There is a beaker of water with ice floating on top of it. If I add oil (lighter than water and ice) to it such that it forms a layer above water and ice is suspended between those layers. Does the ice rise or fall from its previous level?   specific gravity of  ice = 2/3 specific gravity of oil = 1/2
Suppose we have a ball floating in liquid A. Then we pour a liquid B on top of the ball (Assuming that the liquid is poured such that the ball doesn't oscillate) . The ball rises up. Why?  According to me, the liquid on top (liquid B) will exert a force downward on the ball and it should go lower. The particles in the liquid B will exert an normal force which is acting downwards on the ball. How can adding liquid B make it rise up, that is, provide an upward force?  Taking a specific case, density of B > density of ball > density of A   what would happen?
Let $P$ be a prime. Show that $\exists$ $x \in \mathbb{N}$ such that $f(x) = p$ then $\exists$ $y \in \mathbb{N}$ such that $g(y) = p$ What is given?  $$\text{Let P be a prime}$$ $$\text{Let} \space f(x)= 3x+1$$ $$\text{Let} \space g(x)= 6x+1$$  Show that: If there exists $x \in \mathbb{N}$ such that $f(x) = P $ , then there exists a $y \in \mathbb{N}$ , such that $g(y) = P$  What have I tried?   Given the question says 'show that', that leads me to believe I need to do a proof of sorts. I'm guessing it might be a proof by either strong induction, or just induction (or maybe just a direct proof?).  I don't even know how to set this up If $f(x) =$ $3x + 1 = $ prime...for any integer $x$, then there exists an integer $y$ such that $g(y) = 6y + 1 = $ prime?   Any hints or starting points would be greatly appreciated.   $$\text{Thanks in advance!}$$ apt-get cannot connect to Ubuntu archives I just installed a fresh version of Ubuntu 19.10 today, and the first thing I decided to do was to install GNU Screen, using sudo apt-get install screen. my output was: Reading package lists... Done Building dependency tree        Reading state information... Done E: Unable to locate package screen  I figured the package archives were not quite updated as it was a clean install so I tried sudo apt-get update. This is what I got: Get:1 http://security.ubuntu.com/ubuntu eoan-security InRelease [97.5 kB]       Err:2 http://au.archive.ubuntu.com/ubuntu eoan InRelease   Cannot initiate the connection to au.archive.ubuntu.com:80 (2001:388:30bc:cafe::beef). - connect (101: Network is unreachable) Could not connect to au.archive.ubuntu.com:80 (202.158.214.106), connection timed out Err:3 http://au.archive.ubuntu.com/ubuntu eoan-updates InRelease      Cannot initiate the connection to au.archive.ubuntu.com:80 (2001:388:30bc:cafe::beef). - connect (101: Network is unreachable) Err:4 http://au.archive.ubuntu.com/ubuntu eoan-backports InRelease   Cannot initiate the connection to au.archive.ubuntu.com:80 (2001:388:30bc:cafe::beef). - connect (101: Network is unreachable) Fetched 97.5 kB in 31s (3,106 B/s) Reading package lists... Done Building dependency tree        Reading state information... Done All packages are up to date. W: Failed to fetch http://au.archive.ubuntu.com/ubuntu/dists/eoan/InRelease  Cannot initiate the connection to au.archive.ubuntu.com:80 (2001:388:30bc:cafe::beef). - connect (101: Network is unreachable) Could not connect to au.archive.ubuntu.com:80 (202.158.214.106), connection timed out W: Failed to fetch http://au.archive.ubuntu.com/ubuntu/dists/eoan-updates/InRelease  Cannot initiate the connection to au.archive.ubuntu.com:80 (2001:388:30bc:cafe::beef). - connect (101: Network is unreachable) W: Failed to fetch http://au.archive.ubuntu.com/ubuntu/dists/eoan-backports/InRelease  Cannot initiate the connection to au.archive.ubuntu.com:80 (2001:388:30bc:cafe::beef). - connect (101: Network is unreachable) W: Some index files failed to download. They have been ignored, or old ones used instead.  I'm really not sure why it would be doing this, other than maybe some first-use hiccups? The computer is definitely connected to the internet, as I am writing this from that machine, and I have tried running the commands after rebooting and it still gives the errors. I can't seem to be able to connect to au.archive.ubuntu.com at all. ping au.archive.ubuntu.com just sits there without a response. If anyone could help, that would be great! If any extra information is required I'd be happy to edit it into the question. Fréchet differentiability from Gâteaux differentiability Let $X$ be a Banach space and $\Omega \subset X$ be open.  The functional $f$ has a Gâteaux derivative $g \in X'$ at $u \in \Omega$ if, $\forall h\in X,$ $$\lim_{t \rightarrow 0}[f(u+th)-f(u)- \langle g,th \rangle]=0$$   How can I prove the following:  If $f$ has a continuous Gâteaux derivative on $\Omega$, then $f \in C^1(\Omega,\mathbb R)$. Show that $1^{p-1} + 2^{p-1} +\ldots + (p-1)^{p-1} \equiv -1 \mod p$ Show that    $$1^{p-1} + 2^{p-1} +\ldots + (p-1)^{p-1} \equiv -1 \mod p$$   So, I use Fermat's little theorem, that is if $p$ does not divide $a$, then $a^{p-1}$ is congruent to $1$ (mod $p$). But I'm still confused. If $1^{p-1}$ is congruent to $1$ (mod $p$), $2^{p-1}$ is congruent to $1$ (mod $p$), and $p$ does not divide $p-1$, so $(p-1)^{p-1}$ is congruent to $1$ (mod $p$). How does that equal $-1$ (mod $p$)? Does the shape of a containter affect the apparent mass of liquid inside it when measured by a scale? For both containers it is true that:  $p_{bottom}=p_{atm}+\rho gh$ and $F_{bottom}=a*p_{bottom} \iff F_{bottom}=a*p_{atm} + a\rho gh$  I do not know how a scale functions, but I guess that the only force it can count is the force by the liquid on the bottom of each container (the only surface that is in touch with the scale is the bottom of the container). Thus, if we measured the mass of these two masses of water (lets assume that the mass of the containers is negligible) it would appear to be the same in each case. Am I right? Custom Adapter for List View Preventing itemize environment to insert initial vertical space I'm using itemize and tables like the following  \begin{tabularx}{\textwidth}{&gt;{\ttfamily}llX}  \toprule  \textbf{\rmfamily Operator} &amp; \textbf{Typen} &amp; \textbf{Regel} \\  \midrule  = &amp; Alle Typen &amp;   \begin{itemize}    \item This is how we roll    \item Maybe we can talk?   \end{itemize} \\  \bottomrule \end{tabularx}   But it keeps inserting a vertical space before the itemized list. How can I prevent it from doing so? How to enable hibernation? How do I backup all files? (.dbus and dconf backup problems) What are /home/user/.cache/dconf and /home/user/.dbus? Why does their backup fail?  I tried following  (see code below) but while it allowed me to backup the above files, now I am told it couldn't backup /home/[user]/.dbus/session-bus.    $ man chown $ sudo chown user .cache/dconf   [sudo] password for user:  $ sudo chown user .dbus $ man chgrp $ sudo chgrp user .cache/dconf $ sudo chgrp user .dbus   How do I fix it so it will not take my time closing an error message after each backup? Should I exclude these folders ? Why are they not needed? Why would someone want to back them up? Making a logic circuit with only NAND GATES? Getting shapefile of river from OpenStreetMap? I do not want to download the whole map of an area. I just want rivers and other waterbodies present in an area in shapefiles so that I can use them in QGIS. I tried geofabrik but it is giving whole map and that also of full country while i need data of a city. Right now I am trying JOSM. will update you when it will works. Complex zeros of the polynomials $\sum_{k=0}^{n} z^k/k!$, inside balls mysql select distinct rows into a comma delimited list column Good 1st PDE book for self study Extreme Points of a Convex Hull How do I detect what .NET Framework versions and service packs are installed? A similar question was asked , but it was specific to .NET 3.5. Specifically, I'm looking for the following:   What is the correct way to determine which .NET Framework versions and service packs are installed? Is there a list of registry keys that can be used? Are there any dependencies between Framework versions? I find myself frequently needing to rename different files with mv that are in deep directories:  mv /a/long/path/to/a/file.txt /a/long/path/to/a/file.txt.bak   But I don't want to retype the full path name. Is there a quick shorthand or alias that I could use? I.e.:  $ mv /a/long/path/to/a/file.txt file.txt.bak $ ls /a/long/path/to/a/file.txt.bak a/long/path/to/a/file.txt.bak   (note: this is for straightforward, single file renames in different directories at different times, not for ) Executing SSIS 2012 package that has script components from external application I'm using Xubuntu 14.04 64 bit. I know that question been asked a billion times, but all of them consist of editing following file:  /etc/polkit-1/localauthority/50-local.d/com.ubuntu.enable-hibernate.pkla   That file does not exist on my system. I tried com.xubuntu.enable-hibernate.pkla but doesn't exist either, locate command doesn't find it. Split file into multiple files based on pattern I have a large file that's a couple hundred lines long. This file is partitioned into many parts by a specific identifier, lets say 'ABC'. This line 'ABC' appears 6 times so I want 6 output files. I'm familiar with split and awk but can't seem to create a command line that will do what I've described, any ideas?  Here's an example  ABC line 1 line 2 line 3 ABC line 1 line 2 ABC line1   I'd like three files where ABC is the first line in the new file and it ends before the next ABC is encountered.