Define $P=\{M\subset G: M\unlhd G, M\neq G\}$.  Let $\mathscr{C}$ be a chain of $P$.  I have shown that $\bigcup \mathscr{C}$ is a normal subgroup of $G$, but i don't know how to prove $\bigcup\mathscr{C}\neq G$. Moreover, i'm not sure whether this is true.  Is it true? If so, how do i prove it is not $G$?
$G$ is an infinite group.  Is it necessary true that there exists a subgroup $H$ of $G$ and $H$ is maximal ?  Is it possible that there exists such series $H_1 &lt; H_2 &lt; H_3 &lt;\cdots &lt;G $ with the property that for every $H_i$ there exists $H_{i+1}$ such that $H_i &lt; H_{i+1}$?
How to prove that $d \sin(x)/dx = \cos(x)$ without circular logic such as L'Hôpital's rule? $a_1x_1+a_2x_2+a_3x_3+...+a_nx_n=$ is called a linear equation because it represents the equation of a line in an n dimensional space. So "linear" comes from the word "line".Basically there should not be any higher power of x failing which the graph of the function will not be a straight line.  simillarly  $a(x)y+b(x)y'+c(x)y"+d(x)y'''+...+q(x)=0$ is also called linear differential equation because all the derivatives have power equal to 1 which is similar to the above definition of a linear equation.  A function f is called linear if: $f(x+y)=f(x)+f(y)$ and $f(cx)=cf(x)$. Here c is a constant. In this definition of linearity of function "$f$" what does the word linear means? How does it relate to a straight line?  Finally what does the term linear means in case of linear vector spaces? Where is the reference to a straight line?  So, whether linear is just a word used in different contexts? Does it have different meaning in different situation? Or linearity refers to some relation to a straight line? At Least please explain how the linearity of function f and linear vector space relate to the equation of a line. I accidentally issued some commands I thought were safe in the terminal, and I ended up with no System Settings and lots of other missing functionality, but the OS still boots normally. I've decided I want to install Lubuntu instead of fixing Ubuntu for a fresh start and since my laptop is running on a 1GHz dual core AMD CPU.  I created a bootable USB as I usually do and placed Lubuntu 17.10 amd64 using Rufus 2.14.1086. I have GRUB that was installed with Ubuntu upon first install I still have Win8.1 which I still want to keep.  PS: I also tried creating a startup disk from Ubuntu  The question is: How can I boot from USB with GRUB2 so I can overwrite Ubuntu? I am trying to classify using randomForest classification in GEE . However, I always get the error User memory limit exceeded, so I assume the number  of the feature are too large. Nevertheless, I saw many developers can classify even bigger dataset, so I am wondering if there is a better option there? The code can be accessed via  var fire= ee.FeatureCollection('users/spatola/recovery_rf_1712').map(function(feature){     var num = ee.Number.parse(feature.get('CLC_311'));     return feature.set('CLC_311', num);   }) var indices= ['EVI_mean','Dnbr','NBR2_mean','NBR2_min','NBR2_max','EVI_min','EVI_max']; var fire1=  fire.filter(ee.Filter.notNull(indices)); var fire2= fire1.select('Class_R','Dnbr','EVI_max','NBR2_max','imageId') var bands= ['NBR2_max', 'EVI_max']; var classProperty= 'Class_R'; var random= fire2.randomColumn('random'); var split_train= 0.8; var split_test= 0.2; var trainPartition= random.filter(ee.Filter.lt('random',split_train)); var testPartition= random.filter(ee.Filter.gte('random',split_test));  var RFtrained_c= ee.Classifier.smileRandomForest({  numberOfTrees: 100,  minLeafPopulation:2,  seed: 123,  }) .setOutputMode('CLASSIFICATION') .train({     features: trainPartition,     inputProperties: bands,     classProperty: classProperty,     }) print(RFtrained_c); var dict =  RFtrained_c.explain(); var test = testPartition.classify(RFtrained_c) print(test.first()) How do you fix a bad merge, and replay your good commits onto a fixed merge? I accidentally committed an unwanted file (filename.orig while resolving a merge) to my repository several commits ago, without me noticing it until now. I want to completely delete the file from the repository history.   Is it possible to rewrite the change history such that filename.orig was never added to the repository in the first place? Becoming Thieves Guild Guildmaster, am I bugged? I've completed the Thieves Guild main questline () and chosen my special power (Shadowcloak of Nocturnal). It says completed in my quest log.  I also completed the  in each city you can give Thieves guild influence to (Markarth, Whiterun, Windhelm, and Solitude) and done their special jobs from Delvin. They show completed in my quest log. I've even done at least one of each kind of job from Vex and Delvin (but not in each city).  I got  in the Ragged Flagon and have done the quest for Tonilia (delivering the Moon Sugar to a caravan) and Vekel (getting the 4 books). I've also upgrade one of my pieces of Thief Armor.  I talk to  to become guild master and I get no dialogue options but he just says he's busy and we'll talk another time. There is a guild chest in front of his desk but it's locked and I don't have the key.  What am I missing? Is there some bug I've encountered? If A is invertible, prove that $\lambda \neq 0$, and $\vec{v}$ is also an eigenvector for $A^{-1}$, what is the corresponding eigenvalue? If A is invertible, prove that $\lambda \neq 0$, and $\vec{v}$ is also an eigenvector for $A^{-1}$, what is the corresponding eigenvalue?  I don't really know where to start with this one. I know that $p(0)=det(0*I_{n}-A)=det(-A)=(-1)^{n}*det(A)$, thus if both $p(0)$ and $det(0) = 0$ then $0$ is an eigenvalue of A and A is not invertible. If neither are $0$, then $0$ is not an eigenvalue of A and thus A is invertible. I'm unsure of how to use this information to prove $\vec{v}$ is also an eigenvector for $A^{-1}$ and how to find a corresponding eigenvalue. Number of elements in a finite $\sigma$-algebra Disk storage graphic and usage amount inconsistent? What happens to the second Marty at the end of Back to the Future? In  (the original movie), Marty comes back from 1955 10 mins early to prevent Doc from being shot (which he fails at). He then rolls up to the Lone Pine Mall and sees himself hit 88 mph and shoot back to 1955 (again). Marty then carries on with this life as usual. What happens to the Marty that goes back to 1955 at the end of the film?  This obviously is a recursive problem where it would create another timeline and Marty would keep coming back 10 mins early to view another Marty zip back to 1955? Draw Lewis structures like a book I want to draw a Lewis structure as in the picture. I tried to do with chemfig but does not perform the separation of the points and a minus in the superscript. what is the real use case of dependency injection? How can anything ever fall into a black hole as seen from an outside observer? The event horizon of a black hole is where gravity is such that not even light can escape. This is also the point I understand that according to Einstein time dilation will be infinite for a far-away-observer.  If this is the case how can anything ever fall into a black hole.  In my thought experiment I am in a spaceship with a powerful telescope that can detect light at a wide range of wavelengths.  I have it focused on the black hole and watch as a large rock approaches the event horizon.  Am I correct in saying that from my far-away-position the rock would freeze outside the event horizon and would never pass it? If this is the case how can a black hole ever consume any material, let alone grow to millions of solar masses.  If I was able to train the telescope onto the black hole for millions of years would I still see the rock at the edge of the event horizon?  I am getting ready for the response of the object would slowly fade.  Why would it slowly fade and if it would how long would this fading take?  If it is going to red shift at some point would the red shifting not slow down to a standstill? This question has been bugging me for years!  OK - just an edit based on responses so far.  Again, please keep thinking from an observers point of view.  If observers see objects slowly fade and slowly disappear as they approach the event horizon would that mean that over time the event horizon would be "lumpy" with objects invisible, but not passed through?  We should be able to detect the "lumpiness" should we not through? Losing permissions by adding capability? I observed the following phenomenon that I can not explain. After adding the CAP_SYS_ADMIN capability, unshare is no longer able to write to /proc/self/setgroups.   In fact, writing to this file requires the capability, but that is achieved by changing the user namespace. So why does adding the capability to the parent process prevents writing to this file?  me@myhost:~$ unshare -r root@myhost:~# exit logout  me@myhost:~$ sudo setcap cap_sys_admin=ep /usr/bin/unshare me@myhost:~$ unshare -r unshare: cannot open /proc/self/setgroups: Permission denied  me@myhost:~$ sudo setcap cap_sys_admin= /usr/bin/unshare me@myhost:~$ unshare -r root@myhost:~#   By the way: I am running Ubuntu 16.04.4 LTS with kernel version 4.4 and the version of util-linux (including unshare) is 2.27.1. About compactness and completeness of set $X = \mathbb{R}$ with the metric $d(x, y) = \frac{|x-y|}{1+|x-y|}$ Ubuntu 16.10 Wifi Trouble - RT2561/RT61 I've used Ubuntu before, but I'm still very new. I just installed 16.10 on a new computer yesterday, and the Wi-Fi has been considerably slower than my Windows laptop. I put the computer together mainly to play Dota 2, and I experience decent ping but constant 5%-10% packet loss while playing. Occasionally, after the computer has been left on for a while, the Wi-Fi connection will drop completely- I can still see networks in the drop down menu, including my home network, but my computer will not reconnect without a reboot. I've looked all over for a solution, but my inexperience has prevented me from getting anything to work. My Wi-Fi card is a Ralink RT2561/RT61, here's some info  The output of sudo lshw -C network   *-network                          description: Wireless interface         product: RT2561/RT61 802.11g PCI         vendor: Ralink corp.         physical id: 6         bus info: pci@0000:01:06.0         logical name: wlp1s6         version: 00         serial: 00:0e:2e:de:df:c0         width: 32 bits         clock: 33MHz         capabilities: pm bus_master cap_list ethernet physical wireless         configuration: broadcast=yes driver=rt61pci driverversion=4.8.0-22-generic firmware=0.8 ip=192.168.1.107  latency=64 link=yes multicast=yes wireless=IEEE 802.11         resources: irq:17 memory:dfef8000-dfefffff   Here is the output of sudo iwconfig:   *-network                          description: Wireless interface         product: RT2561/RT61 802.11g PCI         vendor: Ralink corp.         physical id: 6         bus info: pci@0000:01:06.0         logical name: wlp1s6         version: 00         serial: 00:0e:2e:de:df:c0         width: 32 bits         clock: 33MHz         capabilities: pm bus_master cap_list ethernet physical wireless         configuration: broadcast=yes driver=rt61pci driverversion=4.8.0-22-generic firmware=0.8 ip=192.168.1.107  latency=64 link=yes multicast=yes wireless=IEEE 802.11         resources: irq:17 memory:dfef8000-dfefffff   I'm at a loss, I'm very bad at this. If any additional info is needed I can provide it StackExchange drop down hiding chat, blog and log out, concerns "Back" and "Install Now" buttons grayed out in 14.04 installer The "Back" and "Install Now" buttons are grayed out in the 14.04 installer. How do I restore them? What are some of the rule changes that Pathfinder brings in comparison to D&amp;D 3.5?  I am curious what changes were made so I can explain to my group what it has to offer. Can WMS be used to get a local copy of raster data?