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Why do we have an elementary charge $e$ in physics but no elementary mass? Is an elementary mass ruled out by experiment or is an elementary mass forbidden by some theoretical reason? Positive integer multiples of an irrational mod 1 are dense I'm not sure how to solve this one. Thank you!  $2.$ For any $\alpha\in \mathbb R$ we define $$\lfloor \alpha \rfloor = \max_{n\in\mathbb Z}\{\,n\mid n\leq \alpha\,\}$$ and $$\alpha\bmod 1 = \alpha - \lfloor \alpha \rfloor$$ Let $\alpha$ be irrational. (a) Given $n\in\mathbb N$ show that $\{\,k\alpha\bmod1\mid k\in\mathbb N\,\}\cap\left[0,\frac{1}{n}\right]\neq\emptyset$ (b) Prove that $\{\,n\alpha\bmod 1\mid n\in\mathbb N\,\}$ is dense in $[0,1]$. What is the best way to install Guest Additions with all the resize and clipboard and file sharing features on Ubuntu 16.04? Your Mileage May Vary In the United States we have a saying, "your mileage may vary", which means "your experience may be different".  In English-speaking countries that don't use Imperial miles, is there an equivalent expression? Command with percent symbols not running in crontab The following crontab does not work:  # TEST LINE DOES DOT RUN */1 * * * * /bin/echo 'test '`/bin/date +%Y-%m-%d` &gt;&gt; /tmp/test   I also tried starting it with:  SHELL=/bin/bash   Update: I thought the backtick characters ` were the villains, but as the answer below clarifies the percentage % was the culprit! USB Charge and data separate cables Is it possible to split micro USB to USB cable so two cables (data + and data -) go to the PC with USB and two other power cables go to power brick? What is the SKU of a motor? I came across an 'N20' motor, and I don't know what that means. As far as I can tell (and please correct me if I'm wrong), N20 is the SKU of the motor. What is the SKU? Can I SSH into a specific directory? I SSH into a multitude of different servers every day and some of them have different filesystem structures for their websites. Each time I change server, I have to work out where I'm going to get to the files I really want (because it's very rarely that I want to be in ~). I use Fabric files to automate all my actions (pushing to git, automated SSH to the server to pull from git, restarting the web application, changing dependencies, etc), so I was wondering if I couldn't also tell it that I want to SSH into the right directory so I can run fab ssh and be in a normal SSH session, just in the right place and start working immediately. This is not about Fabric though; please ignore it. Fabric is just running a batch of commands. My argument is with the ssh command. That's what I'll be telling Fabric to run. So how I can get ssh to cd into a specific directory on login? Note: this cannot be a server or user-wide thing. I have multiple websites under one user on some servers so sticking something in ~/.bashrc would not work for me. "implements Runnable" vs "extends Thread" in Java From what time I've spent with threads in Java, I've found these two ways to write threads:  With implements Runnable:  public class MyRunnable implements Runnable {     public void run() {         //Code     } } //Started with a "new Thread(new MyRunnable()).start()" call   Or, with extends Thread:  public class MyThread extends Thread {     public MyThread() {         super("MyThread");     }     public void run() {         //Code     } } //Started with a "new MyThread().start()" call   Is there any significant difference in these two blocks of code ? Dates and times: "on", "in", "at"? I’m often confused when I speak about times and dates. What is the rule for using on, in, and at in the following sentences?   I will do it ___ Tuesday. We married ___ March. He returned ___ the same day. Every day ___ the same time, I walk the dog. Database synchronization between dev/staging and production Why images are blury when extracting image sequence from video clip? I have an issue when generating an image sequence from a video clip. Pictures 1,2,and 3 are three consecutive images from a video clip. Picture 4 is a close-up of the heel of one of them. The camera is on a tripod. As you can see there is no issue with the brick wall, but with any moving part there is. The camera is a Panasonic HDC-SD100. The clip is AVCHD 1920/1080. If I use VLC to extract the image sequence there is no issue. I use all the default settings except for the rendering resolution which I set to 100% and image compression to 0%. I have the issue with Blender 2.79b and 2.80. With this image issue it is impossible to do camera tracking. Any ideas what I need to do in order to solve this. Best regards. My PS3 controllers have died and are not charging For the Fibonacci numbers, show for all $n$: $F_1^2+F_2^2+\dots+F_n^2=F_nF_{n+1}$ D&D 5e and "Theatre of the Mind" in combat Our target play style for combat encounters would be to use battle boards roughly half the time (when there is enough of interest in play that the tactics are fun to play out), and to skip that with faster "cinematic" combat otherwise.  We have not been 100% successful. The problem is the numbers-driven descriptions in the rules. Needing to be "adjacent" (within 5 feet) for abilities such as Protection style fighting, ranged weapon ranges, spell effects and movement speeds are all much easier to resolve on a grid. Running those rules without a board rapidly devolves into sketches with numbers written on them and/or "mother, may I?" style of play because the DM needs to rule who is next to who, and which monsters are in the spell radius. These rulings also quickly get out of sync with player expectations when there is no map and we want to run combats quickly.  It doesn't help that the numbers involved come in pretty much all multiples of the base 5 feet. There is no obvious close/short/medium for effects and ranges. If there were such then I think it would be much easier for the DM to categorise and hand-wave things.  I was hoping that both board-based and no-board style of play would feel equally viable. But the focus on numbers seems to push toward board playing, at least for my group.  We would still like to pay heed to the differences in movement, range, area-of-effect etc of the rules in play. At least in the general sense of creature X is faster than creature Y, or weapon/spell A has better range than weapon/spell B.  Going beyond "just don't sweat the details", how can a group like mine play with the D&amp;D 5e rules as written, without a map, but without leaning on the DM to resolve all our questions of "So, am I next to the Paladin?", or "Is it in range?"      Clarifying: "just don't sweat the details" is a welcome approach to solving the problem, but please give some examples of how this works in practice; i.e. which details can be removed and how? note: This is a question about relativistic effects. I've included some detail about the spacecraft and its orbit for background, but the question is about relativistic effects and their observability.     When the Parker Solar Probe passes within 8.5 solar radii of the Sun, it will be moving really fast.  A perihelion and aphelion of about 6.6 and 109 million kilometers suggests a semi-major axis of 57.8 million kilometers. The standard gravitational parameter for the Sun is:  $$GM_{Sun} \ = 1.327\times 10^{20} m^3/s^2$$  So using the :  $$v^2 \ = \ GM_{Sun}\left(\frac{2}{r}-\frac{1}{a}\right)$$  the orbital velocity at perihelion will be about 195,000 m/s, or about 25 times faster than a satellite in LEO. This is close to one part per thousand of the speed of light! I think this will be a new record for a spacecraft velocity relative to the solar system.  Depth of a gravitational potential well for an object of mass $m$ is approximately given by:  $$u_G \ = \ -GM_{Sun}\frac{1}{r}$$  At 6.6 million kilometers the spacecraft will be 23 times deeper in the Sun's gravitational well than it would be at 1 AU.  Will there be any particularly unique relativistic effects detectable for the Parker Solar Probe during it's close flyby of the Sun?  Just for example, during one close flyby, how much time will its on-board clock gain or lose compared to some convenient standard solar system timescale? I know there are several different choices, "GPS time" might be one but I don't want to arbitrarily choose an inconvenient one.     above: Parker Solar Probe cropped, from .    above: screen shot from NASA's      note: the link seems to be dead now, but there is plenty of related information available at     above: Illustration of the Parker Solar Probe's orbit, from . How to get Intel Dual Band Wireless-AC 3168 work on Ubuntu 16.04? I searched many solutions on this site but most of them were referring to Broadcom Wifi drivers while mine has Intel one.   Output of lspci -knn | grep Net -A2  arijit@HP-AU134TX:~$ lspci -knn | grep Net -A2  05:00.0 Network controller [0280]: Intel Corporation Device [8086:24fb] (rev 10) DeviceName: Intel Dual BandWireless-AC 3168 802.11 ac 1x1 WiFi + BT 4.0 Combo Adapter Subsystem: Intel Corporation Device [8086:2110]   Output of rfkill list  arijit@HP-AU134TX:~$ rfkill list  arijit@HP-AU134TX:~$    i.e No Output  Output of uname -a; dmesg | grep iwl  arijit@HP-AU134TX:~$ uname -a; dmesg | grep iwl  Linux HP-AU134TX 4.4.0-31-generic #50-Ubuntu SMP Wed Jul 13 00:07:12 UTC 2016 x86_64 x86_64 x86_64 GNU/Linux Find the condition on $a$ and $b$ so that the two tangents drawn to the parabola $y^2=4ax$ from a point are normals to the parabola $x^2=4by$ Find the condition on $a$ and $b$ so that the two tangents drawn to the parabola $y^2=4ax$ from a point are normals to the parabola $x^2=4by.$  The required condition is $a^2&gt;8b^2$.I dont know how to prove it.I tried.  Let $(h,k)$ be the point from where tangents are drawn to the parabola $y^2=4ax$.Let $m_1,m_2$ be the slopes of the tangents.Equations of tangents are $y=m_1x+\frac{a}{m_1}$ and $y=m_2x+\frac{a}{m_2}$ and equations of normal to the parabola $x^2=4by$ is $y=m_1x-2bm_1-bm^3_1$ and $y=m_2x-2bm_2-bm^3_2$.  As $y=m_1x+\frac{a}{m_1}$ and $y=m_1x-2bm_1-bm^3_1$  are the same lines.Therefore,$\frac{a}{m_1}=-2bm_1-bm^3_1$  $bm^4+2bm^2+a=0$  And I am stuck.I cannot get the desired condition.Please help me. How can I deactivate Bluetooth on system startup? How can I deactivate Bluetooth on system startup? Is time continuous or discrete?