I was looking at the gravitational inverse-square law: $$ F_G = G \frac{Mm}{r^2} $$  This law comes from some experimental data? Why it is an exact inverse-square law? Could it be $$ F_G = G \frac{Mm}{r^{2.00000000000000001}} $$ or there is a mathematical method to find exactly this law?
When I was in the university (in the late 90s, circa 1995) I was told there had been research investigating the $2$ (the square of distance) in the Newton's law of universal gravitation.  $$F=G\frac{m_1m_2}{r^2}.$$  Maybe a model like   $$F=G\frac{m_1m_2}{r^a}$$  with $a$ slightly different from $2$, let say $1.999$ or $2.001$, fits some experimental data better?  Is that really true? Or did I misunderstand something?
Meaning of telegraph code in the song "Planetes" The song Planetes in Guilty Crown begins with some telegraph encoding, which I think is . Here is the . My transcription is: -•-• -••-- •-- --•-- •- --•-• •-•-- -•--•  which decodes to this:  ni-yu-ya a-i-shi-te-ru  Niyuya didn't make sense. Did I get something wrong? Neutrino project in Tamil-Nadu India In , the Indian government is going to start a neutrino project.  Questions:   Is it harmful for people who stay there?   Will this project affect any living beings? Does it cause any pollution? I am writing a script, which I would like to spice up a bit.  What I need, is a loop, which takes one letter at a time, including space, and then write it on the screen, like an old typewriter would do.  I would write some kind of pause/sleep between the typing, so it would look retro like. PC started having screen glitches after update I'm not sure what's caused it. I haven't changed any system settings. The only thing I've done that may have broken something is updating my software. My update was: upgrade linux-generic-hwe-20.04:amd64 5.8.0.48.54~20.04.32 5.8.0.49.55~20.04.33 upgrade linux-image-generic-hwe-20.04:amd64 5.8.0.48.54~20.04.32  upgrade linux-headers-generic-hwe-20.04:amd64 5.8.0.48.54~20.04.32  upgrade linux-image-extra-virtual:amd64 5.4.0.70.73 5.4.0.71.74 upgrade linux-image-generic:amd64 5.4.0.70.73 5.4.0.71.74 upgrade linux-libc-dev:amd64 5.4.0-70.78 5.4.0-71.79 upgrade libseccomp2:amd64 2.4.3-1ubuntu3.20.04.3 2.5.1-1ubuntu1~20.04.1 upgrade nomachine:amd64 7.3.2-3 7.4.1-3```  Removing nomachine did not solve the problem. It was far worse before, I'm not sure how to describe exactly, but my entire screen's UI was glitching out. So I rebooted and that stopped, but now the terminal text looks like this:  I haven't changed any terminal settings. This seems to be occurring in the entire OS. Here is another example:  Not sure how to go about troubleshooting this? I don't have the time to reinstall the OS, nor do I think I should have to because this is just ridiculous and shouldn't happen. I'm not aware of anything I've done to make the system unstable. Edit Earlier I also tried booting the PC. It failed and got stuck on some command in a terminal (didn't manage to take a photo of what it said). So I rebooted into recovery mode with both available kernels. I was getting strange glitches like this:  That didn't work. So I rebooted the PC normally, and now I somehow logged in fine. Now I'm getting popups saying to report a system error. It says this:  Environment  Ubuntu 20.04.2 LTS GNOME 3.36.8  sudo lshw -class display   *-display        description: VGA compatible controller        product: Xeon E3-1200 v3/4th Gen Core Processor Integrated Graphics Controller        vendor: Intel Corporation        physical id: 2        bus info: pci@0000:00:02.0        version: 06        width: 64 bits        clock: 33MHz        capabilities: msi pm vga_controller bus_master cap_list rom        configuration: driver=i915 latency=0        resources: irq:33 memory:f7800000-f7bfffff memory:e0000000-efffffff ioport:f000(size=64) memory:c0000-dffff  I also did , which passed with 0 errors. Let me know if more details are required. Change Default Scrolling Behavior of UITableView Section Header Run VirtualBox in background, without a window? Continuity of $L^1$ functions with respect to translation A closed form of $\int_0^1\frac{\ln\ln\left({1}/{x}\right)}{x^2-x+1}\mathrm dx$ This integral has been bugging me since yesterday:     $$\int_0^1\frac{\ln\ln\left({1}/{x}\right)}{x^2-x+1}\mathrm dx$$   I've tried substitution $y={1}/{x}$ and $e^y={1}/{x}$, but those didn't help much. Wolfram Alpha gives me result: $-0.67172$. Could anyone here please help me to obtain the closed form of the integral preferably (if possible) with elementary ways (high school methods)? Any help would be greatly appreciated. Thank you. I observe the following strange characters ^@ in syslog at the time when my machine freezes.  Then i have to hardboot my machine.  What could be wrong any ideas?  Oct 12 12:20:52 nanak-P570WM anacron[1074]: Normal exit (1 job run) ^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@  Oct 12 12:24:28 nanak-P570WM kernel: imklog 5.8.11, log source = /proc/kmsg started. Oct 12 12:24:28 nanak-P570WM rsyslogd: [origin software="rsyslogd" swVersion="5.8.11" x-pid="543" x-info="http://www.rsyslog.com"] start I'm trying to prove by induction that for all $a,b \in \mathbb{Z}$ and $n \in \mathbb{N}$, that $(a-b) \mid (a^n-b^n)$. The base case was trivial, so I started by assuming that $(a-b) \mid (a^n-b^n)$. But I found that: \begin{align*} (a-b)(a^{n-1}+a^{n-2}b + a^{n-3}b^2+...+b^{n-1}) &amp;= a^n-b^n. \end{align*}  Doesn't this imply that $(a-b) \mid (a^n-b^n)$ as $a^{n-1}+a^{n-2}b+\dots+b^{n-1}$ is clearly an integer? This obviously isn't a proof by induction, but is there anything wrong with taking this approach to prove this result, other than the fact that it isn't what is being asked?   Aside, I'm having trouble getting started with the proof by induction. I've tried making the above equivalence useful during the induction step, but it doesn't seem to be helpful, so any hints would be greatly appreciated! How do I integrate the following? $\int{\frac{(1+x^{2})\mathrm dx}{(1-x^{2})\sqrt{1+x^{4}}}}$ What is the meaning of p values and t values in statistical tests? Is it possible to use more than 20 layers? Is it possible to use more than 20 Scene layers or Render layers in Blender? What is the product of the empty set? Give:  $fn(S)=\prod_{x\in S}x$  what is:  $fn(\emptyset)$  I can see reason that it would be defined as 1 or 0.    Note: I thought about restricting the domain of $S$ but that would make the problem less general. If there is no general answer then answers for restricted domains would be valid. If $f'(c)$ exists then $\lim_{x\to c} f(x)=f(c)$ If $f'(c)$ exists then $\lim_{x\to c} f(x)=f(c)$   My attempt is below. Did I do wrong? Since I don't really understand it. I thought $f'(x) = f'(c)$ so $f'(x)-f'(c)=0$. Am I on the right track? Many thanks!  \begin{align} f(x)-f(c)&amp;=\frac{f(x)-f(c)}{x-c}\cdot(x-c) \\ \lim_{x\to c}[f(x)-f(c)]&amp;=\lim_{x\to c}\left[\frac{f(x)-f(c)}{x-c}\cdot(x-c)\right] \\ \lim_{x\to c}[f(x)-f(c)]&amp;=0 \\ \lim_{x\to c} f(x)&amp;=f(c) \end{align} Is sudo required for installing VMware Tools? I'm trying to install VMware Tools in Ubuntu 16.04. Should I do this by using sudo? Why? Prove that $\left (\frac{a^2 + b^2 +c^2}{a+b+c} \right) ^ {(a+b+c)} > a^a b^b c^c$ Prove that $\left (\dfrac{a^2 + b^2 +c^2}{a+b+c} \right) ^ {(a+b+c)} &gt; a^a b^b c^c$ if $a$, $b$ and $c$ are distinct natural numbers.  Is it possible using induction? How can I search the bash history and rerun a command? Can I search history in bash and run the result? Select 1px aliased border When I choose Select > Modify > Border and select a 1px border - the border I get is anti-aliased, hence if I fill my selection a 3px border is affected.  Is there a way to select a 1px aliased border? Firefox suspicious subprocess (turned out to be legitimate FF behavior) I've discovered this yesterday, as it was generating system load on otherwise idle Ubuntu 14.04 Desktop machine.  Today, I've confirmed such behavior also on another Ubuntu instance (17.10 in Virtualbox VM)  The sub-process in command line looks like below ( I've put this as picture below intentionally to prevent askubuntu.com from changing/escaping the content )   For me it looks like exploit/malware. Or at least there is something to hide.  this happens even for url