Why does gravitational force of the Earth decrease as we move towards the centre of the Earth? Where as inverse square rule says that distance is less than gravitational force is more.
Would the effect of gravity on me change if I were to dig a very deep hole and stand in it? If so, how would it change? Am I more likely to be pulled downwards, or pulled towards the edges of the hole? If there would be no change, why not?
GPU compute grayed out How do snaps work exactly and why are they mounted? Recently I started learning more about snap packages and what intrigued me was that snap packages are mounted together with their own file system and dependencies (correct me if I'm wrong). Now, with all the great things that snaps bring to the table, what I'd like to know is:  Are snap packages always going to be mounted? This seems kind of awkward, since they appear in places like lsblk and give the impression of not being well integrated into my system.   Also, could they ever replace .deb and .rpm packages completely, or are they a different type of animal and are only complementary to .deb and.rpm packages?  As a side-note, I noticed that some applications are so self-contained that they don't respect any of my desktop settings like the mouse cursor theme or icon and menu themes. $(n+1)$th prime $p_{n+1}$ less than or equal to $p_1p_2\dots p_n+1$ Can a virtualized machine have the CPU and RAM resources of multiple underlying physical machines? We are studying implementing some virtualized servers here, but we don't know what will be better suitable for us. Some folks are saying better have two huge servers, and others are saying have like a ten middle-end servers.  We have a legacy Visual Foxpro application, which nowadays run on Dual Xeon E5405 @ 2GHz and 16Gb of RAM. The currently server its getting too slow due the number of active users and process running on it. Virtualizing this server will give us the benefit of an faster disaster recovery.  So the question is, having like ten physical servers running at 1.7GHz and 4Gb of RAM, we could virtualize one server into 4 machines, and have one virtualized server running at 6.8GHz and 16Gb of memory? If yes, there is some how if one machine stops, automatically manage this virtual machine to another one, and execute the appropriate maintenance on it, and later back to it again? What is the capacity of this circuit? I simply can't figure out what the capacity of this circuit is.  I can do the math myself, I just need a hint how to create an equivalent circuit where it is obvious what is parallel and what is in series... What are the differences between "to talk" and "to speak"? Ubuntu does not offers me "install alongside Windows" Any of the partitions makes in Windows was not recognised by Ubuntu, or Zorin OS, or LinuxMint. It's like a blank hard disc, 250 gb free. I'm a new Linux user, but what confuses me is that I was able to install Ubuntu alongside Windows on my laptop with no problems, but on PC... no way! Occasionally I see constructs like this in English sentences:        Not only is this true in this case, but ...      What puzzles me is the "is this" part of it. The sentence is not a question, yet it seems to have a question-like order. Isn't this correct/preferred:        Not only this is true in this case, but ...      Are there other cases where non-question sentences contain question-like parts (so to speak)? I am looking for a closed form (ideally expressed as elementary functions) of the function $\exp_n(x) = \sum_{k=0}^n x^k / k!$.  I am already aware of expressing it in terms of the gamma function. Background / Motivation When counting combinations of objects with generating functions, it is useful to be able to express the partial sum $1 + x + \cdots + x^n$ as $\frac{1-x^{n+1}}{1-x}$. For example, to count the number of ways to pick 5 marbles from a bag of blue, red, and green marbles where we pick at most 3 blue marbles and at most 2 red marbles, we can consider the generating function $f(x) = (1+x+x^2+x^3)(1+x+x^2)(1+x+x^2+\cdots)$. By using the partial sum identity, we can express it as $f(x) = \left(\frac{1-x^4}{1-x}\right)\left(\frac{1-x^3}{1-x}\right)\left(\frac{1}{1-x}\right)$.  Simplify, express as simpler product of series, and find the coefficient of the $x^5$ term. I want to be able to do the same for a generating function in the form $g(x) = \exp_{n_1}(x)^{p_1} \exp_{n_2}(x)^{p_2} \cdots \exp_{n_j}(x)^{p_j}$ The easiest way to extract the coefficient of a given term $x^p / p!$ would be to use a similar closed form expression for $\exp_n(x)$ and a similar technique to $f$. Attempted Solutions Differential equation Recall that the way to prove the identity $1+x+x^2+\cdots+x^n = \frac{1-x^{n+1}}{1-x}$ is to define $S = 1 + x + x^2 + \cdots + x^n$ and notice that: $S - Sx = 1 - x^{n+1}$. Likewise, notice that $y(x) = \exp_n(x)$ satisfies $y - y' = x^n/n!$.  Via SAGE, the solution is $y(x) = \frac{c+\Gamma(n+1,x)}{n!}e^x$.  Our initial condition $y(0) = 0$ so $c=0$.  By , $\Gamma(n+1,x) = n! e^{-x} \exp_n(x)$ so we have $y(x) = \exp_n(x)$. Recurrence Relation Notice that $\exp_n(x) = \exp_{n-1}(x) + x^n/n!$. Using the unilateral Z-Transform and related properties, we find that $\mathcal{Z}[\exp_n(x)] = (z e^{x/z})/(z-1)$. Therefore, $\exp_n(x) = \mathcal{Z}^{-1}\left[(z e^{x/z})/(z-1)\right] = \frac{1}{2 \pi i} \oint_C z^n e^{x/z}/(z-1)\;dz$. $(z^n e^{x/z})/(z-1)$ has two singularities: $z = 1$ and $z = 0$.  The point $z = 1$ is a pole of order one with residue $e^x$.  To find the residue at $z = 0$ consider the product $z^n e^{x/z} (-1/(1-z)) = -z^n \left( \sum_{m=0}^\infty x^m z^{-m} / m! \right) \left( \sum_{j=0}^\infty x^j \right)$.  The coefficient of the $z^{-1}$ term is given when $n - m + j = -1$. The residue of the point $z=0$ is then $-\sum_{m,j} x^m / m! = -\sum_{m=n+1}^\infty x^m / m!$. Let $C$ by the positively oriented unit circle centered at the origin.  By Cauchy's Residue Theorem, $\frac{1}{2 \pi i} \oint_C z^n e^{x/z}/(z-1)\;dz = \frac{1}{2 \pi i} 2 \pi i \left(e^x - \sum_{m=n+1}^\infty x^m / m!\right) = \exp_n(x)$. Finite Calculus I've tried to evaluate the sum using finite calculus, but can't seem to make much progress. How can i use ArcGIS server 10 mapservices in OpenLayers2? How do I fix the GPG error "NO_PUBKEY"? What is the meaning of ./ before a given path? Dynamic select list in the form (dependent dropdown) Module 'pdftexcmds' not found after update Yesterday I updated my distribution (tlmgr update --all) and now I cannot use TikZ with lualatex anymore.  Minimal working example:  \documentclass{article} \usepackage{tikz} \begin{document} a \end{document}   Compilation output:  &gt; lualatex a This is LuaTeX, Version 1.10.0 (TeX Live 2019)                                                                                                                                                                         restricted system commands enabled.                                                                                                                                                                                  (./a.tex                                                                                                                                                                                                              LaTeX2e &lt;2019-10-01&gt; patch level 3                                                                                                                                                                                     luaotfload | main : initialization completed in 0.098 seconds                                                                                                                                                         (/opt/texlive/2019/texmf-dist/tex/latex/base/article.cls                                                                                                                                                              Document Class: article 2019/10/25 v1.4k Standard LaTeX document class                                                                                                                                                (/opt/texlive/2019/texmf-dist/tex/latex/base/size10.clo))                                                                                                                                                             (/opt/texlive/2019/texmf-dist/tex/latex/pgf/frontendlayer/tikz.sty                                                                                                                                                    (/opt/texlive/2019/texmf-dist/tex/latex/pgf/basiclayer/pgf.sty                                                                                                                                                        (/opt/texlive/2019/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty              ...  (/opt/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/tikz.code.tex (/opt/texlive/2019/texmf-dist/tex/generic/pgf/libraries/pgflibraryplothandlers. code.tex) (/opt/texlive/2019/texmf-dist/tex/generic/pgf/modules/pgfmodulematrix.code.tex)  (/opt/texlive/2019/texmf-dist/tex/generic/pgf/frontendlayer/tikz/libraries/tikz librarytopaths.code.tex))) No file a.aux. ABD: EveryShipout initializing macros (/opt/texlive/2019/texmf-dist/tex/context/base/mkii/supp-pdf.mkii [Loading MPS to PDF converter (version 2006.09.02).] ) (/opt/texlive/2019/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty[\direc tlua]:1: module 'pdftexcmds' not found:         no field package.preload['pdftexcmds']         [kpse lua searcher] file not found: 'pdftexcmds'         [kpse C searcher] file not found: 'pdftexcmds' stack traceback:         [C]: in function 'require'         [\directlua]:1: in main chunk. l.165       \directlua{require("pdftexcmds")}  ?    It seems that a Lua package is missing.  Is this a bug? Am I missing a package? I am debating whether or not to get the Potato Sack (yes, I realize I've been debating for several weeks now) bundle through steam.  I already own: Defense Grid: The Awakening, Super Meat Boy, and Audiosurf, all of which are part of the sack pack. What happens if I buy the Potato Sack?  Do I get 2nd copies of the three above mentioned games to gift elsewhere?  Is the price of the bundle reduced by the games I already have?  Do I get nothing special for already owning those 3 games? What is the best practice for multiple button placement and emphasis? Usually, we only have 2 buttons at the bottom of a form or a page that requires an action: Submit &amp; Cancel.  This one is quite easy to style. However, what if we are dealing with multiple buttons at the bottom of a page? For example:   ← Back | Send Email | Booking Paid | Check-in Guest   I'm thinking of making "Back" just a link (not a button) and making "Send Email" / "Booking Paid" secondary buttons (light gray). Then I'd make "Check-in Guest" as the primary button (green).  I still feel that there's a better way of doing it or dealing with multiple buttons / actions. What do you think? Failed upgrade from Ubuntu 19.10 to 20.04 "Please update the microcode" While installing the Ubuntu 20.04 updates during upgrade from 19.10 to 20.04, my screen suddenly went completely white and displayed a message saying  Please logout and try again.  But I could not locate my cursor and so I switched off my laptop. After restarting it again, it was showing an error message saying  Please update the microcode  Why did this happen? How can I solve this problem? How to interpret regression coefficients in logistic regression? ICS devices support creating backups using adb backup, and restoring them using adb restore. Using adb backup, one can select that everything should be backed up, or only a certain package.  However, adb restore lacks all the options of adb backup. When I call adb restore backup.ab, where backup.ab is only the backup of a single package, the phone shows me a warning dialog saying that all settings are going to be lost after restoring the backup. Is it true that restoring a single package backup wipes all data on the phone? If yes, is it somehow possible to restore a single package using adb restore? Sometimes, on the internet, particularly in online games, I see people using the  indefinite article before someone's name: "I see a Joey" or "I hug a Polly". I know some of these people and I'm positive that they are from English-speaking countries, which means it is not a mistake, they do it intentionally.  I was always taught that articles are not used in front of people's names. Maybe it is some sort of slang? What does it mean, when the indefinite article is put in front of someone's name?  Edit: There is a group of people standing somewhere. And somebody, named Joey, appears. They all know Joey. And someone in this group says "I see a Joey". So it is used when someone sees his or her friend. But I don't understand why they use the indefinite article here.