Heterogeneous vs. inhomogeneous
Heterogeneous vs. inhomogeneous
What's the rationale for requiring that a field be a $\boldsymbol{non}$-$\boldsymbol{trivial}$ ring? The title pretty much says it all.  Of course, one answer (IMO unsatisfactory) to such questions goes something like "a definition is a definition, period."  In my experience, mathematical definitions are rarely completely arbitrary.  Therefore I figure there must be a good reason for insisting that a field be a non-trivial ring (among other things), but it's not obvious to me.  I realize that a trivial ring would make for a very boring field, but it is also a rather boring ring, and yet it is not disallowed as such.  EDIT: I found a hint of a rationale in the statement that "the zero ring ... does not behave like a finite field," but I could not find in the  for this assertion exactly how the zero ring fails to behave like a finite field. How to recursively download a folder via FTP on Linux I'm trying to ftp a folder using the command line ftp client, but so far I've only been able to use 'get' to get individual files. Is it safe to answer "erase disk and install Ubuntu" on a virtual machine? Why isn't my Minecraft LAN server working? Imagine this directory structure:  app/    __init__.py    sub1/       __init__.py       mod1.py    sub2/       __init__.py       mod2.py   I'm coding mod1, and I need to import something from mod2. How should I do it?    I tried from ..sub2 import mod2 but I'm getting an "Attempted relative import in non-package".  I googled around but found only "sys.path manipulation" hacks. Isn't there a clean way?      Edit: all my __init__.py's are currently empty    Edit2: I'm trying to do this because sub2 contains classes that are shared across sub packages (sub1, subX, etc.).  Edit3: The behaviour I'm looking for is the same as described in  (thanks John B) I'm an avid book collector, and I was wondering what house has the most bookshelves in Skyrim. I currently only own the Whiterun house, which has 2 bookshelves of 2 shelves each.  Is there a house I could buy that has more bookshelves?  Edit  Since not all bookshelves hold the same number of books, I'm looking for the house that has the most spaces on bookshelves for books, not necessarily the most bookshelves Comment character counter does not trim, while the server does I need to find a period-3 orbit for a tent map defined by $$T(x)=\begin{cases}2x &amp; 0\le x\le0.5 \\[2ex] 2-2x &amp;0.5 &lt; x \le 1 \end{cases}$$  So, what is a fast and efficient way of finding it? I only know how to do it by guessing-and-checking. For example, I picked a random point, $x = \frac{2}{7}$ and found that $$T\left(\frac{2}{7}\right)=\frac{4}{7}\\[4ex]T\left(\frac{4}{7}\right)=\frac{6}{7}\\[4ex]T\left(\frac{6}{7}\right)=\frac{2}{7}$$ which is a period-3 orbit. Is there any better way of finding a period-3 orbit , more specifically, a period-3 point? What is the command to open the shutdown dialog? Radius of Convergence of a given Power series Has there ever been a warp / non-warp collision? Like something going at warp hits a planet, star, another ship, moon, satellite, alien, tribble, etc.... Basically, can stuff in subspace (warped stuff) hit stuff in real space (not warp)?  Star Trek: The Voyage Home shows us clearly that entering warp speed within Earth's atmosphere has no adverse consequences for the starship or the environment.  HMS Bounty went through miles of atmosphere, dust, birds, and clouds and nothing entered  to hit her. Enterprise did this again by dropping out of warp in Titan's atmosphere.  which explains some tech used, but now we have to ask if there is even a hazard of collision at warp. Has there ever been a collision in the franchise - any normal object being hit at warp speed? (This obviously doesn't include things traveling at warp together, like photon torpedoes from one ship to another).  Closest example I found: In  Enterprise attempts warp before balancing their warp drives and creates a wormhole instead. They nearly collide with an asteroid which was already inside the wormhole, but neither the ship not the asteroid were traveling at warp.  Compressing Space Real-world (non-canon) attempts to explain ST warp technology rely on compressing space itself in front of the vessel. The image sequence below from Star Trek: Into Darkness (canon) shows Enterprise warping past the USS Vengeance and confirms that Enterprise actually gets smaller (or compresses the space around it). The light on Vengeance shows that at the instant the two ships are side-by-side Enterprise looks like a toy beside it. (Large image needed for detail - sorry):    Thus objects at warp occupy less real space - possibly none at - because , which according to Memory-Alpha is implied to be the medium which FTL travel happens. Normally subspace and real space do not interact. When they do it is called a .  Also, objects inside the warp bubble are not moved by the warp drive:     The observer(spaceship) is still immersed in the interior of the warp bubble and this bubble is carried   out by the spacetime ”stream” at faster than light velocities with the observer at the rest with respect to   its local neighborhoods  (p. 25)   It is my belief that to be consistent with warp bubble physics derived from the Einstein Field Equations in general relativity, the ST interpretation of warp speed travel should not allow collisions between n-space objects and objects within a warp bubble and remove the phenomenon from any plotlines such that warp travel is intrinsically safe for the traveller, but possibly greatly impactful for the normal world outside.  But has it happened?  Points of clarification raised by comments  Deflector Shields: Yes, ships have them. Gene Roddenberry envisioned Navigational Shields which would deflect anything in front of the ship, from a single hydrogen atom to an asteroid (see his notes below).    It's important to note that much of what he had in this guide was changed in final production. In fact, based on this guide we could write an entire episode on a simple deflector sabotage resulting in them hitting a hydrogen atom. Real science calculations tell us that even hitting a photon of light would release incalculable energy.  These notes very much explain the many questions asking why ships don't go to warp inside a solar system. Solar systems are FULL of particles and debris, which would require a great deal of energy to deflect. Interstellar space simply draws less power because the shields work less. I don't see advantages of using events over delegates, other than being syntactical sugar. Perhaps I am misunderstanding, but it seems that event is just a placeholder for delegate.  Would you explain to me the differences and when to use which?  What are the advantages and disadvantages? Our code is heavily rooted with events, and I want to get to the bottom of it.  When would you use delegates over events and vice versa? Please state your real world experience with both, say in the production code. Why shouldn't we allow NULLs? Python: Uniqueness for list of lists Group theory proof of Euler's theorem ($a^{\phi(m)} \equiv 1\mbox{ }(\mbox{mod }m)$ if $\gcd(a,m)=1$) From A Classical Introduction to Modern Number Theory by Ireland and Rosen, page 33:     Corollary 1 (Euler's Theorem). If $(a,m) = 1$, then $a^{\phi(m)} \equiv 1\,(m)$.      Proof. The units in $\mathbb{Z}/m\mathbb{Z}$ form a group of order $\phi(m)$. If $(a,m) = 1$, $\bar{a}$ is a unit. Thus $\bar{a}^{\phi(m)} = \bar{1}$ or $a^{\phi(m)} \equiv 1\,(m)$.   If I'm interpreting this correctly, this proof implicitly uses the fact(?) that if $G$ is a group and $x\in G$, then $x^{|G|}=1$, where $1$ is the identity element of $G$.  If this is indeed true, can someone explain why? (I have had no prior exposure to group theory.) Any site that I visit can see my local IP (e.g. 192.168.1.102). Is there any reliable way to block this behavior? (browser setting, OS setting, router setting, etc.) I am particularity interested in Google Chrome and Mozilla Firefox, but any modern open source browser will do. If it is a browser plugin, it should be open source with a solid user base.  By reliable I mean, that I do not need to check the leakage again after every software update.  More info / what I have read so far:   There was a Chrome plugin that is not working anymore Local IP is obtained using JavaScript / WebRTC   Live Demo here:  Source code:   Firefox developers is not going to fix the leak (bug is open from 2014):  In Private / Incognito mode does not prevent this leak. If you use static IPs, and have non typical local IP (e.g. 10.66.77.88 or 192.168.212.121), then there is no need for . Your external IP + internal IP is perfect fingerprint.   Update 1: It is not duplicate of  - I want to prevent private / local IP leak, not disable all WebRTC.  Update 2: No, it is not related to header X-Forwarded-For (HTTP_X_FORWARDED_FOR), that is added when user/administrator chooses to do so. In my provided demo it is not your choice... It is Google's and Mozilla's choice to leak your private IP even when using private browsing mode.  Update 3: Ability to determine my local IP allows malicious JavaScript to quickly identify my local subnet and print on my local network printers from any website in the world without me doing anything.  This  finds my LAN printers using JavaScript.  Update 4: It is not a good idea to reveal private IP to every website. By seeing your private IP, for example 192.168.88.101, I can guess with very high probability that you are using MikroTik router. Also, I can track particular user event if user uses different browsers because local and remote IP stays the same for some time (hours or even days). I am using the algorithmic package to typeset an algorithm. I would like to have connecting lines between the beginning and end of a loop or conditional like in the algorithm2e package. Is there an easy way to accomplish this, or am I better off switching to algorithm2e? What is Deviance? (specifically in CART/rpart) What is "Deviance," how is it calculated, and what are its uses in different fields in statistics?  In particular, I'm personally interested in its uses in CART (and its implementation in rpart in R).   I'm asking this since the  seems somewhat lacking and your insights will be most welcomed. Where does the damage of relocated dashes land? The end location of a dash or jump can be changed by an ability that pushes () or pulls () you. So for example I try to  on the adc, but a Blitzcrank pulls me, preventing me from landing on my chosen spot.  Will I deal my AoE damage around the adc, where I tried to land, or around the Blitzcrank, where I ended up landing? $\mathbb{Z}[X]/(X^2-1)$ is not isomorphic with $\mathbb{Z}\times \mathbb{Z}$ I have to show that the ring $\mathbb{Z}[X]/(X^2-1)$ is not isomorphic with $\mathbb{Z}\times \mathbb{Z}$.    I know that $(\mathbb{Z}\times\mathbb{Z})^*=\{(\pm1,\pm1)\}$, so I thought I should be looking for elements which have inverses in $\mathbb{Z}[X]/(X^2-1)$ and hopefully find more or less than 4. But I didnt succeed, so I need hints. Thanks.