If $R$ be a Union of zero measure (lebesgue) sets , what can we say about the cardinal of index set? Does this question related to continuum hypothesis? Thanks.
$\aleph_1$ is the cardinality of the countable ordinals. It is the least cardinal number greater than $\aleph_0$, and assuming the continuum hypothesis it's equal to $\mathfrak{c}$, the cardinality of the the real numbers.    My question is, is it possible for all $\aleph_1$ subsets of $\Bbb{R}$ to have Lebesgue measure $0$?  This is, of course, impossible assuming the continuum hypothesis, because then all sets of real numbers would have measure $0$, which is absurd.  But is $\mathsf{ZFC}$ + $\lnot\mathsf{CH}$ + "all subset of $\Bbb{R}$ of cardinality $\aleph_1$ have Lebesgue measure $0$" consistent?  If not, what if we replaced Lebesgue measure with some other measure?  It may be worth noting that, although there may be subsets of $\Bbb{R}$ with cardinality $\aleph_1$, there are no subsets of $\Bbb{R}$ which have the order-type $\omega_1$ (the order-type of the countable ordinals) under the usual ordering on $\Bbb{R}$.  Any help would be greatly appreciated.  Thank you in Advance.
Is there a way to make rigid body interact with softbody physics? i want to add watermark on some videos but problem is because image in some videos its too small and in some videos its too big, (i think it depends on input link resulution)   so all my videos have output '640:360' so how can i make watermark image to be same in all videos not on some videos smaller in some bigger   ffmpeg -i "link" -i "/var/www/logo/logo1.png" -filter_complex 'overlay=10:10' -vcodec h264 -crf 25 -preset superfast -maxrate 600k -bufsize 600k -aspect '640:360' -s '640:360' .... continuous functions on $\mathbb R$ such that $g(x+y)=g(x)g(y)$ Let $g$ be a function on $\mathbb R$ to $\mathbb R$ which is not identically zero and which satisfies the equation $g(x+y)=g(x)g(y)$ for $x$,$y$ in $\mathbb R$.   $g(0)=1$. If $a=g(1)$,then $a&gt;0$ and $g(r)&gt;a^r$ for all $r$ in $\mathbb Q$.   Show that the function is strictly increasing if $g(1)$ is greater than $1$, constant if $g(1)$ is equal to $1$ or strictly decreasing if $g(1)$ is between zero and one, when $g$ is continuous. Ubuntu 19.04 and Gnome desktop icons I'm trying to configure a ubuntu server network. [Edit] My ifconfig: lo: flags=73&lt;UP,LOOPBACK,RUNNING&gt; mtu 65536 inet 127.0.0.1 netmask 255.0.0.0 inet6 ::1 prefixlen 128 scopeid 0x10&lt;host&gt; loop txqueuelen 1000 (Loopback Local) RX packets 352 bytes 26160 (26.1KB) RX erors 0 dropped 0 overruns 0 frame 0 TX packets 352 bytes 26160 (26.1KB) TX errors 0 dropped 0 overruns 0 carrier 0 collisions 0  I edited the /etc/network/interfaces file and here is how it is: auto lo auto eth0 iface eth0 inet static address 192.168.15.141 netmask 255.255.255.0 gateway 192.168.15.1  I'm trying to restart the server and I'm using: sudo ifdown --exclude=lo -a &amp;&amp; sudo ifup --exclude=lo -a  This command is an answer from this link:  And I'm getting the following error: Cannot find device &quot;eth0&quot; Failed to bring up eth0.  I'm pretty lay in configuring a server, so I have no idea what's going on. Can someone help me please? How does tangzhong (water roux) make bread softer? As I understand it, tangzhong in relation to bread making is an Asian technique in which a portion of the flour in the recipe is pre-cooked with water to make a roux, before being cooled and added to the rest of the ingredients. It is supposed to make the resulting bread very soft, often for days longer than non-tangzhong breads. I thought maybe it works by binding the proteins so gluten development is restricted, but I'd have thought that would inhibit rise, which seems not to be an issue.  Disclaimer: I have not yet made a tangzhong bread, but I believe I have eaten a few... My first attempt is currently doing a bulk rise.  How does tangzhong work exactly? How does it actually change the dough to achieve the distinct softness for which it is so famous? Why are my villagers attempting to breed (they have the red hearts) but then stop breeding suddenly and gray "angry" particles pop up? I set up a villager breeder and set down 4 beds with 2 villagers in the middle and a carrot farmer supplying them with carrots. The first time both of those villagers bred the breeder worked perfectly fine. After that, however, the gray angry particles popped up every time they would try to breed (they would engage in breeding at a normal pace, just could not actually produce a baby). I even killed the one villager they produced in case it took a bed and even then, the two villagers would not breed anymore. Is there something I have done wrong? This seems odd considering they were only able to breed once and even by eliminating their offspring the villagers could not breed again. Edit: version is 1.16.4 I am trying to prove:     $a\mid b \land b\mid c \Rightarrow a\mid c $   $a\mid b$ means that a divides b if there is an integer k, that $b=k\cdot a$  Please give me a hint on how to start, because I have no idea. How to capture the scale of a scene? My framing in the following photo completely failed to capture the sheer height of the waterfall.  If you don't know I was standing at the top of the waterfall, and the "splinters" at the bottom are full-size tree trunks 2 feet in diameter, you wouldn't have a clue how tall it is.  It just looks like a smaller, shorter waterfall shot with a slower shutter speed.    My question isn't about this picture or waterfalls in particular, but about framing to capture the scale of your subject in general.  What are techniques to show just how big something really is?  Or conversely, something really small?  Especially things like rocks or water that look roughly the same at any size. I am visiting a friend in Switzerland for ten days. I have already got visitor's visa for the same on my Indian passport. Particulars mentioned on my visa are:  1. 'Etats Schengen' , 2. Single entry 3. Dates of visit. &amp; 4. Type of visa : C.  I understand that having already got ‘Etats Schengen' visa, I can travel to all Schengen countries.  I am also planning to visit Paris for a day during my stay in Switzerland. However, I never mentioned this in my visa application. I am planning to take a train from Geneva to Paris and book a hotel in Paris for one night. Can I possibly plan my Paris trip this way as I have Schengen visa, or I am not allowed to travel out of Switzerland as I never mentioned this in my visa application?   Also what does Single entry on my Schengen visa means: does it mean that I can enter and leave Switzerland only once or it means that I can enter and leave the Schengen region only once. My host is x64 bits Windows 8.1.  I downloaded the latest Virtual Box (4.3) and I'm trying to create a VM with a 64 bits Ubuntu OS (ubuntu-12.04.3-desktop-amd64).  When I go to New VM wizard, it doesn't give me option to select "Ubuntu (x64)" as I have seen in other people's screenshots, only just "Ubuntu". As a result, the ISO can't boot. I tried in another PC and Virtual Box gives the x64 variants to most listed OS...  Control Panel shows x64 OS, x64 processor. My host laptop is a Sony Vaio VPCZ22UGX/N, Intel® Core™ i7-2640M processor. CPUz shows Vx-t is available on my processor, of course.  Here is what I tried so far:   I enabled IO APIC as required in the docs. I have virtualization enabled in the BIOS. It works fine in VMware. Check that Hyper-V is not running or even installed on my Windows. Same for VMware. I also tried running the command:  VBoxManage modifyvm [vmname] --longmode on    for that VM, but no change.. I think the issue is really that I can't select x64 variant of the Ubuntu OS for that VM. Other people seem to indicate that's a requirement, but I don't get that option for some reason.  I spent a lot of time and can't find what's wrong... Anyone knows what could be missing here?  Thank you very much!!  Eduardo My book uses the argument that the multiplicities of a few macrostates in a macroscopic object take up an extraordinarily large share of all possible microstates, such that even over the entire lifetime of the universe, if each microstate had an equal chance of being accessed, fluctuations away from equilibrium would never occur.  My question to this statistical proof is this: In the real world, is there really an infinitesimal but nonzero chance that macroscopic systems could access some of the more unlikely macrostates (e.g. heat transfer from a cold object to a hot object)? Prove that $\phi(n) \geq \sqrt{n}/2$ Change the URL in the browser without loading the new page using JavaScript String output: format or concat in C#? Let's say that you want to output or concat strings. Which of the following styles do you prefer?   var p = new { FirstName = "Bill", LastName = "Gates" }; Console.WriteLine("{0} {1}", p.FirstName, p.LastName); Console.WriteLine(p.FirstName + " " + p.LastName);   Do you rather use format or do you simply concat strings? What is your favorite? Is one of these hurting your eyes?  Do you have any rational arguments to use one and not the other?  I'd go for the second one. The external Flash is not showing up in the configuration Menu when using Pixel TF-321 hot shoe. Is this the wrong adapter for that camera, as there is no X connection? I want to trigger a studio flash via sync connection. Can you recommend me a working hot shoe? Thanks! When can I use a forward declaration? Is it possible to create matter? Is it possible to create matter? In a recent discussion I had, it was suggested that with enough energy in the future, "particles" could be created.   It seems like this shouldn't be possible due to conservation but perhaps I could be wrong. Would any of you Physics masters care to elaborate?  (Note... I will understand the basics, I am by no means an expert.) Alternate property name while deserializing Problem with packages subfiles and import.sty after using TexLive Utility for updating Everything worked fine before using TexLive for updating my packages three days ago. Now it is not possible anymore to use the package subfiles   Here are the MWE  The minimum Preamble.tex file  \documentclass{amsart} \usepackage{subfiles}   The MWE.tex file  \documentclass[MWE.tex]{subfiles} \begin{document} MWE \end{document}   When I compile the MWE file I get on the console  This is pdfTeX, Version 3.14159265-2.6-1.40.20 (TeX Live 2019) (preloaded &gt;format=pdflatex)  restricted \write18 enabled. entering extended mode  (./MWE.tex  LaTeX2e &lt;2020-02-02&gt; patch level 5  L3 programming layer &lt;2020-02-25&gt; (/usr/local/texlive/2019/texmf-dist/tex/latex/subfiles/subfiles.cls  Document Class: subfiles 2020/02/14 v16 Multi-file projects (class)  Preamble taken from file `MWE.tex'  (/usr/local/texlive/2019/texmf-dist/tex/latex/tools/verbatim.sty)  (/usr/local/texlive/2019/texmf-dist/tex/latex/import/import.sty)  ! LaTeX Error: File `import.sty' not found.  Type X to quit or &lt;RETURN&gt; to proceed, or enter new name. (Default extension: sty)  Enter file name:   I checked and the import.sty file is in the right directory, and everything worked fine before. Do you have any idea about what happened (apart from the fact that I should not upgrade stuff before the official releases of TexLive I suppose  ...)?