How to prove $(\sqrt2 n^2)_{n\in \mathbb{N}}$ is uniformly distributed mod 1.  I tried using Weyl's criterion but I seem to go nowhere.   Thank You very much.
The Weyl equidistribution theorem states that the sequence of fractional parts ${n \xi}$, $n = 0, 1, 2, \dots$ is uniformly distributed for $\xi$ irrational.   This can be proved using a bit of ergodic theory, specifically the fact that an irrational rotation is uniquely ergodic with respect to Lebesgue measure. It can also be proved by simply playing with trigonometric polynomials (i.e., polynomials in $e^{2\pi i k x}$ for $k$ an integer) and using the fact they are dense in the space of all continuous functions with period 1.  In particular, one shows that if $f(x)$ is a continuous function with period 1, then for any $t$,  $\int_0^1 f(x) dx = \lim \frac{1}{N} \sum_{i=0}^{N-1} f(t+i \xi)$. One shows this by checking this (directly) for trigonometric polynomials via the geometric series.  This is a very elementary and nice proof.  The general form of Weyl's theorem states that if $p$ is a monic integer-valued polynomial, then the sequence ${p(n \xi)}$ for $\xi$ irrational is uniformly distributed modulo 1.  I believe this can be proved using extensions of these ergodic theory techniques -- it's an exercise in Katok and Hasselblatt.  I'd like to see an elementary proof.  Can the general form of Weyl's theorem be proved using the same elementary techniques as in the basic version?
How do I install "non-free firmware" (wifi driver) from USB? (Debian) Proof Check: Every Cauchy Sequence is Bounded Sorry if I keep asking for proof checks. I'll try to keep it to a minimum after this. I know this has a well-known proof. I understand that proof as well but I thought I'd do a proof that made sense to me and seemed, in some ways, simpler. Trouble is I'm not sure if it's totally correct. It's quite short though. I was just hoping someone could look it over and see if it is a valid proof. Thank you!  Lemma: Every Cauchy sequence is bounded.  Proof: Let $(a_{n})$ be Cauchy. We choose $ 0&lt;\epsilon_{0}$. So $ \forall \; n&gt;m\geq N_{0}$ we have that $\vert a_{n}-a_{m} \vert &lt; \epsilon_{0}$. Therefore $(a_{n})$ is bounded for all $ m \geq N_{0} $ by $ \epsilon_{0} $. Since $ \mathbb{N}_{N_{0}}$ is finite, it is bounded. So, for all $ m&lt;N_{0} $, $ (a_{n})$ is bounded. Therefore $(a_{n})$ is bounded.  I realize I haven't said what the bounds are but I think that's sort of irrelevant. So long as we know it's bounded. Any help is much appreciated! Should it be /about_us or /about-us?  From usability point of view, I personally think /about-us is much better for end-user yet Google and most other websites (and javascript frameworks) use underscore naming pattern. Is it just matter of style? Are there any compatibility issues with dashes? Replace environment variables in a file with their actual values? Is there an easy way to substitute/evaluate environment variables in a file? Like let's say I have a file config.xml that contains:  &lt;property&gt;     &lt;name&gt;instanceId&lt;/name&gt;     &lt;value&gt;$INSTANCE_ID&lt;/value&gt; &lt;/property&gt; &lt;property&gt;     &lt;name&gt;rootPath&lt;/name&gt;     &lt;value&gt;/services/$SERVICE_NAME&lt;/value&gt; &lt;/property&gt;   ...etc. I want to replace $INSTANCE_ID in the file with the value of the INSTANCE_ID environment variable, $SERVICE_NAME with the value of the SERVICE_NAME env var. I won't know a priori which environment vars are needed (or rather, I don't want to have to update the script if someone adds a new environment variable to the config file). Thanks! What's the purpose (if any) of "javascript:" in event handler tags? I've been making a concerted effort to improve my javascript skills lately by reading as much javascript code as I can. In doing this I've sometimes seen the javascript: prefix appended to the front of event handler attributes in HTML element tags. What's the purpose of this prefix? Basically, is there any appreciable difference between:  onchange="javascript: myFunction(this)"   and  onchange="myFunction(this)"   ? How to extrude and scale with an even offset? I'm quite new to Blender, so forgive me if this is a dumb question. I don't even know hot to describe it properly, so:      Basically I want to extrude a rectangle and then scale it along two axis, but without keeping the proportions; I want to reduce them of the same quantity, so that I get something similar to the frame of a picture. Is there a way to target the arrow from an OP bow via command block? I am trying to make an effect trail for a specific weapon in MC, a bow with power 120.  I want to use a command block to target any arrow with the damage it should be doing (63) and summon a trail of splash potions.  I also want to kill any arrow entities that have inGround=True  Is there any way to do this or do I have to come up with something else?  Attempt:  /execute @e[type=Arrow,damage=63,inGround=0] ~ ~ ~ summon ThrownPotion ~ ~ ~ {Potion:{id:potion,Damage:8197}} /kill @e[type=Arrow,inGround=True]   Each unfortunately just affects all arrows.  Is this even possible? Given a $10×10$ grid with $9$ red blocks and $91$ white blocks, in each step we color one red block black and after that one white block red until there are $91$ black blocks and $9$ red blocks. Prove that there exits a step in which a black block shares an edge with a white block. I ran a few trials and couldn't get past the fifth step without adjacency. But I don't know how to build a proof. This discussion informed our process for seven years, but eventually required updates and augmentation. For discussion regarding the successors to this process, please see:    Stack Exchange has some of the best Moderators around — seriously — and that is due in no small part to the communities who scrutinize and vote in our Moderator elections.  We have a rather formal process for electing moderators, but up until now  has been quite informal. On those rare, rare occasion when things just aren't working out, you can see why it becomes such a sticky business to remove someone from a democratically elected position. Our current process is one of &quot;we've got your back.&quot; The Community Team routinely monitors moderator activities, so if a moderator violates the  or fails to uphold the , we can and do intervene. If the disputes and problems persist without resolution, eventually we can ask someone to step down. That's not good enough. We need a much more transparent and prescribed process for when — and more importantly how — to intervene when on-going problems are going unresolved. In urgent situations, we'll step in to protect a site, but ideally the process should be impartial and not subject to accusations of Stack Exchange simply applying undue influence over a community when the whim strikes. The Question: Before we start bouncing around ideas internally, I'd like to hear what you think. What process should be in place to add some checks and balances to deal with higher issues like possible impeachment of a moderator? When and how would such a process come into play? How should such situations be resolved? How do we keep it from being abused? Have at it. This is a round table vetting of ideas. How to make Arduino (Snap) access serial port /dev/ttyUSB0 (or /dev/ttyACM0)? I have Ubuntu 18.04 LTS installed.  I installed Arduino IDE v1.8.5 as a snap with  snap install arduino-mhall119   I have added my user to dial-out group and rebooted:  sudo usermod -a -G dialout $USER  $ groups  username adm dialout cdrom sudo dip plugdev lpadmin sambashare   I connected my Arduino Nano to the laptop, it is detected on port /dev/ttyUSB0 (ch341-uart).  $ ls /dev/ttyUSB0 -al crw-rw---- 1 root dialout 188, 0 авг 13 11:59 /dev/ttyUSB0   I tried to upload the simpliest Blink sketch and get the following errors in the terminal:  $ arduino-mhall119.arduino    avrdude: ser_open(): can't open device "/dev/ttyUSB0": Permission denied   Problem uploading to board.  See http://www.arduino.cc/en/Guide/Troubleshooting#upload for suggestions.**   Also I have installed arduino 1.8.5 with umake - it can access serial port in usual way.  Note: we already have  here, but the solution was to use , not fix for the Snap.  So my question is: How to make Arduino (Snap) access serial port /dev/ttyUSB0 (or /dev/ttyACM0)?  Update: connecting serial-port to the snap fails with the following message  $ snap connect arduino-mhall119:serial-port  error: snap "core" has no "serial-port" interface slots   Current interfaces are:  $ snap interfaces | grep arduino :home                            arduino-mhall119,gedit,organize-my-files,vlc :network                         arduino-mhall119,gedit,organize-my-files,pulsemixer,vlc :unity7                          arduino-mhall119,gedit,organize-my-files,vlc :x11                             arduino-mhall119,organize-my-files,pulsemixer,vlc -                                arduino-mhall119:serial-port   Changing permissions of serial port to 777 with sudo chmod 777 /dev/ttyUSB0 does not help.  The following do not work too (because of syntax errors):  $ sudo snap connect arduino-mhall119:arduino error: snap "arduino-mhall119" has no plug named "arduino"  $ snap connect arduino-mhall119:/dev/ttyUSB0 error: snap "arduino-mhall119" has no plug named "/dev/ttyUSB0"  $ snap connect arduino-mhall119 error: cannot resolve connection, plug snap name is empty   Information about YAML-manifest:  $ ls -l /snap/arduino-mhall119/current/meta/snap.yaml -rw-r--r-- 1 root root 353 июл 28 00:20 /snap/arduino-mhall119/current/meta/snap.yaml  $ cat /snap/arduino-mhall119/current/meta/snap.yaml |grep -A 12 slots  $ cat /snap/arduino-mhall119/current/meta/snap.yaml |grep -A 5 plug     plugs:     - x11     - unity7     - home     - network     - serial-port I found Minecraft's marvelous large worlds extremely slow to navigate, even with a quad core and meaty graphics card.  I assume Minecraft's slowness comes from:   Java, as spatial partitioning and memory management are faster in native C++. Weak world partitioning.   I could be wrong on both assumptions. However, this got me thinking about the best way to manage large voxel worlds. As it is a true 3D world, where a block can exist in any part of the world, it is basically a big 3D array [x][y][z], where each block in the world has a type (i.e BlockType.Empty = 0, BlockType.Dirt = 1, etc.)  I assume that to make this sort of world perform well, you would need to:   Use a tree of some variety (//) to split all the cubes out; it seems like an oct/kd would be the better option, as you can just partition on a per cube level not a per triangle level. Use some algorithm to work out which blocks can currently be seen, as blocks closer to the user could obfuscate the blocks behind, making it pointless to render them. Keep the block object themselves lightweight, so it is quick to add and remove them from the trees.   I guess there is no right answer to this, but I would be interested to see peoples' opinions on the subject.  How would you improve performance in a large voxel-based world? I have lost functionality in the tool shelf. When I click on a selector or something in the bottom part of the tool self that you can hide and unhide, nothing happens. I can see the options but when I click it doesn't work. See screenshot where it says "Add Cylinder". RELATED,ESTABLISHED state in iptables What do the words 'RELATED' and 'ESTABLISHED' individually signify in RELATED,ESTABLISHED state in iptables? In transformers, the ratio of the voltages equals the ratio of the turns - so double the output coil's turns and the output voltage doubles. Then, in order to conserve energy, current halves.  This makes perfect sense in terms of $\mathrm{P=VI}$, but what happened to $\mathrm{V=IR}$? Doubling voltage and halving the current seems to completely contradict this basic law. That is, of course, unless the resistance in the output circuit changes, with R proportional to $\mathrm{V^2}$ - but I don't see how this is possible.  So how can a transformer obey both laws? Can resistance change or am I missing something else? Deleting comments and review actions cause layout to be broken, AJAX response contains full page Wireframe iOS design apps for iPad? Is there an app for iPad to assist in wireframe design for iPad and iPhone apps?  I don't mean actual development, just placing standard widgets and the things you usually do with photoshop and associated kits. How to revert to factory settings? Wilson and Fermat's Theorem Question!     Show that the pair of positive integers $n$ and $n+2$ are twin primes if and only if $$4[(n-1)! + 1] + n = 0 \ (\text{mod} \ n(n+2)),$$ where $n$ does not equal $1$. Consider numbers of the type $n=2^m+1$. Prove that such an n is prime only if $n=F_K$ for some $ k \in N$, where $F_k$ is a Fermat Prime. Consider numbers of the type $n=2^m+1$. Prove that such an $n$ is prime only if $n=F_K$ for some $ k \in N$, where $F_k$ is a Fermat Prime. Consider numbers of the type $n=a^m-1$ where $a&gt;1$ is a natural number. Prove that such an $n$ is prime only if $n=M_p$ for p-prime, where $M_p$ is a Mersennt Prime.   I do not understand what the question is asking me for. Can I get some clarification? List questions: Community Wiki?