after the information from the other door is reach.   Under this interpretation, the observer on the train would observe the train getting hit by the doors. What if, by some means, this event can be explained in terms of a stationary observer too? Everyone always concludes that the train remains untouched by the doors, but really the only condition that needs to be met is that both observers must agree. Why can't they both agree that the train was hit?  So, to summarize.   Why is the door that is farther away from the train observed to close first? Why can't the other possible conclusion (both see a hit train) be observed? Imagine you're trapped inside a free falling elevator. Would you decrease your impact impulse by jumping during the fall? When? How to add D-link dwm-156 to usb switch mode for automatic eject of driver CD / switch to modem mode? I am running Ubuntu 13.10, 32-bit.  I had every time need to go terminal to "eject" the windows driver CD,nor gone to disk &amp; eject windows driver CD to make the d-link dwm-156 gsm modem functioning through network manager.  It seems that usb mode switch is not working.  But on my friend's computer with ubuntu 14.04 ,usb mode switch is working and i can connect d-link dwm-156 to mobile broadband through network manager as soon as i get in.  Is there any way to make usb mode switch to eject the windows driver CD of d-link dwm-156 in my ubuntu 13.10,32-bit at start up?   The lsusb output is Bus 002 Device 013: ID 2001:a706 D-Link Corp before ejecting the CD (modem as storage device of windows driver).  The lsusb output is Bus 002 Device 005: ID 2001:7d01 D-Link Corp after ejecting the CD (as modem).   I don't want to use the driver in the CD,as i need to send sms &amp; find the data usage. Positioning the arrows on a PCA biplot I am looking to implement a biplot for principal component analysis (PCA) in JavaScript. My question is, how do I determine the coordinates of the arrows from the $U,V,D$ output of the singular vector decomposition (SVD) of the data matrix?  Here is an example biplot produced by R:  biplot(prcomp(iris[,1:4]))     I tried looking it up in the  but it's not very useful. Or correct. Not sure which. Word for lights on police cars, etc How to approach the modeling of duct work, piping, wiring? I have been facing hard time understanding meaning of "random sample" as well as "iid random variable". I tried to find out the meaning from several sources, but just got more and more confused. I am posting here what I tried and got to know:  Degroot's Probability &amp; Statistics says:     Random Samples / i.i.d. / Sample Size : Consider a given probability distribution on the real line that can be represented by either a p.f. or a p.d.f. $f$. It is said that $n$ random variables $X_1 , . . . , X_n$ form a random sample from this distribution if these random variables are independent and the marginal p.f. or p.d.f. of each of them is $f$. Such random variables are also said to be independent and identically distributed, abbreviated i.i.d. We refer to the number n of random variables as the sample size.   But one of the other statistics book I have says:       In a Random Sampling, we guarantee that every individual unit in the population gets an equal chance(probability) of being selected.   So, I have a feeling that i.i.d.s are elements that construct random sample, and the procedure to have random sample is random sampling. Am I right?   P.S.: I am very confused about this topic, so I will appreciate elaborate reply. Thanks. Prediction interval for binomial random variable How can I use 3rd party libraries on QGIS plugins on Windows?   I've developed a plugin that uses rasterio and numpy for a customer, but he's having problems installing rasterio and numpy.  Actually rasterio and numpy were installed in it's main system Python (C:\Python27), but I need QGIS Python to recognize it. I know that the product of two Gaussians is a Gaussian, and I know that the convolution of two Gaussians is also a Gaussian. I guess I was just wondering if there's a proof out there to show that the convolution of two Gaussians is a Gaussian. Editing older revision (not most recent) no longer includes older title and tags, and automatic rollback note